Thomas Aquinas

In libros Aristotelis De caelo et mundo expositio

translated by Fabian R.Larcher and Pierre H. Conway


Introduction by Thomas Aquinas
Lecture 1The things it pertains to natural science to consider
Lecture 2The perfection of the universe both as body and as containing all
Lecture 3Preliminary notions for showing the parts perfecting the universe
Lecture 4Five reasons why, besides the elements, there must be another simple body
Lecture 5Difference of the body moved circularly as to light and heavy
Lecture 6The fifth body not subject to other motions
Lecture 7The heavenly body is not subject to growth and decrease, or to alteration
Lecture 8Only five simple bodies required. No motion contrary to circular
Lecture 9The need for treating of the infinity of the universe
Lecture 10The second and third reasons proving the circularly moved body not infinite
Lecture 11Three additional reasons why the body moving circularly cannot be infinite.
Lecture 12Various reasons why a body moving in a straight line is not infinite
Lecture 13A natural and demonstrative argument showing no natural body can be infinite
Lecture 14No sensible body is infinite—from action and passion, which follow upon motion
Lecture 15Logical reasons why no body is infinite
Lecture 16Two arguments for one universe, taken from lower bodies
Lecture 17A third argument from lower bodies. Natural bodies have determinate places
Lecture 18Exclusion of the opinion that natural bodies are not moved naturally to determined places. Unity of the world from higher bodies
Lecture 19Solution of the argument seeming to justify several worlds
Lecture 20The universe shown to consist of every natural and sensible body as its matter
Lecture 21Outside the heaven there is no place, time etc., consequent upon sensible bodies
Lecture 22Whether the universe is infinite by eternal duration
Lecture 23A Platonic evasion rejected. Two remaining opinions disproved
Lecture 24Various meanings of "generable" and "ungenerable," "corruptible" and "incorruptible"
Lecture 25How something is said to be "possible" and "impossible"
Lecture 26Everything eternal is indestructible and ungenerated
Lecture 27Nothing eternal generated and corrupted, and conversely
Lecture 28Generated and corruptible, ungenerated and incorruptible, follow on each other
Lecture 29Refutation of corruptible ungenerated and incorruptible generated. Argument from natural science
Lecture 1The heaven is eternal and its motion endless and without labor. Contrary opinions excluded
Lecture 2Diversity of parts of the heaven as to position. Opinion of Pythagoras
Lecture 3How the differences of position befit the parts of the heaven according to the Philosopher's opinion
Lecture 4The reason why there are in the heaven several spheres moved with a circular motion
Lecture 5The spherical shape of the heaven shown from the fact that it is the first of figures
Lecture 6The heaven must be spherical, because this shape is most fitting
Lecture 7Why the circular motion of the heaven is in one direction rather than another
Lecture 8The regularity, or uniform velocity, of the heaven's motion shown by two arguments
Lecture 9Two other arguments proving no irregularity in the motion of the heaven
Lecture 10On the nature of the stars
Lecture 11Proof that the stars move, not of themselves, but as carried by the motion of the spheres, from a comparison with their circles
Lecture 12That the stars do not move themselves concluded from the motions proper to the spherical shape
Lecture 13From their shape the stars shown not to move themselves. No sense power in the heavenly bodies
Lecture 14Indirect and direct proof that heavenly bodies do not produce sounds
Lecture 15Swiftness and slowness in the motion of the planets is proportionate to their distance from the first sphere and the earth
Lecture 16By reason, and by what sensibly appears, the stars are proved to be spherical in shape
Lecture 17Two difficulties proposed in connection with what has been determined about the stars
Lecture 18The first difficulty, concerning the number of motions of the stars, is solved. The number shown to agree with modern astronomers
Lecture 19The second difficulty of Lecture 17 is resolved
Lecture 20Opinions of the philosophers as to the site of the earth. Pythagorean theory of fire in the center is rejected
Lecture 21Different opinions of the motion, rest, and shape of the earth
Lecture 22The problem about the earth's rest
Lecture 23The cause of the earth's rest is not supporting air
Lecture 24Earth's rest not from gyration of the heaven
Lecture 25Earth's rest not explained by supposing that all directions being alike to earth, nothing induces it to be moved in one direction rather than another
Lecture 26Proof of the earth's rest in the middle
Lecture 27Proof of the earth's spherical shape, from motion
Lecture 28Proofs of the earth's sphericity from the angle of motion of its parts, and from astronomy
Lecture 1What has gone before and what remains to be treated
Lecture 2Opinions of the ancients on the generation of things
Lecture 3Bodies not generated from surfaces, proved mathematically and naturally
Lecture 4Other natural arguments against Plato's opinion. Pythagorean opinion refuted
Lecture 5Natural motion in natural bodies. Leucippus & Democritus
Lecture 6Refutation of Plato's opinion of disordered motion before the world
Lecture 7Every body moving naturally in a straight line has gravity or lightness. Natural and violent motions
Lecture 8Everything not generated. Elements and their existence

[The numbers in brackets refer to the passages in the text of Aristotle.]

Subject matter of this book
and its relation to the subject matter of natural science in general

Sicut philosophus dicit in I Physic., tunc opinamur cognoscere unumquodque, cum causas cognoscimus primas, et principia prima, et usque ad elementa. Ex quo manifeste philosophus ostendit in scientiis esse processum ordinatum, prout proceditur a primis causis et principiis usque ad proximas causas, quae sunt elementa constituentia essentiam rei. Et hoc est rationabile: nam processus scientiarum est opus rationis, cuius proprium est ordinare; unde in omni opere rationis ordo aliquis invenitur, secundum quem proceditur ab uno in aliud. Et hoc patet tam in ratione practica, cuius consideratio est circa ea quae nos facimus, quam in ratione speculativa, cuius consideratio est circa ea quae sunt aliunde facta. 1. As the Philosopher says in Physics I, "We judge that we know a thing when we know the first causes and the first. principles down to the elements." Plainly from this the Philosopher shows that in sciences there is an orderly process, a procedure from first causes and principles to the proximate causes, which are the elements constituting the essence of a thing. And this is reasonable: For the method pursued in sciences is a work of reason, whose prerogative it is to establish order; wherefore, in every work of reason is found some order according to which one goes from one thing to another. And this shows up not only in the practical reason, which considers things that we make, but in the speculative reason as well, which considers things made by some other source.
Invenitur autem processus de priori ad posterius in consideratione practicae rationis secundum quadruplicem ordinem: primo quidem secundum ordinem apprehensionis, prout artifex primo apprehendit formam domus absolute, et postea inducit eam in materiam; secundo secundum ordinem intentionis, secundum quod artifex intendit totam domum perficere, et propter hoc facit quidquid operatur circa partes domus; tertio secundum ordinem compositionis, prout scilicet prius dolat lapides, et postea compingit eos in unum parietem; quarto secundum ordinem sustentationis artificii, prout artifex primo iacit fundamentum, super quod ceterae partes domus sustentantur. Similiter etiam invenitur quadruplex ordo in consideratione rationis speculativae. Primus quidem secundum quod proceditur a communibus ad minus communia. Et hic ordo respondet proportionaliter primo ordini, quem diximus apprehensionis: universalia enim considerantur secundum formam absolutam, particularia vero secundum applicationem formae ad materiam; sicut philosophus in I de caelo dicit quod qui dicit caelum, dicit formam, qui autem dicit hoc caelum, dicit formam in materia. Secundus ordo est secundum quod proceditur a toto ad partes. Et hic ordo proportionaliter respondet ordini quem diximus intentionis, prout scilicet totum est prius in consideratione quam partes, non qualescumque, sed partes quae sunt secundum materiam et quae sunt individui; sicut semicirculus, in cuius definitione ponitur circulus (est enim semicirculus media pars circuli), et acutus angulus, in cuius definitione ponitur rectus (est enim acutus angulus minor recto). Accidit autem circulo et recto angulo sic dividi: unde huiusmodi non sunt partes speciei. Huiusmodi enim partes sunt priores in consideratione quam totum, et ponuntur in definitione totius, sicut carnes et ossa in definitione hominis, ut dicitur in VII Metaphys. 2. The process from prior to subsequent is found in the act of the practical reason with respect to a fourfold order: first, according to the order of apprehension, inasmuch as an artisan first apprehends the form of a house absolutely and then realizes it in matter; secondly, according to the order of intention, inasmuch as an artisan intends to complete the house and for that purpose does whatever he does to the parts of the house; thirdly, according to the order of combining, inasmuch as he first trims the stones and then joins them into one wall; fourthly, according to the order of supporting the edifice, inasmuch as the artisan first lays the foundation, upon which the other parts of the house are supported.
In like manner, a fourfold order is found in the consideration of speculative reason. First, because there is a process from the general to the less general.. And this order corresponds to the first order which we have called "the order of apprehension," for universals are considered according to an absolute form, but particulars by applying form to matter, as the Philosopher in On the Heavens says, that the word "heaven" signifies a form, and "this heaven" signifies a form in matter.
The second order is that according to which one goes from the whole to the parts. And this corresponds to "the order of intention," inasmuch as, namely, the whole is considered prior to the parts, not just any parts but parts which are according to matter and which are of the individual — as in the case of a semi-circle, in the definition of which "circle" is used (for it is "half a circle") and of an acute angle, in the definition of which "right angle" is used (for an acute angle is an angle "less than a right angle"). To be divided in that manner is incidental to a circle and to a right angle; hence, neither is a part of the species of a circle or right angle. For parts of this sort [i.e. parts of the species] are prior in consideration to the whole and are used in the definition of the whole, as are flesh and bones in the definition of man, as is said in Metaphysics VII.
Tertius autem ordo est secundum quod proceditur a simplicibus ad composita, inquantum composita cognoscuntur per simplicia, sicut per sua principia. Et hic ordo comparatur tertio ordini, quem diximus compositionis. Quartus autem ordo est secundum quod principales partes necesse est prius considerare, sicut cor et hepar quam arterias et sanguinem. Et hic proportionatur practico ordini, secundum quod fundamentum prius iacitur. The third order is that according to which one goes from the simple to the combined, inasmuch as composites are known in terms of the simple, as through their principles. And this order is compared to the third order, which is the "order of combining." But the fourth order is the one that calls for the principal parts to be considered first, as are the heart and liver before the arteries and blood. And this corresponds in the practical order to that order according to which the foundation is laid first.
Et hic quadruplex ordo consideratur etiam in processu scientiae naturalis. Nam primo determinantur communia naturae in libro physicorum, in quo agitur de mobili inquantum est mobile. Unde restat in aliis libris scientiae naturalis huiusmodi communia applicare ad propria subiecta. Subiectum autem motus est magnitudo et corpus: quia nihil movetur nisi quantum. This fourfold order is also considered in the procedure of natural science. For, first of all, things common to nature are determined in the book of the Physics, in which mobile being is treated insofar as it is mobile. Hence what remains in the other books of natural science is to apply these common things to their proper subjects. The subject of motion, however, is a magnitude and body, because nothing is moved except what is quantified.
In corporibus autem est attendere tres alios ordines: uno quidem modo secundum quod totum universum corporeum est prius in consideratione quam partes eius; alio modo secundum quod simplicia corpora prius considerantur quam mixta; tertio secundum quod inter simplicia corpora prius necesse est de priori considerare, scilicet de caelesti corpore, per quod omnia alia firmantur. Et haec tria in hoc libro aguntur, qui apud Graecos intitulatur de caelo. Traduntur enim in hoc libro quaedam pertinentia ad totum universum, sicut patet in primo libro; quaedam pertinentia ad corpus caeleste, sicut patet in secundo; quaedam pertinentia ad alia simplicia corpora, sicut patet in tertio et quarto. Et ideo rationabiliter hic liber ordinatur primus post librum physicorum. Et propter hoc statim in principio huius libri agitur de corpore, cui necesse est applicari omnia quae tradita sunt de motu in libro physicorum. Now it is in bodies that the three other orders are considered: in one way, insofar as the entire corporeal universe is prior in consideration to its parts; in another way, insofar as simple bodies are considered before the mixed; thirdly, insofar as, among the simple bodies, the first must be considered first, i.e., the heavenly body, through which all the others are sustained. And these three are treated in this book, which the Greeks entitle On the Heavens. For in this book are treated certain things that pertain to the entire universe, as is plain in Book I; and things that pertain to the heavenly body, as is plain in Book II; and things that pertain to the simple bodies, as is plain in Books III-IV. Consequently, it is with good reason that this book is first in order after the book of the Physics. For this reason the first topic of discussion in the very beginning of this book is body, to which must be applied all that was set forth about motion in the Physics.
Quia igitur diversa in hoc libro traduntur, dubium fuit apud antiquos expositores Aristotelis de subiecto huius libri. Alexander enim opinatus est quod subiectum de quo principaliter in hoc libro agitur, sit ipsum universum. Unde, cum caelum tripliciter dicatur, quandoque ipsa ultima sphaera, quandoque totum corpus quod circulariter movetur, quandoque autem ipsum universum, asserit hunc librum intitulari de caelo, quasi de universo vel de mundo: in cuius assertionem assumit quod philosophus in hoc libro determinat quaedam ad totum universum pertinentia, puta quod sit finitum, quod sit unum tantum, et alia huiusmodi. Because diverse things are treated in this book, there was among the early expositors of Aristotle a question about the subject of this book. For Alexander believed that the subject principally treated herein is the universe. Hence, since "the heavens" is subject to a threefold meaning — for sometimes it refers to the outermost sphere, sometimes to the whole body moved circularly, and sometimes to the entire universe — he asserts that this book is entitled On the Heavens as though meaning On the Universe or On the World. In asserting this he assumes that the Philosopher is here determining certain matters pertaining to the entire universe, for example, that it is finite, that it is unique, and things of this sort.
E contrario autem aliis videtur quod subiectum de quo principaliter in hoc libro intenditur, est corpus caeleste quod circulariter movetur; et propter hoc intitulatur de caelo. De aliis autem corporibus determinatur in hoc libro vel ex consequenti, inquantum continentur a caelo et eius influentiam recipiunt, sicut Iamblichus dixit; vel per accidens, inquantum aliorum corporum notitia assumitur ad manifestandum ea quae dicuntur de caelo, ut dixit Syrianus. Sed hoc non videtur probabile: quia postquam philosophus in secundo libro determinavit de caelo, in tertio et quarto subiungit considerationem de aliis simplicibus corporibus, quasi principaliter de eis intendens. Non enim consuevit philosophus principalem partem alicuius scientiae assignare his quae per accidens assumuntur. On the other hand, it seems to some that the main subject handled in this book is the heavenly body which is moved circularly, for which reason it is entitled On the Heavens. Other bodies, however, are discussed therein consequentially, insofar as they are contained by the heavens and influenced by them, as Iamblichus said; or only incidentally, insofar as a knowledge of other bodies is assumed in order to explain what is being said of the heavens, as Syrianus says. But this does not seem probable, for after the Philosopher has finished his discussion of the heavens in Book II, he treats in Books III and IV of the other simple bodies as though they were his main subject. Now the Philosopher is not wont to assign a principal part in some science to things that are brought up only incidentally.
Et ideo aliis visum est, sicut Simplicius dixit, quod intentio philosophi in hoc libro est determinare de simplicibus corporibus, inquantum conveniunt in communi intentione simplicis corporis: et quia inter simplicia corpora principalius est caelum, a quo alia dependent, ideo denominatur totus liber a caelo. Et, sicut dicit, non obstat quod in hoc libro determinantur quaedam quae pertinent ad totum universum: quia huiusmodi conditiones conveniunt universo inquantum conveniunt caelesti corpori, scilicet esse finitum et sempiternum, et alia huiusmodi. Si autem intentio principalis philosophi esset determinare de universo, sive de mundo, oporteret quod Aristoteles considerationem suam extenderet ad omnes partes mundi, etiam usque ad plantas et animalia, sicut Plato in Timaeo. Therefore it seemed to others, as Simplicius said, that the intention of Aristotle in this book is to determine about simple bodies inasmuch as they share in the common notion of simple body; and because among simple bodies. The chief is the heavens, on which the others depend, the entire book gets its name from the heavens. And, so he says, it makes no difference that in this book things pertaining to the whole universe are considered, for the conditions in question belong to the universe insofar as they belong to the heavenly body, i.e., to be finite and eternal, and so on. But if the principal intention of the Philosopher were to determine about the universe or the world, then he would have had to extend his consideration to all the parts of the world, even down to plants and animals, as Plato does in the Timaeus.
Sed eadem ratione possumus arguere contra Simplicium: quia si in hoc libro principaliter intenderet de corporibus simplicibus, oporteret quod omnia quae pertinent ad corpora simplicia in hoc libro traderentur; nunc autem in hoc libro traduntur solum ea quae pertinent ad levitatem et gravitatem ipsorum, alia vero traduntur in libro de generatione. But the same argument could be used against Simplicius, because if Aristotle in this book intended to treat principally of the simple bodies, then in this book he would have had to mention everything that pertains to the simple bodies, whereas he discusses only what pertains to their lightness and heaviness, while he treats the other aspects in the book, On Generation.
Et ideo rationabilior videtur sententia Alexandri, quod subiectum huius libri sit ipsum universum, quod dicitur caelum vel mundus; et quod de simplicibus corporibus determinatur in hoc libro, secundum quod sunt partes universi. Constituitur autem universum corporeum ex suis partibus secundum ordinem situs: et ideo de illis solum partibus universi determinatur in hoc libro, quae primo et per se habent situm in universo, scilicet de corporibus simplicibus. Unde et de quatuor elementis non determinatur in hoc libro secundum quod sunt calida vel frigida, vel aliquid huiusmodi; sed solum secundum gravitatem et levitatem, ex quibus determinatur eis situs in universo. Aliis autem partibus universi, puta lapidibus, plantis et animalibus, non determinatur situs secundum se, sed secundum simplicia corpora: et ideo de his non erat in hoc libro agendum. Et hoc consonat ei quod consuevit apud Latinos dici, quod in hoc libro agitur de corpore mobili ad situm, sive secundum locum: qui quidem motus communis est omnibus partibus universi. 5. Accordingly, the opinion of Alexander appears more reasonable, i.e., that the subject of this book is the universe itself, which is called "the heavens" or "the world," and that determination is made concerning simple bodies in this book accordingly as they are parts of the universe. Now, the corporeal universe is composed of its parts according to an order of position [situs]; consequently this book determines only concerning those parts of the universe that primarily and per se have position in the universe, namely, the simple bodies. That is why the four elements are not dealt with in this book from the aspect of their being hot or cold or something of that sort, but only with respect to their heaviness and lightness, from which their position in the universe is determined. Other parts of the universe, such as stones, plants and animals, have a determined place [situs] in the universe not according to what they are in themselves but according to the simple bodies; consequently, they are not treated in this book. And this agrees with what is usually said among the Latins, that this book discusses body that is mobile with respect to position or place, such motion being common to all the parts of the universe.


Lecture 1
The things it pertains to natural science to consider.

Chapter 1
(268a.) Ἡ περὶ φύσεως ἐπιστήμη σχεδὸν ἡ πλείστη φαίνεται περί τε σώματα καὶ μεγέθη καὶ τὰ τούτων οὖσα πάθη καὶ τὰς κινήσεις, ἔτι δὲ περὶ τὰς ἀρχάς, ὅσαι τῆς τοιαύτης οὐσίας εἰσίν τῶν γὰρ φύσει συνεστώτων τὰ μέν ἐστι σώματα καὶ μεγέθη, τὰ δ' ἔχει σῶμα καὶ μέγεθος, τὰ δ' ἀρχαὶ τῶν ἐχόντων εἰσίν. 1 THE science which has to do with nature clearly concerns itself for the most part with bodies and magnitudes and their properties and movements, but also with the principles of this sort of substance, as many as they may be. For of things constituted by nature some are bodies and magnitudes, some possess body and magnitude, and some are principles of things which possess these.
Quia igitur in hoc libro primo incipit applicare Aristoteles ad corpora, ea quae communiter dicta sunt de motu in libro physicorum, ideo primo prooemialiter ostendit quod ad scientiam naturalem pertinet determinare de corporibus et magnitudinibus; secundo incipit prosequi suum propositum, ibi: continuum quidem et cetera. 6. In this first book Aristotle begins for the first time to apply to bodies the things that were said about motion in a general way in the book of the Physics. For that reason he first shows by way of introduction that it pertains to natural science to determine about bodies and magnitudes; Secondly, he begins to carry out his proposal (Lecture 2).
Circa primum ponit talem rationem. Res naturales sunt corpora et magnitudines, et quae ad haec pertinent: sed scientia naturalis est de rebus naturalibus: ergo scientia naturalis consistit circa corpora et magnitudines. With respect to the first he presents this argument: Natural things are bodies and magnitudes and whatever pertains to these. But natural science is about natural things. Therefore, natural science consists in treating of bodies and magnitudes.
Primo ergo ponit conclusionem, dicens quod scientia quae est de natura, fere plurima, idest in maiori parte, videtur esse existens circa corpora et magnitudines, idest lineas et superficies. De quibus tamen aliter considerat naturalis quam geometra. Naturalis quidem considerat de corporibus inquantum sunt mobilia, de superficiebus autem et lineis inquantum sunt termini corporum mobilium: geometra autem considerat de eis prout sunt quaedam quanta mensurabilia. Et quia ad scientiam pertinet non solum considerare subiecta, sed etiam passiones, ut dicitur in I Poster., ideo subiungit quod naturalis scientia existit circa praedictorum passiones et motus: ut per passiones intelligantur alterationes et alii motus consequentes, secundum quos alteratur aliquid in substantia rei: subdit autem et motus, quasi procedens a speciali ad commune. Vel per motus intelligit specialiter motus locales, qui sunt perfectiores in genere motuum. Vel per passiones intelligit proprietates, per motus autem operationes rerum naturalium, quae non sunt sine motu. Et quia in qualibet scientia oportet considerare principia, subiungit quod naturalis scientia est circa quaecumque principia praedictae substantiae; scilicet corporeae mobilis. Per quod datur intelligi quod ad naturalem pertinet praecipue considerare de corpore inquantum est in genere substantiae, sic enim est subiectum motus: ad geometram autem inquantum est in genere quantitatis, sic enim mensuratur. 7. First [1] therefore, he posits the conclusion, saying that the science which treats of nature seems to be "for the most part" concerned with bodies, and "magnitudes," i.e., lines and surfaces. However, the natural philosopher considers these in a different way from the geometer. For the former treats of bodies insofar as they are mobile, and of surfaces and lines insofar as they are the boundaries of mobile bodies; the geometer, on the other hand, considers them insofar as they are measurable quantities. And because a science should consider not only subjects but also their passions, as is said in Post. Anal. I, he therefore adds that natural science is concerned with the passions and motions of the aforesaid — by "passions" meaning alterations and other consequent motions, with respect to which something is altered in the substance of a thing; and he adds, "and motions," as though going from the particular to the general. Or perhaps by "motions" he specifically understands local motions, which are the more perfect in the genera of motions. Or by "passions" is meant the properties, and by "motions" the operations of natural things, which do not occur without motion. And because, in every science, principles must be considered, he adds that natural science is concerned with any and all the principles of the afore-mentioned substance, namely, mobile corporeal substance. By this we are given to understand that it pertains to natural science primarily to consider body insofar as it is in the genus of substance, for it is in this respect that it is the subject of motion; whereas it pertains to the geometer to consider it insofar as it is in the genus of quantity, for thus it is measured.
Et quia minor est manifesta, scilicet quod scientia naturalis sit de rebus naturalibus, subiungit maiorem, dicens quod ideo scientia naturalis existit circa praedicta, quia eorum quae sunt secundum naturam, quaedam sunt corpora et magnitudines, sicut lapides et alia inanimata; quaedam habent corpus et magnitudinem, sicut plantae et animalia, quorum principalior pars est anima (unde magis sunt id quod sunt secundum animam quam secundum corpus); quaedam vero sunt principia habentium corpus et magnitudinem, sicut anima, et universaliter forma, et materia. Since the minor premise is plain, namely, that natural science is concerned with natural things, he adds the major, saying that the reason why natural science is concerned with the aforementioned is that among things which are according to nature, some are bodies and magnitudes, e.g. stones and other inanimate things; and some have body and magnitude, as do plants and animals, whose principal part is the soul (hence they are what they are more with respect to soul than with respect to body); finally, some things are principles of things having body and magnitude — for example, the soul, and universally form, and matter.
Et ex hoc apparet quare dixit quod scientia de natura fere plurima existit circa corpora et magnitudines: quaedam enim pars eius est circa habentia corpus et magnitudines; est etiam circa principia horum; est etiam circa quaedam quae non sunt in natura, quae aliqui attribuerunt corporibus et magnitudinibus, scilicet circa vacuum et infinitum. From this is clear why he said that the science of nature is "for the most part" concerned with bodies and magnitudes: for one part of this science is concerned with things having body and magnitude; it is also concerned with the principles of these; it is further concerned with some things which do not exist in nature but which some have attributed to bodies and magnitudes, namely, the void and the infinite.

Lecture 2
The perfection of the universe both as body and as containing all.
Chapter 1 cont.
Συνεχὲς μὲν οὖν ἐστι τὸ διαιρετὸν εἰς ἀεὶ διαιρετά, 2 Now a continuum is that which is divisible into parts always capable of subdivision,
σῶμα δὲ τὸ πάντῃ διαιρετόν. 3 and a body is that which is every way divisible.
Μεγέθους δὲ τὸ μὲν ἐφ' ἓν γραμμή, τὸ δ' ἐπὶ δύο ἐπίπεδον, τὸ δ' ἐπὶ τρία σῶμα 4 A magnitude if divisible one way is a line, if two ways a surface, and if three a body.
καὶ παρὰ ταῦτα οὐκ ἔστιν ἄλλο μέγεθος διὰ τὸ τὰ τρία πάντα εἶναι καὶ τὸ τρὶς πάντῃ. 5 Beyond these there is no other magnitude, because the three dimensions are all that there are, and that which is divisible in three directions is divisible in all.
Καθάπερ γάρ φασι καὶ οἱ Πυθαγόρειοι, τὸ πᾶν καὶ τὰ πάντα τοῖς τρισὶν ὥρισται τελευτὴ γὰρ καὶ μέσον καὶ ἀρχὴ τὸν ἀριθμὸν ἔχει τὸν τοῦ παντός, ταῦτα δὲ τὸν τῆς τριάδος. 6 For, as the Pythagoreans say, the world and all that is in it is determined by the number three, since beginning and middle and end give the number of an 'all', and the number they give is the triad.
Διὸ παρὰ τῆς φύσεως εἰληφότες ὥσπερ νόμους ἐκείνης, καὶ πρὸς τὰς ἁγιστείας χρώμεθα τῶν θεῶν τῷ ἀριθμῷ τούτῳ. 7 And so, having taken these three from nature as (so to speak) laws of it, we make further use of the number three in the worship of the Gods.
Ἀποδίδομεν δὲ καὶ τὰς προσηγορίας τὸν τρόπον τοῦτον τὰ γὰρ δύο ἄμφω μὲν λέγομεν καὶ τοὺς δύο ἀμφοτέρους, πάντας δ' οὐ λέγομεν, ἀλλὰ κατὰ τῶν τριῶν ταύτην τὴν κατηγορίαν κατάφαμεν πρῶτον. Ταῦτα δ', ὥσπερ εἴρηται, διὰ τὸ τὴν φύσιν αὐτὴν οὕτως ἐπάγειν ἀκολουθοῦμεν. 8 Further, we use the terms in practice in this way. Of two things, or men, we say 'both', but not 'all': three is the first number to which the term 'all' has been appropriated. And in this, as we have said, we do but follow the lead which nature gives.
Ὥστ' ἐπεὶ τὰ πάντα καὶ τὸ πᾶνκαὶ τὸ τέλειον οὐ κατὰ τὴν ἰδέαν διαφέρουσιν ἀλλήλων, ἀλλ' εἴπερ, ἐν τῇ ὕλῃ καὶ ἐφ' ὧν λέγονται, τὸ σῶμα μόνον ἂν εἴη τῶν μεγεθῶν τέλειον μόνον γὰρ ὥρισται τοῖς τρισίν, τοῦτο δ' ἐστὶ πᾶν. Τριχῇ δὲ ὂν διαιρετὸν πάντῃ διαιρετόν ἐστιν τῶν δ' ἄλλων τὸ μὲν ἐφ' ἓν τὸ δ' ἐπὶ δύο ὡς γὰρ τοῦ ἀριθμοῦ τετυχήκασιν, οὕτω καὶ τῆς διαιρέσεωςκαὶ τοῦ συνεχοῦς τὸ μὲν γὰρ ἐφ' ἓν συνεχές, τὸ δ' ἐπὶ δύο, τὸ δὲ πάντῃ τοιοῦτον. 9 Therefore, since 'every' and 'all' and 'complete' do not differ from one another in respect of form, but only, if at all, in their matter and in that to which they are applied, body alone among magnitudes can be complete. For it alone is determined by the three dimensions, that is, is an 'all'. But if it is divisible in three dimensions it is every way divisible, while the other magnitudes are divisible in one dimension or in two alone: for the divisibility and continuity of magnitudes depend upon the number of the dimensions, one sort being continuous in one direction, another in two, another in all.
Ὅσα μὲν οὖν διαιρετὰ τῶν μεγεθῶν, καὶ συνεχῆ ταῦτα εἰ δὲ καὶ τὰ συνεχῆ πάντα διαιρετά, οὔπω δῆλον ἐκ τῶν νῦν. Ἀλλ' ἐκεῖνο μὲν δῆλον, ὡς οὐκ (268b.) ἔστιν εἰς ἄλλο γένος μετάβασις, ὥσπερ ἐκ μήκους εἰς ἐπιφάνειαν, εἰς δὲ σῶμα ἐξ ἐπιφανείας οὐ γὰρ ἂν ἔτι τὸ τοιοῦτον τέλειον εἴη μέγεθος ἀνάγκη γὰρ γίγνεσθαι τὴν ἔκβασιν κατὰ τὴν ἔλλειψιν, οὐχ οἷόν τε δὲ τὸ τέλειον ἐλλείπειν πάντῃ γάρ ἐστιν. 10 All magnitudes, then, which are divisible are also continuous. Whether we can also say that whatever is continuous is divisible does not yet, on our present grounds, appear. One thing, however, is clear. We cannot pass beyond body to a further kind, as we passed from length to surface, and from surface to body. For if we could, it would cease to be true that body is complete magnitude. We could pass beyond it only in virtue of a defect in it; and that which is complete cannot be defective, since it has being in every respect.
Τῶν μὲν οὖν ἐν μορίου εἴδει σωμάτων κατὰ τὸν λόγον ἕκαστον τοιοῦτόν ἐστιν πάσας γὰρ ἔχει τὰς διαστάσεις ἀλλ' ὥρισται πρὸς τὸ πλησίον ἁφῇ διὸ τρόπον τινὰ πολλὰ τῶν σωμάτων ἕκαστόν ἐστιν. 11 Now bodies which are classed as parts of the whole are each complete according to our formula, since each possesses every dimension. But each is determined relatively to that part which is next to it by contact, for which reason each of them is in a sense many bodies.
Τὸ δὲ πᾶν οὗ ταῦτα μόρια, τέλειον ἀναγκαῖον εἶναι καὶ καθάπερ τοὔνομα ημαίνει πάντῃ, καὶ μὴ τῇ μὲν τῇ δὲ μή. 12 But the whole of which they are parts must necessarily be complete, and thus, in accordance with the meaning of the word, have being, not in some respect only, but in every respect.
Postquam philosophus ostendit prooemialiter quod determinandum est de corporibus et magnitudinibus in scientia naturali, hic incipit prosequi principale propositum. Et quia, ut supra dictum est, in hoc libro principaliter intendit Aristoteles determinare de universo corporeo et principalibus partibus eius, quae sunt corpora simplicia, inter quae potissimum est corpus caeleste, ideo dividitur liber iste in partes tres: After showing by way of introduction that bodies and magnitudes are to be studied in natural science, the Philosopher here begins to carry out his main resolve. And because, as was said above, Aristotle in this book is mainly concerned with determining about the corporeal universe and its principal parts which are the simple bodies, among which the most important is the heavenly body, the book therefore is divided into three parts:

in prima determinat de universo corporeo;

in secunda determinat de corpore caelesti, et hoc in secundo libro, ibi: quod quidem igitur neque factum est etc.;

in tertia parte determinat de aliis simplicibus corporibus, scilicet de gravi et levi, in tertio libro, ibi: de primo quidem igitur caelo et cetera.

In the first he determines concerning the corporeal universe;

In the second concerning the heavenly body, in Book II;

In the third about other simple bodies, i.e., heavy and light, in Book III.

Circa primum duo facit: With respect to the first he does two things:

primo ostendit perfectionem universi;

secundo determinat quasdam conditiones seu proprietates ipsius, ibi: sed quoniam manifestum de his et cetera.

First he shows the perfection of the universe;

Secondly, he determines certain of its conditions or properties (L. 13. 9).

Circa primum duo facit: About the first he does two things:

primo ostendit perfectionem universi;

secundo ostendit ex quibus partibus eius perfectio integretur, ibi: de totius quidem igitur natura et cetera.

First he shows the perfection of the universe;

Secondly, he explains of what parts it is composed (L. 13).

Circa primum duo facit: As to the first he does two things:

primo ostendit perfectionem universi quam habet secundum communem rationem sui generis, inquantum scilicet est corpus;

secundo probat perfectionem propriam ipsius, ibi: partialium quidem igitur corporum et cetera.

First he shows the perfection which the universe has in virtue of the common notion of its genus, i.e., inasmuch as it is a body, at 9.

Secondly, he proves the perfection proper to it, at 18.

Circa primum tria facit: About the first he does three things:

primo manifestat definitionem corporis, qua utitur ad propositum ostendendum;

secundo probat propositum, ibi: itaque quoniam omne et totum etc.;

tertio ostendit quid ex praemissis possit esse manifestum, ibi: quaecumque quidem igitur et cetera.

First he explains the definition of body, to be used in proving his proposition, at 10.

Secondly, he proves the proposition, at 15;

Thirdly, he shows what could be clear from the foregoing, at 16.

Circa primum duo facit: As to the first he does two things:

primo definit continuum, quod est genus corporis;

secundo manifestat corporis definitionem, ibi: corpus autem et cetera.

First he defines "continuum," which is the genus of body;

Secondly, he clarifies the definition of body, at 10.

Circa primum considerandum est quod continuum invenitur a philosopho dupliciter definitum. Uno modo definitione formali, prout dicitur in praedicamentis quod continuum est cuius partes copulantur ad unum communem terminum: unitas enim continui est quasi forma ipsius. Alio modo definitione materiali, quae sumitur ex partibus, quae habent rationem materiae, ut dicitur in II Physic.: et sic definitur hic, quod continuum est quod est divisibile in semper divisibilia. Nulla enim pars continui potest esse indivisibilis: quia ex indivisibilibus non componitur aliquod continuum, ut probatur in VI Physic. Et satis convenienter haec definitio ponitur hic, alia autem in praedicamentis: quia consideratio naturalis versatur circa materiam, consideratio autem logici circa rationem et speciem. With regard to the first [2], we must consider that the continuum is found defined in two ways by the Philosopher. In one way with a formal definition, where it is said in the Predicaments (c.4) that the continuum is "that whose parts are joined at one common term"; for the unity of a continuum is, as it were, its form. In another way, with a material definition taken from the parts, which have the aspect of matter, as is said in Physics II — and it is thus that the continuum is defined here, namely, as "what is divisible into parts always divisible." For no part of a continuum can be indivisible, because no continuum is composed of indivisibles, as is proved in Physics VI. And it is fitting that this latter definition be used here, and the other one in the Predicaments, because the consideration of natural science is concerned with matter, while that of logic is concerned with notions and species.
Deinde cum dicit: corpus autem etc., definit corpus. 10.   Then at [3] he defines "body."

Et primo proponit definitionem, dicens quod corpus est continuum quod est divisibile omniquaque, idest ad omnem partem, vel secundum omnem dimensionem.

Secundo ibi: magnitudinis autem etc., probat propositam definitionem tali ratione.

First he proposes the definition, saying that body is "a continuum which is divisible in every way," i.e., at every part or according to every dimension.

Secondly, at [4] he proves the proposed definition with this argument:

Corpus dividitur secundum tres dimensiones: quod autem dividitur secundum tres dimensiones, dividitur secundum omnes: ergo corpus est divisibile secundum omnes dimensiones. Body is divided according to three dimensions. But what is divided according to three dimensions is divided according to all. Therefore, body is divisible according to all the dimensions.
Primo ergo manifestat minorem, quasi per divisionem. Nam magnitudinum quaedam est quae dividitur ad unam partem, et haec dicitur linea: quaedam autem est quae dividitur ad duas partes, et haec dicitur planum, idest superficies: quaedam autem est quae dividitur secundum tres dimensiones; et cum talis magnitudo non sit linea neque superficies, sequitur quod sit corpus. First, therefore, he explains the minor proposition as though by division. For among magnitudes there is one which is divided with respect to one dimension, and this is called "line"; another is divided with respect to two dimensions, and this is called "plane," i.e., a surface; still another is divided according to three dimensions, and since such a magnitude is neither line nor surface, it follows that it is body.
Maiorem propositionem ponit ibi: et praeter has et cetera. Et primo ponit eam: et dicit quod praeter has magnitudines seu dimensiones non est alia magnitudo seu dimensio, propter hoc quod tria habent rationem ut sint omnia, quia habent rationem cuiusdam totalitatis; et quod est ter, videtur esse omniquaque, vel omnino, idest secundum omnem modum. The major proposition he gives at [5]. First he mentions it and says that, besides these magnitudes or dimensions, there is no other magnitude or dimension, on the ground that "three" has the property of being all, because it implies a certain totality, and because whatever is thrice seems to be "in all ways" and "entirely," i.e., according to every mode.
Secundo ibi: quemadmodum enim etc., probat quod dixerat tripliciter. 11. Secondly, at [6] he proves what he had said in three ways.
Primo quidem secundum rationem Pythagoricorum, qui dixerunt quod id quod dicitur totum et omne, determinatur ternario numero. Principium enim et medium et consummatio, idest finis, habent numerum qui convenit toti et omni: in rebus enim divisibilibus prima pars non sufficit ad integritatem totius, quod constituitur per ultimum, ad quod a principio pervenitur per medium. Haec autem, scilicet principium, medium et finis, habent numerum ternarium: et sic patet quod numerus ternarius convenit omni et toti. First, according to the teaching of Pythagoras who said that what is called "whole" and "all" is determined by the number 3. For the beginning and the middle and the "consummation," i.e., the end, have a number which befits what is "whole" and "all" — for in things divisible, the first part is not enough to complete the whole, which is completed by the ultimate that is reached by passing from the beginning through the middle. But these three, namely, beginning, middle and end, have 3 as their number. Consequently, it is clear that the number 3 belongs to the "all" and "whole."
Secundo ibi: propter quod a natura etc., probat idem per ea quae in cultu divino observantur. Utimur enim numero hoc, scilicet ternario, ad sanctificationes deorum (quos scilicet gentiles colebant), idest in sacrificiis et laudibus ipsorum, ac si acceperimus a natura leges et regulas ipsius: ut scilicet, sicut natura perficit omnia ternario numero, ita illi qui instituerunt cultum divinum, volentes Deo attribuere omne quod perfectum est, attribuunt ei ternarium numerum. 12. Secondly, at [7] he proves the same by means of what is observed in divine worship. For we use this number 3 "in the worship of the gods" (whom, namely, the gentiles worshipped), i.e., in sacrifices and praises for them, as though we should receive from nature its laws and rules: just as nature completes all things with the number 3, so those who established the divine worship have, in their desire to attribute to God everything perfect, attributed to Him the number 3.
Tertio ibi: assignamus autem etc., probat idem per communem usum loquendi. Et dicit quod etiam assignamus vocabula rebus secundum modum praedictum, quo scilicet perfectio competit ternario. Si enim aliqua sunt duo, dicimus quod sint ambo, et duos homines dicimus ambos: non autem de his dicimus omnes, sed primo hoc vocabulo utimur circa tres. Et istum modum loquendi sequimur communiter omnes, propter hoc quod natura ad hoc nos inclinat. Ea enim quae sunt propria singulis in modo loquendi, videntur provenire ex propriis conceptionibus uniuscuiusque: sed id quod observatur communiter apud omnes, videtur ex naturali inclinatione provenire. 13. Thirdly, he proves at [8] the same by appealing to the general way we speak. And he says that we even assign names to things according to the aforementioned method, in which perfection agrees with the number 3. For when there are two things, we say "both," — thus we speak of two men as "both" — but we do not say "all," which we use for the first time in the case of three. And we all in general use this way of speaking, because nature so inclines us. For whatever is peculiar to individuals in their way of speaking seems to arise from the particular conceptions of each, but what is generally observed among all would seem to arise from natural inclination.
Est autem attendendum quod nusquam alibi Aristoteles invenitur Pythagoricis rationibus utens ad propositum ostendendum; neque invenitur alibi per numerorum proprietates aliquid de rebus concludere: et forte hoc hic facit propter affinitatem numerorum ad magnitudines, de quibus hic agitur. 14. Now, it should be noted that nowhere else does Aristotle either use the arguments of Pythagoras to explain a proposition, or from the properties of numbers conclude anything about things. And perhaps he does so here on account of the affinity of numbers to magnitudes, which he is now considering.
Videtur tamen quod haec probatio non sit efficax: non enim magis videtur sequi quod dimensiones sint tres, propter hoc quod ternarius est numerus totius et omnis: alioquin sequeretur per eandem rationem quod essent solum tria elementa, vel tres digiti manus. Be that as it may, the proof here given does not seem valid, for it does not seem, if 3 is the number corresponding to "whole" and "all" that it follows there are three dimensions. Otherwise, it would follow according to the same reasoning that there would be only three elements or only three fingers on the hand.
Sed sciendum est quod, sicut dicit Simplicius in commento, Aristoteles non procedit hic demonstrative, sed secundum probabilitatem: et hic modus sufficiens est post demonstrationes praemissas, vel praesuppositas ab alia scientia. Manifestum est autem quod determinare de dimensionibus corporum inquantum huiusmodi, per se pertinet ad mathematicum: naturalis autem assumit a mathematico ea quae circa dimensiones considerat. Et ideo probare demonstrative esse solum tres dimensiones, pertinet ad mathematicum: sicut Ptolomaeus probat per hoc quod impossibile est coniungi simul lineas perpendiculares plures quam tres super idem punctum; omnis autem dimensio mensuratur secundum aliquam lineam perpendicularem. Huius igitur demonstrationem Aristoteles supponens a mathematico, utitur testimonio et signis, sicut consuevit facere post demonstrationes a se inductas. But it should be known that, as Simplicius says in his Commentary 13, Aristotle is not here proceeding demonstratively but according to probability, and this is sufficient after previous demonstrations or ones supposed from another science. Now, it is plain that the task of deciding about the dimensions of bodies as such pertains to mathematics; and whatever the natural philosopher considers with dimensions, he takes from mathematics. Therefore, to prove demonstratively that there are just three dimensions pertains to mathematics — thus Ptolemy proves it by showing that it is impossible for more than three perpendicular lines to meet at the same point, while each dimension is measured according to a perpendicular line. Supposing such a demonstration from mathematics, Aristotle here uses testimony and signs, just as he customarily does after his own demonstrations.
Deinde cum dicit: itaque quoniam omne etc., ex eo quod ostensum est, procedit ad principale propositum ostendendum. Et dicit quod haec tria, omne et totum et perfectum, non differunt ab invicem secundum speciem, idest secundum formalem rationem, quia omnia important integritatem quandam: sed si in aliquo differant, differunt in materia et subiecto, inquantum de diversis dicuntur. Nam hoc quod dicitur omne, utimur in discretis, sicut dicimus omnem hominem: utimur etiam eo in continuis quae sunt propinqua divisioni, sicut dicimus omnem aquam et omnem aerem. Totum autem dicitur et in his et in continuis: dicimus enim totum populum et totum lignum. Perfectum autem dicimus et in his et in formis: dicimus enim perfectam albedinem et perfectam virtutem. Quia igitur omne et perfectum est idem, consequens est quod corpus sit perfectum inter magnitudines: quia solum corpus est determinatum tribus dimensionibus, et hoc habet rationem omnis, ut supra ostensum est: cum enim sit tribus modis divisibile, sequitur quod sit divisibile omniquaque, idest secundum omnem dimensionem. Sed inter alias magnitudines aliquid est divisibile secundum duas dimensiones, scilicet superficies; aliud autem secundum unam, scilicet linea. Ut enim numerum adepta sunt, idest sicut magnitudines habent numerum dimensionum, ita habent divisionem et continuitatem: ita scilicet quod aliqua magnitudo est continua secundum unum modum, scilicet linea; alia est continua duobus modis, scilicet superficies; corpus autem est continuum secundum omnem modum. Unde patet quod corpus est magnitudo perfecta, quasi habens omnem modum continuitatis. 15. Then at [9] he goes on to manifest the main proposition from what has been shown. And he says that these three, namely, "all," "whole," and "perfect," do not differ from one another according to species, i.e., according to their formal notion, because all imply a certain completeness; but if they do differ in any way, it is in matter and subject, insofar as they are said of diverse things. For we use "all" in discrete things, as we say "all men"; we use it also with respect to continua which are easily divided, as we say "all water" and "all air." "Whole," however, is used both with these and with all continua, as we say "the whole people" and "the whole world." But "perfect" is used with respect to these and forms: for we say "perfect whiteness" and "perfect virtue." Therefore, because "all" and "perfect" are the same, the consequence is that among magnitudes the perfect one is body, because only a body is determined by three dimensions, and this carries with it the notion of "all," as has been shown above, for since it is divisible in three ways, it follows that it is divisible in every way, i.e., according to every dimension. But among other magnitudes, there is one divisible according to two dimensions, namely, a surface; and another according to one, namely, a line. "Now according to the number that it has," i.e., the number of dimensions that a magnitude has, so is it divisible and continuous. Thus one magnitude is continuous in one way, namely, a line; another in two ways, namely, a surface; but a body is continuous in every way. Hence it is plain that body is a perfect magnitude, as possessing all ways of being continuous.
Deinde cum dicit: quaecumque quidem igitur etc., ostendit quid ex praemissis manifestum sit vel non: et ponit tria. Quorum primum secundum se manifestum est, scilicet quod quaecumque magnitudo est divisibilis, sit continua: si enim non esset continua, non haberet rationem magnitudinis, sed potius numeri. Secundum autem est conversum huius, scilicet quod omne continuum sit divisibile, sicut in definitione fuit positum. Et hoc quidem manifestum est ex his quae probata sunt in VI Physic., ut supra dictum est. Non est autem manifestum ex his quae nunc dicta sunt: quia quod continuum sit divisibile, hic supposuit, non probavit. Tertium est manifestum ex praemissis, scilicet quod non fit transitus a corpore in aliud genus magnitudinis, sicut fit transitus ex longitudine in superficiem, et ex superficie in corpus. Et utitur modo loquendi quo utuntur geometrae, imaginantes quod punctus motus facit lineam, linea vero mota facit superficiem, superficies autem corpus. A corpore autem non fit transitus ad aliam magnitudinem: quia talis exitus, sive processus, ad aliud genus magnitudinis, est secundum defectum eius a quo transitur (unde etiam motus naturalis est actus imperfecti). Non est autem possibile quod corpus, quod est perfecta magnitudo, deficiat secundum hanc rationem, quia est continuum secundum omnem modum: et ideo non potest fieri transitus a corpore in aliud genus magnitudinis. 16. Then at [10] he shows what is or is not plain from the foregoing. And he mentions three things. The first of these is plain in itself, namely, that any magnitude that is divisible is continuous; for if it were not continuous, it would not be a magnitude but a number. The second is the converse of this, namely, that every continuum is divisible, as was indicated in the definition. And this is plain from what was proved in Physics VI, as was said above. But it is not plain from what was just said, however, because here he supposes, but does not prove, that a continuum is divisible. The third thing is plain from the foregoing, namely, that unlike the passing from length to surface and from surface to body, there is no passing from body to another kind of magnitude. And he uses a way of speaking employed by geometers imagining that a point in motion makes a line, and a line in motion a surface, and a surface a body. But from body there is no transition to another magnitude, because such a passing, i.e., to another kind of magnitude is due to a defect in that from which the process beings — that is why natural motion is the act of an imperfect thing. But it is not possible that body, which is perfect magnitude, should be defective in this way, because it is continuous in every way. Consequently, no transition from body to another kind of magnitude is possible.
Deinde cum dicit: partialium quidem etc., manifestat propriam perfectionem universi, per differentiam ad corpora particularia. Et primo ponit qualiter particularia corpora se habeant ad perfectionem. Et dicit quod unumquodque particularium corporum, secundum rationem communem corporis, est tale, idest perfectum, inquantum habet omnes dimensiones: sed tamen terminatur ad proximum corpus, inquantum contingit ipsum. Et ita unumquodque talium corporum quodammodo est multa, idest perfectum, inquantum habet omnes dimensiones, et imperfectum, inquantum habet aliud corpus extra se ad quod terminatur. Vel est multa secundum contactum ad diversa corpora: vel est multa, quia sunt plura unius speciei propter imperfectionem; quod non contingit de universo. 17. Then at [113 he manifests the proper perfection of the universe based on its difference from particular bodies. First he mentions how particular bodies are related to perfection. And he says that each particular body, according to the common notion of body, is such, i.e., perfect, inasmuch as it has three dimensions; nevertheless, it is terminated at an adjacent body, inasmuch as it touches it. And thus every such body is in a certain way "many," i.e., perfect, in having three dimensions, but imperfect in having another body outside it at which it is terminated. Or it is "many" according to contact with diverse bodies; or it is "many" because there are more than one in one species due to imperfection, whereas such is not the case with the universe.
Secundo ibi: totum autem etc., ostendit quomodo universum se habeat ad perfectionem. Et dicit quod totum, idest universum, cuius partes sunt particularia corpora, necesse est quod sit perfectum omnibus modis; et sicut ipsum nomen universi significat, omniquaque, idest omnibus modis, perfectum, et non secundum unum modum ita quod non secundum alium: quia et habet omnes dimensiones, et comprehendit in se omnia corpora. 18. Secondly at [12] he shows how the universe is related to perfection. And he says that "the whole," i.e., the universe, which has particular bodies as its parts, must be perfect in all ways, for the word "universe" signifies perfect "in all ways," and not in one way to the exclusion of some other way, and it both has all the dimensions, and includes in itself all bodies.

Lecture 3:
Preliminary notions for showing the parts perfecting the universe.
Chapter 2
Περὶ μὲν οὖν τῆς τοῦ παντὸς φύσεως, εἴτ' ἄπειρός ἐστι κατὰ τὸ μέγεθος εἴτε πεπέρανται τὸν σύνολον ὄγκον, ὕστερον ἐπισκεπτέον περὶ δὲ τῶν κατ' εἶδος αὐτοῦ μορίων νῦν λέγωμεν ἀρχὴν ποιησάμενοι τήνδε. 13 The question as to the nature of the whole, whether it is infinite in size or limited in its total mass, is a matter for subsequent inquiry. We will now speak of those parts of the whole which are specifically distinct. Let us take this as our starting-point.
Πάντα γὰρ τὰ φυσικὰ σώματα καὶ μεγέθη καθ' αὑτὰ κινητὰ λέγομεν εἶναι κατὰ τόπον τὴν γὰρ φύσιν κινήσεως ἀρχὴν εἶναί φαμεν αὐτοῖς. 14 All natural bodies and magnitudes we hold to be, as such, capable of locomotion; for nature, we say, is their principle of movement.
Πᾶσα δὲ κίνησις ὅση κατὰ τόπον, ἣν καλοῦμεν φοράν, ἢ εὐθεῖα ἢ κύκλῳ ἢ ἐκ τούτων μικτή 15 But all movement that is in place, all locomotion, as we term it, is either straight or circular or a combination of these two,
ἁπλαῖ γὰρ αὗται δύο μόναι. Αἴτιον δ' ὅτι καὶ τὰ μεγέθη ταῦτα ἁπλᾶ μόνον, ἥ τ' εὐθεῖα καὶ ἡ περιφερής. 16 which are the only simple movements. And the reason of this is that these two, the straight and the circular line, are the only simple magnitudes.
Κύκλῳ μὲν οὖν ἐστιν ἡ περὶ τὸ μέσον, 17 Now revolution about the centre is circular motion,
εὐθεῖα δ' ἡ ἄνω καὶ κάτω. Λέγω δ' ἄνω μὲν τὴν ἀπὸ τοῦ μέσου, κάτω δὲ τὴν ἐπὶ τὸ μέσον. 18 while the upward and downward movements are in a straight line, 'upward' meaning motion away from the centre, and 'downward' motion towards it.
Ὥστ' ἀνάγκη πᾶσαν εἶναι τὴν ἁπλῆν φορὰν τὴν μὲν ἀπὸ τοῦ μέσου, τὴν δ' ἐπὶ τὸ μέσον, τὴν δὲ περὶ τὸ μέσον. 19 All simple motion, then, must be motion either away from or towards or about the centre.
Καὶ ἔοικεν ἠκολουθηκέναι κατὰ λόγον τοῦτο τοῖς ἐξ ἀρχῆς τό τε γὰρ σῶμα ἀπετελέσθη ἐν τρισὶ καὶ ἡ κίνησις αὐτοῦ. 20 This seems to be in exact accord with what we said above: as body found its completion in three dimensions, so its movement completes itself in three forms.
Ἐπεὶ δὲ τῶν σωμάτων τὰ μέν ἐστιν ἁπλᾶ τὰ δὲ σύνθετα ἐκ τούτων (λέγω δ' ἁπλᾶ μὲν ὅσα κινήσεως ἀρχὴν ἔχει κατὰ φύσιν, οἷον πῦρ καὶ γῆν καὶ τὰ τούτων εἴδη καὶ τὰ συγγενῆ τούτοις), ἀνάγκη καὶ τὰς κινήσεις εἶναι τὰς μὲν ἁπλᾶς τὰς δὲ μικτάς πως, (269a.) καὶ τῶν μὲν ἁπλῶν ἁπλᾶς, μικτὰς δὲ τῶν συνθέτων, κινεῖσθαι δὲ κατὰ τὸ ἐπικρατοῦν. 21 Bodies are either simple or compounded of such; and by simple bodies I mean those which possess a principle of movement in their own nature, such as fire and earth with their kinds, and whatever is akin to them. Necessarily, then, movements also will be either simple or in some sort compoundᾰsimple in the case of the simple bodies, compound in that of the compositeᾰand in the latter case the motion will be that of the simple body which prevails in the composition.
Postquam philosophus ostendit universum esse perfectum et ratione suae corporeitatis et ratione suae universitatis, hic ostendit ex quibus partibus eius perfectio integratur. After showing that the universe is perfect by reason both of its corporeity and its universalness, the Philosopher here shows from which parts its perfection is made up.

Et primo dicit de quo est intentio;

secundo ostendit propositum, ibi: omnia enim physica corpora et cetera.

First he expresses his intention;

Secondly, he proves his proposition, at 20.

Circa primum considerandum est quod, sicut dicitur in III Physic., antiqui dixerunt infinitum esse extra quod nihil est. Quia igitur probavit universum esse perfectum ex hoc quod nihil est extra ipsum, sed omnia complectitur, posset aliquis suspicari ipsum esse infinitum. Et ideo huic opinioni occurrens, concludit subdens quod posterius intendendum est quantum ad naturam totius universi, si est infinitum secundum magnitudinem, sive finitum secundum totam suam molem. Interim tamen, antequam hoc tractetur, dicendum est de partibus eius quae sunt secundum speciem, in quibus scilicet integritas speciei ipsius consistit, cuiusmodi sunt simplicia corpora. Nam animalia et plantae et alia huiusmodi sunt secundariae partes eius, quae magis pertinent ad bene esse ipsius quam ad primam eius integritatem. Et hanc considerationem inchoabimus a principio infra posito. With respect to the first [13] it should be considered that, as is said in Physics III, the ancients described the infinite as "that outside of which there is nothing." Now, since he has proved that the universe is perfect on the ground that nothing is outside it, but that it embraces all things, one might think it to be infinite. Accordingly, meeting this opinion, he concludes by adding that later on, in discussing the nature of the whole universe, there will be treated the question of whether it is infinite in magnitude, or finite with respect to its total mass. But meanwhile, before treating of this, something must be said about those parts of it that are "according to the species," namely, those parts in which the integrity of its species consists, and which are the simple bodies. For animals and plants and other such are its secondary parts, and pertain more to the well-being of the universe than to its basic integrity. And we shall begin this consideration from a principle given below.
Deinde cum dicit: omnia enim physica etc., ostendit propositum, scilicet ex quibus partibus principalibus perfecta species universi integretur. 20. Then at [14] he starts to manifest the proposition stating of which principal parts the perfect species of the universe is made.

Et primo ostendit quod praeter quatuor elementa, necesse est esse aliud corpus simplex;

secundo ostendit quod praeter quinque corpora simplicia non est aliud corpus, ibi: manifestum autem ex dictis et cetera.

First he shows that in addition to the four elements, another simple body must exist;

Secondly, that there is no simple body other than these five (L. 8).

Circa primum duo facit: About the first he does two things:

primo ostendit esse quintum corpus praeter quatuor elementa;

secundo ostendit differentiam eius ad quatuor elementa, ibi: quoniam autem haec quidem supponuntur et cetera.

First he shows that there is a fifth body besides the four elements;

Secondly, how it differs from the four elements (L. 5).

Circa primum duo facit: With respect to the first he does two things:

primo praemittit quaedam quae sunt necessaria ad propositum ostendendum;

secundo argumentatur ad propositum, ibi: si quidem igitur est simplex motus et cetera.

First he mentions some preliminary facts needed in proving his proposition;

Secondly, he argues to the proposition (L. 4).

Circa primum duo facit: About the first he does two things:

primo praemittit quaedam quae pertinent ad motus;

secundo ponit quaedam quae pertinent ad corpora mobilia, ibi: quoniam autem corporum haec quidem et cetera.

First he premises facts regarding motion;

Secondly, facts pertaining to mobile bodies, at 32.

Circa primum duo facit: About the first he does two things:

primo praemittit continuitatem motus localis ad corpora naturalia;

secundo ponit distinctionem motuum localium, ibi: omnis autem et cetera.

First he mentions the connection between local motion and mobile bodies;

Secondly, he distinguishes the kinds of local motion, at 23.

Dicit ergo primo quod omnia corpora physica, idest naturalia, dicimus esse mobilia secundum locum secundum seipsa, idest secundum sui naturam; et similiter alias magnitudines naturales, puta superficies et lineas, prout sunt termini naturalium corporum; ita tamen quod corpora per se moventur, aliae tamen magnitudines per accidens, motis corporibus. Et ad huius probationem inducit definitionem naturae, quae est principium motus in eis in quibus est, ut dicitur in II Physic. Ex hoc autem sic argumentatur. Corpora naturalia sunt quae habent naturam: sed natura est principium motus in eis in quibus est: ergo corpora naturalia habent principium motus in seipsis. Sed quaecumque moventur quocumque motu, moventur localiter, non autem e converso, ut patet in VIII Physic., eo quod motus localis est primus motuum. Omnia ergo corpora naturalia moventur naturaliter motu locali, non autem omnia aliquo aliorum motuum. 21. He says therefore first [14] that all physical, i.e., natural, bodies are said to be mobile with respect to place according to themselves, i.e., according to their very natures, and the same is true for other natural magnitudes, e.g. planes and lines, insofar as they are the boundaries of natural bodies. And this is true in the sense that bodies are moved per se, but the other magnitudes per accidens, when the bodies are moved. In proof of this he adduces the definition of nature, which is "the principle of motion in those things in which it exists," as is said in Physics II. From this he argues thus: Natural bodies are ones that have a nature, but nature is a principle of motion in things in which it is; therefore, natural bodies have a principle of motion in them. But whatever is moved with any sort of motion is moved locally, but not conversely, as is plain in Physics VII, because local motion is the first of motions. Therefore all natural bodies are naturally moved with a local motion, but not all of them with all of the other motions.
Sed videtur hoc esse falsum: caelum enim est corpus naturale, nec tamen eius motus videtur esse a natura, sed magis ab aliquo intellectu, sicut ex his quae determinantur in VIII Physic. et XII Metaphysic. patet. This, however, seems to be false: for the heavens are a natural body, but their motion seems to be due, not to nature but to intellect, as is plain from what has been determined in Physics VIII and Metaphysics XII.
Sed dicendum est quod duplex est principium motus: unum quidem activum, quod est ipse motor, et tale principium motus animalium est anima: aliud autem est principium motus passivum, scilicet secundum quod corpus habet aptitudinem ut sic moveatur, et huiusmodi principium motus est in gravibus et levibus. Non enim componuntur ex movente et moto, ut philosophus dicit in VIII Physic.: quod quidem, inquit, nihil horum, scilicet gravium et levium, ipsum movet seipsum, manifestum est: sed motus habent principium, non movendi neque faciendi, sed patiendi. Sic igitur dicendum est quod principium activum motus caelestium corporum est intellectualis substantia: principium autem passivum est natura illius corporis, secundum quam natum est tali motu moveri. Et esset simile in nobis si anima non moveret corpus nostrum nisi secundum naturalem inclinationem eius, scilicet deorsum. But it must be said that there are two kinds of principles of motion: one is active, i.e., the mover, as the soul is the active principle of the motion of animals; the other is a passive principle of motion, i.e., a principle according to which a body has an aptitude to be thus moved, and such a principle of motion exists in the heavy and the light. For these are not composed of a mover and a moved, because, as the Philosopher says in Physics VIII, "it is plain that none of these — i.e., the heavy and the light — moves itself, but each has, with respect to its motion, a principle not of causing motion or of acting, but of being acted upon." Consequently, it must be said that the active principle of the motion of heavenly bodies is an intellectual stance, but the passive principle is that body's nature according to which it is apt to be moved with such a motion. And the same situation would prevail in us, if the soul did not move our body in any way other than according to its natural inclination, namely, down.
Deinde cum dicit: omnis autem motus etc., ponit distinctionem localium motuum. 23. Then at [15] he distinguishes local motions.

Et primo distinguit communiter motus locales tam compositos quam simplices;

secundo distinguit motus simplices, ibi: circulatio quidem igitur et cetera.

First he distinguishes in a general way both composite and simple local motions;

Secondly, he distinguishes simple motions, at 27.

Circa primum duo facit. With respect to the first he does two things:
Primo proponit quod intendit, scilicet quod omnis motus localis (qui vocatur latio) aut est circularis, aut rectus, aut mixtus ex his, sicut motus obliquus eorum quae hac illacque feruntur. First at [15] he proposes what he intends, namely, that every local motion — which is called latio — is either circular, or straight, or composed of these, as is the oblique motion of things that are borne this way and that.
Secundo ibi: simplices enim etc., probat quod dixerat, per hoc quod motus simplices non sunt nisi duo, scilicet rectus et circularis. Et huius causam assignat ex hoc quod solae sunt duae magnitudines simplices, scilicet recta et circularis: motus autem localis secundum loca specificatur, sicut et quilibet alius motus secundum suos terminos. Secondly, at [16] he proves what he had said, on the ground that there are just two simple motions, the straight and the circular. And the reason for this, he says, is that there exist just two simple magnitudes, namely, the straight and the circular: but local motion is specified according to places, just as every other motion is specified according to its termini.
Sed videtur quod probatio Aristotelis non sit conveniens: quia, ut dicitur in I Poster., transcendentem in aliud genus non contingit demonstrare. Inconvenienter igitur per divisionem magnitudinum, quae pertinet ad mathematicum, concluditur aliquid circa motus, qui pertinent ad naturalem. 24. But it seems that Aristotle's proof is not suitable, because, as is said in Post. Anal. I, one does not demonstrate who crosses into another genus. Consequently, it seems unfitting to use the division of magnitudes, which pertain to mathematics, in order to reach a conclusion about motion, which pertains to natural science.
Sed dicendum quod scientia quae se habet ex additione ad aliam, utitur principiis eius in demonstrando, sicut geometria utitur principiis arithmeticae: magnitudo enim addit positionem supra numerum, unde punctus dicitur esse unitas posita. Similiter autem corpus naturale addit materiam sensibilem supra magnitudinem mathematicam: et ideo non est inconveniens si naturalis in suis demonstrationibus utatur principiis mathematicis: non enim est omnino aliud genus, sed quodammodo sub illo continetur. But it must be said that a science which is by addition to some other science uses the latter's principles in demonstrating, as geometry uses the principles of arithmetic — for magnitude adds position to number, and thus a point is said to be "a positioned unit." In like manner, natural body adds sensible matter to mathematical magnitude. Consequently, it is not unfitting for the natural philosopher in his demonstrations to use the principles of mathematics — for the latter is not of a completely different genus but is in a certain way contained under the former.
Item videtur esse falsum quod solae duae magnitudines sint simplices, scilicet recta et circularis. Elix enim videtur esse una linea simplex, quia omnis pars eius est uniformis; et tamen linea elica nec est recta nec est circularis. Likewise, it seems to be false that only two magnitudes are simple, namely, the straight and the circular. For a helix [spiral] seems to be one simple line, because every one of its parts is uniform, and yet a helical line [such as a screw thread] is neither straight nor circular.
Sed dicendum quod elix, si quis eius originem consideret, non est linea simplex, sed mixta ex recta et circulari. Causatur enim elix ex duobus motibus imaginatis, quorum unus est lineae circumeuntis columnam, alius autem est puncti moti per lineam: si enim uterque motus simul et regulariter perficiatur, constituetur elica linea per motum puncti in linea mota. But it must be said that a helix, if one considers its origin, is not a simple line, but a combination of straight and circular. For a helix is produced by two imaginary motions, one of which is the motion of a line moving round a cylinder, and the other of a point moving through the line: if two such motions take place in a regular manner at the same time, a helix will be formed by the motion of the point in the moving line.
Item videtur quod motus circularis non sit simplex. Partes enim sphaerae circulariter motae non uniformiter moventur, sed pars quae est circa polos vel circa centrum, movetur tardius, quia peragit minorem circulum in eodem tempore: et ita motus sphaerae videtur compositus ex tardo et veloci. Likewise, it seems that circular motion is not simple. For the parts of a sphere that is in circular motion are not in uniform motion but the parts near the poles or near the center are moved more slowly, because they traverse a smaller circle in a given time; consequently, the motion of a sphere seems to be composed of fast and slow motions.
Sed dicendum quod continuum non habet partes in actu, sed solum in potentia: quod autem non est actu, non movetur actu: unde partes sphaerae, cum sint corpus continuum, non moventur actu. Unde non sequitur quod in motu sphaerico vel circulari sit diversitas actualis, sed solum potentialis; quae non repugnat simplicitati de qua nunc loquimur; omnis enim magnitudo habet pluralitatem potentialem. But it must be said that a continuum does not have parts in act but only in potency. Now, what is not in act is not in actual motion. Hence the parts of a sphere, since they are a continuous body, are not actually being moved. Hence it does not follow that, in a spherical or circular motion, there is actual diversity, but this is only potentially. This does not conflict with the simplicity about which we are now speaking, for every magnitude possesses potential plurality.
Deinde cum dicit: circulatio quidem igitur etc., distinguit motus simplices. 27. Then at [17] he distinguishes simple motions.

Et primo ponit unum, scilicet circularem;

secundo ponit duos rectos, ibi: rectus autem etc.;

tertio concludit numerum ternarium simplicium motuum, ibi: itaque necesse et cetera.

First he mentions one, namely, the circular;

Secondly, he mentions two that are straight, at 29;

Thirdly, he concludes that the number of simple motions is three, at 30.

Dicit ergo primo quod circulatio, idest motus circularis, dicitur qui est circa medium. Et est intelligendum circa mundi medium: rota enim, quae movetur circa medium sui, non movetur proprie circulariter; sed motus eius est compositus ex elevatione et depressione. He says therefore first [17] that circulation, i.e., circular motion is around the middle. And this is to be understood as around the middle of the world: for a wheel which is in motion around its own middle is not in circular motion in the proper sense of the word, but its motion is composed of ups and downs.
Sed videtur secundum hoc quod non omnia corpora caelestia circulariter moveantur: nam, secundum Ptolomaeum, motus planetarum est in excentricis et epicyclis; qui quidem motus non sunt circa medium mundi, quod est centrum terrae, sed circa quaedam alia centra. But it seems according to this that not all heavenly bodies are in circular motion: for according to Ptolemy, the motion of the planets is in eccentrics and epicycles, which are motions, not around the middle of the world, which is the earth's center, but around certain other centers.
Dicendum est autem quod Aristoteles non fuit huius opinionis, sed existimavit quod omnes motus caelestium corporum sunt circa centrum terrae, ut ponebant astrologi sui temporis. Postmodum autem Hipparchus et Ptolomaeus adinvenerunt motus excentricorum et epicyclorum, ad salvandum ea quae apparent sensibus in corporibus caelestibus. Unde hoc non est demonstratum, sed suppositio quaedam. Si tamen hoc verum sit, nihilominus omnia corpora caelestia moventur circa centrum mundi secundum motum diurnum, qui est motus supremae sphaerae revolventis totum caelum. But it must be said that Aristotle was not of this opinion, but thought that all motions of the heavenly bodies are about the center of the earth, as did all the astronomers of his time. But later, Hipparchus and Ptolemy hit upon eccentric and epicyclic motions to save what appears to the senses in heavenly bodies. Hence this is not a demonstration, but a certain assumption. Yet if it be true, all the heavenly bodies are still in motion about the center of the world with respect to the diurnal motion, which is the motion of the supreme sphere that revolves the entire heaven.
Deinde cum dicit: rectus autem etc., distinguit motum rectum in duos, scilicet in eum qui est sursum, et in eum qui est deorsum: et describit utrumque per habitudinem ad medium mundi, sicut descripserat motum circularem, ut sit uniformis descriptio. Et dicit quod motus sursum est qui est a medio mundi; motus autem deorsum qui est ad medium mundi. Quorum primus est motus levium, secundum motus gravium. 29. Then at [18] he distinguishes straight motion into two: namely, one which is up, and one that is down, and describes each in relation to the middle of the world, as he had described circular motion, in order to keep the description uniform. And he says that an upward motion is one from the middle of the world, but a downward motion is one to the middle of the world. The first of these is the motion of light things, the second of heavy things.
Deinde cum dicit: itaque necesse etc., concludit numerum simplicium motuum. Et primo inducit conclusionem intentam: et dicit quod necesse est simplicem lationem, idest motum localem, quendam esse a medio, et hic est motus sursum corporum levium; quendam vero esse ad medium, et hic est motus deorsum corporum gravium; alium vero esse circa medium, et huiusmodi est motus circularis corporum caelestium. 30. Then at [19] he concludes to the number of simple motions. First he expresses the conclusion he intended, and says that as to simple latio, i.e., simple local motion, one must be from the middle, and this is the upward motion of light bodies; another must be to the middle, and this is the downward motion of heavy bodies; still another must be about the middle, and such is the circular motion of heavenly bodies.
Secundo ibi: et videtur sequi etc., ostendit hanc conclusionem supra dictis congruere. Et dicit quod hoc quod dictum est de numero simplicium motuum, videtur consequenter se habere ad id quod supra dictum est de perfectione corporis: sicut enim perfectio corporis consistit in tribus dimensionibus, ita et motus simplices corporis in tres distinguuntur. Hoc autem dicit esse secundum rationem, idest secundum probabilitatem quandam: non enim proprie tres motus coaptantur tribus dimensionibus. 31. Secondly, at [20] he shows that this conclusion agrees with what has been said above. And he says that what has just been said about the number of simple motions seems to be a consequence of what was said above about the perfection of body, for just as the perfection of body consists in three dimensions, so the simple motions of body are distinguished into three kinds. But he says that this is "according to reason," i.e., according to a certain probability: for three motions are not properly equated to three dimensions.
Deinde cum dicit: quoniam autem corporum etc., ponit quaedam ex parte corporum mobilium. Circa quod sciendum est quod, sicut habitum est in III Physic., motus est actus mobilis; actus autem proportionatur perfectibili; unde oportet motus proportionari corporibus mobilibus. Sunt autem corporum quaedam simplicia, quaedam composita. Simplex autem corpus est quod habet principium alicuius naturalis motus in seipso; sicut patet de igne, qui est simpliciter levis, et de terra, quae est simpliciter gravis, et de speciebus horum (sicut flamma dicitur esse quaedam species ignis, et bitumen quaedam species terrae). Addit autem et cognata his, propter media elementa; quorum aer habet maiorem affinitatem cum igne, aqua vero cum terra. Et per consequens necesse est corpus mixtum esse quod non habet in se secundum propriam naturam principium alicuius motus simplicis. 32. Then at [21] he gives some reflections about mobile bodies. In regard to this it must be known that, as was stated in Physics III, motion is an act of a mobile. Now an act is proportionate to the thing to be perfected. Hence motions ought to be proportionate to mobile bodies. But some bodies are simple, some composite. A simple body is one that has a principle of some natural motion in it, as is plain in the case of fire, which is light simply, and in that of earth, which is heavy simply, and in their species — as a flame is said to be a species of fire, and bitumen a species of earth. He adds the phrase, "and those related to them," on account of the intermediate elements, of which air has a greater affinity to fire, and water to earth. As a consequence, a mixed body must be one that, according to its proper nature, does not have in itself the principle of some simple motion.
Et ex hoc concludit quod necesse est motuum quosdam esse simplices, quosdam autem aliqualiter mixtos: sive ita quod motus mixtus non sit unus, sed habens diversas partes, sicut ille qui componitur ex elevatione et depressione, aut ex pulsu et tractu; sive ita quod motus mixtus sit unus, sicut patet de motu qui in obliquum tendit, et de motu qui est super lineam elicam. Unde simplicium corporum necesse est esse simplices motus: mixtorum autem, mixtos, ut patet de motu pluviae aut alicuius huiusmodi corporis, in quo non totaliter gravitas aut levitas dominatur. Et si aliquando contingat quod corpus mixtum moveatur motu simplici, hoc erit secundum elementum in eo praedominans; sicut ferrum movetur deorsum secundum motum terrae, quae in eius mixtione dominatur. And from this he concludes that some motions must be simple and some mixed: whether the mixed motion is not one but has diverse parts, as one composed of elevation and depression, or of a push and a pull, or whether the mixed motion is one, as is plain in oblique motion and motion upon a helical line. Accordingly, the motions of simple bodies must be simple and those of mixed bodies mixed, as seen in the motion of rain, or any body of this kind in which neither heaviness nor lightness totally predominates. And if it sometimes happens that a mixed body is moved with a simple motion, that will be due to the element predominant in it, as iron is moved downwards according to the motion of earth which is predominant in its composition.

Lecture 4:
Five reasons why, besides the elements, there must be another simple body
Chapter 2 cont.
Εἴπερ οὖν ἐστιν ἁπλῆ κίνησις, ἁπλῆ δ' ἡ κύκλῳ κίνησις, καὶ τοῦ τε ἁπλοῦ σώματος ἁπλῆ ἡ κίνησις καὶ ἡ ἁπλῆ κίνησις ἁπλοῦ σώματος (καὶ γὰρ ἂν συνθέτου ᾖ, κατὰ τὸ ἐπικρατοῦν ἔσται), ἀναγκαῖον εἶναί τι σῶμα ἁπλοῦν ὃ πέφυκε φέρεσθαι τὴν κύκλῳ κίνησιν κατὰ τὴν ἑαυτοῦ φύσιν βίᾳ μὲν γὰρ ἐνδέχεται τὴν ἄλλου καὶ ἑτέρου, κατὰ φύσιν δὲ ἀδύνατον, εἴπερ μία ἑκάστου κίνησις ἡ κατὰ φύσιν τῶν ἁπλῶν. 22 Supposing, then, that there is such a thing as simple movement, and that circular movement is an instance of it, and that both movement of a simple body is simple and simple movement is of a simple body (for if it is movement of a compound it will be in virtue of a prevailing simple element), then there must necessarily be some simple body which revolves naturally and in virtue of its own nature with a circular movement. By constraint, of course, it may be brought to move with the motion of something else different from itself, but it cannot so move naturally, since there is one sort of movement natural to each of the simple bodies.
Ἔτι εἰ ἡ παρὰ φύσιν ἐναντία τῇ κατὰ φύσιν καὶ ἓν ἑνὶ ἐναντίον, ἀνάγκη, ἐπεὶ ἁπλῆ ἡ κύκλῳ, εἰ μὴ ἔσται κατὰ φύσιν τοῦ φερομένου σώματος, παρὰ φύσιν αὐτοῦ εἶναι. Εἰ οὖν πῦρ ἢ ἄλλο τι τῶν τοιούτων ἐστὶ τὸ κύκλῳ φερόμενον, ἐναντία ἡ κατὰ φύσιν αὐτοῦ φορὰ ἔσται τῇ κύκλῳ. Ἀλλ' ἓν ἑνὶ ἐναντίον ἡ δ' ἄνω καὶ κάτω ἀλλήλαις ἐναντίαι. Εἰ δ' ἕτερόν τί ἐστι σῶμα τὸ φερόμενον κύκλῳ παρὰ φύσιν, ἔσται τις αὐτοῦ ἄλλη κίνησις κατὰ φύσιν. Ἀλλὰ τοῦτ' ἀδύνατον εἰ μὲν γὰρ ἡ ἄνω, πῦρ ἔσται ἢ ἀήρ, εἰ δ' ἡ κάτω, ὕδωρ ἢ γῆ. 23 Again, if the unnatural movement is the contrary of the natural and a thing can have no more than one contrary, it will follow that circular movement, being a simple motion, must be unnatural, if it is not natural, to the body moved. If then (1) the body, whose movement is circular, is fire or some other element, its natural motion must be the contrary of the circular motion. But a single thing has a single contrary; and upward and downward motion are the contraries of one another. If, on the other hand, (2) the body moving with this circular motion which is unnatural to it is something different from the elements, there will be some other motion which is natural to it. But this cannot be. For if the natural motion is upward, it will be fire or air, and if downward, water or earth.
Ἀλλὰ μὴν καὶ πρώτην γε ἀναγκαῖον εἶναι τὴν τοιαύτην φοράν. Τὸ γὰρ τέλειον πρότερον τῇ φύσει τοῦ ἀτελοῦς, ὁ δὲ κύκλος τῶν τελείων, εὐθεῖα δὲ γραμμὴ οὐδεμία οὔτε γὰρ ἡ ἄπειρος (ἔχοι γὰρ ἂν πέρας καὶ τέλος) οὔτε τῶν πεπερασμένων οὐδεμία (πασῶν γάρ ἐστί τι ἐκτός αὐξῆσαι γὰρ ἐνδέχεται ὁποιανοῦν). Ὥστ' εἴπερ ἡ μὲν προτέρα κίνησις προτέρου τῇ φύσει σώματος, ἡ δὲ κύκλῳ προτέρα τῆς εὐθείας, ἡ δ' ἐπ' εὐθείας τῶν ἁπλῶν σωμάτων ἐστί (τό τε γὰρ πῦρ ἐπ' εὐθείας ἄνω φέρεται καὶ τὰ γεηρὰ κάτω πρὸς τὸ μέσον), ἀνάγκη καὶ τὴν κύκλῳ κίνησιν τῶν ἁπλῶν τινος εἶναι σωμάτων τῶν γὰρ μικτῶν τὴν φορὰν ἔφαμεν εἶναι κατὰ τὸ ἐπικρατοῦν ἐν τῇ μίξει τῶν ἁπλῶν. 24 Further, this circular motion is necessarily primary. For the perfect is naturally prior to the imperfect, and the circle is a perfect thing. This cannot be said of any straight line:—not of an infinite line; for, if it were perfect, it would have a limit and an end: nor of any finite line; for in every case there is something beyond it, since any finite line can be extended. And so, since the prior movement belongs to the body which naturally prior, and circular movement is prior to straight, and movement in a straight line belongs to simple bodies—fire moving straight upward and earthy bodies straight downward towards the centre—since this is so, it follows that circular movement also must be the movement of some simple body. For the movement of composite bodies is, as we said, determined by that simple body which preponderates in the composition. These premises clearly give the conclusion that there is in nature some bodily substance other than the formations we know, prior to them all and more divine than they.
Ἔκ τε δὴ τούτων φανερὸν ὅτι πέφυκέ τις οὐσία σώματος ἄλλη παρὰ τὰς ἐνταῦθα συστάσεις, θειοτέρα καὶ προτέρα τούτων ἁπάντων, κἂν εἴ τις ἔτι λάβοι πᾶσαν εἶναι κίνησιν ἢ κατὰ φύσιν ἢ παρὰ φύσιν, καὶ τὴν ἄλλῳ παρὰ φύσιν ἑτέρῳ κατὰ φύσιν, οἷον ἡ ἄνω καὶ ἡ κάτω πέπονθεν ἡ μὲν γὰρ τῷ πυρί, ἡ δὲ τῇ γῇ παρὰ φύσιν καὶ κατὰ φύσιν (269b.) ὥστ' ἀναγκαῖον καὶ τὴν κύκλῳ κίνησιν, ἐπειδὴ τούτοις παρὰ φύσιν, ἑτέρου τινὸς εἶναι κατὰ φύσιν. 25 But it may also be proved as follows. We may take it that all movement is either natural or unnatural, and that the movement which is unnatural to one body is natural to another—as, for instance, is the case with the upward and downward movements, which are natural and unnatural to fire and earth respectively. It necessarily follows that circular movement, being unnatural to these bodies, is the natural movement of some other.
Πρὸς δὲ τούτοις εἰ μέν ἐστιν ἡ κύκλῳ τινὶ φορὰ κατὰ φύσιν, δῆλον ὡς εἴη ἄν τι σῶμα τῶν ἁπλῶν καὶ πρώτων, ὃ πέφυκεν, ὥσπερ τὸ πῦρ ἄνω καὶ ἡ γῆ κάτω, ἐκεῖνο κύκλῳ φέρεσθαι κατὰ φύσιν. Εἰ δὲ παρὰ φύσιν φέρεται τὰ φερόμενα κύκλῳ τὴν πέριξ φοράν, θαυμαστὸν καὶ παντελῶς ἄλογον τὸ μόνην εἶναι συνεχῆ ταύτην τὴν κίνησιν καὶ ἀΐδιον, οὖσαν παρὰ φύσιν φαίνεται γὰρ ἔν γε τοῖς ἄλλοις τάχιστα φθειρόμενα τὰ παρὰ φύσιν. Ὥστ' εἴπερ ἐστὶ πῦρ τὸ φερόμενον, καθάπερ φασί τινες, οὐδὲν ἧττον αὐτῷ παρὰ φύσιν ἡ κίνησίς ἐστιν αὕτη ἢ ἡ κάτω πυρὸς γὰρ κίνησιν ὁρῶμεν τὴν ἀπὸ τοῦ μέσου κατ' εὐθεῖαν. Διόπερ ἐξ ἁπάντων ἄν τις τούτων συλλογιζόμενος πιστεύσειεν ὡς ἔστι τι παρὰ τὰ σώματα τὰ δεῦρο καὶ περὶ ἡμᾶς ἕτερον κεχωρισμένον, τοσούτῳ τιμιωτέραν ἔχον τὴν φύσιν ὅσῳπερ ἀφέστηκε τῶν ἐνταῦθα πλεῖον. 26 Further, if, on the one hand, circular movement is natural to something, it must surely be some simple and primary body which is ordained to move with a natural circular motion, as fire is ordained to fly up and earth down. If, on the other hand, the movement of the rotating bodies about the centre is unnatural, it would be remarkable and indeed quite inconceivable that this movement alone should be continuous and eternal, being nevertheless contrary to nature. At any rate the evidence of all other cases goes to show that it is the unnatural which quickest passes away. And so, if, as some say, the body so moved is fire, this movement is just as unnatural to it as downward movement; for any one can see that fire moves in a straight line away from the centre. On all these grounds, therefore, we may infer with confidence that there is something beyond the bodies that are about us on this earth, different and separate from them; and that the superior glory of its nature is proportionate to its distance from this world of ours.
Postquam philosophus praemisit quaedam necessaria ad propositum ostendendum, hic incipit arguere ad propositum; et hoc quinque rationibus. Quarum prima talis est. Motus circularis est motus simplex: motus autem simplex est primo et per se simplicis corporis (quia etsi contingat quod aliquis motus simplex sit alicuius corporis compositi, hoc erit secundum corpus simplex quod in eo praedominatur; sicut in lapide praedominatur terra, secundum cuius naturam movetur deorsum): ergo necesse est esse aliquod corpus simplex, quod moveatur naturaliter secundum motum circularem. 33. After stating in advance certain things necessary for showing the proposition, the Philosopher here begins to reason toward the proposition, and this with five arguments. The first [22] is this: Circular motion is a simple motion. But a simple motion belongs primarily and per se to a simple body — because even though a simple motion might occur in a composite body, this will be with respect to the simple body that is predominant in it; for example, in a stone, earth predominates, according to whose nature it is moved down. Therefore, there must be a simple body which is naturally moved according to a circular motion.
Posset autem aliquis huic rationi obviare, dicendo quod, licet simplex motus sit simplicis corporis, non tamen oportet quod illud simplex corpus quod movetur circulariter, sit aliud a corpore simplici quod movetur motu simplici recto. Et ideo hoc excludit, subdens quod nihil prohibet quin diversa corpora moveantur uno motu non naturaliter, ita scilicet quod unum corpus moveatur per violentiam motu alterius; sed quod unum corpus moveatur secundum naturam motu naturali alterius corporis, est impossibile. Necesse enim est esse unum motum simplicem naturalem unius simplicis corporis, et diversos diversorum. Unde, si motus circularis est simplex, et alius a motibus rectis, necesse est quod sit naturalis corpori simplici, quod sit aliud a corporibus simplicibus quae moventur motu recto. Now, someone could object to this argument and say that, although a simple motion belongs to a simple body, yet that simple body which is circularly moved would not necessarily be different from the simple body that is moved with a simple straight motion. Accordingly, he rejects this by adding that nothing prevents diverse bodies from being moved unnaturally with some one motion, as when one body might be moved violently with the motion of another; but that one body be moved according to nature with the natural motion of some other body is impossible. For one simple natural motion must belong to one simple body, and diverse to diverse. Hence, if circular motion is simple and distinct from straight motions, then it must belong to a natural simple body that is different from the simple bodies that are moved with a straight motion.
Sed videtur hoc esse falsum, quod unus motus simplex sit solum unius corporis simplicis: motus enim deorsum est naturalis aquae et terrae, et motus sursum est naturalis igni et aeri. 34. But this seems to be false, namely, that one simple motion belongs to just one simple body, for downward motion is natural to both water and earth, and upward motion to fire and air.
Sed dicendum quod motus localis attribuitur elementis, non secundum calidum et frigidum, humidum et siccum, secundum quae distinguuntur quatuor elementa, ut patet in II de Generat.: haec enim sunt principia alterationum. Motus autem localis attribuitur elementis secundum gravitatem et levitatem. Unde duo corpora gravia comparantur ad motum localem sicut unum corpus; et similiter duo corpora levia. Humidum enim et siccum, secundum quae differunt terra et aqua vel ignis et aer, accidentalem habitudinem habent ad motum localem. Et tamen in gravi et levi differentia quaedam est: nam ignis est levis simpliciter et absolute, terra autem gravis; aer autem est levis per comparationem ad duo elementa, et similiter aqua est gravis. Unde non omnino est idem secundum speciem motus aquae et terrae, vel ignis et aeris: quia non sunt idem termini, secundum quos specificantur eorum motus: aer enim natus est moveri ad locum qui subsidet igni, aqua autem ad locum qui supereminet terrae. But it must be said that local motion is attributed to the elements, not according to hot and cold, moist and dry, with respect to which the four elements are distinguished — as is plain in On Generation II — for these four properties are principles of alterations. But local motion is attributed to the elements with respect to heaviness and lightness. Hence the two heavy bodies are compared to local motion as one body; and the same for the two light bodies. For moist and dry, according to which earth and water, or fire and air, differ, have an incidental relationship to local motion. Yet in the realm of heavy and light there is a difference, for fire is light simply and absolutely, and earth heavy; while air is light compared to two elements and likewise water is heavy. Hence the motions of water and earth, or fire and air, are not completely the same according to species, because the termini according to which their motions are specified are not the same: for air is apt to be moved to a place below fire, and water to a place above earth.
Item videtur quod non sit necessarium, si corporis simplicis est unus motus simplex, quod propter hoc aliquis motus simplex sit alicuius corporis simplicis: sicut etiam non est necessarium quod tot sint corpora composita quot sunt motus compositi, qui diversificantur in infinitum. 35. Likewise it seems not necessary, if of one simple body there is one simple motion, that on this account any simple motion should belong to some [different?] simple body, any more than it is necessary that there be as many composite bodies as there are composite motions, which are infinitely diverse.
Sed dicendum est quod, sicut motus simplex localis non respondet corpori simplici quantum ad calidum et frigidum, humidum et siccum, ita etiam neque motus compositus respondet corpori mixto secundum gradus mixtionis praedictarum qualitatum, sed secundum compositionem gravis et levis; secundum cuius diversitatem diversificatur obliquatio corporis mixti a simplici motu gravis vel levis. Utraque autem diversitas non tendit in infinitum secundum speciem, sed solum secundum numerum. But it must be said that just as simple local motion does not correspond to a simple body with respect to hot and cold, and moist and dry, so neither does composite motion correspond to mixed body according to the degrees of mixture of those qualities, but rather according to a composition of heavy and light, according to the diversity of which is diversified the obliquity of a mixed body from the simple motion of the heavy or light. Neither of these diversities tends to infinity with respect to species, but only with respect to number.
Item videtur quod secundum hoc sint multa corpora simplicia: quia sicut motus sursum et deorsum videntur esse motus simplices, ita motus qui est dextrorsum vel sinistrorsum, et qui est ante et retro. 36. Likewise it seems that according to this there are many simple bodies. For just as motions upward and downward seem to be simple motions, so too motions to the right and to the left, and those ahead and to the rear.
Et dicendum est quod, cum corpora simplicia sint essentiales et primae partes universi, oportet quod motus simplices, qui sunt naturales corporibus simplicibus, attendantur secundum conditionem universi. Quod cum sit sphaericum, ut infra probabitur, oportet quod motus eius attendatur per comparationem ad medium, quod est immobile: quia omnis motus fundatur supra aliquod immobile, ut dicitur in libro de causa motus animalium. Et ideo oportet esse solum tres motus simplices, secundum diversas habitudines ad medium: scilicet eum qui est a medio, et eum qui est ad medium, et eum qui est circa medium. Dextrum autem et sinistrum, ante et retro, considerantur in animalibus, et non in toto universo, nisi secundum quod ponuntur in caelo, ut in secundo dicetur: et secundum hoc motus circularis caeli est secundum dextrum et sinistrum, ante et retro. It must be stated therefore that, since simple bodies are the essential and first parts of the universe, the simple motions which are natural to simple bodies must be considered in relation to the condition of the universe. Since this latter is spherical, as will be proved later, its motion must be considered in relation to the middle, which is immobile, because every motion is founded upon something immobile, as is stated in the book, On the Cause of the Motion of Animals. Consequently, there must be but three simple motions, according to their diverse relations to the middle [center]: i.e., one which is from the center, one which is to the center, and one which is around the center. To the right and left, ahead and to the rear, are considered in animals but not in the whole universe, except in the sense that" they are placed in the heavens, as will be said in Book II. And according to this the circular motion of the heavens is with respect to right and left, ahead and to the rear.
Item videtur quod motus rectus et circularis non sint eiusdem rationis. Est enim motus rectus corporis nondum habentis complementum suae speciei, ut in quarto dicetur, et existentis extra proprium locum: motus autem circularis est corporis habentis complementum suae speciei, et in loco proprio existentis. Unde non videtur quod secundum eandem rationem motus simplices corporales sint simplicium corporum; sed quod alii motus sint corporum prout sunt in fieri, circularis autem prout sunt in facto esse. 37. In like manner it seems that straight motion and circular are not of the same kind. For a straight motion belongs to a body not yet having its completeness of species, as will be said in Book IV, and existing outside its proper place, while a circular motion belongs to a body that has completeness of species and is existing in its proper place. Hence simple bodily motions do not seem to belong to simple bodies according to a same notion, but some motions seem to belong to bodies inasmuch as they are coming into being, while circular motion insofar as they have complete existence.
Sed dicendum quod, quia motus proportionatur mobili tanquam actus eius, conveniens est quod corpori quod est separatum a generatione et a corruptione, et non potest per violentiam expelli a proprio loco, debeatur motus circularis, qui est corporis in suo loco existentis: corporibus autem aliis generabilibus et corruptibilibus debetur motus extra proprium locum, qui est absque complemento speciei. Non tamen ita quod corpus quod movetur naturaliter motu recto, non habeat primum complementum suae speciei, quod est forma; hanc enim sequitur talis motus: sed quia non habet ultimum complementum, quod est in consecutione finis, qui est locus conveniens et conservans. But it must be said that, since a motion is proportionate to the mobile as being its act, it is fitting that a body which is separated from generation and corruption and cannot be expelled from its proper place by violence should have a circular motion, which is proper to a body existing in its own place; but to other bodies that can be generated and corrupted there belongs a motion outside their proper place and which is incomplete in species. But this is not in the sense that a body which is naturally moved with a straight motion lacks the first complement of its species, namely, form, for it is the form that such a motion is consequent upon; but in the sense that it does not have its final complement which consists in attaining the end, which is a place that agrees with it and conserves it.
Secundam rationem ponit ibi: adhuc si qui praeter naturam etc.: in qua praesupponit duo principia. Quorum unum est quod motus qui est praeter naturam, idest violentus, contrarietur motui naturali; sicut terra movetur deorsum secundum naturam, sursum autem contra naturam. Secundum principium est quod unum uni est contrarium, ut probatum est in X Metaphys. Oportet autem et tertium supponere, quod sensu videtur, scilicet esse aliquod corpus circulariter motum. Et si quidem ille motus sit illi corpori naturalis, habemus propositum secundum praemissam rationem, quod scilicet illud corpus naturaliter motum circulo, sit aliud a quatuor corporibus simplicibus. Si vero motus huiusmodi non sit ei naturalis, oportet quod sit ei contra naturam. 38. The second argument he gives at [23] and in it he presupposes two principles: one of which is that a motion which is outside nature, i.e., violent, is contrary to a natural motion, as earth is according to nature moved downward but upward against its nature. The second principle is that one thing is contrary to one thing, as is proved in Metaphysics X. A third also must be presupposed from sense experience, namely, that there exists a body which is moved circularly. Now, if that motion is natural to that body, we have the proposition, in keeping with the previously given reason, namely, that that body which is moved in a circle naturally is distinct from the four simple elements. But if such a motion is not natural to it, it must be against its nature.
Ponatur ergo primo quod illud corpus circulariter motum sit ignis, ut quidam dicunt, vel quodcumque aliud quatuor elementorum. Oportebit ergo quod motus naturalis ignis, qui est moveri sursum, sit contrarius motui circulari. Sed hoc non potest esse: quia uni unum est contrarium, motui autem sursum contrariatur motus deorsum, et sic non potest ei contrariari motus circularis. Et eadem ratio est de aliis tribus elementis. Et similiter, si detur quod illud corpus quod contra naturam movetur circulariter, sit quodcumque aliud corpus praeter quatuor elementa, oportebit quod habeat aliquem alium motum naturalem. Sed hoc est impossibile: quia si sit ei naturalis motus qui est sursum, erit ignis aut aer; si autem motus qui est deorsum, erit aqua aut terra; positum est autem quod sit extra quatuor elementa. Sic ergo necesse est corpus quod movetur circulariter, naturaliter hoc motu moveri. Let us therefore first assume that that body in circular motion is fire, as some claim, or any of the other four elements. Then the natural motion of fire, which is to be moved upward, will have to be contrary to the circular motion. But this cannot be, for to one thing, one thing is contrary, and the motion contrary to an upward motion is a downward one; consequently, circular motion cannot be contrary to it. And the same holds for the other three elements. Likewise, if it be assumed that the body which is being moved circularly against its nature is a body other than the four elements, it would have to have some other natural motion. But this is impossible, because if its natural motion is up, it will be fire or air; if its motion is down, it will be water or earth. But we supposed that it is not one of the four elements. Accordingly it must be that the body moved in circular motion is being moved naturally with this motion.
Videtur autem Aristoteles, secundum ea quae hic dicit, contrarius esse Platoni, qui posuit corpus quod circulariter fertur, esse ignem. Sed secundum veritatem eadem est circa hoc utriusque philosophi opinio. Plato enim corpus quod circulariter fertur, ignem vocat propter lucem, quae species ignis ponitur; non quod sit de natura ignis elementaris. Unde et posuit quinque corpora in universo, quibus adaptavit quinque figuras corporales quas geometrae tradunt, quintum corpus aetherem nominans. Now according to what he says here Aristotle seems to be contrary to Plato who assumed that the body which is circularly moved is fire. But with respect to the truth, the opinion of both philosophers is the same on this point. For Plato calls the body which is being circularly moved "fire" on account of light, which is posited as a form of fire, but not as being of the nature of elemental fire. Hence he posited five bodies in the universe, and to these he adapted five bodily figures which geometers teach, calling the fifth body "aether."
Sed ulterius, quod hic dicitur, ignem moveri circulariter esse praeter naturam, videtur contrarium ei quod dicitur in I Meteor., ubi ipse Aristoteles ponit quod hypeccauma, idest ignis, et superior pars aeris feruntur circulariter motu firmamenti, sicut patet per motum stellae comatae. 39. But further, what is said here, namely, that for fire to be moved circularly is outside nature seems to be contrary to what is said in Meteorology I, where Aristotle himself sets forth that hypeccauma, i.e., fire, and the upper portion of the air, are carried along circularly by the motion of the firmament, as is plain in the motion of a comet.
Sed dicendum est quod illa circulatio ignis vel aeris non est eis naturalis, quia non causatur ex principio intrinseco; neque iterum est per violentiam, sive contra naturam; sed est quodammodo supra naturam, quia talis motus inest eis ex impressione superioris corporis, cuius motum ignis et aer sequuntur secundum completam circulationem, quia haec corpora sunt caelo propinquiora; aqua vero secundum circulationem incompletam, scilicet secundum fluxum et refluxum maris; terra autem, velut remotissima a caelo, nihil de tali permutatione participat, nisi secundum solam alterationem partium ipsius. Quod autem inest inferioribus corporibus ex impressione superiorum, non est eis violentum nec contra naturam: quia naturaliter apta sunt moveri a superiori corpore. But it must be said that that circulation of fire or air is not natural to them, because it is not caused from an intrinsic principle. Neither is it through violence or against nature, because such a motion is in them from the influence of a higher body, whose motion fire and air follow according to a complete circulation because these bodies are closer to the heavens, but water according to an incomplete circulation, i.e., according to the ebb and flow of the sea. Earth, however, as being most remote from the heavens, suffers no such change except with respect to the sole alteration of its parts. Now whatever is present in lower bodies from the impression of the higher is not violent for them or against nature, for they are naturally apt to be moved by the higher body.
Item videtur falsum esse quod hic dicitur, unum uni esse contrarium: uni enim vitio contrariatur et virtus et vitium oppositum, sicut illiberalitati prodigalitas et liberalitas. 40. Likewise it seems to be false, as here stated, that to one thing one thing is contrary, for to one vice both a virtue and the opposite vice are contrary, as to illiberality both prodigality and liberality are opposed.
Dicendum est autem quod eidem secundum idem est unum tantum contrarium; nihil tamen prohibet quin uni secundum diversa sint plura contraria, sicut si sit idem subiectum dulce et album, contrariabitur ei nigrum et amarum. Sic igitur illiberalitati contrariatur virtus liberalitatis sicut ordinatum inordinato; prodigalitas autem sicut superabundantia defectui. Non potest autem dici quod uterque motus, scilicet qui est sursum et qui est deorsum, contrarietur motui circulari secundum communem rationem recti. Rectum enim et circulare non sunt contraria: pertinent enim ad figuram, cui nihil est contrarium. But it must be said that there is only one contrary to one thing according to the same aspect, although from different aspects nothing forbids one thing from having several contraries: thus, if the same subject is sweet and white, black and bitter will be contrary to it. Accordingly, the virtue of liberality is contrary to illiberality as what is well ordered to what is disordered, but prodigality is contrary to it as superabundance is to defect. Now, it cannot be said that both motions, namely, the one that is upward and the one that is downward, are contrary to circular motion according to the common aspect of straightness. For straight and circular are not contrary, for they pertain to figure, to which nothing is contrary.
Tertiam rationem ponit ibi: sed adhuc et primam et cetera. Circa quam primo ostendit quod motus circularis sit primus inter motus locales. Est enim comparatio motus circularis ad motum rectum, qui est sursum vel deorsum, sicut comparatio circuli ad lineam rectam. Probatur autem quod circulus, idest linea circularis, sit prior linea recta, quia perfectum naturaliter est prius imperfecto; circulus autem sive linea circularis est perfecta, quia quidquid in ea accipitur, est principium et finis et medium; unde non recipit alicuius exterioris additionem. Linea autem recta nulla est perfecta. Quod patet et quantum ad lineam infinitam, quae imperfecta est quia fine caret, ex quo denominatur aliquid perfectum in Graeco: et idem patet in linea finita, quia quamlibet lineam finitam contingit augeri, idest accipere maiorem quantitatem, et sic est aliquid extra eam. Et sic linea circularis naturaliter est prior quam recta. Ergo et motus circularis est prior naturaliter motu recto. 41. He gives the third argument at [24]. With regard to this he first shows that circular motion is the first of local motions. For circular motion is related to straight motion, such as up or down, as circle is compared to straight line. A circle, i.e., a circular line, is proved to be prior to a straight line because the perfect is naturally prior to the imperfect. But a circle, or circular line, is perfect, because whatever is taken in it is a beginning and middle and end. Hence it does not suffer the addition of anything from without. But no straight line is perfect, whether it be an infinite line, which is imperfect because it lacks an end, from which things are called perfect in Greek, or a finite line, because every finite line can be increased, i.e., receive more quantity and so there is something outside it. Consequently a circular line is naturally prior to the straight. Therefore circular motion, too, is naturally prior to straight motion.
Sed prior motus est naturaliter prioris corporis. Motus autem rectus est naturaliter alicuius simplicium corporum, sicut ignis movetur sursum, et terra deorsum et ad medium: et si contingat quod motus rectus sit corporum mixtorum, hoc erit secundum naturam simplicis corporis dominantis in mixtione. Cum igitur corpus simplex sit naturaliter prius mixto, consequens est quod motus circularis est proprius et naturalis alicuius corporis simplicis, quod est prius corporibus elementaribus quae sunt hic apud nos. Et ita ex his patet quod, praeter substantias corporales quae hic sunt apud nos, nata est esse quaedam substantia corporalis, quae est dignior et prior omnibus corporibus quae sunt apud nos. But a prior motion naturally belongs to a prior body. Now straight motion naturally belongs to some one or other of the simple bodies, as fire is moved upward and earth downward and toward the middle. And if it happens that a straight motion is found in mixed bodies, that will be due to the nature of the simple body predominant in it. Since, therefore, a simple body is naturally prior to the mixed, the consequence is that circular motion is proper and natural to some simple body which is prior to the elementary bodies that exist here among us. Thus it is clear from these facts that besides the bodily substances that exist here among us, there must be some bodily substance which is nobler and prior to all the bodies that exist among us.
Videtur autem esse falsum quod nulla linea recta sit perfecta. Si enim perfectum est quod habet principium, medium et finem, ut supra habitum est, videtur quod linea recta finita, quae habet principium et medium et finem, sit perfecta. 42. But the assertion that no straight line is perfect seems to be false. For if the perfect is what has a beginning, middle and end, as we held above, it seems that a straight finite line, which has beginning, middle and end, is perfect.
Sed dicendum est quod ad hoc quod aliquid sit perfectum partialiter, oportet quod habeat principium, medium et finem in seipso: sed ad rationem perfecti simpliciter, requiritur quod non sit aliquid extra ipsum. Et hic modus perfectionis competit primo et supremo corpori, quod est omnium corporum contentivum: et secundum hunc modum linea recta dicitur esse imperfecta, circularis vero perfecta. But it should be stated that in order for something to be partially perfect it must have the beginning, middle and end in itself; but to be completely perfect it is required that there be nothing outside it. And this mode of perfection belongs to the first and supreme body which contains all bodies; and with respect to this mode a straight line is said to be imperfect and a circular line perfect.
Item videtur quod etiam secundum hunc modum aliqua linea recta sit perfecta: quia diameter caeli non potest additionem accipere. Yet it seems that even according to this mode some straight lines are perfect, because the diameter of a circle cannot suffer addition.
Sed dicendum est quod hoc ei accidit inquantum est in tali materia, non autem hoc habet ex hoc quod est linea recta: secundum hoc enim non impediretur ne ei possit additio fieri. Sed circulus ex propria ratione circuli habet quod non sit additionis susceptivus. But it must be said that this happens to it insofar as it is in such and such a matter, and not insofar as it is a straight line, from which aspect there is nothing to prevent additions being made. But a circle, precisely as circle, cannot suffer such addition.
Videtur quod secundum hoc concludi non possit quod motus circularis sit perfectus: additionem enim recipit, cum sit continuus et sempiternus, secundum Aristotelem. 43. But it seems that, if this is so, one cannot conclude that circular motion is perfect, because it does receive addition, since it is continuous and eternal, according to Aristotle.
Ad quod dicendum est quod una circulatio habet complementum suae speciei, cum redierit ad principium a quo incoepit. Unde non fit additio ad eandem circulationem: sed quod sequitur, ad aliam circulationem pertinet. To this it should be said that one revolution is complete in species when it returns to the beginning from which it started. Hence no addition is being made to the same revolution, but whatever follows pertains to another revolution.
Item, si hoc solum perfectum dicitur, cui non potest fieri additio, sequitur quod neque homo neque aliquid aliud finitum in corporibus sit perfectum, cum eis possit additio fieri. Yet if only a thing to which no addition can be made is to be called perfect, it follows that neither man nor any finite thing in bodies is perfect, since additions can be made to them.
Et dicendum quod huiusmodi dicuntur esse perfecta secundum speciem, inquantum non potest eis fieri additio alicuius quod pertineat ad rationem speciei ipsorum: lineae autem rectae fit additio eius quod pertinet ad speciem suam, et pro tanto dicitur imperfecta inquantum est linea. And it should be answered that things of this kind are said to be perfect with respect to their species, inasmuch as they can suffer no addition of anything pertaining to the notion of their species; but to a straight line can be added something that pertains to its species, and to that extent it is said to be imperfect insofar as it is a line.
Praeterea videtur quod circulus non sit perfectus. Perfectum enim est in magnitudinibus quod habet tres dimensiones: hoc autem lineae circulari non competit. But still it seems that a circle is not perfect. For a perfect thing among magnitudes is something having three dimensions; which our circular line certainly lacks.
Et dicendum est quod linea circularis non est simpliciter magnitudo perfecta, quia non habet quidquid pertinet ad rationem magnitudinis: est tamen quoddam perfectum in linea, quia linealiter aliquid ei addi non potest. To this it should be responded that a circular line is not an absolutely perfect magnitude, because it does not have everything that pertains to the notion of a magnitude, Yet it is perfect in the realm of lines, because linearly something cannot be added to it.
Videtur etiam falsum esse quod perfectum sit prius imperfecto. Simplex enim est prius composito, cum tamen compositum se habeat ad simplicia ut perfectum ad imperfecta. Ad quod dicendum quod perfectum ad imperfectum se habet sicut actus ad potentiam: qui quidem simpliciter est prior potentia in diversis; in uno autem et eodem, quod movetur de potentia ad actum, potentia est prior actu tempore, sed actus est prior secundum naturam; quia scilicet hoc est quod primo et principaliter natura intendit. Non autem philosophus hic intendit quod perfectum sit prius imperfecto in uno et eodem, sed in diversis: nec etiam quod sit prius tempore, sed natura, sicut expresse dicit. 44. It also seems false that the perfect is prior to the imperfect. For the simple is prior to the composite and yet the latter is to the former as perfect to imperfect. To this it must be said that perfect is to imperfect as act to potency, and simply speaking, act is prior to potency in things that are diverse, although in one and the same thing that is moved from potency to act, potency is prior to act in time, but act is prior to potency according to nature, for this is what nature intends first and principally. Now the Philosopher does not mean here that the perfect is prior to the imperfect in one and the same thing, but in diverse things, nor does he intend to say that it is prior in time but in nature, as he expressly states.
Item, videtur quod philosophus inconvenienter argumentetur. Procedit enim ex perfectione lineae circularis ad probandum perfectionem circularis motus; ex cuius perfectione procedit ad probandum perfectionem circularis corporis; et sic videtur eius probatio esse circularis, quia linea circularis non videtur esse alia quam quae est ipsius corporis quod circulariter movetur. Et dicendum est quod motus circularis probatur esse perfectus ex perfectione lineae circularis absolute; ex perfectione autem motus circularis in communi, probatur hoc corpus quod circulariter movetur, esse perfectum; et sic non proceditur ab eodem in idem, sed ex communi ad proprium. 45. Moreover it seems that the Philosopher is arguing in an unsuitable manner. For he proceeds from the perfection of a circular line to prove the perfection of a circular motion, and from the latter perfection he goes on to prove the perfection of a circular body. And so his proof seems to be circular, because a circular line does not seem to be anything other than that of the very body that is being moved circularly. And it should be said that a circular motion is proved to be perfect on account of the perfection of the circular line absolutely; then from the perfection of circular motion in common one proves that this body which is moved circularly is perfect. Thus one does not go from the same to the same, but from common to proper.
Quartam rationem ponit ibi: et utique si quis etc.: quae quidem procedit ex duabus propositionibus suppositis. Quarum prima est, quod omnis motus simplex aut est secundum naturam, aut praeter naturam. Secunda est, quod motus qui est praeter naturam uni corpori, est alii corpori secundum naturam; sicut patet in motu qui est sursum, qui est secundum naturam igni et praeter naturam terrae; et in motu qui est deorsum, qui est naturalis terrae et praeter naturam igni. Manifestum est autem quod motus circularis inest alicui corpori, quod ad sensum circulariter movetur. Et si quidem talis motus sit ei naturalis, habebimus propositum, scilicet quod praeter quatuor elementa sit quoddam aliud corpus, quod circulariter movetur. Si autem motus circularis sit praeter naturam corpori quod circulariter fertur, sequitur ex praemissa suppositione quod sit alicuius alterius corporis secundum naturam: quod consequenter erit aliud in natura a quatuor elementis. 46. The fourth argument is given at [25], and it proceeds from two assumptions. The first is that every simple motion is either according to nature or outside nature. The second is that a motion which is outside nature for one body is according to nature for another, as is clear in the upward motion which, for fire, is according to nature, and for earth is outside nature; and in the downward motion which is natural to earth, but outside nature for fire. Now it is manifest that a circular motion is present in some body, which the senses observe is moved circularly. And if such a motion is natural to it, we will have the conclusion, namely, that, besides the four elements, there is an additional body which is moved circularly. But if the circular motion is outside the nature of the body that is moved circularly, it follows from the foregoing assumption that for some other body it is according to nature, which body, consequently, will be of a different nature from the four elements.
Videtur autem Aristoteles sibi ipsi esse contrarius: nam supra probavit quod motus circularis non est praeter naturam corpori quod circulariter fertur, hic autem supponit contrarium. 47. Aristotle here seems to be at odds with himself, for above he proved that circular motion is not outside the nature of the body in circular motion, but here he supposes the contrary.
Dicunt igitur quidam quod philosophus supra accepit praeter naturam pro eo quod est contra naturam: sic enim oportet quod motus contra naturam alicuius corporis, sit contrarius motui etiam naturali eiusdem, ut supra procedebat. Hic autem accipit praeter naturam communius, secundum quod praeter naturam idem est quod non secundum naturam. Sic autem in se comprehendit tam id quod est contra naturam, quam id quod est supra naturam: et hoc modo supponit hic quod aliquod corpus potest circulariter praeter naturam moveri; sicut dictum est supra quod ignis in sua sphaera circulariter movetur praeter naturam, delatus a motu caeli. Accordingly some say that above the Philosopher was taking "outside nature" in the sense of "against nature" — for then a motion against the nature of some body would also be contrary to its natural motion, as he proceeded above. But here he takes "outside nature" in the more general sense of "not according to nature." Thus it includes both what is against nature and what is above nature, and it is in this sense that he assumes here that a body can be moved circularly outside its nature, just as it was said above that fire in its sphere is moved circularly outside its nature under the influence of the motion of the heavens.
Sed hoc videtur esse contra intentionem Aristotelis. Eodem enim modo videtur utrobique accipere praeter naturam: quia tam hic quam supra exemplificat de motu qui est sursum et deorsum, qui est uni corpori contra naturam et alteri secundum naturam. Et ideo dicendum est, et melius, quod Aristoteles in prima ratione probavit quod aliquod corpus secundum naturam circulariter movetur. Et quia posset aliquis dicere quod corpus quod videtur circulariter moveri, movetur hoc motu contra naturam, dupliciter contra hoc argumentatur: uno modo ostendendo quod iste motus non est contra naturam, ut patet in secunda ratione et etiam in tertia; alio modo ostendendo quod etiam si moveatur contra naturam, adhuc sequitur esse aliud corpus, quod secundum naturam movetur circulariter. Sic ergo quod supra negavit secundum veritatem propriae opinionis loquens, hic negat quasi utens suppositione adversariorum. But this seems to be against the intention of Aristotle. For he seems to take "outside nature" in the same sense in both cases, because both here and above he uses the example of motion which is upward and downward, which is against nature for one body, and according to nature for another. Therefore it is better to say that Aristotle in the first argument proved that some body is being moved circularly according to nature. And because someone could say that that body which is seen to be moved circularly is being moved against nature by this movement, he argues against this in two ways: in one way, by showing that that motion is not against nature, as is clear in the second argument and also in the third; in another way, by showing that, even if it is being moved against nature, it still follows that there is some other body which is moved circularly according to nature. Consequently what he denied above when speaking according to the truth of his own opinion, he here denies by using, so to speak, the assumptions of his adversaries.
Item, non videtur sequi quod, si aliquis motus sit praeter naturam alicui corpori, quod sit alteri corpori naturalis. Potest enim ignis, vel quodcumque aliud corpus, multiformiter moveri: nec tamen propter hoc oportet quod huiusmodi motus omnes sint naturales aliquibus corporibus. 48. Likewise it does not seem to follow that, if some motion is outside nature for one body, it is natural to some other body. For fire or any other body can be moved in a number of ways — yet this does not prove that such motions are natural to certain bodies.
Est autem advertendum quod philosophus hic loquitur de simplici motu, ad quem natura corporis simplicis inclinat sicut ad aliquid unum: motus autem diversimode variati magis videntur ex arte dispositi, quae potest esse principium diversorum. Est etiam considerandum quod, licet motus qui est alicui corpori praeter naturam, sit alteri corpori secundum naturam, non tamen oportet quod omne corpus cui est aliquis motus secundum naturam, habeat aliquem motum praeter naturam: quia omne corpus quod est susceptivum alienae impressionis, habet aliquid sibi proprium et connaturale; non autem omne corpus potest extraneam impressionem recipere, ut sic possit naturalem motum habere. But it should be noted that the Philosopher is here speaking of simple motion, to which the nature of a simple body is inclined as to one definite thing, whereas motions diversely various seem to be rather brought about by art, which can be a principle of diverse things. It should also be considered that, although a motion which for one body is beside nature is according to nature for another, yet it is not necessary that every body for which some motion is natural should have a motion that is beside nature: for every body which can suffer an impression from without has something proper and connatural to it, yet not every body can receive an impression from without so as to be able to have a natural motion.
Quintam rationem ponit ibi: adhuc autem etc., quae talis est. Conclusum est ex praemissa ratione quod si corpus quod ad sensum circulariter movetur, moveatur praeter naturam, oportet quod talis motus sit alteri corpori secundum naturam. Quod quidem si concedatur, scilicet quod circularis motus sit alicui corpori secundum naturam, manifestum est quod erit aliquod corpus simplex et primum quod circulariter movetur, propter simplicitatem et prioritatem circularis motus, ut ex praemissis rationibus patet, sicut ignis movetur sursum et terra deorsum. Si autem non concedatur processus praecedentis rationis, sed dicatur quod omnia quae moventur circulariter secundum peripheriam, idest secundum circumferentiam, moventur praeter naturam, ita quod hic motus nulli corpori sit secundum naturam: hoc videtur esse mirabile, immo omnino irrationabile. Ostensum est enim in VIII Physic. quod solum motum circularem contingit esse continuum et sempiternum: irrationabile autem est quod id quod est sempiternum, sit praeter naturam, et motus non sempiternus sit secundum naturam. Videmus enim quod ea quae sunt praeter naturam, citissime transeunt et corrumpuntur, sicut calefactio aquae et proiectio lapidis in altum: ea vero quae sunt secundum naturam, videntur diutius permanere. Sic ergo oportet omnino motum circularem esse alicui corpori naturalem. 49. The fifth argument is at [26], and it is this. The conclusion of the foregoing argument was that if a body observed to be in circular motion is being moved outside its nature, then such a motion must be according to nature for some other body. And if this is granted, namely, that circular motion is according to nature for some body, then it is clear that there will be some first and simple body which is being moved circularly, on account of the simplicity and priority of circular motion, as is plain from the foregoing arguments, just as fire is moved upward and earth downward. But if the procedure of the foregoing argument is not admitted, and it is stated rather that all things in circular motion with respect to a periphery, i.e., a circumference, are being moved outside their nature, in such a way that this motion is not natural to any body, then such a thing seems to be marvelous and, indeed, wholly unreasonable. For it was proved in Physics VIII that only circular motion can be continuous and eternal. Now it is unreasonable that what is eternal should be outside nature, and that a non-eternal motion should be according to nature. For we see that things which are outside nature quickly pass and cease to be, as in the case of the heat of water and the projecting of a stone into the air, while things that are according to nature are seen to last a longer time. Thus it is wholly necessary that circular motion be natural to some body.
Si ergo istud corpus quod videmus circulariter ferri, est de natura ignis, ut quidam dicunt, motus iste erit ei praeter naturam, sicut et motus qui est deorsum: videmus enim quod motus naturalis ignis est sursum secundum rectam lineam. Et sic, sicut motus qui est deorsum est alteri corpori naturalis, scilicet terrae, ita erit motus circularis alicui alii corpori naturalis. If therefore the body which is observed to be carried along circularly is of the nature of fire, as some say, that motion will be beside its nature, just as a downward motion is. For we see that the natural motion of fire is upward according to a straight line. Accordingly, just as a downward motion is natural for another body, namely, earth, so a circular motion will be natural to some other body.
Ultimo autem epilogando concludit, quod si aliquis ex omnibus praemissis syllogizaverit per modum praedictum, credet, idest firmiter assentiet, quod sit aliquod corpus praeter corpora quae sunt hic circa nos (idest quatuor elementa et ex his composita), separatum ab eis, et in natura tanto habens nobiliorem naturam, quanto est magis elongatum secundum loci distantiam ab his quae sunt hic: corpora enim continentia in universo se habent ad corpora contenta sicut forma ad materiam et actus ad potentiam, ut dictum est in IV Physic. 50. Finally, in summary, he concludes that if someone should reason from all the foregoing in the aforesaid manner, he will believe, i.e., firmly assent, that there is a body over and above the bodies which exist among us (i.e., the four elements and composites of them), a body that is separated from them and of a nature that is more noble than they to the extent that it is farther separated from them in space. For in the universe the bodies that contain are to contained bodies as form to matter, and act to potency, as was said in Physics IV.

Lecture 5:

Difference of the body moved circularly as to light and heavy

Chapter 3
Ἐπεὶ δὲ τὰ μὲν ὑπόκειται τὰ δ' ἀποδέδεικται τῶν εἰρημένων, φανερὸν ὅτι οὔτε κουφότητα οὔτε βάρος ἔχει σῶμα ἅπαν, 28 In consequence of what has been said, in part by way of assumption and in part by way of proof, it is clear that not every body either possesses lightness or heaviness.
δεῖ δὲ ὑποθέσθαι τί λέγομεν τὸ βαρὺ καὶ τὸ κοῦφον, νῦν μὲν ἱκανῶς ὡς πρὸς τὴν παροῦσαν χρείαν, ἀκριβέστερον δὲ πάλιν, ὅταν ἐπισκοπῶμεν περὶ τῆς οὐσίας αὐτῶν. Βαρὺ μὲν οὖν ἔστω τὸ φέρεσθαι πεφυκὸς ἐπὶ τὸ μέσον, κοῦφον δὲ τὸ ἀπὸ τοῦ μέσου, 29 As a preliminary we must explain in what sense we are using the words 'heavy' and 'light', sufficiently, at least, for our present purpose: we can examine the terms more closely later, when we come to consider their essential nature. Let us then apply the term 'heavy' to that which naturally moves towards the centre, and 'light' to that which moves naturally away from the centre.
βαρύτατον δὲ τὸ πᾶσιν ὑφιστάμενον τοῖς κάτω φερομένοις, κουφότατον δὲ τὸ πᾶσιν ἐπιπολάζον τοῖς ἄνω φερομένοις. 30 The heaviest thing will be that which sinks to the bottom of all things that move downward, and the lightest that which rises to the surface of everything that moves upward.
Ἀνάγκη δὴ πᾶν τὸ φερόμενον ἢ κάτω ἢ ἄνω ἢ κουφότητ' ἔχειν ἢ βάρος ἢ ἄμφω, μὴ πρὸς τὸ αὐτὸ δέ πρὸς ἄλληλα γάρ ἐστι βαρέα καὶ κοῦφα, οἷον ἀὴρ πρὸς ὕδωρ, καὶ πρὸς γῆν ὕδωρ. Τὸ δὲ κύκλῳ σῶμα φερόμενον ἀδύνατον ἔχειν βάρος ἢ κουφότητα οὔτε γὰρ κατὰ φύσιν οὔτε παρὰ φύσιν ἐνδέχεται αὐτῷ κινηθῆναι ἐπὶ τὸ μέσον ἢ ἀπὸ τοῦ μέσου. Κατὰ φύσιν μὲν γὰρ οὐκ ἔστιν αὐτῷ ἡ ἐπ' εὐθείας φορά μία γὰρ ἦν ἑκάστου τῶν ἁπλῶν, ὥστ' ἔσται τὸ αὐτὸ τῶν οὕτω τινὶ φερομένων. Παρὰ φύσιν δ' ἐνεχθέντος, εἰ μὲν ἡ κάτω (270a.) παρὰ φύσιν, ἡ ἄνω ἔσται κατὰ φύσιν, εἰ δ' ἡ ἄνω παρὰ φύσιν, ἡ κάτω κατὰ φύσιν ἔθεμεν γὰρ τῶν ἐναντίων ᾧ ἡ ἑτέρα παρὰ φύσιν, τὴν ἑτέραν εἶναι κατὰ φύσιν. Now, necessarily, everything which moves either up or down possesses lightness or heaviness or both—but not both relatively to the same thing: for things are heavy and light relatively to one another; air, for instance, is light relatively to water, and water light relatively to earth. The body, then, which moves in a circle cannot possibly possess either heaviness or lightness. For neither naturally nor unnaturally can it move either towards or away from the centre. Movement in a straight line certainly does not belong to it naturally, since one sort of movement is, as we saw, appropriate to each simple body, and so we should be compelled to identify it with one of the bodies which move in this way. Suppose, then, that the movement is unnatural. In that case, if it is the downward movement which is unnatural, the upward movement will be natural; and if it is the upward which is unnatural, the downward will be natural. For we decided that of contrary movements, if the one is unnatural to anything, the other will be natural to it.
Ἐπεὶ δ' εἰς τὸ αὐτὸ φέρεται τὸ ὅλον καὶ τὸ μόριον κατὰ φύσιν, οἷον πᾶσα γῆ καὶ μικρὰ βῶλος, συμβαίνει πρῶτον μὲν μήτε κουφότητ' ἔχειν αὐτὸ μηδεμίαν μήτε βάρος (ἢ γὰρ ἂν πρὸς τὸ μέσον ἢ ἀπὸ τοῦ μέσου ἠδύνατο φέρεσθαι κατὰ τὴν ἑαυτοῦ φύσιν), ἔπειθ' ὅτι ἀδύνατον κινηθῆναι τὴν κατὰ τόπον κίνησιν ἢ ἄνω ἢ κάτω κατασπώμενον οὔτε γὰρ κατὰ φύσιν ἐνδέχεται κινηθῆναι κίνησιν αὐτῷ ἄλλην οὔτε παρὰ φύσιν, οὔτ' αὐτῷ οὔτε τῶν μορίων οὐδενί ὁ γὰρ αὐτὸς λόγος περὶ ὅλου καὶ μέρους. But since the natural movement of the whole and of its part of earth, for instance, as a whole and of a small clod—have one and the same direction, it results, in the first place, that this body can possess no lightness or heaviness at all (for that would mean that it could move by its own nature either from or towards the centre, which, as we know, is impossible); and, secondly, that it cannot possibly move in the way of locomotion by being forced violently aside in an upward or downward direction. For neither naturally nor unnaturally can it move with any other motion but its own, either itself or any part of it, since the reasoning which applies to the whole applies also to the part.
Postquam philosophus ostendit quod est corpus quoddam aliud a corporibus quae sunt hic, scilicet a quatuor elementis et his quae componuntur ex eis, hic ostendit differentiam huius corporis ad corpora quae sunt hic. 51. After showing that there is a body distinct from those that are here, namely, from the four elements, and from things composed of them, the Philosopher here shows the difference of this body from those which exist here.
Et primo per comparationem ad motum localem; secundo secundum alios motus, ibi: similiter autem rationabile et cetera.

First by comparing them with respect to local motion; Secondly, with respect to other motions (L. 6);

Circa primum tria facit: About the first he does three things:

primo proponit quod intendit;

secundo ostendit propositum, ibi: oportet autem supponere etc.;

tertio excludit quandam obviationem, ibi: quoniam autem in idem feruntur et cetera.

First he proposes what he intends;

Secondly, he proves the proposition, at 52;

Thirdly, he dismisses an objection, at 56.

Dicit ergo primo quod, quia eorum quae dicta sunt quaedam sunt supposita (scilicet quod unum uni sit contrarium, et quod sint solae duae simplices magnitudines, scilicet recta et circularis, et si qua alia sunt huiusmodi), quaedam autem sunt demonstrata ex quibusdam praemissis (puta quod sint tres motus simplices, et quod motus circularis sit naturalis alicui corpori quod est aliud in natura a corporibus quae sunt hic), manifestum potest esse ex praedictis quod totum corpus illud quod circulariter movetur, non habet gravitatem neque levitatem, quae sunt principia quorundam motuum localium. He says therefore first [28] that, since some of the foregoing statements were supposed (namely, that one thing has one contrary, and that there are but two simple magnitudes, the straight line and the circle, and any other such suppositions) and others were demonstrated from certain premises (for example, that there are three simple motions, and that circular motion is natural to some body which is different in nature from the bodies that exist here), it can be plain from the foregoing that that entire body which is being moved circularly has neither heaviness nor lightness, which are principles of certain local motions.
Deinde cum dicit: oportet autem supponere etc., ostendit propositum. Et quia principium demonstrationis est quod quid est, ut dicitur in libro Poster., 52. Then at [29] he manifests his proposition. And because the principle of demonstration is "that which something is," as is said in Post. Anal. II,

primo supponit definitiones gravis et levis;

secundo ex his argumentatur ad propositum, ibi: necesse autem et cetera.

he first supposes the definitions of heavy and light, at 52;

Secondly, from these he argues to his proposition, at 54.

Circa primum duo facit: primo describit quid est grave et quid est leve; secundo describit quid est gravissimum et quid levissimum, ibi: gravissimum autem et cetera. Dicit ergo primo quod ad propositum ostendendum, oportet supponere quid dicamus grave et quid leve. Ideo autem dicit supponere, quia non perfecte investigat hic eorum definitiones; sed utitur eis ut suppositionibus, quantum sufficit ad necessitatem praesentis demonstrationis. Diligentius autem considerabitur de eis in quarto huius, ubi exponetur substantia, sive natura, ipsorum. Definit ergo grave, quod natum est moveri ad medium: leve autem, quod natum est moveri a medio. He says therefore first [29] that in order to prove the proposition we ought to suppose what it is that we call "heavy" and what "light." And he says "suppose" because he is not perfectly investigating their definitions here, but he uses them as suppositions to the extent that the present demonstration requires. But they will be considered more carefully in Book IV, where their "substance," or nature, will be explained. Accordingly, he defines heavy as "That which is apt to be moved to the middle," and the light as "that which is apt to be moved from the middle."
Utitur autem tali modo definiendi, ut observet se a contrarietate Platonis, qui dicebat quod in mundo secundum se non est sursum et deorsum, propter rotunditatem mundi: corpus enim rotundum est undique uniforme. Dicebat autem quod sursum et deorsum est in mundo solum quoad nos, qui nominamus sursum id quod est supra caput nostrum, deorsum autem id quod est sub pedibus nostris: si autem essemus e contrario situati, e contrario nominaremus sursum et deorsum. Sic ergo Plato non accipit id quod est sursum et deorsum, secundum rei naturam, sed quoad nos. 53. He uses this mode of defining in order to keep himself from the contrary position of Plato, who said that in the world according to itself there is no "up" and "down," on account of the rotundity of the world: for a rotund body is everywhere uniform. He said that there is "up" and "down" in the world only with respect to us, who call "up" that which is above our head, and "down" that which is below our feet, so that if we were contrarily situated, we would call "up" and "down" in a contrary manner. Consequently, Plato does not admit an "up" and "down" in the very nature of things but only with respect to us.
Aristoteles autem utitur his nominibus secundum communem modum loquendi, prout dicit in II Topic. quod nominibus utendum est ut plures: unde sursum et deorsum appellat in mundo id quod communiter ab hominibus appellatur sursum et deorsum. Nec tamen est distinctum solum quoad nos, sed etiam secundum naturam. Sicut enim in nobis distinguitur dextrum et sinistrum secundum diversam habitudinem ad motum animalem qui est secundum locum, ita sursum et deorsum determinatur in mundo secundum habitudinem ad motus simplicium corporum, quae sunt principales partes mundi. Et propter hoc ipse dicit quod sursum est locus in quem feruntur levia, deorsum autem locus in quem feruntur gravia. Et hoc rationabiliter: nam sicut in nobis nobilior pars est quae est sursum, ita in mundo corpora levia sunt nobiliora, quasi formaliora. Hic tamen, ut sine calumnia procedat ad propositum ostendendum, definit grave et leve per habitudinem ad medium. Aristotle, however, uses these names according to the common way of speaking, in keeping with his statement in Topics II, that names are to be used as they are used for the most part; hence he calls "up" and "down" in the world what are generally called "up" and "down" by men. Yet they are distinct not only with respect to us, but also according to nature. For just as we distinguish "right" and "left" in ourselves according to the diverse relationship to animal motion which is with respect to place, so too "up" and "down" in the world are distinguished with relation to the motions of the simple bodies which are the principal parts of the world. On this account he says that "up" is the place where light things are carried, and "down" the place where heavy things are carried. And this is reasonable: for just as in us the nobler part is that which is above, so in the world, light bodies are more noble, as if more formal. But here in order to proceed without calumny to the proof of the proposition, he defines "heavy" and "light" by their relation to the middle.
Deinde cum dicit: gravissimum autem etc., definit gravissimum et levissimum. Et dicit quod gravissimum est quod substat omnibus quae deorsum feruntur: levissimum autem est quod supereminet omnibus quae sursum feruntur. Et est intelligendum inter ea quae sursum et deorsum feruntur: nam caelum non est levissimum, quamvis omnibus superemineat, quia non sursum fertur. Est autem attendendum quod hic iam utitur eo quod est sursum et deorsum, tanquam sursum et deorsum esse accipiat ad quae terminatur motus qui est a medio, vel ad medium. 54. Then at [30] he defines "heaviest" and "lightest." And he says that the heaviest is "that which stands under all things that are carried downward," while the lightest is "that which is at the top of all things that are carried up." And this must be understood as concerning those things that are carried upward and downward — for the heaven is not the lightest, even though it is above all, because it is not carried upward. Now it should be recognized that here he is already using "up" and "down" as though "up" and "down" arise as being where a motion from the middle, or to the middle, is terminated.
Deinde cum dicit: necesse autem etc., ostendit propositum ex praemissis, dicens necessarium esse quod omne corpus quod fertur deorsum aut sursum, habeat absolute gravitatem, tanquam gravissimum, sicut terra, quae substat omnibus; aut quod habeat levitatem absolute, sicut ignis, qui superstat omnibus; aut habeat ambo, non quidem respectu eiusdem, sed respectu diversorum. Media enim elementa, scilicet aer et aqua, sunt ad invicem gravia et levia: sicut aer est levis per respectum ad aquam, quia superfertur ei, et eadem ratione aqua ad terram; aer vero ad ignem quidem est gravis, quia substat ei, et similiter aqua ad aerem. Corpus autem quod circulariter movetur, impossibile est quod habeat gravitatem aut levitatem. Neque enim potest moveri ad medium vel a medio secundum naturam, neque praeter naturam. 55. Then at [31] he proves his proposition from the foregoing, and says that every body carried up or down must have heaviness absolutely, as does the heaviest, namely, earth, which stands under all, or must have lightness absolutely, as does fire, which is above all, or must have both, not in respect to the same, but in respect to diverse things. For the intermediate elements, namely, air and water, are mutually heavy and light, as air is light with respect to water, because it is carried above it, and the same is true of water with respect to earth; meanwhile, air with respect to fire is heavy, because it exists under it, and similarly water with respect to air. But the body that is moved circularly can have neither heaviness nor lightness. For it cannot be moved to the middle or from the middle, either according to nature, or outside nature.
Et quod non possit secundum naturam hoc modo moveri, manifestat per hoc quod motus rectus, qui est ad medium vel a medio, est naturalis quatuor elementis: dictum est autem supra quod unus motus est naturalis uni simplicium corporum: ergo sequeretur quod corpus quod circulariter fertur, sit eiusdem naturae cum aliquo corporum quod movetur motu recto; cuius contrarium est supra ostensum. Similiter non potest dici quod motus rectus praeter naturam conveniat corpori quod circulariter fertur. Quia si unus contrariorum motuum inest alicui corpori praeter naturam, alius motus erit ei secundum naturam, ut ex supra dictis patet. Si ergo motus deorsum sit quinto corpori praeter naturam, motus sursum erit ei secundum naturam, et e converso. Utrumque autem eorum est falsum, ut patet per praecedentem rationem. Sequitur ergo quod corpus quintum, quod circulariter fertur, non moveatur a medio vel ad medium, neque secundum naturam neque praeter naturam. Omne autem corpus habens gravitatem aut levitatem, movetur uno horum motuum secundum naturam, et altero praeter naturam. Ergo corpus quintum neque habet gravitatem neque levitatem. And, that it cannot be so moved according to nature, is clear from the fact that a straight motion, which is to the middle, or from the middle, is natural to the four elements. But it was said above that one motion is natural to one of the simple bodies. Therefore it would follow that the body which is moved circularly would be of the same nature as one of the bodies that is moved in a straight line, the contrary of which was proved above. Similarly it cannot be said that a straight motion outside nature belongs to the body that is moved circularly. For if one of a pair of contrary motions is present in a body outside its nature, the other will be for it according to nature, as is plain from what has been said above. Therefore, if downward motion is outside nature for the fifth body, upward motion will be for it according to nature, and conversely. But both are false, as is plain from the preceding argument. It follows therefore that the fifth body, which is carried circularly, is not carried from the middle or to the middle, either according to nature or outside its nature. But every body having lightness or heaviness is moved according to nature by one of these motions, and outside its nature by the other. Therefore, the fifth body has neither heaviness nor lightness.
Deinde cum dicit: quoniam autem in idem etc., excludit quandam obviationem. Dicebant enim quidam quod partes elementorum sunt corruptibiles, ita quod extra proprium locum existentes, moventur naturaliter motu recto: ipsa autem elementa secundum suam totalitatem sunt incorruptibilia, et nunquam extra proprium locum esse possunt: unde in locis suis moventur circulariter. Et sic corpus quod circulariter movetur in suo loco secundum suam totalitatem, non oportet quod careat gravitate et levitate. 56. Then at [32] he excludes a certain objection. For some said that the parts of the elements are perishable, so that when existing outside their proper place they are naturally moved with a straight motion, while the elements themselves according to their totality are imperishable and cannot ever be outside their proper place — whence they are being moved circularly in their places. Consequently a body that is being moved circularly in its place according to its totality need not lack heaviness and lightness.
Ad hoc igitur excludendum, philosophus proponit quod in eundem locum feruntur naturaliter pars et totum, sicut tota terra et unus bolus eius. Et hoc patet ex quiete: quia unumquodque movetur naturaliter ad locum in quo quiescit naturaliter, in eodem autem loco quiescit naturaliter tota terra et pars eius. Unde manifestum est quod tota terra habet inclinationem naturalem quod moveatur ad medium, si esset extra suum locum. To exclude this the Philosopher proposes that part and whole are naturally carried to the same place, as, for example, in the case of the whole earth and one clod. And this is clear from the state of rest: for each thing is naturally moved to the place in which it is naturally at rest, and it is in the same place that the whole earth and part of it naturally rest. Hence it is clear that the whole earth has a natural inclination to be moved to the center, should it be outside its own place.
Sic ergo ex praemissis duo sequuntur. Quorum primum est quod totum corpus quintum nullam levitatem neque gravitatem habet: quia, ut patet ex ratione praedicta, moveretur naturaliter ad medium vel a medio. Secundo sequitur ex suppositione nunc inducta, quod si aliqua pars detraheretur a corpore caelesti, non moveretur neque sursum neque deorsum: quia cum sit eadem ratio de toto et partibus, non convenit neque toti quinto corpori neque alicui parti eius quod moveatur vel secundum naturam vel praeter naturam alio motu quam circulari. Therefore from the foregoing two things follow: The first of these is that the whole fifth body has no lightness or heaviness — for, as is clear from the aforesaid reason, it would be moved naturally to or from the middle. Secondly, it follows from the supposition now introduced that, if any part were detached from a heavenly body it would be moved neither up nor down, for, since the whole and part are of the same nature, it does not befit either the entire fifth body, or any part of it, to be moved either according to its nature or outside it with any motion other than the circular.

Lecture 6:
The fifth body not subject to other motions
Chapter 3 cont.
Ὁμοίως δ' εὔλογον ὑπολαβεῖν περὶ αὐτοῦ καὶ ὅτι ἀγένητον καὶ ἄφθαρτον καὶ ἀναυξὲς καὶ ἀναλλοίωτον, 33 It is equally reasonable to assume that this body will be ungenerated and indestructible and exempt from increase and alteration,
διὰ τὸ γίγνεσθαι μὲν ἅπαν τὸ γιγνόμενον ἐξ ἐναντίου τε καὶ ὑποκειμένου τινός, καὶ φθείρεσθαι ὡσαύτως ὑποκειμένου τέ τινος καὶ ὑπ' ἐναντίου καὶ εἰς ἐναντίον, καθάπερ ἐν τοῖς πρώτοις εἴρηται λόγοις τῶν δ' ἐναντίων καὶ αἱ φοραὶ ἐναντίαι. Εἰ δὴ τούτῳ μηδὲν ἐναντίον ἐνδέχεται εἶναι διὰ τὸ καὶ τῇ φορᾷ τῇ κύκλῳ μὴ εἶναι ἄν τιν' ἐναντίαν κίνησιν, ὀρθῶς ἔοικεν ἡ φύσις τὸ μέλλον ἔσεσθαι ἀγένητον καὶ ἄφθαρτον ἐξελέσθαι ἐκ τῶν ἐναντίων ἐν τοῖς ἐναντίοις γὰρ ἡ γένεσις καὶ ἡ φθορά. 34 since everything that comes to be comes into being from its contrary and in some substrate, and passes away likewise in a substrate by the action of the contrary into the contrary, as we explained in our opening discussions. Now the motions of contraries are contrary. If then this body can have no contrary, because there can be no contrary motion to the circular, nature seems justly to have exempted from contraries the body which was to be ungenerated and indestructible. For it is in contraries that generation and decay subsist.
Postquam philosophus ostendit differentiam quinti corporis ad alia corpora quae sunt hic, ex parte levitatis et gravitatis, secundum quod corpora habent inclinationem ad motum localem; hic ostendit differentiam quinti corporis ad corpora quae sunt hic, secundum alios motus; ostendens scilicet quod illud corpus non subiicitur aliis motibus, quibus haec corpora subiiciuntur. 58. After having shown the difference between the fifth body and the other bodies that exist here from the standpoint of lightness and heaviness, according to which bodies have an inclination to local motion, the Philosopher here shows how the fifth body differs from bodies that exist here from the standpoint of other motions, and shows that the former is not subject to the other motions to which these bodies are subject.

Et primo ostendit hoc per rationem;

secundo per signa, ibi: videtur autem et ratio et cetera.

First he shows this by an argument;

Secondly, by signs (L. 7);

Circa primum duo facit. With respect to the first he does two things:
Primo proponit quod intendit: et dicit quod sicut dictum est de quinto corpore quod caret gravitate et levitate, similiter rationabile est aestimare de ipso quod sit ingenitum et incorruptibile et inaugmentabile et inalterabile, idest non subiectum generationi et corruptioni, neque augmento neque alterationi.

Secundo ibi: propter fieri quidem etc., probat propositum:

First he proposes what he intends [33] and says that just as it has been pointed out above that the fifth body lacks heaviness and lightness, in like manner it is reasonable to believe that it is unproduced and imperishable, and incapable of increase and alteration, i.e., that it is not subject to generation and ceasing-to-be, or to growth or alteration.
Secondly [34], he proves the proposition:

et primo ostendit corpus caeleste esse ingenerabile et incorruptibile;

secundo quod est inaugmentabile, ibi: at vero et augmentabile etc.;

tertio quod non est alterabile, ibi: si autem est et inaugmentabile et cetera.

First he shows that the heavenly body is incapable of being generated or corrupted;

Secondly, that it cannot be increased (L. 7);

Thirdly, that it cannot be altered (also in L. 7).

Circa primum ponit talem rationem. Omne generabile fit ex contrario et subiecto quodam, sive materia: nam ex contrario fit aliquid sicut ex non permanente, ex subiecto autem sicut ex permanente, ut patet in I Physic. Et similiter etiam omne corruptibile corrumpitur existente aliquo subiecto. Est etiam omnis corruptio a contrario activo: omnis etiam corruptio terminatur in contrarium, sicut dictum est in primis sermonibus, idest in I Physic. Sed corpori quinto non est aliquid contrarium: ergo nec est generabile nec corruptibile. Mediam probat per hoc quod contrariorum contrarii sunt motus, sicut leve movetur sursum et grave deorsum: sed motui naturali quinti corporis, qui est motus circularis, nullus motus est contrarius, ut infra probabitur: ergo huic corpori nihil est contrarium. Et ita recte videtur natura fecisse, eximens hoc corpus a contrarietate, tanquam futurum, idest debens esse, ingenitum et incorruptibile. 59. With regard to the first he presents the following argument: Whatever can be generated comes to be from a contrary and a certain subject or matter — for something comes to be from a contrary as from something non-permanent, but from a subject as from something permanent, as is plain in Physics I. Likewise, every body that is perishable ceases to be while some subject [continues to] exist Also every case of ceasing-to-be is from a contrary active principle, for every ceasing-to-be is terminated at a contrary, as was said in the first discussions, i.e., in Physics I. But nothing is contrary to the fifth body. Therefore, it can be neither generated nor destroyed. He proves the middle [minor] proposition through the fact that the motions of contraries are contrary, as the light is moved upward and the heavy downward; but the fifth body's natural motion, which is circular motion, has no contrary motion, as will be proved later. Therefore nothing is contrary to this body. Thus nature seems to have acted rightly, exempting this body from contrariety as destined to be, i.e., having to be, unproduced and imperishable.
Sed circa ea quae hic Aristoteles dicit, duplex consideratio occurrit: una quidem circa positionem eius, qua ponit corpus caeli esse ingenerabile et incorruptibile; alia autem est circa rationem ipsius. 60. But two thoughts come to mind regarding what Aristotle says here: one is about his assumption that the body of the heaven is incapable of being generated and destroyed; the other is about the reason for it.
Sciendum est autem circa primum, quod quidam posuerunt corpus caeli esse generabile et corruptibile secundum suam naturam, sicut Ioannes grammaticus, qui dictus est Philoponus. Et ad suam intentionem adstruendam, primo utitur auctoritate Platonis, qui posuit caelum esse genitum et totum mundum. Secundo inducit talem rationem. Omnis virtus corporis finiti est finita, ut probatur in VIII Physic.: sed virtus finita non potest se extendere ad durationem infinitam (unde per virtutem finitam non potest aliquid moveri tempore infinito, ut ibidem probatur): ergo corpus caeleste non habet virtutem ut sit infinitum tempore. Tertio obiicit sic. In omni corpore naturali est materia et privatio, ut patet ex I Physic.: sed ubicumque est materia cum privatione, est potentia ad corruptionem: ergo corpus caeleste est corruptibile. Si quis autem dicat quod non est eadem materia caelestium corporum et inferiorum, obiicit in contrarium: quia secundum hoc oporteret quod materia esset composita, ex eo scilicet quod est commune utrique materiae, et ex eo quod facit diversitatem inter materias. Now it should be known, with regard to the first, that some supposed the body of the heaven to be generable and perishable according to its very nature, as did John the Grammarian, called Philoponus. And in support of his contention he uses first the authority of Plato who supposed that the heavens and the entire world were generated. Secondly, he presents this argument: Every power of a finite body is finite, as was proved in Physics VIII; but a finite power cannot extend itself unto infinite duration (that is why something cannot be moved for an infinite time through a finite power, as was proved in the same book); therefore, a heavenly body does not have the power to be infinite in time. Thirdly, he forms the following objection: In every natural body there is matter and privation, as is plain from Physics I; but wherever there is matter with privation, there is potency to cease to be; therefore, the heavenly body is perishable. And if anyone says that the matter of heavenly bodies is not the same as that of inferior bodies, he objects to the contrary — for, according to this, matter would have to be composite, made out of what is common to both matters, and out of what produces diversity between matters.
Sed haec necessitatem non habent. Quod enim Plato posuit caelum genitum, non intellexit ex hoc quod est generationi subiectum, quod Aristoteles hic negare intendit: sed quod necesse est ipsum habere esse ab aliqua superiori causa, utpote multitudinem et distensionem in suis partibus habens; per quod significatur esse eius a primo uno causari, a quo oportet omnem multitudinem causari. 61. But these statements lack necessity. For the fact that Plato posited the heavens as generated was not drawn from an understanding that they were subject to generation, which Aristotle intends here to deny, but because it was necessary for them to have their existence from a higher cause, as composed of parts multiple and extended — which meant that their existence was caused by some one first thing, from which all multiplicity must be caused.
Quod autem obiicit virtutem corporis caelestis esse finitam, solvit Averroes dicendo quod in corpore caelesti est virtus sive potentia ad motum secundum locum, non est autem virtus sive potentia ad esse, neque finita neque infinita. 62. The objection that the power of a heavenly body is finite Averroes solved by saying that in a heavenly body there is a power for local motion, but no power, either finite or infinite, respecting existence.
Sed in hoc manifeste dixit contra Aristotelem, qui infra in hoc eodem libro ponit in sempiternis virtutem ad hoc quod sint semper. Fuit autem deceptus per hoc quod existimavit virtutem essendi pertinere solum ad potentiam passivam, quae est potentia materiae; cum magis pertineat ad potentiam formae, quia unumquodque est per suam formam. Unde tantum et tamdiu habet unaquaeque res de esse, quanta est virtus formae eius. Et sic non solum in corporibus caelestibus, sed etiam in substantiis separatis est virtus essendi semper. But in this he is clearly going against Aristotle who later on in the same book supposes in sempiternal things a power to exist forever. But Averroes was deceived by supposing that the power respecting existence pertains solely to the passive power, which is the potency of matter; but the truth is that it pertains more to the power of the form, because everything exists through its form. Hence a thing has as much and as long an existence as the power of its form. Thus there is a power to exist forever, not only in heavenly bodies, but also in separated substances.
Dicendum est ergo quod id quod requirit virtutem infinitam, oportet esse infinitum. Infinitum autem, secundum philosophum in I Physic., pertinet ad quantitatem; ita quod id quod quantitate caret, neque finitum neque infinitum est. Motus autem quantitatem habet, quae mensuratur tempore et magnitudine, ut patet in VI Physic.: et ideo virtus quae potest in motum sempiternum, potest in effectum infinitum: et propter hoc talem virtutem oportet esse infinitam. Ipsum autem esse alicuius rei secundum se consideratum non est quantum: non enim habet partes, sed totum est simul. Accidit autem ei quod sit quantum, uno quidem modo secundum durationem, inquantum est subiectum motui et per consequens tempori, sicut esse rerum variabilium: unde virtus cuiuslibet rei corporalis cuius esse subiectum est variationi, non potest nisi in durationem finitam. Alio autem modo esse alicuius rei potest per accidens dici quantum, ex parte subiecti, quod habet determinatam quantitatem. Dicendum est ergo quod esse caeli non est subiectum variationi nec tempori: unde non est quantum quantitate durationis, et per consequens neque finitum neque infinitum. Est autem quantum secundum quantitatem corporis extensi; et secundum hoc est finitum. Sic igitur dicendum est quod virtus essendi corporis caelestis est finita: nec tamen sequitur quod sit ad essendum tempore finito; quia finitum et infinitum temporis accidit ipsi esse rei, quod non est subiectum varietati temporis. Non tamen posset huiusmodi virtus causare esse in infinita magnitudine, vel etiam in maiori quam sit magnitudo caelestis corporis. Therefore it should be said that whatever requires infinite power must be infinite. But the infinite, according to the Philosopher in Physics I, pertains to quantity, so that what lacks quantity is neither finite nor infinite. Now motion does have a quantity that is measured by time and magnitude, as is plain in Physics VI, and therefore the power which is capable of eternal motion is capable of an infinite effect — and consequently such a power must be infinite. But a thing's existence considered in itself is not a quantity, for it has no parts, but is entire and all at once. Rather it is accidental to it that it is quantified in one sense according to duration, insofar as it is subject to motion, and consequently to time, just as is the existence of changeable things. That is why the power of any bodily thing whose existence is subject to change cannot go beyond a finite duration. In another way the existence of a thing can be called quantified per accidens on the part of the subject, which has a definite quantity. Therefore it must be said that the existence of the heavens is not subject either to variation or time; hence it is not quantified by a quantity of duration, and consequently is neither finite nor infinite in this respect. But it is quantified according to the quantity of an extended body, and in this respect it is finite. Consequently. it must be said that the power of existing of a heavenly body is finite, but that does not mean that it is limited to existing in a finite time, because temporal finiteness or infinity are accidental to a thing's existence, which is not subject to the variation of time. Nevertheless a power of this kind could not cause existence in an infinite magnitude nor even in a magnitude greater than the magnitude of the heavenly body.
Similiter tertium quod obiicit, Averroes solvit per interemptionem. Negat enim corpus caeleste habere materiam: sed dicit corpus caeleste esse subiectum actu ens, ad quod comparatur anima eius sicut forma ad materiam. Et si quidem intelligat quod corpus caeleste non habeat materiam secundum quod dicitur materia in ordine ad motum vel mutationem, verum dicit: sic enim etiam Aristoteles in VIII et XII Metaphys. ponit corpus caeleste habere materiam non ad esse sed ad ubi; quia scilicet non est subiecta transmutationi quae est secundum esse, sed ei quae est secundum ubi. Si vero intelligat quod corpus caeleste nullo modo habet materiam, vel quodcumque subiectum, manifeste dicit falsum. Patet enim quod corpus illud est actu ens: alioquin non ageret in haec inferiora. Omne autem quod est actu ens, vel est actus, vel est habens actum. Non potest autem dici quod corpus caeleste sit actus: quia sic esset forma subsistens, et esset aliquid intellectum in actu, non autem sensu apprehensum. Oportet ergo in corpore caelesti ponere aliquod subiectum suae actualitati. Similarly Averroes solves the third objection by destroying it. For he denies that a heavenly body has matter, but says that a heavenly body is a subject that is actual being, to which its soul is compared as form to matter. Now if in stating that a heavenly body does not have matter he should mean matter in relation to motion or change, then it is true — for thus does Aristotle also say in Physics VIII and Metaphysics XII, namely, that a heavenly body has matter not with respect to existence but to "where," for the simple reason that this matter is not subject to a change according to being but to one according to "where." But if he means that a heavenly body has matter in no way at all or no subject at all, then plainly he is wrong. For it is clear that that body is a being in act; otherwise it would not act on the lower bodies. But whatever is a being in act is either act itself, or has act. Now it cannot be said that a heavenly body is act, for then it would be a subsistent form, and something understood in act but not apprehended by sense. Therefore in a heavenly body there must be something which is the subject of its actuality.
Non tamen oportet quod istud subiectum vel materia habeat privationem: quia privatio nihil aliud est quam absentia formae quae est nata inesse, huic autem materiae vel subiecto non est nata inesse alia forma, sed forma sua replet totam potentialitatem materiae, cum sit quaedam totalis et universalis perfectio. Quod patet ex hoc, quod virtus activa eius est universalis, non particularis sicut virtus inferiorum corporum; quorum formae, tanquam particulares existentes, non possunt replere totam potentialitatem materiae; unde simul cum una forma remanet in materia privatio formae alterius, quae est apta nata inesse. Sicut etiam videmus quod corpora inferiora sunt susceptiva diversarum figurarum: sed corpus caeli non est figurabile alia figura. Sic igitur in corpore caelesti non est privatio alicuius formae, sed solum privatio alicuius ubi. Unde non est mutabile secundum formam per generationem et corruptionem; sed solum secundum ubi. Ex quo patet quod materia caelestis corporis est alia et alterius rationis a materia inferiorum corporum, non quidem per aliquam compositionem, sicut Philoponus existimavit; sed per habitudinem ad diversas formas, quarum una est totalis et alia partialis: sic enim potentiae diversificantur secundum diversitatem actuum ad quos sunt. However this subject or matter does not need to have privation, for privation is nothing but the absence of a form which is apt to exist in the matter; but in this matter or subject there is no other form apt to be — rather its form fills out the entire potentiality of the matter, since it is a certain total and universal perfection. And this is clear from the fact that its active power is universal, and not particular like the power of the lower bodies, whose forms as being particular cannot exhaust the entire potentiality of the matter; hence, together with one form there remains in matter the privation of another form which is apt to be in it. Similarly, we see that the lower bodies are subject to diverse shapes, but the heavenly body not. Accordingly, in a heavenly body there is not privation of any form but only privation of some "where." Consequently, it is not changeable with respect to form through generation and ceasing to be, but only with respect to "where." From this it is plain that the matter of the heavenly body is distinct and of a different nature from the matter of lower bodies, not on account of some composition, as Philoponus supposed, but on account of their relationship to diverse forms, of which one is total and the other partial — for thus potencies are diversified, namely, according to the diversity of acts to which they are in potency.
Manifestum est igitur ex his quod corpus caeli secundum suam naturam non est subiectum generationi et corruptioni, utpote primum in genere mobilium, et propinquissimum rebus immobilibus. Et inde est quod minimum habet de motu. Movetur enim solum motu locali, qui nihil variat intrinsecum rei. Et inter motus locales habet motum circularem, qui etiam minimum variationis habet: quia in motu sphaerico totum non mutat suum ubi subiecto, sed solum ratione, ut probatur in VI Physic.; sed partes mutant ubi diversum etiam subiecto. 64. Therefore it is manifest from the foregoing that the body of the heavens according to its nature is not subject to generation and ceasing-to-be, as being first in the genus of mobiles, and the closest to immobile things. That is why it has a minimum of motion. For it is moved only with local motion, which varies nothing intrinsic to a thing. And among local motions it has a circular motion, which also has a minimum of variation, because in spherical motion the whole does not vary its "where" as to subject, but only in conception, as was proved in Physics VI; but the parts change their "where" even as to subject.
Non tamen dicimus secundum fidem Catholicam, quod caelum semper fuerit, licet dicamus quod semper sit duraturum. Nec hoc est contra demonstrationem Aristotelis hic positam: non enim dicimus quod incoeperit esse per generationem, sed per effluxum a primo principio, a quo perficitur totum esse omnium rerum, sicut etiam philosophi posuerunt. A quibus tamen in hoc differimus, quod illi ponunt Deum produxisse caelum coaeternum sibi; nos autem ponimus caelum esse productum a Deo secundum totam sui substantiam ab aliquo determinato principio temporis. However, we do not say according to the Catholic faith that the heavens always existed, although we say that they will endure forever. Nor is this against Aristotle's demonstration here, for we do not say that they began to be through generation, but through an efflux from the first principle, by whom is perfect the entire existence of all things, as even the philosophers posited. From whom, however, we differ in this, that they suppose God to have produced the heavens co-eternal to Himself, but we posit that the heavens were produced by God according to their whole substance at some definite beginning of time.
Contra quod tamen obiicit Simplicius, Aristotelis Commentator, super hunc locum, tripliciter. Primo quidem quia Deus produxit caelum secundum suum esse, non per aliquid aliud additum: unde, cum esse suum sit aeternum et invariabile, semper caelum ab ipso processit. Item, si bonitas Dei est causa rerum, fuisset bonitas Dei otiosa et vacans antequam mundus esset, si ex aliquo determinato principio temporis incoepit. Item, omne quod incipit esse in aliqua determinata parte temporis cum prius non fuerit, hoc contingit ei ex ordine alicuius superioris motus, ex quo contingit quod hoc nunc incoepit et non prius; sicut homo incoepit esse nunc et non prius, secundum ordinem revolutionis caelestis corporis. Non est autem dare aliquam superiorem revolutionem aut motum ultra corpus caeleste. Non ergo potest dici quod corpus caeli ita nunc incoeperit quod prius non fuerit. 65. Against this, however, Simplicius, a commentator on Aristotle, at this passage, objects on three counts. First, since God produced the heavens, therefore, through His essence and not through something added, since His essence is eternal and unchanging, the heavens have always proceeded from Him. Again, if the goodness of God is the cause of things, the goodness of God would have been idle and disengaged before the world existed, if the latter began to exist from some definite beginning of time. Again, whatever begins to exist in some determined part of time after previously not existing, this happens to it from the ordination of some higher motion from which it happens that this being begins now and not before, as a man begins to be now and not previously, according to the order of the revolution of the heavenly body. But there is no higher revolution or motion beyond the heavenly body. Therefore it cannot be said that the body of the heavens began to be now, so as not to have been before.
Sed haec necessitatem non habent. Quod enim primo dicitur, quod Deus agit per suum esse et non per aliquid superadditum, verum est: sed esse suum non est distinctum a suo intelligere, sicut in nobis, nec etiam a suo velle: unde producit secundum intelligere et velle suum. In his autem quae producuntur ab aliquo agente inquantum est intelligens et volens, oportet esse illud quod producitur, hoc modo sicut est intellectum a producente; non autem eo modo quo est ipse producens secundum suum esse. Unde, sicut non oportet quod id quod est productum a Deo producente secundum suum esse, sit in aliis conditionibus tale quale est esse divinum, sed quale est determinatum per eius intelligere; ita non est necessarium quod id quod est productum a Deo, sit tam diuturnum quantum Deus, sed quantum determinatum est per intellectum ipsius. But these lack necessity. For the first statement that God acts through His essence and not through something superadded is true, but His essence is not distinct from His understanding, as in us, nor from His willing. Hence He produces according to His understanding and His willing. Now in things produced by an agent acting in virtue of his understanding and will, that which is produced must be as it was understood by the producer, and not as the producer is in his being. Hence, just as what is produced by God acting through His essence does not have to be, in other respects, in the same way as the divine essence, but such as it is determined by His understanding, so too it is not necessary that what is produced by God be as long-lasting as God, but only to the degree determined by His understanding.
Et hoc etiam potest dici circa quantitatem dimensivam caeli. Quod enim caelum habeat tantam quantitatem et non maiorem, provenit ex determinatione intellectus divini determinantis sibi talem quantitatem, et coaptantis ei naturam proportionatam tali quantitati: sicut etiam exemit ipsum a contrariis, ut esset ingenitum et incorruptibile, ut dicitur in littera. Quod enim dicit recte fecisse naturam, importat actionem intellectus agentis propter aliquem finem: non enim alia natura superior exemit eum a contrariis nisi divina. And this applies also to the dimensive quantity of the heavens. For the fact that the heavens have such-and-such a quantity, and no greater, is a result of a determination of the divine intellect determining such a quantity for them, and adapting to them a nature proportionate to such quantity, just as He frees them from contraries so that they may be ungenerated and incorruptible, as stated in the text. The phrase in the text that "nature acted rightly" implies the action of an intellect acting for an end, for it is no nature other than the divine that has freed them from contraries.
Similiter, quod dicit bonitatem divinam fuisse vacantem et otiosam ante productionem mundi, non habet rationem. Otiosum enim dicitur quod non consequitur finem ad quem est: bonitas autem Dei non est propter creaturas. Unde creaturae essent otiosae si non consequerentur divinam bonitatem: divina autem bonitas non esset otiosa, etiam si nullam unquam creaturam produxisset. Similarly, the statement that the divine goodness would have been idle and disengaged before the production of the world does not have any weight. For a thing is called "idle" that does not attain the end for which it is. But the goodness of God is not for the sake of creatures. Hence creatures would be idle if they did not attain to the divine goodness, but the divine goodness would not be idle even if It never produced a creature.
Similiter etiam quod tertio obiicit, locum habet in agente particulari, quod praesupponit tempus et in parte temporis aliquid facit: et ita oportet quod id quod fit, proportionetur ab agente et ad aliam partem temporis et ad totum tempus, vel etiam ad causam totius temporis. Sed nunc agimus de agente universali, quod producit ipsum totum tempus simul cum his quae sunt in tempore. Et ideo non habet hic locum ut quaeratur quare nunc et non prius: quasi praesupponatur alia pars temporis praecedens, vel aliqua alia causa universalior causans totum tempus. Sed habet hic locum quaestio, quare agens universale, scilicet Deus, voluit tempus non esse semper et ea quae sunt in tempore. Et hoc dependet ex determinatione intellectus ipsius: sicut et in domo artifex quantitatem alicuius partis domus accipit secundum proportionem ad aliam partem vel ad totam domum; sed quantitatem totius domus determinat secundum suum intellectum et voluntatem. Again, the third objection applies to a particular agent, which supposes time and works in time. In this way what comes to be must be proportioned by the agent both to some part of time, and to the whole of time, or even to the cause of the whole of time. But we are dealing now with a universal agent who produces the whole time together with the things in time. So there is no place here for the question of why now and not before, as though there were presupposed some other preceding part of time, or some more general cause producing all of time. But the pertinent question here is why the universal agent, namely, God, willed time and the things in time not always to exist. And this depends on a determination of His intellect, just as in a house the artisan determines the size of one part of the house in relation to another part or to the whole house, but the size of the entire house he himself determines according to his understanding and will.
Restat autem alia consideratio circa demonstrationem Aristotelis, contra quam obiicit Ioannes grammaticus: quia si nihil generatur et corrumpitur nisi quod habet contrarium, cum substantiae non sit aliquid contrarium, quod maxime manifestum est in animalibus et plantis (similiter etiam nec figuris et relationibus est aliquid contrarium), nihil horum generabitur aut corrumpetur. 67. Another point remains to be considered about Aristotle's demonstration against which John the Grammarian objects: if nothing but what has a contrary can be generated and cease to be, then since there is no contrary of a substance, as is plain in animals and plants (similarly, nothing is contrary to a figure or a relation), none of these will be generated and cease to be.
Respondet autem ad hoc Simplicius quod hoc est intelligendum de contrario communiter dicto, prout includit etiam contrarietatem privationis et speciei: sic enim Aristoteles loquitur de contrario in I Physic., quo nos remittit. Et hoc modo contrarium invenitur in omnibus praedictis, sicut informe est contrarium formato, et infiguratum figurato: privatio autem non habet locum in corporibus caelestibus, ut dictum est. To this Simplicius responds that this is to be understood about a contrary in the general sense as including even contrariety of privation and species, for that is Aristotle's meaning when he speaks of contraries in Physics I. And that is the way in which contrariety is found in all the foregoing, as the unformed is contrary to the formed, and the unfigured to the figured; but privation has no place in heavenly bodies, as has been said.
Haec autem responsio, etsi sit vera, non tamen habet locum in proposito. Aristoteles enim dicit contrarietatem motuum localium respondere contrarietati corporum; cum tamen certum sit quod privationi non respondet aliquis motus localis. Unde dicendum est quod, sicut ipse etiam post dicet, substantiae nihil est contrarium secundum compositum, vel secundum materiam, vel secundum formam substantialem: est tamen aliquid sibi contrarium secundum propriam dispositionem ad talem formam, sicut ignis dicitur esse contrarius aquae contrarietate calidi et frigidi. Et talis contrarietas requiritur in omnibus quae generantur et corrumpuntur. Huiusmodi autem contrarietatem consequitur contrarietas motuum secundum grave et leve: per quorum subtractionem intelligitur corpus caeleste esse exemptum ab omnibus aliis contrariis quae comitantur grave et leve. But this response, although true, is not ad rem, For Aristotle says that contrariety of local motions corresponds to contrariety of bodies; and it is certain that no local motion corresponds to a privation. Consequently, it must be said that, as he himself will say later, nothing is contrary to substance with respect to its being a composite, or according to matter or substantial form; but there is something contrary to it according to its proper disposition to such a form, as fire is said to be contrary to water by reason of the contrariety of hot and cold. And such contrariety is required in all things that are generated and cease to be. But it is upon such contrariety that contrariety of motions according to heavy and light follow: through the absence of which, a heavenly body is understood to be free of all the contraries that accompany the heavy and the light.
Item videtur, secundum hoc quod contrarietati corporum dicit respondere contrarietatem motuum, quod ignis magis sit contrarius terrae quam aquae, cum qua convenit in una qualitate, scilicet in siccitate. 68. Likewise, since he says that contrariety of motions corresponds to contrariety of bodies, it seems that fire is contrary more to earth than to water, because fire agrees with the former in respect of one quality, namely, dryness.
Et dicendum est quod philosophus in hoc libro agit de corporibus simplicibus secundum situm: sic enim constituunt universum ut partes. Et secundum hoc maior est contrarietas ignis ad terram quam ad aquam: licet ad aquam sit maior contrarietas ignis secundum qualitates activas et passivas, quod pertinet ad considerationem libri de generatione. And it must be said that in this book the Philosopher is discussing simple bodies with respect to their position; for it is under this aspect that they are parts making up the universe. And according to this, the contrariety of fire to earth is greater than its contrariety to water. Yet it is true that fire has a greater contrariety to water from the viewpoint of active and passive qualities, which consideration belongs to the book On Generation.
Videtur etiam non ex necessitate sequi quod corpori caelesti nihil sit contrarium, ex eo quod motui circulari, quo movetur, nihil sit contrarium: quia etiam ignis in propria sphaera, et suprema pars aeris circulariter moventur, ut in I Meteor. dicitur; aeri tamen et igni est aliquid contrarium. 69. Again, it does not seem to follow of necessity that nothing is contrary to a heavenly body just because nothing is contrary to the circular motion with which it is moved, because fire also in its own sphere, and the upper region of air, are moved circularly, as is said in Meteorology I, and yet there is a contrary to fire and air.
Sed dicendum est quod ignis et aer non moventur circulariter quasi proprio motu, sed deferuntur per motum caeli: corpora autem caelestia moventur circulariter proprio motu: unde non est similis ratio. But it should be said that fire and air are not moved circularly as though by their own motion; rather they are carried along by the motion of the heavens. The heavenly bodies, however, are moved circularly by their own motion; consequently, the case is not the same.
Item videtur quod contrarietas motuum non attestetur contrarietati mobilium. Eadem enim substantia numero, quae sibi non contrariatur, est susceptiva contrariorum, ut dicitur in praedicamentis; et ita movetur motibus contrariis, qui sunt ad contraria, puta dealbatione et denigratione et similibus motibus. Praeterea aer movetur sursum in loco aquae existens, deorsum autem existens in loco ignis: idem ergo contrariis motibus movetur, et sic contrarietas motuum non consequitur contrarietatem mobilium. Adhuc etiam videmus quod eadem anima movetur motu virtutis et vitii, qui sunt contrarii motus. 70. Again, it seems that contrariety of motions does not attest to contrariety of mobiles. For the same numerical substance, which is not contrary to itself, is subject to contraries, as is said in the Predicaments; thus it is moved by contrary motions which are terminated at contraries: for example, a substance is moved by whitening and blackening and similar motions. Moreover, air existing in the place of water is moved upward, but in the place of fire downwards. Therefore the same thing is moved by contrary motions, and, consequently, contrariety of motions does not follow upon contrariety of mobiles. Furthermore, we see that the same soul is moved by the motions of vice and virtue, which are contrary motions.
Est autem circa hoc considerandum quod philosophus utitur hac propositione: quod si motus non sint contrarii, quod etiam mobilia non sunt contraria. Non autem ponit e converso quod si mobilia non sunt contraria, quod motus non sint contrarii (quia posset aliquis dicere quod omnium corporum contrarietatem habentium sint contrarii motus, non autem omnes contrarii motus sunt contrariorum): contra quod praedictae obiectiones procedunt. Tamen, secundum rei veritatem, contrarietas motuum naturalium consequitur proprietatem principiorum activorum sive formalium, ad quae consequitur motus; non autem contrarietatem principiorum passivorum sive materialium, quia eadem materia susceptiva est contrariorum. Et ideo alterationes quae fiunt ex principiis extrinsecis, nihil prohibet esse circa idem subiectum, quamvis sint contrariae. Si qua vero est alteratio ex intrinseco principio proveniens, sicut sanatio quando fit per naturam, oportet quod contrarietas talium alterationum consequatur contrarietatem mobilium. Et eadem ratio est de motibus localibus, de quibus nunc intendit: huiusmodi enim motus consequuntur principia formalia intrinseca. With regard to this it must be considered that the Philosopher uses this proposition, namely, that if motions are not contrary, the mobiles also are not contrary. But he does not state the converse, that if the mobiles are not contrary the motions are not contrary (because someone could say that the motions of all bodies having contrariety are contrary, but not that all contrary motions involve contrary things): against which the foregoing objection is directed. Yet in truth contrariety of natural motions follows upon what is proper to the active or formal principles (which the motion follows upon), and not upon the contrariety of the passive or material principles, because the same matter is subject to contraries. And therefore nothing prevents the same subject from being affected by alterations caused by extrinsic principles, even though such alterations be contrary. But if an alteration arises from an intrinsic principle, as when health is restored by the nature, then the contrariety of such alterations follows upon the contrariety of the mobiles. And the same holds for local motions, which we are now discussing: for such motions follow upon intrinsic formal principles
Ad id vero quod obiicitur de aere, dicendum quod contradictio quae includitur in omnibus oppositis, habet in sui ratione quod sit secundum idem et respectu eiusdem. Motus autem aeris naturalis non est sursum et deorsum respectu eiusdem; sed sursum quidem respectu aquae et terrae, deorsum vero respectu ignis. Unde huiusmodi motus non sunt contrarii: non enim sunt ad contraria loca, sed ad eundem locum, qui scilicet supereminet aquae et subsidet igni. Now, regarding the objection about air, it must be said that the contradiction which is included in all opposites requires in its very notion that it be with respect to the same thing and according to the same aspect. But the natural motion of air is not up and down with respect to the same thing; rather it is upward with respect to water and earth, and downward with respect to fire. Consequently such motions are not contrary, for they are not tending to contrary places but to the same place, i.e., the place which is above water and below fire.
Quod autem dicitur de motu animae secundum virtutem et vitium, non est ad propositum: quia huiusmodi motus non sunt naturales, sed voluntarii. What is said about the motion of the soul according to virtue and vice is not ad rem — for such motions are not natural but voluntary.

Lecture 7:

The heavenly body is not subject to growth and decrease, or to alteration.

Chapter 3 cont.
Ἀλλὰ μὴν καὶ τὸ αὐξανόμενον ἅπαν αὐξάνεται [καὶ τὸ φθῖνον φθίνει] ὑπὸ συγγενοῦς προσιόντος καὶ ἀναλυομένου εἰς τὴν ὕλην τούτῳ δ' οὐκ ἔστιν ἐξ οὗ γέγονεν. 35 Again, that which is subject to increase increases upon contact with a kindred body, which is resolved into its matter. But there is nothing out of which this body can have been generated.
Εἰ δ' ἐστὶ καὶ ἀναύξητον καὶ ἄφθαρτον, τῆς αὐτῆς διανοίας ἐστὶν ὑπολαβεῖν καὶ ἀναλλοίωτον εἶναι. Ἔστι μὲν γὰρ ἡ ἀλλοίωσις κίνησις κατὰ τὸ ποιόν, τοῦ δὲ ποιοῦ αἱ μὲν ἕξεις καὶ διαθέσεις οὐκ ἄνευ τῶν κατὰ τὰ πάθη γίγνονται μεταβολῶν, οἷον ὑγίεια καὶ νόσος. Κατὰ δὲ πάθος ὅσα μεταβάλλει τῶν φυσικῶν σωμάτων, ἔχονθ' ὁρῶμεν πάντα καὶ αὔξησιν καὶ φθίσιν, οἷον τά τε τῶν ζῴων σώματα καὶ τὰ μόρια αὐτῶν καὶ τὰ τῶν φυτῶν, ὁμοίως δὲ καὶ τὰ τῶν στοιχείων ὥστ' εἴπερ τὸ κύκλῳ σῶμα μήτ' αὔξησιν ἔχειν ἐνδέχεται μήτε φθίσιν, εὔλογον καὶ ἀναλλοίωτον εἶναι. (270b.) Διότι μὲν οὖν ἀΐδιον καὶ οὔτ' αὔξησιν ἔχον οὔτε φθίσιν, ἀλλ' ἀγήρατον καὶ ἀναλλοίωτον καὶ ἀπαθές ἐστι τὸ πρῶτον τῶν σωμάτων, εἴ τις τοῖς ὑποκειμένοις πιστεύει, φανερὸν ἐκ τῶν εἰρημένων ἐστίν. 36 And if it is exempt from increase and diminution, the same reasoning leads us to suppose that it is also unalterable. For alteration is movement in respect of quality; and qualitative states and dispositions, such as health and disease, do not come into being without changes of properties. But all natural bodies which change their properties we see to be subject without exception to increase and diminution. This is the case, for instance, with the bodies of animals and their parts and with vegetable bodies, and similarly also with those of the elements. And so, if the body which moves with a circular motion cannot admit of increase or diminution, it is reasonable to suppose that it is also unalterable. The reasons why the primary body is eternal and not subject to increase or diminution, but unaging and unalterable and unmodified, will be clear from what has been said to any one who believes in our assumptions.
Ἔοικε δ' ὅ τε λόγος τοῖς φαινομένοις μαρτυρεῖν καὶ τὰ φαινόμενα τῷ λόγῳ πάντες γὰρ ἄνθρωποι περὶ θεῶν ἔχουσιν ὑπόληψιν, καὶ πάντες τὸν ἀνωτάτω τῷ θείῳ τόπον ἀποδιδόασι, καὶ βάρβαροι καὶ Ἕλληνες, ὅσοι περ εἶναι νομίζουσι θεούς, δῆλον ὅτι ὡς τῷ ἀθανάτῳ τὸ ἀθάνατον συνηρτημένον ἀδύνατον γὰρ ἄλλως. Εἴπερ οὖν ἔστι τι θεῖον, ὥσπερ ἔστι, καὶ τὰ νῦν εἰρημένα περὶ τῆς πρώτης οὐσίας τῶν σωμάτων εἴρηται καλῶς. 37 Our theory seems to confirm experience and to be confirmed by it. For all men have some conception of the nature of the gods, and all who believe in the existence of gods at all, whether barbarian or Greek, agree in allotting the highest place to the deity, surely because they suppose that immortal is linked with immortal and regard any other supposition as inconceivable. If then there is, as there certainly is, anything divine, what we have just said about the primary bodily substance was well said.
Συμβαίνει δὲ τοῦτο καὶ διὰ τῆς αἰσθήσεως ἱκανῶς, ὥς γε πρὸς ἀνθρωπίνην εἰπεῖν πίστιν ἐν ἅπαντι γὰρ τῷ παρεληλυθότι χρόνῳ κατὰ τὴν παραδεδομένην ἀλλήλοις μνήμην οὐθὲν φαίνεται μεταβεβληκὸς οὔτε καθ' ὅλον τὸν ἔσχατον οὐρανὸν οὔτε κατὰ μόριον αὐτοῦ τῶν οἰκείων οὐθέν. 38 The mere evidence of the senses is enough to convince us of this, at least with human certainty. For in the whole range of time past, so far as our inherited records reach, no change appears to have taken place either in the whole scheme of the outermost heaven or in any of its proper parts.
Ἔοικε δὲ καὶ τοὔνομα παρὰ τῶν ἀρχαίων παραδεδόσθαι μέχρι καὶ τοῦ νῦν χρόνου, τοῦτον τὸν τρόπον ὑπολαμβανόντων ὅνπερ καὶ ἡμεῖς λέγομεν οὐ γὰρ ἅπαξ οὐδὲ δὶς ἀλλ' ἀπειράκις δεῖ νομίζειν τὰς αὐτὰς ἀφικνεῖσθαι δόξας εἰς ἡμᾶς. Διόπερ ὡς ἑτέρου τινὸς ὄντος τοῦ πρώτου σώματος παρὰ γῆν καὶ πῦρ καὶ ἀέρα καὶ ὕδωρ, αἰθέρα προσωνόμασαν τὸν ἀνωτάτω τόπον, ἀπὸ τοῦ θεῖν ἀεὶ τὸν ἀΐδιον χρόνον θέμενοι τὴν ἐπωνυμίαν αὐτῷ. Ἀναξαγόρας δὲ καταχρῆται τῷ ὀνόματι τούτῳ οὐ καλῶς ὀνομάζει γὰρ αἰθέρα ἀντὶ πυρός. 39 The common name, too, which has been handed down from our distant ancestors even to our own day, seems to show that they conceived of it in the fashion which we have been expressing. The same ideas, one must believe, recur in men's minds not once or twice but again and again. And so, implying that the primary body is something else beyond earth, fire, air, and water, they gave the highest place a name of its own, aither, derived from the fact that it 'runs always' for an eternity of time. Anaxagoras, however, scandalously misuses this name, taking aither as equivalent to fire.
Postquam philosophus ostendit quod corpus quintum non est subiectum generationi et corruptioni, hic ostendit quod non est subiectum augmento et deminutioni. Et utitur tali ratione. Omne corpus augmentabile est quantum ad aliquid subiectum generationi et corruptioni. Ad cuius manifestationem proponit quod omne corpus augmentabile augetur per appositionem alicuius connaturalis advenientis; quod quidem, cum prius esset dissimile, factum est simile per resolutionem in propriam materiam, quae, deposita forma priori, formam corporis augmentandi assumpsit; sicut panis, resolutus in materiam, accipit formam carnis, et ita per additionem ad carnem praeexistentem facit augmentum. Unde ubicumque est augmentum, ibi oportet quod sit generatio et corruptio in aliquid. Corpori autem caelesti non est dare aliquid ex quo sit generatum, ut ostensum est. Ergo non potest esse augmentabile vel deminuibile. After showing that the fifth body is not subject to generation and corruption, the Philosopher here shows that it is not subject to increase and diminution [35] and uses this argument: Every augmentable body is, with respect to something, subject to generation and corruption. To explain this, he proposes that every augmentable body is increased by the addition of something connatural that comes to it. This, indeed, while being first unlike, has become like by being resolved into its proper matter which, doffing its previous form, has assumed the form of the body to be increased — as bread, after being resolved into matter, receives the form of flesh, and thus, through being added to pre-existing flesh, produces increase. Hence wherever there is growth there must be generation and corruption into something. But there is nothing from which a heavenly body can be generated, as has been shown. Therefore it cannot be augmentable or decreasable.
Deinde cum dicit: si autem est etc., ostendit quod non sit subiectum alterationi. Posset autem videri alicui quod brevis via removendi alterationem a corpore caelesti, esset per remotionem contrarietatis: sicut enim generatio est ex contrariis, ita et alteratio. Sed advertendum quod Aristoteles removit contrarietatem a quinto corpore removendo ab eo contrarietatem motus: alteratio autem videtur fieri non solum secundum contrarietatem cui respondent contrarii motus locales, quae est gravis et levis et eorum quae assequuntur; sed etiam secundum alia contraria quae ad hoc non pertinent, puta secundum album et nigrum: et ideo utitur alia via, quae sumitur ex parte augmenti. 72. Then at [36] he shows that it is not subject to alteration. Now it might seem to someone that an easy way to remove alteration from the heavenly body would be by removing contrariety, for just as generation occurs from contraries, so too, does alteration. But it should be observed that Aristotle removed contrariety from the fifth body by removing from it contrariety of motion. Alteration, however, seems to occur not only according to the contrariety to which contrary local motions correspond, namely, heavy and light and whatever results from them, but also according to other contraries which do not pertain to this, for example, according to black and white. Accordingly, he uses another way, based on increase.
Et dicit quod eiusdem rationis est aestimare quod corpus caeleste non sit alterabile, et quod non sit augmentabile seu corruptibile. Quia alteratio est motus secundum qualitatem, ut dictum est in V Physic. Alteratio autem, ut in VII Physic. ostensum est, proprie fit secundum tertiam speciem qualitatis, quae est passio et passibilis qualitas: quamvis enim habitus et dispositio pertineant ad genus qualitatis, non tamen causantur sine transmutatione quae fit secundum passiones; sicut sanitas et languor proveniunt ex transmutatione frigidi et calidi, humidi et sicci. Omnia autem corpora naturalia quae transmutantur secundum passionem vel passibilem qualitatem, per consequens videntur habere augmentum et decrementum; sicut patet de corporibus animalium et de partibus eorum, et etiam de plantis, in quibus proprie est augmentum. Ita etiam est de elementis: quae quidem secundum transmutationem calidi et frigidi rarefiunt et condensantur, et per consequens transmutantur in maiorem vel minorem quantitatem, quod est quodammodo augeri et deminui. Sic igitur patet quod, si corpus quod circulariter movetur, non subiacet augmento vel decremento, quod etiam non subiaceat alterationi. And he says that it is for the same reason that we estimate a heavenly body not to be alterable and not to be augmentable or perishable. For alteration is a motion affecting quality, as has been said in Physics V. But alteration, as was shown in Physics VII, properly takes place according to the third species of quality, which is "passion and passible quality": for although "habit and disposition" pertain to [the first species of] the genus of quality, they are not produced without a change made according to the passions, just as health and languor result from a change of cold and hot, moist and dry. Now all natural bodies that are changed with respect to passion or passible quality seem as a consequence to have growth and decrease, as is clear from the bodies of animals and their parts and even of plants, in which growth properly exists. The same applies also to the elements, which rarefy and condense with respect to a change in hot and cold, from which results a change into larger or smaller quantity which is in a sense the same as being increased and decreased. Thus it is plain that if a body which is moved circularly is not subject to increase or decrease it is not subject to alteration.
Ultimo autem epilogando concludit manifestum esse ex dictis, si quis velit assentire prioribus demonstrationibus, non proterve contradicendo, quod corpus primum, quod scilicet movetur motu primo et perfecto, idest circulari, est sempiternum, quasi non subiacens generationi et corruptioni; neque etiam habet augmentum neque decrementum; et non subiacet senectuti, neque alterationi, neque passioni. Finally, in summary he concludes that it is plain from the foregoing — if anyone wants to assent to the previous demonstrations without wantonly contradicting — that the first body, which, namely, is moved with the first and perfect motion, i.e., circular motion, is sempiternal (as not being subject to generation and corruption), that it undergoes neither increase nor decrease, and that it is not subject to aging or alteration or passion.
Potest autem obiici contra hanc Aristotelis rationem dupliciter. Primo quidem contra conclusionem. Videtur enim esse falsum quod corpus caeleste non alteretur: manifeste enim apparet lunam a sole illuminari, et per umbram terrae obscurari. 73. Nevertheless objections can be leveled against this argument of Aristotle on two counts. First of all against the conclusion. For it seems to be false that a heavenly body is not altered, for it is plainly evident that the moon is illumined by the sun and obscured by the shadow of the earth.
Dicendum est autem quod duplex est alteratio. Una quidem passiva, secundum quam ita aliquid adiicitur, quod etiam aliquid aliud abiicitur; sicut cum aliquid alteratur de calido in frigidum, amittit calorem et recipit frigiditatem: et talem alterationem, quae fit secundum passiones, intendit hic philosophus excludere a corpore caelesti. Est autem alia alteratio perfectiva, quae fit secundum quod aliquid ab alio perficitur absque alterius abiectione, qualem alterationem ponit philosophus in II de anima etiam in potentia sensitiva: et talem alterationem nihil prohibet esse in corporibus caelestibus, quorum quaedam recipiunt virtutes ab aliis secundum coniunctiones et varios aspectus, absque hoc quod aliquod eorum propriam virtutem amittat. But it must be said that alteration is of two kinds. One is passive and according to it things are so added that something else is cast off, as, when something is altered from hot to cold, it loses heat and receives coldness. It is that kind of alteration, which takes place according to passions, that the Philosopher is here excluding from heavenly body. But there is another kind of alteration which is perfecting, which occurs insofar as something is perfected by something else without loss to the former — this is the kind of alteration that the Philosopher in On the Soul II posits even in a sense power. Such an alteration nothing prevents from being in heavenly bodies, some of which receive virtues from others according to conjunctions and various aspects, but without any of them losing their own virtue.
Secundo obiicitur contra processum rationis hic inductae: non enim videtur esse verum quod quaecumque alterantur, augmentum et decrementum suscipiant. Augmentum enim et decrementum fit per additionem alicuius quod est conversum in substantiam eius quod augetur, ut dicitur in libro de Generat. et in II de anima; et etiam hoc supra dictum est. Hic autem motus augmenti non est nisi in animalibus et plantis: nam ea quae rarefiunt et condensantur, non augentur ex aliquo addito, ut probatur in IV Physic. Inconvenienter igitur videtur hic Aristoteles attribuere motum augmenti non solum animalibus et plantis et partibus eorum, sed etiam elementis. 74. The second objection is directed against the procedure of his argument: for it does not seem to be true that whatever is altered receives increase and decrease. For these result from the addition of something that is converted into the substance of what is increased, as is said in the book On Generation and in On the Soul II, and as was said above. Now the motion of increase does not exist except in animals and plants, for things that rarefy and condense are not increased by the addition of anything, as was proved in Physics V. Consequently, it seems unsuitable for Aristotle here to attribute the motion of increase not only to animals and plants and their parts, but to the elements as well.
Dicendum est autem quod Aristoteles hic loquitur de augmento pro quolibet motu quo aliquid proficit in maiorem quantitatem. Nondum enim perfecte explicaverat naturam motus augmenti: est autem suae consuetudinis ut ante manifestationem veritatis, utatur opinionibus communibus. Nec impedit virtutem probationis eius, quod supra exclusit augmentum a corpore caelesti per exclusionem additionis corporis in ipsum quod augetur transmutati: quia sicut quod augetur per additionem, non est omnino liberum a generatione et corruptione, ita etiam quod augetur per rarefactionem. But it should be said that Aristotle is here speaking of increase in the sense of any motion by which something proceeds to greater quantity. For he has not yet perfectly explained the nature of the motion of increase and it is his custom, before he has shown the true view, to use common opinions. But the force of his proof is not impeded by his having excluded increase from a heavenly body by excluding addition of a body changed into what is increased: for just as anything increased by addition is not utterly free of generation and corruption, so, too, what is increased by rarefaction.
Est autem considerandum quod signanter in hac ratione mentionem facit de corporibus physicis: quia in corporibus mathematicis potest esse augmentum sine alteratione, puta cum quadratum crevit apposito gnomone, sed non est alteratum, ut dicitur in praedicamentis; et e converso potest aliquid alterari sine hoc quod augeatur, sicut cum fit triangulus aequalis quadrato. However it is to be noted that in this proof he makes mention of Physical bodies advisedly, because in mathematical bodies increase can occur without alteration — for example, a square grows by adding to it a gnomon, but it is not altered, as is said in the Predicaments; conversely, a thing can be altered without being increased, as when a triangle is made equal to a square.
Deinde cum dicit: videtur autem etc., manifestat propositum per signa. Et dicit quod ratio et ea quae apparent probabiliter videntur in materia ista sibi invicem testificari. Et ponit tria signa. Quorum primum est ex communi hominum opinione, qui ponunt multos deos, vel unum Deum, cui alias substantias separatas deservire dicunt; et omnes sic opinantes attribuunt supremum locum, scilicet caelestem, Deo, sive sint barbari sive Graeci, quicumque scilicet putant esse res divinas. Sic autem attribuunt caelum divinis substantiis, quasi adaptantes immortalem locum immortalibus et divinis rebus; ut sic habitatio Dei in caelo intelligatur esse secundum similitudinis adaptationem, quia scilicet hoc corpus inter cetera corpora magis accedit ad similitudinem spiritualium substantiarum et divinarum. Est enim impossibile quod aliter Deo habitatio caeli attribuatur, quasi indigeat loco corporali a quo comprehendatur. Si igitur ponendae sint res divinae, immo quia pro certo ponendae sunt, consequens est quod bene sint dicta ea quae dicta sunt de prima substantia corporali, scilicet de corpore caelesti, quod scilicet est ingenitum et impassibile. 75. Then at [37] he manifests the proposition through signs. And he says that both reason and things that appear to be probable seem to support one another on this point. And he gives three signs. The first of which is taken from the general opinion of men, who posit many gods or one God, whom the other separated substances serve. All who believe thus, whether Greeks or barbarians, assign the highest place, namely, the heavenly, to God, namely, all those who believe there are divine beings. But they assign the heavens to the divine substances as though adapting an immortal place to immortal and divine beings. In this way God's habitation in the heavens is understood as appropriate according to likeness, that is, that among all other bodies this body more closely approaches to a likeness to spiritual and divine substances. For it is impossible for the habitation of the heavens to be assigned to God for any other reason, as though He should need a bodily place by which He is comprehended. If therefore divine beings are to be posited, and since, indeed, they certainly must, the consequence is that the statements made about the first bodily substance, namely, the heavenly body, were well made, namely, that the heavenly body is ungenerated and unalterable.
Quamvis autem existimant homines templa esse locum Dei, hoc tamen non existimant ex parte ipsius Dei, sed ex parte colentium Deum, quos oportet in aliquo loco Deum colere. Unde templa corruptibilia sunt proportionalia hominibus corruptibilibus, caelum autem incorruptioni divinae. Although men suppose that temples are the place of God, they do not suppose this from God's viewpoint but from that of the worshippers, who must worship Him in some place. That is why perishable temples are proportioned to perishable men, but the heavens to the divine imperishability.
Secundum signum ponit ibi: accidit autem hoc et per sensum etc.: quod quidem accipitur ab experientia longi temporis. Et dicit quod id quod probatum est per rationem et per communem opinionem, accidit, idest consequitur, sufficienter; non quidem simpliciter, sed sicut potest dici per comparationem ad humanam fidem, idest quantum homines possunt testificari de his quae parvo tempore et a remotis viderunt. Secundum enim memoriam quam sibi invicem tradiderunt astrologi, dispositiones et motus caelestium corporum observantes, in toto praeterito tempore non videtur aliquid transmutatum esse neque secundum totum caelum, neque secundum aliquam propriam partem eius. Quod quidem non esset si caelum generabile et corruptibile esset: quaecumque enim generantur et corrumpuntur, paulatim et successive ad perfectum statum perveniunt, et ex eo paulatim recedunt: quod quidem non posset tanto tempore latere in caelo, si naturaliter generationi et corruptioni subiaceret. 76. The second sign he gives at [38] and it is taken from long experience. And he says that what has been proved by reason and common opinion occurs, i.e., follows, sufficiently — i.e., not absolutely but to the extent of human faith, i.e., so far as men can testify to what they have seen for a short time and from afar. For according to the tradition which astronomers have passed on concerning their observations of the dispositions and motions of heavenly bodies, in the whole time past there does not seem to have been any change affecting either the entire heavens or any of its own parts. Now this would not be, if the heaven were generable or perishable — for things subject to generation and corruption arrive at their perfect state little by little and step by step, and then gradually depart from that state, and this could not have been concealed in the heavens for such a long time, if they were naturally subject to generation and corruption.
Nec tamen hoc est necessarium, sed probabile. Quanto enim aliquid est diuturnius, tanto maius tempus requiritur ad hoc quod eius mutatio deprehendatur; sicut transmutatio hominis non deprehenditur in duobus vel tribus annis, in quibus deprehenditur transmutatio canis, vel alicuius alterius animalis breviorem vitam habentis. Posset igitur aliquis dicere quod, etsi caelum sit naturaliter corruptibile, est tamen tam diuturnum, quod totum tempus cuius memoria potest haberi, non sufficit ad deprehendendam eius transmutationem. However, this is not necessary but probable. For the more lasting something. is, the greater the time required for its change to be noted, just as change in a man is not noticed in two or three years, as it is in a dog or other animals having a shorter life-span. Consequently someone could say that, even though the heavens are naturally corruptible, nevertheless they are so lasting that the whole extent of human memory is not sufficient to observe their change.
Tertium signum ponit ibi: videtur autem et cetera. Quod quidem sumitur a nomine imposito ab antiquis, quod durat usque ad praesens tempus; per quod datur intelligi quod ipsi etiam hoc modo opinabantur caelum esse incorruptibile, sicut nos opinamur. Et ne aliquis contra hoc obiiceret quod aliqui ante suum tempus, caelum generabile et corruptibile posuerunt, subiungit quod opiniones verae renovatae sunt secundum diversa tempora non semel aut bis, sed infinities, supposita infinitate temporis. Destruuntur enim studia veritatis per diversas mutationes in his inferioribus accidentes: sed quia mentes hominum naturaliter inclinantur ad veritatem, cessantibus impedimentis, renovantur studia, et homines tandem perveniunt ad opiniones veras quae prius fuerant: opiniones autem falsas non necesse est renovari. 77. The third sign is given at [39] and is based on a name given by the ancients, which endures to the present, and which gives us to understand that they thought the heaven to be imperishable just as we do. And lest anyone object that some before their time thought the heavens were subject to generation and corruption, he adds that true opinions are revived according to diverse times not once or twice but infinitely, supposing that time is infinite. For the studies of truth are destroyed by various changes occurring in these lower things, but because the minds of men are naturally inclined to truth, then when obstacles are removed, studies are renewed and men at last arrive at the true opinions which previously flourished, but false opinions need not be revived.
Et ideo antiqui, opinantes quod primum corpus, scilicet caeli, esset alterius naturae praeter quatuor elementa, nominaverunt supremum locum mundi aethera, ponentes scilicet ei nomen ab eo quod semper currit sempiterno tempore: thein enim in Graeco idem est quod currere. Sed Anaxagoras male interpretatus est hoc nomen, attribuens ipsum igni, quasi caeleste corpus sit igneum: aethein enim in Graeco idem est quod ardere, quod est proprium ignis. Sed quod caeleste corpus non sit igneum, patet ex supra dictis. Consequently the ancients, supposing that the first body, namely, the heaven, to be of a nature different from the four elements, named the highest place of the world the "aether," thus applying to it a name based on the fact that it always runs for an eternity of time — for thein in Greek is the same as "to run." But Anaxagoras misinterpreted this name, attributing it to fire, as though the heavenly body were fiery — for aether in in Greek is the same as "to burn," which is proper to fire. But that a heavenly body is not of fire is plain from what has been said above [in L. 4].

Lecture 8:

Only five simple bodies required. No motion contrary to circular.

Chapter 3 cont.
Φανερὸν δ' ἐκ τῶν εἰρημένων καὶ διότι τὸν ἀριθμὸν ἀδύνατον εἶναι πλείω τὸν τῶν λεγομένων σωμάτων ἁπλῶν τοῦ μὲν γὰρ ἁπλοῦ σώματος ἀνάγκη τὴν κίνησιν ἁπλῆν εἶναι, μόνας δὲ ταύτας εἶναί φαμεν ἁπλᾶς, τήν τε κύκλῳ καὶ τὴν ἐπ' εὐθείας, καὶ ταύτης τὰ δύο μόρια, τὴν μὲν ἀπὸ τοῦ μέσου, τὴν δ' ἐπὶ τὸ μέσον. 41 It is also clear from what has been said why the number of what we call simple bodies cannot be greater than it is. The motion of a simple body must itself be simple, and we assert that there are only these two simple motions, the circular and the straight, the latter being subdivided into motion away from and motion towards the centre.
Chapter 4
Ὅτι δ' οὐκ ἔστι τῇ κύκλῳ φορᾷ ἐναντία ἄλλη φορά, πλεοναχόθεν ἄν τις λάβοι τὴν πίστιν πρῶτον μὲν ὅτι τῇ περιφερεῖ τὴν εὐθεῖαν ἀντικεῖσθαι μάλιστα τίθεμεν τὸ γὰρ κοῖλον καὶ τὸ κυρτὸν οὐ μόνον ἀλλήλοις ἀντικεῖσθαι δοκεῖ (271a.) ἀλλὰ καὶ τῷ εὐθεῖ, συνδυαζόμενα καὶ λαβόντα σύνθεσιν ὥστ' εἴπερ ἐναντία τίς ἐστι, τὴν ἐπὶ τῆς εὐθείας μάλιστα ἀναγκαῖον ἐναντίαν εἶναι πρὸς τὴν κύκλῳ κίνησιν. Αἱ δ' ἐπὶ τῆς εὐθείας ἀλλήλαις ἀντίκεινται διὰ τοὺς τόπους τὸ γὰρ ἄνω κάτω τόπου τέ ἐστι διαφορὰ καὶ ἐναντίωσις. 42 That there is no other form of motion opposed as contrary to the circular may be proved in various ways. In the first place, there is an obvious tendency to oppose the straight line to the circular. For concave and convex are a not only regarded as opposed to one another, but they are also coupled together and treated as a unity in opposition to the straight. And so, if there is a contrary to circular motion, motion in a straight line must be recognized as having the best claim to that name. But the two forms of rectilinear motion are opposed to one another by reason of their places; for up and down is a difference and a contrary opposition in place.
Ἔπειτ' εἴ τις ὑπολαμβάνει τὸν αὐτὸν εἶναι λόγον ὅνπερ ἐπὶ τῆς εὐθείας καὶ ἐπὶ τῆς περιφεροῦς (τὴν γὰρ ἀπὸ τοῦ Α πρὸς τὸ Β φορὰν ἐναντίαν εἶναι τῇ ἀπὸ τοῦ Β πρὸς τὸ Α), τὴν ἐπὶ τῆς εὐθείας λέγει αὕτη γὰρ πεπέρανται, περιφερεῖς δ' ἄπειροι ἂν εἶεν περὶ τὰ αὐτὰ σημεῖα. 43 Secondly, it may be thought that the same reasoning which holds good of the rectilinear path applies also the circular, movement from A to B being opposed as contrary to movement from B to A. But what is meant is still rectilinear motion. For that is limited to a single path, while the circular paths which pass through the same two points are infinite in number.
Ὁμοίως δὲ καὶ ἐπὶ τοῦ ἡμικυκλίου τοῦ ἑνός, οἷον ἀπὸ τοῦ Γ ἐπὶ τὸ Δ καὶ ἀπὸ τοῦ Δ ἐπὶ τὸ Γ ἡ γὰρ αὐτὴ τῇ ἐπὶ τῆς διαμέτρου ἐστίν ἀεὶ γὰρ ἕκαστον ἀπέχειν τὴν εὐθεῖαν τίθεμεν. 44 Even if we are confined to the single semicircle and the opposition is between movement from C to D and from D to C along that semicircle, the case is no better. For the motion is the same as that along the diameter, since we invariably regard the distance between two points as the length of the straight line which joins them.
Ὁμοίως δὲ κἂν εἴ τις κύκλον ποιήσας τὴν ἐπὶ θατέρου ἡμικυκλίου φορὰν ἐναντίαν θείη τῇ ἐπὶ θατέρου, οἷον ἐν τῷ ὅλῳ κύκλῳ τὴν ἀπὸ τοῦ Ε πρὸς τὸ Ζ τοῦ Η ἡμικυκλίου τῇ ἀπὸ τοῦ Ζ πρὸς τὸ Ε ἐν τῷ Θ ἡμικυκλίῳ. 45 It is no more satisfactory to construct a circle and treat motion 'along one semicircle as contrary to motion along the other. For example, taking a complete circle, motion from E to F on the semicircle G may be opposed to motion from F to E on the semicircle H.
Εἰ δὲ καὶ αὗται ἐναντίαι, ἀλλ' οὔτι γε αἱ ἐπὶ τοῦ ὅλου κύκλου φοραὶ ἀλλήλαις διὰ τοῦτο ἐναντίαι. *—* 46 But even supposing these are contraries, it in no way follows that the reverse motions on the complete circumference contraries.
Ἀλλὰ μὴν οὐδ' ἡ ἀπὸ τοῦ Α ἐπὶ τὸ Β κύκλῳ φορὰ ἐναντία τῇ ἀπὸ τοῦ Α ἐπὶ τὸ Γ ἐκ ταὐτοῦ γὰρ εἰς ταὐτὸ ἡ κίνησις, ἡ δ' ἐναντία διωρίσθη φορὰ ἐκ τοῦ ἐναντίου εἰς τὸ ἐναντίον. 46bis Nor again can motion along the circle from A to B be regarded as the contrary of motion from A to C: for the motion goes from the same point towards the same point, and contrary motion was distinguished as motion from a contrary to its contrary.
Εἰ δὲ καὶ ἦν ἡ κύκλῳ τῇ κύκλῳ ἐναντία, μάτην ἂν ἦν ἡ ἑτέρα *ἐπὶ τὸ αὐτὸ γάρ, ὅτι ἀνάγκη τὸ κύκλῳ φερόμενον ὁποθενοῦν ἀρξάμενον εἰς πάντας ὁμοίως ἀφικνεῖσθαι τοὺς ἐναντίους τόπους (εἰσὶ δὲ τόπου ἐναντιότητες τὸ ἄνω καὶ κάτω καὶ τὸ πρόσθιον καὶ ὀπίσθιον καὶ τὸ δεξιὸν καὶ ἀριστερόν), αἱ δὲ τῆς φορᾶς ἐναντιώσεις κατὰ τὰς τῶν τόπων εἰσὶν ἐναντιώσεις* εἰ μὲν γὰρ ἴσαι ἦσαν, οὐκ ἂν ἦν κίνησις αὐτῶν, εἰ δ' ἡ ἑτέρα κίνησις ἐκράτει, ἡ ἑτέρα οὐκ ἂν ἦν. Ὥστ' εἰ ἀμφότερα ἦν, μάτην ἂν θάτερον ἦν σῶμα μὴ κινούμενον τὴν αὑτοῦ κίνησιν μάτην γὰρ ὑπόδημα τοῦτο λέγομεν, οὗ μή ἐστιν ὑπόδεσις. Ὁ δὲ θεὸς καὶ ἡ φύσις οὐδὲν μάτην ποιοῦσιν. 47 And even if the motion round a circle is the contrary of the reverse motion, one of the two would be ineffective: for both move to the same point, because that which moves in a circle, at whatever point it begins, must necessarily pass through all the contrary places alike. (By contrarieties of place I mean up and down, back and front, and right and left; and the contrary oppositions of movements are determined by those of places.) One of the motions, then, would be ineffective, for if the two motions were of equal strength, there would be no movement either way, and if one of the two were preponderant, the other would be inoperative. So that if both bodies were there, one of them, inasmuch as it would not be moving with its own movement, would be useless, in the sense in which a shoe is useless when it is not worn. But God and nature create nothing that has not its use.
Postquam philosophus ostendit necesse esse aliquod corpus praeter quatuor elementa, hic ostendit quod praeter ista corpora non requirit integritas universi aliquod aliud corpus. 78. After showing the necessity of some body besides the four elements, the Philosopher here shows that the integrity of the universe requires no other body besides these five.

Et primo ostendit propositum;

secundo probat quoddam quod supposuerat, ibi: quod autem non est circulationi et cetera.

First he shows his proposition;

Secondly, he proves something he had assumed, at 79.

Dicit ergo primo quod ex dictis, quibus probatum est esse quintum corpus praeter corpora gravia et levia, potest etiam manifestari quod impossibile est esse maiorem numerum simplicium corporum. Quia, sicut supra dictum est, necesse est quod cuiuslibet simplicis corporis sit aliquis motus simplex. He says therefore first [40] that from what was said in proving that there exists a fifth body in addition to heavy and light bodies, it can be shown that it is impossible for a greater number of simple bodies to exist. For as was said above, for each simple body there must be some simple motion.
Sed non est alius motus simplex praeter praedictos, quorum unus est circularis et alius est rectus, qui in duas partes dividitur: nam motuum rectorum unus quidem est a medio, qui dicitur motus sursum; alius autem est ad medium, qui dicitur motus deorsum. Horum autem motuum ille qui est ad medium, est corporis gravis, scilicet terrae et aquae; ille autem qui est a medio, est corporis levis, scilicet ignis et aeris; ille autem qui est circularis, est primi et supremi corporis. Unde relinquitur quod praeter praedicta corpora simplicia non sit aliquod aliud corpus simplex: et ita integritas universi ex istis quinque corporibus consistit. But there is no simple motion other than the ones previously mentioned: one of which is circular and the other straight, the latter being divided into two kinds, one of which is from the middle and is called "upward motion" and the other toward the middle and is called "downward motion." Of the latter two, the one which is toward the middle belongs to a heavy body, namely, to earth and water, while the one from the middle belongs to a light body, namely, to fire and air. Finally, the circular motion is assigned to the first and supreme body. Hence what remains is that there is no other simple body besides the ones mentioned. Consequently, the wholeness of the universe consists of these five bodies.
Deinde cum dicit: quod autem non est circulationi etc., probat quoddam quod supposuerat, scilicet quod motui circulari non sit aliquis motus contrarius. Et hoc quidem supposuerat in demonstratione qua probavit corpus caeli non esse subiectum generationi et corruptioni: sed ideo non statim ibi probavit, sed distulit probationem usque huc, quia hoc etiam valet ad ostendendum quod non sit maior numerus simplicium corporum. Si enim motui circulari esset aliquis motus contrarius, posset dici quod sicut est duplex corpus quod movetur motu recto, propter contrarietatem huius motus, ita etiam est duplex corpus quod movetur motu circulari. Hoc autem non continget, si constet quod corpori circulari non sit aliquis motus contrarius. 79. Then at [41] he proves something he had assumed, namely, that there is not a motion contrary to circular motion. This he had assumed in the discussion in which he proved that the body of the heavens is not subject to generation and corruption. But the reason why he did not prove it right away, but waited until now, is that it is also useful in proving that there is not a greater number of simple bodies. For if there were a motion contrary to circular motion, it could be held that just as there are two bodies moved with straight motion on account of the contrariety of this motion, so there are also two bodies moved with circular motion. But this will not occur if it is plain that there is no motion contrary to circular motion. Therefore, on this point,
Circa hoc ergo primo proponit quod intendit. Et dicit quod per multas rationes potest aliquis accipere fidem quod motui circulari non sit aliquis motus localis contrarius. First he proposes what he intends, and says that there are many reasons to induce one to believe that there is not a circular motion contrary to circular motion.
Secundo ibi: primum quidem etc., ostendit propositum. Circa quod considerandum est quod, si in motu circulari sit contrarietas, oportet hoc esse altero trium modorum: quorum unus est ut motui circulari rectus sit contrarius, alius modus est ut sit aliqua contrarietas in ipsis partibus motus circularis, tertius est ut uni motui circulari alius motus circularis contrarietur. 80. Secondly, he establishes the proposition. In regard to this it must be noted that if there exists contrariety in circular motion, it must be in one of three ways: one is that a straight motion be contrary to circular motion; the second is that there be some sort of contrariety in the parts themselves of circular motion; the third is that one circular motion have some other circular motion contrary to it.

Primo ergo ostendit quod motui circulari non contrariatur motus rectus;

secundo ostendit quod non sit contrarietas in partibus motus circularis, ibi: deinde si quis existimat etc.;

tertio quod non sit contrarietas in toto motu circulari, unius scilicet motus circularis ad alium, ibi: at vero neque quae ab a et cetera.

First therefore he shows that a straight motion is not contrary to circular motion;

Secondly, he shows that there is no contrariety in the parts of circular motion, at 10.83;

Thirdly, that there is no contrariety between complete circular motions, i.e., of one to another, at 89.

Dicit ergo primo quod maxime circulari videtur opponi rectum. Linea enim recta nullam fractionem habet; figura autem angularis habet quandam fractionem, non per totum, sed in angulis; sed figura circularis videtur per totum habere fractionem, ac si totum esset angulus. Et secundum hoc rectum et circulare videntur esse contraria quasi maxime distantia. 81. He says therefore first [42] that what seems most opposite to something circular is something straight. For a straight line has no break, while an angular line does have a break, not through the whole, but in the angles; meanwhile a circular figure seems to have breaks throughout, as if the whole were an angle. According to this the straight and the circular seem to be contraries, as though at the farthest extremes.
Et quia posset aliquis dicere quod circulari non opponitur rectum, sed concavo opponitur convexum sive gibbosum, ad hanc obviationem excludendam, subiungit quod concavum et gibbosum, idest convexum, non solum videntur habere oppositionem ad invicem, sed etiam ad rectum. Ad se invicem autem videntur habere oppositionem sicut combinata et iuxta se posita, idest secundum relationem: nam concavum dicitur respectu eorum quae intra sunt, gibbosum autem respectu eorum quae sunt extra. Et sic omni modo rectum contrariatur circulari, sive accipiatur sub ratione concavi, sive sub ratione convexi. And because someone could say that it is not the straight that is opposed to the circular, but rather the convex or "gibbous" which is opposed to the concave, to reject this objection, he adds that concave and "gibbous," i.e., convex, are seen to be opposed not only to one another, but to the straight as well. They seem to be mutually opposed after the manner of the combined and the juxtaposed, i.e., in terms of relation: for "concave" is said in relation to things that are inside [a circle or sphere], but "gibbous" with respect to things outside. Consequently, from every aspect, the straight is contrary to the circular, whether taken as concave or as convex.
Et quia contrarietas motuum videtur esse secundum contrarietatem eorum in quibus est motus, videtur esse consequens quod si aliquis motus sit contrarius motui circulari, maxime sit ei contrarius motus rectus, qui scilicet est super lineam rectam. Sed motus recti contrariantur ad invicem, propter loca contraria (motus enim qui est sursum, contrariatur ei qui deorsum est, quia sursum et deorsum important differentiam et contrarietatem loci): et sic uni motui recto contrariabitur alius motus rectus, et circularis. Hoc autem est impossibile: quia uni unum est contrarium. Ergo impossibile est quod motui circulari sit aliquis motus contrarius. And because the contrariety of motions is seen to follow the contrariety of the things in which the motion is, the consequence seems to be that if there is a motion contrary to circular motion, it should be most of all straight motion which, namely, is over a straight line. But straight motions are contrary to one another because of contrary places — for upward motion is contrary to downward because "up" and "down" imply a difference and contrariety of place. Consequently, one straight motion will have as its contrary some other straight motion, and a circular one. This, however, is impossible, for to one thing there is one contrary. Therefore, it is impossible for any motion to be contrary to circular.
Potest autem aliquis obiicere contra hoc quod dicitur, quod circulari maxime contrariatur rectum. Dictum est enim in praedicamentis quod figurae nihil est contrarium: rectum autem et circulare sunt differentiae figurarum. 82. But someone could object to the statement that the straight is most contrary to the circular. For it is stated in the Predicaments that nothing is contrary to figure, whereas "straight" and "circular" are differences in figure.
Potest autem dici quod philosophus hic ex hypothesi loquitur, et non simpliciter. Si enim aliquid esset contrarium circulari, maxime contrariaretur sibi rectum, ratione supra dicta. But it can be said that the Philosopher is here speaking hypothetically and not categorically. For if anything were contrary to the circular, it would be the straight most of all, for the reason given above.
Potest etiam dici quod in quolibet genere invenitur contrarietas differentiarum, ut patet X Metaphys., licet non sit in omni genere contrarietas specierum: etsi enim rationale et irrationale sint contrariae differentiae, non tamen homo et asinus sunt contrariae species. Sic igitur ponitur contrarietas inter rectum et circulare, non sicut inter species, sed sicut inter differentias eiusdem generis. Huiusmodi autem contrarietas, quae posset attendi in motibus secundum differentiam recti et circularis, non est contrarietas corruptiva, qualem intendit hic philosophus excludere a corpore caelesti, sicut est contrarietas calidi et frigidi: contrarietatem autem secundum differentias aliquorum generum nihil prohibet in corpore caelesti esse, puta sicut par vel impar, vel secundum aliquid huiusmodi. It can also be said that in every genus there is found a contrariety of differences, as is plain from Metaphysics X, although there is not a contrariety of species in every genus: for although "rational" and "irrational" are contrary differences, "man" and "ass" are not contrary species. Consequently, there is a contrariety between straight and circular not as between species, but as between differences of the same genus. Such contrariety, which can be discerned in motions on the basis of the difference between straight and circular, is not a corruptive contrariety, of the sort, namely, which the Philosopher here intends to exclude from the heavenly body, such as is the contrariety of hot to cold. But nothing forbids contrariety according to the differences of certain genera from being in a heavenly body, for example, that of equal and unequal, or something of that kind.
Obiicit autem Ioannes grammaticus contra id quod philosophus videtur ponere concavum et gibbosum opponi secundum relationem: quia relativa videntur simul esse, concavum autem et gibbosum non sunt simul ex necessitate: potest enim esse aliquod corpus sphaericum exterius convexum absque hoc quod sit interius concavum. Sed in hoc deceptus fuit: quia philosophus hic loquitur de concavo et convexo secundum quod inveniuntur in linea circulari, non autem secundum quod inveniuntur in corpore sphaerico, in quo unum potest esse sine altero, non autem in linea. John the Grammarian, however, objects against the Philosopher's seeming to state that concave and gibbous are opposed according to a relation: because relative things seem to be co-existent, but concave and gibbous are not necessarily together, for a spherical body can be exteriorly convex without being interiorly concave. But in this he has been deceived, for the Philosopher is here speaking of concave and convex as found in a circular line, and not as found in a spherical body, in which latter one can indeed exist without the other, but not in a line.
Deinde cum dicit: deinde si quis existimat etc., ostendit non esse contrarietatem in partibus motus circularis. 83. Then at [43] he shows that there is no contrariety in the parts of circular motion.

Et primo excludit contrarietatem a partibus huius motus;

secundo ostendit quod contrarietas partium non sufficeret ad contrarietatem totius, ibi: si autem et istae contrariae et cetera.

First he excludes contrariety from the parts of this motion;

Secondly, he shows that contrariety of parts would not be enough for contrariety of the whole, at 88.

Circa primum tria facit: Regarding the first he does three things:

primo ostendit quod non est contrarietas in partibus motus circularis quae accipiuntur secundum diversas portiones circuli, quae designantur inter duo puncta;

secundo ostendit quod non est contrarietas in partibus motus circularis quae accipiuntur secundum eundem semicirculum, ibi: similiter autem et quae in semicirculo etc.;

tertio ostendit quod non est contrarietas in partibus motus circularis quae accipiuntur secundum duos semicirculos, ibi: similiter autem et utique et cetera.

First he shows that there is not contrariety in the parts of circular motion if the parts are taken according to diverse portions of the circle which are designated between two points;

Secondly, he shows that there is not contrariety in the parts of circular motion, if the parts are taken according to the same semicircle, at 85;

Thirdly, if the parts are taken according to two semicircles, at 87.

Dicit ergo primo quod posset aliquis existimare quod eadem sit ratio contrarietatis in motu qui est per lineam circularem, et in motu qui est per lineam rectam. Si enim designetur una linea recta inter duo puncta quae sunt a et b, manifestum est quod motus localis qui fiet super lineam rectam ab a in b, contrarius erit motui locali qui fiet e converso a b in a. Sed non est similis ratio si describatur una linea circularis super duo puncta quae sunt a et b: quia inter duo puncta non potest esse nisi una linea recta, sed inter duo puncta possunt describi infinitae lineae curvae, quae sunt diversae portiones circulorum. Sequeretur igitur, si motui qui est ab a in b per lineam circularem, esset contrarius motus qui est a b in a secundum lineam circularem, quod infiniti motus essent contrarii uni. He says therefore first [43] that someone could think that the aspect of contrariety in motion upon a circular line, and that in motion upon a straight line, are the same. For if one straight line between two points, A and B, be designated, it is evident that the local motion occurring on the straight line from A to B will be contrary to the local motion from B to A. But the notion is not the same if a circular line be described through the two points, A and B, because between two points there can be but one straight line, but an infinity of curved lines, which are diverse portions of circles. Therefore it would follow that, if the motion from A to B over a circular line were contrary to the one which is from B to A over a circular line, an infinitude of motions would be contrary to one.
Est autem attendendum quod, loco huius quod debuit dicere, quod linea recta est una inter duo puncta, dixit quod lineae rectae sunt finitae: quia si accipiamus in diversis locis duo puncta, erunt inter ea lineae rectae finitae; sed inter quaelibet duo puncta poterunt describi lineae curvae infinitae. But it should be observed that, in place of what he ought to have said, namely, that the straight line between two points is one, he said that straight lines are "finite" — because if we take two points in diverse places, there will be between them finite straight lines [i.e., in finite number], but between any two points there could be described an infinitude of curved lines.
Obiicit autem contra hanc rationem Ioannes grammaticus, quia non videtur sequi quod uni motui sint infiniti motus contrarii, sed infiniti infinitis: quia secundum unamquamque portionem circuli qui describitur super duo puncta, erunt duo motus sibi invicem contrarii. Item videtur quod sit idem inconveniens quod sequitur ex contrarietate motuum rectorum. Manifestum est enim quod sicut inter duo puncta possunt describi infinitae lineae curvae, ita a centro mundi ad circumferentiam possunt describi infinitae lineae rectae. 84. Against this argument John the Grammarian objects, since it does not seem to follow that to one motion there is an infinitude of contrary motions, but that to an infinite number there is. For with respect to each portion of the circle described between two points there will be two motions contrary one to the other. Likewise, the same difficulty seems to follow from the contrariety of straight motions. For it is manifest that just as an infinitude of curved lines can be described between two points, so from the center of the world to the circumference there can be described an infinitude of straight lines.
Sed dicendum est ad primum quod, si contrarietas sit motuum qui fiunt per lineas curvas secundum contrarietatem terminorum, sicut accidit in motibus rectis, sequitur ex hac suppositione quod quilibet motus qui fit a b in a per quamcumque linearum curvarum, sit contrarius motui qui est ab a in b: et sic sequetur quod non solum uni motui sint infiniti motus contrarii, sed quod cuilibet infinitorum motuum ex una parte incipientium, contrarientur infiniti motus qui incipiunt ex parte contraria. But in regard to the first it must be said that if the contrariety of motions that occur through curved lines is to be according to the contrariety of the termini as happens in straight motions, then, from this supposition it follows that every motion from B to A through any of the curved lines is contrary to a motion from A to B. Thus it will follow not only that there is an infinitude of motions contrary to one motion, but also that to each of the infinite motions starting from one end there will be contrary the infinitude of motions beginning from the contrary end.
Ad secundum dicendum quod omnes infinitae lineae rectae quae sunt a centro ad circumferentiam, sunt aequales, et ideo designant eandem distantiam inter contrarios terminos; et ideo in omnibus est eadem ratio contrarietatis, quae importat maximam distantiam. Sed omnes lineae curvae infinitae quae describuntur super eadem puncta, sunt inaequales: unde non est in eis eadem ratio contrarietatis, quia non est una et eadem distantia accepta secundum quantitatem lineae curvae. In regard to the second it must be said that all the infinitude of straight lines from the center to the circumference are equal, and therefore designate the same distance between contrary termini — therefore in all of them is present the same aspect of contrariety, which implies maximum distance. But all the infinitude of curved lines described between the same points are unequal; hence the same aspect of contrariety is not present in them, for the distance taken with regard to the quantity of the curved line is not the same in every case.
Deinde cum dicit: similiter autem et quae in semicirculo etc., ostendit quod non sit contrarietas in motu circulari secundum unum et eundem semicirculum. Posset enim aliquis dicere quod motui qui est super unam lineam curvam ab a in b, non contrariatur quilibet motus qui est a b in a per quamcumque lineam curvam, sed per unam et eandem, puta per unum semicirculum. Sit autem semicirculus gd, et sit ita quod motus qui est per semicirculum a g ad d, contrarietur motui qui est super eundem semicirculum a d ad g. 85. Then at [44] he shows that there is not contrariety in circular motion according to one and the same semicircle. For someone could say that the motion upon one curved line from A to B has as its contrary not a motion from R to A through just any curved line but through one and the same — for example, through one semicircle. Let GD be that semicircle, such that the motion through it from G to D is contrary to the one through it from D to G.
Sed contra hoc procedit Aristoteles ex hoc quod eadem distantia reputatur quae est inter g et d per semicirculum, illi distantiae quae accipitur per diametrum: non quod semicirculus sit aequalis diametro, sed quia omnem distantiam mensuramus per lineam rectam. Cuius ratio est, quia omnis mensura debet esse certa et determinata et minima: inter duo autem puncta mensura lineae rectae est certa et determinata, quia non potest esse nisi una; et est minima omnium linearum quae sunt inter duo puncta. Lineae vero curvae inter duo puncta describi possunt infinitae, quae omnes sunt maiores linea recta inter eadem puncta descripta. Unde distantia quae est inter duo puncta, mensuratur per lineam rectam, et non per lineam curvam semicirculi, seu cuiuslibet alterius portionis circuli, aut maioris aut minoris circuli. Cum igitur de ratione contrarietatis sit quod habeat maximam distantiam, ut dicitur in X Metaphys., cum distantia quae est inter duo puncta non mensuretur secundum lineam curvam sed secundum rectam, consequens est quod contrarietas terminorum non faciat contrarietatem in motibus qui sunt super semicirculum, sed solum in motibus qui sunt super diametrum. But Aristotle proceeds against this on the ground that the semicircular distance from G to D is computed in terms of the diametric distance, not in the sense that the semicircle is equal to the diameter, but because we measure every distance by a straight line. The reason for this is that every measure ought to be certain and determinate and the smallest. Now between two points the length of a straight line is certain and determinate, because it can be but one, and it is the smallest of all the lines between the two points. But an infinitude of curved lines can be drawn between two points, and all are greater than the straight line drawn between the two given points. Hence the distance between two points is measured by a straight line, and not by the curved line of a semicircle or any other portion of the circle, either of a larger or a smaller circle. Therefore, since it belongs to the very notion of contrariety that it have maximum distance, as is said in Metaphysics X, then, since the distance between two points is not measured according to a curved line but according to a straight, the consequence is that a contrariety of termini does not bring about a contrariety in motions upon a semi-circle, but only in motions upon the diameter.
Obiicit autem contra hoc Ioannes grammaticus, quia non solum geometrae et astrologi accipiunt quantitatem lineae curvae per lineam rectam, sed etiam e converso: probant enim quantitatem chordae per arcum, et quantitatem arcus per chordam. 86. But John the Grammarian objects against this, because not only do geometers and astronomers reckon the quantity of a curved line by a straight line, but they also do the converse: for they prove the quantity of a chord by means of the arc and that of the arc by the chord.
Sed in hoc deficit ab intellectu Aristotelis. Non enim hoc intendit Aristoteles, quod linea curva mensuretur per rectam; sed quod distantia quae est inter quaelibet duo puncta, mensuretur per lineam rectam, ratione iam dicta. But in this he departs from the intent of Aristotle. For Aristotle does not intend to maintain that a curved line is measured by a straight, but the distance between any two given points is measured by a straight line, for the reason just given.
Obiicit etiam quod maxima distantia est in caelo, quae est inter duo puncta opposita, puta inter principium arietis et principium librae: et tunc, si contrarietas est maxima distantia, potest secundum hanc distantiam attendi contrarietas in motu circulari. He [John] objects too that in the heavens there is a greatest distance between two opposite points: for example, between the beginning of Aries and the beginning of Libra; consequently, if contrariety is the greatest distance, then according to this distance, contrariety can be found in circular motion.
Sed dicendum est quod ista distantia maxima attenditur secundum quantitatem diametri, et non secundum quantitatem semicirculi: alioquin plus distaret principium arietis a principio sagittarii, quod respicit trino aspectu, quam a principio librae, quod respicit aspectu rectae oppositionis. But to this it should be said that that greatest distance is reckoned according to the quantity of the diameter and not according to the quantity of the semicircle — otherwise the beginning of Aries would be farther from the beginning of Sagittarius, to which it has a trinary aspect, than from the beginning of Libra to which it has the aspect of right opposition.
Deinde cum dicit: similiter autem et utique etc., ostendit non esse contrarietatem in motu circulari secundum duos semicirculos. Et dicit quod similis est ratio, si quis describens circulum totum, ponat motum qui est in uno semicirculo, contrarium ei qui est in alio semicirculo. Sit enim circulus cuius diameter sit ez, dividens ipsum in duos semicirculos, in uno quorum describatur I, in alio t. Posset ergo aliquis dicere quod motus qui est ab e ad z per semicirculum I, contrariatur motui qui est a z ad e per semicirculum t. Sed hoc improbatur eadem ratione qua et primum: quia scilicet distantia quae est inter e et z, non mensuratur semicirculo, sed diametro. Et adhuc alia ratio est: quia unus motus continuus est, qui incipiens ab e, venit in z per I semicirculum, et iterum per t semicirculum redit a z in e; duo autem motus contrarii non possunt sibi invicem continuari, ut patet in VIII Physic. 87. Then at [45] he shows that there is not contrariety in circular motion according to two semicircles. And he says that the reasoning is similar to describing a whole circle and positing that the motion in one semicircle is contrary to a motion in the other. For let a circle have a diameter EZ dividing it into two semicircles called I and T respectively. Now someone could say that a motion from E to Z through semicircle I is contrary to the motion from Z to E through semicircle T. But this is disproved by the same argument as the first case: namely, because the distance between E and Z is not measured by a semicircle but by the diameter. But there is still another reason: namely, the motion which begins at E and proceeds to Z through I, and then returns from Z to E through semicircle T, is one continuous motion; but two motions that are contrary cannot be continuous with one another, as is plain in Physics VIII.
Deinde cum dicit: si autem et istae etc., ostendit quod etiam si istae partes motuum circularium essent contrariae, non tamen propter hoc sequeretur quod contrarietas esset in motibus circularibus secundum totum: non enim sequitur ad contrarietatem partium contrarietas totius. Et sic patet quod id quod iam ostendit philosophus de contrarietate partium motus circularis, ex abundanti prosecutus est, ut totaliter a motu circulari contrarietatem excluderet. 88. Then at [46] he shows that even if those parts of circular motions were contrary, that would be no reason for concluding that there would be contrariety in circular motions as a whole; for contrariety of parts is no proof for the contrariety of the whole. Consequently, it is plain that what the Philosopher has just showed about contrariety of the parts of circular motion has been done for added measure in order to exclude contrariety entirely from circular motion.
Deinde cum dicit: at vero etc., ostendit quod toti motui circulari non est alius totus motus circularis contrarius: et hoc duabus rationibus. Quarum prima sumitur ex consideratione ipsius motus circularis in communi. Sit ergo unus circulus, super quem in tribus punctis describantur a et b et g. Super hunc autem circulum intelligantur duo motus circulares, quorum unus incipiat ab a, et per b vadat in g, et sic revertatur ad a; alius autem motus e converso, incipiens ab a, primo vadat ad g, et sic transiens per b revertatur ad a. Dicit ergo hos duos motus non esse contrarios. Uterque enim horum motuum ab eodem incipit, scilicet ab a, et in idem terminatur, scilicet in ipsum a; et sic patet quod isti duo motus non incipiunt a contrario, neque terminantur ad contrarium; contrarius autem motus localis est qui est a contrario in contrarium. Patet ergo praedictos motus circulares non esse contrarios. 89. Then at [46 bis] he shows that to one complete circular motion there is not another circular motion contrary: and this for two reasons. The first of these is based on considering circular motion in general. Therefore, take a circle upon which A, B and G are described at three points. Suppose two circular motions occur upon this circle, one beginning at A through B to G and back to A; conversely, let the other start at A through G to B and back to A. He says then that these two motions are not contrary. For each begins at the same term A and terminates at the same term, namely, A; consequently, they neither begin at terms that are contrary nor end at terms that are contrary. But a contrary local motion is one that goes from contrary to contrary. Therefore, the two circular motions in question are not contrary.
Obiicit autem contra hoc iterum Ioannes grammaticus. Primo quidem quia in diversis videtur esse diversa ratio contrarietatis. Moveri enim a contrario in contrarium determinat contrarietatem in motibus rectis: unde non oportet, si talis contrarietas non est in motibus circularibus, quod propter hoc nulla contrarietas in eis esse possit. Item, sicut est de ratione motus contrarii in motibus rectis quod sit de contrario in contrarium, ita est de ratione motus quod sit de uno in aliud. Per hoc autem quod motus circularis est ab eodem in idem, non solum excluditur quod non sit de contrario in contrarium, sed etiam quod non sit de uno in aliud. Ergo non solum excluditur a motibus circularibus quod non sint contrarii, sed etiam quod penitus non sint motus. 90. The objector against this is once more John the Grammarian. First on the ground that the notion of contrariety in diverse things is seen to be diverse. For to be moved from contrary to contrary determines contrariety in straight motions; hence it is not necessary, if such contrariety is not present in circular motions, that on this account no contrariety may exist therein. Likewise, just as it is of the very nature of contrary motion in straight motions to be from contrary to contrary, so it is of the very nature of motion to be from one thing to another. Now, by the very fact that circular motion is from the same to the same, not only is it not from contrary to contrary, but it is not from one thing to another. Therefore there is excluded from circular motions not only that they be contrary, but that they be motions at all.
Dicendum est autem ad primum quod esse a contrario in contrarium non est ratio contrarietatis propria in motibus localibus qui sunt secundum lineam rectam; sed est communis ratio contrarietatis in omnibus motibus, ut patet in V Physic. Et huius ratio est, quia contrarietas est differentia secundum formam, ut ostenditur in X Metaphys.; motus autem habet formam seu speciem ex suo termino; et ideo in nullo motu potest esse contrarietas absque contrarietate terminorum. To the first objection it should be replied that to be from contrary to contrary is not a special property of the contrariety found in local motions in a straight line, but it is a common property of contrariety in all motions, as is plain in Physics V. And the reason for this is that contrariety is a difference according to form, as is shown in Metaphysics X. Now a motion possesses form or species from its terminus. Therefore, there can be contrariety in no motion, unless there is contrariety of termini.
Ad secundum dicendum quod motus circularis, quia est primus motuum, minimum habet de diversitate et plurimum de uniformitate. Et hoc quidem apparet proportionaliter in mobili et in motu. In mobili quidem, quia non mutat suum ubi secundum totum subiecto, sed solum ratione: pars vero quaelibet mutat suum ubi etiam subiecto, ut ostensum est in VI Physic. Et similiter etiam pars motus circularis est de uno in aliud subiecto differens: totus autem motus circularis est quidem de eodem in idem secundum subiectum, sed est de uno in aliud differens sola ratione. Si enim accipiatur circulatio una quae ab a redit in a, ipsum a, quod est terminus a quo et in quem, est idem subiecto, sed differt ratione, inquantum accipitur ut principium et finis. Et ideo, quia motus circularis plurimum habet de unitate, est natura eius longinqua a contrarietate, quae est maxima distantia. Et ideo talis motus competit primis corporibus, quae sunt propinquissima substantiis simplicibus, quae penitus contrarietate carent. To the second it must be said that circular motion, because it is the first of motions, has a minimum of diversity and a maximum of uniformity. And this even appears proportionally in the mobile and in the motion. In the mobile, indeed, because it does not change its "where" with respect to the whole subject, but only in conception, whereas each part changes its "where" even as to subject, as was shown in Physics VI. And similarly a part of a circular motion is from one to another with a difference as to subject; but the whole circular motion is indeed from the same to the same according to subject, but from one thing to another that differs only in conception. For if we take one circular motion from A to A, the A which is the terminus a quo and the terminus ad quem is the same as to subject, but differs in conception, insofar as it is taken now as beginning and now as end. And therefore, because circular motion has the most unity, its nature is very far from contrariety, which is a maximum distance. That is why such motion belongs to the first bodies which are the nearest to the simple substances which completely lack contrariety.
Secundam rationem ponit ibi: si autem et esset et cetera. Et haec quidem ratio sumitur per applicationem circularis motus ad corpora naturalia. Quae quidem ratio talis est. Si unus motus circularis esset contrarius alii, oporteret quod alter eorum esset frustra; sed nihil est frustra in natura; ergo non sunt duo motus circulares contrarii. 91. The second argument is at [47], and this argument is based on applying circular motion to natural bodies. And this is the argument: If one circular motion were contrary to another, then one of them would have to be in vain. But nothing in nature is in vain. Therefore, there are not two contrary circular motions.
Conditionalem autem probat sic. Si essent duo motus circulares contrarii, oporteret quod corpora quae moverentur illis duobus motibus, transirent per eadem signa in circulo signata: et hoc ideo, quia contrarietas motus localis exigit contrarietatem locorum, quae attingit utrumque mobilium. Si ergo essent motus circulares contrarii, oporteret quod loca aliqua designarentur contraria in circulo. In recta quidem linea designantur sola duo loca contraria, quae scilicet maxime distant: alia vero loca signata per lineam rectam, quae sunt infra duo loca extrema, cum non maxime distent, non habent contrarietatem ad invicem. Sed in circulo cuiuslibet puncti est accipere maximam distantiam ad aliquod aliud punctum circuli: quia a quolibet puncto signato in circulo contingit ducere aliquam diametrum, quae est maxima linearum rectarum cadentium in circulo; dictum est autem quod omnis distantia mensuratur secundum lineam rectam. Quia igitur ea quae moventur contrariis motibus, necesse est attingere contraria loca, necesse est, si motus circulares sint contrarii, quod utrumque corpus circulariter motum, a quovis puncto circuli moveri incipiat, perveniat ad omnia loca circuli, quae omnia sunt contraria. Nec est inconveniens si in circulo describantur loca contraria secundum omnem partem: quia contrarietates loci accipiuntur non solum secundum sursum et deorsum, sed etiam secundum ante et retro, et dextrum et sinistrum; dictum est autem quod contrarietates motus localis accipiuntur secundum contrarietates locorum; et sic, si motus circulares sunt contrarii, necesse est accipi contrarietates in circulo secundum praedicta. The truth of the conditional proposition he proves in the following manner: If there were two contrary circular motions, then the bodies subject to them ought to pass through the same signs marked on a circle. The reason for this is that contrariety of local motion demands contrariety of the places, which affect both mobiles. Consequently, if there were contrary circular motions, then contrary places should be able to be designated on the circle. Now on a straight line only two contrary places are designated, namely, those the greatest distance apart, while other places designated on that line, since they are within the extreme places, are not contrary to one another. But on a circle any point at random can be at a greatest distance from some other point on the circle: because from any point on the circle a diameter can be drawn, which is the greatest of the straight lines falling in the circle. And it has been said that every distance is measured according to a straight line. Therefore, because things in contrary motions must reach contrary places, then if circular motions are contrary, it is necessary that each body in circular motion, no matter from which point of the circle its motion begins, reach all the places of the circle, all of which are contrary. (Nor is it unfitting that in a circle places be marked as in every way contrary — for contrariety of place is taken not only with respect to up and down, but according to ahead and to the rear, and left and right.) But it has been said that the contraries of local motion are based on contrariety of places. And thus, if circular motions are contrary, the contrarieties in the circle must be taken according to the forementioned.
Ex his autem sequitur quod alterum motuum vel corporum esset frustra. Quia si aequales essent magnitudines motae, idest aequalis virtutis, neutra ipsarum moveretur; quia una totaliter impediret alteram, cum oporteret utramque transire per eadem loca. Si vero alter motus dominaretur propter praeeminentiam virtutis in altero mobilium vel moventium, consequens est quod alter motus esse non posset; quia totaliter impediretur per motum fortiorem. Itaque, si ambo corpora essent, quae essent nata moveri contrariis motibus circularibus, frustra esset alterum ipsorum corporum, quod non posset moveri illo motu qui impediretur per fortiorem: unumquodque enim dicimus esse frustra, quod non potest habere suum usum, sicut dicimus calceamentum esse frustra, quo non potest aliquis calceari. Et similiter corpus erit frustra, quod non poterit moveri proprio motu: et etiam motus erit frustra, quo nihil potest moveri. Now from all this it follows that one of the motions or of the bodies would be in vain. For if the magnitudes moved were equal, i.e., of equal power, neither would be moved, because one would totally obstruct the other, since both would have to traverse the same places. But if one motion dominated on account of a greater power in one of the mobiles or movers, then the other motion could not exist, because it would be totally obstructed by the stronger motion. Therefore, if both were bodies apt to be moved with contrary circular motions, one of them would exist in vain, for it could not be moved with that motion which was obstructed by the stronger. For we say that a thing is "in vain" when it does not realize its usefulness, as we say that a shoe is in vain if no one can wear it. In like manner, a body would be in vain, if it could not be moved with its proper motion; and likewise a motion would be in vain if nothing could be moved with it.
Sic ergo patet quod, si sint duo motus circulares contrarii, necesse est aliquid esse frustra in natura. Sed quod hoc sit impossibile, probat sic. Omne quod est in natura, vel est a Deo, sicut primae res naturales; vel est a natura sicut a secunda causa, puta inferiores effectus. Sed Deus nihil facit frustra, quia, cum sit agens per intellectum, agit propter finem. Similiter etiam natura nihil facit frustra, quia agit sicut mota a Deo velut a primo movente; sicut sagitta non movetur frustra, inquantum emittitur a sagittante ad aliquid certum. Relinquitur ergo quod nihil in natura sit frustra. Consequently, it is plain that if there are two contrary circular motions, there would have to be something in vain in nature. But that this is impossible he now proves: Whatever exists in nature is either from God, as are the first natural things, or from nature as from a second cause, as, for example, lower effects. But God makes nothing in vain, because, since He is a being that acts through understanding, He acts for a purpose. Likewise nature makes nothing in vain, because it acts as moved by God as by a first mover, just as an arrow is not moved in vain, inasmuch as it is shot by the bowman at some definite thing. What remains, therefore, is that nothing in nature is in vain.
Est autem attendendum quod Aristoteles hic ponit Deum esse factorem caelestium corporum, et non solum causam per modum finis, ut quidam dixerunt. It should be noted that Aristotle here posits God to be the maker of the celestial bodies, and not just a cause after the manner of an end, as some have said.
Obiicit autem contra hanc rationem Ioannes grammaticus, quia pari ratione posset aliquis concludere quod in motibus rectis non sit contrarietas; quia contraria mobilia impediunt se invicem. 92. John the Grammarian objects against this argument that, for the same reason, someone could conclude that there is no contrariety in straight motions, because contrary mobiles obstruct one another.
Sed dicendum quod alia ratio est in motibus rectis et circularibus, propter duo. Primo quidem quia duo corpora moventur contrariis motibus rectis absque eo quod se invicem impediant, eo quod non attenditur contrarietas in motibus rectis nisi secundum extrema linearum rectarum, puta secundum centrum mundi et circumferentiam eius: a centro autem ad circumferentiam possunt infinitae lineae duci, ita quod id quod movetur per unam earum sursum, non impedit id quod movetur per aliam deorsum. Sed in motu circulari eadem ratio contrarietatis est in omnibus partibus circuli: et ideo oportebit quod per eadem loca circuli utrumque transeat; et sic ex necessitate oportet quod motus circulares contrarii se invicem impediant. But it should be said that the case with straight motions is different from that of circular, for two reasons. First, because two bodies are moved with contrary straight motions without mutually obstructing one another, for in straight motions contrariety is not reckoned except with respect to the extremes of straight lines, for example, with respect to the center of the world and its circumference. Now from the center to the circumference an infinitude of lines can be drawn so that what is moved upward through one of them does not obstruct what is being moved downward through another. But in circular motion the same aspect of contrariety is present in all parts of the circle. Therefore it will be necessary that both move through the same places of the circle. And so of necessity contrary circular motions would have to obstruct one another.
Secundo est diversa ratio utrobique, quia corpus quod movetur naturaliter motu recto, sicut naturaliter est aptum corrumpi, ita naturaliter est aptum impediri: unde si impediatur, non est hoc frustra, sicut nec quod corrumpatur. Sed corpus circulariter motum est naturaliter incorruptibile; unde non est natum impediri: unde si in natura esset aliquid impeditivum ipsius, esset frustra. Secondly, in the two cases the aspect is different — for in the case of a body that is being naturally moved with a straight motion, just as it is naturally apt to be corrupted, so it is naturally apt to be obstructed. Hence, if it is obstructed, this is no more in vain than if it be corrupted. But a body circularly moved is naturally incorruptible: hence it is not apt to be obstructed. Hence if there were in nature something to impede it, that impediment would be useless.
Item potest obiici de motu planetarum, qui moventur propriis motibus ab occidente in orientem; quod videtur esse in contrarium motus firmamenti, quod movetur motu diurno ab oriente in occidentem. 93. Likewise, it can be objected about the motion of the planets which are moved with their own motions from west to east, which seems to be contrary to the motion of the firmament which in its diurnal motion is from east to west.
Sed dicendum est quod tales motus habent quidem aliquam diversitatem ad invicem, quae designat aliquo modo diversam naturam mobilium: non tamen est aliqua contrarietas, propter tria. But it must be said that such motions have indeed a certain mutual diversity which somehow designates the diverse natures of the mobiles. But, for three reasons, there is no contrariety:
Primo quidem quia huiusmodi diversitas non est secundum contrarios terminos, sed secundum contrarias vias perveniendi ad eundem terminum; puta quia firmamentum a puncto orientis movetur ad punctum occidentis per hemisphaerium superius, et redit ad punctum orientis per hemisphaerium inferius, planeta autem movetur a puncto occidentis ad orientem per aliud hemisphaerium. First, this is so, because diversity of this kind is not based on contrary termini but on contrary ways of reaching the same terminus: for example, because the firmament is moved from the eastern point to the western point through the upper hemisphere, and returns to the eastern point through the lower hemisphere, while a planet is moved from a western point to the east through another hemisphere.
Moveri autem diversis viis ad eundem finem, non facit contrarietatem actionum vel motuum, sed pertinet ad diversum ordinem motuum et mobilium: quia quod nobiliori via pertingit ad terminum est nobilius, sicut melior medicus est qui efficaciori via sanitatem inducit. Et inde est quod motus primus firmamenti est nobilior secundo motu, qui est planetarum, sicut et supremus orbis est nobilior. Unde et orbes planetarum moventur motu primi orbis absque hoc quod impediantur a suis propriis motibus. But to be moved to the same end by diverse routes does not make for contrariety of actions or passions, but pertains to the diverse order of the motions and mobiles — for what reaches its terminus by a nobler route is nobler, just as a better doctor is one who induces health by a more efficacious way. Hence the first motion of the firmament is nobler than the second motion, i.e., that of the planets, just as the supreme orb is nobler. Wherefore, the orbs of the planets are moved with the motion of the first orb without their being impeded from their own proper motions.
Secunda ratio est, quia quamvis uterque motus sit super idem centrum, est tamen uterque motus super alios et alios polos: unde non sunt contrarii. The second reason is that, although each motion is over the same center, nevertheless they are over other and other poles; hence they are not contrary.
Tertia ratio est, quia non sunt in eodem circulo, sed motus planetarum sunt in inferioribus circulis. The third reason is that they are not in the same circle, but the motions of the planets are in the lower circles.
Oportet autem contrarietatem attendi circa eandem distantiam, sicut patet in motibus rectis, quorum contrarietas consistit in distantia centri et circumferentiae. But contrariety must be reckoned with respect to the same distance, as is plain in straight motions, the contrariety of which consists in the distance from the center to the circumference.

Lecture 9:
The need for treating of the infinity of the universe.
Chapter 5
(271b.) Ἀλλ' ἐπεὶ δῆλον περὶ τούτων, περὶ τῶν λοιπῶν σκεπτέον, καὶ πρῶτον πότερον ἔστι τι σῶμα ἄπειρον, ὥσπερ οἱ πλεῖστοι τῶν ἀρχαίων φιλοσόφων ᾠήθησαν, ἢ τοῦτ' ἔστιν ἕν τι τῶν ἀδυνάτων 48 This being clear, we must go on to consider the questions which remain. First, is there an infinite body, as the majority of the ancient philosophers thought, or is this an impossibility?
τὸ γὰρ οὕτως ἢ ἐκείνως ἔχειν οὔ τι μικρὸν ἀλλ' ὅλον διαφέρει καὶ πᾶν πρὸς τὴν περὶ τῆς ἀληθείας θεωρίαν σχεδὸν γὰρ αὕτη πασῶν ἀρχὴ τῶν ἐναντιώσεων τοῖς ἀποφηναμένοις τι περὶ τῆς ὅλης φύσεως καὶ γέγονε καὶ γένοιτ' ἄν, 49 The decision of this question, either way, is not unimportant, but rather all-important, to our search for the truth. It is this problem which has practically always been the source of the differences of those who have written about nature as a whole. So it has been and so it must be;
εἴπερ καὶ τὸ μικρὸν παραβῆναι τῆς ἀληθείας ἀφισταμένοις γίνεται πόρρω μυριοπλάσιον. Οἷον εἴ τις ἐλάχιστον εἶναί τι φαίη μέγεθος οὗτος γὰρ τοὐλάχιστον εἰσαγαγὼν τὰ μέγιστ' ἂν κινήσειε τῶν μαθηματικῶν. Τούτου δ' αἴτιον ὅτι ἡ ἀρχὴ δυνάμει μείζων ἢ μεγέθει, διόπερ τὸ ἐν ἀρχῇ μικρὸν ἐν τῇ τελευτῇ γίνεται παμμέγεθες. Τὸ δ' ἄπειρον καὶ ἀρχῆς ἔχει δύναμιν καὶ τοῦ ποσοῦ τὴν μεγίστην, ὥστ' οὐδὲν ἄτοπον οὐδ' ἄλογον τὸ θαυμαστὴν εἶναι τὴν διαφορὰν ἐκ τοῦ λαβεῖν ὡς ἔστι τι σῶμα ἄπειρον. Διὸ περὶ αὐτοῦ λεκτέον ἐξ ἀρχῆς ἀναλαβοῦσιν. 50 since the least initial deviation from the truth is multiplied later a thousandfold. Admit, for instance, the existence of a minimum magnitude, and you will find that the minimum which you have introduced, small as it is, causes the greatest truths of mathematics to totter. The reason is that a principle is great rather in power than in extent; hence that which was small at the start turns out a giant at the end. Now the conception of the infinite possesses this power of principles, and indeed in the sphere of quantity possesses it in a higher degree than any other conception; so that it is in no way absurd or unreasonable that the assumption that an infinite body exists should be of peculiar moment to our inquiry. The infinite, then, we must now discuss, opening the whole matter from the beginning.
Ἀνάγκη δὴ πᾶν σῶμα ἤτοι τῶν ἁπλῶν εἶναι ἢ τῶν συνθέτων, ὥστε καὶ τὸ ἄπειρον ἢ ἁπλοῦν ἔσται ἢ σύνθετον. Ἀλλὰ μὴν καὶ ὅτι γε πεπερασμένων τῶν ἁπλῶν ἀνάγκη πεπερασμένον εἶναι τὸ σύνθετον, δῆλον τὸ γὰρ ἐκ πεπερασμένων καὶ πλήθει καὶ μεγέθει συγκείμενον πεπέρανται καὶ πλήθει καὶ μεγέθει τοσοῦτον γάρ ἐστιν ἐξ ὅσων ἐστὶ συγκείμενον. Λοιπὸν τοίνυν ἰδεῖν πότερον ἐνδέχεταί τι τῶν ἁπλῶν ἄπειρον εἶναι τὸ μέγεθος, ἢ τοῦτ' ἀδύνατον. Προχειρισάμενοι δὴ περὶ τοῦ πρώτου τῶν σωμάτων, οὕτω σκοπῶμεν καὶ περὶ τῶν λοιπῶν. 51 Every body is necessarily to be classed either as simple or as composite; the infinite body, therefore, will be either simple or composite. But it is clear, further, that if the simple bodies are finite, the composite must also be finite, since that which is composed of bodies finite both in number and in magnitude is itself finite in respect of number and magnitude: its quantity is in fact the same as that of the bodies which compose it. What remains for us to consider, then, is whether any of the simple bodies can be infinite in magnitude, or whether this is impossible. Let us try the primary body first, and then go on to consider the others.
Ὅτι μὲν τοίνυν ἀνάγκη τὸ σῶμα τὸ κύκλῳ φερόμενον πεπεράνθαι πᾶν, ἐκ τῶνδε δῆλον. 52 The body which moves in a circle must necessarily be finite in every respect, for the following reasons.
Εἰ γὰρ ἄπειρον τὸ κύκλῳ φερόμενον σῶμα, ἄπειροι ἔσονται αἱ ἀπὸ τοῦ μέσου ἐκβαλλόμεναι. Τῶν δ' ἀπείρων τὸ διάστημα ἄπειρον διάστημα δὲ λέγω τῶν γραμμῶν, οὗ μηδὲν ἔστιν ἔξω λαβεῖν μέγεθος ἁπτόμενον τῶν γραμμῶν. Τοῦτ' οὖν ἀνάγκη ἄπειρον εἶναι τῶν γὰρ πεπερασμένων ἀεὶ ἔσται πεπερασμένον. Ἔτι δ' ἀεὶ ἔστι τοῦ (272a.) δοθέντος μεῖζον λαβεῖν, ὥστε καθάπερ ἀριθμὸν λέγομεν ἄπειρον, ὅτι μέγιστος οὐκ ἔστιν, ὁ αὐτὸς λόγος καὶ περὶ τοῦ διαστήματος εἰ οὖν τὸ μὲν ἄπειρον μὴ ἔστι διελθεῖν, ἀπείρου δ' ὄντος ἀνάγκη τὸ διάστημα ἄπειρον εἶναι, οὐκ ἂν ἐνδέχοιτο κινηθῆναι κύκλῳ 53 (1) If the body so moving is infinite, the radii drawn from the centre will be infinite. But the space between infinite radii is infinite: and by the space between the radii I mean the area outside which no magnitude which is in contact with the two lines can be conceived as falling. This, I say, will be infinite: first, because in the case of finite radii it is always finite; and secondly, because in it one can always go on to a width greater than any given width; thus the reasoning which forces us to believe in infinite number, because there is no maximum, applies also to the space between the radii. Now the infinite cannot be traversed, and if the body is infinite the interval between the radii is necessarily infinite: circular motion therefore is an impossibility.
τὸν δ' οὐρανὸν ὁρῶμεν κύκλῳ στρεφόμενον, καὶ τῷ λόγῳ δὲ διωρίσαμεν ὅτι ἐστί τινος ἡ κύκλῳ κίνησις. 54 Yet our eyes tell us that the heavens revolve in a circle, and by argument also we have determined that there is something to which circular movement belongs.
Postquam philosophus ostendit perfectionem universi et ex quibus partibus eius perfectio integretur, hic incipit inquirere de infinitate ipsius; quia, ut dicitur in III Physic., quidam rationem perfecti attribuerunt infinito. 94. After explaining the perfection of the universe and pointing out the parts that make it complete, the Philosopher here begins to inquire into its infinity, because, as is said in Physics III, some have attributed the notion of "perfect" to the infinite.
Potest autem aliquid dici infinitum tripliciter: uno modo secundum magnitudinem, alio modo secundum numerum, tertio modo secundum durationem. Now something can be said to be infinite in three ways: in one way with respect to magnitude; in another with respect to number, and in a third way with respect to duration.

Primo igitur inquirit utrum universum sit infinitum secundum magnitudinem;

secundo utrum sit infinitum secundum multitudinem, utrum scilicet sit unus mundus tantum, vel infiniti seu plures, ibi: quia autem neque plures etc.;

tertio utrum sit infinitum duratione, quasi semper existens, ibi: his autem determinatis et cetera.

First, then, he asks whether the universe is infinite according to magnitude;

Secondly, whether according to multitude, i.e., whether there is just one world, or an infinitude, or many (L. 16);

Thirdly, whether it is infinite in duration, as though ever existing (L.22).

Circa primum duo facit: About the first he does two things:

primo dicit prooemialiter de quo est intentio;

secundo exequitur propositum, ibi: quod quidem igitur necesse et cetera.

First he speaks in a prefatory manner about his intention;

Secondly, he carries out his proposal, at 99.

Circa primum tria facit: About the first he does three things:

primo dicit de quo est intentio;

secundo assignat rationem suae intentionis, ibi: sic enim aut illo modo etc.;

tertio determinat modum agendi, ibi: necesse itaque et cetera.

First he states his intention;

Secondly, he assigns the reason for his intention, at 96;

Thirdly, he decides upon a method of treatment, at 98.

Dicit ergo primo quod, quia manifestum est de praedictis, quod motui circulari non est aliquis motus contrarius, et de aliis quae dicta sunt, oportet nunc intendere ad ea quae residua sunt. Et primo inquirendum est utrum sit aliquod corpus infinitum in actu secundum magnitudinem, sicut plurimi antiquorum philosophorum putaverunt (omnes scilicet qui posuerunt unum principium materiale, puta ignem aut aerem aut aquam aut aliquod medium horum); vel potius hoc est impossibile, quod sit aliquod corpus infinitum in actu, sicut probatum est in III Physic., supponendo tamen quod non sit aliud corpus praeter quatuor elementa, secundum opinionem aliorum. Sed quia iam probavit quod est aliquod corpus praeter quatuor elementa, repetit hanc considerationem, ut universalior sit inquisitio veritatis. 95. He says therefore first [48] that because it is now clear with respect to the foregoing, namely, that there is no motion contrary to circular motion, and as to the other things mentioned, we must now direct our attention to what remains. And first we must inquire whether there exists any body infinite in act with respect to magnitude, as very many of the early philosophers thought (i.e., all those who posited one material principle, such as fire, or air, or water, or something intermediate); or whether it is impossible that there be a body infinite in act, as was proved in Physics III, supposing, however, that there is no body other than the four elements, according to the opinion of others. Since, however, he has just now proved that there is another body besides the four elements, he therefore repeats this consideration in order that the search for the truth may be more universal.
Deinde cum dicit: sic enim aut illo modo etc., assignat rationem suae intentionis, ex diversitate quae accidit propter praedictam positionem. Et primo proponit hanc diversitatem consequentem. Et dicit quod non modicum differt in comparatione ad speculationem veritatis in naturali philosophia, utrum hoc aut illo modo se habeat, scilicet quod sit aliquod corpus infinitum secundum magnitudinem vel non: sed magis inducit differentiam circa totum, idest circa totum universum, et circa omnem considerationem naturalem. Hoc enim quod dictum est, fere fuit in praeterito, et erit in futuro principium omnium contradictionum inter eos qui aliquid enuntiaverunt de tota natura rerum. Illi enim qui posuerunt unum infinitum principium, posuerunt alia fieri quasi per separationem ab illo principio; et sic, propter infinitatem illius principii, dixerunt generationem rerum non deficere; sicut si aliquis diceret quod ex infinita massa possunt fieri panes in infinitum. Illi vero qui posuerunt principia finita, dixerunt fieri res in infinitum per reciprocam congregationem et separationem elementorum. Then at [49] he gives a reason for his intention, from the diversity that happens on account of the aforesaid position. And first he mentions this con sequent diversity, and says that it makes no slight difference to the speculation of truth in natural philosophy whether things are this way or that, i.e., whether or not there exists a body that is infinite according to magnitude. Rather, it does make a difference with respect to the whole universe and every natural consideration. For what has just been said, was in the past, and will be in the future, the source of almost all the contradictions between those who have put forth anything about the whole nature of things. For those who posited one infinite principle assumed that all things come to be by a kind of separation from that principle: thus, on account of the infinitude of that principle, they said that the generation of things does not fail. It is as though someone said that from an infinite mass of dough, loaves of bread could be made ad infinitum. But those who posited finite principles said that things come to be ad infinitum through a reciprocal commingling and separating of the elements.
Deinde cum dicit: siquidem qui modicum etc., assignat causam quare tanta diversitas ex hoc sequatur: quia scilicet qui modicum transgreditur a veritate circa principium, procedens in ulteriora fit magis longe a veritate decies millies. Et hoc ideo, quia omnia subsequentia dependent ex suis principiis. Et hoc maxime apparet in errore viarum: quia qui parum elongatur a recta via, postmodum procedens fit multum longe. Et ponit exemplum de eo quod dictum est, in his qui posuerunt aliquam minimam magnitudinem, sicut Democritus posuit corpora indivisibilia: sic autem introducens aliquid minimum in quantitate, destruit maximas propositiones mathematicorum, puta quod lineam datam contingit secari in duo media. Et huius causa est, quia principium, etsi sit modicum magnitudine, est tamen magnum virtute, sicut ex modico semine producitur magna arbor: et inde est quod illud quod est modicum in principio, in fine multiplicatur, quia pertingit ad totum id ad quod se extendit virtus principii, sive hoc sit verum sive falsum. Infinitum autem habet rationem principii (omnes enim quicumque sunt locuti de infinito, posuerunt infinitum esse principium, ut dictum est in III Physic.); et cum hoc habet maximam virtutem quantum ad quantitatem, quia excedit omnem quantitatem datam. Si igitur principium quod est minimum quantitate, facit magnam differentiam in sequentibus, multo magis infinitum, quod non solum excedit in virtute principii, sed etiam in quantitate. Et ideo neque inconveniens neque irrationabile est, si mirabilis differentia sequatur in scientia naturali ex eo quod sumitur aliquod corpus esse infinitum. Et ideo de hoc dicendum est, resumendo considerationem nostram a principio quod supra accepimus, de differentia simplicium corporum et compositorum. 97. Then at [50] he assigns the cause why such diversity follows from this: it is because one who makes a slight departure from the truth in his principles gets 10,000 times farther from the truth as he goes on. This is so because all things that follow depend on their principles. This is especially clear in an error at the crossroads: for one who at the beginning is only a slight distance from the right road gets very far away from it later on. And he gives, as an example of what he is talking about, the case of those who posited a smallest magnitude, as Democritus posited indivisible bodies. By thus introducing a least quantity, he overthrew the most important propositions of mathematics — for example, that any given line can be cut into two halves. The reason for this effect is that a principle, though small in stature, is nevertheless great in power, just as from a small seed a mammoth tree is produced. Hence it is that what is small in the beginning becomes multiplied in the end, because it reaches unto all that to which the power of the principle extends, whether this be true or false. Now the infinite has the nature of a principle (for all who have spoken about the infinite considered it a principle, as was said in Physics III); besides, the infinite has the greatest force with respect to quantity, because it exceeds every given quantity. If, therefore, a principle which is the least in quantity makes a great difference in what follows from it, then much more is this so of the infinite, which is outstanding not only in virtue of being a principle but also in quantity. Consequently, it is neither inappropriate nor unreasonable that a remarkable difference should follow in natural science from the assumption that some body is infinite. And therefore it must be discussed by resuming our consideration from the principle which we accepted above about the difference between simple and composite bodies.
Deinde cum dicit: necesse itaque etc., ostendit quo ordine agendum sit. Et dicit quod necesse est omne corpus aut de numero simplicium esse aut de numero compositorum corporum: unde oportet quod etiam corpus infinitum aut sit simplex aut compositum. Iterum manifestum est quod, si corpora simplicia essent finita multitudine et magnitudine, necesse est quod compositum sit finitum et multitudine et magnitudine: tantam enim quantitatem habet corpus compositum, quanta est quantitas corporum simplicium ex quibus componitur. Ostensum est autem supra quod corpora simplicia sunt finita multitudine, quia non est aliquod corpus praeter praedicta. Restat igitur videre utrum aliquod corpus simplicium sit infinitum magnitudine, vel si hoc sit impossibile. Et hoc quidem ostendemus primo argumentantes de primo corporum, quod scilicet circulariter movetur; et sic intendemus ad reliqua corpora, quae scilicet moventur motu recto. Then he points out what order must be followed, and says that of necessity every body is either a member of the simple group or of the composite group. Consequently an infinite body must be one or the other. Again, it is plain that if simple bodies are finite in multitude and magnitude, so too must composite body be — for a composite body has as much quantity as the quantity of the simple bodies of which it is composed. However, it has been shown above that simple bodies are finite in multitude, because there is no body other than the ones mentioned. It remains, therefore, to see whether any of the simple bodies is infinite in magnitude, or whether this is impossible. And this we shall show by first arguing from the first body, i.e., the one that is moved circularly; then we shall consider the remaining bodies, namely, those moved with a straight motion.
Deinde cum dicit: quod quidem igitur etc., ostendit quod non sit corpus infinitum: 99. Then at [52] he shows that there is not an infinite body:

et primo propriis rationibus de singulis corporibus;

secundo tribus communibus rationibus de omnibus, ibi: quod quidem igitur non est infinitum corpus et cetera.

First with reasons proper to the individual bodies;

Secondly, with three general reasons applying to all, (L. 13).

Circa primum duo facit: As to the first he does two things:

primo ostendit propositum in corpore quod circulariter movetur;

secundo in corporibus quae moventur motu recto, ibi: sed adhuc neque quod ad medium et cetera.

First he proves the proposition as to the body moved circularly;

Secondly, as to the bodies moved with a straight motion, (L. 12).

Circa primum duo facit. About the first he does two things:

Primo proponit quod intendit: et dicit quod manifestum est ex his quae dicentur, quod necesse est omne corpus quod circulariter fertur, esse finitum (hoc enim est primum corporum).

Deinde cum dicit: si enim infinitum etc., probat propositum sex rationibus: quarum prima talis est. Si aliquod corpus est infinitum, non potest moveri circulariter; sed corpus primum movetur circulariter; ergo non est infinitum.

First he proposes his intention and says that it is plain from what will be said that every circularly moved body must be finite (for this is the first of bodies).

Then at [53] he proves his proposition with six arguments, the first of which is this: If any body is infinite, it cannot be moved circularly; but the first body is moved circularly. Therefore, it is not infinite.

Primo ergo probat conditionalem sic: quia si corpus quod circulariter fertur est infinitum, necesse est quod lineae rectae quae egrediuntur a centro ipsius, sint infinitae; protenduntur enim quamdiu durat corporis quantitas. Distantia autem quae est inter infinitas lineas, est infinita. First, then, he proves the conditional proposition as follows: If a circularly moved body is infinite, then the straight lines proceeding from its center are infinite, for they are extended as far as the quantity of the body. But the distance between the infinite lines is infinite.
Posset autem aliquis dicere quod, etiam si sint lineae infinitae a centro egredientes, tamen inter eas est aliqua distantia finita: quia omnis distantia mensuratur secundum lineam rectam, potest autem aliqua linea finita protrahi infra duas praedictas lineas, puta in propinquitate ad centrum. Sed manifestum est quod extra illam lineam poterit alia linea recta maior protrahi inter illas lineas de quibus primo loquebamur. Et ideo dicit quod non loquitur de distantia quam mensurant tales lineae; sed illam distantiam dicit esse infinitam, quae mensuratur per lineam extra quam non est sumere aliquam aliam lineam maiorem, quae tangat utramque primarum linearum. Now someone might say that even if there are infinite lines from the center, yet the distance between them is finite, because every distance is measured according to a straight line, and a finite line can be drawn between two such radii, for example, very close to the center. But it is clear that beyond that line a greater straight line can be drawn between the lines we first mentioned. And therefore he says that he is not speaking of the distance that such lines measure, but that that distance is infinite which is measured by a line beyond which no greater line can be taken, and which touches each of the first lines.
Et talem distantiam probat esse infinitam dupliciter. Primo quidem quia omnis talis distantia finita est inter lineas egredientes a centro finitas: oportet enim quod iidem sint termini linearum egredientium a centro, et lineae finitae mensurantis extremam distantiam inter eas. That this distance is infinite he proves in two ways. First, because every such distance between any finite lines proceeding from the center is finite; for the ends of the lines proceeding from the center and of the finite line measuring the greatest distance between them must coincide.
Secundo probat idem per hoc quod qualibet distantia data inter duas lineas mensuratas egredientes a centro, est accipere aliam maiorem, sicut quolibet numero dato est accipere maiorem: unde sicut est infinitum in numeris, ita est infinitum in tali distantia. Secondly, he proves the same point because it is possible, if the distance between two measured lines proceeding from the center is given, to take another distance which is greater, just as it is possible to take a number greater than a given number. Hence, just as the infinite is in numbers, so is it in this distance under discussion.
Ex hoc sic argumentatur. Infinitum non est pertransire, ut probatum est in VI Physic.; sed si corpus sit infinitum, necesse est quod distantia sit infinita inter lineas egredientes a centro, ut probatum est; ad hoc autem quod fiat motus circularis, oportet quod una linea egrediens a centro pertingat ad situm alterius; sic igitur nunquam contingeret aliquid circulariter moveri. From this he argues as follows: The infinite cannot be traversed, as was proved in Physics VI. But if a body be infinite, the distance between the lines proceeding from the center must be infinite, as was proved. But in order that circular motion occur, one line proceeding from the center must reach the position of another. Consequently, it could never happen that anything be moved circularly.
Secundo ibi: caelum autem videmus etc., probat destructionem consequentis dupliciter: primo quidem quia ad sensum videmus quod caelum circulariter movetur; secundo quia supra per rationem probatum est quod motus circularis est alicuius corporis. Unde relinquitur quod impossibile sit corpus esse infinitum, quod circulariter movetur. 101. Secondly, at [54] he proves in two ways the destruction of the consequent. First, because it is evident to sense that the heavens are moved circularly; secondly, because it was proved above by reason that circular motion belongs to some body. What remains, therefore, is that it is impossible for the circularly moved body to be infinite.

Lecture 10:
The second and third reasons proving the circularly moved body not infinite
Chapter 5 cont.
Ἔτι ἀπὸ πεπερασμένου χρόνου ἐὰν ἀφέλῃς πεπερασμένον, ἀνάγκη καὶ τὸν λοιπὸν εἶναι πεπερασμένον καὶ ἔχειν ἀρχήν. Εἰ δ' ὁ χρόνος ὁ τῆς βαδίσεως ἔχει ἀρχήν, ἔστιν ἀρχὴ καὶ τῆς κινήσεως, ὥστε καὶ τοῦ μεγέθους ὃ βεβάδικεν. Ὁμοίως δὲ τοῦτο καὶ ἐπὶ τῶν ἄλλων. Ἔστω δὴ γραμμὴ ἄπειρος, ἐφ' ᾗ ΑΓΕ, ἐπὶ θάτερα, ᾗ τὸ Ε ἡ δ' ἐφ' ᾗ τὰ ΒΒ, ἐπ' ἀμφότερα ἄπειρος. Εἰ δὴ γράψει κύκλον ἡ τὸ ΑΓΕ ἀπὸ τοῦ Γ κέντρου, τέμνουσά ποτε οἰσθήσεται κύκλῳ τὴν τὰ ΒΒ ἡ τὸ ΑΓΕ πεπερασμένον χρόνον ὁ γὰρ πᾶς χρόνος, ἐν ὅσῳ κύκλῳ ἠνέχθη ὁ οὐρανός, πεπερασμένος. Καὶ ὁ ἀφῃρημένος ἄρα, ὃν ἡ τέμνουσα ἐφέρετο. Ἔσται ἄρα τις ἀρχὴ ᾗ πρῶτον ἡ τὸ ΑΓΕ τὴν τὰ ΒΒ ἔτεμεν. Ἀλλ' ἀδύνατον. Οὐκ ἄρα ἔστι κύκλῳ στραφῆναι τὸ ἄπειρον. Ὥστ' οὐδὲ τὸν κόσμον, εἰ ἦν ἄπειρος. 55 (2) Again, if from a finite time a finite time be subtracted, what remains must be finite and have a beginning. And if the time of a journey has a beginning, there must be a beginning also of the movement, and consequently also of the distance traversed. This applies universally. Take a line, ACE, infinite in one direction, E, and another line, BB, infinite in both directions. Let ACE describe a circle, revolving upon C as centre. In its movement it will cut BB continuously for a certain time. This will be a finite time, since the total time is finite in which the heavens complete their circular orbit, and consequently the time subtracted from it, during which the one line in its motion cuts the other, is also finite. Therefore there will be a point at which ACE began for the first time to cut BB. This, however, is impossible. The infinite, then, cannot revolve in a circle; nor could the world, if it were infinite.
Ἔτι δὲ καὶ ἐκ τῶνδε φανερόν, ὅτι τὸ ἄπειρον ἀδύνατον κινηθῆναι. Ἔστω γὰρ ἡ τὸ Α φερομένη παρὰ τὴν Β, πεπερασμένη παρὰ πεπερασμένην. Ἀνάγκη δὴ ἅμα τήν τε Α τῆς Β ἀπολελύσθαι καὶ τὴν Β τῆς Α ὅσον γὰρ ἡ ἑτέρα ἐπιβάλλει τῆς ἑτέρας, καὶ ἡ ἑτέρα ἐκείνης τοσοῦτον. Εἰ μὲν οὖν ἄμφω κινοῖντο εἰς τοὐναντίον, θᾶττον ἂν ἀπολύοιντο, εἰ δὲ παρὰ μένουσαν φέροιτο, βραδύτερον, τῷ αὐτῷ τάχει κινουμένου τοῦ παραφερομένου. Ἀλλ' ἐκεῖνό γε φανερόν, ὅτι ἀδύνατον τὴν ἄπειρον διελθεῖν ἐν πεπερασμένῳ χρόνῳ. Ἐν ἀπείρῳ ἄρα δέδεικται γὰρ τοῦτο πρότερον ἐν τοῖς περὶ κινήσεως. Διαφέρει δέ γε οὐθὲν ἢ τὴν πεπερασμένην φέρεσθαι παρὰ τὴν ἄπειρον ἢ τὴν ἄπειρον παρ' ἐκείνην ὅταν γὰρ (272b.) ἐκείνη παρ' ἐκείνην, κἀκείνη παραλλάττει ἐκείνην, ὁμοίως κινουμένη καὶ ἀκίνητος πλὴν θᾶττον, ἐὰν κινῶνται ἀμφότεραι, ἀπολυθήσονται. Καίτοι γ' ἐνίοτ' οὐθὲν κωλύει τὴν κινουμένην παρ' ἠρεμοῦσαν θᾶττον παρελθεῖν ἢ τὴν ἀντικινουμένην, ἐάν τις ποιήσῃ τὰς μὲν ἀντικινουμένας ἀμφοτέρας φερομένας βραδέως, τὴν δὲ παρὰ τὴν ἠρεμοῦσαν πολλῷ ἐκείνων θᾶττον φερομένην. Οὐδὲν οὖν πρὸς τὸν λόγον ἐμπόδιον ὅτι παρ' ἠρεμοῦσαν, ἐπείπερ κινουμένην ἐνδέχεται τὴν Α παρὰ κινουμένην τὴν Β βραδύτερον παρελθεῖν. Εἰ οὖν ἄπειρος ὁ χρόνος ὃν ἡ πεπερασμένη ἀπολύεται κινουμένη, καὶ ἐν ᾧ ἡ ἄπειρος τὴν πεπερασμένην ἐκινήθη ἀνάγκη ἄπειρον εἶναι. Ἀδύνατον ἄρα τὸ ἄπειρον κινεῖσθαι ὅλον ἐὰν γὰρ καὶ τοὐλάχιστον κινηθῇ, ἀνάγκη ἄπειρον γίγνεσθαι χρόνον. Ἀλλὰ μὴν ὅ γ' οὐρανὸς περιέρχεται καὶ στρέφεται ὅλος κύκλῳ ἐν πεπερασμένῳ χρόνῳ, ὥστε περίεισιν ἅπασαν τὴν ἐντός, οἷον τὴν ΑΒ πεπερασμένην. Ἀδύνατον ἄρα ἄπειρον εἶναι τὸ κύκλῳ. 56 (3) That the infinite cannot move may also be shown as follows. Let A be a finite line moving past the finite line, B. Of necessity A will pass clear of B and B of A at the same moment; for each overlaps the other to precisely the same extent. Now if the two were both moving, and moving in contrary directions, they would pass clear of one another more rapidly; if one were still and the other moving past it, less rapidly; provided that the speed of the latter were the same in both cases. This, however, is clear: that it is impossible to traverse an infinite line in a finite time. Infinite time, then, would be required. (This we demonstrated above in the discussion of movement.) And it makes no difference whether a finite is passing by an infinite or an infinite by a finite. For when A is passing B, then B overlaps A and it makes no difference whether B is moved or unmoved, except that, if both move, they pass clear of one another more quickly. It is, however, quite possible that a moving line should in certain cases pass one which is stationary quicker than it passes one moving in an opposite direction. One has only to imagine the movement to be slow where both move and much faster where one is stationary. To suppose one line stationary, then, makes no difficulty for our argument, since it is quite possible for A to pass B at a slower rate when both are moving than when only one is. If, therefore, the time which the finite moving line takes to pass the other is infinite, then necessarily the time occupied by the motion of the infinite past the finite is also infinite. For the infinite to move at all is thus absolutely impossible; since the very smallest movement conceivable must take an infinity of time. Moreover the heavens certainly revolve, and they complete their circular orbit in a finite time; so that they pass round the whole extent of any line within their orbit, such as the finite line AB. The revolving body, therefore, cannot be infinite.
Praemissa prima ratione, quae procedebat ad ostendendum corpus non esse infinitum quod circulariter fertur, ex hoc quod distantia quae est inter duas lineas a centro egredientes erit infinita et impertransibilis, hic ponit secundam rationem, ex hoc quod lineae descriptae imaginatae in corpore infinito, sive in eius loco, non possunt se invicem intersecare. 102. After setting forth the first argument showing that the circularly moved body is not infinite, on the ground that the distance between two lines proceeding from the center will be infinite and untraversable, the Philosopher now presents the second argument, based on the fact that imaginary lines described in an infinite body, or in its place, cannot intersect.
Et praemittit in hac ratione quoddam principium, scilicet quod si a tempore finito subtrahatur tempus finitum, quod relinquitur necesse est esse finitum: quia pars finiti non potest esse infinita, alioquin totum esset minus sua parte. Et si illud residuum temporis est finitum, consequens est quod habeat principium: hoc enim tempus dicimus esse finitum, quod habet principium et finem. Demonstratum est autem in VI Physic. quod tempus et motus et mobile consequuntur se invicem in hoc quod est esse finitum vel infinitum. Unde si tempus mensurans incessum sive motum, est finitum et habens principium, necesse est quod motus sit finitus et quod habeat principium, et quod etiam magnitudo mota sit finita et habens principium. Et sicut hoc dicimus in motu caeli, similiter oportet se habere in aliis motibus et mobilibus. And in this argument he sets forth the principle that if a finite time is subtracted from a finite time, the remainder will be finite, because part of a finite cannot be infinite; otherwise the whole would be less than the part. And if that remainder of time is finite, it has a beginning, for we say a time is said to be finite, if it has a beginning and end. But it has been demonstrated in Physics VI that time and motion and mobile follow one another in respect to being finite or infinite. Hence if the time which measures a starting out or motion is finite and has a beginning, then the motion must be finite and have a beginning, and so also must be the magnitude moved. And just as this applies to celestial motion, so too to other motions and mobiles.
Istis igitur praemissis tanquam principiis, procedit ad demonstrandum propositum. Supponatur ergo quod a centro corporis infiniti quod est a, protrahatur quaedam linea, scilicet age, quae sit infinita ad aliam partem, scilicet ex parte e; et intelligatur ista linea circumvolvi secundum motum totius corporis, et quod secundum punctum g describat quendam circulum suo motu. Imaginemur etiam in spatio imaginato in quo revolvitur corpus infinitum, quandam lineam stantem immobilem, quae non transeat per centrum, sed sit infinita ex utraque parte, et sit linea bb. Si ergo, sicut dictum est, linea quae est age, sua incessione describat circulum a g, idest cuius semidiameter sit ag, continget quod linea age, circumeundo circulum praedictum, secabit totam lineam bb in tempore finito. Manifestum est enim quod semidiameter circuli non potest volvi in circuitu nisi incidat vel secet successive totam lineam immobilem imaginatam in circulo extra centrum. Et quod tempus sit finitum in quo linea quae educitur a centro, secet lineam infinitam quae describitur extra centrum, manifestat per hoc quod totum tempus in quo caelum movetur, est finitum, sicut patet ad sensum: unde consequens est quod pars illius temporis, quod aufertur a toto tempore, sit finita, in quo scilicet linea age incidit lineam bb. Vel potius sequitur illud tempus esse finitum, in quo illa linea incidens fertur usque ad lineam quae inciditur; et hoc oportet auferri a toto tempore finito, ut residui temporis accipiatur quoddam principium, secundum principium supra positum. Sequitur ergo quod sit aliquod principium temporis, in quo linea age incipit incidere lineam bb. Hoc autem est impossibile: quia, cum unam partem incidat ante aliam, si sit dare principium temporis in quo incipit incidere, esset dare principium aliquod in linea infinita, quod est contra rationem infiniti. Having set these things down as principles, he proceeds to demonstrate the proposition. Suppose that, from the center of an infinite body A, there is drawn the line AGE, which is infinite in one direction, namely, toward E; and let that line be revolving with the motion of the whole body, and that, with respect to the point G its motion describes a circle. Let us imagine also, in that imaginary space in which the infinite body is revolved, a certain immobile fixed line BB which does not cross the center but is nevertheless infinite. If then, as has been said, the line AGE by its motion describes a circle from G, i.e., whose radius is AG, it will turn out that the line AGE in making a revolution will cross the entire line BB in finite time. For it is manifest that the radius of a circle cannot be revolved in its circuit without covering or cutting successively the whole fixed immobile line imagined to be in the circle and not passing through the center. And that it is in a finite time that the line drawn from the center cuts the infinite line not passing through the center is manifest from the fact that the whole time in which the heaven is moved is finite, as is evident to our senses. Consequently, a part of that time which is subtracted from the whole time is finite, namely, the time in which AGE falls on line BB. Or rather it follows that that time is finite in which that cutting line is moved to the line which is cut; and this is the time that must be subtracted from all of finite time, so that the remaining time has a beginning, in keeping with the principle enunciated above. It follows, therefore, that the time in which AGE begins to cut BB has a beginning. However, this is impossible, because since it cuts one part before another, then if there is a beginning of the time in which it begins to cut, there would be a beginning in the infinite line, and that is contrary to the notion of infinite.
Sic ergo patet quod corpus infinitum non contingit revolvi circulariter. Unde si mundus sit infinitus, sequitur quod non moveatur circulariter. Videmus autem firmamentum moveri circulariter: non ergo est infinitum. In this way, then, it is plain that an infinite body cannot be revolved circularly. Hence if the world is infinite, it follows that it is not moved circularly. However, we do observe that the firmament is moved circularly. Hence it is not infinite.
Tertiam rationem ponit ibi: adhuc autem et ex his etc.: et sumitur haec ratio ex infinitate totius corporis quod ponitur circulariter moveri. Dicit ergo quod ex his etiam quae sequuntur, manifestum est quod impossibile est corpus infinitum moveri circulariter. Praemittit autem quod si sint duae lineae finitae, quarum una sit a et alia b, ita quod a feratur iuxta b quiescentem, ex necessitate sequitur quod simul linea mota quae est a, separetur a linea stante quae est b, et e contra linea stans quae est b, separetur a linea mota quae est a. Et huius ratio est, quia quantam partem una earum accipit de alia, tantam e converso alia accipit de ipsa. Sed tamen si ambae moveantur una contra aliam, velocius separabuntur lineae ab invicem; si autem una moveatur iuxta aliam quiescentem, tardius separabuntur lineae ab invicem; dummodo sit aequalis velocitas duarum motarum contra se invicem, et unius motae iuxta aliam stantem. Et hoc ideo praemisit, quia idem est tempus quo una linea pertransit aliam, et quo alia pertransit ipsam. 104. The third argument is given at [56] and is based on the infinity of the whole body which is posited as moving circularly. He says, therefore, that also from what follows it is manifestly impossible for an infinite body to be moved circularly. As a premise he says that if A and B are two finite lines so that A is in motion beside B which is stationary, it follows of necessity that as A moves along it departs from the stationary line B, and conversely that the stationary line B is separated from the moved line A. The reason for this is that each of them overlaps the other to the same extent. But now if both are moved in contrary directions, the lines will separate more quickly. If, however, one is in motion beside the other which is stationary, they will separate more slowly — provided, of course, that they have the same speeds when both are in a separating motion and when one alone is in motion. The reason for presenting this as a premise is that the time in which one line traverses the other is the same as that in which the other traverses it.
Et postquam hoc manifestavit per lineas finitas, applicat hoc ad lineas infinitas, de quibus intendit. Et dicit manifestum esse quod impossibile est lineam infinitam pertransiri tempore finito a linea finita; unde relinquitur quod linea finita pertranseat infinitam tempore infinito; quod quidem ostensum est prius in his quae de motu, idest in VI Physic. Sicut autem apparet ex his quae dicta sunt de lineis finitis, nihil differt quod linea finita moveatur per infinitam, et quod infinita moveatur super finitam: cum enim linea infinita moveatur per lineam finitam, similis ratio est si linea finita moveatur vel non moveatur; manifestum est autem quod si moveatur linea finita sicut et infinita, utraque earum pertransibit aliam. After manifesting this point with respect to finite lines, he applies it to the infinite lines he is discussing. And he says that it is manifestly impossible for an infinite line to be traversed by a finite line in finite time. Hence it remains that a finite line traverses an infinite line in infinite time, and this was shown previously in the treatise on motion, i.e., in Physics VI. But as appears from what has been said about finite lines, it makes no difference whether it is a finite line being moved through an infinite or an infinite line being moved over a finite, for when an infinite line is being moved through a finite line, the same reasoning holds, whether the finite line is being moved or not. However, it is manifest that if the finite line is being moved as well as the infinite, each traverses the other.
Unde manifestum est quod etiam si non moveatur linea finita, simile erit quod pertransitur a linea infinita, ac si pertransiret illam. Hence it is manifest that even if the finite line is not being moved, being traversed by the infinite line will be similar to traversing it.
Sed quia dixerat quod similiter se habet sive moveatur altera sive non, ostendit in quo circa hoc posset esse differentia: quia si utraque linearum moveatur una contra aliam, velocius separabuntur ab invicem. Sed hoc intelligendum est, si sit eadem velocitas, sicut supra dictum est: aliquando tamen nihil prohibet quin linea quae movetur iuxta quiescentem, velocius pertranseat eam, quam si moveretur iuxta lineam in contrarium motam; puta quando duae lineae quae contra se moverentur, haberent motum lentum, illa vero quae moveretur iuxta quiescentem, haberet motum velocem. Sic igitur patet quod nullum impedimentum est quantum ad rationem istam, quod linea infinita moveatur iuxta lineam finitam quietam: quia contingit quod linea mota quae est a, tardius pertransit lineam b motam, quam si non moveretur, dummodo ponatur quod, linea b quiescente, linea a velocius moveretur. But because he had said that the situation is similar whether the other is moved or not, he now shows wherein there could be a difference: if each of the lines is being moved in a contrary direction, they will separate more swiftly. But this must be understood if the speed is the same, as was said above. For sometimes nothing prevents the line which is being moved next to a stationary one from traversing it more quickly than if it were moved next to a line in contrary motion; for example, when the two lines in contrary motion would have a slow motion, while the one in motion next to the stationary one would have a swift motion. Accordingly, it is no obstacle, so far as the argument is concerned, that the infinite line be moved next to a stationary finite line — since it happens that the moving line A more slowly traverses the moving line B than if the latter were not in motion, provided, of course, that in this second case, while B is stationary, line A is being moved more swiftly.
Sic igitur ostenso quod nihil differt lineam infinitam moveri iuxta finitam quiescentem, ab eo quod linea finita moveretur supra infinitam, ex hoc argumentatur quod, si tempus quo linea finita pertransit lineam infinitam, est infinitum, consequens est quod tempus quo linea infinita movetur per lineam finitam, sit infinitum. Sic igitur patet quod impossibile est totum corpus infinitum moveri per totum spatium infinitum, in quo imaginamur motum eius, tempore scilicet finito: quia si infinitum moveretur etiam per minimum spatium finitum, sequeretur quod tempus esset infinitum: probatum est enim quod infinitum movetur per finitum tempore infinito, sicut et finitum per infinitum. Videmus autem quod caelum circuit totum spatium suum tempore finito. Unde manifestum est quod pertransit tempore finito aliquam lineam finitam, puta quae continet interius totum circulum descriptum circa centrum eius, scilicet lineam ab: quod non contingeret si esset infinitum. Impossibile est igitur corpus quod circulariter fertur, esse infinitum. 105. Thus, having shown that it makes no difference whether the infinite line is moved next to a stationary finite line, or whether the finite line is moved against the infinite, he argues from this that if the time in which the finite line traverses the infinite line is infinite, the consequence is that the time in which an infinite line is moved through a finite line is also infinite. Accordingly, it is plainly impossible for an entire infinite body to be moved through an entire infinite space — in which we imagine its motion to occur — in finite time: because if the infinite were moved even through the slightest finite space, it would follow that the time would be infinite, for it has been proved that the infinite is moved through the finite in infinite time, just as the finite through the infinite. But we observe that the heaven circles all its space in finite time. Hence it is manifest that it traverses some finite line in finite time, for example, the line containing within itself the whole circle described about its center, namely, the line AB. Now this would not happen if it were infinite. It is impossible, therefore, that the circularly moved body be infinite.

Lecture 11:

Three additional reasons why the body moving circularly cannot be infinite.

Chapter 5 cont.
Ἔτι ὥσπερ γραμμὴν ᾗ πέρας ἐστὶν ἀδύνατον εἶναι ἄπειρον, ἀλλ' εἴπερ, ἐπὶ μῆκος, καὶ ἐπίπεδον ὡσαύτως ᾗ πέρας οὐκ ἐνδέχεται ὅταν δ' ὁρισθῇ, οὐθαμῇ, οἷον τετράγωνον ἄπειρον ἢ κύκλον ἢ σφαῖραν, ὥσπερ οὐδὲ ποδιαίαν ἄπειρον. Εἰ οὖν μήτε σφαῖρα [μήτε τετράγωνον] μήτε κύκλος ἐστὶν ἄπειρος, μὴ ὄντος δὲ κύκλου οὐδ' ἂν ἡ κύκλῳ εἴη φορά, ὁμοίως δὲ μηδ' ἀπείρου ὄντος οὐκ ἂν εἴη ἄπειρος, εἰ μηδ' ὁ κύκλος ἄπειρός ἐστιν, οὐκ ἂν κινοῖτο κυκλικῶς ἄπειρον σῶμα. 57 (4) Again, as a line which has a limit cannot be infinite, or, if it is infinite, is so only in length, so a surface cannot be infinite in that respect in which it has a limit; or, indeed, if it is completely determinate, in any respect whatever. Whether it be a square or a circle or a sphere, it cannot be infinite, any more than a foot-rule can. There is then no such thing as an infinite sphere or square or circle, and where there is no circle there can be no circular movement, and similarly where there is no infinite at all there can be no infinite movement; and from this it follows that, an infinite circle being itself an impossibility, there can be no circular motion of an infinite body.
Ἔτι εἰ τὸ Γ κέντρον, ἡ δὲ τὸ ΑΒ ἄπειρος καὶ ἡ τὸ Ε πρὸς ὀρθὴν ἄπειρος καὶ ἡ τὸ ΓΔ κινουμένη, οὐδέποτ' ἀπολυθήσεται τῆς Ε, ἀλλ' ἀεὶ ἕξει ὥσπερ ἡ ΓΕ τέμνει γὰρ ᾗ τὸ Ζ. Οὐκ ἄρα περίεισι κύκλῳ ἡ ἄπειρος. 58 (5) Again, take a centre C, an infinite line, AB, another infinite line at right angles to it, E, and a moving radius, CD. CD will never cease contact with E, but the position will always be something like CE, CD cutting E at F. The infinite line, therefore, refuses to complete the circle.
Ἔτι εἴπερ ἄπειρος ὁ οὐρανός, κινεῖται δὲ κύκλῳ, ἐν πεπερασμένῳ χρόνῳ ἄπειρον ἔσται διεληλυθώς. Ἔστω γὰρ ὁ μὲν μένων οὐρανὸς ἄπειρος,ὁ δ' ἐν τούτῳ κινούμενος ἴσος. Ὥστ' εἴπερ περιελήλυθε κύκλῳ ἄπειρος ὤν, ἄπειρον τὸν ἴσον αὑτῷ διελήλυθεν ἐν πεπερασμέ— (273a.) νῳ χρόνῳ. Ἀλλὰ τοῦτ' ἦν ἀδύνατον. 59 (6) Again, if the heaven is infinite and moves in a circle, we shall have to admit that in a finite time it has traversed the infinite. For suppose the fixed heaven infinite, and that which moves within it equal to it. It results that when the infinite body has completed its revolution, it has traversed an infinite equal to itself in a finite time. But that we know to be impossible.
Ἔστι δὲ καὶ ἀντεστραμμένως εἰπεῖν, ὅτι εἰ πεπερασμένος ὁ χρόνος ἐν ᾧ περιεστράφη, καὶ τὸ μέγεθος ὃ διελήλυθεν ἀναγκαῖον εἶναι πεπερασμένον ἴσον δ' αὑτῷ διελήλυθεν πεπέρανται ἄρα καὶ αὐτός. Ὅτι μὲν οὖν τὸ κύκλῳ κινούμενον οὐκ ἔστιν ἀτελεύτητον οὐδ' ἄπειρον, ἀλλ' ἔχει τέλος, φανερόν. 60 (7) It can also be shown, conversely, that if the time of revolution is finite, the area traversed must also be finite; but the area traversed was equal to itself; therefore, it is itself finite. We have now shown that the body which moves in a circle is not endless or infinite, but has its limit.
Praemissis tribus rationibus ad probandum quod corpus quod circulariter movetur, non possit esse infinitum, hic ponit quartam, quae talis est. Impossibile est lineam esse infinitam, cuius est aliquis finis, nisi forte ad alteram partem habeat finem et ad alteram partem sit infinita. Et simile etiam est de superficie, quod si habeat finem ad unam partem, quod non contingit eam esse infinitam ad illam partem. Sed quando ad omnem partem determinatur, nullo modo potest esse infinita; sicut patet quod non contingit esse tetragonum, idest quadratum, infinitum, neque circulum, qui est superficialis figura, neque sphaeram, quae est figura corporea; haec enim sunt nomina figurarum, figura autem est quae termino vel terminis comprehenditur. Et sic patet quod nulla superficies figurata est infinita. Si ergo neque sphaera est infinita neque quadratum neque circulus, manifestum est quod non potest esse motus circularis infinitus. Sicut enim si non est circulus, non potest esse motus circularis, ita si non sit infinitus circulus, non potest esse infinitus motus circularis. Sed si corpus infinitum moveatur circulariter, necesse est motum circularem esse infinitum: non est ergo possibile quod corpus infinitum circulariter moveatur. 106. Having given three arguments to prove that the body in circular motion cannot be infinite, he now gives a fourth [47] which is this. It is impossible for a line having an end to be infinite, unless it have an end at one extremity and be infinite at the other. The same is true of a surface: if it has an end at one part, it is not infinite at that part. But when it is limited from every part, it is in no sense infinite. Thus, it is clear that no tetragon, i.e., square, is infinite, nor is a circle which is a plane figure, nor a sphere which is a solid — for these are names of figures, and a figure is something bounded by a terminus or by termini. Thus it is clear that no figured plane is infinite. If, therefore, neither a sphere nor a square nor a circle is infinite, it is clear that there cannot be circular motion that is infinite. For just as there can be no circular motion unless there is a circle, so, if there is no infinite circle, there cannot be an infinite circular motion. But if an infinite body were moved circularly, there would have to be a circular motion that is infinite. Therefore, it is not possible for an infinite body to be moved circularly.
Quintam rationem ponit ibi: adhuc autem si g etc., quae talis est. Supponatur quod corporis infiniti circulariter moti centrum sit g; ducatur autem per hoc centrum linea ad utramque partem infinita, quae sit linea ab; ducatur autem alia linea praeter centrum, cadens ad rectos angulos super lineam ba, in puncto scilicet e, et sit etiam haec linea infinita ex utraque parte; et hae duae lineae sint stantes, quasi imaginatae in spatio in quo corpus infinitum movetur circulariter. Sit etiam tertia linea egrediens a centro, quae sit linea dg, infinita ex parte d (nam ex parte g oportet eam esse finitam): haec autem linea moveatur per motum corporis, utpote in eo descripta. Quia igitur linea e est infinita, nunquam absolvetur, idest separabitur, ab ea: quia non potest eam pertransire, cum sit infinita, sed semper se habebit quemadmodum ge, idest semper continget vel secabit lineam e, sicut secabat eam in principio a quo incoepit moveri, puta quando linea gd superponebatur lineae ba et secabat lineam e perpendiculariter in puncto e. Recedens enim ab hoc situ incidet lineam e in puncto z, et sic semper in alio et alio puncto secabit illam: nunquam tamen totaliter poterit ab ea separari. Impossibile est autem quod motus circularis compleatur, nisi linea gd dimittat lineam e: quia oportebit, antequam compleatur motus circularis, quod linea gd pertranseat partem circuli quae est in opposito lineae e. Sic patet ergo quod linea infinita nullo modo potest circuire circulum, ita scilicet quod totus motus circularis compleatur. Et ita sequitur quod corpus infinitum non possit circulariter moveri. 107. The fifth argument is presented at [58] and it is this. Let G be the center of the infinite body in circular motion. Then through this center let a line AB be drawn which is infinite in both directions; then draw another infinite line not passing through the center but perpendicular to BA at E. Imagine these two lines as stationary in the space in which the infinite body is moved circularly. Draw a third line DG from the center and let it be infinite in the direction of D — for in the direction G it has to be finite. Finally suppose that this third line is in motion by the motion of the body. Because the line E is infinite, it will never be separated from it, because it cannot traverse it, since it is infinite; rather it will always maintain itself as GE, i.e., it will always touch or cut line E just as it cut it in the beginning when it began to be moved — for example, when the line GD was superimposed on the line BA and cut the line E perpendicularly at point E. For leaving this position it will cut the line E at the point Z, and so it will cut point after point in it; yet it will never be able to be entirely separated from it. It is impossible, however, for the circular motion to be completed, unless the line GD departs from the line E: because before the circular motion can be completed, the line GD will have to traverse that part of the whole that is opposite to the line E. And so it is plain that an infinite line can in no way traverse the circle in such a way that the entire circular motion be completed. Consequently, an infinite body cannot be moved circularly.
Sextam rationem ponit ibi: adhuc si quidem et cetera. Et hanc quidem rationem format dupliciter: primo ducendo ad impossibile hoc modo. Sit caelum infinitum, sicut tu ponis. Manifestum est autem ad sensum quod movetur circumquaque tempore finito: videmus enim eius revolutionem perfici in viginti quatuor horis. Ex hoc ergo sequetur quod infinitum sit pertransitum tempore finito: et hoc ideo, quia necesse est imaginari aliquod spatium aequale caelo, in quo caelum movetur. Hoc autem spatium imaginamur ut quiescens: sic igitur oportebit quod sit quoddam caelum manens infinitum, idest ipsum spatium in quo caelum movetur; et quod sit corpus caeli quod movetur in hoc spatio, aequale dicto spatio, quia oportet corpus aequari spatio in quo est. Si igitur caelum infinitum existens circulariter motum est tempore finito, consequens est quod pertransiverit infinitum tempore finito. Hoc autem est impossibile, scilicet infinitum pertransire tempore finito, ut probatum est in VI Physic. Impossibile est igitur quod corpus infinitum circulariter moveatur. The sixth argument is presented at [59] and he forms his argument in two ways. The first is by leading to an impossibility as follows: Suppose, as you say, that the heaven is infinite. Now it is manifest to us that it moves around in finite time — for we see that its revolution is completed in 24 hours. Therefore it will follow that the infinite is traversed in a finite time. This is so because it is necessary to imagine a space equal to the heaven in which the heaven is moved. But we imagine this space as stationary: thus there will have to be an infinite space in which the heaven is moved and a heavenly body equal to the space in which it is moved, because the body must be equal to the space in which it is. If, then, the infinite heaven has been circularly moved in finite time, the consequence is that it traversed the infinite in finite time. But this is impossible, i.e., to traverse the infinite in finite time, as was proved in Physics VI. It is, therefore, impossible for an infinite body to be moved circularly.
Secundo ibi: est autem et convertibiliter etc., format rationem e converso, ut sit probatio ostensiva. Et dicit quod possumus e converso dicere quod, ex quo tempus est finitum in quo caelum revolutum est, sicut ad sensum patet, consequens est quod magnitudo quae est pertransita, sit finita. Manifestum est autem quod spatium pertransitum est aequale ipsi corpori pertranseunti. Sequitur ergo corpus quod circulariter movetur, esse finitum. 109. Then at [60] he forms his argument conversely in order to make it an ostensive proof. And he says that we can say conversely that, from the fact that the time in which the heaven is revolved is finite (as is plain to the senses), it follows that the magnitude traversed is finite. Now it is plain that the space traversed is equal to the body traversing it. Therefore, the body which is moved circularly is finite.
Sic ergo epilogando concludit manifestum esse quod corpus quod circulariter movetur, non est interminatum, idest carens termino quasi infiguratum: et per consequens non est infinitum, sed habet finem. Therefore he concludes in summary that it is plain that the body which is being moved circularly is not unterminated, i.e., it does not lack a terminus as though it were devoid of shape. Consequently, it is not infinite, but has an ending.

Lecture 12:
Various reasons why a body moving in a straight line is not infinite.
Chapter 6
Ἀλλὰ μὴν οὐδὲ τὸ ἐπὶ τὸ μέσον οὐδὲ τὸ ἀπὸ τοῦ μέσου φερόμενον ἄπειρον ἔσται 61 Further, neither that which moves towards nor that which moves away from the centre can be infinite.
ἐναντίαι γὰρ αἱ φοραὶ ἡ ἄνω καὶ ἡ κάτω, αἱ δ' ἐναντίαι εἰς ἐναντίους τόπους. Τῶν δ' ἐναντίων εἰ θάτερον ὥρισται, καὶ θάτερον ὡρισμένον ἔσται. Τὸ δὲ μέσον ὥρισται εἰ γὰρ ὁποθενοῦν φέροιτο κάτω τὸ ὑφιστάμενον, οὐκ ἐνδέχεται πορρωτέρω διελθεῖν τοῦ μέσου. Ὡρισμένου οὖν τοῦ μέσου, καὶ τὸν ἄνω τόπον ἀνάγκη ὡρίσθαι. Εἰ δ' οἱ τόποι ὡρισμένοι καὶ πεπερασμένοι, καὶ τὰ σώματα ἔσται πεπερασμένα. 62 For the upward and downward motions are contraries and are therefore motions towards contrary places. But if one of a pair of contraries is determinate, the other must be determinate also. Now the centre is determined; for, from whatever point the body which sinks to the bottom starts its downward motion, it cannot go farther than the centre. The centre, therefore, being determinate, the upper place must also be determinate. But if these two places are determined and finite, the corresponding bodies must also be finite.
Ἔτι εἰ τὸ ἄνω καὶ τὸ κάτω ὥρισται, καὶ τὸ μεταξὺ ἀνάγκη ὡρίσθαι. Εἰ γὰρ μὴ ὥρισται, ἄπειρος ἂν εἴη ἡ κίνησις τοῦτο δ' ὅτι ἀδύνατον, δέδεικται πρότερον. Ὥρισται ἄρα τὸ μέσον, ὥστε καὶ τὸ ἐν τούτῳ σῶμα ἢ ὂν ἢ γενέσθαι δυνατόν. 63 Further, if up and down are determinate, the intermediate place is also necessarily determinate. For, if it is indeterminate, the movement within it will be infinite; and that we have already shown to be an impossibility. The middle region then is determinate, and consequently any body which either is in it, or might be in it, is determinate.
Ἀλλὰ μὴν τὸ ἄνω καὶ κάτω φερόμενον σῶμα δύναται ἐν τούτῳ γενέσθαι πέφυκε γὰρ τὸ μὲν ἀπὸ τοῦ μέσου κινεῖσθαι, τὸ δ' ἐπὶ τὸ μέσον. Ἔκ τε δὴ τούτων φανερὸν ὅτι οὐκ ἐνδέχεται σῶμα εἶναι ἄπειρον, 64 But the bodies which move up and down may be in it, since the one moves naturally away from the centre and the other towards it. From this alone it is clear that an infinite body is an impossibility;
καὶ πρὸς τούτοις εἰ βάρος μή ἐστιν ἄπειρον, οὐδ' ἂν τούτων τῶν σωμάτων οὐθὲν εἴη ἄπειρον ἀνάγκη γὰρ τοῦ ἀπείρου σώματος ἄπειρον εἶναι καὶ τὸ βάρος. (Ὁ δ' αὐτὸς λόγος ἔσται καὶ ἐπὶ τοῦ κούφου εἰ γάρ ἐστιν ἄπειρος βαρύτης, ἔστι καὶ κουφότης, ἐὰν ἄπειρον ᾖ τὸ ἐπιπολάζον). 65 but there is a further point. If there is no such thing as infinite weight, then it follows that none of these bodies can be infinite. For the supposed infinite body would have to be infinite in weight. (The same argument applies to lightness: for as the one supposition involves infinite weight, so the infinity of the body which rises to the surface involves infinite lightness.)
Δῆλον δ' ἐκ τῶνδε. Ἔστω γὰρ πεπερασμένον, καὶ εἰλήφθω τὸ μὲν ἄπειρον σῶμα ἐφ' ᾧ τὸ ΑΒ, τὸ δὲ βάρος αὐτοῦ ἐφ' ᾧ τὸ Γ. Ἀφῃρήσθω οὖν ἀπὸ τοῦ ἀπείρου πεπερασμένον μέγεθος ἐφ' ᾧ τὸ ΒΔ καὶ τὸ βάρος αὐτοῦ ἔστω ἐφ' ᾧ τὸ Ε. Τὸ δὴ Ε τοῦ Γ ἔλαττον ἔσται τὸ γὰρ τοῦ ἐλάττονος βάρος ἔλαττον. Καταμετρείτω δὴ τὸ ἔλαττον ὁποσακισοῦν, (273b.) καὶ ὡς τὸ βάρος τοὔλαττον πρὸς τὸ μεῖζον, τὸ ΒΔ πρὸς τὸ ΒΖ γεγενήσθω ἐνδέχεται γὰρ ἀφελεῖν τοῦ ἀπείρου ὁποσονοῦν. Εἰ τοίνυν ἀνάλογον τὰ μεγέθη τοῖς βάρεσι, τὸ δ' ἔλαττον βάρος τοῦ ἐλάττονός ἐστι μεγέθους, καὶ τὸ μεῖζον ἂν εἴη τοῦ μείζονος. Ἴσον ἄρα ἔσται τὸ τοῦ πεπερασμένου καὶ τὸ τοῦ ἀπείρου βάρος. 66 This is proved as follows. Assume the weight to be finite, and take an infinite body, AB, of the weight C. Subtract from the infinite body a finite mass, BD, the weight of which shall be E. E then is less than C, since it is the weight of a lesser mass. Suppose then that the smaller goes into the greater a certain number of times, and take BF bearing the same proportion to BD which the greater weight bears to the smaller. For you may subtract as much as you please from an infinite. If now the masses are proportionate to the weights, and the lesser weight is that of the lesser mass, the greater must be that of the greater. The weights, therefore, of the finite and of the infinite body are equal.
Ἔτι δ' εἰ τοῦ μείζονος σώματος μεῖζον τὸ βάρος, τὸ τοῦ ΗΒ μεῖζον ἔσται βάρος ἢ τὸ τοῦ ΖΒ, ὥστε τὸ τοῦ πεπερασμένου ἢ τὸ τοῦ ἀπείρου [μεῖζον ἔσται βάρος]. 67 Again, if the weight of a greater body is greater than that of a less, the weight of GB will be greater than that of FB; and thus the weight of the finite body is greater than that of the infinite.
Καὶ τὸ τῶν ἀνίσων δὲ μεγεθῶν ταὐτὸν ἔσται βάρος ἄνισον γὰρ τῷ πεπερασμένῳ τὸ ἄπειρον. 68 And, further, the weight of unequal masses will be the same, since the infinite and the finite cannot be equal.
Οὐθὲν δὲ διαφέρει τὰ βάρη σύμμετρα εἶναι ἢ ἀσύμμετρα καὶ γὰρ ἀσυμμέτρων ὄντων ὁ αὐτὸς ἔσται λόγος οἷον εἰ [τὸ Ε] τρίτον ὑπερβάλλει μετροῦν τὸ βάρος τῶν γὰρ ΒΔ μεγεθῶν τριῶν ὅλων ληφθέντων μεῖζον ἔσται τὸ βάρος ἢ τὸ ἐφ' ᾧ τὸ Γ. Ὥστε τὸ αὐτὸ ἔσται ἀδύνατον. 69 It does not matter whether the weights are commensurable or not. If (a) they are incommensurable the same reasoning holds. For instance, suppose E multiplied by three is rather more than C: the weight of three masses of the full size of BD will be greater than C. We thus arrive at the same impossibility as before.
Ἔτι δὲ καὶ ἐγχωρεῖ σύμμετρα λαβεῖν οὐδὲν γὰρ διαφέρει ἄρχεσθαι ἀπὸ τοῦ βάρους ἢ ἀπὸ τοῦ μεγέθους οἷον ἐὰν ληφθῇ σύμμετρον βάρος τῷ Γ τὸ ἐφ' ᾧ τὸ Ε, καὶ ἀπὸ τοῦ ἀπείρου ἀφαιρεθῇ τὸ ἔχον τὸ ἐφ' ᾧ Ε βάρος, οἷον τὸ ΒΔ, εἶτα ὡς τὸ βάρος πρὸς τὸ βάρος, τὸ ΒΔ πρὸς ἄλλο γένηται μέγεθος, οἷον πρὸς τὸ ΒΖ ἐνδέχεται γὰρ ἀπείρου ὄντος τοῦ μεγέθους ὁποσονοῦν ἀφαιρεθῆναι τούτων γὰρ ληφθέντων σύμμετρα ἔσται καὶ τὰ μεγέθη καὶ τὰ βάρη ἀλλήλοις. 70 Again (b) we may assume weights which are commensurate; for it makes no difference whether we begin with the weight or with the mass. For example, assume the weight E to be commensurate with C, and take from the infinite mass a part BD of weight E. Then let a mass BF be taken having the same proportion to BD which the two weights have to one another. (For the mass being infinite you may subtract from it as much as you please.) These assumed bodies will be commensurate in mass and in weight alike.
Οὐδὲ δὴ τὸ μέγεθος ὁμοιοβαρὲς εἶναι ἢ ἀνομοιοβαρὲς οὐδὲν διοίσει πρὸς τὴν ἀπόδειξιν ἀεὶ γὰρ ἔσται λαβεῖν ἰσοβαρῆ σώματα τῷ ΒΔ, ἀπὸ τοῦ ἀπείρου ὁποσαοῦν ἢ ἀφαιροῦντας ἢ προστιθέντας. Ὥστε δῆλον ἐκ τῶν εἰρημένων ὅτι οὐκ ἔσται τοῦ ἀπείρου σώματος πεπερασμένον τὸ βάρος. Ἄπειρον ἄρα. Εἰ τοίνυν τοῦτ' ἀδύνατον, καὶ τὸ ἄπειρόν τι εἶναι σῶμα ἀδύνατον. 71 Nor again does it make any difference to our demonstration whether the total mass has its weight equally or unequally distributed. For it must always be Possible to take from the infinite mass a body of equal weight to BD by diminishing or increasing the size of the section to the necessary extent. From what we have said, then, it is clear that the weight of the infinite body cannot be finite. It must then be infinite. We have therefore only to show this to be impossible in order to prove an infinite body impossible.
Ἀλλὰ μὴν ὅτι ἄπειρόν τι εἶναι βάρος ἀδύνατον, ἐκ τῶνδε φανερόν. 72 But the impossibility of infinite weight can be shown in the following way.
Εἰ γὰρ τοσόνδε βάρος τὴν τοσήνδε ἐν τῷδε τῷ χρόνῳ κινεῖται, τὸ τοσοῦτον καὶ ἔτι ἐν ἐλάττονι, 73 A given weight moves a given distance in a given time; a weight which is as great and more moves the same distance in a less time,
καὶ τὴν ἀναλογίαν ἣν τὰ βάρη ἔχει, οἱ χρόνοι ἀνάπαλιν (274a.) ἕξουσιν, οἷον εἰ τὸ ἥμισυ βάρος ἐν τῷδε, τὸ διπλάσιον ἐν ἡμίσει τούτου. 74 the times being in inverse proportion to the weights. For instance, if one weight is twice another, it will take half as long over a given movement.
Ἔτι τὸ πεπερασμένον βάρος ἅπασαν πεπερασμένην δίεισιν ἔν τινι χρόνῳ πεπερασμένῳ. 75 Further, a finite weight traverses any finite distance in a finite time.
Ἀνάγκη ἄρα ἐκ τούτων, εἴ τι ἔστιν ἄπειρον βάρος, κινεῖσθαι μὲν ᾗ τοσόνδε ὅσον τὸ πεπερασμένον καὶ ἔτι, μὴ κινεῖσθαι δέ, ᾗ ἀνάλογον μὲν δεῖ κατὰ τὰς ὑπεροχὰς κινεῖσθαι, ἐναντίως δὲ τὸ μεῖζον ἐν τῷ ἐλάττονι. Λόγος δ' οὐθείς ἐστι τοῦ ἀπείρου πρὸς τὸ πεπερασμένον, τοῦ δ' ἐλάττονος χρόνου πρὸς τὸν μείζω πεπερασμένον ἀλλ' ἀεὶ ἐν ἐλάττονι. Ἐλάχιστος δ' οὐκ ἔστιν. 76 It necessarily follows from this that infinite weight, if there is such a thing, being, on the one hand, as great and more than as great as the finite, will move accordingly, but being, on the other hand, compelled to move in a time inversely proportionate to its greatness, cannot move at all. The time should be less in proportion as the weight is greater. But there is no proportion between the infinite and the finite: proportion can only hold between a less and a greater finite time. And though you may say that the time of the movement can be continually diminished, yet there is no minimum.
Οὐδ' εἰ ἦν, ὄφελός τι ἂν ἦν ἄλλο γὰρ ἄν τι πεπερασμένον ἐλήφθη ἐν τῷ αὐτῷ λόγῳ, ἐν ᾧ τὸ ἄπειρον πρὸς ἕτερον, μεῖζον, ὥστ' ἐν ἴσῳ χρόνῳ τὴν ἴσην ἂν ἐκινεῖτο τὸ ἄπειρον τῷ πεπερασμένῳ. 77 Nor, if there were, would it help us. For some finite body could have been found greater than the given finite in the same proportion which is supposed to hold between the infinite and the given finite; so that an infinite and a finite weight must have traversed an equal distance in equal time. But that is impossible.
Ἀλλ' ἀδύνατον. Ἀλλὰ μὴν ἀνάγκη γε, εἴπερ ἐν ὁπηλικῳοῦν χρόνῳ πεπερασμένῳ δὲ κινεῖται τὸ ἄπειρον, καὶ ἄλλο ἐν τῷ αὐτῷ τούτῳ πεπερασμένον βάρος κινεῖσθαί τινα πεπερασμένην. 78 Again, whatever the time, so long as it is finite, in which the infinite performs the motion, a finite weight must necessarily move a certain finite distance in that same time.
Postquam philosophus ostendit quod corpus circulariter motum non est infinitum, hic ostendit idem de corpore quod movetur motu recto, vel a medio vel ad medium. 110. After showing that the circularly moved body is not infinite, the Philosopher here shows the same for the body moved with a straight motion, whether from the middle [center] or to the middle [center].

Et primo proponit quod intendit: dicens quod sicut corpus quod circulariter fertur non potest esse infinitum, ita corpus quod fertur motu recto, vel a medio vel ad medium, non potest esse infinitum.

Secundo ibi: contrariae enim lationes etc., ostendit propositum:

First he proposes what he intends and says that just as the circularly moved body cannot be infinite, so, too, the body which is moved with a straight motion, whether from the middle or to the middle, cannot be infinite;

Secondly, he shows the proposition, at 111, and this:

et primo ex parte locorum quae sunt huiusmodi corporibus propria;

secundo ex parte gravitatis et levitatis, per quae huiusmodi corpora in propria loca moventur, ibi: et adhuc si gravitas et cetera.

First on the part of the places which are proper to such bodies;

Secondly, on the part of heaviness and lightness, through which such bodies are moved to their proper places, at 114.

Circa primum duo facit: About the first he does two things:

primo ostendit propositum quantum ad corpora extrema, quorum unum est simpliciter grave, scilicet terra, et aliud simpliciter leve, scilicet ignis;

secundo quantum ad corpora media, quae sunt aer et aqua, ibi: adhuc si sursum et cetera.

First he shows the proposition as to the extreme bodies, of which one is absolutely heavy, namely, earth, and the other absolutely light, namely, fire, at 111;

Secondly, as to the intermediate bodies, which are air and water, 112.

Proponit ergo primo quod huiusmodi motus qui sunt sursum et deorsum, vel a medio et ad medium, sunt motus contrarii: contrarii autem motus locales sunt, qui sunt ad loca contraria, ut supra dictum est, et est ostensum in V Physic.: relinquitur ergo quod loca propria in quae feruntur huiusmodi corpora, sint contraria. 111. He proposes therefore first [62] that motions of the kind that are up and down, or from the middle and to the middle, are contrary motions. For contrary local motions are ones to contrary places, as has been said, and as was shown in Physics V. It remains, therefore, that the proper places to which such bodies are carried are contrary.
Ex hoc autem statim concludere posset huiusmodi loca esse determinata: contraria enim sunt quae maxime distant; maxima autem distantia locorum non potest esse nisi sint loca determinata, quia maxima distantia est qua non est alia maior, in infinitis autem semper est maiorem ac maiorem distantiam accipere; unde si loca essent infinita, cessaret locorum contrarietas. Sed Aristoteles, praetermissa hac probatione tanquam manifesta, procedit per alium modum. Verum est enim quod, si unum contrariorum est determinatum, quod aliud erit determinatum, eo quod contraria sunt unius generis. Medium autem mundi, quod est medius terminus motus deorsum, est determinatum: ex quacumque enim parte caeli aliquid feratur deorsum (quod scilicet substat superiori parti quae est versus caelum), non continget longius pertransire recedendo a caelo quam quod perveniat ad medium: si enim pertransiret medium, iam fieret propinquius caelo, et sic moveretur sursum. Sic igitur patet quod medius locus est determinatus. Patet etiam ex praedictis quod, determinato medio, quod est locus deorsum, necesse est et determinatum esse locum qui est sursum, cum sint contraria. Si autem ambo loca sunt determinata et finita, necesse est quod corpora quae sunt nata esse in his locis, sint finita. Unde patet huiusmodi corpora extrema, quae moventur motu recto, esse finita. Now, we could have at once concluded from this that such places are determinate: for contraries are things which are most distant; but places that are the greatest distance apart are determinate, for the greatest distance is such that none is greater, whereas in infinites a greater and greater distance is always possible. Hence if the places were infinite, contrariety of places would cease. However, Aristotle passes over this argument as manifest and proceeds by another tack. For it is true that if one contrary is determinate, so too the other, because contraries are members of one genus. But the middle of the world which is the midway terminus of a downward motion is determinate — for from whatever part of the heavens something is moved downward (which exists under the upper part facing the heavens) it can travel no farther in its journey from the heavens than the middle: for if it should go beyond the middle, it would now get closer to the heavens and thus would be moved upward. Accordingly, it is clear that the middle place is determinate. It is likewise clear from the aforesaid that the middle having been determined, i.e., the downward place, then the upward place is also necessarily determinate, because they are contraries. And if both are determinate, then the bodies which are apt to be in these places must be finite. Hence it is clear that the extreme bodies subject to straight motion are finite.
Deinde cum dicit: adhuc si sursum etc., ostendit idem quantum ad media corpora. 112. Then at 13 [63] he shows the same thing for the intermediate bodies.
Et primo proponit quandam conditionalem, scilicet quod, si sursum et deorsum sunt determinata, necesse est quod locus intermedius sit determinatus. Et hoc probat duplici ratione. Quarum prima est: si, primis existentibus determinatis, medium non sit determinatum, sequetur quod motus qui est ab uno extremo in aliud, sit infinitus, utpote medio existente infinito. Quod autem hoc sit impossibile, ostensum est prius in his quae dicta sunt de motu circulari, ubi ostensum est quod motus qui est per infinitum, non potest compleri. Sic ergo patet quod locus medius est determinatus. Et ita, cum locatum commensuretur loco, consequens est quod corpus sit finitum quod actu existit in hoc loco, vel quod potest ibi existere. First he proposes a conditional, namely, that if up and down are determinate, the intermediate place must be determinate. And he proves this with two arguments, the first of which is this: If, when the extremes were determinate, the intermediate should not be determinate, it would follow that a motion from one extreme to the other would be infinite, on account of the infinite intermediate. But that this is impossible has been shown previously in the discussion about circular motion where it was pointed out that motion through the infinite cannot be completed. Consequently, the intermediate place is determinate. Thus, since the thing in place is commensurate with the place, it follows that the body actually existing in this place or that can exist there, is finite.
Secundam rationem ponit ibi: sed et adhuc etc.: quae talis est. Corpus quod fertur sursum vel deorsum, potest pervenire ad hoc quod sit factum existens in loco tali. Quod quidem patet per hoc quod tale corpus natum est moveri a medio vel ad medium, idest habet naturalem inclinationem ad hunc vel illum locum; naturalis autem inclinatio non potest esse frustra, quia Deus et natura nihil frustra faciunt, ut supra habitum est. Sic igitur omne quod movetur naturaliter sursum vel deorsum, potest motus eius terminari ad hoc quod sit sursum vel deorsum. Sed hoc non posset esse si locus medius esset infinitus. Est ergo locus medius finitus, et corpus in eo existens finitum. 113. He gives the second argument at [64] and it is this: A body that is moved up or down can reach the state of existing in such a place. This is clear from the fact that such a body is apt to be moved from the middle or to the middle, i.e., it has a natural inclination to this or that place. Now a natural inclination cannot be in vain, because God and nature do nothing in vain, as was had above. Consequently whatever is naturally moved upward or downward can have its own motion terminated so as to be up or down. But this could not be, if the intermediate place were infinite. Consequently, the intermediate place is finite; so, too, is the body existing in it.
Ex praemissis igitur epilogando concludit, manifestum esse quod non contingit aliquod corpus esse infinitum. Therefore in summary he concludes from the foregoing that it is clear that no body can be infinite.
Deinde cum dicit: et adhuc si gravitas etc., ostendit non esse corpus grave vel leve infinitum, ratione sumpta ex gravitate vel levitate: quae talis est. Si est corpus grave vel leve infinitum, necesse est quod sit gravitas vel levitas infinita: sed hoc est impossibile: ergo et primum. Then at [65] he shows that there is no infinite heavy or light body by an argument based on heaviness or lightness. It is this: If a heavy or a light body be infinite, then heaviness or lightness must be infinite. But this is impossible. Therefore, the first supposition [of non-infinity] must be true.
Circa hoc ergo duo facit: With respect to this, then, he does two things:

primo probat conditionalem;

secundo probat destructionem consequentis, ibi: sed adhuc quoniam infinitam et cetera.

First he proves the conditional proposition, at 114;

Secondly, he proves the destruction of the consequent, at 119.

Circa primum duo facit. As to the first he does two things:
Primo proponit quod intendit, dicens: si non est gravitas infinita, nullum erit corporum horum, scilicet gravium, infinitum: et hoc ideo, quia necesse est infiniti corporis infinitam esse gravitatem. Et eadem ratio est de corpore levi: quia si infinita est gravitas corporis gravis, necesse est quod etiam levitas sit infinita, si supponatur corpus leve, quod sursum fertur, esse infinitum. First he proposes what he intends and says[65]: If there is no infinite heaviness, none of these, i.e., no heavy body, will be infinite, for the heaviness of an infinite body must be infinite. And the same goes for a light body — for if the heaviness of a heavy body is infinite, the lightness, too, must be infinite, if one supposes some light body carried upward to be infinite.
Secundo ibi: palam autem etc., probat quod supposuerat: Secondly, at [66] he proves what he had supposed.

et primo ponit probationem;

secundo excludit obviationes quasdam, ibi: nihil autem differt gravitates et cetera.

First he presents the proof, at 115;

Secondly, he dismisses some objections, at 116.

Ponit ergo primo rationem ducentem ad impossibile, quae talis est. Si non est verum quod supra dictum est, supponatur quod corporis infiniti sit gravitas finita: et sit corpus infinitum ab, gravitas autem eius finita sit g. A corpore igitur infinito praedicto auferatur aliqua pars eius finita quae est magnitudo bd, quam necesse est esse multo minorem toto corpore infinito. Minoris autem corporis minor est gravitas: sic ergo gravitas corporis bd est minor quam sit gravitas g, quae est gravitas totius corporis infiniti; et sit ista minor gravitas e. Haec autem minor gravitas, scilicet e, mensuret maiorem gravitatem finitam quae est g, quotiescumque, idest secundum quemcumque numerum, puta secundum tria, ut scilicet dicatur quod e est tertia pars totius g. Accipiatur autem a corpore infinito aliqua pars, quae superaddatur corpori finito bd, secundum proportionem qua g excedit e, et hoc corpus excedens sit bz; ita scilicet quod, sicut gravitas minor quae est e se habet ad maiorem quae est g, ita corpus bd se habeat ad bz. Et quod hoc fieri possit, probat quia a corpore infinito potest auferri quantumcumque oportuerit; eo quod, sicut dicitur in III Physic., infinitum est cuius quantitatem accipientibus semper est aliquid extra accipere. First, then, he presents an argument leading to an impossibility [66] and it is this: If what was said above is not true, then suppose that the heaviness of an infinite body is finite, and let AB be the infinite body and G its finite heaviness. From this infinite body take away a finite part which is the magnitude BD, which is necessarily much less than the whole infinite body. Now the heaviness of this smaller body is less; consequently, the heaviness of BD is less than the heaviness G which is the heaviness of the whole infinite body body. Let this lesser heaviness be E. Now let E be a measure of the greater but finite heaviness G —for example, E is a third part of the whole G. Now take from the infinite body a part to be added to the finite body BD, according to the proportion by which G exceeds E, and let this exceeding body be BZ, in such a way that the ratio between the lesser heaviness E to the greater G is the same as that between the body BD and body BZ. That this can be done is proved by the fact that from an infinite body can be taken away as much as is needed, since, as is said in Physics II, the infinite is that whose quantity is such that, as much as is taken away, there always remains something beyond to be taken.
His igitur praesuppositis, argumentatur ducendo ad tria inconvenientia: primo quidem sic. Eadem est proportio magnitudinum gravium, quae est ipsarum gravitatum: videmus enim quod minor gravitas est minoris magnitudinis, et maior maioris. Sed quae est proportio e ad g, minoris scilicet gravitatis ad maiorem, eadem est proportio bd ad bz, minoris scilicet corporis ad maius, ut suppositum est: cum igitur e sit gravitas corporis bd, sequetur quod g sit gravitas corporis bz. Supponebatur autem quod esset gravitas totius corporis infiniti: ergo aequalis numero eadem erit gravitas corporis finiti et infiniti. Quod est inconveniens, quia sequetur quod totum residuum corporis infiniti nihil habeat gravitatis. Ergo et primum est impossibile, scilicet quod corporis infiniti sit gravitas finita. Therefore, with these presuppositions, he now argues to three incompatible consequences. First he reasons in this manner. The ratio of heavy magnitudes is the same as the ratio of their heaviness — for we see that a larger body has more heaviness and a smaller body less. But the ratio of E to G, i.e., of the lesser to the more heavy is the same as that of BD to BZ, i.e., of the smaller body to the larger, was was supposed. Therefore, since E is the heaviness of BD, it will follow that G is the heaviness of the body BZ. But G was assumed to be the heaviness of the whole infinite body. Therefore the numerical value of the heaviness of the finite and of the infinite body will be the same. But this is unacceptable, because it will follow that the whole remainder of the infinity body will have no heaviness. Therefore, the first is impossible, namely, that the heaviness of an infinite body be finite.
Secundo ibi: adhuc autem si maioris etc., ducit ad aliud inconveniens. Quia enim a corpore infinito potest accipi quantumcumque quis voluerit, ut dictum est, accipiatur adhuc aliqua pars corporis infiniti, quae superaddatur corpori bz, et sit unum corpus bi finitum maius corpore finito quod est bz. Maioris autem corporis maior est gravitas, ut supra dictum est: ergo gravitas corporis bi est maior quam gravitas g, quae concludebatur gravitas esse corporis bz. Sed primo supponebatur quod g erat gravitas totius corporis infiniti. Ergo gravitas corporis finiti erit maior quam gravitas corporis infiniti, quod est impossibile. Ergo et primum, scilicet quod gravitas corporis infiniti sit finita. Secondly, at [67] he leads to another unacceptable consequence. For since it is possible to take from an infinite body as much as one wishes, as has been said, let yet another part be taken from the infinite body and added to the body BZ. And let G be one finite body greater than the finite body BZ. Now the heaviness of larger body is greater, as was said above. Therefore, the heaviness of BI is greater than the heaviness G which was proved to be the heaviness of the body BZ. But it was assumed in the beginning that G was the heaviness of the whole infinite body. Therefore, the heaviness of a finite body will be greater than that of an infinite body. This is impossible. Therefore, the first is impossible, namely, that the heaviness of an infinite body be finite.
Tertio ibi: et inaequalium etc., ducit ad tertium inconveniens, scilicet quod inaequalium magnitudinum sit eadem gravitas. Quod manifeste sequitur ex praemissis, quia infinitum est inaequale finito, cum sit maius eo. Unde, cum haec sint impossibilia, impossibile est corporis infiniti esse gravitatem finitam. Thirdly, at [68] he leads to the third incompatibility, namely, that the heaviness of unequal magnitudes would be the same. This clearly follows from the foregoing, because the infinite is not equal to the finite, because it is greater than it. Hence, since these conclusions are impossible, it is impossible for the heaviness of an infinite body to be finite.
Deinde cum dicit: nihil autem differt etc., excludit duas obviationes contra praemissam rationem: 116. Then at 14 [69] he dismisses two objections against the foregoing argument:

primo primam;

secundo secundam, ibi: nec utique magnitudinem et cetera.

First, the first;

Secondly, the second, at 118.

Prima autem obviatio est, quia supposuerat in praecedenti ratione quod gravitas minor quae est e, mensuret secundum aliquem numerum gravitatem maiorem quae est g: quod quidem aliquis posset negare: non enim omne maius mensuratur a minori, quia linea trium palmarum non mensurat lineam octo palmarum. The first objection is that he had supposed in the preceding argument that the lesser heaviness E is a numerical measure of the greater heaviness G. Now this can be denied, for not every greater is measured by a smaller, because a line of 3 hands' length is not a measure of a line of hands' length.
Hanc autem obviationem excludit philosophus dupliciter. Primo quidem quia nihil differt ad propositum utrum duae praedictae gravitates, scilicet maior et minor, sint commensuratae, ita scilicet quod minor mensuret maiorem; vel incommensuratae, scilicet quod minor maiorem non mensuret: eadem enim ratio sequitur utrobique. Necesse est enim quod minus aliquoties sumptum aut mensuret maius aut excedat ipsum; sicut binarius ter sumptus mensurat senarium (ter enim duo sunt sex), quinarium autem non mensurat sed excedit. Sic igitur, si gravitas e non mensuret gravitatem g, sit ita quod ter sumpta mensuret quandam maiorem gravitatem, quae excedit gravitatem g. Et ex hoc sequitur inconveniens sicut prius. Quia si assumpserimus ex corpore infinito tres magnitudines secundum quantitatem bd, magnitudinis ex his tribus compositae erit tripla gravitas gravitatis e, quae ponitur esse gravitas corporis bd. Gravitas autem tripla ad e est maior secundum praedicta quam gravitas g, quae est gravitas corporis infiniti. Quare sequitur idem impossibile quod prius, scilicet quod maior sit gravitas corporis finiti quam infiniti. But the Philosopher excludes this objection in two ways. First, because it makes no difference, so far as the conclusion is concerned, whether the two heavinesses, namely, the greater and the less, in question are commensurate, so that the less measures the greater, or not, for the same reasoning holds in either case. For it is necessary that the lesser, taken a certain number of times, either measure or exceed the greater: for example, the product of 2 taken 3 times measures 6 [for 3 times 2 equals 6], while it does not measure 5, but exceeds it. Accordingly, if the heaviness E does not measure the heaviness G, let E be such that 3 times E measures a heaviness greater than the heaviness G. And so in this case the same impossibility as before results, because, if we had taken from the infinite body three magnitudes of quantity BD, the heaviness of such a magnitude will be 3 times that of heaviness E, which is assumed to be the heaviness of the body BD. But a heaviness that is 3 times E is greater (according to our assumptions) than the heaviness G which is the heaviness of the infinite body. Wherefore, the same impossibility as before follows, namely, that the heaviness of a finite body exceed that of an infinite body.
Secundo ibi: adhuc autem etiam contingit etc., excludit eandem obviationem alio modo. Et dicit quod possumus sumere in demonstratione praedicta quod duae gravitates sint commensuratae, ita scilicet quod e commensuret g. Supra enim primo sumpta est magnitudinis pars, scilicet bd, cuius gravitatem diximus esse e: et ideo dici poterat quod e non mensurat g. Nihil autem differt ad propositum utrum incipiamus a gravitate, accipiendo partem eius quamcumque volumus, aut a magnitudine sic sumpta; puta si, incipiendo a gravitate, sumatur quaedam pars eius, scilicet e, quae mensuret totum, scilicet g; et consequenter ab infinito corpore accipiamus aliquam partem, scilicet bd, cuius gravitas sit e; et deinde procedamus ut supra, ut scilicet sicut se habet gravitas e ad gravitatem g, ita se habeat magnitudo bd ad aliam magnitudinem maiorem quae est bz. Et hoc ideo, quia ex quo magnitudo totius corporis est infinita, contingit auferri ex ea quantumcumque placuerit. Hoc igitur modo sumptis partibus gravitatis et magnitudinis, sequetur quod et magnitudines et gravitates erunt invicem commensuratae; ita scilicet quod minor gravitas mensurabit maiorem, et similiter minor magnitudo maiorem. 117. Secondly, at [70] he excludes the same objection in another way. And he says that we can assume in the demonstration under discussion that the two heavinesses are commensurate, in such a way that E is commensurate to G. For above we first took from the magnitude a part BD whose heaviness we called E, and this was grounds for saying that E does not measure G. But it makes no difference, so far as the proposition is concerned, whether we begin with the heaviness (by taking any part we want) or with the magnitude so taken. For example, we might begin with the heaviness and take a part of it, namely, E, which measures the whole, namely, G, and then we can take from the infinite body a part BD whose heaviness is E and proceed as above, so that as the heaviness E is to heaviness G, so the magnitude BD is to a greater magnitude BZ. This we can do, because, since the magnitude of the whole body is infinite, as much can be taken from it as we please. By taking the parts of the heaviness and of the magnitude in this way, it will follow that the magnitudes and the heavinesses will be mutually commensurate, i.e., the lesser heaviness will measure the greater, and the smaller magnitude the larger.
Deinde cum dicit: nec utique magnitudinem etc., excludit secundam obviationem. Supposuerat enim esse magnitudines proportionales gravitatibus. Quod quidem necesse est in corpore similium partium; cum enim sit undique per totum similis gravitatis, necesse est quod in maiori parte sit maior gravitas: sed in corpore dissimilium partium hoc non est necesse, quia potest esse quod gravitas minoris partis excedat gravitatem maioris, sicut minor pars terrae est gravior maiori parte aquae. 118. Then at [71] he excludes a second objection. For he had supposed that the magnitudes are proportional to the heavinesses. Now this is true in bodies having similar parts, for where there is like heaviness throughout the whole, there must be more heaviness in the larger part. But in a body of unlike parts this is not necessarily so, because the heaviness of a smaller part could be greater than that of a larger part, just as a smaller part of earth is heavier than a larger part of water.
Hanc ergo obviationem excludit, dicens quod nihil differt ad demonstrationem praemissam utrum magnitudo infinita de qua loquimur, quantum ad gravitatem sit homoeomera, idest similium partium, vel anomoeomera, idest dissimilium partium. Quia a corpore infinito possumus sumere quantumcumque voluerimus, vel apponendo vel subtrahendo; ita quod accipiamus aliquas partes habere aequalem gravitatem parti primo sumptae, scilicet bd, sive illae partes posterius assumptae sint maiores in magnitudine sive minores. Si enim primo acceperimus quod bd sit tricubitum, habens gravitatem e; et accipiamus alias multas partes, puta decem cubitorum, habentes aequalem gravitatem; idem erit ac si sumeretur alia pars aequalis habens aequalem gravitatem. Sic igitur sequitur idem inconveniens. This objection he therefore excludes by saying that it makes no difference to the aforesaid demonstration whether the infinite magnitude in question is homogeneous, i.e., of similar parts, or heterogeneous, i.e., of dissimilar parts. For from an infinite body we can take as much as we wish, either adding or subtracting; hence we can assume certain parts to have a heaviness equal to the part taken first, namely BD, whether the parts taken later are larger or smaller in magnitude. For if we should first take BD as having 3 Cubits and having heaviness E, and then take many other parts, for example, of 10 cubits, to make an equal heaviness, it will be the same as if we had taken another equal part having the same heaviness. Consequently, the same impossibility follows.
Praemissa igitur demonstratione, et exclusis obviationibus, concludit ex dictis quod infiniti corporis non potest esse finita gravitas. Relinquitur ergo quod sit infinita. Si ergo impossibile est esse gravitatem infinitam, ut statim probabit, consequens est quod impossibile sit esse aliquod corpus infinitum. Therefore, having presented his demonstration and excluded the objections thereto, he concludes from the foregoing that the heaviness of an infinite body cannot be finite. Therefore, it must be infinite. If then, as he will immediately prove, infinite heaviness is impossible, the consequence is that it is impossible for there to be an infinite body.
Deinde cum dicit: sed adhuc quoniam infinitam etc., ostendit quod supposuerat, scilicet quod non possit esse gravitas infinita: et in hoc destruit consequens praemissae conditionalis. Circa hoc autem duo facit. 119. Then at [72] he proves what he had supposed, namely, that there cannot be infinite heaviness. And in this he destroys the consequent of the previously posited conditional. Concerning this he does two things:

Primo proponit quod intendit: et dicit quod adhuc oportet manifestare ex his quae subsequuntur, quod impossibile sit gravitatem infinitam esse.

Secundo ibi: si enim tanta etc., probat propositum.

First he proposes what he intends, and says that we must still show from what will follow that infinite heaviness is impossible.

Secondly, at [73] he proves the proposition.

Et primo praemittit quasdam suppositiones;

secundo ex his argumentatur ad propositum, ibi: necesse igitur ex his etc.;

tertio excludit quandam obiectionem, ibi: neque si esset et cetera.

First he lays down certain presuppositions;

Secondly, he uses them in his argument, at 121;

Thirdly, he excludes an objection, at 122.

Ponit autem primo tres suppositiones. Quarum prima est quod, si gravitas tanta, idest alicuius determinatae mensurae, movet tantam, idest per determinatam magnitudinem spatii, in hoc tempore, scilicet determinato, necesse est quod tanta et adhuc, idest quod gravitas maior quae habet tantam quantam minor et adhuc amplius, moveat per tantam magnitudinem spatii in minori tempore: quia quanto virtus motiva est fortior, tanto motus eius est velocior, et ita pertransit aequale spatium in minori tempore, ut probatum est in VI Physic. First, then, he presents three suppositions. The first of these [73] is that, if such a heaviness, i.e., of some certain amount, moves so much, i.e., throughout a definite magnitude of space, in this time, i.e., a determined time, then necessarily as much and more, i.e., a greater heaviness that has as much and something more than a lesser, will move through as great a magnitude in less time — for by as much as a moving power is stronger, by that much is its motion swifter. Consequently, it will traverse an equal distance in less time, as is proved in Physics VI.
Secundam suppositionem ponit ibi: et analogiam etc.: et haec sequitur ex prima. Si enim maior gravitas movet in minori tempore, consequens est quod eadem sit analogia, idest proportio, gravitatum et temporum, tamen e converso; ita scilicet quod, si media gravitas movet in tanto tempore, duplum gravitatis movet in medietate eius, scilicet temporis. The second supposition is at [74] and follows from the first. For if a greater heaviness moves in less time, then the analogy, i.e., proportion between heavinesses and times is the same, but inversely so, i.e., if half the heaviness moves something in a certain amount of time, then double that amount moves in its half, i.e., in half the time.
Tertiam suppositionem ponit ibi: adhuc finita et cetera. Et dicit quod finita gravitas movet per finitam magnitudinem spatii in quodam tempore finito. The third supposition, at [75], states that a finite heaviness moves through a finite magnitude of space in a certain finite time.
Deinde cum dicit: necesse igitur ex his etc., argumentatur ex praemissis. Si enim sit gravitas infinita, sequentur duo contradictoria; scilicet quod aliquid moveatur secundum eam, et quod non moveatur. Quod moveatur quidem, sequitur ex prima suppositione; quia, si tanta gravitas movet in tanto tempore, maior movebit velocius, scilicet in minori tempore. Quia ergo infinita gravitas est maior quam finita, si finita movet secundum determinatum tempus per determinatum spatium, ut tertia suppositio dicebat, consequens est quod infinita moveat tantum et adhuc amplius, idest vel per maius spatium in aequali tempore, vel per aequale spatium in minori tempore, quod est velocius moveri. 121. Then at [76] he argues from these premises. If an infinite heaviness should exist, two contradictories will follow: namely, that something would be moved according to it, and not moved. That it would be moved follows, indeed, from the first supposition — for if a certain heaviness moves in a certain amount of time, a greater will move more swiftly, i.e., in less time. Since, therefore, an infinite weight is greater than a finite, then, if a finite moves through a definite distance in a definite time, as the third supposition says, the consequence is that an infinite heaviness will move as much and more, i.e., either through a greater distance in the same time, or through an equal distance in less time, which is to be moved more swiftly.
Sed quod aliquid non moveatur secundum infinitam gravitatem, sequitur ex secunda suppositione. Oportet enim proportionaliter aliquid moveri secundum excellentias gravitatis e contrario, scilicet quod maior gravitas moveat in minori tempore. Nulla autem proportio potest esse infinitae gravitatis ad finitam: minoris autem temporis ad maius, dummodo sit finitum, est aliqua proportio. Sic igitur non erit aliquod tempus dare in quo infinita gravitas moveat; sed semper erit accipere aliquid moveri in minori tempore quam sit tempus in quo movet gravitas infinita; non est autem dare minimum tempus in quo gravitas infinita moveat, ita quod possit dici quod non potest aliquid in minori tempore moveri. Ideo autem non est minimum tempus accipere, quia, cum omne tempus sit divisibile, sicut et quodlibet continuum, quolibet tempore est accipere aliquod minus, partem scilicet temporis divisi. Sic igitur non potest esse gravitas infinita. But that something is not moved according to infinite heaviness follows from the second supposition. For a thing must be moved in proportion to the greatness of the weight in inverse proportion, i.e., the greater weight will move in less time. But there can be no proportion between an infinite and a finite weight, although there is a proportion between less time and more time, provided the time is finite. Consequently, there can be no time given in which an infinite weight can move, but something will always be able to be taken as moved in less time than the time in which an infinite weight moves, for there can be taken no least time in which an infinite weight can move in the sense that it would be impossible for something to be moved in a lesser time. Now the reason why no such least time can be assumed is that since all time is divisible, as is any continuum, it is always possible to take a time smaller than any given time, i.e., a part of the divided time. Consequently, an infinite heaviness cannot exist.
Deinde cum dicit: neque si esset etc., excludit quandam obviationem. Posset enim aliquis dicere aliquod esse minimum tempus, scilicet indivisibile, in quo movet gravitas infinita; sicut et quidam posuerunt aliquas magnitudines esse minimas et indivisibiles. Sed hanc obviationem excludit: 122. Then at [77] he excludes an objection. For someone could say that there is a least time, namely, an indivisible time, in which the infinite heaviness moves, just as some have posited certain minimum and indivisible magnitudes. But he excludes this objection:

et primo ostendit quod inconveniens sequatur si ponatur minimum tempus, et quod in hoc infinita gravitas movet;

secundo ostendit idem inconveniens sequi si in quocumque tempore, etiam non minimo, infinita gravitas moveat, ibi: sed adhuc necesse et cetera.

First he shows that an impossibility follows upon assuming a minimum time and that an infinite heaviness moves in that time;

Secondly, he shows that the same impossibility follows if an infinite heaviness should move in any amount of time, even not the minimum, at 123.

Dicit ergo primo quod, etiam si esset tempus minimum, nulla utilitas ex hoc esset ponenti gravitatem infinitam, ad vitandum inconveniens. Quamvis enim ponamus minimum tempus, non tamen excludimus quin sit aliqua proportio huius minimi temporis ad tempus maius, eo quod hoc tempus minimum erit pars maioris temporis; sicut unitas est pars numeri, unde est aliqua proportio eius ad omnem numerum. Illud autem indivisibile non habet proportionem ad divisibile, quod non est pars eius; sicut punctum non est pars lineae, et ideo non est aliqua proportio puncti ad lineam. Accipiatur ergo alia gravitas finita e contrario, tanto maior gravitate finita quae movebat in maiori tempore quam gravitas infinita, in qua proportione tempus minimum gravitatis infinitae se habet ad tempus maius alterius gravitatis finitae. Puta, sit gravitas infinita e, tempus minimum in quo movet b, gravitas autem finita g, quae movet in maiori tempore quam b, scilicet in tempore d: accipiatur ergo alia gravitas tanto maior quam g, in qua proportione d excedit b, et sit haec gravitas f. Sic ergo, cum minoratio temporis sit secundum additionem gravitatis, sequetur quod gravitas f, quae est finita, moveat in eodem tempore cum gravitate infinita: quod est impossibile. He says therefore first [77] that even if there were a minimum time, it would not help in escaping the impossibility that follows from the assumption of infinite heaviness. For although we posit a minimum time, we do not exclude a ratio of this time to a greater time, for this minimum time will be a part of a greater time, just as one is part of number, which allows it to have a ratio to every number. But an indivisible which is not part of a divisible has no proportion to it, just as a point is not a part of a line and therefore there is no proportion of a point to a line. So let us take another heaviness which is finite and as much heavier proportionally than the finite heaviness that moved something in more time than the infinite heaviness, as the minimum time of the infinite heaviness is less than the greater time of the other finite heaviness. For example, let E be the infinite heaviness, and B the minimum time in which it moves, and let G be the finite heaviness that moves something in more time D than time B. Then let F be the other heaviness which is greater than G in the proportion that D exceeds B. Then, since the lessening of time corresponds to the increasing of heaviness, it will follow that the heaviness F, which is finite, will move something in the same time as the infinite heaviness. But this is impossible.
Est autem attendendum quod, sicut non est proportio puncti ad lineam, ita etiam non est proportio instantis ad tempus; quia instans non est pars temporis. Sic ergo solum ista ratio tolleretur, si quis poneret quod gravitas infinita moveret in instanti: sed hoc est impossibile, ut probatum est in VI Physic., scilicet quod aliquis motus sit in instanti. It should be noted that, just as there is no proportion between a point and a line, so there is none between an instant and time, because an instant is not a part of time. Consequently, the only way Aristotle's argument could be destroyed, would be by positing that an infinite heaviness should move in an instant. But that is impossible, as was proved in Physics VI, namely, that any motion should occur in an instant.
Deinde cum dicit: sed adhuc necesse etc., ostendit quod idem inconveniens sequitur in quocumque tempore ponamus gravitatem infinitam movere, etiam in tempore non minimo. Et hoc est quod dicit, quod si in qualicumque tempore finito, etiam non minimo, gravitas infinita movet, adhuc necesse est quod in ipso tempore aliqua gravitas finita moveat per finitum spatium; quia erit accipere excessum gravitatis secundum deminutionem temporis, ut praedictum est. Sic igitur patet quod impossibile est esse gravitatem infinitam: et eadem ratio est de levitate. Then at [78] he shows that the same impossibility follows in whatever time we assume an infinite heaviness to move. And this is what he says, namely, that if an infinite heaviness should move in any finite time whatever, even though it be not the minimum, it is still necessary that in that time a finite heaviness could move through a finite distance — for one will be able to take an excess of weight corresponding to a lessening of the time, as was said above. Consequently, it is clearly impossible for an infinite heaviness to exist; and the same argument holds for lightness.

Lecture 13:
A natural and demonstrative argument showing no natural body can be infinite
Chapter 6 cont.
Ἀδύνατον ἄρα ἄπειρον εἶναι βάρος, ὁμοίως δὲ καὶ κουφότητα. Καὶ σώματα ἄρ' ἄπειρον βάρος ἔχοντα καὶ κουφότητα ἀδύνατον. Ὅτι μὲν οὖν οὐκ ἔστιν ἄπειρον σῶμα, δῆλον διά τε τῶν κατὰ μέρος θεωροῦσι τοῦτον τὸν τρόπον, καὶ καθόλου σκοπουμένοις μὴ μόνον κατὰ τοὺς λόγους τοὺς ἐν τοῖς περὶ τὰς ἀρχὰς εἰρημένους ἡμῖν (διωρίσθη γὰρ κἀκεῖ καθόλου πρότερον περὶ ἀπείρου πῶς ἔστι καὶ πῶς οὐκ ἔστιν) ἀλλὰ καὶ νῦν ἄλλον τρόπον. 79 Infinite weight is therefore impossible, and the same reasoning applies also to infinite lightness. Bodies then of infinite weight and of infinite lightness are equally impossible. That there is no infinite body may be shown, as we have shown it, by a detailed consideration of the various cases.
80 But it may also be shown universally, not only by such reasoning as we advanced in our discussion of principles (though in that passage we have already determined universally the sense in which the existence of an infinite is to be asserted or denied), but also suitably to our present purpose in the following way.
Μετὰ δὲ ταῦτ' ἐπισκεπτέον κἂν εἰ μὴ ἄπειρον μὲν τὸ σῶμα τὸ πᾶν, οὐ μὴν ἀλλὰ τοσοῦτόν γε ὥστ' εἶναι πλείους οὐρανούς τάχα γὰρ ἄν τις τοῦτ' ἀπορήσειεν, ὅτι καθάπερ ὁ περὶ ἡμᾶς κόσμος συνέστηκεν, οὐδὲν κωλύει καὶ ἑτέρους εἶναι πλείους μὲν ἑνός, μὴ μέντοι γε ἀπείρους. Πρῶτον δ' εἴπωμεν καθόλου περὶ τοῦ ἀπείρου. 81 That will lead us to a further question. Even if the total mass is not infinite, it may yet be great enough to admit a plurality of universes. The question might possibly be raised whether there is any obstacle to our believing that there are other universes composed on the pattern of our own, more than one, though stopping short of infinity. First, however, let us treat of the infinite universally.
Chapter 7
Ἀνάγκη δὴ σῶμα πᾶν ἤτοι ἄπειρον εἶναι ἢ πεπερασμένον, καὶ εἰ ἄπειρον, ἤτοι ἀνομοιομερὲς ἅπαν ἢ ὁμοιομερές, κἂν εἰ ἀνομοιομερές, ἤτοι ἐκ πεπερασμένων εἰδῶν ἢ ἐξ ἀπείρων. Ὅτι μὲν τοίνυν οὐχ οἷόν τε ἐξ ἀπείρων, φανερόν, εἴ τις ἡμῖν ἐάσει μένειν τὰς πρώτας ὑποθέσεις πεπερασ(274b.) μένων γὰρ τῶν πρώτων κινήσεων οὐσῶν, ἀνάγκη καὶ τὰς ἰδέας τῶν ἁπλῶν σωμάτων εἶναι πεπερασμένας. Ἁπλῆ μὲν γὰρ ἡ τοῦ ἁπλοῦ σώματος κίνησις, αἱ δ' ἁπλαῖ πεπερασμέναι κινήσεις εἰσίν ἀνάγκη δὲ κίνησιν ἔχειν σῶμα πᾶν φυσικόν. 82 Every body must necessarily be either finite or infinite, and if infinite, either of similar or of dissimilar parts. If its parts are dissimilar, they must represent either a finite or an infinite number of kinds.
83 That the kinds cannot be infinite is evident, if our original presuppositions remain unchallenged. For the primary movements being finite in number, the kinds of simple body are necessarily also finite, since the movement of a simple body is simple, and the simple movements are finite, and every natural body must always have its proper motion.
Ἀλλὰ μὴν εἴ γε ἐκ πεπερασμένων ἔσται τὸ ἄπειρον, ἀνάγκη καὶ τῶν μορίων ἕκαστον εἶναι ἄπειρον, λέγω δ' οἷον τὸ ὕδωρ ἢ τὸ πῦρ. Ἀλλ' ἀδύνατον δέδεικται γὰρ ὅτι οὔτε βάρος οὔτε κουφότης ἐστὶν ἄπειρος. 84 Now if the infinite body is to be composed of a finite number of kinds, then each of its parts must necessarily be infinite in quantity, that is to say, the water, fire, etc., which compose it. But this is impossible, because, as we have already shown, infinite weight and lightness do not exist.
Ἔτι ἀναγκαῖον ἀπείρους τῷ μεγέθει εἶναι καὶ τοὺς τόπους αὐτῶν, ὥστε καὶ τὰς κινήσεις ἀπείρους εἶναι πάντων. Τοῦτο δ' ἀδύνατον, εἰ θήσομεν ἀληθεῖς εἶναι τὰς πρώτας ὑποθέσεις, καὶ μήτε τὸ κάτω φερόμενον εἰς ἄπειρον ἐνδέχεσθαι φέρεσθαι μήτε τὸ ἄνω κατὰ τὸν αὐτὸν λόγον. Ἀδύνατον γὰρ γίνεσθαι ὃ μὴ ἐνδέχεται γενέσθαι, ὁμοίως ἐπὶ τοῦ τοιόνδε καὶ τοσόνδε καὶ τοῦ ποῦ. Λέγω δ', εἰ ἀδύνατον γενέσθαι λευκὸν ἢ πηχυαῖον ἢ ἐν Αἰγύπτῳ, καὶ γίνεσθαί τι τούτων ἀδύνατον. Ἀδύνατον ἄρα καὶ φέρεσθαι ἐκεῖ οὗ μηθὲν δυνατὸν ἀφικέσθαι φερόμενον. 85 Moreover it would be necessary also that their places should be infinite in extent, so that the movements too of all these bodies would be infinite. But this is not possible, if we are to hold to the truth of our original presuppositions and to the view that neither that which moves downward, nor, by the same reasoning, that which moves upward, can prolong its movement to infinity. For it is true in regard to quality, quantity, and place alike that any process of change is impossible which can have no end. I mean that if it is impossible for a thing to have come to be white, or a cubit long, or in Egypt, it is also impossible for it to be in process of coming to be any of these. It is thus impossible for a thing to be moving to a place at which in its motion it can never by any possibility arrive.
Ἔτι εἰ καὶ διεσπασμένα ἐστίν, οὐδὲν ἧττον ἐνδέχοιτ' ἂν τὸ ἐξ ἁπάντων [πῦρ] ἄπειρον εἶναι. Ἀλλὰ σῶμα ἦν τὸ πάντῃ διάστασιν ἔχον ὥστε πῶς οἷόν τε πλείω μὲν ἀνόμοια, ἕκαστον δ' αὐτῶν ἄπειρον εἶναι; πάντῃ γὰρ ἕκαστον δεῖ ἄπειρον εἶναι. 86 Again, suppose the body to exist in dispersion, it may be maintained none the less that the total of all these scattered particles, say, of fire, is infinite.
87 But a body we saw to be that which has extension every way. How can there be several dissimilar elements, each infinite? Each would have to be infinitely extended every way.
Ἀλλὰ μὴν οὐδὲ πᾶν ὁμοιομερὲς ἐνδέχεται τὸ ἄπειρον εἶναι. Πρῶτον μὲν γὰρ οὐκ ἔστιν ἄλλη παρὰ ταύτας κίνησις. Ἕξει οὖν μίαν τούτων. Εἰ δὲ τοῦτο, συμβήσεται ἢ βάρος ἄπειρον ἢ κουφότητα εἶναι ἄπειρον. Ἀλλὰ μὴν οὐδ' οἷόν τε τὸ κύκλῳ σῶμα φερόμενον [εἶναι ἄπειρον]. Ἀδύνατον γὰρ τὸ ἄπειρον φέρεσθαι κύκλῳ οὐθὲν γὰρ διαφέρει τοῦτο λέγειν ἢ τὸ τὸν οὐρανὸν φάναι ἄπειρον εἶναι, τοῦτο δὲ δέδεικται ὅτι ἀδύνατον. 88 It is no more conceivable, again, that the infinite should exist as a whole of similar parts. For, in the first place, there is no other (straight) movement beyond those mentioned: we must therefore give it one of them. And if so, we shall have to admit either infinite weight or infinite lightness. Nor, secondly, could the body whose movement is circular be infinite, since it is impossible for the infinite to move in a circle. This, indeed, would be as good as saying that the heavens are infinite, which we have shown to be impossible.
Ἀλλὰ μὴν οὐδ' ὅλως γε τὸ ἄπειρον ἐνδέχεται κινεῖσθαι. Ἢ γὰρ κατὰ φύσιν κινηθήσεται ἢ βίᾳ καὶ εἰ βίᾳ, ἔστιν αὐτῷ καὶ ἡ κατὰ φύσιν, ὥστε καὶ τόπος ἄλλος ἴσος εἰς ὃν οἰσθήσεται. Τοῦτο δ' ἀδύνατον. 89 Moreover, in general, it is impossible that the infinite should move at all. If it did, it would move either naturally or by constraint: and if by constraint, it possesses also a natural motion, that is to say, there is another place, infinite like itself, to which it will move. But that is impossible.
Postquam philosophus ostendit de singulis corporibus naturalibus quod nullum eorum sit infinitum, hic ostendit communi ratione quod nullum corpus naturale sit infinitum: probatio enim quae est per medium commune, perfectiorem scientiam causat. 124. After showing that no single natural body is infinite, the Philosopher here shows by a general argument that no natural body is infinite — for a proof through a common medium causes more perfect science.
Circa hoc ergo duo facit: About this, therefore, he does two things:

primo dicit de quo est intentio;

secundo ostendit propositum, ibi: necesse itaque corpus omne et cetera.

First he mentions his intention;

Secondly, he proves his proposition, at 128.

Circa primum tria facit. 125. As to the first he does three things:
Primo ostendit quasi epilogando quid prius sit dictum; dicens quod praedicto modo considerantibus manifestum est quod non est corpus infinitum, per ea quae sunt secundum partem, idest secundum proprias rationes singularium partium universi, scilicet corporis quod movetur circulariter, et quod movetur sursum aut deorsum. First he shows (as if summarizing) what has been previously said. And he states that for those who think according to the lines already laid down, it is clear that there is no infinite body, "by a detailed consideration of the various cases," i.e., on account of the reasons applied to the individual parts of the universe, namely, to the body that is moved circularly and to the bodies that are move upward or downward.
Secundo ibi: et universaliter intendentibus etc., ostendit quid immediate restet dicendum. Et dicit quod idem potest esse manifestum si aliquis intendat universaliter, idest per medium commune. Et hoc non solum secundum illas rationes communes quae positae sunt in libro physicorum, ubi determinatum est de principiis communibus corporum naturalium (in tertio enim physicorum determinatur universaliter de infinito quomodo sit et quomodo non sit: ostensum est enim ibi quod infinitum est in potentia, sed non in actu). Nunc autem determinandum est alio modo de infinito, ostendendo scilicet universaliter quod nullum corpus sensibile potest esse infinitum in actu. 126. Secondly, at [801 he shows what immediately remains to be said. And he says that the same thing can be clear if someone looks at it universally, i.e., by a common medium. And this is in addition to those general arguments given in the book of the Physics where the common principles of all natural bodies were discussed — for in Physics III is a universal treatment of the infinite, as to how it exists and how not, it being shown there that the infinite exists in potency but not in act. Now, however, the infinite has to be treated in another way, by showing universally that no sensible body can be infinite in act.
Tertio ibi: post haec autem intendendum etc., ostendit quid sit determinandum immediate post ista. Et dicit quod postquam ostenderimus hoc quod dictum est, intentio nostra erit inquirere, supposito quod totum corpus universi non sit infinitum, utrum tamen totum corpus sit tantae quantitatis, quod possint ex eo esse plures caeli, idest plures mundi. Forte enim potest de hoc aliquis dubitare, an sit possibile quod, sicut iste mundus est constitutus circa nos, ita etiam sint alii mundi plures uno, non tamen infiniti. Sed antequam hoc pertractemus, dicemus universaliter de infinito, ostendendo scilicet communibus rationibus quod nullum corpus sit infinitum. 127. Thirdly, at [81] he shows what must be determined immediately after these questions. And he says that after proving what has been proposed, our aim will be to inquire (on the supposition that the whole body of the universe is not infinite) whether the whole body is of such size that there can be made from it several heavens, i.e., many worlds. For perhaps someone could wonder whether it is possible that, just as our world is established about us, there might be other worlds, i.e., more than one though not an infinitude. But before dealing with that question, we shall speak universally of the infinite and show that from common reasons no body is infinite.
Deinde cum dicit: necesse itaque etc., ostendit propositum: 128. Then at [82] he proves the proposition:

et primo per rationes naturales demonstrativas;

secundo per rationes logicas, ibi: rationabilius autem et cetera.

First by natural demonstrative arguments;

Secondly, by logical arguments (L. 15).

Dico autem rationes demonstrativas et naturales, quae sumuntur ex propriis principiis scientiae naturalis; cuius consideratio consistit circa motum, et actionem et passionem, quae in motu consistunt, ut dicitur in III Physic. Now I call "demonstrative" and "natural" those arguments that are taken from the proper principles of natural science, whose consideration concerns motion, and action and passion which reside in motion, as is said in Physics III.

Primo ergo ostendit nullum corpus esse infinitum, ex parte motus localis, qui est primus et communissimus motuum;

secundo universaliter ex parte actionis et passionis, ibi: quod autem omnino impossibile et cetera.

First, therefore, at [82] he shows that no body is infinite from the side of local motion, which is the first and most common of motions;

Secondly, universally on the part of action and passion (L. 14).

Circa primum duo facit: As to the first he does two things:

primo praemittit quasdam divisiones;

secundo prosequitur singula membra, ibi: quod quidem igitur et cetera.

First he presents certain divisions, at 129;

Secondly, he examines the members individually, at 130.

Praemittit ergo primo tres divisiones. Quarum prima est, quod necesse est omne corpus aut esse finitum aut infinitum. Et si quidem sit finitum, habemus propositum: si autem sit infinitum, restat secunda divisio, scilicet quod aut est totum anomoeomerum, idest dissimilium partium, sicut corpus animalis, quod componitur ex carnibus, ossibus et nervis; aut est totum homoeomerum, idest similium partium, sicut aqua, cuius quaelibet pars est aqua. Si vero sit totum dissimilium partium, restat tertia divisio: utrum scilicet species partium talis corporis sint finitae numero aut infinitae. Si ergo probetur quod non sunt infinitae, neque iterum sunt finitae; et quod iterum nullum corpus similium partium sit infinitum: probatum erit quod nullum corpus universaliter est infinitum. 129. Therefore he first [82] presents three divisions: The first of these is that every body must be either finite or infinite. If it is finite, we have our proposition; but if it is infinite, a second division remains, namely, that it be a heterogeneous whole, i.e., having dissimilar parts, as an animal body which is composed of flesh, bones and sinews; or it is a homogeneous whole, i.e., having like parts, such as water, each part of which is water. But if it is a whole of dissimilar parts, a third division remains, namely, whether the species of the parts of such a body are finite or infinite in number. If it is proved that they are not infinite, nor again finite, and further that no body of parts that are alike is infinite, it will have been proved universally that no body is infinite.
Deinde cum dicit: quod quidem igitur etc., prosequitur singula praedictorum. Et circa hoc tria facit: 130. Then at [83] he pursues each member. He does three things about this:

primo ostendit quod non est possibile corporis dissimilium partium esse infinitas species partium eius;

secundo ostendit quod non est possibile esse corpus infinitum dissimilium partium, ita quod species partium sint finitae, ibi: sed tamen si quidem etc.;

tertio ostendit quod non est possibile esse aliquod corpus infinitum similium partium, ibi: sed adhuc neque totum et cetera.

First he shows that it is not possible in a body of unlike parts for the species of its parts to be infinite;

Secondly, that it is not possible for an infinite body of unlike parts to be such that the species of its parts be finite, at 131;

Thirdly, he shows that there can be no infinite body having parts that are alike, at 135.

Dicit ergo primo quod manifestum est quod non est possibile ex infinitis speciebus partium constitui aliquod corpus infinitum, si quis permittat manere in sua veritate primas hypotheses, idest suppositiones prius factas, scilicet quod sint solae tres species motuum simplicium. Si enim primi motus, scilicet simplices, sunt finiti, necesse est quod species corporum simplicium sint finitae: et hoc ideo, quia motus ipsius corporis simplicis est simplex, ut supra habitum est. Dictum est autem supra quod simplices motus sunt finiti: sunt enim tres, scilicet motus qui est ad medium, et motus qui est a medio, et motus qui est circa medium. Ideo autem oportet quod, si motus simplices sunt finiti, quod corpora simplicia sint finita, quia necesse est quod omne corpus naturale habeat proprium motum: si autem essent infinitae species corporum, motibus existentibus finitis, oporteret esse aliquas species corporum, quae non haberent motus: quod est impossibile. He says therefore first [83] that it is plainly not possible for an infinite body to be constituted from an infinite species of parts, so long as one is loyal to the "first hypotheses," i.e., the previously made suppositions that there are only three species of simple motion. For if the first motions, i.e. the simple motions, are finite, then the species of simple bodies must be finite, for the motion of a simple body is itself simple, as was had above. But it was also held above that simple motions are finite: for there are three, namely, motion to the middle, motion from the middle, and motion around the middle. Now the reason why simple bodies are finite, if simple motions are finite, is that every natural body must have its own proper motion — but if there were an infinite species of bodies, while the number of motions was finite, there would have to be some species of bodies without motions, which is impossible.
Sic igitur ex hoc quod motus simplices sunt finiti, sufficienter probatur quod species corporum simplicium sint finitae. Omnia autem corpora mixta componuntur ex simplicibus. Unde si esset aliquod totum dissimilium partium, quod componeretur ex infinitis speciebus corporum mixtorum, tamen oporteret quod species primorum componentium sint finitae: quamvis etiam hoc non videatur possibile, quod finitorum elementorum diversificentur commixtiones in infinitum. Nec tamen aliquod corpus mixtum potest dici omnium similium partium: quia, etsi partes eius quantitativae sint similes specie, sicut quaelibet pars lapidis est lapis, partes tamen essentiales eius sunt diversae secundum speciem: componitur enim substantia corporis mixti ex corporibus simplicibus. Consequently, from the fact that simple motions are finite, it is sufficiently proved that the species of simple bodies are finite. Now it is from simple bodies that mixed bodies are composed. Hence, if there were a whole having unlike parts and composed of an infinite species of mixed bodies, the species of the first components would still have to be finite — though it does not even seem possible that mixtures from finite elements should be infinitely diversified. Neither can any compound body be called a mixture of all like parts, because, even if its quantitative parts be specifically alike, as each part of a stone is stone, yet its essential parts are specifically diverse, for the substance of a mixed body is composed of simple bodies.
Deinde cum dicit: sed tamen si quidem etc., ostendit quod non est possibile esse corpus infinitum dissimilium partium, ita quod species partium sint finitae. Et ad hoc inducit quatuor rationes. Quarum prima est quod, si corpus dissimilium partium, infinitum existens, ex partibus finitis specie componeretur, oporteret quod quaelibet partium eius esset infinita secundum magnitudinem: puta, si aliquod corpus mixtum esset infinitum, elementis existentibus finitis, oporteret aerem esse infinitum et aquam et ignem. Sed hoc est impossibile: quia, cum quodlibet eorum sit grave vel leve, sequeretur secundum praemissa quod gravitas eius vel levitas esset infinita; ostensum est autem quod nulla gravitas vel levitas potest esse infinita. Ergo non est possibile quod corpus infinitum dissimilium partium componatur ex finitis speciebus partium. 131. Then at [84] he shows that it is impossible to have an infinite body of unlike parts, the species of which parts are finite. And he arrives at this with four arguments. The first is that, if a body of unlike parts is infinite and composed of parts that are finite with respect to species, each of the parts would have to be infinite in magnitude. For example, if a mixed body were infinite and composed of elements that were finite, air would have to be infinite, and so would the water and the fire. But this is impossible, because, since each of these is either heavy or light, it would follow according to what was previously said that its heaviness or lightness would be infinite. But it has been proved that no heaviness or lightness can be infinite. Therefore, it is not possible for an infinite body of unlike parts to be composed of a finite species of parts.
Potest autem aliquis obiicere quod non sequitur, hac ratione facta, quod unaquaeque partium sit infinita: esset enim possibile totum esse infinitum, una parte existente infinita secundum magnitudinem, et aliis existentibus finitis. Sed hoc reprobatum est in III Physic.: si enim una pars esset infinita, consumeret alias partes finitas propter excessum virtutis. Potest tamen dici quod, etiam hoc posito, sequetur idem inconveniens, scilicet quod sit gravitas vel levitas infinita; et ideo de hoc Aristoteles non curavit. However, someone could object that it does not follow from this argument that each of the parts is infinite: for it could be possible for the whole to be infinite if one part were infinite in magnitude and the others finite. But this was rejected in Physics III — for if one part were infinite it would consume the other finite parts on account of its excessive power. Likewise it can be said that even in that case the same impossibility will follow, namely, that there would be an infinite heaviness or lightness. And therefore Aristotle was not concerned with it.
Secundam rationem ponit ibi: adhuc necessarium et cetera. Si enim partes totius infiniti sint infinitae secundum magnitudinem, oportet etiam quod loca earum essent infinita secundum magnitudinem; quia loca oportet esse aequalia locatis. Sed motus mensuratur secundum magnitudinem loci in quem pertransit, ut probatur in VI Physic. Ergo sequitur quod motus omnium harum partium sint infiniti. Sed hoc est impossibile, si sint vera ea quae supra supposuimus, scilicet quod non contingit aliquid moveri deorsum in infinitum, neque etiam sursum; quia deorsum est determinatum, cum sit medium, et eadem ratione sursum est determinatum (si enim unum contrariorum est determinatum, et aliud). 132. The second argument is presented at [85]. For if the parts of an infinite whole were infinite in magnitude, their places would have to be infinite in magnitude, because places are necessarily equal to the things in them. But motion is measured according to the magnitude of the place into which it passes, as is proved in Physics VI. Therefore it follows that the motions of all these parts would be infinite. But this is impossible, if what we supposed above is true, namely, that nothing can be moved downward infinitely, nor upward either — because "down" is determinate, since it is the middle, and for the same reason "up" is determinate (for if one contrary is determinate, so is the other).
Et hoc etiam hic ostendit per id quod est commune omnibus motibus. Videmus enim in transmutatione quae est secundum substantiam, quod impossibile est fieri illud quod non potest esse factum; sicut non potest fieri asinus rationalis, quia impossibile est asinum esse talem. Et simile est in tali, idest in motu qui est secundum qualitatem, et in tanto, idest in motu qui est secundum quantitatem, et in ubi, idest in motu qui est secundum locum. Si enim impossibile est quod aliquid nigrum sit factum album, sicut corvus, impossibile est quod fiat album; et si aliquid impossibile est quod sit cubitale, sicut formica, impossibile est quod ad hoc moveatur; et si impossibile est quod aliquid sit in Aegypto, puta Danubius, impossibile est quod illuc moveatur. Et huius ratio est, quia natura nihil facit frustra: esset autem frustra si moveret ad id ad quod impossibile est pervenire. Sic igitur impossibile est quod aliquid moveatur localiter illuc quo non est pervenire. Non est autem pertransire locum infinitum. Si igitur loca essent infinita, nullus esset motus. Quod cum sit impossibile, non potest esse quod partes corporis infiniti dissimilium partium, sint infinitae in magnitudine. And he also proves this by what is common to all motions. For in the transmutation according to substance, we see that it is impossible for a thing to become what it cannot be, as, for example, there cannot be made a rational ass, since it is impossible for an ass to be such. And the same goes for a motion in "such," i.e., with respect to quality and for a motion in "so much," i.e., with respect to quantity, and for a motion in "where," i.e., with respect to place. For if it is impossible for something black ever to have been made white, as a raven, it is impossible for it ever to become white. And if it is impossible for anything to be a foot long, as an ant, it is impossible for it to be moving toward that; and if it is impossible for something to be in Egypt, as the Danube, it is impossible for it to be moving thither. The reason for this is that nature does nothing in vain. But it would be in vain for a thing to be tending to what is impossible for it to reach. Consequently, it is impossible for a thing to be locally moved to a place where it cannot arrive. But it is impossible to traverse an infinite place. If, therefore, places were infinite, there would be no motion. But since that is impossible, it cannot be that the parts of an infinite body of unlike parts be infinite in magnitude.
Tertiam rationem ponit ibi: adhuc si et discerpta et cetera. Posset enim aliquis dicere quod non est unum continuum infinitum, sunt tamen quaedam partes discerptae, idest disiunctae et non continuae, infinitae; sicut Democritus posuit infinita corpora indivisibilia, et sicut Anaxagoras posuit infinitas partes consimiles. 133. He presents the third argument at [86]. For someone could say that there is no infinite continuous unit, but that there are yet certain parts, disconnected and not continued, which are infinite, as Democritus posited infinite indivisible bodies, and as Anaxagoras posited infinite parts all similar to each other.
Sed ipse dicit quod ex hac positione nihil minus sequitur inconveniens: quia, si sint infinitae partes ignis non continuae, nihil prohibet illas omnes coniungi, et sic fieri ex omnibus unum ignem infinitum. But Aristotle says that this position leads no less to an impossibility: for if infinite parts of fire are not joined, there is nothing to prevent all of them from joining and thus making one infinite fire from all of them.
Quartam rationem ponit ibi: sed corpus est et cetera. Cum enim aliquid dicitur esse infinitum, oportet quod infinitum accipiatur secundum propriam eius rationem: puta, si dicamus lineam esse infinitam, intelligimus eam esse infinitam secundum longitudinem; si vero dicamus superficiem esse infinitam, intelligimus quod sit infinita secundum longitudinem et latitudinem. Corpus autem distenditur ad omnem partem, quia habet omnes dimensiones, ut supra dictum est: et sic, si corpus dicatur infinitum, oportet quod sit infinitum ad omnem partem; et ita ex nulla parte erit aliquid extra ipsum. Non ergo est possibile quod in corpore infinito sint plura dissimilia, quorum unumquodque sit infinitum: quia non est possibile esse plura infinita, secundum praedicta. 134. The fourth argument he presents at [87]. For when something is said to be infinite, the term should be taken according to its proper meaning. For example, if we say that a line is infinite, we understand it to be infinite in length; while, if we say that a surface is infinite, we understand that it is infinite in length and width. But a body stretches in every direction, because it has three dimensions, as was said above. Consequently, if a body is said to be infinite, it will have to be infinite in every direction, and so in no direction will there be anything outside it. It is therefore not possible that there be in an infinite body many things that are unlike, each of which is infinite, for according to the foregoing it is not possible for there to be a number of infinites.
Deinde cum dicit: sed adhuc neque totum etc., ostendit quod corpus infinitum non potest esse similium partium: et hoc duabus rationibus. Quarum prima est, quia cuiuslibet corporis naturalis oportet esse aliquem motum localem; non est autem alius motus praeter istos qui supra dicti sunt, quorum scilicet unus est circa medium, alius a medio, et tertius ad medium; sequitur igitur quod habeat unum istorum motuum. Sed hoc est impossibile: quia si moveatur sursum vel deorsum, erit grave vel leve; et ita accidet gravitatem et levitatem esse infinitam, quod est impossibile secundum praemissa. Similiter etiam non est possibile quod moveatur circulariter, quia est impossibile infinitum circumferri: nihil enim differt hoc dicere, quam si dicatur caelum infinitum, quod impossibile est, ut supra ostensum est. Non ergo contingit totum corpus infinitum esse homoeomerum. 135. Then at [88] he shows that there cannot be an infinite body having like parts — and this with two arguments. The first of these is that every natural body must have some local motion; but there is no other except those mentioned above, one of which is around the middle, another from the middle, and a third to the middle. It follows, therefore, that it has one of these. But this is impossible — for if it moves upward or downward, it will be heavy or light, and, consequently, its heaviness or lightness will be infinite, which is impossible according to what has gone before. Likewise it cannot be moved circularly, because it is impossible for the infinite to turn in a circle. For there is no difference between saying this and saying that the heaven is infinite — which is impossible, as was proved above. Therefore a whole infinite body cannot be homogeneous.
Secundam rationem ponit ibi: sed adhuc neque omnino etc.; quae sequitur ex communi ratione motus localis. Si enim sit corpus similium partium infinitum, sequitur quod nullo modo possit moveri. Quia si movetur, aut movebitur secundum naturam, aut secundum violentiam. Si autem sit ei aliquis motus violentus, sequitur quod etiam sit ei aliquis motus naturalis: quia motus violentus contrariatur motui naturali, ut supra habitum est. Si autem aliquis sit ei motus naturalis, sequitur quod etiam sit ei aliquis locus aequalis sibi, in quem naturaliter fertur: quia motus naturalis est eius quod fertur in proprium locum. Hoc autem est impossibile: quia sequeretur quod sint duo corporalia loca infinita; quod est aeque impossibile sicut quod sint duo corpora infinita; quia sicut corpus infinitum est undique infinitum, ita et locus infinitus. Non est igitur possibile quod corpus infinitum moveatur. Si ergo omne corpus naturale movetur, sequitur quod nullum corpus naturale sit infinitum. 136. The second argument is set down at [89] and it follows from the common notion of local motion. For if there should be an infinite body of parts that are alike, it follows that it cannot be moved at all. If it is moved, it will be moved either according to nature, or by compulsion. But if it has a compulsory motion, then must be a motion natural to it, because a compulsory motion is contrary to a natural motion, as was had above. But if there is a motion natural to it, it follows that there is a place equal to it, into which it is naturally moved, for natural motion belongs to what is moved to its own place. This, however, is impossible, because it would follow that there would be two infinite corporeal places, which is as impossible as that there should be two infinite bodies, for, just as an infinite body is infinite in every direction, so too is an infinite place. Therefore it is not possible for an infinite body to be moved. But if every natural body is moved, it therefore follows that no natural body is infinite.
Est tamen attendendum quod haec ratio non procedit nisi de motu recto: nam id quod movetur circulariter, non mutat totum locum subiecto, sed solum ratione, ut probatur in VI Physic. Sed quod corpus infinitum non possit moveri circulariter, supra multipliciter est ostensum. It should be noted that this argument applies only to straight motion, for what is moved circularly does not change its place as to subject, but only in conception, as is proved in Physics VI. But that an infinite body cannot be moved circularly has already been proved above in many ways.

Lecture 14:
No sensible body is infinite — from action and passion, which follow upon motion.
Chapter 7 cont.
Ὅτι δ' ὅλως ἀδύνατον ἄπειρον ὑπὸ πεπερασμένου παθεῖν τι ἢ ποιῆσαι τὸ πεπερασμένον, ἐκ τῶνδε φανερόν. Ἔστω (275a.) γὰρ ἄπειρον ἐφ' οὗ Α, πεπερασμένον ἐφ' οὗ Β, χρόνος ἐν ᾧ ἐκίνησέ τι ἢ ἐκινήθη Γ. 90 That in general it is impossible for the infinite to be acted upon by the finite or to act upon it may be shown as follows.
Εἰ δὴ ὑπὸ τοῦ Β τὸ Α ἐθερμάνθη ἢ ὤσθη ἢ ἄλλο τι ἔπαθεν ἢ καὶ ὁτιοῦν ἐκινήθη ἐν τῷ χρόνῳ ἐφ' οὗ Γ, ἔστω τὸ Δ τοῦ Β ἔλαττον, καὶ τὸ ἔλαττον ἐν τῷ ἴσῳ χρόνῳ ἔλαττον κινείτω ἔστω δὲ τὸ ἐφ' ᾧ Ε ὑπὸ τοῦ Δ ἠλλοιωμένον. Ὃ δή ἐστι τὸ Δ πρὸς τὸ Β, τὸ Ε ἔσται πρὸς πεπερασμένον τι. Ἔστω δὴ τὸ μὲν ἴσον ἐν ἴσῳ χρόνῳ ἴσον ἀλλοιοῦν, τὸ δ' ἔλαττον ἐν τῷ ἴσῳ ἔλαττον, τὸ δὲ μεῖζον μεῖζον, τοσοῦτον δὲ ὅσον ἀνάλογον ἔσται ὅπερ τὸ μεῖζον πρὸς τὸ ἔλαττον. Οὐκ ἄρα τὸ ἄπειρον ὑπ' οὐδενὸς πεπερασμένου κινηθήσεται ἐν οὐθενὶ χρόνῳ ἔλαττον γὰρ ἄλλο ἐν τῷ ἴσῳ χρόνῳ ὑπὸ ἐλάττονος κινηθήσεται, πρὸς ὃ τὸ ἀνάλογον πεπερασμένον ἔσται τὸ γὰρ ἄπειρον πρὸς τὸ πεπερασμένον ἐν οὐθενὶ λόγῳ ἐστίν. 91 (1. The infinite cannot be acted upon by the finite.) Let A be an infinite, B a finite, C the time of a given movement produced by one in the other. Suppose, then, that A was heated, or impelled, or modified in any way, or caused to undergo any sort of movement whatever, by in the time C. Let D be less than B; and, assuming that a lesser agent moves a lesser patient in an equal time, call the quantity thus modified by D, E. Then, as D is to B, so is E to some finite quantum. We assume that the alteration of equal by equal takes equal time, and the alteration of less by less or of greater by greater takes the same time, if the quantity of the patient is such as to keep the proportion which obtains between the agents, greater and less. If so, no movement can be caused in the infinite by any finite agent in any time whatever. For a less agent will produce that movement in a less patient in an equal time, and the proportionate equivalent of that patient will be a finite quantity, since no proportion holds between finite and infinite.
Ἀλλὰ μὴν οὐδὲ τὸ ἄπειρον ἐν οὐθενὶ χρόνῳ κινήσει τὸ πεπερασμένον. Ἔστω γὰρ ἐφ' ᾧ τὸ Α ἄπειρον, τὸ δὲ Β πεπερασμένον, χρόνος ἐν ᾧ τὸ Γ. Οὐκοῦν τὸ Δ ἐν τῷ Γ ἔλαττον τοῦ Β κινήσει ἔστω τὸ Ζ. Ὃ δή ἐστι τὸ ΒΖ ὅλον πρὸς τὸ Ζ, τὸ Ε ἔχον τὸν λόγον τοῦτον ἔστω πρὸς τὸ Δ. Κινήσει ἄρα τὸ Ε τὸ ΒΖ ἐν τῷ Γ. Τὸ πεπερασμένον τοίνυν καὶ τὸ ἄπειρον ἐν τῷ ἴσῳ χρόνῳ ἀλλοιώσει. Ἀλλ' ἀδύνατον ἐν ἐλάττονι γὰρ τὸ μεῖζον ὑπέκειτο. Ἀλλ' ἀεὶ ὁ ληφθεὶς χρόνος ταὐτὸ ποιήσει, ὥστ' οὐκ ἔσται χρόνος οὐθεὶς ἐν ᾧ κινήσει. 92 (2. The infinite cannot act upon the finite.) Nor, again, can the infinite produce a movement in the finite in any time whatever. Let A be an infinite, B a finite, C the time of action. In the time C, D will produce that motion in a patient less than B, say F. Then take E, bearing the same proportion to D as the whole BF bears to F. E will produce the motion in BF in the time C. Thus the finite and infinite effect the same alteration in equal times. But this is impossible; for the assumption is that the greater effects it in a shorter time. It will be the same with any time that can be taken, so that there will no time in which the infinite can effect this movement.
Ἀλλὰ μὴν ἐν ἀπείρῳ γε οὐκ ἔστι κινῆσαι οὐδὲ κινηθῆναι πέρας γὰρ οὐκ ἔχει, ἡ δὲ ποίησις καὶ τὸ πάθος ἔχει. 93 And, as to infinite time, in that nothing can move another or be moved by it. For such time has no limit, while the action and reaction have.
Οὐδ' ἄπειρον δὴ ὑπ' ἀπείρου ἐνδέχεται οὐθὲν παθεῖν. Ἔστω γὰρ τὸ Α ἄπειρον καὶ τὸ Β, χρόνος δ' ἐν ᾧ ἔπαθε τὸ Β ὑπὸ τοῦ Α, ἐφ' ᾧ ΓΔ. Τὸ δὴ ἐφ' ᾧ τὸ Ε τοῦ ἀπείρου μέρος, ἐπεὶ ὅλον πέπονθε τὸ Β, οὐκ ἐν ἴσῳ χρόνῳ τὸ αὐτό ὑποκείσθω γὰρ ἐν ἐλάττονι κινεῖσθαι τὸ ἔλαττον χρόνῳ. Ἔστω τὸ Ε κεκινημένον ὑπὸ τοῦ Α ἐν τῷ Δ. Ὃ δὴ τὸ Δ πρὸς τὸ ΓΔ, τὸ Ε ἐστὶ πρός τι τοῦ Β πεπερασμένον. Τοῦτο τοίνυν ἀνάγκη ὑπὸ τοῦ Α κινηθῆναι ἐν τῷ ΓΔ χρόνῳ ὑπὸ γὰρ τοῦ αὐτοῦ ὑποκείσθω ἐν τῷ πλείονι καὶ ἐλάττονι (275b.) χρόνῳ τὸ μεῖζον καὶ τὸ ἔλαττον πάσχειν, ὅσα ἀνάλογον τῷ χρόνῳ διῄρηται. Ἐν οὐδενὶ ἄρα χρόνῳ δυνατὸν πεπερασμένῳ ἄπειρον ὑπ' ἀπείρου κινηθῆναι ἐν ἀπείρῳ ἄρα. Ἀλλ' ὁ μὲν ἄπειρος χρόνος οὐκ ἔχει τέλος, τὸ δὲ κεκινημένον ἔχει. 94 (3. There is no interaction between infinites.) Nor can infinite be acted upon in any way by infinite. Let A and B be infinites, CD being the time of the action A of upon B. Now the whole B was modified in a certain time, and the part of this infinite, E, cannot be so modified in the same time, since we assume that a less quantity makes the movement in a less time. Let E then, when acted upon by A, complete the movement in the time D. Then, as D is to CD, so is E to some finite part of B. This part will necessarily be moved by A in the time CD. For we suppose that the same agent produces a given effect on a greater and a smaller mass in longer and shorter times, the times and masses varying proportionately. There is thus no finite time in which infinites can move one another. Is their time then infinite? No, for infinite time has no end, but the movement communicated has.
Εἰ τοίνυν πᾶν σῶμα αἰσθητὸν ἔχει δύναμιν ποιητικὴν ἢ παθητικὴν ἢ ἄμφω, ἀδύνατον σῶμα ἄπειρον αἰσθητὸν εἶναι. 95 If therefore every perceptible body possesses the power of acting or of being acted upon, or both of these, it is impossible that an infinite body should be perceptible.
Ἀλλὰ μὴν καὶ ὅσα γε σώματα ἐν τόπῳ, πάντα αἰσθητά. Οὐκ ἔστιν ἄρα σῶμα ἄπειρον ἔξω τοῦ οὐρανοῦ οὐθέν. Ἀλλὰ μὴν οὐδὲ μέχρι τινός. Οὐθὲν ἄρα ὅλως σῶμα ἔξω τοῦ οὐρανοῦ. Εἰ μὲν γὰρ νοητόν, ἔσται ἐν τόπῳ τὸ γὰρ ἔξω καὶ ἔσω τόπον σημαίνει. Ὥστ' ἔσται αἰσθητόν. Αἰσθητὸν δ' οὐθὲν μὴ ἐν τόπῳ. 96 All bodies, however, that occupy place are perceptible. There is therefore no infinite body beyond the heaven. Nor again is there anything of limited extent beyond it. And so beyond the heaven there is no body at all. For if you suppose it an object of intelligence, it will be in a place—since place is what 'within' and 'beyond' denote—and therefore an object of perception. But nothing that is not in a place is perceptible.
Postquam philosophus ostendit corpus sensibile non esse infinitum, ratione accepta ex parte motus localis, hic ostendit idem ratione accepta ex parte actionis et passionis, quae consequuntur omnem motum. Et circa hoc duo facit: 137. After showing that a sensible body is not infinite with a reason based on local motion, the Philosopher here shows the same thing with a reason based on action and passion, which follow upon every motion. Concerning this he does two things:

primo ostendit propositum;

secundo excludit quandam obviationem, ibi: sed tamen et quaecumque et cetera.

First he demonstrates the proposition;

Secondly, he excludes an objection, at 144.

Circa primum ponit talem rationem. Nullum corpus infinitum habet virtutem activam aut passivam aut utramque; sed omne corpus sensibile habet virtutem activam aut passivam aut utramque; ergo nullum corpus sensibile est infinitum. Circa hoc ergo duo facit: 138. With regard to the first, he gives the following argument: No infinite body has active or passive power or both; but every sensible body has active or passive power or both. Therefore, no sensible body is infinite. Then with regard to this he does two things:

primo probat maiorem;

secundo ponit minorem et conclusionem, ibi: si igitur omne corpus et cetera.

First he proves the major premise;

Secondly, he presents the minor and conclusion, at 143.

Circa primum duo facit: About the first he does two things:
primo proponit quod intendit, et dicit manifestum esse ex his quae dicentur, quod non solum impossibile est infinitum moveri localiter, sed universaliter est impossibile infinitum pati aliquid, vel etiam agere aliquid in corpus finitum. Secundo ibi: sit enim infinitum etc., probat propositum.

First he proposes what he intends and says that it is clear from what will be said that not only is it impossible for something infinite to be moved locally but that universally it is impossible for something infinite to be acted upon or to act upon a finite body.

Secondly, at [91] he proves his proposition.

Et primo ostendit quod infinitum non patitur a finito;

secundo ostendit quod finitum non patitur ab infinito, ibi: sed adhuc neque infinitum etc.;

tertio ostendit quod infinitum non patitur ab infinito, ibi: neque infinitum utique et cetera.

First he shows that the infinite is not acted upon by the finite, 139;

Secondly, that the finite is not acted upon by the infinite, at 140;

Thirdly, that the infinite is not acted upon by the infinite, at 142.

Dicit ergo primo quod, si corpus infinitum patitur a finito, sit corpus infinitum in quo est a, corpus autem finitum in quo est b: et quia omnis motus est in tempore, sit tempus g in quo b movit aut a motum est. Si ergo ponamus quod a quod est corpus infinitum, a b quod est corpus finitum, sit alteratum, puta calefactum, aut latum, idest motum secundum locum, aut aliquid aliud passum, puta infrigidatum aut humectatum aut quocumque modo motum, in tempore g: accipiamus unam partem b moventis, quae sit d (et nihil referret ad propositum si d esset quoddam aliud corpus minus quam b). Manifestum est autem quod minus corpus movet minus mobile in aequali tempore (hoc tamen supposito, quod in minori corpore sit minor virtus; quod oportet dicere si sit corpus similium partium; minor autem virtus in aequali tempore movet minus mobile). Sit ergo corpus e, quod alteratur aut qualitercumque movetur a d in tempore g; ita quod intelligamus corpus e esse partem totius infiniti quod est a. Sed quia tam d quam b est finitum, et quorumlibet duorum finitorum corporum est aliqua proportio ad invicem; secundum illam proportionem quam habet d ad b, accipiatur proportio corporis e ad quodcumque corpus maius finitum, puta quod sit f. 139. He says therefore first [91] that if an infinite body is acted upon by a finite, let A be an infinite body and B a finite body and, since every motion occurs in time, let G be the time in which B moves or A has been moved. If, therefore, we posit that A, which is the infinite body, is altered by B, which is the finite body, say heated or carried, i.e., moved locally, or affected in any other way, e.g., cooled or moistened, or moved in any way, in time G, let us take one part of the mover B, i.e., a part D (and it makes no difference, so far as the proposition is concerned, if D be some other body less than B). Now it is clear that a smaller body moves a smaller mobile in an equal time (supposing, of course, that there is in the smaller body less power — which must be said, if it is a body of like parts — hence the lesser power moves a smaller body in an equal time). Therefore, let E be a body which is altered or any other way moved by D in the time G, taking E as a part of the infinite whole A. But since both D and B are finite, and since any two finite bodies are mutually proportionate, then, according to the ratio of D to B, let there be taken the proportion of E to any other larger finite body, for example, F.
Hac ergo positione facta, ponit quasdam suppositiones. Quarum prima est, quod alterans aequale in magnitudine et virtute, in aequali tempore alterabit aequale corpus. Secunda est, quod minus corpus alterans in aequali tempore alterabit minus; ita scilicet quod tantum erit corpus motum minus altero corpore moto, quantum erit analogum quodcumque maius ad minus, idest, quanta erit proportio excessus maioris corporis moventis ad minus. Having posited these preliminaries, he makes some suppositions. The first of these is that an altering cause which is equal in magnitude and power will alter an equal body in equal time. A second is that a smaller altering body will alter a smaller in equal time, the result being that one moved body will be less than the other moved body according to a ratio of something greater to something less, i.e., in the same proportion that the larger moving body exceeds the smaller moving body.
Ex praemissis igitur concludit quod infinitum a nullo finito potest moveri secundum quodcumque tempus. Quia aliquid minus quam infinitum movebitur in aequali tempore ab illo minori quam sit corpus movens infinitum; scilicet e, quod est minus quam a, movebitur a d, quod est minus quam b, secundum praemissa. Id autem quod est analogum ad e, idest quod in eadem proportione se habet ad e sicut b ad d, est quoddam finitum: non enim potest dici quod ipsum infinitum quod est a, se habeat ad e sicut b se habet ad d, quia infinitum ad finitum nullam proportionem habet. Supposito autem quod aliquod finitum se habeat ad e sicut b ad d, erit commutatim dicere quod sicut d se habet ad e, ita b se habet ad illud finitum. Sed d movet e in tempore g: ergo b movet finitum in tempore g. Sed in hoc tempore positum est quod movet totum infinitum quod est a: ergo finitum in eodem tempore movebit finitum et infinitum. From these preliminaries, therefore, he concludes that the infinite cannot be moved by any finite in any time. For something less than the infinite will in an equal time be moved by that body which is less than the body moving the infinite; in other words, E, which is less than A, will be moved by D, which is less than B, according to our suppositions. But what is "analogous" to E, i.e., in the same ratio to E, as B to D, is finite, for it cannot be said that A, which is infinite, is to E, as B is to D, because the infinite has no proportion to the finite. Now on the assumption that something finite is to E as B is to D, then commutatively B is to that finite, as D is to E. But D moves E in time G; therefore B moves the finite in time G. But G was the time in which it was supposed that B moved the infinite whole A. Therefore the finite will move a finite and an infinite in the same time.
Deinde cum dicit: sed adhuc neque infinitum etc., probat quod infinitum corpus non movet corpus finitum in aliquo tempore: 140. Then at [92] he proves that an infinite body does not move a finite body in any time.

et primo ostendit quod non movet in tempore finito;

secundo quod non movet in tempore infinito, ibi: sed adhuc in infinito et cetera.

First he shows that it does not move it in finite time;

Secondly, not in infinite time, at 141.

Dicit ergo primo quod neque etiam corpus infinitum movebit corpus finitum in nullo tempore, scilicet determinato. Si enim detur contrarium, sit corpus infinitum in quo est a, corpus vero finitum quod ab eo movetur sit b vel bz, tempus autem in quo movetur sit g. D autem sit quaedam pars finita corporis infiniti quod est a: et quia minus in aequali tempore minus movet, consequens est quod corpus finitum quod est d, in g tempore moveat minus corpus eo quod est b; et sit id minus z, quod est pars eius. Quia igitur totum bz habet aliquam proportionem ad z, accipiatur quod sicut totum bz se habet ad z, ita e se habet ad d, quorum uterque est pars infiniti. Ergo commutatim quae est proportio d ad z, eadem est proportio e ad bz. Sed d movet z in g tempore: ergo e movebit bz in tempore g. Sed in hoc tempore, bz movebatur a corpore infinito quod est a: sequitur igitur quod infinitum et finitum alterent vel qualitercumque moveant in eodem tempore unum et idem mobile. Sed hoc est impossibile: supponebatur enim supra quod maius movens movet aequale mobile in minori tempore, quia velocius movet. Sic igitur impossibile est quod finitum moveatur ab infinito in tempore g; et idem sequitur quodcumque aliud tempus finitum sumatur. Nullum ergo tempus finitum est dare, in quo infinitum moveat finitum. He says therefore first [92] that neither will an infinite body move a finite body in any time, namely, determinate time. For if the contrary should be the case, let Abe the infinite body, and B or BZ the finite body moved by it, and G the time in which it is being moved. Let D be a finite part of the infinite body A. And because a lesser moves a smaller in equal time, then a finite body D in time G moves Z, a body smaller than B, but a part of B. Now because the whole BZ is proportionate to Z, let it be taken that the whole BZ is to Z, as E is to D, each of which is part of the infinite. Therefore, commutatively, E is to BZ in the same proportion as D is to Z. But D moves Z in time G; therefore E will move BZ in time G. But in time G, BZ was being moved by the infinite body A. It follows, therefore, than an infinite and a finite are altering or somehow moving one and the mane mobile in the same amount of time. But this is impossible — for it was supposed above that a greater mover moves an equal mobile in less time, because it moves more swiftly. Consequently, it is impossible for the finite to be moved by the infinite in time G; and the same follows no matter what finite time is taken. Hence there is no finite time possible in which the infinite moves the finite.
Deinde cum dicit: sed adhuc in infinito etc., ostendit quod neque hoc potest esse in tempore infinito. Non enim contingit quod in tempore infinito aliquid moverit vel motum sit: quia tempus infinitum non habet finem, omnis autem actio vel passio habet finem: nihil enim agit vel patitur nisi ut perveniat ad aliquem finem. Relinquitur ergo quod infinitum non moveat finitum in tempore infinito. 141. Then at 1[93] he shows that this cannot occur in infinite time. For it is not possible that in an infinite time something shall have moved or shall have been moved — because infinite time has no end, whereas every action or passion does have an end, for nothing acts or is acted upon except in order to reach some end. What remains, therefore, is that an infinite does not move a finite in infinite time.
Deinde cum dicit: neque infinitum utique etc., probat quod infinitum non moveat infinitum. Et dicit quod infinitum non contingit aliquid pati ab infinito secundum quamcumque speciem motus. Alioquin, sit corpus infinitum agens in quo est a, et corpus infinitum patiens in quo est b, tempus autem in quo b passum est ab a sit in quo dg; sit autem e pars infiniti mobilis quod est b. Quia ergo totum b passum est ab a in toto tempore quod est dg, manifestum est quod e, quod est pars eius, non movetur in toto hoc tempore: oportet enim supponere quod ab eodem movente minus mobile moveatur in minori tempore; quanto enim mobile magis vincitur a movente, tanto velocius movetur ab ipso. Sit ergo quod e, quod est minus quam b, moveatur ab a in tempore d, quod est pars totius temporis gd. D autem ad gd est aliqua proportio, cum utrumque sit finitum: accipiamus autem quod eandem proportionem habeat e ad aliquam partem ipsius mobilis infiniti maiorem, quam scilicet d habet ad gd. Sic ergo illud finitum maius quam e, necesse est quod moveatur ab a in gd tempore: oportet enim supponere quod ab eodem movente moveatur maius et minus mobile in maiori et minori tempore, ita quod divisio mobilium sit secundum proportionem temporum. Quia igitur proportio illius finiti ad e, est sicut proportio totius temporis gd ad d, oportet commutatim dicere quod proportio totius temporis gd ad illud mobile finitum maius, sit sicut proportio temporis d ad mobile e. Sed e movetur ab a in tempore d: ergo illud finitum maius movebitur ab a in tempore gd: et sic in eodem tempore movebitur finitum et infinitum, quod est impossibile. Et idem inconveniens sequitur, quodcumque tempus finitum accipiatur. Sic igitur impossibile est quod infinitum moveatur ab infinito in tempore finito. 142. Then at [94] he proves that the infinite does not move the infinite. And he says that an infinite cannot undergo anything from an infinite with respect to any species of motion at all. Otherwise let A be the infinite body which is acting, and B the infinite body acted upon, and DG the time in which B underwent something from A, and let E be a part of the infinite mobile B. Now, since the entire B has been modified by A in the entire time DG, it is clear that E, which is part of B, was not being moved in this whole time. For we must suppose that a smaller mobile is moved in less time by the same mover — for to the extent that a mobile is more overcome by a mover, the more swiftly is it moved by it. So let E, which is less than B, be moved by A in a time D which is part of the whole time GD. Now D is proportionate to GD, since both are finite. Let us assume, therefore, that E has the same ratio to some larger part of the infinite mobile as D has to GD. Then that finite mobile greater than E must be moved by A in time GD, for we must suppose that a larger and a smaller mobile are moved in more and less time when the same mover is acting, in such a way that the division of the mobiles corresponds to the ratio of the times. Since, therefore, the ratio of that finite to E equals the ratio of the entire time DG to D, then commutatively, we must say that the ratio of the entire time DG to that larger finite mobile is as the ratio of time D to mobile E. But E is moved by A in time D; therefore, that greater finite mobile will be moved by A in time DG. Hence the finite and the infinite will be moved in the same amount of time — which is impossible. And the same impossibility follows whatever be the finite time assumed. Consequently it is impossible for an infinite to be moved by an infinite in finite time.
Relinquitur igitur, si moveatur, quod moveatur in infinito tempore. Sed hoc est impossibile, ut supra ostensum est, quia infinitum tempus non habet finem, omne autem quod movetur, habet finem sui motus: quia etsi totus motus caeli non haberet finem, una tamen circulatio habet finem. Sic igitur manifestum est quod infinitum non habet neque virtutem activam neque passivam. It remains, therefore, that if it is moved, it is moved in infinite time. But that, too, is impossible, as was proved above, because infinite time has no end, but everything which is moved has an end to its motion — for although the whole motion of the heaven does not have an end, one revolution does. It is therefore plain that the infinite has neither active nor passive power.
Deinde cum dicit: si igitur etc., assumpta minori, infert conclusionem: dicens quod omne corpus sensibile habet virtutem activam aut passivam aut utramque. Dicitur autem hic corpus sensibile ad differentiam corporis mathematici: ita quod corpus sensibile dicatur omne corpus naturale, quod inquantum huiusmodi, natum est movere et moveri. Sic ergo concludit quod impossibile est aliquod corpus sensibile esse infinitum. 143. Then at [95], assuming the minor premise, he draws the conclusion and says that every sensible body has active or passive power or both.. He says "sensible body" here to differentiate from "mathematical body," so that the former means every natural body which, as such, is apt to cause motion or be moved. Thus he concludes that it is impossible for a sensible body to be infinite.
Deinde cum dicit: sed tamen et quaecumque etc., excludit quandam obviationem: quia posset aliquis dicere quod sit aliquod corpus extra caelum intelligibile, quod sit infinitum. 144. Then at [96] he excludes a certain objection — for someone could say that there is outside the heavens an "intelligible body" which is infinite.
Et dicit quod omnia corpora quae sunt in loco, sunt sensibilia. Non enim sunt corpora mathematica, quia talibus non debetur locus nisi secundum metaphoram, ut dicitur in I de Generat.: locus enim non quaeritur nisi propter motum, ut dicitur in IV Physic.; non autem moventur nisi corpora sensibilia et naturalia, nam mathematica sunt extra motum. Sic igitur manifestum est quod quaecumque corpora sunt in loco, sunt sensibilia. And he says that all bodies in place are sensible. For they are not mathematical bodies, because these do not have place except in a metaphorical sense, as is said in On Generation I. Now place is not needed except for motion, as is said in Physics IV, and only sensible and natural bodies are subject to motion — for mathematical things are outside of motion. Consequently, it is plain that all bodies in place are sensible.
Et ex hoc concludit quod corpus infinitum non sit extra caelum; immo universalius, quod nullum corpus sit extra caelum, neque simpliciter, scilicet corpus infinitum, neque secundum quid (vel usque ad aliquid), idest corpus finitum; cum enim corpus omne sit finitum vel infinitum, sequitur quod nullum omnino corpus sit extra caelum. Quia si dicas quod sit intellectuale, sequetur quod sit in loco, ex quo ponitur extra caelum: extra enim et intra significant locum. Sic igitur sequitur quod, si aliquod corpus sit extra caelum, finitum vel infinitum, quod sit sensibile; eo quod nullum sensibile corpus est, quod non sit in loco (quia etiam caelum quodammodo est in loco, ut patet in IV Physic.). From this he concludes that there is not an infinite body outside the heaven; and indeed, more universally, that no body exists outside the heaven, either absolutely, i.e., namely, an infinite body, or in a certain respect (or up to a certain point), i.e., a finite body. Since bodies are either finite or infinite, it follows that no body at all exists outside the heaven. For if you should say that this body is intellectual, it will follow that it is in a place on account of your assuming that it is outside the heaven — because "outside" and "within" imply place. Consequently, it follows that if there is a body outside the heaven, then, whether it is finite or infinite, it is sensible, since there is no sensible body which does not exist in a place —for even the heaven is somehow in place, as is plain from Physics IV.
Manifestum est autem secundum haec verba quod nullum corpus intelligibile, neque finitum neque infinitum, est extra caelum; quia extra significat locum, nihil autem est in loco nisi corpus sensibile. Manifestum est etiam quod nullum corpus infinitum sensibile est extra caelum: ostensum est enim supra quod nullum corpus sensibile est infinitum. Quod autem nullum corpus sensibile finitum sit extra caelum, non videtur hic probari, sed supponi: nisi forte per hoc quod omne corpus sensibile est in loco, omnia autem loca continentur infra caelum, quae determinantur tribus motibus localibus supra positis, scilicet qui sunt circa medium, a medio, et ad medium. So it is manifest according to these words that no intelligible body, finite or infinite, is outside the heaven, because "outside of" signifies place, and nothing is in place except a sensible body. It is also manifest that no infinite sensible body exists outside the heavens, for it was shown above that no sensible body is infinite. But the fact that no sensible finite body exists outside the heavens he does not prove here but supposes it, unless perhaps it is proved by the fact that every sensible body is in a place, and all places are contained within the heavens and determined by the three local motions mentioned above, namely, those around the middle, from the middle, and to the middle.

15 Lecture 15:
Logical reasons why no body is infinite.
Chapter 7 cont.
Λογικώτερον δ' ἔστιν ἐπιχειρεῖν καὶ ὧδε. Οὔτε γὰρ κύκλῳ οἷόν τε κινεῖσθαι τὸ ἄπειρον ὁμοιομερὲς ὄν μέσον μὲν γὰρ τοῦ ἀπείρου οὐκ ἔστι, τὸ δὲ κύκλῳ περὶ τὸ μέσον κινεῖται. Ἀλλὰ μὴν οὐδ' ἐπ' εὐθείας οἷόν τε φέρεσθαι τὸ ἄπειρον δεήσει γὰρ ἕτερον εἶναι τοσοῦτον τόπον ἄπειρον εἰς ὃν οἰσθήσεται κατὰ φύσιν, καὶ ἄλλον τοσοῦτον εἰς ὃν παρὰ φύσιν. 97 The question may also be examined in the light of more general considerations as follows. The infinite, considered as a whole of similar parts, cannot, on the one hand, move in a circle. For there is no centre of the infinite, and that which moves in a circle moves about the centre. Nor again can the infinite move in a straight line. For there would have to be another place infinite like itself to be the goal of its natural movement and another, equally great, for the goal of its unnatural movement.
Ἔτι εἴτε φύσει ἔχει κίνησιν τοῦ εἰς εὐθὺ εἴτε βίᾳ κινεῖται, ἀμφοτέρως δεήσει ἄπειρον εἶναι τὴν κινοῦσαν ἰσχύν ἥ τε γὰρ ἄπειρος ἀπείρου καὶ τοῦ ἀπείρου ἄπειρος ἡ ἰσχύς ὥστ' ἔσται καὶ τὸ κινοῦν ἄπειρον (λόγος δ' ἐν τοῖς περὶ κινήσεως ὅτι οὐθὲν ἔχει ἄπειρον δύναμιν τῶν πεπερασμένων, οὐδὲ τῶν ἀπείρων πεπερασμένην). Εἰ οὖν τὸ κατὰ φύσιν καὶ παρὰ φύσιν ἐνδέχεται κινηθῆναι, ἔσται δύο ἄπειρα, τό τε κινοῦν οὕτω καὶ τὸ κινούμενον. 98 Moreover, whether its rectilinear movement is natural or constrained, in either case the force which causes its motion will have to be infinite. For infinite force is force of an infinite body, and of an infinite body the force is infinite. So the motive body also will be infinite. (The proof of this is given in our discussion of movement, where it is shown that no finite thing possesses infinite power, and no infinite thing finite power.) If then that which moves naturally can also move unnaturally, there will be two infinites, one which causes, and another which exhibits the latter motion.
Ἔτι τὸ κινοῦν τὸ ἄπειρον τί ἐστιν; εἰ μὲν γὰρ αὐτὸ ἑαυτό, ἔμψυχον ἔσται. Τοῦτο δὲ πῶς δυνατόν, ἄπειρον εἶναι ζῷον; εἰ δ' ἄλλο [τι] τὸ κινοῦν, δύο ἔσται ἄπειρα, τό τε κινοῦν καὶ τὸ κινούμενον, διαφέροντα τὴν μορφὴν καὶ τὴν δύναμιν. 99 Again, what is it that moves the infinite? If it moves itself, it must be animate. But how can it possibly be conceived as an infinite animal? And if there is something else that moves it, there will be two infinites, that which moves and that which is moved, differing in their form and power.
Εἰ δὲ μὴ συνεχὲς τὸ πᾶν, ἀλλ' ὥσπερ λέγει Δημόκριτος καὶ Λεύκιππος, διωρισμένα τῷ κενῷ, μίαν ἀναγκαῖον εἶναι πάντων τὴν κίνησιν. Διώρισται μὲν γὰρ τοῖς σχήμασιν τὴν δὲ φύσιν φασὶν αὐτῶν εἶναι μίαν, ὥσ(276a.) περ ἂν εἰ χρυσὸς ἕκαστον εἴη κεχωρισμένος. Τούτων δέ, καθάπερ λέγομεν, ἀναγκαῖον εἶναι τὴν αὐτὴν κίνησιν ὅπου γὰρ μία βῶλος, καὶ ἡ σύμπασα γῆ φέρεται, καὶ τό τε πᾶν πῦρ καὶ σπινθὴρ εἰς τὸν αὐτὸν τόπον. Ὥστ' οὔτε κοῦφον ἁπλῶς οὐθὲν ἔσται τῶν σωμάτων, εἰ πάντ' ἔχει βάρος εἰ δὲ κουφότητα, βαρὺ οὐδέν. 100 If the whole is not continuous, but exists, as Democritus and Leucippus think, in the form of parts separated by void, there must necessarily be one movement of all the multitude. They are distinguished, we are told, from one another by their figures; but their nature is one, like many pieces of gold separated from one another. But each piece must, as we assert, have the same motion. For a single clod moves to the same place as the whole mass of earth, and a spark to the same place as the whole mass of fire. So that if it be weight that all possess, no body is, strictly speaking, light: and if lightness be universal, none is heavy.
Ἔτι εἰ βάρος ἔχει ἢ κουφότητα, ἔσται ἢ ἔσχατόν τι τοῦ παντὸς ἢ μέσον. Τοῦτο δ' ἀδύνατον ἀπείρου γ' ὄντος. 101 Moreover, whatever possesses weight or lightness will have its place either at one of the extremes or in the middle region. But this is impossible while the world is conceived as infinite.
Ὅλως δ', οὗ μή ἐστι μέσον μηδ' ἔσχατον, μηδὲ τὸ μὲν ἄνω τὸ δὲ κάτω, τόπος οὐθεὶς ἔσται τοῖς σώμασι τῆς φορᾶς. Τούτου δὲ μὴ ὄντος κίνησις οὐκ ἔσται ἀνάγκη γὰρ κινεῖσθαι ἤτοι κατὰ φύσιν ἢ παρὰ φύσιν, ταῦτα δ' ὥρισται τοῖς τόποις τοῖς τ' οἰκείοις καὶ τοῖς ἀλλοτρίοις. 102 And, generally, that which has no centre or extreme limit, no up or down, gives the bodies no place for their motion; and without that movement is impossible. A thing must move either naturally or unnaturally, and the two movements are determined by the proper and alien places.
Ἔτι εἰ οὗ παρὰ φύσιν τι μένει ἢ φέρεται, ἀνάγκη ἄλλου τινὸς εἶναι τοῦτον τὸν τόπον κατὰ φύσιν (τοῦτο δὲ πιστὸν ἐκ τῆς ἐπαγωγῆς), ἀνάγκη δὴ μὴ πάντα ἢ βάρος ἔχειν ἢ κουφότητα, ἀλλὰ τὰ μὲν τὰ δὲ μή. 103 Again, a place in which a thing rests or to which it moves unnaturally, must be the natural place for some other body, as experience shows. Necessarily, therefore, not everything possesses weight or lightness, but some things do and some do not.
Ὅτι μὲν τοίνυν οὐκ ἔστι τὸ σῶμα τοῦ παντὸς ἄπειρον, ἐκ τούτων φανερόν. From these arguments then it is clear that the body of the universe is not infinite.
Postquam philosophus ostendit universaliter non esse corpus infinitum rationibus physicis, idest quae sumuntur ex propriis scientiae naturalis, hic ostendit idem rationibus logicis, idest quae sumuntur ex aliquibus communioribus principiis, vel ex aliquibus probabilibus et non necessariis. Et hoc est quod dicit: est, idest contingit, conari ad propositum ostendendum rationabilius, idest magis per viam logicam, sic, idest secundum rationes sequentes. Unde alia littera planior est quae sic habet: magis autem logice est argumentari et sic. 145. After showing universally with Physical reasons, i.e., with arguments taken from facts proper to natural science, that there is no infinite body, the Philosopher here shows the same thing with logical reasons, i.e., arguments taken from certain common principles or from things that are probable but not necessary. And this is what he says [97]: "It is," i.e., it is possible, "to try," to prove the proposition "more from reason, i.e., more according to the logical mode, "thus," i.e., according to the following arguments. Hence another MS is more plain when it says: "One can argue more logically [i.e., dialectically] as follows."

Primo autem ostendit propositum de corpore infinito continuo;

secundo de infinito non continuo, ibi: si autem non continuum et cetera.

First he proves the proposition about an infinite continuous body;

Secondly, about one that is not continuous, at 150.

Circa primum duo facit. 146. Concerning the first he does two things:
Primo ostendit quod corpus infinitum, similium partium existens, non potest moveri circulariter. Quod quidem probat per hoc, quod infiniti non est aliquod medium, sicut nec extremum: motus autem circularis est circa medium, ut supra habitum est: ergo et cetera. First he shows that an infinite body of like parts cannot be moved circularly. This he proves on the ground that there is neither a middle nor a boundary in an infinite body. But circular motion is around a middle, as was had above. Therefore....
Secundo ostendit tribus rationibus quod non est possibile quod tale corpus infinitum moveatur motu recto. Quarum prima talis est. Omne corpus quod movetur motu recto, potest moveri naturaliter et per violentiam. Quod autem movetur per violentiam, habet aliquem locum in quem movetur violenter; et omne quod movetur naturaliter, habet aliquem locum in quem movetur naturaliter. Locus autem omnis est aequalis locato. Sic ergo sequetur quod sint duo loca tanta quantum est corpus infinitum, in quorum unum movetur violenter, et in alium naturaliter. Hoc autem est impossibile, scilicet quod sint duo loca infinita, sicut et quod sint duo infinita corpora, ut supra habitum est. Relinquitur ergo quod nullum corpus naturale sit infinitum. 147. Secondly, with three arguments he shows that it is not possible for such an infinite body to be moved with a straight motion. The first of these is this: Every body that is moved with a straight motion can be moved naturally and through force. Now what is moved by force has a place to which it is forcefully moved, and whatever is moved naturally has a place to which it is moved naturally. But every place is equal to the thing in place. Consequently, it will follow that there are two places as large as the infinite body, to one of which it is forcefully moved, and to the other of which naturally. But it is no more possible that there be two infinite places than that there be two infinite bodies, as was had prove. It remains, therefore, that no natural body is infinite.
Dicitur autem utraque ratio logica esse, quia procedit ex eo quod contingit corpori infinito inquantum est infinitum, sive sit mathematicum sive sit naturale, scilicet non habere medium, et non habere aliquid aequale extra se. Supra autem posuit aliqua similia, sed non tanquam principalia, sed tanquam assumpta ad manifestationem aliorum. Both of these reasons are called "logical," because they proceed from what occurs to an infinite body as infinite; whether it be mathematical or natural, namely, to have no middle and nothing equal to it outside of it. Above he posited similar statements, not, however, as principal premises but as assumptions used to manifest other things.
Secundam rationem ponit ibi: adhuc sive natura habet etc.: quae talis est. Sive dicatur quod corpus infinitum moveatur motu recto naturaliter, sive per violentiam, utroque modo oportet dicere quod sit potentia movens corpus infinitum: ostensum est enim in VII et VIII Physic. quod omne quod movetur ab alio movetur, non solum in his quae moventur per violentiam, de quibus magis est manifestum, sed etiam in his quae moventur naturaliter, sicut corpora gravia et levia moventur a generante vel a removente prohibens. Cum autem fortius non moveatur a debiliori, impossibile est quod infinitum, cuius virtus est infinita, moveatur a potentia finita moventis: unde relinquitur quod oportet potentiam moventis esse infinitam. Manifestum est autem quod, si potentia sit infinita, erit rei infinitae: et e converso, si corpus sit infinitum, oportet quod virtus eius sit infinita. 148. The second argument is given at [98], and is as follows: Whether it be said that an infinite body is moved naturally with straight motion or by force, in either case there must be posited a power moving the infinite body — for it was shown in Physics VII and VIII that whatever is moved is moved by another, not only in things that are moved by force (where the principle is more evident), but also in things that are moved naturally, as heavy and light bodies are moved by the generator [or agent producing them], or by whatever removes an obstacle. But since the stronger is not moved by the weaker, it is impossible for an infinite, whose power is infinite, to be moved by the finite power of some mover. Hence it remains that the power of the mover must be infinite.
Si ergo est corpus infinitum quod movetur, necesse est quod corpus movens sit etiam infinitum. Probatum est enim in his quae de motu, idest in VIII Physic., quod nullum finitorum habet virtutem infinitam, nec aliquod infinitorum habet virtutem finitam. Sic igitur patet quod, si sit corpus infinitum quod movetur motu recto, oportet quod moveatur a corpore infinito. But it is manifest that if a power is infinite, it will belong to an infinite thing; conversely, if a body is infinite, its power must be infinite. Therefore, if an infinite body is being moved, then the body moving it must be infinite. For it was proved "in the discussion on motion," i.e., in Physics VIII, that no finite thing has infinite virtue, and that no infinite has finite virtue. Consequently, it is plain that if an infinite body is being moved with straight motion, it must be being moved by an infinite body.
Si ergo ponamus quod hoc corpus infinitum contingit moveri et secundum naturam et praeter naturam, similiter continget secundum utrumque motum quod sint duo infinita, scilicet illud quod movet sic, idest naturaliter vel violenter, et aliud quod movetur. Hoc autem est impossibile, quod sint duo corpora infinita, ut supra ostensum est. Ergo non est possibile esse corpus infinitum quod moveatur motu recto. Now, if we assume that this infinite body can be moved both according to nature and beside its nature, it will likewise happen, with respect to each motion, that there are two infinites, namely, one that moves thus, i.e., causes natural or compulsory motion, and one that is moved. But this is impossible, namely, that there be two infinite bodies, as was proved above. Therefore, it is not possible for an infinite body to be moved with a straight motion.
Haec etiam ratio logica est, quia procedit ex communi proprietate infiniti corporis, quod scilicet non habeat extra se aliud corpus aequale. This argument is called "logical" because it proceeds from a common property of an infinite body, namely, that it does not have outside it another body equal to it.
Potest autem ex hac ratione concludi non solum quod sint duo infinita, sed plura. Nam si corpus infinitum movetur naturaliter, corpus naturaliter ipsum movens erit infinitum; et quia contingit ipsum moveri violenter, corpus quod movet ipsum violenter erit infinitum; et sic erunt tria infinita. It can be concluded from this argument not only that there would be two infinites but more still. For if the infinite body is moved naturally, the body moving it naturally will be infinite; and because it can be moved by force, the body that moves it by force will be infinite. Thus there will be three infinites.
Rursus, quia motus qui est violentus uni, est naturalis alteri, ut supra dictum est, sequetur etiam quod sit aliud corpus infinitum, quod naturaliter hoc modo moveatur a virtute infinita. Again, since a motion which is compulsory for one thing, is natural to another, as was stated above, it will follow, too, that there is another infinite body that is moved naturally in the aforesaid way by an infinite power.
Tertiam rationem ponit ibi: adhuc movens et cetera. Et haec quidem ratio inducitur ad excludendum obviationem quandam ad praedictam rationem. Posset enim aliquis dicere quod corpus infinitum movetur naturaliter non quidem ab alio, sed a seipso, sicut animalia dicuntur seipsa movere: et sic non sequetur esse duo corpora infinita, quod praemissa ratio concludebat. 149. The third argument he gives at [99]. And this argument is adduced in order to exclude an objection to the preceding argument. For someone could say that an infinite body is naturally moved not by some other body but by itself, as animals are said to move themselves. Consequently, it will not follow that there are two infinite bodies, as the preceding argument concluded.
Et ideo proponit quod necesse est dicere, si sit corpus infinitum, quod movens ipsum sit aliquid aliud: quia si moveret seipsum, esset animatum (hoc enim est proprium animalium, quod seipsa moveant). Si ergo corpus infinitum sit movens seipsum, sequetur quod sit animal infinitum. Sed hoc non videtur esse possibile, quia omne animal habet determinatam figuram et determinatam proportionem partium ad totum, quod non competit infinito. Sic igitur non potest dici quod infinitum moveat seipsum. Si autem dicatur quod aliquid aliud moveat ipsum, sequetur quod sint duo infinita, scilicet movens et quod movetur. Et ex hoc sequitur quod differunt secundum speciem et virtutem: quia movens comparatur ad mobile sicut actus ad potentiam. Hoc autem est impossibile, sicut prius dictum est. And therefore he proposes that it is necessary to say, that if there is an infinite body, whatever moves it is distinct from it. For if it moved itself, it would be animate—for it is proper to animals to move themselves. Consequently if the infinite body should move itself, it will be an infinite animal. But this does not seem possible, because every animal has a definite shape and a definite ratio between its parts and the whole, which factors do not belong to an infinite. Consequently, it cannot be said that the infinite moves itself. But if it be said that something else moves it, it will follow that there are two infinites, namely, the mover and the moved. And from this it follows that they differ in kind and in power: because the mover is related to the mobile as act to potency. But this is impossible, as was previously shown.
Deinde cum dicit: si autem non continuum etc., ostendit non esse infinitum non continuum, sed distinctum per interpositionem vacui, sicut posuerunt Democritus et Leucippus. Et hoc ostendit tribus rationibus. Circa quarum primam dicit quod, si infinitum non sit unum totum continuum, sed, sicut dicunt Democritus et Leucippus, distinguatur vacuo intermedio (ponebant enim quod corpora indivisibilia non possunt invicem coniungi nisi vacuo mediante); secundum autem horum opinionem sequitur quod necessarium sit omnium esse unum motum. Dicebant enim quod illa corpora indivisibilia infinita sunt determinata, idest distincta ad invicem, solummodo per figuras, inquantum scilicet unum eorum est pyramidale, aliud sphaericum, aliud cubicum, et sic de aliis; et tamen dicunt naturam omnium eorum esse unam, sicut si aliquis dicat quod unumquodque eorum, per se separatum, sit de natura auri. Si autem eorum est una natura, necesse est quod sit unus et idem motus eorum, non obstante quod sint minimae partes corporum; quia idem est motus totius et partis, sicut totius terrae et unius boli (idest unius particulae), et totius ignis et unius scintillae. 150. Then at [100] he shows that there is no infinite which is non-continuous but distinguished by the interposition of voids, as Democritus and Leucippus posited. This he proves with three arguments. With regard to the first he says that if an infinite is not one continuous whole but is, as Democritus and Leucippus maintain, distinguished by an intermediate void — for they posited that the indivisible bodies cannot be mutually joined without an intervening void — then according to their opinion it follows that for all of them there is one motion. For they said that those infinite indivisible bodies are determined, i.e., mutually distinguished, only by their shape, namely, insofar as one is pyramidal, another spherical, another cubic, and so on. Yet they say that all of them are one with respect to their nature, as if, for example, someone said that each of them in isolation had the nature of gold. But if the nature of all is one, then, necessarily, all have one and the same motion in spite of their being the minimal parts of bodies — because the motion of the whole and of the part is the same, as is the motion of the whole earth and one clod, and of all fire and one spark.
Si ergo omnia sunt eiusdem naturae et habent eundem motum, aut omnia moventur deorsum quasi habentia gravitatem, et sic nullum corpus erit simpliciter leve, cum omnia corpora dicantur esse ex his composita; aut omnia moventur sursum quasi habentia levitatem, et sic nullum corpus erit grave; quod est impossibile. Therefore, if all are of the same nature and have the same motion, then all are either moved downward as though having gravity — and thus there will be no body that is absolutely light, since all bodies are said to be composed of these; or else all are moved upward, as though having lightness, and thus no body will be heavy — which is impossible.
Secundam rationem ponit ibi: adhuc si gravitatem etc.: quae talis est. Omne corpus grave movetur ad medium, omne autem corpus leve movetur ad extremum. Si ergo aliquod vel quodlibet praedictorum indivisibilium corporum haberet gravitatem aut levitatem, sequeretur quod totius spatii contenti ex indivisibilibus corporibus et vacuis intermediis, sit aliquod extremum aut medium. Sed hoc est impossibile, cum totum istud spatium sit infinitum. Relinquitur ergo hanc positionem esse impossibilem. 151. The second argument, given at [101], is this: Every heavy body is moved to the middle and every light body to the boundary. If, therefore, some or each of the aforesaid indivisible bodies had heaviness or lightness, it would follow that there would be a boundary and a center of that whole space contained by the indivisible bodies and the intermediate voids. But that is impossible, since all that space is infinite. It remains, therefore, that this position is impossible.
Et quia haec ratio valet ad destruendum infinitum, qualitercumque infinitum ponatur, sive sicut continuum sive sicut non continuum, ideo hanc eandem rationem universalius ponit cum subdit: totaliterque et cetera. Et dicit quod universaliter possumus dicere quod ubi non est medium et extremum, ibi non est sursum, quod est extremum, neque deorsum, quod est medium. Quibus subtractis, nullus locus erit quo corpora ferantur motu recto: feruntur enim sursum vel deorsum. Sublato autem loco, nullus erit motus: quia omne quod movetur necesse est moveri aut secundum naturam aut praeter naturam, quod quidem determinatur per loca propria et aliena (nam motus naturales dicuntur quibus corpora moventur ad loca propria, motus autem violenti dicuntur quibus moventur ad loca aliena). Hoc autem est impossibile, quod motus auferatur a corporibus: ergo impossibile est ponere infinitum. 152. And since this argument effectively destroys the infinite howsoever assumed, i.e., whether continuous or non-continuous, he therefore presents this same argument in a more universal way at [102]. And he says that we can say universally that where there is no middle and no extreme boundary, there is no "up," which is the boundary, and no "down," which is the middle. And if these are removed, there is no place where bodies can be moved with straight motion; for they are moved upward or downward. But if place is removed, there will be no motion — for whatever is moved, must be moved either according to its nature or outside its nature, and this is judged by places that are proper and alien — for natural motions are those in which bodies are moved to their proper places, while compulsory motions are those in which they are moved to alien places. But this is impossible, namely, that motion be taken away from bodies. Therefore, it is impossible to posit an infinite.
Tertiam rationem ponit ibi: adhuc si ubi et cetera. Et dicit quod locus ad quem movetur aliquid praeter naturam, vel in quo quiescit praeter naturam, necesse est quod sit cuiusdam alterius secundum naturam, ad quem scilicet naturaliter moveatur, et in quo naturaliter quiescat. Et hoc credibile fit ex inductione: nam terra movetur sursum praeter naturam, ignis vero secundum naturam; et e converso ignis deorsum praeter naturam, terra vero secundum naturam. Videmus autem quaedam moveri deorsum et quaedam sursum. Si autem illa quae moventur sursum, moventur praeter naturam, oportebit dicere aliqua alia esse quae moventur sursum secundum naturam; et similiter, si ponatur quod ea quae moventur deorsum, moventur praeter naturam, necesse est ponere alia quae moventur deorsum secundum naturam. Unde neque omnia habent gravitatem, neque omnia levitatem, secundum positionem praedictam: sed haec quidem habent gravitatem quae naturaliter moventur deorsum; haec autem non, quae naturaliter moventur sursum. 153. The third argument is given at [103]. And he says that the place to which something is moved outside its nature, or in which it rests outside its nature, must be according to nature for something else which is moved to it naturally and rests in it naturally. And this becomes credible by induction: for earth is moved upward outside its nature but fire according to nature; conversely, fire is moved downward outside its nature but earth according to nature. Now we observe certain things being moved downward and others upward. If the things being moved upward are moved outside their nature, we will be obliged to say that there are other things which are moved upward according to nature; likewise, if the things being moved downward are assumed to be moved outside their nature, it is necessary to posit other things that are moved downward according to nature. Hence not all things have heaviness and not all have lightness according to the foregoing position, but those naturally moved downward have heaviness, while those naturally moved upward do not have it.
Ultimo autem epilogando concludit manifestum esse ex praedictis quod omnino non est corpus infinitum, scilicet infinitum continuum neque infinitum distinctum per interpositionem vacui. Finally in summary he concludes [104] that it is manifest from the foregoing that there is no infinite body at all, i.e., no infinite that is continuous and none that is distinguished by intervals of void.
Dicuntur autem hae ultimae rationes logicae, quia procedunt ex quibusdam probabilibus nondum plene probatis. And these last arguments are called "logical," because they proceed. from probabilities not yet completely proved.

Lecture 16:
Two arguments for one universe, taken from lower bodies.
Chapter 8
Διότι δ' οὐδὲ πλείους οἷόν τ' οὐρανοὺς εἶναι, λέγωμεν τοῦτο γὰρ ἔφαμεν ἐπισκεπτέον, εἴ τις μὴ νομίζει καθόλου δεδεῖχθαι περὶ τῶν σωμάτων ὅτι ἀδύνατον ἐκτὸς εἶναι τοῦ κόσμου τοῦδε ὁτιοῦν αὐτῶν, ἀλλὰ μόνον ἐπὶ τῶν ἀορίστως κειμένων εἰρῆσθαι τὸν λόγον. 105 We must now proceed to explain why there cannot be more than one heaven—the further question mentioned above. For it may be thought that we have not proved universal of bodies that none whatever can exist outside our universe, and that our argument applied only to those of indeterminate extent.
Ἅπαντα γὰρ καὶ μένει καὶ κινεῖται καὶ κατὰ φύσιν καὶ βίᾳ, καὶ κατὰ φύσιν μέν, ἐν ᾧ μένει μὴ βίᾳ, καὶ φέρεται, καὶ εἰς ὃν φέρεται, καὶ μένει ἐν ᾧ δὲ βίᾳ, καὶ φέρεται βίᾳ, καὶ εἰς ὃν βίᾳ φέρεται, βίᾳ καὶ μένει. Ἔτι εἰ βίᾳ ἥδε ἡ φορά, ἡ ἐναντία κατὰ φύσιν. 106 Now all things rest and move naturally and by constraint. A thing moves naturally to a place in which it rests without constraint, and rests naturally in a place to which it moves without constraint. On the other hand, a thing moves by constraint to a place in which it rests by constraint, and rests by constraint in a place to which it moves by constraint. Further, if a given movement is due to constraint, its contrary is natural.
Ἐπὶ δὴ τὸ μέσον τὸ ἐνταῦθα εἰ βίᾳ οἰσθήσεται ἡ γῆ ἐκεῖθεν, ἐντεῦθεν οἰσθήσεται ἐκεῖ κατὰ φύσιν 107 If, then, it is by constraint that earth moves from a certain place to the centre here, its movement from here to there will be natural,
καὶ εἰ μένει ἐνταῦθα ἡ ἐκεῖθεν μὴ βίᾳ, καὶ οἰσθήσεται δεῦρο κατὰ φύσιν. Μία γὰρ ἡ κατὰ φύσιν. 108 and if earth from there rests here without constraint, its movement hither will be natural. And the natural movement in each case is one.
Ἔτι ἀνάγκη πάντας τοὺς κόσμους ἐκ τῶν αὐτῶν εἶναι σωμάτων, ὁμοίους γ' ὄντας τὴν φύσιν. Ἀλλὰ μὴν καὶ τῶν σωμάτων ἕκαστον ἀναγκαῖον τὴν αὐτὴν (276b.) ἔχειν δύναμιν, οἷον λέγω πῦρ καὶ γῆν καὶ τὰ μεταξὺ τούτων εἰ γὰρ ὁμώνυμα ταῦτα καὶ μὴ κατὰ τὴν αὐτὴν ἰδέαν λέγονται τἀκεῖ τοῖς παρ' ἡμῖν, καὶ τὸ πᾶν ὁμωνύμως ἂν λέγοιτο κόσμος. Δῆλον τοίνυν ὅτι τὸ μὲν ἀπὸ τοῦ μέσου φέρεσθαι πέφυκε, τὸ δ' ἐπὶ τὸ μέσον αὐτῶν, εἴπερ πᾶν ὁμοειδὲς τὸ πῦρ τῷ πυρὶ καὶ τῶν ἄλλων ἕκαστον, ὥσπερ καὶ τὰ ἐν τούτῳ μόρια τοῦ πυρός. 109 Further, these worlds, being similar in nature to ours, must all be composed of the same bodies as it. Moreover each of the bodies, fire, I mean, and earth and their intermediates, must have the same power as in our world. For if these names are used equivocally, if the identity of name does not rest upon an identity of form in these elements and ours, then the whole to which they belong can only be called a world by equivocation. Clearly, then, one of the bodies will move naturally away from the centre and another towards the centre, since fire must be identical with fire, earth with earth, and so on, as the fragments of each are identical in this world.
Ὅτι δ' ἀναγκαῖον οὕτως ἔχειν, ἐκ τῶν περὶ τὰς κινήσεις ὑποθέσεων φανερόν αἵ τε γὰρ κινήσεις πεπερασμέναι, ἕκαστόν τε τῶν στοιχείων λέγεται καθ' ἑκάστην τῶν κινήσεων. Ὥστ' εἴπερ καὶ αἱ κινήσεις αἱ αὐταί, καὶ τὰ στοιχεῖα ἀναγκαῖον εἶναι πανταχοῦ ταὐτά. 110 That this must be the case is evident from the principles laid down in our discussion of the movements, for these are limited in number, and the distinction of the elements depends upon the distinction of the movements. Therefore, since the movements are the same, the elements must also be the same everywhere.
Πέφυκεν ἄρα φέρεσθαι καὶ ἐπὶ τόδε τὸ μέσον τὰ ἐν ἄλλῳ κόσμῳ τῆς γῆς μόρια, καὶ πρὸς τόδε τὸ ἔσχατον τὸ ἐκεῖ πῦρ. Ἀλλ' ἀδύνατον τούτου γὰρ συμβαίνοντος ἀνάγκη φέρεσθαι ἄνω μὲν τὴν γῆν ἐν τῷ οἰκείῳ κόσμῳ, τὸ δὲ πῦρ ἐπὶ τὸ μέσον, ὁμοίως δὲ καὶ τὴν ἐντεῦθεν γῆν ἀπὸ τοῦ μέσου φέρεσθαι κατὰ φύσιν πρὸς τὸ ἐκεῖ φερομένην μέσον, διὰ τὸ τοὺς κόσμους οὕτω κεῖσθαι πρὸς ἀλλήλους. Ἢ γὰρ οὐ θετέον τὴν αὐτὴν εἶναι φύσιν τῶν ἁπλῶν σωμάτων ἐν τοῖς πλείοσιν οὐρανοῖς, ἢ λέγοντας οὕτως τὸ μέσον ἓν ποιεῖν ἀνάγκη καὶ τὸ ἔσχατον τούτου δ' ὄντος ἀδύνατον εἶναι κόσμους πλείους ἑνός. 111 The particles of earth, then, in another world move naturally also to our centre and its fire to our circumference. This, however, is impossible, since, if it were true, earth must, in its own world, move upwards, and fire to the centre; in the same way the earth of our world must move naturally away from the centre when it moves towards the centre of another universe. This follows from the supposed juxtaposition of the worlds. For either we must refuse to admit the identical nature of the simple bodies in the various universes, or, admitting this, we must make the centre and the extremity one as suggested. This being so, it follows that there cannot be more worlds than one.
Τὸ δ' ἀξιοῦν ἄλλην εἶναι φύσιν τῶν ἁπλῶν σωμάτων, ἂν ἀποσχῶσιν ἔλαττον ἢ πλεῖον τῶν οἰκείων τόπων, ἄλογον τί γὰρ διαφέρει τὸ τοσονδὶ φάναι μῆκος ἀπέχειν ἢ τοσονδί; Διοίσει γὰρ κατὰ λόγον, ὅσῳ πλεῖον μᾶλλον, τὸ δ' εἶδος τὸ αὐτό. 112 To postulate a difference of nature in the simple bodies according as they are more or less distant from their proper places is unreasonable. For what difference can it make whether we say that a thing is this distance away or that? One would have to suppose a difference proportionate to the distance and increasing with it, but the form is in fact the same.
Postquam philosophus ostendit quod universum non est infinitum magnitudine, hic ostendit quod non sunt plures mundi numero, nedum quod sint infiniti. 154. After showing that the universe is not infinite in magnitude, the Philosopher here shows that there are not numerically many worlds, much less an infinitude of them.

Et primo dicit de quo est intentio;

secundo exequitur propositum, ibi: omnia enim et manent et cetera.

First he mentions his intention;

Secondly, he pursues his proposition, at 155.

Dicit ergo primo quod, quia ostensum est quod corpus totius universi non est infinitum, restat dicendum quod non est possibile esse plures caelos, idest plures mundos: iam enim supra diximus quod de hoc erat intendendum. He says therefore first [105] that because it has been proved that the body of the whole universe is not infinite, there remains for us to say that it is not possible that there be many heavens, i.e., many worlds: for we had already mentioned above that this was to be discussed.
Est autem considerandum quod supra philosophus fecit mentionem quod extra caelum non est aliquod corpus neque finitum neque infinitum; ex quo sequitur quod non sit alius mundus praeter istum; esset enim aliquod corpus extra caelum. Et ideo, si sufficienter esset supra probatum quod extra caelum non sit aliquod corpus neque finitum neque infinitum, nihil restaret probandum. Sed si quis non putat quod in superioribus sit ostensum universaliter de corporibus, quod scilicet impossibile sit quodcumque eorum esse extra mundum, sed solum quod ratio supra sit inducta de corporibus quae ponuntur esse infinita; secundum hoc adhuc restat videndum an sit possibile esse plures caelos, sive plures mundos. It should be noted that above the Philosopher mentioned that outside the heavens there is no body either finite or infinite; from which it follows that there is not another world besides it, for that would put a body outside the heavens. Consequently, if it were sufficiently proved above that outside the heavens there exists no body either finite or infinite, nothing would remain to be proved. But if someone does not consider that it was proved for bodies universally, namely, that it is impossible for any of them to be outside the world, but considers that the argument given above refers only to bodies assumed infinite, then, according to this, it still remains to be seen whether it is possible that there be many heavens, i.e., many worlds.
Deinde cum dicit: omnia enim et manent etc., probat propositum: 155. Then at [106] he proves his proposition:

et primo ostendit quod sit tantum unus mundus;

secundo inquirit an possibile sit esse plures mundos, ibi: quod autem non solum unus et cetera.

First he shows that there is but one world;

Secondly, he inquires whether it is possible that there be many worlds (L. 19).

Circa primum duo facit: As to the first he does two things:

primo ostendit esse tantummodo unum mundum, ratione sumpta ex inferioribus corporibus, ex quibus omnes ponebant mundum consistere;

secundo ostendit idem communiter ex utrisque corporibus, tam inferioribus quam caelestibus, ibi: adhuc autem et per eas et cetera.

First he shows that there is only one world and takes his argument from the lower bodies, of which everyone supposed the world to consist, at 156;

Secondly, he shows the same with a general argument based on both the lower and the celestial bodies (L. 18).

Circa primum duo facit: About the first he does two things:

primo inducit rationes ad propositum ostendendum;

secundo probat quoddam quod supposuerat, ibi: quod autem est aliquid et cetera.

First he adduces arguments to prove his proposition;

Secondly, he proves something he had presupposed (L. 17).

Circa primum ponit tres rationes:

secunda incipit ibi: adhuc necesse etc.;

tertia ibi: sed adhuc et cetera.

With regard to the first he gives three arguments:

The second one begins at 159;

The third one in Lecture 17.

Circa primum duo facit. 156. Regarding the first he does two things:
Primo praemittit tres suppositiones. Quarum prima est, quod omnia corpora quiescunt et moventur tam secundum naturam, quam etiam secundum violentiam. Quod quidem habet veritatem in corporibus inferioribus, quae cum sint generabilia et corruptibilia, sicut per vim fortioris agentis possunt permutari a sua specie, ita etiam possunt removeri a suo loco per motum violentum vel quietem: in corporibus autem caelestibus nihil potest esse violentum et extra naturam, cum sint incorruptibilia. First he presents three suppositions. The first is that all bodies rest and are moved both according to nature and according to compulsion. This of course is true in lower bodies which, since they can be generated and corrupted, can not only be transmuted from their species by the power of a stronger agent, but can be removed from their place by a violent motion or by violent rest. But in celestial bodies, since they are incorruptible, nothing can be violent and outside their nature.
Secunda suppositio est, quod in quocumque loco aliqua corpora manent secundum naturam et non per violentiam, in illum locum per naturam feruntur: et in quemcumque locum e converso aliqua per naturam feruntur, in illo loco naturaliter quiescunt. Et idem dicendum est circa violentiam: quia in quo loco aliqua quiescunt per violentiam, in illum locum feruntur per violentiam; et e converso, si ad aliquem locum feruntur per violentiam, in illo loco per violentiam quiescunt. Et huius suppositionis ratio est quia, cum quies in loco sit finis motus localis, oportet motum proportionari quieti, sicut finis proportionatur his quae sunt ad finem. The second supposition is that in whatever place certain bodies remain according to nature and not through compulsion, they are moved thither by nature, and into whatever place things are carried by nature they naturally rest there. And the same is to be said about violence: in whatever place things rest through violence, they are carried to that place by violence; conversely, if they are carried to a place through violence, they are at rest there through violence. The reason for this supposition is that since rest in a place is the end of local motion, the motion must be proportionate to the rest, just as the end is proportionate to the means.
Tertia suppositio est, quod si aliqua loci mutatio sit per violentiam alicui corpori, contraria est ei secundum naturam, sicut patet ex his quae supra dicta sunt. The third supposition is that if any change of place is accomplished by violence to a body, the contrary change is according to nature for that body, as is plain from what was said above.
Secundo ibi: ad medium itaque etc., ex praedictis suppositionibus argumentatur ad propositum. Primo quidem ex parte motus. Si enim sunt duo mundi, oportet esse in utroque aliquam terram. Terra ergo quae est in alio mundo, aut feretur ad medium huius mundi per naturam, aut per violentiam. Si per violentiam, oportebit dicere, secundum tertiam suppositionem, quod contraria loci mutatio, quae est ab isto mundo in medium illius mundi, sit ei secundum naturam. Et hoc patet esse falsum, quia a medio istius mundi nunquam terra movetur secundum naturam: ergo et primum est falsum, scilicet quod sint plures mundi. 157. Secondly, at [107] from these suppositions he argues to his proposition. First on the part of motion. For if there are two worlds, there must be earth in both. Therefore the earth in that other world will be moved to the middle of this world either by nature or by compulsion. If by the latter, we shall have to say, according to the third supposition, that the contrary change of place, i.e., from this world to the middle of that world is natural to it. And this is plainly false, since earth is never naturally moved from the middle of this world. Therefore, the first is also false, namely, that there is more than one world.
Secundo ibi: et si manet etc., argumentatur ad idem ex parte quietis. Sicut enim manifestum est quod natura terrae non patitur quod moveatur secundum naturam a medio huius mundi, ita etiam terrae natura hoc habet, quod in medio huius mundi quiescat naturaliter. Si ergo inde huc delata terra manet hic non per violentiam, sed per naturam, sequitur per secundam suppositionem quod ab illo medio feretur huc secundum naturam. Et hoc ideo, quia unus est motus, vel una loci mutatio terrae secundum naturam: unde non potest esse quod uterque motus sit terrae naturalis, scilicet ab illo medio ad istud, et ab isto ad illud. 158. Secondly, at [108] he argues to the same on the part of rest. For just as it is plain that the nature of earth does not allow being moved naturally from the middle of this world, so, too, the nature of earth has this quality, that it be naturally at rest in the middle of this world. If then earth brought here from that world remains here not by violence but by nature, it follows, according to the second supposition, that it will be brought from that middle to here according to nature. And this is so because there is but one motion, or one change of place, that is according to nature for earth; hence both motions cannot be natural to earth, namely, from that middle to this or from this to that.
Deinde cum dicit: adhuc necesse etc., ponit secundam rationem, quae excludit quendam defectum quem posset aliquis imponere primae rationi: posset enim aliquis ad primam rationem respondere quod terra quae est in illo mundo, est alterius naturae quam terra quae est in hoc mundo. 159. Then at [109] he presents a second argument which excludes a certain defect which someone can claim in the first argument: for someone could answer to the first that the earth in that world is different in nature from that in this world.

Primo ergo Aristoteles hoc excludit;

secundo ex hoc argumentatur ad propositum, ibi: natae sunt igitur ferri etc.;

tertio excludit quandam obviationem, ibi: dignificare autem et cetera.

First, then, Aristotle dismisses this at 160;

Secondly, from this he argues to his proposition, at 162;

Thirdly, he excludes an objection, at 163.

Ostendit autem terram quae est in alio mundo, esse eiusdem naturae cum terra quae est in hoc mundo, He shows that the earth in the other world is of the same nature as that of this world:

primo quidem ratione accepta ex parte mundi;

secundo ratione accepta ex parte motus, ibi: quod autem necesse sit et cetera.

First with an argument taken on the part of the world, at

Secondly, with one based on motion, at 161.

Dicit ergo primo quod, si plures mundi qui ponuntur sint similis naturae, necesse est quod sint ex eisdem corporibus: et adhuc ulterius necesse est quod unumquodque illorum corporum habeat eandem virtutem cum corpore quod est in hoc mundo: et sic oportet ignem et terram esse eiusdem virtutis in quolibet illorum mundorum, et eadem ratio est de intermediis corporibus, quae sunt aer et aqua. Quia si corpora quae sunt ibi in alio mundo, dicuntur aequivoce cum corporibus quae sunt apud nos in hoc mundo, et non secundum eandem ideam, idest non secundum eandem speciem, consequens erit quod etiam ipsum totum constans ex huiusmodi partibus aequivoce dicatur mundus: ex partibus enim diversis in specie necesse est et totum diversum in specie componi. Hoc autem non videntur intendere qui ponunt plures mundos; sed univoce utuntur nomine mundi. Unde sequitur secundum eorum intentionem quod corpora quae sunt in diversis mundis, habeant eandem virtutem. Et ita manifestum est quod etiam in aliis mundis, sicut et in isto, aliquod ipsorum corporum ex quibus constituitur mundus, natum sit ferri a medio, quod competit igni, aliud autem ad medium, quod competit terrae; si hoc verum est, quod omnis ignis omni igni est eiusdem speciei, in quocumque mundo sit ignis, sicut et diversae partes ignis in hoc mundo existentis sunt unius speciei. Et eadem est ratio de aliis corporibus. 160. He says therefore first [109] that if the several worlds posited are of a like nature, they must be composed of the same bodies; further, each of those bodies must have the same virtue as the body of this world. Consequently, fire and earth must have the same virtue in each of those worlds, and the same goes for the intermediate bodies, air and water. For if the bodies that are there in another world are spoken of equivocally in relation to the bodies that exist among us in this world and are not according to the same "idea," i.e., not of the same species, the consequence will be that the entire world consisting of such bodies will be only equivocally called a world. For wholes that are composed of parts diverse in species are themselves diverse. But this does not seem to be the intention of those who posit many worlds; rather they use the word "world" univocally. Hence it follows according to their intention that the bodies in these different worlds possess the same virtue. And thus it is manifest that even in those worlds, just as in this, some one of the bodies constituting the world is apt to be moved from the middle, which belongs to fire, and some other to the middle, which belongs to earth, if it is true that all fire is akin in species to all other fire in whatever world it exists, just as the various parts of fire in this world are of one species. And the same holds for the other bodies.
Deinde cum dicit: quod autem necesse etc., ostendit idem ratione accepta ex parte motus. Et dicit manifestum esse quod necesse sit sic se habere sicut dictum est, de uniformitate corporum quae sunt in diversis mundis; et hoc ex suppositionibus quae accipiuntur circa motus. Vocat autem suppositiones ea quibus utitur ad propositum ostendendum, propter hoc quod hic supponuntur sicut principia, licet quaedam eorum supra fuerint probata. Est autem una suppositio quod motus sunt finiti, idest determinati secundum species: non enim sunt infinitae species motuum simplicium, sed tres tantum, ut supra probatum est. Secunda suppositio est quod quodlibet elementorum dicitur secundum quod habet naturam ad unum aliquem motuum; sicut terra dicitur gravis propter habitudinem ad motum deorsum, ignis dicitur levis propter aptitudinem ad motum sursum. 161. Then at [110] he shows the same thing with an argument taken from motion. And he says that it is manifestly necessary that things be as we have said concerning the uniformity of the bodies which are in the various worlds; and this from the suppositions which are assumed with respect to motions. And he calls "suppositions" the statements which he uses to show the proposition, because here they are being assumed as principles, although some of them have been previously proved. Now one of the suppositions is that motions are finite, i.e., determinate with respect to species; for there are not infinite species of simple notions, but three only, as was proved above. A second supposition is that each of the elements is described in terms of having a natural tendency toward some one of the motions; as earth is described as heavy on account of its tendency to downward motion, and fire light on account of its aptitude for upward motion.
Quia igitur sunt determinatae species motus, necesse est quod sint iidem motus secundum speciem in quolibet mundo. Et quia unumquodque elementorum dicitur secundum aliquem motuum, necesse est ulterius quod elementa sint eadem secundum speciem ubique, idest in quolibet mundo. Hence, since the species of motion are determinate, the same specific motions must exist in every world. And because each of the elements is described with respect to some motion, it is further necessary that the elements are specifically the same everywhere, i.e., in each world.
Deinde cum dicit: natae sunt igitur etc., ex praemissis argumentatur ad propositum. Si enim corpora quae sunt in quolibet mundo, sunt eiusdem speciei; videmus autem quod omnes partes terrae quae sunt in hoc mundo, feruntur ad hoc medium huius mundi, et omnes partes ignis ad extremum huius; consequens erit quod etiam omnes partes terrae quae sunt in quocumque alio mundo, feruntur ad medium huius mundi; et omnes partes ignis quae sunt in quocumque alio mundo, feruntur ad extremum huius mundi. Sed hoc est impossibile. Si enim hoc accideret, necesse esset quod terra quae est in alio mundo, ferretur sursum in proprio suo mundo, et quod ignis in illo mundo ferretur ad medium. Et simili ratione terra quae est in hoc mundo, ferretur secundum naturam a medio huius mundi in medium illius mundi. 162. Then at [111] from these premises he argues to the proposition. For if the bodies in every world are of the same species, and we see that all the parts of earth in this world are carried to the middle of this world, and all parts of fire to its boundary, then the consequence will be that also all the parts of earth in any other world are moved to the middle of this world, and all the parts of fire in any other world to the boundary of this world. But this is impossible. For if this should happen, the earth in another world would have to be carried upward in its own world and fire in that world would have to be carried to its middle. Similarly, the earth in this world would be naturally carried from the center of this world to the center of that world.
Et hoc necesse est sequi propter dispositionem mundorum, qui talem situm habent ut medium unius mundi sit distans a medio alterius; et sic non potest terra ad medium alterius mundi moveri, nisi recedat a medio sui mundi mota versus extremum, quod est moveri sursum. Similiter, quia extrema diversorum mundorum habent diversum situm, necesse est quod si ignis debeat ferri ad extremum alterius mundi, quod recedat ab extremo proprii mundi, quod est moveri deorsum in proprio mundo. Haec autem sunt inconvenientia: quia aut ponendum est quod non sit eadem natura simplicium corporum in pluribus mundis, quod supra improbatum est; aut si dicamus esse eandem naturam, et velimus vitare praedicta inconvenientia quae sequuntur ex diversitate mediorum et extremorum, necesse est ponere unum solum medium, ad quod feruntur omnia gravia ubicumque sint, et unum extremum, ad quod feruntur omnia levia ubicumque sint. Quo posito, impossibile est esse plures mundos; quia ad unitatem medii et extremi sequitur unitas circuli seu sphaerae. And this must follow on account of the disposition of the worlds which have such a position that the middle of one world is at a distance from the middle of another; consequently, earth cannot be moved to the middle of another world without leaving the middle of its own world and moving to the boundary, which is to be moved upward. Likewise, because the boundaries of various worlds have different positions, then if fire is to be carried to the boundary of another world, it must leave the boundary of its own world, which is to be moved downward in its own world. But all these things are untenable — for either we must posit that the natures of the simple bodies are not the same in the several worlds (which was disproved above), or, if we say that they are of the same nature and wish to avoid the aforesaid inconsistencies which follow upon a diversity of middles and boundaries, we must admit but one middle to which all heavy bodies, wherever they are, are moved, and one boundary to which are moved all light things wherever they be. On this assumption, it is impossible that there be many worlds, because one middle and one boundary imply one circle or sphere.
Deinde cum dicit: dignificare autem etc., excludit quandam obviationem, qua posset aliquis dicere quod corpora quae sunt in alio mundo, non moventur ad medium et extremum huius mundi, propter distantiam. 163. Then at [112] he excludes an objection, since someone could say that the bodies in another world are not moved to the center and boundaries of this world on account of the distance.
Sed ipse hoc excludens dicit quod irrationabile est dignum reputare quod sit alia natura simplicium corporum, propter hoc quod distent plus vel minus a propriis locis, ita scilicet quod ad propria loca moveantur de propinquo et non de remoto. Non enim videtur differre quantum ad naturam corporis, quod per tantam longitudinem distet a suo loco vel per tantam: quia differentia mathematicorum non diversificat naturam. Est enim secundum rationem quod quanto plus corpus appropinquat ad suum locum, tanto magis velociter moveatur; ita tamen quod species sit eadem et motus et mobilis. Differentia enim velocitatis est secundum quantitatem, non secundum speciem; sicut et differentia longitudinis. But he rejects this and says that it is unreasonable to accept the postulate that the natures of simple bodies vary on the ground of their being more or less distant from their places, so as to be moved to their places when they are near but not when they are far away. For it does not seem to make any difference to the nature of the body whether it is this far or that far from its place, because mathematical differences do not vary the nature. For it is according to reason that the closer a body gets to its place the more swiftly is it moved, but yet the species of its motion and of the mobile are not varied. For a difference in velocity is according to quantity, not according to species, just as is a difference in length.

Lecture 17:
A third argument from lower bodies. Natural bodies have determinate places
Chapter 8 cont.
Ἀλλὰ μὴν ἀνάγκη γ' εἶναί τινα κίνησιν αὐτῶν ὅτι μὲν γὰρ κινοῦνται, φανερόν. Πότερον οὖν βίᾳ πάσας ἐροῦμεν κινεῖσθαι καὶ τὰς ἐναντίας; ἀλλ' ὃ μὴ πέφυκεν ὅλως κινεῖσθαι, ἀδύνατον τοῦτο κινεῖσθαι βίᾳ. Εἰ τοίνυν ἐστί τις κίνησις αὐτῶν κατὰ φύσιν, ἀνάγκη τῶν ὁμοειδῶν καὶ τῶν καθ' ἕκαστον πρὸς ἕνα ἀριθμῷ τόπον ὑπάρχειν τὴν κίνησιν, οἷον πρὸς τόδε τι μέσον καὶ πρὸς τόδε τι ἔσχατον. 113 Moreover, the bodies must have some movement, since the fact that they move is quite evident. Are we to say then that all their movements, even those which are mutually contrary, are due to constraint? No, for a body which has no natural movement at all cannot be moved by constraint. If then the bodies have a natural movement, the movement of the particular instances of each form must necessarily have for goal a place numerically one, i.e. a particular centre or a particular extremity.
Εἰ δὲ πρὸς εἴδει ταὐτά, (277a.) πλείω δέ, διότι καὶ τὰ καθ' ἕκαστα πλείω μέν, εἴδει δ' ἕκαστον ἀδιάφορον, οὐ τῷ μὲν τῷ δ' οὐ τοιοῦτον ἔσται τῶν μορίων, ἀλλ' ὁμοίως πᾶσιν ὁμοίως γὰρ ἅπαντα κατ' εἶδος ἀδιάφορα ἀλλήλων, ἀριθμῷ δ' ἕτερον ὁτιοῦν ὁτουοῦν. Λέγω δὲ τοῦτο, ὅτι εἰ τὰ ἐνταῦθα μόρια πρὸς ἄλληλα καὶ τὰ ἐν ἑτέρῳ κόσμῳ ὁμοίως ἔχει, καὶ τὸ ληφθὲν ἐντεῦθεν οὐδὲν διαφερόντως πρὸς τῶν ἐν ἄλλῳ τινὶ κόσμῳ μορίων καὶ πρὸς τῶν ἐν τῷ αὐτῷ, ἀλλ' ὡσαύτως διαφέρουσι γὰρ οὐθὲν εἴδει ἀλλήλων. 114 If it be suggested that the goal in each case is one in form but numerically more than one, on the analogy of particulars which are many though each undifferentiated in form, we reply that the variety of goal cannot be limited to this portion or that but must extend to all alike. For all are equally undifferentiated in form, but any one is different numerically from any other. What I mean is this: if the portions in this world behave similarly both to one another and to those in another world, then the portion which is taken hence will not behave differently either from the portions in another world or from those in the same world, but similarly to them, since in form no portion differs from another.
Ὥστ' ἀναγκαῖον ἢ κινεῖν ταύτας τὰς ὑποθέσεις, ἢ τὸ μέσον ἓν εἶναι καὶ τὸ ἔσχατον. Τούτου δ' ὄντος ἀνάγκη καὶ τὸν οὐρανὸν ἕνα μόνον εἶναι καὶ μὴ πλείους, τοῖς αὐτοῖς τεκμηρίοις τούτοις καὶ ταῖς αὐταῖς ἀνάγκαις. 115 The result is that we must either abandon our present assumption or assert that the centre and the extremity are each numerically one. But this being so, the heaven, by the same evidence and the same necessary inferences, must be one only and no more.
Ὅτι δ' ἔστι τι οὗ πέφυκεν ἡ γῆ φέρεσθαι καὶ τὸ πῦρ, δῆλον καὶ ἐκ τῶν ἄλλων. 116 A consideration of the other kinds of movement also makes it plain that there is some point to which earth and fire move naturally.
Ὅλως γὰρ τὸ κινούμενον ἔκ τινος εἴς τι μεταβάλλει, καὶ ταῦτα ἐξ οὗ καὶ εἰς ὃ εἴδει διαφέρει πᾶσα δὲ πεπερασμένη μεταβολή οἷον τὸ ὑγιαζόμενον ἐκ νόσου εἰς ὑγίειαν καὶ τὸ αὐξανόμενον ἐκ μικρότητος εἰς μέγεθος. Καὶ τὸ φερόμενον ἄρα καὶ γὰρ τοῦτο γίνεταί ποθέν ποι. Δεῖ ἄρα εἴδει διαφέρειν ἐξ οὗ καὶ εἰς ὃ πέφυκε φέρεσθαι, ὥσπερ τὸ ὑγιαζόμενον οὐχ οὗ ἔτυχεν, οὐδ' οὗ βούλεται ὁ κινῶν. Καὶ τὸ πῦρ ἄρα καὶ ἡ γῆ οὐκ εἰς ἄπειρον φέρονται, ἀλλ' εἰς ἀντικείμενα ἀντίκειται δὲ κατὰ τόπον τὸ ἄνω τῷ κάτω, ὥστε ταῦτα ἔσται πέρατα τῆς φορᾶς. 117 For in general that which is moved changes from something into something, the starting-point and the goal being different in form, and always it is a finite change. For instance, to recover health is to change from disease to health, to increase is to change from smallness to greatness. Locomotion must be similar: for it also has its goal and starting-point—and therefore the starting-point and the goal of the natural movement must differ in form—just as the movement of coming to health does not take any direction which chance or the wishes of the mover may select. Thus, too, fire and earth move not to infinity but to opposite points; and since the opposition in place is between above and below, these will be the limits of their movement.
Ἐπεὶ καὶ ἡ κύκλῳ ἔχει πως ἀντικείμενα τὰ κατὰ διάμετρον, τῇ δ' ὅλῃ οὐκ ἔστιν ἐναντίον οὐδέν, ὥστε καὶ τούτοις τρόπον τινὰ ἡ κίνησις εἰς ἀντικείμενα καὶ πεπερασμένα. Ἀνάγκη ἄρα εἶναί τι τέλος καὶ μὴ εἰς ἄπειρον φέρεσθαι. 118 (Even in circular movement there is a sort of opposition between the ends of the diameter, though the movement as a whole has no contrary: so that here too the movement has in a sense an opposed and finite goal.) There must therefore be some end to locomotion: it cannot continue to infinity.
Τεκμήριον δὲ τοῦ μὴ εἰς ἄπειρον φέρεσθαι καὶ τὸ τὴν γῆν μέν, ὅσῳ ἂν ἐγγυτέρω ᾖ τοῦ μέσου, θᾶττον φέρεσθαι, τὸ δὲ πῦρ, ὅσῳ ἂν τοῦ ἄνω. Εἰ δ' ἄπειρον ἦν, ἄπειρος ἂν ἦν καὶ ἡ ταχυτής, εἰ δ' ἡ ταχυτής, καὶ τὸ βάρος καὶ ἡ κουφότης ὡς γὰρ <εἰ> τῷ κατωτέρω ταχὺ ἦν τι, ἕτερον τῷ βάρει ἂν ἦν ταχύ, οὕτως εἰ ἄπειρος ἦν ἡ τούτου ἐπίδοσις, καὶ ἡ τῆς ταχυτῆτος ἐπίδοσις ἄπειρος ἂν ἦν. 119 This conclusion that local movement is not continued to infinity is corroborated by the fact that earth moves more quickly the nearer it is to the centre, and fire the nearer it is to the upper place. But if movement were infinite speed would be infinite also; and if speed then weight and lightness. For as superior speed in downward movement implies superior weight, so infinite increase of weight necessitates infinite increase of speed.
Praemissis duabus rationibus ad ostendendum unitatem mundi, hic Aristoteles ponit tertiam rationem ad idem; quae quidem addit quoddam aliud, quod videbatur deficere ad primam rationem. Posset enim aliquis dicere quod corporibus non inest moveri naturaliter ad aliqua loca determinata: vel, si ad aliqua loca determinata moventur, ea quae sunt unius speciei et diversa secundum numerum, moventur ad loca diversa secundum numerum, quae conveniunt in specie; non autem ad eundem locum secundum numerum, sicut prima ratio supponebat. Ad haec igitur certificanda philosophus inducit hanc tertiam rationem. Circa quam tria facit: 164. Having given two arguments showing that the world is one, Aristotle here gives a third argument for the same. And this argument adds something which seemed to be lacking in the first argument. For someone could say that it is not inherent in bodies to be naturally moved to certain definite places, or, if they are moved to definite places, those that are one in species and diverse in number are moved to numerically diverse places, which agree in species. But they are not moved to the same numerical place as the first argument supposed. Therefore, in order to make these things sure, Aristotle adduces this third argument. With respect to this he does three things:

primo ponit rationem;

secundo excludit quandam obviationem, ibi: si autem ad specie eadem etc.;

tertio infert principalem conclusionem, ibi: itaque necessarium et cetera.

First he gives the argument, at 165;

Secondly, he excludes an objection, at 166;

Thirdly, he infers the main conclusion, at 169.

Dicit ergo primo necessarium esse quod sit aliquis motus praedictorum corporum. Manifestum est autem quod moventur: quod quidem apparet et per sensum et per rationem, quia huiusmodi sunt corpora naturalia, quibus competit moveri. Potest ergo dubitatio remanere, utrum sit dicendum quod corpora naturalia moveantur per violentiam omnibus motibus quibus moventur, etiam si sint contrarii; puta quod ignis inducatur et sursum et deorsum per violentiam. Sed hoc est impossibile: quia quod non est omnino natum moveri, idest quod nullum motum habet ex sua natura, impossibile est quod moveatur per violentiam. Hoc enim dicimus violentiam pati, quod per vim fortioris agentis removetur a propria inclinatione: si igitur corporibus non inesset aliqua naturalis inclinatio ad quosdam motus, violentia in eis locum non haberet; sicut si animal non esset natum videre, non attribueretur ei caecitas. Oportet igitur dicere quod istorum corporum quae sunt partes mundi, sit aliquis motus secundum naturam. Eorum igitur quorum est una natura, est unus motus. Unus autem motus dicitur, qui est ad unum terminum, ut patet in V Physic. Necesse est ergo quod motus singulorum quae sunt unius speciei, sit ad unum numero locum: videlicet, si sint gravia, ad hoc medium quod est huius mundi; et si sint levia, ad hoc extremum huius mundi. Et ad hoc sequitur esse unum mundum. 165. He says therefore first [113] that the above-mentioned bodies must have some motion. For it is manifest that they are moved — this, indeed, is evident to sense and to reason, because such are natural bodies, i.e., bodies which it befits to be moved. Therefore there can remain the doubt whether it is to be said that natural bodies are moved violently with all the motions with which they are moved, even if they are contrary motions — for example, that fire is moved both upward and downward by compulsion. But this is impossible, because what is not apt to be moved at all, i.e., what of its nature has no motion cannot be moved by compulsion. For we say that a thing suffers compulsion if it is removed from its proper inclination by the force of a stronger agent. If, therefore, there is not a natural inclination to certain motions in bodies, compulsion has no place in them — any more than blindness would be attributed to an animal if it had no capacity to see. Consequently, we must admit that those bodies which are parts of the world have a motion according to nature, and among the bodies having a nature, the motion is one. Now motion is called "one" inasmuch as it is to one terminus, as is plain in Physics V. Therefore the motion of each thing belonging to the same species must be to one numerical place: namely, if they are heavy, it is to the middle, which is of this world; if they are light, it is to the boundary which is of this world. And upon this it follows that there is one world.
Deinde cum dicit: si autem ad specie eadem etc., excludit quandam obviationem. Posset enim aliquis dicere quod omnia corpora quae habent eundem motum naturalem, moventur ad loca quae sunt eadem specie, sed plura numero: quia etiam ipsa singularia, idest singulae partes unius corporis naturalis, puta terrae vel aquae, sunt plura numero, sed non differunt specie. Non videtur autem plura requirere unitas naturae mobilium quae sunt unius speciei, quam quod eorum motus sit unus secundum speciem; ad quod videtur sufficere quod loca ad quae terminatur, sint similia in specie. 166. Then at [114] he excludes an objection. For someone could say that all bodies having the same natural motion are moved to places that are the same in species, but several numerically — since even the singulars, i.e., the individual parts of one natural body, e.g., earth or water, are numerically many but do not differ in species. But oneness of nature in the mobiles that are of the same species does not seem to require any more than that their motion be one in species; in keeping with this, it would seem to be enough if the places at which the motion is terminated were alike in species.
Sed ipse ad hoc excludendum dicit quod tale accidens, scilicet moveri ad eadem loca secundum speciem, non videtur convenire huic partium, huic autem non (ut scilicet quaedam partes similes specie moveantur ad eundem locum numero, quaedam vero ad eundem locum secundum speciem); sed similiter oportet quod conveniat omnibus (ut scilicet vel omnes partes similes specie moveantur ad unum locum secundum numerum, vel omnes huiusmodi partes moveantur ad unum locum similem specie, numero tamen differentem); quia omnes huiusmodi similiter se habent quantum ad hoc quod non differunt specie ab invicem, sed unumquodque differt ab altero secundum numerum. Hoc autem ideo dicit, quia partes alicuius corporis, puta terrae, quae sunt in hoc mundo, similiter se habent ad invicem et cum partibus terrae quae sunt in alio mundo, ex quo terra hic et ibi est eiusdem speciei. Si ergo hinc, idest ex isto mundo, sumatur aliqua pars, puta terrae, nihil differt si comparetur ad aliquam partium quae sunt in aliquo alio mundo, vel si comparatur ad eas quae sunt in hoc mundo, sed similis est comparatio ad utrasque; quia non differunt specie ad invicem partes terrae quae sunt in hoc mundo, et quae sunt in alio mundo. Et eadem ratio est de aliis corporibus. Videmus autem quod omnes partes terrae quae sunt in hoc mundo, moventur ad unum numero locum; et similiter est in aliis corporibus. Ergo omnes partes terrae, in quocumque mundo sint, naturaliter moventur ad hoc medium huius mundi. 167. But in order to exclude this he says that such an accident, namely, being moved to the same specific places, does not seem to be congruent to one set of parts and not to another (i.e., such that some parts alike in species would be moved to the same numerical place and others to the same specific place); rather it should be congruent to all alike (i.e., either all the parts alike in species be moved to the same numerical place, or all such parts be moved to one place specifically similar but numerically different) — for all such parts are alike in not differing specifically, but each differs from the other numerically. The reason he says this is that the parts of any body, for example, earth, which are in this world are similarly related both to the parts of earth in this world and to the parts in another world, since earth here and earth there are specifically the same. If, then, a part, e.g., of earth, be taken hence, i.e., from this world, it makes no difference whether it is compared to parts in some other world or to parts in this world; rather the relationships are the same in both cases. For the parts of earth in this world and those in some other world do not differ in species. And the same holds for other bodies. But we see that all parts of earth in this world are moved to one numerical place; similarly for other bodies. Therefore all the parts of earth in whatever world they exist are naturally moved to this middle of this world.
Ipsa igitur naturalis inclinatio omnium corporum gravium ad unum numero medium, et omnium levium corporum ad unum numero extremum, manifestat unitatem mundi. Non enim potest dici quod in pluribus mundis ordinentur corpora secundum diversa media et extrema, sicut et in pluribus hominibus sunt media et extrema diversa numero, sed in eadem specie. Quia natura membrorum hominis vel cuiuslibet animalis non determinatur secundum ordinem ad aliquem locum, sed magis secundum ordinem ad aliquem actum; talis autem situs partium animalis congruit decentiae operationis membrorum. Sed natura gravium et levium determinatur ad certa loca; ita scilicet quod omnia quae habent eandem naturam, ad unum numero locum unam numero habent naturalem inclinationem. 168. Therefore the very natural inclination of all heavy bodies to one numerical middle, and of all light bodies to numerically one boundary, manifests the unity of the world. For it cannot be said that in the many worlds, bodies would be arranged according to diverse middles and boundaries, as happens in the case of men in whom the centers and boundaries are numerically diverse but specifically the same. For the nature of man's members or those of any other animal is not determined with respect to their relationship to some place but rather with respect to their relationship to some act; indeed, the position occupied by the parts of animals is in keeping with a suitable operation of the members. But the nature of heavy and of light things is determined to definite places, such that all having the same nature also have numerically one natural inclination to numerically one place.
Deinde cum dicit: itaque necessarium etc., infert principalem conclusionem. Cum enim conclusio secundum formam debitam infertur ex praemissis, necesse est vel conclusionem concedere, vel praemissas negare. Concludit ergo quod aut est necesse amovere, idest negare, has suppositiones, idest principia ex quibus conclusit propositum; aut necesse est concedere conclusionem, quod scilicet sit unum medium, ad quod feruntur omnia gravia, et unum extremum, ad quod feruntur omnia levia. Quo existente vero, necesse est ex consequenti quod sit unum caelum, idest unus mundus, et non plures; et hoc per argumenta, idest signa, praedicta, et per necessitates, idest necessarias rationes, praedictas. 169. Then at [115] he infers the principal conclusion. For when a conclusion according to due form is inferred from premises, either the conclusion must be concluded or the premises denied. He concludes, therefore, that either it is necessary to deny these suppositions, i.e., the principles from which he concluded the proposition, or to concede the conclusion, namely, that there is one middle to which all heavy things are moved, and one boundary to which all light things are carried. If this is true, then it is necessary as a consequence that there be one heaven, i.e., one world and not several, and this on account of the above-given "arguments," i.e., signs and "necessities," i.e., necessary arguments.
Deinde cum dicit: quod autem est aliquid etc., ostendit quoddam quod supposuerat, scilicet quod corpora naturalia habent loca determinata, ad quae naturaliter ferantur. 170. Then at [116] he proves something he had assumed, namely, that natural bodies have definite places to which they are naturally borne.

Et primo ostendit propositum;

secundo destruit opinionem contrariam, ibi: sed adhuc neque ab alio et cetera.

First he proves the proposition;

Secondly, he rejects a contrary opinion (L. 18).

Circa primum duo facit: About the first he does two things:

primo ostendit propositum per rationem naturalem;

secundo per signum, ibi: argumentum autem et cetera.

First he shows the proposition by a natural argument;

Secondly, by a sign, at 173.

Circa primum tria facit. As to the first he does three things:
Primo proponit quod intendit: et dicit manifestum esse tam ex aliis rationibus quam ex praemissis (vel etiam ex aliis motibus) quod est aliquis locus determinatus, quo naturaliter terra fertur. Et similiter dicendum est de aqua et de quolibet aliorum corporum. First he proposes what he intends, and says it is clear from other arguments than the foregoing — or even from other motions — that there is a definite place whither earth is naturally borne. And the same is to be said of water and of any of the other bodies.
Secundo ibi: omnino enim quod movetur etc., ponit rationem: dicens omnino, idest universaliter, hoc esse verum, quod omne quod movetur, transmutatur ex quodam determinato in quoddam determinatum: dicitur enim in I Physic. quod album fit non ex quolibet non albo, sed ex nigro. Haec autem duo, scilicet ex quo motus procedit et in quod terminatur, differunt specie: sunt enim contraria, ut patet in V Physic.; contrarietas autem est differentia secundum formam, ut dicitur in X Metaphys. 171. Secondly, at [117] he gives his argument, saying that entirely, i.e., universally, this is true, that whatever is moved is changed from something determinate to something determinate: for it is said in Physics I that something white comes to be, not from any non-white at random, but from black. Now these two factors, namely, that from which a motion proceeds, and that into which it is terminated, differ in species — for they are contrary, as is plain in Physics V; but contrariety is a difference respecting form, as is said in Metaphysics X.
Hoc autem quod dictum est, probat per hoc, quod omnis transmutatio est finita, ut probatur in VI Physic., et etiam per ea quae supra dicta sunt, scilicet quod nihil movetur ad id ad quod non potest pervenire; nihil autem potest pervenire ad infinitum; unde oportet omnem mutationem esse finitam. Si autem non esset aliquod determinatum in quod tendit motus, differens specie ab eo a quo motus incipit, oporteret motum esse infinitum: nulla enim ratio esset quare motus magis terminaretur hic quam alibi; sed eadem ratione qua incoepit illinc moveri, inciperet moveri et hinc. He proves what he has said by the fact that every change is finite, as was proved in Physics VI, and also by the facts cited above, namely, that nothing is moved to what it cannot attain; but nothing can attain to the infinite; hence every change must be finite. But if there were not something definite toward which a motion tends and something specifically different from that, at which it begins, the motion would have to be infinite; for there would be no reason why the motion should end here rather than elsewhere, but, for the same reason that it began to be moved thence, it would also begin to be moved hence.
Manifestat etiam per exemplum quod dictum est. Illud enim quod sanatur, movetur ex infirmitate in sanitatem; et illud quod augmentatur, movetur ex parvitate in magnitudinem: oportet igitur etiam illud quod fertur, idest quod movetur secundum locum, moveri a quodam determinato in quoddam determinatum; et haec sunt locus unde incipit motus, et locus quo tendit. Sic igitur oportet quod specie differat locus a quo aliquid movetur localiter, et in quem naturaliter fertur; sicut id quod sanatur non tendit ubicumque contingit, quasi a casu, neque ex sola voluntate moventis, sed ad aliquid determinatum, ad quod natura inclinatur. Sic igitur et ignis et terra et alia corpora naturalia non feruntur ad infinitum, idest ad aliquod indeterminatum, sicut posuit Democritus; sed feruntur in loca opposita locis in quibus prius erant. Contrariatur autem sursum secundum locum ei quod est deorsum. Sequitur ergo quod sursum et deorsum sunt termini naturalium motuum corporum simplicium. He also explains what was said, by an example. For what is healed is moved from sickness to health; what is increased is moved from small to large. Hence, too, what is carried, i.e., moved according to place, is moved from something definite to something definite, and these are the place at which a motion begins, and the place to which it tends. Consequently, there must be a specific difference between the place from which something is locally moved and the place into which it is naturally borne, just as what is healed does not tend to just anything at random, as though by chance, or solely according to the will of the mover, but to something definite, to which it is inclined by nature. In the same way, therefore, fire and earth and other natural bodies are not borne ad infinitum, i.e., to something indefinite, as Democritus held; rather they are borne to places opposite to those in which they previously found themselves. But "up" is contrary to "down" in the realm of place. It follows, therefore, that "up" and "down" are the termini of the natural motions of simple bodies.
Tertio ibi: quoniam autem et qui in circuitu etc., excludit quandam obviationem, qua posset aliquis obviare ex motu circulari, qui non videtur esse ex opposito in oppositum, sed magis ex eodem in idem. 172. Then at [118 he excludes an objection by which someone could object that circular motion does not seem to be from opposite to opposite, but more from the same to the same.
Sed ipse dicit quod etiam motus circularis aliqualiter habet oppositum in termino. Dicit autem aliqualiter, propter duo. Primo quidem quia non invenitur oppositio in motu circulari secundum aliqua puncta in circulo designata, prout sunt puncta ipsius circuli, sed solum prout sunt extrema diametri, secundum quam mensuratur maxima distantia in circulo, ut supra dictum est: unde subdit: ea quae secundum diametrum, scilicet extrema, opposita sunt. Secundo quia, sicut totum corpus sphaericum non mutat locum subiecto sed solum ratione, partes autem eius variant locum etiam subiecto; ita si accipiatur totus motus circularis, non invenitur aliqua oppositio in terminis nisi secundum rationem, prout scilicet idem, a quo et in quod est motus circularis, accipitur ut principium et ut finis; sed accipiendo partes motus circularis, accipitur ibi oppositio secundum lineam rectam, ut supra dictum est; et ideo subdit quod toti circulationi non est aliquid contrarium. Sic ergo patet quod etiam in his quae circulariter feruntur, mutatio est aliquo modo in opposita et finita. But he says that even circular motion somehow involves opposition of termini. He says "somehow" for two reasons. First, because opposition in circular motion is not found with respect to points designated on the circle insofar as they are points of the circle, but only insofar as they are the extremities of the diameter — on the basis of which a maximum distance is reckoned in a circle, as was said above. Hence he adds: "What are according to the diameter," i.e., the extremities of the diameter, "are opposite." Secondly, because just as the whole spherical body does not change place as to subject but only in conception, although the parts change their place even as to subject, so also, if the entire circular motion is taken, there is no opposition in termini, except conceptually, namely, in the sense that the same [point], from which and to which circular motion is, is taken now as the beginning and now as the end. But if we take the parts of circular motion, we find opposition with respect to a straight line, as has been said. And therefore he adds that there is nothing contrary to a whole revolution. Consequently, it is plain that even in things circularly moved, the change is in a certain way toward things opposite and determinate.
Et sic universaliter concludit quod intendit, scilicet quod necesse est esse aliquem finem motus localis; non autem in infinitum fertur corpus naturale, idest ad aliquod indeterminatum, sicut posuit Democritus motum atomorum. Thus he concludes universally to what he intended, namely, that there is necessarily an end involved in local motion and that a natural body is not moved in infinitum [i.e., to nothing definite], as Democritus posited about the motion of atoms.
Deinde cum dicit: argumentum autem etc., probat idem per signum: quam quidem probationem vocat argumentum, eo quod talis probatio est quasi coniecturalis. Et dicit quod argumentum eius quod corpus naturale non feratur in infinitum sed ad aliquod certum, est quod terra, quanto magis appropinquat ad medium, velocius fertur (quod potuit deprehendi ex maiori eius impulsu, prout scilicet a gravi cadente fortius impellitur aliquid iuxta terminum sui motus): et eadem ratio est de igne, quod motus eius in tanto est velocior, quanto magis appropinquat ad locum sursum. Si ergo in infinitum ferretur terra vel ignis, in infinitum posset velocitas eius augeri. 173. Then at [119] he proves the same thing through a sign. This proof he calls an "argument" in the sense that it is, so to speak, conjectural. And he says that the argument for claiming that a natural body is not moved to infinity but to something certain is that earth, the closer it approaches the middle, the more swiftly it is moved (which can be perceived from its greater impetus, namely, as something is more strongly impelled by the heavy in its fall as it nears the terminus of its motion); and the same holds for fire whose motion is swifter, the closer it approaches an upward place. If, therefore, earth or fire were moved to infinity, their speed could increase indefinitely.
Et ex hoc concludit quod in infinitum posset augeri gravitas vel levitas corporis naturalis. Sicut enim velocitas corporis gravis est maior, quanto grave corpus amplius descendit, quod quidem corpus grave est velox per suam gravitatem; sic etiam ita poterit esse additio infinita ad velocitatem, si sit additio infinita ad gravitatem vel levitatem. Ostensum est autem supra quod non potest esse gravitas vel levitas infinita, et quod non potest aliquid moveri ad id ad quod non potest pertingere. Sic igitur additio gravitatis non potest esse in infinitum; et per consequens nec additio velocitatis. Unde nec motus corporum naturalium potest esse in infinitum. And from this he concludes that the heaviness or lightness of a natural body could be increased infinitely. For just as the speed of a heavy body is greater according as the heavy body descends farther ( and a heavy body is swift on account of its heaviness), so, too, an indefinite addition could be made to the speed if an infinite addition were made to heaviness or lightness. But it was shown above that there cannot be an infinite heaviness or lightness, and that nothing can be moved toward what it cannot attain. Consequently, addition of heaviness ad infinitum cannot occur, and, as a result, neither can addition of speed. Hence neither can the motion of natural bodies be tending toward what is infinite.
Sciendum est autem quod causam huius accidentis, quod terra velocius movetur quanto magis descenderit, Hipparchus assignavit ex parte moventis per violentiam; a quo quantum elongatur motus, tanto minus remanet de virtute moventis, et sic motus fit tardior; unde motus violentus in principio quidem intenditur, in fine autem remittitur intantum quod finaliter grave non potest plus sursum ferri, sed incipit moveri deorsum, propter parvitatem eius quod remanserat de virtute motoris violenti; quae quanto magis minoratur, tanto motus contrarius fit velocior. 174. It should be noted that the cause of this accident that earth is moved more swiftly the more it descends was explained by Hipparchus in terms of an agent causing motion by compulsion. The farther the motion is from such an agent the less remains of that agent's power, so that the motion becomes slower. Hence in the beginning, a compulsory motion is intense but in the end it is weakened, until finally the heavy body can no longer be borne upward, but begins to be moved downward due to the small amount of the agent's virtue that remains, which, the less it becomes, so much the swifter becomes the contrary motion.
Sed ista ratio est particularis solum in his quae moventur naturaliter post motum violentum; non autem habet locum in his quae moventur naturaliter eo quod generantur extra propria loca. But this explanation is applicable only to things that are moved naturally after a compulsory motion; it does not apply to things that are moved naturally on account of being generated outside their proper places.
Alii vero assignaverunt huius causam ex quantitate medii per quod fit motus, puta aeris, qui minor restat quanto plus proceditur in motu naturali; et ideo minus potest impedire motum naturalem. Sed et haec ratio non minus competeret in motibus violentis quam naturalibus; in quibus tamen contrarium accidit, ut infra dicetur. Others explained this phenomenon in terms of the amount of the medium through which the motion takes place (for example, the amount of air): in such a motion, if it is natural, the farther a thing has been moved, the less is the amount remaining — and, therefore, the less is it able to impede a natural motion. But this explanation also, applies no less to compulsory motions than to natural motions, in which, nevertheless, the contrary happens, as will be said below.
Et ideo dicendum est cum Aristotele quod causa huius accidentis est, quod quanto corpus grave magis descendit, tanto magis confortatur gravitas eius, propter propinquitatem ad proprium locum. Et ideo argumentatur quod si cresceret in infinitum velocitas, quod cresceret etiam in infinitum gravitas. Et eadem ratio est de levitate. Therefore, it must be said with Aristotle that the cause of this phenomenon is that, to the extent that a heavy body descends more, to that extent is its heaviness the more strengthened on account of its proximity to its proper place. And therefore he argues that if the speed increased infinitely, the heaviness, too, would increase indefinitely. And the same holds for lightness.

Lecture 18:
Exclusion of the opinion that natural bodies are not moved naturally to determined places. Unity of the world from higher bodies.
Chapter 8 cont.
Ἀλλὰ (277b.) μὴν οὐδ' ὑπ' ἄλλου φέρεται αὐτῶν τὸ μὲν ἄνω τὸ δὲ κάτω οὐδὲ βίᾳ, ὥσπερ τινές φασι τῇ ἐκθλίψει. 120 Further, it is not the action of another body that makes one of these bodies move up and the other down; nor is it constraint, like the 'extrusion' of some writers.
Βραδύτερον γὰρ ἂν ἐκινεῖτο τὸ πλεῖον πῦρ ἄνω καὶ ἡ πλείων γῆ κάτω νῦν δὲ τοὐναντίον ἀεὶ τὸ πλεῖον πῦρ θᾶττον φέρεται καὶ ἡ πλείων γῆ εἰς τὸν αὑτῶν τόπον. 121 For in that case the larger the mass of fire or earth the slower would be the upward or downward movement; but the fact is the reverse: the greater the mass of fire or earth the quicker always is its movement towards its own place.
Οὐδὲ θᾶττον ἂν πρὸς τῷ τέλει ἐφέρετο, εἰ βίᾳ καὶ ἐκθλίψει πάντα γὰρ τοῦ βιασαμένου πορρωτέρω γιγνόμενα βραδύτερον φέρεται, καὶ ὅθεν βίᾳ, 122 Again, the speed of the movement would not increase towards the end if it were due to constraint or extrusion; for a constrained movement always diminishes in speed as the source of constraint becomes more distant,
ἐκεῖ φέρεται οὐ βίᾳ. Ὥστ' ἐκ τούτων θεωροῦσιν ἔστι λαβεῖν τὴν πίστιν περὶ τῶν λεγομένων ἱκανῶς. 123 and a body moves without constraint to the place whence it was moved by constraint. A consideration of these points, then, gives adequate assurance of the truth of our contentions.
Ἔτι δὲ καὶ διὰ τῶν ἐκ τῆς πρώτης φιλοσοφίας λόγων δειχθείη ἄν, καὶ ἐκ τῆς κύκλῳ κινήσεως, ἣν ἀναγκαῖον ἀΐδιον ὁμοίως ἐνταῦθά τ' εἶναι καὶ ἐν τοῖς ἄλλοις κόσμοις. 124 The same could also be shown with the aid of the discussions which fall under First Philosophy, as well as from the nature of the circular movement, which must be eternal both here and in the other worlds.
Δῆλον δὲ κἂν ὧδε γένοιτο σκοπουμένοις ὅτι ἀνάγκη ἕνα εἶναι τὸν οὐρανόν. Τριῶν γὰρ ὄντων τῶν σωματικῶν στοιχείων, τρεῖς ἔσονται καὶ οἱ τόποι τῶν στοιχείων, εἷς μὲν ὁ τοῦ ὑφισταμένου σώματος ὁ περὶ τὸ μέσον, ἄλλος δὲ ὁ τοῦ κύκλῳ φερομένου, ὅσπερ ἐστὶν ἔσχατος, τρίτος δ' ὁ μεταξὺ τούτων ὁ τοῦ μέσου σώματος. Ἀνάγκη γὰρ ἐν τούτῳ εἶναι τὸ ἐπιπολάζον. Εἰ γὰρ μὴ ἐν τούτῳ, ἔξω ἔσται ἀλλ' ἀδύνατον ἔξω. Τὸ μὲν γὰρ ἀβαρὲς τὸ δ' ἔχον βάρος, κατωτέρω δὲ ὁ τοῦ βάρος ἔχοντος σώματος τόπος, εἴπερ ὁ πρὸς τῷ μέσῳ τοῦ βαρέος. Ἀλλὰ μὴν οὐδὲ παρὰ φύσιν ἄλλῳ γὰρ ἔσται κατὰ φύσιν, ἄλλο δ' οὐκ ἦν. Ἀνάγκη ἄρα ἐν τῷ μεταξὺ εἶναι. Τούτου δ' αὐτοῦ τίνες εἰσὶ διαφοραί, ὕστερον ἐροῦμεν. 125 It is plain, too, from the following considerations that the universe must be one. The bodily elements are three, and therefore the places of the elements will be three also; the place, first, of the body which sinks to the bottom, namely the region about the centre; the place, secondly, of the revolving body, namely the outermost place, and thirdly, the intermediate place, belonging to the intermediate body. Here in this third place will be the body which rises to the surface; since, if not here, it will be elsewhere, and it cannot be elsewhere: for we have two bodies, one weightless, one endowed with weight, and below is place of the body endowed with weight, since the region about the centre has been given to the heavy body. And its position cannot be unnatural to it, for it would have to be natural to something else, and there is nothing else. It must then occupy the intermediate place. What distinctions there are within the intermediate itself we will explain later on.
Περὶ μὲν οὖν τῶν σωματικῶν στοιχείων, ποῖά τ' ἐστὶ καὶ πόσα, καὶ τίς ἑκάστου τόπος, ἔτι δ' ὅλως πόσοι τὸ πλῆθος οἱ τόποι, δῆλον ἡμῖν ἐκ τῶν εἰρημένων. We have now said enough to make plain the character and number of the bodily elements, the place of each, and further, in general, how many in number the various places are.
Postquam ostendit philosophus quod corpora naturalia moventur naturaliter ad determinata loca, hic excludit opinionem contrariam. 175. After showing that natural bodies are by nature moved to definite places, the Philosopher here excludes a contrary opinion.

Et primo proponit quod intendit;

secundo probat propositum, ibi: tardius enim et cetera.

First he proposes what he intends;

Secondly, he proves his proposition, at 176.

Quia vero per hoc quod falsitas excluditur, veritas comprobatur, inducit hic philosophus exclusionem erroris quasi quandam veritatis demonstrationem; dicens quod adhuc etiam quod dictum est manifestatur per hoc, quod corpora naturalia non feruntur sursum et deorsum neque sicut ab alio exteriori mota. Now, since truth is established by excluding falsehood, the Philosopher here induces the exclusion of an error as a certain demonstration of the truth. He says, therefore, that what has been said is manifested by the fact that natural bodies are not borne upward and downward as though moved by some external agent.
Per quod quidem intelligendum est quod removet exteriorem motorem, qui per se huiusmodi corpora moveat postquam sunt formam specificam sortita. Moventur enim levia quidem sursum, gravia autem deorsum a generante quidem, inquantum dat eis formam quam consequitur talis motus; sed a removente prohibens, per accidens et non per se. Quidam vero posuerunt quod postquam speciem sunt adepta huiusmodi corpora, indigent ab aliquo extrinseco moveri per se: quod hic philosophus removet. By this is to be understood that he rejects an external mover which would move these bodies per se after they obtained their specific form. For light things are indeed moved upward, and heavy bodies downward, by the generator inasmuch as it gives them the form upon which such motion follows, but they are moved per accidens, and not per se, by whatever removes an obstacle to their motion. However, some have claimed that after bodies of this kind have received their form, they need to be moved per se by something extrinsic. It is this claim that the Philosopher rejects here.
Neque etiam dicendum est quod huiusmodi corpora moveantur per violentiam; sicut quidam dixerunt quod moveantur per quandam extrusionem, inquantum scilicet unum corpus truditur ab alio fortiori. Ponebant enim quod omnium corporum erat naturaliter unus motus: sed dum quaedam eorum ab aliis impelluntur, fit quod quaedam eorum moventur sursum, quaedam autem deorsum. Neither should it be said that these bodies are moved by compulsion, which is the opinion of those who said that they are moved by a certain "extrusion," in the sense that one body is displaced by another, stronger, one. For they assumed that there was one motion natural to all bodies, but since some are given momentum by others, it comes to pass that a certain number are moved upward and a certain number downward.
Deinde cum dicit: tardius enim etc., probat propositum tribus rationibus. Quarum prima principaliter inducitur ad ostendendum quod huiusmodi corpora in suis naturalibus motibus non moventur ab exterioribus motoribus. Manifestum est enim quod tanto tardior est motus, quanto movens minus vincit super mobile. Eadem autem virtus moventis minus vincit maius mobile quam minus. 176. Then at [121] he proves his proposition with three arguments. The first of these is adduced mainly to show that bodies of this kind in their natural motions are not moved by external movers. For it is clear that a motion is slower to the extent that the mover overcomes the mobile less. But a given virtue of the mover overcomes a larger mobile less than a smaller.
Si ergo huiusmodi corpora moverentur ab aliquo exteriore movente, tardius moveretur maior ignis sursum et maior terra deorsum. Nunc autem contrarium accidit, quod maior ignis et maior terra velocius feruntur in propria loca. Per quod datur intelligi quod huiusmodi corpora habent intrinsecus principia sui motus; quorum virtutes motivae tanto sunt maiores, quanto corpora fuerint maiora; et ideo velocius feruntur. Sic ergo patet quod huiusmodi corpora suis motibus naturalibus moventur non per virtutem exteriorem, sed per virtutem intrinsecam, quam acceperunt a generante. If, then, these bodies were moved by an external mover, a greater amount of fire would be moved upward more slowly and a larger amount of earth downward more slowly. But just the opposite happens, for a greater quantity of fire and a greater quantity of earth are moved more swiftly to their places. This gives us to understand that these bodies have the principles of their motion within themselves, and their motive powers are greater according as the bodies are greater, and that is why they are moved more swiftly. Consequently, it is plain that such bodies in their natural motions are not moved by an exterior power but by an intrinsic one, which they have received from their generator.
Secundam rationem ponit ibi: neque velocius etc.; quae quidem principaliter ad hoc inducitur, quod motus horum corporum non est per violentiam. Videmus enim quod omnia quae per violentiam moventur, tanto tardius feruntur, quanto magis elongantur a motore qui vim intulit; sicut patet in his quae proiiciuntur, quod eorum motus in fine est remissior, et tandem totaliter deficit. Si ergo corpora gravia et levia moverentur per violentiam, quasi mutuo se trudentia, sequeretur quod eorum motus ad propria loca non esset velocior in fine, sed magis tardior; cuius contrarium ad sensum apparet. 177. At [122] he gives a second argument which is adduced mainly to show that motion of these bodies is not through compulsion. For we see that all things moved by compulsion are moved more slowly according as their distance from the mover increases, as is plain in projectiles, whose motion slackens near the end and finally fails. If, then, heavy and light bodies were moved by compulsion as though mutually pushing one another, it would follow that their motion toward their proper places would not be faster but slower in the end. But the contrary of this is plain to our senses.
Tertiam rationem ponit ibi: et unde vi etc.; quae potest respicere ad utrumque. Videmus enim quod nullum corpus illuc fertur per violentiam, unde per violentiam removetur. Ex hoc enim aliquod corpus a loco aliquo per violentiam removetur, quia natum est ibi esse: unde illuc naturaliter, et non per violentiam fertur. Si ergo ponatur quod motus aliqui corporum gravium et levium, quibus ab aliquibus locis removentur, sint violenti, non potest dici quod motus contrarii, quibus ad illa loca feruntur, sint violenti. Et ita non est verum quod omnes motus horum corporum sint ab alio et per violentiam. 178. He gives at [123] the third argument which can regard both. For we see that no body is moved by violence to a place whence it can be removed by violence. For it is because a body is apt to be in a certain place that it can be moved thence by violence; hence it was originally brought there naturally and not by violence. If, therefore, it is assumed that some motions of heavy and light bodies are violent by which they are moved from certain places, it cannot be said that the contrary motions which brought them there are violent. Thus it is not true that all the motions of these bodies are caused by another and by violence.
Concludit autem ex dictis epilogando, quod per speculationem horum contingit accipere fidem de his quae dicta sunt. He concludes from the foregoing, in summary, that speculation on these points will testify to the truth of what has been said.
Deinde cum dicit: adhuc autem et per eas etc., ostendit unitatem mundi per corpora superiora, quae circulariter feruntur: 179. Then at [124] he shows through the higher bodies which are moved circularly that the world is one:

et primo specialiter per corpora superiora;

secundo communiter per superiora et inferiora, ibi: palam autem utique et cetera.

First in a special way by the higher bodies;

Secondly, in a general way by the higher and the lower, at 181.

Dicit ergo primo quod adhuc ostendi potest quod sit solum unus mundus, per rationes sumptas ex prima philosophia, idest per ea quae determinata sunt in metaphysica, et per hoc quod ostensum est in VIII Physic., quod motus circularis est sempiternus, quod quidem habet naturalem necessitatem et in hoc et in aliis mundis. Conclusit enim philosophus sempiternitatem motus caeli in VIII Physic. per ordinem mobilium et moventium; quod quidem necesse est similiter se habere in quolibet mundo, si mundus univoce dicatur. Si autem motus caeli sit sempiternus, oportet quod moveatur a virtute infinita, quae non sit virtus in magnitudine, ut probatur in VIII Physic. Talis autem virtus est immaterialis, et per consequens una numero, cum sit tantum forma et species, multiplicatio autem individuorum eiusdem speciei est per materiam. Et sic oportet quod virtus quae movet caelum, sit una numero. Unde oportet quod et caelum sit unum numero, et per consequens totus mundus. He says therefore first [124] that there is still another way of proving that there is but one world, by arguments taken from first philosophy, i.e., by using what has been determined in the Metaphysics, and from what has been shown in Physics VIII, namely, that circular motion is eternal, which, both in this and in other worlds, has a natural necessity. For the Philosopher concluded to the eternity of celestial motion in Physics VIII by considering the order between mobiles and movers, which must be similar in any world, if "world" is taken univocally. Now if celestial motion is eternal, it must be moved by an infinite power, such as cannot exist in a magnitude, as was proved in Physics VIII. Such a power is non-material and consequently numerically one, since it is a form and species only, whereas it is through matter that individuals are multiplied in the same species. Consequently, the power that moves the heavens must be numerically one. Hence the heavens too must be numerically one, and, consequently, the whole world.
Potest autem aliquis dicere hanc rationem non ex necessitate concludere. Primum enim movens movet caelum sicut desideratum, ut dicitur in XII Metaphys.; nihil autem prohibet idem a pluribus desiderari; et ita videtur quod ex unitate primi moventis non possit ex necessitate concludi unitas caeli. 180. But someone can say that this argument does not conclude with necessity. For the first mover moves the heaven as that which is desired, as is said in Metaphysics XII. But there is nothing to prevent the same thing from being accidentally many. So it seems that we cannot from the unity of the first mover conclude necessarily to the unity of the heavens.
Sed dicendum est quod multa possunt unum desiderare, non quidem quasi de pari, eo quod uni primo non immediate adiungitur absoluta multitudo; sed secundum quendam ordinem possunt multa desiderare unum, quaedam propinquius et quaedam remotius, quorum coordinatio in ordine ad unum ultimum, facit unitatem mundi. But it must be said that many can desire one thing, but not indeed in an identical way, since an absolute multitude is not joined immediately to one thing that is first; but many things can desire one thing according to a certain order, some being closer and some more remote, the coordination of which to one ultimate objective makes the unity of the world.
Deinde cum dicit: palam autem utique etc., probat propositum ratione sumpta communiter ex corporibus superioribus et inferioribus. Et dicit quod etiam sic intendendo sicut dicetur, necesse est esse unum caelum, idest unum mundum. Ad quod probandum assumit quod, sicut sunt tria corporalia elementa, scilicet caelum et terra et medium, ita sunt et tria loca eis correspondentia: unus quidem locus qui est circa medium, qui est corporis subsistentis, idest corporis gravissimi quod substat omnibus, scilicet terrae; alius autem locus qui est extremus in altitudine, qui est corporis quod movetur circulariter; tertius autem locus qui est intermedius horum, qui est medii corporis. 181. Then at [125] he proves his proposition with an argument taken generally from higher and lower bodies. And he says that even the following consideration will show that it is necessary for the heaven, i.e., the world, to be one. To prove this he assumes that, just as there are three bodily elements, namely, heaven and earth and an intermediate, so there are three places corresponding to them: one is the place about the middle, that of the subsisting body, i.e., the place of the heaviest body which supports all, namely, earth; another is the place which is the highest in altitude, that of the circularly moved body; the third place is intermediate and corresponds to the intermediate body.
Circa quae quidem verba primo considerandum est quod etiam caelum inter elementa computat, cum tamen elementum sit ex quo componitur res, ut dicitur in V Metaphys. With regard to these words it should be noted that Aristotle here reckons the heaven among the elements, although an element is something out of which things are composed, as is said in Metaphysics V.
Caelum autem, etsi non veniat in compositionem corporis mixti, venit tamen in compositionem totius universi, quasi quaedam pars eius. Vel elementa large nominat quaecumque simplicia corpora: quae quidem vocat corporalia elementa, ad differentiam materiae primae, quae est elementum, non tamen corporale, sed absque omni forma, prout in se consideratur. However the heaven, even though it does not enter into the composition of a mixed body enters into the composition of the whole universe, as being a part of it. Or he is using the word "element" in a wide sense to designate any of the simple bodies which he calls "bodily elements" to distinguish them from prime matter, which, though an element, is not a bodily element, for considered in itself it is without any form.
Secundo autem considerandum est de hoc quod dicit tria esse loca. Cum autem locus sit terminus corporis continentis, ut dicitur in IV Physic., satis potest esse manifestum quid sit locus medii elementi; quia superficies supremi corporis continentis ipsum. De primo autem corpore quomodo sit in loco, ostensum est in IV Physic. Sed quomodo medium, quod non habet rationem continentis sed contenti, sit locus corporis gravis, videtur dubitationem habere. Secondly, we should consider his statement that there are three places. Now since place is the boundary of a containing body, as is said in Physics IV, it can be clear what the place of the intermediate element is — for it is the surface of the supreme, body containing it. How the first body is in place has been explained in Physics IV. But how the middle [i.e., the center], which seems to be not a container but a contained, is the place of the heavy body seems to offer difficulty.
Sed dicendum est quod, sicut dictum est in IV Physic., superficies corporis continentis non habet rationem loci secundum quod est superficies talis corporis, sed secundum ordinem situs quem habet ad primum continens, prout scilicet magis vel minus ei appropinquat. Corpus autem grave in sua natura est maxime elongatum a corpore caelesti propter eius materialitatem; et ideo debetur ei locus remotissimus a primo continente, qui est propinquissimus medio; et ita superficies continens corpus grave dicitur locus eius secundum propinquitatem ad medium. Unde signanter dicit quod locus qui est circa medium est corporis subsistentis. But it should be said that, as has been said in Physics IV, the surface of the containing body does not have the notion of place because it is the surface of such a body but with respect to the position it has in relation to the first container accordingly, namely, as it is nearer or farther from it. Now the heavy body in its nature is at a maximum distance from the celestial body on account of its materiality; therefore there is due it a place farthest from the first container and nearest to the middle. Consequently the surface containing the heavy body is called its place according to its nearness to the center. Hence he said advisedly that the place located around the middle is the place of the subsisting body.
Ex his autem quae proposita sunt procedit ad propositum ostendendum ex corpore levi, sicut supra processerat ex corpore gravi. Necesse est enim corpus leve quod superfertur, esse in hoc loco medio: quia, cum omne corpus sit in aliquo loco, si corpus leve non esset in hoc loco medio, esset extra ipsum; quod est impossibile, quia extra hunc locum medium ex una parte est corpus caeleste, quod est sine gravitate et levitate, ex alia autem parte est corpus terrestre, quod habet gravitatem. Non autem potest dici quod sit aliquis locus magis deorsum quam locus qui est corporis habentis gravitatem; quia locus qui est apud medium, est proprius eius. Ex hoc autem patet quod impossibile est esse alium mundum quia oporteret ibi esse aliquod corpus leve; et sic, si mundus ille esset supra hunc mundum, corpus leve esset supra locum caeli; si autem esset infra hunc mundum, corpus leve esset infra locum corporis gravis, quod est impossibile. 182. From what has been set forth he goes on to prove his proposition from a light body, just as above he had proceeded from a heavy body. For it is necessary that a light body which is borne upwards be in this intermediate place: because, since every body is in some place, if the light body were not in this intermediate place, it would be outside it. But that is impossible, because outside this intermediate place there is, on the one side, celestial body which has no heaviness or lightness, and on the other side, terrestrial body which has heaviness. Now it cannot be said that there is a place more downward than the place of the body having heaviness, because the place about the middle is proper to it. But from this it is plainly impossible for another world to exist, because some light body would have to be there and thus, if that world were above this world, a light body would exist above the place of the heavens; if that world were below this world, a light body would be below the place of the heavy body — which is impossible.
Sed huic rationi posset aliquis obviare, dicendo quod corpus leve est extra hunc locum medium, non secundum naturam, sed praeter naturam. Sed ad hoc excludendum, subdit quod neque etiam praeter naturam possibile est corpus leve esse extra hunc medium locum. Quia omnis locus qui est alicuius corporis praeter naturam, est alicuius corporis secundum naturam: non enim Deus vel natura fecit aliquem locum frustra, in quo scilicet non sit natum esse aliquod corpus. Non autem invenitur in rerum natura aliquod aliud corpus praeter ista tria, quibus tria loca praedicta deputantur, ut ex dictis patet. Unde neque secundum naturam, neque praeter naturam, potest esse corpus leve extra hunc medium locum: et sic impossibile est esse multos mundos. 183. But to this argument someone could object that the light body would be outside this intermediate place not according to nature but outside its nature. To exclude this he adds that not even outside its nature can a light body be outside this intermediate place. Because every place that is outside nature for some body is according to nature for some other body. For neither God nor nature has made any place in vain, i.e., a place in which no body is apt to be. Now, no other body is found in nature except the three mentioned and to which the aforesaid places are deputed, as is plain from what has been said. Hence neither according to nature nor beside nature can a light body exist outside this intermediate place. Consequently, it is impossible that there be many worlds.
Quia vero locutus fuerat de medio elemento quasi de uno quodam corpore, subiungit quod posterius, scilicet in tertio et quarto, dicetur quae sunt differentiae istius medii. Dividitur enim in ignem, aerem et aquam, quae etiam est levis per respectum ad terram. Since he had spoken of an intermediate element as if it were one certain body, he adds that later, i.e., in the third and fourth books, he will speak about; the differences in that intermediate. For it is divided into fire, air and water, which is also light in relation to earth.
Ultimo epilogando concludit quod ex dictis manifestum est de corporeis elementis, quae et quot sint, et quis sit locus cuiuslibet eorum, et universaliter quot sint loca corporalia. Finally in summary he concludes that from the foregoing it is manifest about the bodily elements, which and how many they are, and what is the place of each of them and, in general, how many bodily places exist.

Lecture 19:
Solution of the argument seeming to justify several worlds.
Chapter 9
Ὅτι δ' οὐ μόνον εἷς ἐστίν, ἀλλὰ καὶ ἀδύνατον γενέσθαι πλείους, ἔτι δ' ὡς ἀΐδιος ἄφθαρτος ὢν καὶ ἀγένητος, λέγωμεν, πρῶτον διαπορήσαντες περὶ αὐτοῦ. 127 We must show not only that the heaven is one, but also that more than one heaven is and, further, that, as exempt from decay and generation, the heaven is eternal. We may begin by raising a difficulty.
Δόξειε γὰρ ἂν ὡδὶ σκοπουμένοις ἀδύνατον ἕνα καὶ μόνον εἶναι αὐτόν ἐν ἅπασι γὰρ καὶ τοῖς φύσει καὶ τοῖς ἀπὸ τέχνης συνεστῶσι καὶ γεγενημένοις ἕτερόν ἐστιν αὐτή τε καθ' αὑτὴν ἡ μορφὴ καὶ μεμιγμένη μετὰ τῆς ὕλης οἷον τῆς σφαίρας ἕτερον τὸ εἶδος (278a.) καὶ ἡ χρυσῆ καὶ ἡ χαλκῆ σφαῖρα, καὶ πάλιν τοῦ κύκλου ἑτέρα ἡ μορφὴ καὶ ὁ χαλκοῦς καὶ ὁ ξύλινος κύκλος τὸ γὰρ τί ἦν εἶναι λέγοντες σφαίρᾳ ἢ κύκλῳ οὐκ ἐροῦμεν ἐν τῷ λόγῳ χρυσὸν ἢ χαλκόν, ὡς οὐκ ὄντα ταῦτα τῆς οὐσίας ἂν δὲ τὴν χαλκῆν ἢ χρυσῆν, ἐροῦμεν, καὶ ἐὰν μὴ δυνώμεθα νοῆσαι μηδὲ λαβεῖν ἄλλο τι παρὰ τὸ καθ' ἕκαστον. Ἐνίοτε γὰρ οὐθὲν κωλύει τοῦτο συμβαίνειν, οἷον εἰ μόνος εἷς ληφθείη κύκλος οὐθὲν γὰρ ἧττον ἄλλο ἔσται τὸ κύκλῳ εἶναι καὶ τῷδε τῷ κύκλῳ, καὶ τὸ μὲν εἶδος, τὸ δ' εἶδος ἐν τῇ ὕλῃ καὶ τῶν καθ' ἕκαστον. 128 From one point of view it might seem impossible that the heaven should be one and unique, since in all formations and products whether of nature or of art we can distinguish the shape in itself and the shape in combination with matter. For instance the form of the sphere is one thing and the gold or bronze sphere another; the shape of the circle again is one thing, the bronze or wooden circle another. For when we state the essential nature of the sphere or circle we do not include in the formula gold or bronze, because they do not belong to the essence, but if we are speaking of the copper or gold sphere we do include them. We still make the distinction even if we cannot conceive or apprehend any other example beside the particular thing. This may, of course, sometimes be the case: it might be, for instance, that only one circle could be found; yet none the less the difference will remain between the being of circle and of this particular circle, the one being form, the other form in matter, i.e. a particular thing.
Ἐπεὶ οὖν ἐστὶν ὁ οὐρανὸς αἰσθητός, τῶν καθ' ἕκαστον ἂν εἴη τὸ γὰρ αἰσθητὸν ἅπαν ἐν τῇ ὕλῃ ὑπῆρχεν. 129 Now since the universe is perceptible it must be regarded as a particular; for everything that is perceptible subsists, as we know, in matter.
Εἰ δὲ τῶν καθ' ἕκαστον, ἕτερον ἂν εἴη τῷδε τῷ οὐρανῷ εἶναι καὶ οὐρανῷ ἁπλῶς. Ἕτερον ἄρα ὅδε ὁ οὐρανὸς καὶ οὐρανὸς ἁπλῶς, καὶ τὸ μὲν ὡς εἶδος καὶ μορφή, τὸ δ' ὡς τῇ ὕλῃ μεμιγμένον. 130 But if it is a particular, there will be a distinction between the being of 'this universe' and of 'universe' unqualified. There is a difference, then, between 'this universe' and simple 'universe'; the second is form and shape, the first form in combination with matter;
Ὧν δ' ἐστὶ μορφή τις καὶ εἶδος, ἤτοι ἔστιν ἢ ἐνδέχεται πλείω γενέσθαι τὰ καθ' ἕκαστα. 131 and any shape or form has, or may have, more than one particular instance.
Εἴτε γὰρ ἔστιν εἴδη, καθάπερ φασί τινες, ἀνάγκη τοῦτο συμβαίνειν, εἴτε καὶ χωριστὸν μηθὲν τῶν τοιούτων, οὐθὲν ἧττον ἐπὶ πάντων γὰρ οὕτως ὁρῶμεν, ὅσων ἡ οὐσία ἐν ὕλῃ ἐστίν, πλείω καὶ ἄπειρα ὄντα τὰ ὁμοειδῆ. 132 On the supposition of Forms such as some assert, this must be the case, and equally on the view that no such entity has a separate existence. For in every case in which the essence is in matter it is a fact of observation that the particulars of like form are several or infinite in number.
Ὥστε ἤτοι εἰσὶ πλείους οὐρανοὶ ἢ ἐνδέχεται πλείους εἶναι. Ἐκ μὲν δὴ τούτων ὑπολάβοι τις ἂν καὶ εἶναι καὶ ἐνδέχεσθαι πλείους εἶναι οὐρανούς 133 Hence there either are, or may be, more heavens than one. On these grounds, then, it might be inferred either that there are or that there might be several heavens.
σκεπτέον δὲ πάλιν τί τούτων λέγεται καλῶς καὶ τί οὐ καλῶς. Τὸ μὲν οὖν ἕτερον εἶναι τὸν λόγον τὸν ἄνευ τῆς ὕλης καὶ τὸν ἐν τῇ ὕλῃ τῆς μορφῆς καλῶς τε λέγεται, καὶ ἔστω τοῦτ' ἀληθές. Ἀλλ' οὐδὲν ἧττον οὐδεμία ἀνάγκη διὰ τοῦτο πλείους εἶναι κόσμους, οὐδ' ἐνδέχεται γενέσθαι πλείους, εἴπερ οὗτος ἐξ ἁπάσης ἐστὶ τῆς ὕλης, ὥσπερ ἔστιν. 134 We must, however, return and ask how much of this argument is correct and how much not. Now it is quite right to say that the formula of the shape apart from the matter must be different from that of the shape in the matter, and we may allow this to be true. We are not, however, therefore compelled to assert a plurality of worlds. Such a plurality is in fact impossible if this world contains the entirety of matter, as in fact it does.
Ὡδὶ δὲ μᾶλλον ἴσως τὸ λεγόμενον ἔσται δῆλον. Εἰ γάρ ἐστιν ἡ γρυπότης καμπυλότης ἐν ῥινὶ ἢ σαρκί, καὶ ἔστιν ὕλη τῇ γρυπότητι ἡ σάρξ, εἰ ἐξ ἁπασῶν τῶν σαρκῶν μία γένοιτο σὰρξ καὶ ὑπάρξειεν ταύτῃ τὸ γρυπόν, οὐθὲν ἂν ἄλλ' οὔτ' εἴη γρυπὸν οὔτ' ἐνδέχοιτο γενέσθαι. Ὁμοίως δὲ καὶ εἰ τῷ ἀνθρώπῳ ἐστὶν ὕλη σάρκες καὶ ὀστᾶ, εἰ ἐκ πάσης τῆς σαρκὸς καὶ πάντων τῶν ὀστῶν ἄνθρωπος γένοιτο ἀδυνάτων ὄντων διαλυθῆναι, οὐκ ἂν ἐνδέχοιτο εἶναι ἄλλον ἄνθρωπον. Ὡσαύ(278b.) τως δὲ καὶ ἐπὶ τῶν ἄλλων ὅλως γὰρ ὅσων ἐστὶν ἡ οὐσία ἐν ὑποκειμένῃ τινὶ ὕλῃ, τούτων οὐδὲν ἐνδέχεται γίγνεσθαι μὴ ὑπαρχούσης τινὸς ὕλης. 135 But perhaps our contention can be made clearer in this way. Suppose 'aquilinity' to be curvature in the nose or flesh, and flesh to be the matter of aquilinity. Suppose further, that all flesh came together into a single whole of flesh endowed with this aquiline quality. Then neither would there be, nor could there arise, any other thing that was aquiline. Similarly, suppose flesh and bones to be the matter of man, and suppose a man to be created of all flesh and all bones in indissoluble union. The possibility of another man would be removed. Whatever case you took it would be the same. The general rule is this: a thing whose essence resides in a substratum of matter can never come into being in the absence of all matter.
Ὁ δ' οὐρανὸς ἔστι μὲν τῶν καθ' ἕκαστα καὶ τῶν ἐκ τῆς ὕλης ἀλλ' εἰ μὴ ἐκ μορίου αὐτῆς συνέστηκεν ἀλλ' ἐξ ἁπάσης, τὸ μὲν εἶναι αὐτῷ οὐρανῷ καὶ τῷδε τῷ οὐρανῷ ἕτερόν ἐστιν, οὐ μέντοι οὔτ' ἂν εἴη ἄλλος οὔτ' ἂν ἐνδέχοιτο γενέσθαι πλείους, διὰ τὸ πᾶσαν τὴν ὕλην περιειληφέναι τοῦτον. 136 Now the universe is certainly a particular and a material thing: if however, it is composed not of a part but of the whole of matter, then though the being of 'universe' and of 'this universe' are still distinct, yet there is no other universe, and no possibility of others being made, because all the matter is already included in this.
Postquam philosophus ostendit quod est unus solus mundus, hic ostendit quod impossibile est esse plures. Et hoc necessarium fuit ostendere: quia nihil prohibet aliquid esse falsum, quod tamen contingit esse verum. Circa hoc autem tria facit: 184. After showing that there is but one world, the Philosopher here shows that it is impossible for there to be many. And it was necessary to prove this, because nothing prevents the possibility of something's being false [now] which can yet be true [later]. Concerning this he does three things:

primo ponit obiectionem, ex qua videtur ostendi quod possibile sit esse plures mundos;

secundo solvit eam, ibi: considerandum autem iterum etc.;

tertio probat quod in solutione supposuerat, ibi: hoc ipsum igitur restat ostendere et cetera.

First he presents an objection which seems to show that it is possible that many worlds exist;

Secondly, he answers it, at 194;

Thirdly, he proves something he had presupposed in his answer (L. 20).

Circa primum duo facit: About the first he does two things:

primo dicit de quo est intentio, et quo ordine sit agendum;

secundo incipit exequi propositum, ibi: videbitur enim utique et cetera.

First he states his intention and his plan of treatment;

Secondly, he begins to prove his proposition, at 186.

Dicit ergo primo quod post praedicta restat ostendendum quod non solum sit unus mundus, sed quod etiam impossibile sit esse plures: et ulterius quod mundus sit sempiternus, ita scilicet quod sit incorruptibilis, tanquam nunquam desinens esse, et ingenitus, tanquam nunquam esse incipiens, secundum suam opinionem. Et hoc adiungit quia videtur prima consideratio aliqualiter dependere ex secunda. Si enim esset mundus generabilis et corruptibilis per compositionem et dissolutionem, secundum amicitiam et litem, ut Empedocles posuit, possibile esset esse multos mundos, ita scilicet quod, uno corrupto, alius postea generaretur, sicut ipse Empedocles posuit. Et quia tunc vere cognoscitur veritas, quando dubitationes sunt solutae, quae videntur esse contra veritatem; ideo prius oportet ponere dubitationes circa hoc ipsum, ex quibus scilicet videtur quod sint vel possint esse plures mundi; huius enim solutio est confirmatio veritatis. 185. He says therefore first [127] that after the foregoing, we must still prove that not only is there one world but that it is impossible for there to be more, and further that the world is eternal, so as to be imperishable, i.e. never ceasing to be, and unborn, i.e., never beginning to be, according to his opinion. He states this because the first consideration seems somehow to depend on the second. For if the world were generable and perishable by union and separation, according to friendship and strife, as Empedocles said, many worlds would be possible in the sense that when one had perished another would be generated later, as Empedocles believed. And because the truth is truly known when the difficulties which seem to be contrary to it are solved, therefore the first thing to do is bring forth the difficulties concerning this, i.e., which seem to indicate that there are or can be many worlds — for the solution to this difficulty will confirm the truth.
Deinde cum dicit: videbitur enim utique etc., ponit rationem ex qua aliquis potest dubitare, aestimans possibile esse quod sint plures mundi. Unde praemittit quod sic intendentibus, scilicet secundum rationem quae sequitur, videbitur esse impossibile ipsum, scilicet mundum, esse unum et solum: subintelligendum est ex necessitate. Non enim sequens ratio probat quod necesse sit esse plures mundos, quod aequipollet ei quod est impossibile unum solum esse mundum: sed probat quod possibile est esse plures mundos, quod aequipollet ei quod est non necesse esse unum solum mundum. Ad hoc autem ostendendum inducit rationem quae continet duos syllogismos: 186. Then at [128] he presents the argument that could lead one to question whether it is not possible for more than one world to exist. Hence he prefaces the remark that, for those who hold this point of view, i.e., the one coinciding with the argument to follow, it will appear impossible that it, namely, the world, be one and unique, i.e., that there be necessarily just one world. For the following argument does not prove that it is necessary that there be several worlds, which is equivalent to its being impossible that there be but one; rather it proves that it is possible that there be more than one world, which is equivalent to its not being necessary that there be but one. Now in order to show this he induces an argument containing two syllogisms:

quorum primum primo ponit;

secundo secundum, ibi: quorum autem est forma quaedam et cetera.

The first of these is at 187;

The second at 190.

Primus syllogismus talis est. In omnibus sensibilibus quae fiunt ab arte vel a natura, alia est consideratio formae secundum se consideratae, alia est consideratio formae prout est in materia; sed caelum est quoddam sensibile habens formam in materia; ergo alia est consideratio absoluta formae ipsius, prout consideratur in universali, et alia est consideratio formae ipsius in materia, prout consideratur in particulari. The first syllogism is this: In all sensible things that come to be by art or by nature, the consideration of the form considered in itself is one thing and the consideration of the form insofar as it is in matter is another. But the heaven is a sensible thing having a form in matter. Therefore, the absolute consideration of its form, i.e., as considered universally, is one thing, and the consideration of its form in matter, i.e., as considered in particular is another.

Primo ergo ponit maiorem;

secundo minorem, ibi: quoniam igitur est caelum etc.;

tertio infert conclusionem, ibi: si autem singularium et cetera.

First, therefore, he presents the major, at 187;

Secondly, the minor, at 188;

Thirdly, he draws the conclusion, at 189.

Dicit ergo primo quod in omnibus existentibus et generatis, idest factis, vel a natura vel ab arte, alterum est secundum nostram considerationem ipsa forma secundum seipsam considerata; et alterum est ipsa forma mixta cum materia, idest secundum quod accipitur prout est coniuncta cum materia. 187. He says therefore first [128] that in all things that exist and were generated, i.e., made, either by nature or by art, the form considered according to itself is one thing according to our consideration, and the form mixed with matter, i.e., the form taken as joined with matter, is another.
Et hoc primo manifestat per exemplum in mathematicis, in quibus est magis manifestum, eo quod in ratione eorum non ponitur materia sensibilis. Alterum est enim secundum considerationem nostram ipsa species sphaerae, et alterum forma sphaerae in materia sensibili, prout significatur cum dicitur aurea vel aerea sphaera: et similiter aliud est ipsa forma circuli, et aliud est quod dicitur aereus aut ligneus circulus. Et hoc manifestat quia, cum dicimus quod quid erat esse, idest definitivam rationem, sphaerae aut circuli, non ponimus in eius ratione aureum aut aereum; tanquam hoc quod dicimus aureum aut aereum, non sint de eorum substantia, quam scilicet significat definitio. He first explains this by an example in mathematical objects in which it is more evident, because sensible matter does not enter therein. For the species of a sphere is according to our consideration other than the form of the sphere in sensible matter, which is denoted when a sphere is called "golden" or "bronze"; similarly, the form of a circle is one thing, and what is meant by a golden or bronze circle is another. And this is evident, because when we give the quod quid erat esse, i.e., the defining notion, of a circle or a sphere, we make no mention therein of gold or bronze. This implies that to be "golden" or "bronze" does not pertain to their substance [essence], which the definition signifies.
Sed videtur hoc magis esse dubium in rebus naturalibus, quarum formae non possunt esse nec intelligi sine materia sensibili; sicut simum non potest esse nec intelligi sine naso. Sed tamen formae naturales, quamvis non possint intelligi sine materia sensibili in communi, possunt tamen intelligi sine materia sensibili signata, quae est individuationis et singularitatis principium; sicut pes non potest intelligi sine carnibus et ossibus, potest tamen intelligi sine his carnibus et his ossibus. Et ideo subdit quod, si non possumus intelligere neque sumere in nostra consideratione aliquid aliud praeter singulare, idest praeter materiam, quae includitur in ratione singularis, scilicet prout est signata (quia quandoque nihil prohibet hoc accidere, ut scilicet non possit forma intelligi sine materia sensibili, sicut si intelligamus circulum sine materia sensibili): nihilominus tamen in naturalibus, in quibus hoc accidit quod non intelligitur forma sine materia, alia est ratio rei in communi acceptae et in singulari, sicut hominis et huius hominis; puta si dicamus quod aliud est esse circulo et huic circulo, idest alia est ratio definitiva utriusque. Et haec quidem, scilicet ratio rei in communi, est species, idest ipsa ratio speciei: haec autem, scilicet ratio rei in particulari, significat rationem speciei in materia determinata, et est de numero singularium. But there seems to be a difficulty in natural things, whose forms cannot exist or be understood without sensible matter, as "snub" cannot exist and be understood without "nose." Natural forms, however, although they cannot be understood without sensible matter in common, can be understood without signed sensible matter, which is the principle of individuation and of singularity. Thus, "foot" cannot be understood without flesh and bones, but it can be understood without this flesh and these bones. And therefore he adds that if we cannot understand and accept in our consideration anything outside the singular, i.e., outside the matter which is included in the notion of the individual, namely, as it is signate—because sometimes there is nothing to prevent this from happening (namely, that a form be able to be understood without sensible matter) in the same way that we understood a circle without sensible matter; nevertheless, in natural things, in which forms are not understood without matter, the notions of the thing taken in common and taken in the singular are not the same, any more than the notion of "man" and of "this man" are the same. Thus the essence of "circle" and "this circle," i.e., of the notions defining a circle, and this circle, are different. For the notion of a thing in common is the species, i.e., the notion of the species, but the notion of a particular thing signifies the notion of the species as found in determinate matter, and pertains to the singular.
Deinde cum dicit: quoniam igitur est caelum etc., ponit minorem syllogismi inducti. Et dicit quod, cum caelum, idest mundus, sit quoddam sensibile, necesse est quod sit de numero singularium: et hoc ideo, quia omne sensibile habet esse in materia. Id autem quod est forma non in materia, non est sensibile, sed intelligibile tantum: qualitates enim sensibiles sunt dispositiones materiae. 188. Then at [129] he presents the minor of his syllogism. And he says that since the heaven, i.e., the world, is something sensible, it must be among the singulars, for every sensible thing exists in matter. For a form not in matter is not sensible but intelligible only — for sensible qualities are characteristics of matter.
Deinde cum dicit: si autem singularium etc., ponit conclusionem. Et dicit quod si caelum, idest mundus, est de numero singularium, ut ostensum est, alterum erit esse huic caelo singulariter dicto, et caelo simpliciter, idest universaliter sumpto; idest alia erit ratio utriusque. Et sic sequitur quod alterum sit secundum considerationem hoc caelum singulariter dictum, et caelum universaliter sumptum: ita scilicet quod hoc caelum universaliter sumptum sit sicut species et forma; hoc autem, scilicet caelum singulariter sumptum, sit sicut forma coniuncta materiae. Quod non est sic intelligendum quod in ratione rei naturalis universaliter sumptae nullo modo cadat materia; sed quod non cadat ibi materia signata. 189. Then at [130] he presents the conclusion and says that if the heaven, i.e., the world, belongs among the singulars, as has been shown, its notion as a singular will differ from its notion absolutely, i.e., taken universally the two notions will differ. Consequently, it follows that "this heaven" taken singularly will be different in consideration from "heaven" taken universally, i.e., this latter heaven taken universally will be as a species and form, while the other, namely, that taken singularly, will be as form joined to matter. However, this is not to be taken as implying that in the definition of a natural thing taken universally no matter is mentioned at all, but rather that individual matter is not mentioned.
Deinde cum dicit: quorum autem est forma quaedam etc., ponit secundum syllogismum, qui talis est. Quorumcumque est forma in materia, aut sunt aut contingit esse plura individua unius speciei; sed hoc caelum significat formam in materia, ut dictum est; ergo aut sunt aut possunt esse plures caeli. 190. Then at [131] he presents the second syllogism, as follows: Whatever things have their forms in matter, are, or are able to be, several individuals of one species. But "this heaven" signifies a form in matter, as was said. Therefore, there either are, or can be, many heavens.

Circa hoc autem primo ponit maiorem;

secundo manifestat eam, ibi: sive enim sint species etc.;

tertio infert conclusionem, ibi: itaque aut sunt et cetera. Minorem supponit ex praemisso syllogismo.

Now in regard to this he first presents the major;

Secondly, he explains it, at 191;

Thirdly, [having taken the minor from the previous syllogism], he draws the conclusion at 192.

Dicit ergo primo quod omnia illa quorum est forma quaedam et species, idest quae non sunt ipsae formae et species, sed habent formas et species, aut sunt plura singularia unius speciei, aut contingit fieri plura: illa vero quae ipsamet sunt formae et species subsistentes, sicut substantiae separatae, non possunt esse plura unius speciei. He says therefore first [131] that all things of which there is a form and species, i.e., which are not themselves forms and species, but have forms and species, are either many individuals of one species or many can exist. But things that are themselves forms and subsistent species, as are separated substances, cannot have several members of one species.
Deinde cum dicit: sive enim sint species etc., manifestat praedictam propositionem tam secundum opinionem Platonicam, quam secundum opinionem propriam. Et dicit quod sive sint species, idest ideae separatae, sicut Platonici dicunt, necesse est hoc accidere, scilicet quod sint plura individua unius speciei (quia species separata ponitur sicut exemplar rei sensibilis; possibile est autem ad unum exemplar fieri multa exemplata); sive etiam nullum talium, idest nulla specierum, separatim existat; nihilominus plura individua possunt esse unius speciei. Videmus enim in omnibus sic accidere, quorum substantia, idest essentia quam significat definitio, est in materia signata, quod sunt plura, immo infinita individua unius speciei. Et hoc ideo est, quia cum materia signata non sit de ratione speciei, ratio speciei indifferenter potest salvari in hac materia signata et in illa: et ita possunt esse plura individua unius speciei. 191. Then at [132] he explains the foregoing both according to Plato's opinion and according to his own. And he says that whether there are "species," i.e., separated ideas, as the Platonists assume, then this must happen, i.e., there must be several individuals of one species — because the separated species is posited as the exemplar of a sensible thing and it is possible to make many copies according to one exemplar; or whether no such species exist separately, there can still be several individuals of one species. For we see this happen in all things whose substance (i.e., whose essence, which is signified by the definition) exists in signate matter, namely, that there are several individuals, or even an infinitude of individuals, of one species. The reason for this is that, since signate matter does not enter the notion of the species, the notion of the species can be indifferently verified in this individual matter and in that; consequently, there can be many individuals of one species.
Deinde cum dicit: itaque aut sunt etc., infert conclusionem intentam, scilicet quod aut sunt plures caeli, aut contingit esse factos plures caelos. 192. Then at [133] he draws the intended conclusion, namely, that either there are many worlds or many worlds can be made.
Ultimo autem epilogat quod ex praemissis potest aliquis suspicari quod vel sint vel possint esse plures mundi. Finally he says in summary that from the foregoing someone can conjecture that either there are, or can be, many worlds.
Sed videtur hic esse contrarietas inter Aristotelem et Platonem. Nam Plato in Timaeo ex unitate exemplaris probavit unitatem mundi: hic autem Aristoteles ex unitate speciei separatae concludit possibile esse quod sint plures mundi. 193. But there seems to be a conflict here between Aristotle and Plato. For Plato in the Timaeus proved the oneness of the world from the oneness of the exemplar; but here Aristotle from the oneness of the separated species concludes to the possibility of several worlds.
Et potest dupliciter responderi. Uno modo ex parte ipsius exemplaris. Quod quidem si sic sit unum quod unitas sit essentia eius, necesse est exemplatum etiam imitari exemplar in sua unitate. Et tale est primum exemplar separatum: unde et mundum, qui est primum exemplatum, necesse est esse unum: et secundum hoc procedit probatio Platonis. Si vero unitas non sit essentia exemplaris, sed sit praeter essentiam eius, sic exemplatum poterit assimilari exemplari in eo quod pertinet ad eius speciem, puta in ratione hominis vel equi, non autem quantum ad ipsam unitatem: et hoc modo procedit hic ratio Aristotelis. But two answers can be given to this. First on the part of the exemplar, which, if it is one in such a way that oneness is its essence, then the copy must imitate the exemplar in this oneness. But the first separated exemplar is such. Hence also the world, which is the first copy thereof, must be one. This was the way Plato proceeded in his proof. But if oneness is not of the essence of the exemplar but is outside its essence, then the copy could be like the exemplar in respect to what belongs to its species — for example, in the notion of man or horse — but not in respect to oneness. And it is in this way that Aristotle's reasoning proceeds.
Alio modo potest solvi ex parte exemplati, quod tanto est perfectius, quanto magis assimilatur exemplari. Alia ergo exemplata assimilantur exemplari uni secundum unitatem speciei, non secundum unitatem numeralem: sed caelum, quod est perfectum exemplatum, assimilatur suo exemplari secundum unitatem numeralem. Or it can be answered from the viewpoint of the copy, which is more perfect to the extent that it is more faithful to the exemplar. Therefore, some copies are like one exemplar in respect to oneness of species, but not in respect to numerical oneness. But the heaven, which is a perfect copy, is like its exemplar with respect to numerical oneness.
Deinde cum dicit: considerandum autem iterum etc., solvit obiectionem praedictam. 194. Then at [134] he solves this objection:

Et primo ponit solutionem;

secundo manifestat eam, ibi: sic autem forte et cetera.

First he gives the solution;

Secondly, he explains it, at 195.

Dicit ergo primo quod oportet iterum, ad solvendum dubitationem praedictam, considerare quid dicatur bene et quid non bene: si enim omnia praemissa sint vera, necesse est conclusionem esse veram. Dicit igitur quod bene dictum est quod altera sit ratio formae, ea scilicet quae est sine materia, et ea quae est cum materia, He says therefore first that in order to settle the above doubt we must once more consider what was said well and what not well. For if all the premises are true, the conclusion is necessarily true. He says, therefore, that it was correct to say that the notion of form differs, namely, in the case of that which is without matter and in the case of that which is with matter.
et hoc concedatur tanquam verum; et sic concedatur conclusio primi syllogismi, quae est minor secundi. Sed non sequitur ex necessitate propter hoc quod sint multi mundi, vel quod possint esse plures, si verum sit quod iste mundus sit ex tota sua materia, sicuti est verum, ut infra probabitur: maior enim propositio secundi syllogismi, scilicet quod illa quae habent formam in materia possunt esse multa numero unius speciei, non habet veritatem nisi in illis quae non constant ex tota sua materia. This is to be granted as true. Consequently, the first conclusion which is the minor of the second syllogism is conceded. But from this it does not follow of necessity either that there are several worlds, or that there can be several, if it is true that this world consists of all its matter, as is true and as will be proved below. For the major proposition of the second syllogism, namely, that things which have a form in matter can be numerically many in one species, is not true except in things that do not consist of their entire matter.
Deinde cum dicit: sic autem forte etc., manifestat quod dixerat per exemplum. 195. Then at [135] he explains what he had said with an example.

Et primo ponit exempla;

secundo adaptat ad propositum, ibi: caelum autem est quidem singularium et cetera.

First he gives the examples;

Secondly, he adapts them to his proposition, at 196.

Dicit ergo primo quod per ea quae dicentur, magis fiet manifestum quod dictum est. Simitas enim est curvitas in naso aut in carne; et ita caro est materia simitatis. Si ergo ex omnibus carnibus fieret una caro, scilicet unius nasi, et in hac esset simitas, nihil aliud esset simum, neque posset esse. Et eadem ratio est de homine, cum carnes et ossa sint materia hominis, si ex omnibus carnibus et ossibus fieret unus homo, ita scilicet quod nullo modo possent dissolvi, non posset esse aliquis alius homo quam unus (si vero possent dissolvi, possibile esset, illo homine corrupto, alium hominem esse; sicut dissoluta arca, ex eisdem lignis fit alia arca). Et ita etiam est in aliis. Et huius rationem assignat, quia nihil eorum quorum forma est in materia, potest fieri, si non adsit propria materia; sicut domus non posset fieri si non sint lapides et ligna. Et ita, si non sint aliae carnes et ossa praeter ea ex quibus componitur unus homo, non poterit fieri alius homo praeter illum. He says therefore first [135] that what has been said will become clearer from what will be said. For snub-nosedness is curvature in a nose or in flesh; thus flesh is the matter of snub-nosedness. If then from all flesh one flesh were to be made, namely, the flesh of one nose, and snub-nosedness existed in it, nothing else would be snub-nosed nor could be. And the same holds for man, since flesh and bones are the matter of man: if one man were formed from all the flesh and all the bones, so that he could now not be destroyed, there could be no more than one man — but if he could be destroyed, it would be possible, after his corruption, for another man to exist, just as when a box is destroyed, another can be made from the same wood. And the same is true for other things. And the reason for this he assigns, namely, that none of the things whose form is in matter can come into being if the proper matter is not at hand, any more than a house could be made if there were not stones and wood. Consequently, if there were no bones and flesh other than those of which the one man is composed, no other man could come into being but him.
Deinde cum dicit: caelum autem est quidem singularium etc., adaptat ad propositum. Et dicit verum esse caelum esse de numero singularium, et eorum quae ex materia constituuntur: non tamen est ex parte suae materiae, sed ex tota sua materia. Et ideo, quamvis sit alia ratio caeli et huius caeli, non tamen est aut potest esse aliud caelum, propter hoc quod tota materia caeli comprehensa est sub hoc caelo. 196. Then at [136] he adapts this to his proposition. And he says it is true that the heaven is a singular thing and one constituted of matter. But it is not constituted out of part of its matter, but out of all of it. And therefore, although there is a difference between the notions of "heaven" and "this heaven," there neither is, nor can be, another heaven, due to the fact that all the matter of heaven is comprehended under this heaven.
Sciendum est autem quod quidam aliis modis probant possibile esse plures caelos. Uno modo sic. Mundus factus est a Deo; sed potentia Dei, cum sit infinita, non determinatur ad istum solum mundum; ergo non est rationabile quod non possit facere etiam alios mundos. 197. However, it should be realized that some prove the possibility of many worlds in other ways. In one way, as follows: The world was made by God; but the power of God, since it is infinite, is not limited to this world alone. Therefore it is not reasonable to say that He cannot make yet other worlds.
Et ad hoc dicendum est quod, si Deus faceret alios mundos, aut faceret eos similes huic mundo, aut dissimiles. Si omnino similes, essent frustra: quod non competit sapientiae ipsius. Si autem dissimiles, nullus eorum comprehenderet in se omnem naturam corporis sensibilis: et ita nullus eorum esset perfectus, sed ex omnibus constitueretur unus mundus perfectus. To this it must be said that if God were to make other worlds, He would make them either like or unlike this world. If entirely alike, they would be in vain — and that conflicts with His wisdom. If unlike, none of them would comprehend in itself every nature of sensible body; consequently no one of them would be perfect, but one perfect world would result from all of them.
Alio modo potest argui sic. Quanto aliquid est nobilius, tanto eius species est magis virtuosa; mundus autem est nobilior qualibet re naturali hic existente; cum igitur species rei naturalis hic existentis, puta equi aut bovis, possit perficere plura individua, multo magis species totius mundi potest plura individua perficere. In another way, as follows: To the extent that something is more noble, to that extent is its species more powerful. But the world is nobler than any natural thing existing here. Therefore, since the species of a natural thing existing here, for example, of a horse or cow, could perfect many individuals, much more so can the species of the whole world perfect many individuals.
Sed ad hoc dicendum est quod maioris virtutis est facere unum perfectum, quam facere multa imperfecta. Singula autem individua rerum naturalium quae sunt hic, sunt imperfecta; quia nullum eorum comprehendit in se totum quod pertinet ad suam speciem. Sed mundus hoc modo perfectus est: unde ex hoc ipso eius species ostenditur magis virtuosa. But to this it must be answered that it takes more power to make one perfect than to make several imperfect. Now the single individuals of natural things which exist here are imperfect, because no one of them comprehends within itself the total of what, pertains to its species. But it is in this way that the world is perfect; hence, from that very fact its species is shown to be more powerful.
Tertio obiicitur sic. Melius est multiplicari optima, quam ea quae sunt minus bona; sed mundus est optimus; ergo melius est esse plures mundos, quam plura animalia aut plures plantas. Thirdly, one objects thus: It is better for the best to be multiplied than for things not so good. But the world is the best. Therefore, it is better to have many worlds than many animals or many plants.
Et ad hoc dicendum quod hoc ipsum pertinet ad bonitatem mundi, quod sit unus; quia unum habet rationem boni: videmus enim quod per divisionem aliqua decidunt a propria bonitate. To this it must be said that here it pertains to the goodness of the world to be one, because oneness possesses the aspect of goodness. For we see that through being divided some things lose their proper goodness.

Lecture 20:
The universe shown to consist of every natural and sensible body as its matter
Chapter 9 cont.
Λείπεται ἄρα αὐτὸ τοῦτο δεῖξαι, ὅτι ἐξ ἅπαντος τοῦ φυσικοῦ καὶ τοῦ αἰσθητοῦ συνέστηκε σώματος. Εἴπωμεν δὲ πρῶτον τί λέγομεν εἶναι τὸν οὐρανὸν καὶ ποσαχῶς, ἵνα μᾶλλον ἡμῖν δῆλον γένηται τὸ ζητούμενον. 137 It remains, then, only to prove that it is composed of all natural perceptible body. First, however, we must explain what we mean by 'heaven' and in how many senses we use the word, in order to make clearer the object of our inquiry.
Ἕνα μὲν οὖν τρόπον οὐρανὸν λέγομεν τὴν οὐσίαν τὴν τῆς ἐσχάτης τοῦ παντὸς περιφορᾶς, ἢ σῶμα φυσικὸν τὸ ἐν τῇ ἐσχάτῃ περιφορᾷ τοῦ παντός εἰώθαμεν γὰρ τὸ ἔσχατον καὶ τὸ ἄνω μάλιστα καλεῖν οὐρανόν, ἐν ᾧ καὶ τὸ θεῖον πᾶν ἱδρῦσθαί φαμεν. Ἄλλον δ' αὖ τρόπον τὸ συνεχὲς σῶμα τῇ ἐσχάτῃ περιφορᾷ τοῦ παντός, ἐν ᾧ σελήνη καὶ ἥλιος καὶ ἔνια τῶν ἄστρων καὶ γὰρ ταῦτα ἐν τῷ οὐρανῷ εἶναί φαμεν. Ἔτι δ' ἄλλως λέγομεν οὐρανὸν τὸ περιεχόμενον σῶμα ὑπὸ τῆς ἐσχάτης περιφορᾶς τὸ γὰρ ὅλον καὶ τὸ πᾶν εἰώθαμεν λέγειν οὐρανόν. 138 (a) In one sense, then, we call 'heaven' the substance of the extreme circumference of the whole, or that natural body whose place is at the extreme circumference. We recognize habitually a special right to the name 'heaven' in the extremity or upper region, which we take to be the seat of all that is divine. (b) In another sense, we use this name for the body continuous with the extreme circumference which contains the moon, the sun, and some of the stars; these we say are 'in the heaven'. (c) In yet another sense we give the name to all bodies included within extreme circumference, since we habitually call the whole or totality 'the heaven'.
Τριχῶς δὴ λεγομένου τοῦ οὐρανοῦ, τὸ ὅλον τὸ ὑπὸ τῆς ἐσχάτης περιεχόμενον περιφορᾶς ἐξ ἅπαντος ἀνάγκη συνεστάναι τοῦ φυσικοῦ καὶ τοῦ αἰσθητοῦ σώματος διὰ τὸ μήτ' εἶναι μηδὲν ἔξω σῶμα τοῦ οὐρανοῦ μήτ' ἐνδέχεσθαι γενέσθαι. 139 The word, then, is used in three senses. Now the whole included within the extreme circumference must be composed of all physical and sensible body, because there neither is, nor can come into being, any body outside the heaven.
Εἰ γὰρ ἔστιν ἔξω τῆς ἐσχάτης περιφορᾶς σῶμα φυσικόν, ἀνάγκη αὐτὸ ἤτοι τῶν ἁπλῶν εἶναι σωμάτων ἢ τῶν συνθέτων, καὶ ἢ κατὰ φύσιν ἢ παρὰ φύσιν ἔχειν. 140 For if there is a natural body outside the extreme circumference it must be either a simple or a composite body, and its position must be either natural or unnatural.
Τῶν μὲν οὖν ἁπλῶν οὐθὲν ἂν εἴη. Τὸ μὲν γὰρ κύκλῳ φερόμενον δέδεικται ὅτι οὐκ ἐνδέχεται μεταλλάξαι τὸν αὑτοῦ τόπον. Ἀλλὰ μὴν οὐδὲ τὸ ἀπὸ τοῦ μέσου δυνατόν, οὐδὲ τὸ ὑφιστάμενον. Κατὰ φύσιν μὲν γὰρ οὐκ ἂν εἴησαν (ἄλλοι γὰρ αὐτῶν οἰκεῖοι τόποι), 141 But it cannot be any of the simple bodies. For, first, it has been shown that that which moves in a circle cannot change its place. And, secondly, it cannot be that which moves from the centre or that which lies lowest. Naturally they could not be there, since their proper places are elsewhere;
παρὰ φύσιν δ' εἴπερ εἰσίν, ἄλλῳ τινὶ ἔσται κατὰ φύσιν ὁ ἔξω τόπος τὸν γὰρ τούτῳ παρὰ φύσιν ἀναγκαῖον ἄλλῳ εἶναι κατὰ φύσιν. Ἀλλ' οὐκ ἦν ἄλλο σῶμα παρὰ ταῦτα. 142 and if these are there unnaturally, the exterior place will be natural to some other body, since a place which is unnatural to one body must be natural to another: but we saw that there is no other body besides these. Then it is not possible that any simple body should be outside the heaven.
Οὐκ ἄρ' ἐστὶ δυνατὸν οὐθὲν τῶν ἁπλῶν ἔξω εἶναι τοῦ (279a.) οὐρανοῦ σῶμα. Εἰ δὲ μὴ τῶν ἁπλῶν, οὐδὲ τῶν μικτῶν ἀνάγκη γὰρ εἶναι καὶ τὰ ἁπλᾶ τοῦ μικτοῦ ὄντος. 143 But, if no simple body, neither can any mixed body be there: for the presence of the simple body is involved in the presence of the mixture.
Ἀλλὰ μὴν οὐδὲ γενέσθαι δυνατόν ἤτοι γὰρ κατὰ φύσιν ἔσται ἢ παρὰ φύσιν, καὶ ἢ ἁπλοῦν ἢ μικτόν. Ὥστε πάλιν ὁ αὐτὸς ἥξει λόγος οὐδὲν γὰρ διαφέρει σκοπεῖν εἰ ἔστιν ἢ εἰ γενέσθαι δυνατόν. 144 Further neither can any body come into that place: for it will do so either naturally or unnaturally, and will be either simple or composite; so that the same argument will apply, since it makes no difference whether the question is 'does A exist?' or 'could A come to exist?'
Φανερὸν τοίνυν ἐκ τῶν εἰρημένων ὅτι οὔτ' ἔστιν ἔξω οὔτ' ἐγχωρεῖ γενέσθαι σώματος ὄγκον οὐθενός ἐξ ἁπάσης ἄρ' ἐστὶ τῆς οἰκείας ὕλης ὁ πᾶς κόσμος ὕλη γὰρ ἦν αὐτῷ τὸ φυσικὸν σῶμα καὶ αἰσθητόν. Ὥστ' οὔτε νῦν εἰσὶ πλείους οὐρανοὶ οὔτ' ἐγένοντο, οὔτ' ἐνδέχεται γενέσθαι πλείους ἀλλ' εἷς καὶ μόνος καὶ τέλειος οὗτος οὐρανός ἐστιν. 145 From our arguments then it is evident not only that there is not, but also that there could never come to be, any bodily mass whatever outside the circumference. The world as a whole, therefore, includes all its appropriate matter, which is, as we saw, natural perceptible body. So that neither are there now, nor have there ever been, nor can there ever be formed more heavens than one, but this heaven of ours is one and unique and complete.
Posita solutione inducta, hic philosophus probat quod supposuerat, scilicet quod mundus constet ex tota sua materia. 198. Having presented the solution brought forward, the Philosopher here proves what he had presupposed, namely, that the world consists of all its matter.
Et primo dicit de quo est intentio, et quo ordine sit procedendum: dicens quod hoc ipsum restat ostendere ad complementum praemissae solutionis, quod mundus constet ex omni corpore naturali et sensibili, quod est materia eius. Sed antequam hoc ostendamus, oportet primo dicere quid significetur per hoc nomen caelum, et quot modis dicatur, ut illud quod quaeritur magis possit manifestari. First he tells his intention and order of procedure [137] and says that in order to complete the preceding solution, we must show that the world consists of every natural and sensible body, which is its matter. But before showing this, it is necessary to explain what is meant by this word "heaven," and in how many senses it is used, so that our question can be answered more clearly.
Secundo ibi: uno quidem igitur modo etc., exequitur propositum: 199. Secondly he proves his proposition:

et primo ostendit quot modis dicatur caelum;

secundo ostendit principale propositum, ibi: tripliciter autem et cetera.

First he shows the various senses of the word "heaven";

Secondly, he proves the main proposition, at 200.

Circa primum ponit tres significationes caeli. Uno enim modo dicitur caelum substantia quaedam quae est extremae circulationis totius, idest quae in toto universo est extrema, et circulariter movetur. Et quia exposuerat significationem nominis per substantiam, cuius ratio transcendit considerationem naturalem, cum pertineat ad considerationem metaphysici, adhibet aliam expositionem, in eadem tamen significatione, dicens quod caelum est corpus naturale quod est in extrema circumferentia totius: et haec expositio est magis propria scientiae naturali. With regard to the first [138] he gives three senses of heaven. In one way the heaven is called "the substance of the extreme circulation of the whole," i.e., that which is at the boundary of the whole universe and is moved circularly. And because he had explained the meaning of the word in terms of "substance," whose notion transcends natural philosophy, since it pertains to Metaphysics, he adds another explanation having the same meaning, saying that the heaven is "the natural body whose place is at the extreme circumference of the world," which explanation is more befitting to natural science.
Probat autem hanc significationem ex consuetudine loquendi: quia nominibus est utendum ut plures, sicut dicitur in II Topic. Consueverunt enim homines vocare caelum illud quod est extremum totius mundi, et quod maxime est sursum: non quidem secundum quod sursum accipitur in scientia naturali, prout scilicet est terminus motus levium (sic enim nihil magis est sursum quam locus in quem fertur ignis): sed sumitur hic sursum secundum communem modum loquendi, prout id quod est remotius a medio, vocatur sursum. Consuevit etiam vocari sursum id quod est locus omnium divinorum (ut tamen divina non dicantur hic corpora caelestia, quae non omnia sunt in suprema sphaera; sed secundum quod divina dicuntur substantiae immateriales et incorporeae): dictum est enim supra quod omnes homines locum qui est sursum attribuunt Deo. He proves this meaning from the way people speak — since words are to be used in the sense most people use them, as is said in Topics II. For men are more likely to call "heaven" that which is the extreme of the entire world and which is most up, not, indeed, as "up" is taken in natural science, i.e., as being the terminus of the motion of light things (for in this sense nothing is farther "up" than the place to which fire is borne) but as taken according to common parlance, where "up" designates that which is farther from the middle. "Up" also refers to the place of all divine beings (where "divine" signifies not celestial bodies — not all of which are in the outermost sphere — but non-material and incorporeal substances), for it has been said above that all men attribute to God a place that is up.
Secundo modo dicitur caelum non solum suprema sphaera, sed totum corpus quod continuatur cum extrema circumferentia totius universi, idest omnes sphaerae caelestium corporum, in quibus sunt luna et sol et quaedam stellarum, scilicet alii quinque planetae (nam stellae fixae sunt in suprema sphaera secundum opinionem Aristotelis, qui non posuit aliam sphaeram esse supra sphaeram stellarum fixarum). In a second way "heaven" means not only the outermost sphere but "the whole body continuous with the extreme circumference of the whole universe," i.e., all the spheres of celestial bodies, in which exist the moon and sun and certain of the stars, namely, the other five planets (for the fixed stars are in the supreme sphere according to the opinion of Aristotle, who did not posit another sphere above that of the fixed stars).
Et hanc etiam significationem probat per communem usum loquendi: dicimus enim solem et lunam et alios planetas esse in caelo. Dicuntur autem haec corpora continuari cum suprema sphaera, propter convenientiam in natura, quia scilicet sunt incorruptibilia et circulariter mobilia; non autem ita quod ex omnibus sit unum corpus continuum; quia sic eorum non possent esse plures et diversi motus; continuum est enim cuius motus est unus, ut dicitur in V Metaphys. And he proves this meaning also on the basis of common parlance: for we say that the sun and moon and other planets exist in the heaven. Now these bodies are said to be continuous with the extreme sphere, because they are alike in nature, i.e., they are imperishable and movable circularly, and not because one continuous body is formed from all of them — for then they could not have several and different motions, a continuum being something whose motion is one, as is said in Metaphysics V.
Tertio modo dicitur caelum totum corpus quod continetur ab extrema circumferentia, idest a suprema sphaera. Et hoc etiam probat ex usu loquendi: quia consuevimus totum mundum et omne, idest universum, vocare caelum. In a third way "heaven" means "the whole body contained within the extreme circumference," i.e., by the extreme sphere. This, too, he proves from the common use of the word — since we are wont to call the whole world and the totality, i.e., the universe, the "heaven."
Est autem considerandum quod caelum his tribus modis dicitur non aequivoce, sed analogice, scilicet per respectum ad unum primum: primo enim et principaliter dicitur caelum suprema sphaera; secundo autem aliae sphaerae caelestes, ex continuitate quam habent ad supremam sphaeram; tertio modo universitas corporum, secundum quod continetur ab extrema sphaera. It should be noted that "heaven" is here used in these three ways not equivocally but analogically, i.e., in relation to one first. For it is the supreme sphere that is first and principally called "heaven"; secondly, the other celestial spheres from the continuity they have with the supreme sphere; thirdly, the universe of bodies insofar as they are contained by the extreme sphere.
Deinde cum dicit: tripliciter autem etc., ostendit propositum. 200. Then at [139] he proves the proposition.

Et primo ostendit quod non est aliquod corpus sensibile extra caelum tertio modo dictum, idest extra hunc mundum;

secundo ostendit quod non est extra ipsum aliquid eorum quae consequuntur ad corpora naturalia, ibi: simul autem manifestum et cetera.

First he shows that there is no sensible body outside the heaven taken in the third sense, i.e., outside this world;

Secondly, he shows that there is not outside it any of the things that are normally consequent upon natural bodies (L. 21).

Circa primum tria facit: As to the first he does three things:

primo proponit quod intendit;

secundo probat propositum, ibi: si enim est etc.;

tertio concludit principale intentum, ibi: manifestum igitur ex dictis et cetera.

First he proposes what he intends;

Secondly, he proves his proposition, at 201;

Thirdly, he concludes to his main proposition, at 206.

Dicit ergo primo quod, cum tripliciter dicatur caelum, nunc intendimus de caelo tertio modo dicto, secundum quod caelum dicitur totum quod continetur ab extrema circumferentia: et hoc caelum necesse est quod constet ex omni corpore sensibili et naturali (quod est eius materia: et sic constat ex tota sua materia), propter hoc quod extra hoc caelum nullum corpus est, nec contingit esse. He says therefore first [139] that whereas "heaven" is said in three ways, we shall be discussing it now in its third sense, where heaven is taken as "the whole contained by the extreme circumference." Concerning this heaven it is necessary that it consist of every sensible and natural body — which is its matter, and thus it consists of all its matter — due to the fact that outside this heaven no body exists, nor can exist.
Deinde cum dicit: si enim est etc., probat propositum. 201. Then at [140] he proves the proposition.

Et primo ostendit quod nullum corpus est extra caelum;

secundo quod nullum potest ibi esse, ibi: sed et neque factum esse et cetera.

First he shows that there is no body outside the heaven;

Secondly, that none can be there, at 205.

Circa primum duo facit: About the first he does two things:

primo praemittit quandam divisionem, per quam manifestat propositum;

secundo excludit singula membra divisionis, ibi: simplicium quidem igitur et cetera.

First he presents a division through which he manifests the proposition;

Secondly, he excludes each member of the division, at 202.

Dicit ergo primo quod, si est aliquod corpus physicum, idest naturale, extra extremam peripheriam, idest circumferentiam, necesse est quod illud corpus aut sit de numero simplicium corporum, aut de numero compositorum. Item necesse est quod vel sit ibi secundum naturam, vel praeter naturam. He says therefore first [140] that if there is a Physical, i.e., natural, body outside the extreme periphery , i.e., circumference, it has to be either of the number of simple bodies, or of the number of composite bodies. Moreover, it must exist there according to nature, or outside its nature.
Deinde cum dicit: simplicium quidem igitur etc., excludit singula membra praedictae divisionis. 202. Then at [141] he eliminates each member of this division.
Et primo ostendit quod extra extremam sphaeram non est aliquod corpus simplex secundum naturam. Corporum enim simplicium quoddam est circulariter motum; quoddam est quod movetur a medio; quoddam quod movetur ad medium, et in medio subsistit omnibus aliis, ut supra habitum est. Nullum autem horum potest esse extra extremam circumferentiam. Ostensum est enim supra in VI Physic. quod corpus quod circulariter fertur, non permutat proprium locum secundum totum, nisi solum ratione. Sic igitur non est possibile quod corpus quod circulariter fertur, transferatur ad aliquem locum extra eum in quo est. Hoc autem sequeretur si esset aliquod corpus circulariter motum extra extremam circumferentiam, sicut in suo loco naturali. Quia per quam rationem esset naturalis illi corpori circulariter moto, per eandem rationem esset naturalis huic corpori quod in hoc mundo circulariter fertur; omne autem corpus naturaliter fertur ad suum locum naturalem; sequeretur ergo quod istud corpus circulariter motum transferretur extra suum locum ad alium locum, quod est impossibile. First he shows that outside the extreme sphere no simple body exists according to nature. For simple bodies are such that one is moved circularly, one from the middle, and one is moved to the middle and in the middle supports all the others, as was had above. But none of these bodies can exist outside the extreme circumference. For it has been shown above in Physics VI that the circularly moved body does not as to its whole being change its place except in conception. Consequently, it is not possible for that body which is moved circularly to be transferred to a place outside of that in which it exists. But this would follow, if there were a circularly moved body existing outside the extreme circumference as in its natural place. Since the reason that it would be natural to that circularly moved body would also make it natural to the body circularly moved in this world, and every body is naturally borne to its natural place, it would follow, therefore, that that latter circularly moved body would be transferred outside its proper place to another place — which is impossible.
Similiter etiam non est possibile esse extra extremam circumferentiam corpus leve, quod movetur a medio, neque etiam corpus grave, quod substat aliis corporibus in medio. Si enim dicatur quod sint extra extremam circumferentiam naturaliter, hoc esse non potest, quia habent alia loca naturalia, scilicet infra extremam circumferentiam totius; ostensum est autem supra quod omnium gravium est unus numero locus, et similiter omnium levium. Unde non est possibile quod ista corpora sint naturaliter extra extremam circumferentiam totius. Similarly it is not possible for a light body which is moved from the center to be outside the extreme circumference or for a heavy body which supports the other bodies in the center. For if it is maintained that they exist naturally outside the extreme circumference, such a thing cannot be, since they have other natural places, namely, within the extreme circumference of the whole. For it was shown above that there is one numerical place for all heavy bodies and one for all light bodies. Hence it is not possible that those bodies be naturally outside the extreme circumference of the whole.
Et est considerandum quod ista ratio, et quantum ad corpus circulariter motum, et quantum ad corpus quod movetur motu recto, habet necessitatem ex eo quod supra probatum est, quod est tantum unum extremum et unum medium. And it should be noted that this argument, both as to the body circularly moved, and as to the body moved with straight motion, possesses necessity on account of what was proved above, namely, that there is but one extreme and one middle.
Secundo ibi: praeter naturam autem etc., ostendit quod nullum corpus simplex est extra caelum praeter naturam. Si enim esset ibi praeter naturam, ille locus alicui corpori esset naturalis: locus enim qui est uni corpori praeter naturam, necesse est quod sit alii corpori secundum naturam: quia si alicui loco deesset proprium corpus, locus ille esset frustra. Sed non potest esse quod ille locus sit naturalis alicui corpori: non enim est naturalis neque corpori circulariter moto, neque corpori levi aut gravi; ostensum est autem supra quod nullum aliud corpus est praeter ista. Sic igitur patet quod nullum corpus simplex est extra caelum, neque secundum naturam neque praeter naturam. 203. Secondly, at [142] he shows that no simple body is outside the heaven outside its nature. For if it were there in that manner that place would be natural to some other body; for a place outside nature for one body must be according to nature for some other — if a proper body were lacking to a place, that place would exist in vain. But it cannot be said that that place is natural to any body: for it is not natural to a circularly moved body, nor to a light or heavy body. But it has been shown above that there are no other bodies besides these. Consequently, it is plain that no simple body exists outside the heaven, either according to nature or outside nature.
Tertio ibi: si autem non simplicium etc., probat quod non est ibi aliquod corpus mixtum. Quia si non est ibi aliquod simplicium corporum, sequitur quod non sit ibi etiam aliquod corpus mixtum: ubicumque enim est corpus mixtum, necesse est ibi esse corpora simplicia, eo quod corpora simplicia sunt in mixto; et mixtum sortitur locum naturalem secundum corpus simplex quod in eo dominatur. 204. Thirdly, at [143] he proves that there is no mixed body there. For if none of the simple bodies exists there, it follows that no mixed body is. Wherever there is a mixed body, simple bodies must be there, due to the fact that simple bodies are present in the mixed; and a mixed body gets its natural place according to the simple body predominant in it.
Deinde cum dicit: sed et neque factum esse etc., ostendit quod etiam extra caelum non contingit esse aliquod corpus. Unde dicit quod non est possibile fieri aliquod corpus extra caelum. Quia aut esset ibi secundum naturam aut praeter naturam, et iterum aut esset simplex aut mixtum; et quidquid horum detur, erit eadem ratio quae est supra: quia non differt secundum rationes praemissas an sit aliquod corpus extra caelum, vel possit ibi fieri; quia rationes praemissae utrumque concludunt, et quia in sempiternis non differt esse et posse, ut dicitur in III Physic. 205. Then at [144] he shows that outside the heaven there cannot be any body. Hence he says that it is not possible for a body to come to be outside the heaven. For it would be there either according to nature or outside nature; again, it would be either simple or mixed. But no matter which of these is given, the same situation as above would prevail. For according to the above-stated reasons, it makes no difference whether the question concerns the existence of a body outside the heaven, or the possibility of its coming to be there, since the foregoing arguments conclude both, and since in sempiternal things to be and to be able to be do not differ, as is said in Physics III.
Deinde cum dicit: manifestum igitur ex dictis etc., concludit conclusionem principaliter intentam. Et dicit manifestum esse ex dictis quod extra caelum neque est aliqua moles cuiuscumque corporis, neque contingit ibi tale aliquid fieri: quia totus mundus est ex tota materia sua propria, materia autem mundi est corpus naturale sensibile. 206. Then at [145] he draws the conclusion mainly intended. And he says it is manifest from what has been said that outside the heaven no mass of any sort of body exists, nor can exist, since the whole world consists of its entire proper matter and the matter of the world is the sensible natural body.
Nec est intelligendum quod velit probare nullum corpus sensibile esse extra caelum, propter hoc quod est ex tota sua materia; sed potius e converso. Utitur autem illo modo loquendi propter hoc quod ista duo invicem convertuntur. However, it should be not understood that he wishes to prove that no sensible body exists outside the heaven on the ground that it consists of the totality of its matter; but rather the converse. Nevertheless, he uses that manner of speaking because the two are mutually convertible.
Concludit igitur quod neque sunt in praesenti plures caeli, neque fuerunt in praeterito, neque unquam poterunt fieri in futuro: sed istud caelum est unum et solum et perfectum, utpote constans ex omnibus suis partibus, sive ex tota sua materia. He concludes, therefore, that there are not many worlds at present, nor were there many in the past, nor will there ever be able to be in the future. Rather the heaven is one and unique and perfect in the sense of consisting of all its parts or of its total matter.

Lecture 21:
Outside the heaven there is no place, time etc., consequent upon sensible bodies.
Chapter 9 cont.
Ἅμα δὲ δῆλον ὅτι οὐδὲ τόπος οὐδὲ κενὸν οὐδὲ χρόνος ἐστὶν ἔξω τοῦ οὐρανοῦ. 146 It is therefore evident that there is also no place or void or time outside the heaven.
Ἐν ἅπαντι γὰρ τόπῳ δυνατὸν ὑπάρξαι σῶμα 147 For in every place body can be present;
κενὸν δ' εἶναί φασιν ἐν ᾧ μὴ ἐνυπάρχει σῶμα, δυνατὸν δ' ἐστὶ γενέσθαι 148 and void is said to be that in which the presence of body, though not actual, is possible;
χρόνος δὲ ἀριθμὸς κινήσεως κίνησις δ' ἄνευ φυσικοῦ σώματος οὐκ ἔστιν. Ἔξω δὲ τοῦ οὐρανοῦ δέδεικται ὅτι οὔτ' ἔστιν οὔτ' ἐνδέχεται γενέσθαι σῶμα. 149 and time is the number of movement. But in the absence of natural body there is no movement, and outside the heaven, as we have shown, body neither exists nor can come to exist.
Φανερὸν ἄρα ὅτι οὔτε τόπος οὔτε κενὸν οὔτε χρόνος ἐστὶν ἔξω. 150 It is clear then that there is neither place, nor void, nor time, outside the heaven.
Διόπερ οὔτ' ἐν τόπῳ τἀκεῖ πέφυκεν, οὔτε χρόνος αὐτὰ ποιεῖ γηράσκειν, οὐδ' ἐστὶν οὐδενὸς οὐδεμία μεταβολὴ τῶν ὑπὲρ τὴν ἐξωτάτω τεταγμένων φοράν, 151 Hence whatever is there, is of such a nature as not to occupy any place, nor does time age it; nor is there any change in any of the things which lie beyond the outermost motion;
ἀλλ' ἀναλλοίωτα καὶ ἀπαθῆ τὴν ἀρίστην ἔχοντα ζωὴν καὶ τὴν αὐταρκεστάτην διατελεῖ τὸν ἅπαντα αἰῶνα. 152 they continue through their entire duration unalterable and unmodified, living the best and most selfsufficient of lives.
(Καὶ γὰρ τοῦτο τοὔνομα θείως ἔφθεγκται παρὰ τῶν ἀρχαίων. Τὸ γὰρ τέλος τὸ περιέχον τὸν τῆς ἑκάστου ζωῆς χρόνον, οὗ μηθὲν ἔξω κατὰ φύσιν, αἰὼν ἑκάστου κέκληται. Κατὰ τὸν αὐτὸν δὲ λόγον καὶ τὸ τοῦ παντὸς οὐρανοῦ τέλος καὶ τὸ τὸν πάντα χρόνον καὶ τὴν ἀπειρίαν περιέχον τέλος αἰών ἐστιν, ἀπὸ τοῦ αἰεὶ εἶναι τὴν ἐπωνυμίαν εἰληφώς, ἀθάνατος καὶ θεῖος). 153 As a matter of fact, this word 'duration' possessed a divine significance for the ancients, for the fulfilment which includes the period of life of any creature, outside of which no natural development can fall, has been called its duration. On the same principle the fulfilment of the whole heaven, the fulfilment which includes all time and infinity, is 'duration'—a name based upon the fact that it is always—duration immortal and divine.
Ὅθεν καὶ τοῖς ἄλλοις ἐξήρτηται, τοῖς μὲν ἀκριβέστερον τοῖς δ' ἀμαυρῶς, τὸ εἶναί τε καὶ ζῆν. 154 From it derive the being and life which other things, some more or less articulately but others feebly, enjoy.
Καὶ γάρ, καθάπερ ἐν τοῖς ἐγκυκλίοις φιλοσοφήμασι περὶ τὰ θεῖα, πολλάκις προφαίνεται τοῖς λόγοις ὅτι τὸ θεῖον ἀμετάβλητον ἀναγκαῖον εἶναι πᾶν τὸ πρῶτον καὶ ἀκρότατον ὃ οὕτως ἔχον μαρτυρεῖ τοῖς εἰρημένοις. 155 So, too, in its discussions concerning the divine, popular philosophy often propounds the view that whatever is divine, whatever is primary and supreme, is necessarily unchangeable. This fact confirms what we have said.
Οὔτε γὰρ ἄλλο κρεῖττόν ἐστιν ὅ τι κινήσει (ἐκεῖνο γὰρ ἂν εἴη θειότερον) 156 For there is nothing else stronger than it to move it—since that would mean more divine—
οὔτ' ἔχει φαῦλον οὐδέν, οὔτ' ἐνδεὲς τῶν αὑτοῦ καλῶν οὐδενός ἐστιν. 157 and it has no defect and lacks none of its proper excellences.
(279b.) Καὶ ἄπαυστον δὴ κίνησιν κινεῖται εὐλόγως πάντα γὰρ παύεται κινούμενα ὅταν ἔλθῃ εἰς τὸν οἰκεῖον τόπον, τοῦ δὲ κύκλῳ σώματος ὁ αὐτὸς τόπος ὅθεν ἤρξατο καὶ εἰς ὃν τελευτᾷ. 158 Its unceasing movement, then, is also reasonable, since everything ceases to move when it comes to its proper place, but the body whose path is the circle has one and the same place for starting-point and goal.
Postquam philosophus ostendit quod extra caelum non est aliquod corpus sensibile, nec potest esse, hic ostendit quod extra caelum non est aliquod eorum quae consequuntur ad corpora sensibilia. After showing that there neither is, nor can be, any sensible body outside the heaven, the Philosopher here shows that outside the heaven there is none of the things that follow upon sensible bodies.

Et primo ostendit propositum;

secundo ostendit qualia sint quae extra caelum nata sunt esse, ibi: propter quod quidem neque in loco et cetera.

First he proves the proposition;

Secondly, he describes the things that do exist outside the heaven, at 213.

Circa primum tria facit: About the first he does three things:

primo proponit quod intendit;

secundo probat propositum, ibi: in omni enim loco etc.;

tertio infert conclusionem intentam, ibi: manifestum igitur et cetera.

First he proposes what he intends;

Secondly, he proves the proposition, at 208;

Thirdly, he draws the intended conclusion, at 212.

Dicit ergo primo quod simul cum hoc quod probatum est, extra caelum non esse corpus sensibile, manifestum est quod extra caelum neque est locus, neque vacuum, neque tempus: de his enim tribus determinatur in IV Physic. sicut de quibusdam consequentibus corpora naturalia. He says therefore first [146] that with the proof that outside the heaven there is no sensible body, it is also manifest that outside the heaven there is neither place nor void nor time — for these three things were discussed as being concomitants of natural bodies in Physics IV.
Deinde cum dicit: in omni enim loco etc., probat propositum. 208. Then at [147] he proves the proposition.
Primo quidem quantum ad locum. In omni enim loco possibile est existere corpus, alioquin locus esset frustra; sed extra caelum non est possibile existere aliquod corpus, ut probatum est; ergo extra caelum non est locus.

First, as to place: In every place it is possible for a body to exist, otherwise it would be in vain. But outside the heaven it is not possible for any body to exist, as was proved. Therefore, outside the heaven there is no place.

Secundo ibi: vacuum autem etc., probat quod extra caelum non est vacuum. Illi enim qui ponunt vacuum, definiunt vacuum esse locum in quo non existit corpus, sed possibile est esse; sed extra caelum non est possibile corpus esse, ut ostensum est; ergo extra caelum non est vacuum. Secondly, at [148] he proves that outside the heaven there is not a void: Those who posit a void define it to be a place in which a body is not existing but can exist. But outside the heaven it is not possible for a body to exist, as has been shown. Therefore, outside the heaven there is not a void.
Est autem sciendum quod Stoici posuerunt vacuum infinitum, in cuius quadam parte est mundus: et ita relinquitur secundum eos quod extra extremam circumferentiam sit vacuum. Quod quidem tali imaginatione probare volebant. Si enim esset aliquis in extrema circumferentia caeli, aut posset extendere manum suam extra aut non. Si non posset, ergo impediretur ab aliquo extrinseco existente; et redibit eadem quaestio de illo extrinseco, si in extremo eius aliquis existens posset ultra manum porrigere; et ita vel procedetur in infinitum, vel devenietur ad aliquod extremum corpus, ultra quod homo ibi existens posset manum porrigere. Quo dato, sequitur quod extra illud possit esse corpus et non sit; et ita extra erit vacuum. 209. But it should be noted that the Stoics posited an infinite void, in one part of which the world exists. Consequently, according to them, there is a void outside the heaven. They wanted to prove this with the following fantasy: If someone were on the extreme circumference of the heaven, he could either extend his hand beyond or not. If not, then it is being impeded by something existing beyond. The same question will return regarding that thing existing beyond, if anyone could, while on the extremity, reach out his hand beyond. Consequently we must go on infinitely, or come to an extreme body beyond which a man existing there could reach out his hand. In that case it follows that beyond that a body could exist and does not. Hence there will be a void beyond.
Ad hoc autem respondet Alexander, dicens positionem esse impossibilem: cum enim corpus caeli sit impassibile, non est receptivum alicuius extranei. Unde si ex hac impossibili positione sequitur aliquod inconveniens, non est curandum. To this Alexander responds that the position is impossible. For since the body of the heaven cannot undergo anything, it cannot receive anything extraneous. Hence, if from this impossible assumption, something against the thesis follows, one should pay it no heed.
Sed haec responsio non videtur esse sufficiens: quia impossibilitas huius positionis non est ex parte eius quod est extra caelum, sed ex parte ipsius caeli; nunc autem agitur de eo quod est extra caelum. Unde eadem ratio est si totum universum esset terra, in cuius extremo posset esse homo. Et ideo oportet aliter dicere, sicut ipse etiam dicit, quod manum suam extra extendere non posset homo in extrema circumferentia constitutus, non propter aliquod extrinsecum impediens, sed quia de natura omnium corporum naturalium est, quod contineantur infra extremam circumferentiam caeli; alioquin caelum non esset universum. Unde si esset aliquod corpus quod non dependeret a corpore caeli sicut a continente, illud nihil prohiberet esse extra caelum, sicut substantiae spirituales, ut infra dicetur. But this answer does not seem to be sufficient — since the impossibility of this position is not on the part of something outside the heaven but on the part of the heaven itself. But now we are dealing with what is outside the heaven. Hence it is the same argument if the whole universe were the earth, on whose boundary a man could exist. Consequently, we must state otherwise, just as he says, that a man situated on the extreme circumference could not extend his hand beyond, not because of something outside impeding it, but because it is of the very nature of all natural bodies that they be contained within the extreme circumference of the heaven — otherwise the heaven would not be the universe. Hence if there were a body not depending on the body of the heaven as on a container, there would be nothing to prevent it from existing outside the heaven, as in the case of the spiritual substances, as will be said below.
Quod autem non sit vacuum extra caelum, probat Alexander quia aut illud vacuum erit finitum, aut infinitum: si finitum, oportet quod alicubi terminetur, et redibit eadem quaestio, utrum extra illud possit aliquis manum extendere; si autem sit infinitum, erit potens recipere corpus infinitum; aut ergo illa potentia vacui erit frustra, aut oportebit ponere corpus infinitum, quod possit recipi in vacuo infinito. 210. But that there is no void outside the heaven Alexander proves on the ground that such a void is either finite or infinite: If finite, then it is terminated somewhere and the same question will return: Could a person extend his hand beyond that? If it is infinite, it will be capable of receiving an infinite body: then either that power of the void will be in vain or it will be necessary to posit an infinite body capable of being received into the void of the infinite.
Item, si sit vacuum extra mundum, similiter se habet mundus ad quamlibet partem vacui, quia in vacuo nulla est differentia: et ita haec pars vacui in qua est mundus, non est proprius locus eius. Nulla est ergo causa quare in hac parte vacui maneat. Si autem mundus feratur, non feretur magis ad unam partem quam ad aliam, quia in vacuo non est differentia: feretur ergo ad omnem partem; et ita mundus discerpetur. Likewise, if there is a void outside the world, the world will be related to each part of the void in exactly the same way, because in a void there are no differences. Consequently, this part of the void in which the world exists is not its proper place. Therefore there is no cause why it should remain in this part of the void. But if the world is in motion, it will not be moved to one part rather than to another, because in the void there are no differences. Therefore, it will be moved in every direction; and thus the world will be torn asunder.
Tertio, ibi: est autem tempus etc. probat quod extra caelum non sit tempus. Tempus enim est numerus motus, ut patet in IV Physic.; motus autem non potest esse sine corpore naturali, corpus autem naturale nec est nec potest esse extra caelum, ut probatum est; ergo extra caelum non potest esse nec tempus nec motus. 211. Thirdly, at [149] he proves that outside the heaven there is no time. For time is the number of motion, as is plain in Physics IV. But motion cannot exist without a natural body, and a natural body neither exists nor can exist outside the heaven, as has been proved. Therefore, outside the heaven there neither is, nor can be, time.
Deinde cum dicit: manifestum igitur etc., infert conclusionem intentam; concludens manifestum esse ex praedictis quod extra totum mundum nec est locus, neque vacuum, neque tempus. 212. Then at [150] he draws the conclusion intended, and concludes that it is manifest from the foregoing that outside the whole heaven there is neither place nor void nor time.
Deinde cum dicit: propter quod quidem neque in loco etc., ostendit qualia sunt ea quae sunt extra mundum. Et circa hoc duo facit: 213. Then at [151] he describes what type of things are outside the heaven. About this he does two things:

primo concludit ex praemissis eorum qualitatem;

secundo ostendit idem ex his quae communiter dicuntur, ibi: etenim quemadmodum in encycliis et cetera.

First he concludes their condition from the foregoing;

Secondly, he shows the same from common opinion, at 217.

Circa primum duo facit: About the first he does two things:

primo removet ab eis conditionem eorum quae sunt hic;

secundo ostendit propriam conditionem eorum, ibi: sed inalterabilia et cetera.

First he removes from them the condition of things that exist here;

Secondly, he describes their proper condition, at 214.

Dicit ergo primo quod, quia extra caelum non est locus, sequitur quod ea quae ibi sunt nata esse, non sunt in loco. Et hoc quidem Alexander dicit posse intelligi de ipso caelo, quod quidem non est in loco secundum totum, sed secundum partes, ut probatur in IV Physic. He says therefore first [151] that because there is no place outside the heaven, it follows that things by nature apt to be there do not exist in place. And Alexander says that this can be understood about the heaven itself, which is not in place as a whole but with respect to its parts, as is proved in Physics IV.
Et iterum, quia tempus non est extra caelum, sequitur quod non sint in tempore; et ita tempus non facit ea senescere. Quod etiam dicit Alexander posse caelo convenire, quod quidem non est in tempore, secundum quod esse in tempore est quadam parte temporis mensurari, ut dicitur in IV Physic. Et non solum talia non senescunt in tempore, sed neque est aliqua transmutatio eorum quae sunt super illam lationem quae est maxime extra ordinata, idest super motum localem corporum levium: motum enim rectum consuevit vocare lationem. Again, because time does not exist beyond the heaven, it follows that they do not exist in time; consequently, time does not make them grow old. And this, too, according to Alexander, can belong to the heaven, which, indeed, is not in time in the sense that to be in time consists in being measured by some part of time, as is said in Physics IV. Not only do such beings not grow old in time, but no change affects those things which lie "beyond the outermost motion [lationem]," i.e., beyond the local motion of light bodies — for he is accustomed to call rectilinear motion latio.
Sed hoc non videtur esse verum, quod corporum caelestium non sit aliqua transmutatio, cum moveantur localiter: nisi forte exponamus de transmutatione quae est in substantia. Sed haec videtur extorta expositio, cum philosophus universaliter omnem mutationem excludat. Similiter etiam non potest dici proprie quod caelum sit ibi, idest extra caelum. Et ideo convenientius est quod hoc intelligatur de Deo et de substantiis separatis, quae manifeste neque tempore neque loco continentur, cum sint separatae ab omni magnitudine et motu. Huiusmodi autem substantiae dicuntur esse ibi, idest extra caelum, non sicut in loco, sed sicut non contenta nec inclusa sub continentia corporalium rerum, sed totam corporalem naturam excedentia. Et his convenit quod dicitur, quod eorum nulla sit transmutatio: quia superexcedunt supremam lationem, scilicet ultimae sphaerae, quae ordinatur sicut extrinseca et contentiva omnis mutationis. But it does not seem to be true that no change affects heavenly bodies, since they are moved locally, unless perhaps we limit "change" to one affecting the substance. But this seems to be a forced explanation, since the Philosopher excludes all change universally. Likewise, it cannot be properly said that the heaven is there, i.e., outside the heaven. Consequently, it is better to understand his words as applying to God and separated substances which plainly are not contained by time, nor place, since they are separated from all magnitude and motion. Such substances are said to be "there," i.e., outside the heaven, not as in a place, but as not contained nor included under the containment of bodily things, and as exceeding all of corporeal nature. It is such beings that the expression befits, namely, that they undergo no change; because they lie beyond the extreme motion, namely, that of the farthest sphere, which is ordered as extrinsic to and containing all change.
Deinde cum dicit: sed inalterabilia etc., ostendit qualia sunt huiusmodi entia. 214. Then at [152] he explains the qualities of these beings.

Et primo ostendit eorum conditionem;

secundo exponit quoddam nomen quo usus fuerat, ibi: etenim hoc nomen etc.;

tertio ostendit influentiam eorum in alia, ibi: unde et aliis et cetera.

First he describes their condition;

Secondly, he explains a word he used, at 215;

Thirdly, he shows the influence of these beings on others, at 216.

Dicit ergo primo quod illa entia quae sunt extra caelum, sunt inalterabilia et penitus impassibilia, habentia optimam vitam, inquantum scilicet eorum vita non est materiae permixta, sicut vita corporalium rerum. Habent etiam vitam per se sufficientissimam, inquantum non indigent aliquo vel ad conservationem suae vitae, vel ad executionem operum vitae. Habent etiam vitam non temporalem, sed in toto aeterno. He says therefore first [152] that those beings which are outside the heaven are unalterable and wholly impassible. They lead the best of lives, inasmuch as their life is not mingled with matter as is the life of corporeal beings. They also have a life that is most self-sufficient, inasmuch as they do not need anything in order to conserve their life or to perform the works of life, They have a life, too, which is not temporal but in total eternity.
Horum autem quae hic dicuntur, quaedam possunt attribui corporibus caelestibus, puta quod sint impassibilia et inalterabilia: sed alia duo non possunt eis convenire, etiam si sint animata. Non enim habent optimam vitam, cum eorum vita sit ex unione animae ad corpus caeleste: nec etiam habent vitam per se sufficientissimam, cum per motum suum bonum consequantur, ut dicetur in secundo. Now, among the qualities here listed some can be attributed to heavenly bodies — for example, that they are impassible and unalterable. But the other two cannot belong to them, even if they are alive. For they do not have the best life, since their life would be one resulting from the union of a soul to a celestial body; neither do they have a most self-sufficient life, since they attain their good through motion, as will be said in Book II.
Deinde cum dicit: etenim hoc nomen etc., exponit nomen aeterni, quo usus fuerat. Et dicit quod antiqui pronunciaverunt hoc nomen divine, idest convenienter rebus divinis. Hoc enim nomen dupliciter accipitur. 215. Then at [153] he explains the word "eternal" which he had used. And he says that the ancients pronounced this word as divine, i.e., as befitting divine things. Now this word has two meanings.
Uno quidem modo secundum quid, quod scilicet est aeternum vel saeculum alicuius rei: idem enim apud Graecos utrumque significat. Dicit ergo quod aeternum vel saeculum uniuscuiusque rei vocatur finis, idest mensura quaedam terminans, quae continet tempus vitae cuiuslibet rei, ita quod nihil de tempore vitae quae est alicuius rei secundum naturam, est extra illum finem vel mensuram; sicut si dicamus quod spatium centum annorum est saeculum vel aeternum hominis. In one way it is used in a qualified sense as meaning the eternity or age [saeculum] of a thing: for in Greek the same word signifies both. He says, therefore, that the eternity or age of a thing is called an end, i.e., a certain terminal measure which contains the time of any thing's life, in such a way that no time of the life belonging to the thing according to nature exists outside that end or measure. It is like saying that the span of 100 years is the "age" or "eternity" of a man.
Alio modo dicitur aeternum simpliciter, quod comprehendit et continet omnem durationem. Et hoc est quod dicit, quod secundum eandem rationem aeternum dicitur finis totius caeli, idest spatium continens totam durationem caeli, quod est spatium totius temporis. Et secundum hoc dicitur aeternum perfectio quaedam, quae continet omne tempus et omnem infinitatem durationis: non quidem sic quod ipsum aeternum distendatur secundum successionem praeteriti et futuri, sicut spatium temporis quantumcumque sit, quia talis successio sequitur motum, illa autem sunt penitus immobilia quae dixit habere vitam in aeterno; sed aeternum totum simul existens, comprehendit omne tempus et omnem infinitatem. Et denominatur in Graeco ab hoc quod est semper esse. Et talis finis, qui aeternum dicitur, est immortalis, quia vita illa non terminatur morte; et divinus, quia excedit omnem materiam, quantitatem et motum. In another way "eternity" is used in an absolute sense as comprehending and containing all duration. And this is what he says, namely, that according to the same notion, eternity is called the end of the entire heaven, i.e., it is the span containing the entire duration of the heaven, i.e., the span of all of time. In this sense, eternity refers to a certain perfection which contains all time and the entire infinitude of duration — not as though this eternity is stretched out according to the succession of past and future, as in the case of any span of time, because such succession follows upon motion, whereas the things he described as having life in eternity are completely immobile, but this eternity is a whole existing all at once and comprehending all time and all infinitude. (The Greek word [in English "aeon"] is derived from the words for "always existing".) Such an end, which is called "eternal" is immortal, because that life is not ended by death, and "divine," because it is beyond all matter, quantity, and motion.
Deinde cum dicit: unde et aliis etc., ostendit influentiam eorum in alia. Est autem manifestum quod ab eo quod est perfectissimum, fit derivatio ad alia quae sunt minus perfecta; sicut calidum derivatur ab igne ad alia quae sunt minus calida, ut dicitur in II Metaphys. Unde cum ista entia habeant vitam optimam et per se sufficientissimam, et esse sempiternum, consequens est quod inde communicetur aliis esse et vivere. Non tamen aequaliter omnibus: sed his quidem clarius, idest evidentius et perfectius, scilicet his quae habent esse sempiternum eadem numero existentia, et his quae habent vitam rationalem; his autem obscurius, idest debilius et imperfectius, sicut his quae sunt sempiterna non secundum idem numero sed secundum idem specie, et quae habent vitam sensibilem vel nutritivam. 216. Then at [154] he shows the influence of these things on others. Now it is manifest that from what is most perfect there is a flowing to others that are less perfect, just as heat flows from fire to other things that are less hot, as is said in Metaphysics II. Hence, since those beings possess the best and most self-sufficient life and eternal existence, it is from them that existence and life are communicated to other things. But not equally to all; rather, to some "more luminously," i.e., more evidently and more perfectly, namely, to those that have individual eternal existence and to those that have rational life; to others "more darkly," i.e., in a lesser and more imperfect way, namely, to those things that are eternal, not in the same individuals, but according to sameness of species, and which have sense or nutritive life.
Deinde cum dicit: etenim quemadmodum in encycliis etc., manifestat quod dixerat de conditione praedictorum entium quae sunt extra caelum. 217. Then at [155] he manifests what he had said about the condition of the aforesaid beings that exist outside the heaven.

Et primo proponit quod intendit;

secundo inducit rationes, ibi: neque enim aliud et cetera.

First he proposes what he intends;

Secondly, he presents reasons, at 218.

Circa primum considerandum est quod apud philosophos erant duo genera dogmatum. Quaedam enim erant quae a principio secundum ordinem doctrinae multitudini apponebantur, quae quidem vocabantur encyclia: quaedam autem erant magis subtilia, quae proponebantur auditoribus iam provectis, quae vocabantur syntagmatica, idest coordinalia, vel acroamatica, idest auditionalia. Dogmata autem philosophorum dicuntur philosophemata. With respect to the first [155] it should be known that among the philosophers there were two kinds of teachings. For there were some which from the very beginning were proposed according to the order of doctrine to the multitude and these were called "encyclia"; others were more subtle, and were proposed to the more advanced hearers and were called "syntagmatica," i.e., co-ordinal, or "acromatica," i.e., hearable, teachings. The dogmas of the philosophers are called "philosophemata."
Dicit ergo quod in huiusmodi encycliis philosophematibus circa res divinas, multoties philosophi rationibus manifestabant quod necesse est omne divinum esse intransmutabile, quasi non subiectum motui, et primum, quasi non subiectum tempori, et summum, quasi non contentum loco: divinum autem dicebant omnem substantiam separatam. Et hoc attestatur his quae dicta sunt de huiusmodi entibus. He says, therefore, that in the "encyclic" [or popular] philosophic discussions concerning divine things the philosophers very often in their arguments showed that everything divine must be "untransmutable," as not subject to motion, and "first," as not subject to time, and "highest," as not contained by place. And they called every separated substance "divine." And this confirms what has been said about such beings.
Deinde cum dicit: neque enim aliud etc., ponit rationes ad ostendendum quod dixerat, scilicet quod primum et supremum sit intransmutabile. 218. Then at [156] he gives reasons to manifest what he had said, namely, that the first and highest is untransmutable.

Et primo ostendit propositum;

secundo infert quandam conclusionem ex dictis, ibi: et incessabili itaque et cetera.

First he manifests the proposition;

Secondly, he draws a conclusion, at 220.

Circa primum ponit duas rationes: quarum prima talis est. Semper movens et agens est melius moto et passo; sed non est aliquid melius primo et summo divino, quod possit ipsum movere, quia illud esset adhuc divinius; primum ergo divinum non movetur, quia omne quod movetur necesse est ab alio moveri, ut probatur in VII et VIII Physic. In regard to the first he gives two arguments, the first of which is as follows: What is always causing motion and acting is better than what is moved and acted upon. But there is nothing better than the first and highest divinity, so as to be able to move it, because such a mover would be more divine. Therefore, the first divine being is not moved, since whatever is moved must be moved by another, as is proved in Physics VII and VIII.
Secundam rationem ponit ibi: neque habet pravum etc.: quae talis est. Omne quod movetur, aut movetur ad hoc quod evadat aliquod malum, aut ad hoc quod acquirat aliquod bonum; sed primum non habet aliquod malum quod possit evadere, neque indiget aliquo bono quod possit acquirere, quia est perfectissimum; ergo primum non movetur. 219. The second argument is at [157]: Whatever is moved is moved either to avoid an evil or acquire a good. But what is first has no evil to avoid and lacks no good that it could acquire, because it is most perfect. Therefore, the first is not moved.
Potest autem et sic formari ratio. Omne quod movetur, aut movetur ad melius aut ad deterius; sed neutrum potest Deo convenire, secundum ea quae hic dicuntur; ergo Deus nullo modo movetur. Et est attendendum quod haec secunda ratio potest induci ad hoc quod non moveatur a seipso. The argument could also be presented in the following way: Whatever is moved is moved either to better or to worse. But neither of these can belong to God according to what is said here. Therefore, God, is in no way moved. And one should note that this second argument may be introduced to show that He is not moved by Himself.
Deinde cum dicit: et incessabili itaque etc., infert conclusionem ex dictis. Et dicit rationabiliter, idest probabiliter, sequi quod illud primum movens primum mobile, moveat motu incessabili. Quaecumque enim mota quiescunt, tunc quiescunt quando perveniunt ad proprium locum, sicut patet in gravibus et levibus; sed hoc non potest dici in primo mobili, quod circulariter movetur, quia idem est unde incipit motus eius et in quod terminatur; ergo primum mobile movetur a primo motore motu incessabili. 220. Then at [158] from the foregoing he draws a conclusion. And he says it "reasonably," i.e., probably, follows that that first mover of the first mobile acts with unceasable motion. For whatever things, after having been moved, rest, these do so when they reach their appropriate place, as is clear in heavy and light bodies. But this cannot be said of the first mobile which is moved circularly, because where its motion starts is the same as where it ends. Therefore, the first mobile is moved by the first mover with an unceasing motion.
Et est attendendum quod haec ratio non ex necessitate concludit. Potest enim dici quod motus caeli non cessat, non propter naturam loci, sed propter voluntatem moventis. Et ideo non inducit eam tanquam necessariam, sed tanquam probabilem. And it should be noted that this argument does not conclude of necessity. For it can be said that the motion of the heaven does not cease, not on account of the nature of the place, but on account of the will of the mover. Therefore, he does not present this as a necessary, but as a probable, conclusion.

Lecture 22:
Whether the universe is infinite by eternal duration.
Chapter 10
Τούτων δὲ διωρισμένων λέγωμεν μετὰ ταῦτα πότερον ἀγένητος ἢ γενητὸς καὶ ἄφθαρτος ἢ φθαρτός, διεξελθόντες πρότερον τὰς τῶν ἄλλων ὑπολήψεις 159 Having established these distinctions, we may now proceed to the question whether the heaven is ungenerated or generated, indestructible or destructible. Let us start with a review of the theories of other thinkers;
αἱ γὰρ τῶν ἐναντίων ἀποδείξεις ἀπορίαι περὶ τῶν ἐναντίων εἰσίν. 160 for the proofs of a theory are difficulties for the contrary theory.
Ἅμα δὲ καὶ μᾶλλον ἂν εἴη πιστὰ τὰ μέλλοντα λεχθήσεσθαι προακηκοόσι τὰ τῶν ἀμφισβητούντων λόγων δικαιώματα. 161 Besides, those who have first heard the pleas of our adversaries will be more likely to credit the assertions which we are going to make.
Τὸ γὰρ ἐρήμην καταδικάζεσθαι δοκεῖν ἧττον ἂν ἡμῖν ὑπάρχοι καὶ γὰρ δεῖ διαιτητὰς ἀλλ' οὐκ ἀντιδίκους εἶναι τοὺς μέλλοντας τἀληθὲς κρίνειν ἱκανῶς. 162 We shall be less open to the charge of procuring judgement by default. To give a satisfactory decision as to the truth it is necessary to be rather an arbitrator than a party to the dispute.
Γενόμενον μὲν οὖν ἅπαντες εἶναί φασιν, 163 That the world was generated all are agreed,
ἀλλὰ γενόμενον οἱ μὲν ἀΐδιον, οἱ δὲ φθαρτὸν ὥσπερ ὁτιοῦν ἄλλο τῶν συνισταμένων, οἱ δ' ἐναλλὰξ ὁτὲ μὲν οὕτως ὁτὲ δὲ ἄλλως ἔχειν [φθειρόμενον], καὶ τοῦτο αἰεὶ διατελεῖν οὕτως, ὥσπερ Ἐμπεδοκλῆς ὁ Ἀκραγαντῖνος καὶ Ἡράκλειτος ὁ Ἐφέσιος. 164 but, generation over, some say that it is eternal, others say that it is destructible like any other natural formation. Others again, with Empedliocles of Acragas and Heraclitus of Ephesus, believe that there is alternation in the destructive process, which takes now this direction, now that, and continues without end.
Τὸ μὲν οὖν γενέσθαι μὲν ἀΐδιον δ' ὅμως εἶναι φάναι τῶν ἀδυνάτων. Μόνα γὰρ ταῦτα θετέον ὐλόγως ὅσα ἐπὶ πολλῶν ἢ πάντων ὁρῶμεν ὑπάρχοντα, περὶ δὲ τούτου συμβαίνει τοὐναντίον ἅπαντα γὰρ τὰ γινόμενα καὶ φθειρόμενα φαίνεται. 165 Now to assert that it was generated and yet is eternal is to assert the impossible; for we cannot reasonably attribute to anything any characteristics but those which observation detects in many or all instances. But in this case the facts point the other way: generated things are seen always to be destroyed.
Ἔτι δὲ τὸ μὴ ἔχον ἀρχὴν τοῦ ὡδὶ ἔχειν, ἀλλ' ἀδύνατον ἄλλως ἔχειν πρότερον τὸν ἅπαντα αἰῶνα, ἀδύνατον καὶ μεταβάλλειν ἔσται γάρ τι αἴτιον, ὃ εἰ ὑπῆρχε πρότερον, δυνατὸν ἂν ἦν ἄλλως ἔχειν τὸ ἀδύνατον ἄλλως ἔχειν. Εἰ δὲ πρότερον ἐξ ἄλλως ἐχόντων συνέστη ὁ κόσμος, εἰ μὲν ἀεὶ οὕτως ἐχόντων καὶ ἀδυνάτων ἄλλως ἔχειν, οὐκ ἂν ἐγένετο εἰ δὲ γέγονεν, ἀνάγκη δηλονότι κἀκεῖνα δυνατὰ εἶναι ἄλλως ἔχειν καὶ μὴ ἀεὶ οὕτως ἔχειν, ὥστε καὶ συνεστῶτα διαλυθήσεται καὶ διαλελυμένα συνέστη ἔμπροσθεν, καὶ τοῦτ' ἀπειράκις ἢ οὕτως εἶχεν ἢ δυνατὸν ἦν. Εἰ δὲ τοῦτ', οὐκ ἂν εἴη ἄφθαρτος, οὔτ' εἰ ἄλλως εἶχέ ποτε οὔτ' εἰ δυνατὸν ἄλλως ἔχειν. 166 Further, a thing whose present state had no beginning and which could not have been other than it was at any previous moment throughout its entire duration, cannot possibly be changed. For there will have to be some cause of change, and if this had been present earlier it would have made possible another condition of that to which any other condition was impossible. Suppose that the world was formed out of elements which were formerly otherwise conditioned than as they are now. Then (1) if their condition was always so and could not have been otherwise, the world could never have come into being. And (2) if the world did come into being, then, clearly, their condition must have been capable of change and not eternal: after combination therefore they will be dispersed, just as in the past after dispersion they came into combination, and this process either has been, or could have been, indefinitely repeated. But if this is so, the world cannot be indestructible, and it does not matter whether the change of condition has actually occurred or remains a possibility.
Postquam philosophus ostendit quod corpus totius mundi non est infinitum, et quod non est multiplex numero, hic inquirit utrum sit infinitum durationis aeternitate. 221. After the Philosopher showed that the body of the whole universe is not infinite, and that it is not multiple in number, here he inquires whether it is infinite by eternal duration.

Et primo ponit opiniones aliorum;

secundo determinat propositum secundum propriam opinionem, ibi: primum autem dividendum et cetera.

And first he gives the opinions of others.

Secondly, he settles the question according to his own opinion

Circa primum tria facit: Regarding the first, he does three things:

primo dicit de quo est intentio;

secundo ponit opiniones, ibi: genitum quidem igitur etc.; tertio improbat eas, ibi: factum esse quidem et cetera.

First he declares his intention.

Secondly he gives the opinions, at [163].

Thirdly he refutes them.

Circa primum duo facit: primo dicit de quo est intentio, et quo ordine sit agendum. Et dicit quod post determinationem praemissorum, dicendum est postea utrum mundus sit ingenitus aut genitus, idest utrum per generationem incoeperit esse a quodam principio temporis, aut non; et utrum sit incorruptibilis aut corruptibilis, idest utrum per corruptionem post aliquod tempus esse desinat, vel non. Prius tamen quam haec pertractemus secundum nostram opinionem, debemus pertranseuntes, idest breviter, dicere suspiciones aliorum, idest opiniones aliorum philosophorum circa hoc; quas suspiciones vocat, quia ex levibus rationibus ad haec dicenda movebantur. Difficile enim est ad hoc inducere efficaces rationes: unde et ipse Aristoteles dicit in I Topic. quod quaedam problemata sunt de quibus rationes non habemus, ut utrum mundus sit aeternus vel non. 222. Regarding the first, he does two things. First he states his intention and the order of procedure. And he says that after determining the previous matters, he must go on to say whether the universe is ungenerated or generated, that is whether it begin to exist at some beginning point of time or not, and whether it is incorruptible or corruptible, that is whether after some time it will cease to exist through corruption, or not. But before treating these matters according to our opinion, we must first briefly review the surmisals of others, that is the opinions of other philosophers on this matter. He calls them surmisals [suspiciones], because they were moved to these opinions by frivolous reasons. For it is difficult to adduce efficacious reasons; thus Aristotle said in Topics I that there are some problems about which we do not have reasons, such as whether the world is eternal or not.
Secundo ibi: contrariorum enim etc., assignat rationes tres quare hic et alibi aliorum opiniones pertractet. Quarum prima est quia demonstrationes, idest probationes, contrariorum, idest contrariarum opinionum, sunt dubitationes de contrariis, scilicet opinionibus, idest sunt obiectiones ad contrarias opiniones: expedit autem ei qui vult cognoscere aliquam veritatem, ut sciat dubitationes quae sunt contra illam veritatem; quia solutio dubitatorum est inventio veritatis, ut dicitur in III Metaphys. Et ita ad sciendum veritatem multum valet videre rationes contrariarum opinionum. 223. Secondly, at "of contrary things", he assigns three reasons why here and elsewhere he reviews the opinions of others. The first reason is that "demonstrations", that is proofs, "of contrary things", that is, of contrary opinions, are critiques of "contraries", that is, contrary opinions. That is, they are objections against contrary opinions. Whoever wishes to know any truth, must know the critiques against that truth, because the solution of doubts is the finding of truth, as is said in III Metaphysica. And thus, to know the truth, it is very important to see the reasons for contrary opinions.
Secundam rationem ponit ibi: simul autem et cetera. Et dicit quod simul cum praedicta ratione est alia ratio: quia ea quae dicenda sunt magis redduntur credibilia apud illos qui primo audiunt iustificationes, idest rectificationes, sermonum dubitatorum, idest solutiones rationum ex quibus dubitatio emergit: quia quandiu homo dubitat, antequam eius dubitatio solvatur, est mens eius similis ligato, qui non potest ire. 224. As for the second reason, he says that there is an additional reason. Because what must be said is made more credible to people who first hear the justification or defense of disputed opinions, that is solutions of the reasons which gave rise to the dispute. For as long as a man is in doubt, before his doubt is resolved, his mind is like someone bound, who cannot move.
Tertiam rationem ponit ibi: gratis enim condemnare et cetera. Et dicit quod quando nos posuerimus opiniones aliorum, et induxerimus eorum rationes, et solverimus eas, et posuerimus rationes in contrarium, minus inerit nobis quod videamur condemnare dicta aliorum gratis, idest sine debita ratione, sicut qui reprobant dicta aliorum ex solo odio, quod non convenit philosophis, qui profitentur se inquisitores esse veritatis. Oportet enim eos qui volunt sufficienter iudicare de veritate, quod non exhibeant seipsos sicut inimicos eorum de quorum dictis est iudicandum; sed sicut arbitros, et disquisitores pro utraque parte. 225. The third reason is where he says "to condemn without reason" etc. And he says that when we cite the opinions of others and examine and solve their reasons, and give reasons for the contrary, we will not appear guilty of condemning the opinions of others gratuitously, that is, without proper reason, like those who condemn the opinions of others out of mere hatred. This is not becoming to philosophers, who profess to be searchers of the truth. For those who wish to be adequate judges of the truth must not show themselves enemies of those whose statements are to be judged, but as arbiters, and inquirers for both sides.
Deinde cum dicit: genitum quidem igitur etc., ponit opiniones aliorum. Et primo ponit in quo omnes conveniunt: et dicit quod omnes qui fuerunt ante eum, dixerunt quod mundus sit genitus, idest a quodam principio temporis esse incipiens per generationem. 226. Then at [163] he gives the opinions of others. First he shows in what they all agree, and says that all who were before him stated that the world is generated, i.e., at a certain beginning of time it began to exist through generation.
Secundo ibi: sed genitum etc., ponit in quo differunt. Et tangit tres opiniones. Quidam enim dicebant quod, quamvis incoeperit esse ab aliquo principio temporis, tamen in sempiternum durabit; sicut primo dixerunt quidam poetae, ut Orpheus et Hesiodus, qui dicti sunt theologi, quia res divinas poetice et fabulariter tradiderunt; quos in hac positione secutus est Plato, qui posuit mundum generatum, sed indissolubilem. 227. Secondly, he shows in what they differ. And he touches on three opinions. First of all, some said that, although it began to be at a certain beginning of time, yet it will endure forever, as first was said by certain poets, such as Orpheus and Hesiod, who are called "theologians" because they presented divine things under the form of poetry and myths. Plato followed them in this position, holding the world to be generated but indestructible.
Secunda opinio fuit quorundam aliorum, qui posuerunt mundum corruptibilem esse eo modo quo quodlibet aliud generatorum, quae constituuntur ex multis; ita scilicet quod mundus post corruptionem nunquam reparabitur, sicut Socrates post corruptionem nunquam reparatur per naturam. Et haec fuit positio Democriti, qui posuit mundum generari casu per concursum atomorum semper mobilium, et ita etiam per eorum segregationem quandoque esse dissolvendum. The second opinion was that of certain others, that the world is destructible in the same way as any other generated thing composed of many parts, and that after being destroyed, it will never be repaired, just as Socrates, once corrupted, is never restored by nature. And this was the opinion of Democritus, who declared the world to be generated by a fortuitous gathering together of atoms ever mobile, and likewise to be destined to be dissolved at some time by the separation of these atoms.
Tertia opinio est dicentium quod mundus quandoque vicissim generatur et quandoque corrumpitur, et ista vicissitudo semper duravit et durabit. Et hoc dixit Empedocles Agrigentinus: posuit enim quod, amicitia congregante elementa et lite dissolvente ea, mundus generabatur et corrumpebatur. Hoc etiam posuit Heraclitus Ephesius, qui posuit quod quandoque totus mundus exureretur per ignem, et post certos decursus temporum iterum totus mundus generaretur per ignem, quem ponebat esse principium omnium rerum. The third opinion is that of those who say the world is alternately generated and destroyed, and that this alternation has always endured and will always last. Such was the opinion of Empedocles of Agrigenta, for he posited that with friendship assembling the elements and strife separating them, the world was [continuously] generated and destroyed. This, too, was the opinion of Heraclitus of Ephesus, who posited that at some time the world would be consumed by fire and after a certain lapse of time would again be generated by fire, which he supposed was the principle of all things.
Dicunt autem quidam quod isti poetae et philosophi, et praecipue Plato, non sic intellexerunt secundum quod sonat secundum superficiem verborum; sed suam sapientiam volebant quibusdam fabulis et aenigmaticis locutionibus occultare; et quod Aristotelis consuetudo fuit in pluribus non obiicere contra intellectum eorum, qui erat sanus, sed contra verba eorum, ne aliquis ex tali modo loquendi errorem incurreret, sicut dicit Simplicius in commento. Alexander tamen voluit quod Plato et alii antiqui philosophi hoc intellexerunt quod verba eorum exterius sonant; et sic Aristoteles non solum contra verba, sed contra intellectum eorum conatus est argumentari. Quidquid autem horum sit, non est nobis multum curandum: quia studium philosophiae non est ad hoc quod sciatur quid homines senserint, sed qualiter se habeat veritas rerum. 228. Now, some claim that these poets and philosophers, and especially Plato, did not understand these matters in the way their words sound on the surface, but wished to conceal their wisdom under certain fables and enigmatic statements. Moreover, they claim that Aristotle's custom in many cases was not to object against their understanding, which was sound, but against their words, lest anyone should fall into error on account of their way of speaking. So says Simplicius in his Commentary. But Alexander held that Plato and the other early philosophers understood the matter just as the words sound literally, and that Aristotle undertook to argue not only against their words but against their understanding as well. Whichever of these may be the case, it is of little concern to us, because the study of philosophy aims not at knowing what men feel, but at what is the truth of things.
Deinde cum dicit: factum esse quidem etc., improbat praedictas positiones: 229. Then at [165] he refutes these opinions:

et primo primam;

secundo tertiam, ibi: vicissim autem etc.;

tertio secundam, ibi: totaliter autem factum etc. (secunda enim opinio minus habet rationis).

First, the first one;

Secondly, the third one, at 234;

Thirdly, the second one, at 235 (for the second has less of an argument).

Circa primum duo facit: About the first he does two things:

primo improbat positionem;

secundo excludit quandam excusationem, ibi: auxilium autem et cetera.

First he refutes the opinion;

Secondly, he rejects an excusing of it, at 231.

Circa primum ponit duas rationes. Circa quarum primam dicit quod impossibile est mundum esse factum vel genitum ex quodam principio temporis, et quod postmodum in sempiternum duret. Cum enim aliqua volumus sumere rationabiliter, idest probabiliter absque demonstratione, talia oportet ponere quae videmus esse vera in omnibus aut in multis: hoc enim est de ratione probabilis. Sed in proposito accidit contrarium, quia omnia quae generantur, videmus corrumpi. Non ergo est ponendum quod mundus sit generatus, et quod sit incorruptibilis. With respect to the first he presents two arguments, in the first of which he says that it is impossible for the world to have been made or generated from a certain beginning of time and then afterwards to endure forever. For when we want to assume something "reasonably," i.e., probably, without a demonstration, we must posit what we observe to be true in all or in many cases, for this is the very nature of the probable. But in this case the contrary happens, because all things that are generated we see to corrupt. Therefore one should not lay down that the world is generated and indestructible.
Secundam rationem ponit ibi: adhuc autem et cetera. Et inducit primo quoddam principium: et dicit quod, si aliquid est quod non habet in se potentiam quae sit principium eius quod est sic et aliter se habere, sed impossibile est quod aliter se habuerit prius per omnia saecula, impossibile est quod talis res transmutetur. Et hoc probat ducendo ad impossibile. Quia si talis res transmutaretur, erit quando transmutatur aliqua causa faciens eam transmutari, scilicet sua potentia ad transmutationem: quae si prius fuisset, possibile erat illam rem aliter se habere, quae tamen ponebatur impossibile aliter se habere. Si autem prius non habuit potentiam ad hoc quod aliter se haberet, et postea habet eam, hoc ipsum est transmutari illam rem: et sic etiam antequam haberet potentiam transmutandi, erat potens transmutari, ad hoc scilicet quod acciperet potentiam transmutandi. 230. He gives the second argument at [166]. And first he states a principle and says that if a thing is such that it does not have within itself a potency which is a principle of its being thus and otherwise, but it is impossible for it to have been otherwise throughout all preceding ages, then such a thing cannot be transmuted. This he proves by leading to an impossibility. For if such a thing should be transmuted, it would be when it is transmuted by some cause producing its transmutation, i.e., by its potency to transmutation. This potency, if it had existed before, would have made it possible for that thing to be other than it was, which thing, however, was assumed to be incapable of being otherwise. But if it previously lacked this potency to be otherwise, and later has it, that itself would be a transmutation of that thing. Consequently, even before it had the potency to be changed, it was able to be changed, namely, by receiving the power to be changed.
Ex his autem sic argumentatur ad propositum. Si enim mundus constitutus est ex quibusdam rebus, quae priusquam mundus fieret aliter se habebant; si ita sit quod illa ex quibus constitutus est mundus, semper sic se haberent sicut prius se habebant, et impossibile sit aliter ea se habere, non fieret mundus ex eis. Si ergo factus est mundus ex eis, necesse est quod illa ex quibus factus est mundus, sint possibilia aliter se habere, et quod non semper eodem modo se habeant. Unde sequitur quod etiam constantia, idest postquam fuerint adunata ad constitutionem mundi, iterum possunt dissolvi; et quando erant dissoluta, prius fuerunt composita; et quod infinities vicissim haec sic se habebant, aut possibile erat sic se habere. Et si hoc est verum, sequitur quod mundus non sit incorruptibilis, neque unquam erit incorruptibilis, si ea ex quibus constat mundus aliter se habebant, neque etiam si possibile erat quod aliter se haberent: quia ex utroque sequitur quod etiam nunc possibile sit ea aliter se habere. From this he argues thus to his proposition; If the world was made from certain things which, before the world was made, were otherwise constituted, then if it is true that those things from which the world was formed were never otherwise than they always were, and could never be otherwise, the world could not have been formed from them. But if the world was formed from them, then, necessarily, those things from which it was formed could be otherwise and do not remain always the same. Hence it follows that even as constituents, i.e., after being united to form the world, they can be separated again; and, when dispersed, they have been previously united, and they alternated thus infinitely, or could have. And if this is true, it follows that the world is not imperishable, nor ever will be imperishable, if the things of which the world consists were at one time otherwise, or even could have been: for in either case it follows that even now it is possible that they be otherwise.

Lecture 23:
A Platonic evasion rejected. Two remaining opinions disproved.
Chapter 10 cont.
Ἣν δέ τινες βοήθειαν ἐπιχειροῦσι φέρειν ἑαυτοῖς τῶν λεγόντων ἄφθαρτον μὲν εἶναι γενόμενον δέ, οὐκ ἔστιν ἀληθής ὁμοίως γάρ φασι τοῖς τὰ διαγράμματα γράφουσι καὶ σφᾶς εἰρηκέναι περὶ τῆς γενέσεως, οὐχ ὡς γενομένου ποτέ, ἀλλὰ (280a.) διδασκαλίας χάριν ὡς μᾶλλον γνωριζόντων, ὥσπερ τὸ διάγραμμα γιγνόμενον θεασαμένους. 167 Some of those who hold that the world, though indestructible, was yet generated, try to support their case by a parallel which is illusory. They say that in their statements about its generation they are doing what geometricians do when they construct their figures, not implying that the universe really had a beginning, but for didactic reasons facilitating understanding by exhibiting the object, like the figure, as in course of formation.
Τοῦτο δ' ἐστίν, ὥσπερ λέγομεν, οὐ τὸ αὐτό ἐν μὲν γὰρ τῇ ποιήσει τῶν διαγραμμάτων πάντων τεθέντων εἶναι ἅμα τὸ αὐτὸ συμβαίνει, ἐν δὲ ταῖς τούτων ἀποδείξεσιν οὐ ταὐτόν, ἀλλ' ἀδύνατον τὰ γὰρ λαμβανόμενα πρότερον καὶ ὕστερον ὑπεναντία ἐστίν ἐξ ἀτάκτων γὰρ τεταγμένα γενέσθαι φασίν, ἅμα δὲ ἄτακτον εἶναι καὶ τεταγμένον ἀδύνατον, ἀλλ' ἀνάγκη γένεσιν εἶναι τὴν χωρίζουσαν καὶ χρόνον ἐν δὲ τοῖς διαγράμμασιν οὐδὲν τῷ χρόνῳ κεχώρισται. Ὅτι μὲν οὖν ἀδύνατον ἅμ' ἀΐδιον αὐτὸν εἶναι καὶ γενέσθαι, φανερόν. 168 The two cases, as we said, are not parallel; for, in the construction of the figure, when the various steps are completed the required figure forthwith results; but in these other demonstrations what results is not that which was required. Indeed it cannot be so; for antecedent and consequent, as assumed, are in contradiction. The ordered, it is said, arose out of the unordered; and the same thing cannot be at the same time both ordered and unordered; there must be a process and a lapse of time separating the two states. In the figure, on the other hand, there is no temporal separation. It is clear then that the universe cannot be at once eternal and generated.
Τὸ δ' ἐναλλὰξ συνιστάναι καὶ διαλύειν οὐδὲν ἀλλοιότερον ποιεῖν ἐστὶν ἢ τὸ κατασκευάζειν αὐτὸν ἀΐδιον μέν, ἀλλὰ μεταβάλλοντα τὴν μορφήν, ὥσπερ εἴ τις ἐκ παιδὸς ἄνδρα γινόμενον καὶ ἐξ ἀνδρὸς παῖδα ὁτὲ μὲν φθείρεσθαι ὁτὲ δ' εἶναι οἴοιτο δῆλον γὰρ ὅτι καὶ εἰς ἄλληλα τῶν στοιχείων συνιόντων οὐχ ἡ τυχοῦσα τάξις γίγνεται καὶ σύστασις, ἀλλ' ἡ αὐτή, ἄλλως τε καὶ κατὰ τοὺς τοῦτον τὸν λόγον εἰρηκότας, οἳ τῆς διαθέσεως ἑκατέρας αἰτιῶνται τὸ ἐναντίον. Ὥστ' εἰ τὸ ὅλον σῶμα συνεχὲς ὂν ὁτὲ μὲν οὕτως ὁτὲ δ' ἐκείνως διατίθεται καὶ διακεκόσμηται, ἡ δὲ τοῦ ὅλου σύστασίς ἐστι κόσμος καὶ οὐρανός, οὐκ ἂν ὁ κόσμος γίγνοιτο καὶ φθείροιτο, ἀλλ' αἱ διαθέσεις αὐτοῦ. 169 To say that the universe alternately combines and dissolves is no more paradoxical than to make it eternal but varying in shape. It is as if one were to think that there was now destruction and now existence when from a child a man is generated, and from a man a child. For it is clear that when the elements come together the result is not a chance system and combination, but the very same as before—especially on the view of those who hold this theory, since they say that the contrary is the cause of each state. So that if the totality of body, which is a continuum, is now in this order or disposition and now in that, and if the combination of the whole is a world or heaven, then it will not be the world that comes into being and is destroyed, but only its dispositions.
Τὸ δ' ὅλως γενόμενον φθαρῆναι καὶ μὴ ἀνακάμπτειν ὄντος μὲν ἑνὸς ἀδύνατόν ἐστιν πρὶν γὰρ γενέσθαι ἀεὶ ὑπῆρχεν ἡ πρὸ αὐτοῦ σύστασις, ἣν μὴ γενομένην οὐχ οἷόν τ' εἶναί φαμεν μεταβάλλειν ἀπείρων δ' ὄντων ἐνδέχεται μᾶλλον. If the world is believed to be one, it is impossible to suppose that it should be, as a whole, first generated and then destroyed, never to reappear; since before it came into being there was always present the combination prior to it, and that, we hold, could never change if it was never generated. If, on the other hand, the worlds are infinite in number the view is more plausible.
Οὐ μὴν ἀλλὰ καὶ τοῦτο πότερον ἀδύνατον ἢ δυνατόν, ἔσται δῆλον ἐκ τῶν ὕστερον εἰσὶ γάρ τινες οἷς ἐνδέχεσθαι δοκεῖ καὶ ἀγένητόν τι ὂν φθαρῆναι καὶ γενόμενον ἄφθαρτον διατελεῖν, ὥσπερ ἐν τῷ Τιμαίῳ ἐκεῖ γάρ φησι τὸν οὐρανὸν γενέσθαι μέν, οὐ μὴν ἀλλ' ἔσεσθαί γε τὸν λοιπὸν ἀεὶ χρόνον. Πρὸς οὓς φυσικῶς μὲν περὶ οὐρανοῦ μόνον εἴρηται, καθόλου δὲ περὶ ἅπαντος σκεψαμένοις ἔσται καὶ περὶ τούτου δῆλον. 171 But whether this is, or is not, impossible will be clear from what follows. For there are some who think it possible both for the ungenerated to be destroyed and for the generated to persist undestroyed. (This is held in the Timaeus, where Plato says that the heaven, though it was generated, will none the less exist to eternity.) So far as the heaven is concerned we have answered this view with arguments appropriate to the nature of the heaven: on the general question we shall attain clearness when we examine the matter universally.
Praemissis rationibus contra opinionem Platonis, hic philosophus excludit quandam excusationem praedictae opinionis, quam Xenocrates et alii Platonici afferebant. Et circa hoc duo facit: 231. After presenting the arguments against Plato, the Philosopher here rejects a certain excusing of the aforesaid opinion, which Xenocrates and other Platonists proposed. About this he does two things:

primo proponit excusationem;

secundo excludit eam, ibi: hoc autem est, quemadmodum dicimus et cetera.

First he proposes the explanation;

Secondly, he rejects it, at 232.

Dicit ergo primo quod non est verum illud auxilium, idest illa excusatio, quam quidam Platonicorum, dicentium mundum esse incorruptibilem sed tamen factum vel genitum, conantur ferre sibi ipsis, ut non irrationabiliter posuisse videantur. Dicunt enim se dixisse de generatione mundi ad similitudinem eorum qui describunt figuras geometricas, qui primo describunt quasdam partes figurae, puta trianguli, et postea alias, non quasi prius fuerint huiusmodi partes antequam talis figura ex huiusmodi partibus constitueretur, sed ut magis explicite demonstrent ea quae ad figuram requiruntur. Et similiter dicunt Platonem dixisse mundum factum esse ex elementis, non tanquam aliquo tempore determinato mundus sit generatus, sed causa doctrinae; ut facilius instruerentur aliqui de natura mundi, dum prius demonstrantur eis partes mundi, et quid habeant huiusmodi partes ex seipsis, postea demonstratur eis compositio quam habent a causa mundi, quae Deus est. Et ita aspiciunt, idest considerant, mundum esse genitum, ad modum descriptionis qua utuntur geometrae in descriptione figurarum. He says, therefore, first [167] that there is no truth in that "help," i.e., that excusing, by which some Platonists seek to justify their assertion —that the world is imperishable, but yet made or generated — and make it appear not unreasonable. For they say that their description of the world's generation was after the manner of those who describe geometric figures by first drawing certain parts of the figure, e.g., of a triangle, and later other parts, not implying that these parts existed before the figure was formed of them, but doing this in order to demonstrate more explicitly what things are required for the figure. They say that Plato in like manner declared that the world was made from elements, not as though the world was generated at some definite time, but for the purpose of presenting his doctrine, so that, namely, his hearers would be more easily instructed about the nature of the world, if first the parts of the world were demonstrated to them and what these parts possessed of themselves, and later the composition they had from the cause of the world, which is God. Consequently they look on, i.e., consider, the world as generated in the manner of the description which geometers use in describing figures.
Deinde cum dicit: hoc autem est, quemadmodum dicimus etc., improbat quod dictum est. Et dicit quod non eodem modo se habet quod ipsi dicunt circa generationem mundi, et quod geometrae dicunt circa descriptiones figurarum, sicut manifestabitur per ea quae nunc dicemus. Quia in descriptionibus geometricalibus, idem accidit si omnes partes figurae simul accipiantur ut constituunt figuram, et si non accipiantur simul: quia quando non accipiuntur simul, nihil aliud dicitur de eis nisi quod sunt lineae vel anguli; et hoc etiam salvatur in eis quando accipiuntur omnia simul in figura constituta ex eis. Sed in demonstrationibus eorum qui ponunt generationem mundi, non idem accipitur cum sunt simul et cum non sunt simul; sed impossibile est quod idem ex utraque parte accipiatur, sicut impossibile est opposita esse simul; illa enim quae accipiuntur prius, scilicet ante constitutionem mundi, et posterius, scilicet mundo iam constituto, sunt subcontraria, idest habent quandam adiunctam et latentem contrarietatem. 232. Then at [168] he disproves this explanation. And he says that the way the generation of the world is described by them is not in the same manner as the descriptions of figures made by geometers, as will be clear from what we shall now say. For in geometric descriptions the same thing happens whether all the parts are considered together as constituting the figure, or whether they are not taken together. When they are taken separately, no more is said about them than that they are lines or angles, which is also true of them when they are taken all together in the figure made out of them. But in the demonstrations presented by those who posit the generation of the world, the same thing is not taken when the parts are considered together and when they are not. Rather, it is impossible that the same be taken in both instances, just as it is impossible for opposites to be together — for the things taken first, i.e., before the establishing of the world, and those taken later, i.e., after the world is now established, are "subcontraries," i.e., have a certain conjoined and latent contrariety.
Dicunt enim quod ex elementis inordinatis facta sunt ordinata, Deo scilicet reducente inordinationem elementorum ad ordinem, ut Plato in Timaeo dicit: geometrae autem non dicunt quod ex lineis divisis componatur triangulus, sed simpliciter quod ex lineis. Et esset simile si isti solum dicerent quod mundus sit ex elementis: sed dicunt quod mundus ordinatus sit ex elementis inordinatis. Non est autem possibile quod aliquid sit simul ordinatum et inordinatum: sed necesse est dari aliquam generationem, per quam unum eorum ab altero separetur, ut scilicet ante generationem sit inordinatum, post generationem vero ordinatum; et per consequens necesse est dari aliquod tempus distinguens utrumque. Sed in descriptionibus figurarum non requiritur aliqua distinctio temporis: non enim oportet quod linea et triangulus tempore distinguantur, sicut ordinatum et inordinatum. For they say that out of unordered elements, ordered things were made, God reducing the disorder among the elements to order, as Plato says in the Timaeus. But geometers do not say that a triangle is composed out of separated lines but out of lines. The situation would be similar if those in question solely said that the world results from elements, but what they say is that the orderly world came about from disordered elements. Now it is not possible for something to be at once ordered and disordered, but a process of generation is required through which one is separated from the other, so that before generation it is disordered, and after generation ordered. Consequently it is necessary to suppose some time distinguishing the two. But no such distinction of time is required in the descriptions of figures — for it is not necessary that a line and a triangle be distinguished in the order of time as ordered and disordered are.
Volunt autem quidam adhuc excusare Platonem, quasi non posuerit quod inordinatio prius tempore fuerit in elementis mundi, et postea aliquo tempore incoeperint ordinari; sed quia inordinatio semper quantum ad aliquid adiuncta est elementis mundi, licet quantum ad aliquid ordinentur; sicut etiam ipse Aristoteles ponit quod materiae semper adiungitur privatio, quamvis et semper sit secundum aliquid formata. Potest etiam intelligi Platonem dedisse intelligere quid elementa ex se haberent, si non essent ordinata a Deo; non quod prius tempore fuerint inordinata. 233. Still others desire to excuse Plato on the ground that he did not teach that there was a prior disorder in the elements which subsequently, at a later time, began to be ordered, but rather disorder is always present under some aspect in the elements of the world, although under another aspect there is order, as Aristotle himself posits that matter always has a concomitant privation, although it is always in some respect under form. It is also possible to interpret Plato as stressing what the elements would be of themselves if they had not been put in order by God, not that there was ever a time in which they existed disordered.
Sed quidquid Plato intellexerit, Aristoteles, sicut dictum est, obiiciebat contra id quod verba Platonis exprimunt. Concludit ergo ex praemissis quod impossibile sit mundum factum esse per generationem, et tamen eum in sempiternum durare. But whatever Plato may have understood about the matter, Aristotle, as has been said, objected against what Plato's words express. He concludes, therefore, from the foregoing that it is impossible for the world to have been generated and yet able to go on forever.
Deinde cum dicit: vicissim autem etc., prosequitur opinionem Empedoclis, quam tertio posuerat. Et dicit quod illi qui dicunt mundum vicissim componi et dissolvi, nihil aliud faciunt quam quod adstruunt mundum esse sempiternum secundum substantiam, sed se transmutare secundum formam, sive secundum eius dispositionem; sicut si aliquis videns aliquem ex puero factum virum, si ponatur quod videat vicissim eundem ex viro factum puerum, putet eum quandoque fieri et quandoque corrumpi. Et quod secundum hanc opinionem Empedoclis ponatur ipsa substantia mundi sempiterna, manifestat per hoc quod post separationem elementorum per litem, quando iterum convenient elementa, non fiet qualiscumque ordo mundi et qualiscumque eius constitutio, sed eadem quae nunc est. 234. Then at [169] he takes up the opinion of Empedocles which is the third one mentioned. And he says that those who maintain that the world alternates between being assembled and dissolved do nothing more than assert the substantial permanence of the world but its transmutability with respect to its form or its arrangement. It is as though someone seeing a boy becoming a man, if it should be posited that he sees the same person becoming from a man a boy again, should reckon this person as [alternately] at one time coming into existence and at one time ceasing to be. That the opinion of Empedocles is tantamount to positing the substance of the world as eternal, he manifests by the fact that after the elements shall have been separated by strife and later reassembled, it is not just any order and any new arrangement that will ensue but the very same one that now exists.
Et hoc manifestum est et aliter, scilicet per rationem, quia ab eadem causa, scilicet amicitia, congregabuntur tunc elementa, ex qua et prius congregata sunt, et sic eadem constitutio mundi sequetur: sed etiam hoc manifestum est secundum eos qui hanc positionem ponunt, qui asserunt contrarietatem litis et amicitiae, quas ponunt causam contrariae dispositionis in elementis, ut scilicet quandoque sint coniuncta, quandoque separata. Unde concludit quod, si totum corpus mundi, continuum existens, idest coniunctum, quandoque disponatur et aptetur uno modo, quandoque alio modo; cum ipsa consistentia sive substantia omnium corporum vocetur mundus sive caelum, sequitur quod mundus non generetur et corrumpatur, sed solum dispositiones ipsius. And this is made clear "in another way," i.e., by reason, because the very same cause, namely, friendship, will assemble the elements which previously assembled them; consequently, the same arrangement of the world will result. And this is plain also from the teachings of those who hold this position and assert that friendship and strife are contrary and the causes of a contrary disposition in the elements, so that at one time they are assembled and at another separated. Hence he concludes that if the entire body of the world, while remaining "continuous," i.e., conjoined, is now disposed and arranged in one way and later in another way, then, since it is the "combination," or substance, of all bodies that is called the world or heaven, it follows that the world is not generated and destroyed but only its arrangements are.
Deinde cum dicit: totaliter autem factum etc., prosequitur opinionem Democriti, quam supra secundo posuerat. 235. Then at [170] he takes Democritus' opinion, which was the second one mentioned.

Et primo dicit qualiter se habeat ista opinio;

secundo ostendit quid circa hanc postmodum erit manifestum, ibi: sed tamen et cetera.

First he explains this opinion;

Secondly, he shows what will later be clear about it, at 236.

Dicit ergo primo quod, si aliquis ponat quod mundus sit factus, et totaliter corrumpatur absque regressu, ita scilicet quod nunquam iterum fiat, hoc quidem est impossibile, si ponatur unus tantum mundus. Et hoc ideo, quia si sit unus mundus qui quandoque est factus, cum non sit factus ex nihilo, priusquam fieret existebat substantia quae erat ante eum. Aut ergo ponemus quod illa substantia quae praeerat mundo, poterat subiici generationi, aut non. Et si quidem non poterat generationi subiici, non poterat ex ea fieri mundus: et hoc est quod dicit, qua non facta, vel non genita, idest qua non subiecta generationi, impossibile esse dicimus transmutari, idest non possibile esse quod transmutetur, ad hoc ut ex ea fiat mundus. Si vero in sua natura habebat quod posset transmutari, ad hoc quod fieret ex ea mundus, etiam post corruptionem mundi poterit transmutari, ut ex ea iterum fiat mundus. He says therefore first [170] that if someone should maintain that the world was made, and entirely ceases to be without returning, in such a way, namely, that it will never be restored again, such a thing is impossible, if there is but one world. The reason is that if there is but one world, made at some time, then, since it was not made from nothing, there was, previous to its being made, a substance which existed before it. Either we hold that that substance which pre-existed before the world could have been subject to generation, or that it could not. If not, then the world could not have been made from it. And this is what he says, namely, that if it was not made, or not generated, i.e., not subject to generation, we say it to be impossible of transmutation, i.e., not able to be transmuted in order for the world to be made out of it. But if it possessed in its nature the power to be transmuted, so that the world could be made from it, then also after the destruction of the world it could be transmuted and a world made again from it.
Sed si aliquis ponat infinitos mundos, ita scilicet quod ex quibusdam atomis uno modo compositis fiat hic mundus, et ex eisdem vel aliis alio modo compositis fiat alius mundus, et hoc in infinitum; magis poterit sustineri quod dictum est, scilicet quod mundus semel corruptus nunquam iterum generetur; quia ex quo possibile est esse alios mundos, ex illis atomis poterit alius mundus constitui. Sed si non posset esse mundus nisi unus, sequeretur inconveniens: quia materia in quam mundus resolveretur, esset adhuc in potentia ut ex ea fieret mundus; unde si non posset esse alius mundus, oporteret quod idem ipse iterum fieret. But if someone posits infinite worlds, in the sense that from atoms arranged in one way this world comes to be, and from the same or other atoms differently arranged another world comes to be, and so on ad infinitum, such a position would be a better foundation for what was said, namely, that the world once destroyed is never again regenerated, because from the assumption that other worlds are possible, another world could be arranged from those atoms. However, if there could be but one world, something incompatible with the theory follows: the matter into which the world dissolved would still be in potency to have a world made from it. Hence if a different world were impossible, the very same one would have to be produced again.
Deinde cum dicit: sed tamen etc., ostendit quid restet dicendum: et dicit quod ex posterioribus erit manifestum utrum hoc sit possibile vel impossibile. Et si quidem ly hoc referatur ad immediate dictum de opinione ponentium infinitos mundos, non est intelligendum quod posteriora hic nominet ea quae immediate sequuntur, in quibus nulla de hoc fit mentio; sed intelliguntur posteriora ea quae dicentur de opinione Democriti in tertio huius, et in I de generatione. Si vero ly hoc referatur ad totum praecedens, ubi actum est de opinione ponentium mundum esse genitum, per posteriora intelliguntur immediate sequentia. 236. Then at [171] he shows what remains to be said, and says that from what will follow, it will be clear whether this is possible or impossible. And if "this" refers to what was just said of the opinion about infinite worlds, the phrase "what will follow" refers, not to what follows immediately, in which nothing is said about this opinion, but to what will be said about the opinion of Democritus in On the Heavens III and in On Generation I. But if "this" refers to the whole preceding section, where there is treated the opinion of those who posit that the world was generated, then the phrase "what will follow" refers to what immediately follows.
Et ad hoc concordat quod immediate subditur. Sunt enim quidam, quibus videtur esse contingens quod aliquid quod nunquam fuit generatum, quandoque corrumpatur, et quod aliquid de novo genitum, incorruptibile perduret; sicut in Timaeo dicit Plato non solum quod caelum sit factum de novo, sed etiam quod duret de cetero sempiterno tempore; et sic ponit utrumque dictorum, scilicet quod materia inordinata, quae nunquam incoepit esse inordinata, quandoque esse desinat; et quod mundus incipiat, et nunquam desinat. Et contra istos sic ponentes mundum generari, supra circa principium huius libri naturalibus rationibus processum est solum quantum ad caelum, quod probavit esse ingenitum et incorruptibile, tanquam non habens contrarium: sed nunc hoc manifestabitur universali consideratione de omnibus entibus. And this is confirmed by what he at once adds. For there are some who conceive it possible for something which was never generated to perish at some time, and for something newly generated to remain incorruptible, as Plato says in the Timaeus that the heaven was produced in being, but will nevertheless endure for eternity. Thus he posits both statements: that disarranged matter, which never became disarranged, at some time ceases to be, and that the world began, and never ceases to be. Against those who thus posit that the world began through generation, Aristotle argued above near the beginning of this book with natural reasons solely to the effect that the heaven was proved ungenerated and indestructible, on the ground that it has no contrary. But now this will be shown by a universal consideration of all beings.

Lecture 24:
Various meanings of "generable" and "ungenerable," "corruptible" and "incorruptible"
Chapter 11
(280b.) Πρῶτον δὲ διαιρετέον πῶς ἀγένητα καὶ γενητά φαμεν καὶ φθαρτὰ καὶ ἄφθαρτα 172 We must first distinguish the senses in which we use the words 'ungenerated' and 'generated', 'destructible' and 'indestructible'.
πολλαχῶς γὰρ λεγομένων, κἂν μηδὲν διαφέρῃ πρὸς τὸν λόγον, ἀνάγκη τὴν διάνοιαν ἀορίστως ἔχειν, ἄν τις τῷ διαιρουμένῳ πολλαχῶς ὡς ἀδιαιρέτῳ χρῆται ἄδηλον γὰρ κατὰ ποίαν φύσιν αὐτῶν συμβαίνει τὸ λεχθέν. 173 These have many meanings, and though it may make no difference to the argument, yet some confusion of mind must result from treating as uniform in its use a word which has several distinct applications. The character which is the ground of the predication will always remain obscure.
Λέγεται δ' ἀγένητον ἕνα μὲν τρόπον ἐὰν ᾖ τι νῦν πρότερον μὴ ὂν ἄνευ γενέσεως καὶ μεταβολῆς, καθάπερ ἔνιοι τὸ ἅπτεσθαι καὶ τὸ κινεῖσθαι λέγουσιν οὐ γὰρ εἶναι γενέσθαι φασὶν ἁπτόμενον, οὐδὲ κινούμενον. Ἕνα δ' εἴ τι ἐνδεχόμενον γίνεσθαι ἢ γενέσθαι μή ἐστιν ὁμοίως γὰρ καὶ τοῦτο ἀγένητον, ὅτι ἐνδέχεται γενέσθαι. Ἕνα δ' εἴ τι ὅλως ἀδύνατον γενέσθαι, ὥσθ' ὁτὲ μὲν εἶναι ὁτὲ δὲ μή. (Τὸ δ' ἀδύνατον λέγεται διχῶς. Ἢ γὰρ τῷ μὴ ἀληθὲς εἶναι εἰπεῖν ὅτι γένοιτ' ἄν, ἢ τῷ μὴ ῥᾳδίως μηδὲ ταχὺ ἢ καλῶς.) 174 The word 'ungenerated' then is used (a) in one sense whenever something now is which formerly was not, no process of becoming or change being involved. Such is the case, according to some, with contact and motion, since there is no process of coming to be in contact or in motion. (b) It is used in another sense, when something which is capable of coming to be, with or without process, does not exist; such a thing is ungenerated in the sense that its generation is not a fact but a possibility. (c) It is also applied where there is general impossibility of any generation such that the thing now is which then was not. And 'impossibility' has two uses: first, where it is untrue to say that the thing can ever come into being, and secondly, where it cannot do so easily, quickly, or well.
Τὸν αὐτὸν δὲ τρόπον καὶ τὸ γενητὸν ἕνα μὲν εἰ μὴ ὂν πρότερον ὕστερον ἔστιν, εἴτε γινόμενον εἴτ' ἄνευ τοῦ γίνεσθαι, ὁτὲ μὲν μὴ ὄν, πάλιν δ' ὄν. Ἕνα δ' εἰ δυνατόν, εἴτε τῷ ἀληθεῖ διορισθέντος τοῦ δυνατοῦ εἴτε τῷ ῥᾳδίως. Ἕνα δ' ἐὰν ἡ γένεσις αὐτοῦ ἐκ τοῦ μὴ ὄντος εἰς τὸ ὄν, εἴτ' ἤδη ὄντος, διὰ τοῦ γίνεσθαι δ' ὄντος, εἴτε καὶ μήπω ὄντος, ἀλλ' ἐνδεχομένου. 175 In the same way the word 'generated' is used, (a) first, where what formerly was not afterwards is, whether a process of becoming was or was not involved, so long as that which then was not, now is; (b) secondly, of anything capable of existing, 'capable' being defined with reference either to truth or to facility; (c) thirdly, of anything to which the passage from not being to being belongs, whether already actual, if its existence is due to a past process of becoming, or not yet actual but only possible.
Καὶ φθαρτὸν δὲ καὶ ἄφθαρτον ὡσαύτως εἴτε γὰρ πρότερόν τι ὂν ὕστερον ἢ μή ἐστιν ἢ ἐνδέχεται μὴ εἶναι, φθαρτὸν εἶναί φαμεν, εἴτε φθειρόμενόν ποτε καὶ μεταβάλλον, εἴτε μή. Ἔστι δ' ὅτε καὶ τὸ διὰ τοῦ φθείρεσθαι ἐνδεχόμενον μὴ εἶναι φθαρτὸν εἶναί φαμεν, καὶ ἔτι ἄλλως τὸ ῥᾳδίως φθειρόμενον, ὃ εἴποι ἄν τις εὔφθαρτον. 176 The uses of the words 'destructible' and 'indestructible' are similar. 'Destructible' is applied (a) to that which formerly was and afterwards either is not or might not be, whether a period of being destroyed and changed intervenes or not; and (b) sometimes we apply the word to that which a process of destruction may cause not to be; and also (c) in a third sense, to that which is easily destructible, to the 'easily destroyed', so to speak.
Καὶ περὶ τοῦ ἀφθάρτου ὁ αὐτὸς λόγος Ἢ γὰρ τὸ ἄνευ φθορᾶς ὁτὲ μὲν ὂν ὁτὲ δὲ μὴ ὄν, οἷον τὰς ἁφάς, ὅτι ἄνευ τοῦ φθείρεσθαι πρότερον οὖσαι ὕστερον οὐκ εἰσίν. Ἢ τὸ ὂν μέν, δυνατὸν δὲ μὴ εἶναι, ἢ καὶ οὐκ ἐσόμενόν ποτε, νῦν δ' ὄν σὺ γὰρ εἶ, καὶ ἡ ἁφὴ νῦν ἀλλ' ὅμως φθαρτόν, ὅτι ἔσται ποτὲ ὅτε οὐκ ἀληθὲς εἰπεῖν ὅτι εἶ, οὐδὲ ταῦτα ἅπτεσθαι. Τὸ δὲ μάλιστα κυρίως, τὸ ὂν μέν, ἀδύνατον δὲ φθαρῆναι οὕτως ὥστε νῦν ὂν ὕστερον μὴ εἶναι ἢ ἐνδέχεσθαι μὴ εἶναι. Ἢ καὶ τὸ μήπω ἐφθαρμένον, ἐνδεχόμενον δ' ὕστερον μὴ εἶναι. Λέγεται δ' (281a.) ἄφθαρτον καὶ τὸ μὴ ῥᾳδίως φθειρόμενον. 177 Of the indestructible the same account holds good. It is either (a) that which now is and now is not, without any process of destruction, like contact, which without being destroyed afterwards is not, though formerly it was; or (b) that which is but might not be, or which will at some time not be, though it now is. For you exist now and so does the contact; yet both are destructible, because a time will come when it will not be true of you that you exist, nor of these things that they are in contact. Thirdly (c) in its most proper use, it is that which is, but is incapable of any destruction such that the thing which now is later ceases to be or might cease to be; or again, that which has not yet been destroyed, but in the future may cease to be. For indestructible is also used of that which is destroyed with difficulty.
Postquam philosophus prosecutus est opiniones aliorum circa propositam quaestionem de mundo, an sit genitus et corruptibilis, hic prosequitur praedictam quaestionem secundum suam opinionem. 237. After discussing others' opinions about whether the world is generated and destructible [corruptible], the Philosopher here pursues this question according to his own opinion.

Et primo praemittit quaedam quae sunt necessaria ad investigationem propositi;

secundo prosequitur propositam quaestionem, ibi: determinatis autem his et cetera.

First he presents pre-notes needed in his investigation of the question;

Secondly, he pursues the question (L. 26).

Circa primum duo facit: About the first he does two things:

primo distinguit multiplicitatem horum nominum, quibus utitur in quaestione, scilicet geniti et ingeniti, corruptibilis et incorruptibilis;

secundo distinguit multiplicitatem quorundam nominum, quae in praedictorum definitione cadunt, scilicet possibilis et impossibilis, ibi: si itaque haec sic habent et cetera.

First he distinguishes various senses of the following words used in the question: namely, "generated" and "ungenerated," "destructible" and "indestructible";

Secondly, he distinguishes various senses of certain words used in the definitions of the foregoing: namely, "possible" and "impossible" (L. 25).

Circa primum duo facit: About the first he does two things:

primo dicit de quo est intentio;

secundo propositum prosequitur, ibi: dicitur autem ingenitum et cetera.

First he reveals his intention;

Secondly, he carries it out, at 239.

Circa primum duo facit. About the first he does two things:
Primo dicit de quo est intentio: et dicit quod circa inquisitionem praedictae quaestionis, primo oportet distinguere quibus modis aliqua dicuntur generabilia et ingenerabilia, et iterum corruptibilia et incorruptibilia.

First he reveals his intention [172] and says that in investigating the foregoing question it is first of all necessary to distinguish the various ways in which things are said to be "generable" and "ungenerable," "destructible" and "indestructible."

Deinde cum dicit: multipliciter enim dictis etc., assignat rationem suae intentionis. Et dicit quod quando aliqua multipliciter dicuntur, contingit quandoque quod illa multiplicitas nullam differentiam inducat quantum ad rationem quae proponitur, quando scilicet in illa ratione sumitur nomen solum in una significatione: tunc enim multiplicitas differentiam facit in ratione, quando nomen sumitur in diversis significationibus. Sed tamen, licet nulla differentia fiat quantum ad rationem, tamen intellectus audientis confuse se habet, si aliquis utatur nomine quod multipliciter potest distingui, tanquam distingui non posset: quia quando aliquis utitur indistincte nomine multiplici, non est manifestum secundum quam naturam significatam accidit conclusio. 238. Secondly, at [173] he reveals the reason for his intention and says that when things are said in a number of ways, it sometimes happens that this multiplicity produces no difference with regard to the argument proposed, i.e., when a particular word is restricted to one meaning in the course of the argument. But when a particular word is used with different meanings, such a multiplicity does make a difference. Even where there is no difference as to the argument, the intellect of the hearer becomes confused, if someone uses a word which can be distinguished in many ways as though it could not — for when someone uses a word of multiple meaning, it is not evident according to which signified essence the conclusion occurs.
Deinde cum dicit: dicitur autem ingenitum etc., distinguit praedicta nomina: 239. Then at [174] he distinguishes the aforesaid words:

et primo ingenitum et genitum;

secundo corruptibile et incorruptibile, ibi: et corruptibile autem et cetera.

First, "ungenerated" and "generated";

Secondly, "destructible" and "indestructible," at 243.

Circa primum duo facit: About the first he does two things:

primo distinguit hoc nomen ingenitum;

secundo hoc nomen genitum, ibi: eodem autem modo et cetera.

First he distinguishes the word "ungenerated";

Secondly, the word "generated," at 241.

Ponit autem primo quod hoc nomen ingenitum dicitur tribus modis. Quorum primus est prout dicitur aliquid ingenitum, quod quidem nunc est, sed prius non erat, ita tamen quod hoc contingat sine generatione et transmutatione eius quod esse incipit; sicut aliqui ponunt exemplum de eo quod est tangi et moveri; dicunt enim quod tactum et motum non contingit generari. Et hoc probatum est in V Physic., quia, cum generatio sit quaedam species motus sive transmutationis, si motus generaretur, sequeretur quod mutationis esset mutatio. Sic ergo tactus et motus, licet esse incipiant, tamen dicuntur ingenita, quia non generantur, nec nata sunt generari. He declares first [174] that this word "ungenerated" is used in three ways. The first of these is when something is called "ungenerated" which now exists but previously did not, yet this occurs without its having been begotten or transmuted. Some give the example of being touched or moved: for they assert that contact and motion are not generated. And this was proved in Physics V because, since generation is a kind of motion or transmutation, if motion were generated, it would follow that there would be a change of a change. Consequently, contact and motion, although they begin to be, are called "ungenerated," because they are not generated and are not apt to be generated.
Secundo modo dicitur aliquid esse ingenitum, quod quidem contingit fieri vel non fieri, et tamen nondum est factum; sicut hominem qui nascetur cras, contingit in futurum fieri vel non fieri, et tamen dicitur ingenitus, quia nondum est natus. Similiter enim et hoc potest dici ingenitum, quasi non genitum, quod contingit generari, quia nondum est generatum, sicut et illud quod non contingit generari. In a second way something is said to be "ungenerated," if it is able to either come to be or not come to be and still it has not yet come to be. For example, a man to be born tomorrow is able, as far as the future is concerned, to come to be and not come to be, and yet he is said to be "ungenerated," because he has not yet been born. For "ungenerated," in the sense of "not generated," can be applied similarly to what is able to come to be, because it is not yet generated, and to what is not able to be generated.
Tertio modo dicitur aliquid ingenitum, quod omnino impossibile est fieri hoc modo ut quandoque sit et quandoque non sit, sive per generationem sive quocumque alio modo; et secundum hoc ingenita dicuntur quae non possunt esse, vel quae non possunt non esse. Hic autem modus distinguitur in duos: nam impossibile esse seu fieri dicitur dupliciter; uno modo absolute, quando scilicet simpliciter non est verum dicere quod hoc aliquando fiat; secundo modo prout dicitur aliquid impossibile fieri, quia non de facili potest fieri; et hoc quia non cito potest fieri, vel quia non est bene factibile, sicut si dicamus aliquod malum ferrum non esse bene fabricabile. In a third way something is said to be "ungenerated," when it is entirely impossible for it, through generation or any other way, to come into existence as being able either to exist or not exist. In this sense the word "ungenerated" describes things that cannot be or things that cannot not be. Now this third way is distinguished into two other ways, for there are two ways in which something is "impossible" to be or become: first, absolutely, when it is in no sense true to say that this may at some time come to be; secondly, when a thing is described as impossible to come about because it is not easy for it to come about, either because it does not come about quickly or because its coming into existence cannot be conveniently managed, as when we say that bad iron is not easy to fashion.
Ad evidentiam autem horum modorum, considerandum est quod generatio importat aliquid commune, quod est incipere esse; et etiam importat determinatum modum essendi, scilicet per transformationem. Negatio igitur quae importatur hoc nomine ingenitum, uno modo potest negare utrumque, scilicet incoeptionem et modum incipiendi; vel potest solum negare modum incipiendi. Et utrumque contingit dupliciter: uno modo secundum actum, alio modo secundum potentiam. Si igitur praedicta negatio non neget incoeptionem, sed solum modum incipiendi, sic est primus modus, secundum quem dicitur aliquid ingenitum, quod potest incipere esse, sed non per generationem. Si vero neget non potentiam, sed solum actum, ut puta quia potest incipere esse et potest generari, non tamen adhuc incoepit esse vel est generatum, sic est secundus modus. Si vero non solum neget modum incipiendi, sicut in primo modo, nec solum actum generationis, sicut in secundo, sed simul modum incoeptionis et incoeptionem, et quantum ad actum et quantum ad potentiam; sic est tertius modus, qui est perfectissimus, secundum quem proprie et simpliciter dicitur aliquid ingenitum; quamvis et hic modus distinguatur secundum quod possibile dicitur aliquid vel simpliciter vel secundum quid. 240. In order to understand these three ways it should be noted that generation has the common note of something's beginning to exist, and also implies a definite way of existing, namely, through transformation. Therefore the negation implied by the word "ungenerated" may either negate both, namely, both a beginning and the way of beginning, or it can negate only the way of beginning. And both can occur in two ways: in one way in the sphere of act, and in the other in the sphere of potency. Therefore, if the negation does not deny a beginning but merely the manner of beginning, we have the first meaning of the word according to which something is said to be "ungenerated," if it can begin to be but not through generation. But if it does not deny the possibility but merely the actual state, for example, because it can begin to be and can be generated, but has not yet begun to be or been generated, then it is the second sense of the word. But if it does not only deny the manner of its beginning, as in the first sense, or only the actual state of existence, as in the second sense, but both the manner of its beginning and the very beginning itself, both as to its actual state and even its possibility, then it is the third and most perfect sense, according to which something is said to be ungenerated in the strict and absolute sense. This sense, however, is still distinguished according to whether something is said to be "possible" either absolutely or in a qualified sense.
Deinde cum dicit: eodem autem modo etc., distinguit significationem huius nominis genitum. Et dicit quod eodem modo genitum dicitur tribus modis. Quorum primus est si aliquid prius non fuit et postea incoepit esse, sive per generationem, sicut homo, sive sine generatione, sicut tactus; dummodo illud quod dicitur genitum, quandoque non sit, et iterum postea sit. 241. Then at [175] he distinguishes the meanings of the word "generated," and says that it is also in three senses that "generated" is used. The first of these occurs if something previously did not exist and later began to exist, either through generation, as man, or without generation, as contact, provided that the thing described as generated is something that one time is not and later is.
Secundo modo dicitur aliquid genitum, si possibile sit illud incipere esse; sive possibile determinetur per verum, ut scilicet dicatur possibile quod potest esse, sive determinetur per facile, ut scilicet dicatur possibile fieri quod de facili potest. In a second way, something is described as "generated," if it is possible for it to begin to exist, where "possible" refers either to the truth, i.e., to what can exist, or to what is easy, i.e., can easily be made to exist.
Tertio modo dicitur aliquid genitum, cuius potest esse generatio, ut per hoc procedat de non esse in esse: et hoc indifferenter sive iam esse incoeperit, et hoc per fieri, idest per modum generationis; sive nondum esse incoeperit, sed contingat illud esse incipere per modum generationis. In the third way, something is described as "generated," if it can be the subject of generation and proceeds thus from non-existence to existence. In this third sense it makes no difference whether the thing has already begun to be, and this by being made, i.e., through the process of generation, or whether it has not yet begun to be, but may come to be through generation.
Apparet etiam secundum praemissa ratio horum modorum. Quia cum dicitur genitum secundum primum modum, asseritur actualis incoeptio, non autem modus determinatus incipiendi, quem significat generatio. Secundum autem modum secundum, asseritur possibilitas incoeptionis absque determinato modo incipiendi: et hic modus potest distingui in duos secundum distinctionem potentiae. Secundum autem modum tertium, asseritur non solum incoeptio, sed determinatus modus incipiendi: et hic modus potest distingui in duos, quia vel asseritur determinatus modus incipiendi secundum actum, ut quia sit aliquid iam generatum; aut secundum potentiam, ut quia aptum natum sit generari. In keeping with what has been said, the notion of these ways is apparent. Because when something is called "generated" in the first sense, its actual inception is asserted but not a definite mode of inception that the word "generation" signifies. But in the second sense the possibility of inception is asserted without asserting the definite way it began, which sense is distinguished according to the way "potency" is distinguished. However, the third way asserts not only inception but a definite kind of inception. And this third way can be further distinguished into two: for it asserts either a definite kind of inception that is actual, as when something is already generated, or one that is potential, as indicating something is naturally apt to be generated.
Et si quis recte consideret modos quos posuit circa genitum, differunt a modis quos posuit circa ingenitum dupliciter: uno modo secundum distinctionem, alio modo secundum ordinem. 242. Now if anyone rightly considers the senses he has set down of the word "generated," he will see that they differ from the senses of "ungenerated" in two ways: first with respect to distinction, and, secondly, with respect to order.
Secundum distinctionem quidem, quia in distinctione modorum ingeniti, sub alio modo comprehendebatur negatio determinati modi incipiendi secundum potentiam, et in alio secundum actum: nam in primo modo dicebatur ingenitum, quod non poterat incipere per generationem; in secundo autem quod poterat incipere per generationem, sed nondum erat generatum. With respect to distinction: In distinguishing the senses of "ungenerated," the denial of a definite kind of inception as possible was included under one sense and the denial of the same kind of inception as actual, was included under another sense — for in the first sense "ungenerated" referred to what could not begin to be through generation, but in the second it referred to what could begin to be through generation but had not yet been generated.
Sed quantum ad negationem incoeptionis in communi, sub eodem modo comprehendebat negationem potentiae et actus: dicebatur enim tertio modo ingenitum, quod nec incoepit esse, nec potest incipere. Sed circa modos geniti, e converso ex parte incoeptionis in communi distinguit modos secundum potentiam et actum: nam primus modus est quod actu incipit esse quocumque modo; secundus modus est quod potest incipere quocumque modo, licet nondum incoeperit. Sed ex parte determinati modi incipiendi, sub uno modo comprehendit potentiam et actum: dicitur enim tertio modo genitum, quod vel est generatum vel potest generari. Et sic patet quod isti tres modi non directe contraponuntur tribus primis: quia quod ibi distinguebatur, hic remanet indistinctum, et e converso. But in regard to the denial of inception in common, both the possibility and the actuality of inception are included under the same sense — for the third sense of "ungenerated" referred to what has both not begun to be and cannot begin to be. But conversely, in the senses of "generated" it is on the part of a beginning in common that he distinguishes the modalities according to potency and act — for the first sense refers to what actually begins to be in any way whatever, while the second sense refers to what can begin in any way, although it has not yet begun. However, with regard to a definite kind of inception, the actuality and possibility are included under one mode —for in the third sense something is described as "generated" which either has been generated or can be generated. Thus it is plain that the last three senses are not exactly parallel to the first three, because what was distinguished in the first remains undistinguished in the second, and vice versa.
Secundum ordinem autem differunt isti modi. Nam in modis ingeniti praemittebatur id quod pertinet ad determinatum modum incoeptionis, ei quod pertinet ad incoeptionem in communi: sed circa modos geniti praemittitur id quod est ex parte incoeptionis in communi. Et hoc subtili ratione Aristoteles fecit. Voluit enim primo ponere modos imperfectos, et ultimo modos perfectos: differenter autem se habent negatio et affirmatio circa proprium et commune. Nam negatio quae negat proprium, est imperfecta; negatio autem quae negat commune, est perfecta, quia negato communi negatur proprium: et ideo ultimum modum ingeniti quasi perfectum posuit, quo negatur incoeptio in communi. Et quia negatio particularis modi incipiendi est imperfecta, ideo ex hac parte posuit partiales modos distinctos secundum potentiam et actum. With respect to order these senses are different: For in presenting the modes of "ungenerated," that which pertains to a definite kind of inception was placed before that which pertains to inception in common, whereas in presenting the modes of "generated," that which pertains to inception in common was mentioned first. And Aristotle had a subtle reason for so doing. For he wanted to list the imperfect senses first and the perfect ones last. Now denial and affirmation are related to the proper and to the common in different ways: for a denial of what is proper is imperfect, but a denial of what is common is perfect, because when the common is denied, the proper is denied. Consequently, the last sense of "ungenerated" is presented as the perfect sense, because it denies inception in general. And because the denial of a particular kind of inception is imperfect, he presents the partial modes as distinguished according to potency and act.
Sed affirmatio proprii est perfecta, quia posito proprio ponitur commune; affirmatio autem communis est imperfecta: et ideo ultimum modum geniti posuit tanquam perfectum, quod incoepit esse per generationem; et comprehendit sub hoc modo, tanquam sub perfecto, et potentiam et actum. Modos autem pertinentes ad incoeptionem in communi, praemisit tanquam imperfectos: non enim perfecte dicitur aliquid genitum ex hoc solo quod incoepit esse. Et ideo ex hac etiam parte distinxit hos modos, tanquam partiales, secundum potentiam et actum. But the affirming of what is proper is perfect, because in affirming what is proper that which is common is also affirmed, while the affirming of what is common is imperfect. Accordingly, the last sense of "generated" is presented as the perfect one, namely, when something begins to be through generation, and he includes under this sense, as under the perfect sense, both the possibility and the actuality. However, the senses pertaining to inception in general are presented first, as the imperfect senses: for a thing is not said to be "generated" in the perfect sense just because it has begun to be. For this reason he distinguished these modes, as partial, into one referring to possibility and another referring to actuality.
Deinde cum dicit: et corruptibile autem etc., distinguit modos corruptibilis et incorruptibilis: 243. Then at [176] he distinguishes the senses of "destructible" and "indestructible":

et primo corruptibilis;

secundo incorruptibilis, ibi: et de incorruptibili et cetera.

First, "destructible";

Secondly, "indestructible," at 245.

Dicit ergo primo quod corruptibile et incorruptibile similiter dicuntur multipliciter: et ponit tres modos corruptibilis. Ubi considerandum est quod, sicut generatio importat incoeptionem cum determinato modo, ita corruptio importat desitionem cum determinato modo, scilicet transmutationis. Primus ergo modus corruptionis ponit desitionem in communi, absque distinctione potentiae et actus. Et est eadem ratio ordinis quae est supra de genito: sicut enim non dicitur aliquid perfecte genitum ex hoc quod incipit esse, ita non dicitur aliquid perfecte corruptum ex hoc quod desinit esse, nec perfecte corruptibile ex hoc quod potest desinere esse. He says therefore first [176] that "destructible" and "indestructible" are also said in many senses, and he presents three senses of "destructible." Now it should be noted that, just as generation implies inception in a definite way, so, too, destruction implies extinction in a definite way, namely, through transmutation. Consequently, the first sense of "destruction" is that of extinction in common without any distinction between possibility and actuality. And the reason for this order is the same as that used above for the word "generated": just as a thing is not said perfectly to be "generated" just because it begins to be, so a thing is not said perfectly to be "destroyed" just because it ceases to be, nor "destructible" just because it can cease to be.
Est ergo primus modus, secundum quem dicimus aliquid esse corruptibile, quod, cum prius sit aliquid, posterius vel non est vel contingit non esse; sive hoc contingat per corruptionem et transmutationem, sicut homo est corruptibilis; sive non per corruptionem et transmutationem desinat esse, sicut tactus et motus. Therefore the first sense in which we describe something as "destructible" is when it previously existed but later it either is not, or is able not to be, whether this is due to perishing and transmutation, as a man is perishable, or not through perishing and transmutation, as contact and motion cease to be.
Secundo modo dicimus aliquid esse corruptibile, quod contingit non esse, idest quandoque potest desinere esse, per specialem modum corruptionis. In the second sense we describe something as "destructible," if it can not be, i.e., able at some time to cease to be, on account of a specific way of ceasing to be.
Tertio modo dicitur aliquid corruptibile, quod de facili corrumpitur: quod potest dici euphtharton, idest bene corruptibile. In the third sense, something is said to be "destructible," because it is easily destroyed, and can be called "euphtharton," i.e., well destructible.
Est autem considerandum quod, licet modi corruptibilis cum modis geniti conveniant quantum ad ordinem, quia sicut ibi praemittitur generalis incoeptio, ita hic praemittitur generalis desitio; est tamen differentia quantum ad distinctionem. Nam ibi distinguebantur modi secundum actum et potentiam: hic autem distinguuntur modi secundum potentiam absolutam, et perfectam; quod est ultimus modus, tanquam perfectissimus; perfectissime enim corruptibile est quod de facili potest corrumpi. Et huius ratio est, quia genitum dicitur secundum actum, corruptibile autem dicitur secundum potentiam: unde genitum potest intelligi secundum actum et secundum potentiam, sed corruptibile non potest intelligi nisi secundum potentiam. 244. It should be observed that although the senses of "destructible" agree with those of "generated" as far as the order is concerned for just as in the latter there is placed first inception in general, so here there is placed first destruction in general, there is a difference in the way their modes are distinguished. For there the modes were distinguished according to possibility and actuality, but here they are distinguished according to absolute, and perfect, possibility, which latter is the last, as the most perfect mode — for the most perfectly destructible is what is easily destroyed. The reason for this is that "generated" is said according to act, while "destructible" according to potency. Hence "generated" can refer to both actuality and possibility, but "destructible" to possibility only.
Ideo autem posuit genitum secundum actum, et corruptibile secundum potentiam, quia cum generatio sit de non esse in esse, corruptio de esse in non esse, illud quod est generabile nondum est ens, sed solum illud quod iam est genitum: e converso autem id quod est corruptibile est ens, non autem id quod iam est corruptum. Intendit autem philosophus facere quaestionem de entibus, non autem de non entibus: et ideo utitur nomine geniti et corruptibilis. The reason he set down "generated," which is according to act, and "corruptible," which is according to potency, is this: Since generation is from non-being to being, and corruption from being to non-being, that which is "generable" is not yet a being, but only that which has been "generated" is; on the other hand, that which is "corruptible" is a being, but that which has been "corrupted" is no longer a being. Now the intention of the Philosopher is to discuss a question, not of non-beings but of beings. And that is why he employs the words "generated" and "corruptible."
Deinde cum dicit: et de incorruptibili etc., distinguit modos incorruptibilis. Et dicit quod de incorruptibili etiam est eadem distinctionis ratio. Ponit enim tres modos. Quorum primus est secundum negationem determinati modi desitionis; secundum scilicet quod incorruptibile dicitur, quod quidem potest desinere sic quod quandoque sit ens et postmodum non ens, sed hoc sine corruptione; sicut tactus et motus, qui cum primo sint, posterius non sunt, sed hoc est sine corruptione eorum, quia eorum non est corruptio, sicut nec generatio. Unde hic modus respondet primo modo ingeniti. 245. Then at [177] he distinguishes the senses of "indestructible." And he presents three senses. The first of these denies a definite kind of extinguishing process, insofar as that is said to be "indestructible" which can cease to be in such a way that at one time it exists and later does not, but this without corruption. Examples are contact and motion which, after first existing do not later, but this is without their corruption, since things are subject neither to generation nor corruption. Consequently, this sense corresponds to the first sense of "ungenerated."
Secundo modo dicitur aliquid incorruptibile secundum negationem desitionis in communi: et sic dicit quod illud quod nunc est ens, et est impossibile quod postea non sit, vel quandoque non sit futurum, dicitur incorruptibile. Et hic modus incorruptibilitatis non competit alicui rei quae possit desinere esse per corruptionem: tu enim qui potes desinere esse per corruptionem, es nunc in praesenti; et similiter tactus, qui potest desinere esse, sed non per corruptionem, est nunc; sed tamen utrumque horum dicitur aliquo modo corruptibile, scilicet secundum primum modum corruptibilis; quia scilicet erit aliquando quando non erit verum dicere quod tu sis, nec erit verum dicere quod hoc tangatur. Et ideo illud maxime proprie dicitur incorruptibile, quod quidem est ens, sed impossibile est illud corrumpi hoc modo ut, cum modo sit ens, posterius non sit ens aut contingat non esse, et quamvis nondum sit corruptum, tamen contingat postremo illud non esse: illud enim quod non hoc modo se habet, dicitur proprie incorruptibile. In a second sense something is called "indestructible," when extinction in common is denied. And he says that "indestructible" in this sense refers to what is now a being and it is impossible for it later not to be a being, or not to be in the future. This kind of indestructibility does not belong to any thing that can cease to be through corruption. For you, who can cease to be through corruption, exist now, and so does contact, which can cease to be, but not by corruption — yet both of these are called "corruptible" in a certain way, since a time will come when it will not be true to say that you exist, or that this is in contact. And, therefore, that is most properly called "indestructible" which is, indeed, a being but cannot be destroyed in such a way that, while it is a being now, later it will not be, or is able not to be, and although not yet destroyed can, nevertheless, eventually become non-existent. What is not so constituted is properly called "indestructible."
Tertio modo dicitur aliquid incorruptibile, quod non de facili corrumpitur. Quod etiam respondet tertio modo corruptibilis, sicut et secundus secundo, et primus primo. In a third sense, something is called "indestructible" which is not destroyed easily. And this corresponds to the third sense of "destructible," just as the second corresponds to the second, and the first to the first.

Lecture 25:
How something is said to be "possible" and "impossible".
Chapter 11 cont.
Εἰ δὴ ταῦθ' οὕτως ἔχει, σκεπτέον πῶς λέγομεν τὸ δυνατὸν καὶ ἀδύνατον 178 This being so, we must ask what we mean by 'possible' and 'impossible'.
τό τε γὰρ κυριώτατα λεγόμενον ἄφθαρτον τῷ μὴ δύνασθαι ἂν φθαρῆναι, μηδ' ὁτὲ μὲν εἶναι ὁτὲ δὲ μή λέγεται δὲ καὶ τὸ ἀγένητον τὸ ἀδύνατον καὶ μὴ δυνάμενον γενέσθαι οὕτως ὥστε πρότερον μὲν μὴ εἶναι ὕστερον δὲ εἶναι, οἷον τὴν διάμετρον σύμμετρον. 179 For in its most proper use the predicate 'indestructible' is given because it is impossible that the thing should be destroyed, i.e. exist at one time and not at another. And 'ungenerated' also involves impossibility when used for that which cannot be generated, in such fashion that, while formerly it was not, later it is. An instance is a commensurable diagonal.
Εἰ δή τι δύναται κινηθῆναι [στάδια ἑκατὸν] ἢ ἆραι βάρος, ἀεὶ πρὸς τὸ πλεῖστον λέγομεν, οἷον τάλαντα ἆραι ἑκατὸν ἢ στάδια βαδίσαι ἑκατόν (καίτοι καὶ τὰ μόρια δύναται τὰ ἐντός, εἴπερ καὶ τὴν ὑπεροχήν), ὡς δέον ὁρίζεσθαι πρὸς τὸ τέλος καὶ τὴν ὑπεροχὴν τὴν δύναμιν. Ἀνάγκη μὲν οὖν τὸ δυνατὸν καθ' ὑπεροχὴν τοσαδὶ καὶ τὰ ἐντὸς δύνασθαι, οἷον εἰ τάλαντα ἑκατὸν ἆραι, καὶ δύο, κἂν εἰ στάδια ἑκατόν, καὶ δύο δύνασθαι βαδίσαι. 180 Now when we speak of a power to move or to lift weights, we refer always to the maximum. We speak, for instance, of a power to lift a hundred talents or walk a hundred stades—though a power to effect the maximum is also a power to effect any part of the maximum—since we feel obliged in defining the power to give the limit or maximum. A thing, then, which is within it. If, for example, a man can lift a hundred talents, he can also lift two, and if he can walk a hundred stades, he can also walk two. But the power is of the maximum,
Ἡ δὲ δύναμις τῆς ὑπεροχῆς ἐστίν κἂν εἴ τι ἀδύνατον τοσονδὶ καθ' ὑπερβολὴν εἰπόντων, καὶ τὰ πλείω ἀδύνατον, οἷον ὁ χίλια βαδίσαι στάδια μὴ δυνάμενος δῆλον ὅτι καὶ χίλια καὶ ἕν. 181 and a thing said, with reference to its maximum, to be incapable of so much is also incapable of any greater amount. It is, for instance, clear that a person who cannot walk a thousand stades will also be unable to walk a thousand and one.
Μηδὲν δ' ἡμᾶς παρενοχλείτω διωρίσθω γὰρ κατὰ τῆς ὑπεροχῆς τὸ τέλος λεγόμενον τὸ κυρίως δυνατόν. Τάχα γὰρ ἐνσταίη τις ἂν ὡς οὐκ ἀνάγκη τὸ λεχθέν ὁ γὰρ ὁρῶν στάδιον οὐ καὶ τὰ ἐντὸς ὄψεται μεγέθη, ἀλλὰ τοὐναντίον μᾶλλον ὁ δυνάμενος ἰδεῖν στιγμὴν ἢ ἀκοῦσαι μικροῦ ψόφου καὶ τῶν μειζόνων ἕξει αἴσθησιν. 182 This point need not trouble us, for we may take it as settled that what is, in the strict sense, possible is determined by a limiting maximum. Now perhaps the objection might be raised that there is no necessity in this, since he who sees a stade need not see the smaller measures contained in it, while, on the contrary, he who can see a dot or hear a small sound will perceive what is greater.
Ἀλλ' οὐδὲν διαφέρει πρὸς τὸν λόγον διωρίσθω γὰρ ἤτοι ἐπὶ τῆς δυνάμεως ἢ ἐπὶ τοῦ πράγματος ἡ ὑπερβολή. Τὸ γὰρ λεγόμενον δῆλον ἡ μὲν γὰρ ὄψις ἡ τοῦ ἐλάττονος ὑπερέχει, ἡ δὲ ταχυτὴς ἡ τοῦ πλείονος. 183 This, however, does not touch our argument. The maximum may be determined either in the power or in its object. The application of this is plain. Superior sight is sight of the smaller body, but superior speed is that of the greater body.
Postquam philosophus ostendit quot modis dicitur genitum et ingenitum, corruptibile et incorruptibile, hic exponit significationem huius quod dicitur possibile et impossibile. 246. After pointing out the various senses of "generated" and "ungenerated," "destructible" and "indestructible," the Philosopher here explains the meaning of what is called "possible" and "impossible."

Et primo dicit de quo est intentio;

secundo exequitur propositum, ibi: si itaque aliquid potest et cetera.

First he reveals his intention;

Secondly, he executes his plan, at 248.

Circa primum duo facit. About the first he does two things:
Primo dicit de quo est intentio: et dicit quod, cum ita se habeant ea quae dicta sunt circa significationem geniti et ingeniti, corruptibilis et incorruptibilis, oportet considerare quomodo dicatur aliquid possibile et impossibile. First he states what his intention is concerned with [178], and says that since the situation is thus with regard to the meanings of "generated" and "ungenerated," "destructible" and "indestructible," it is necessary to consider how something is said to be "possible" and "impossible."
Secundo ibi: propriissime enim etc., assignat rationem suae intentionis, quia scilicet possibile et impossibile includuntur in ratione praedictorum. Quia, ut supra dictum est, propriissime dicitur aliquid esse incorruptibile, quod non solum non potest corrumpi, sed nec etiam quocumque modo aliquando esse et postea non esse. Et similiter ingenitum proprie dicitur quod est impossibile, scilicet esse et non esse, et quod non potest fieri quocumque tali modo quod prius non sit et postea sit; sicut diametrum quadrati esse symmetrum, idest commensuratum lateri quadrati, est ingenitum, quia nullo modo potest incipere esse. 247. Secondly, at [179] he gives the reason for his intention, namely, that "possible" and "impossible" are included in the definition of the aforesaid. For, as has been said above, that is most appropriately called "indestructible" which not only cannot be destroyed but can in no way exist at one time and later not exist. Similarly, "ungenerated" is appropriately applied to what is impossible, namely, to be and not to be, and which cannot come to be in any way such that previously it does not exist and later does exist. Thus, for the diagonal of a square to be symmetric, i.e., commensurate, to the side, is ungenerated, because it never can begin to be.
Deinde cum dicit: si itaque aliquid potest etc., ostendit quomodo aliquid dicatur possibile et impossibile. Et est notandum quod, sicut dicit philosophus in V Metaphys., possibile et impossibile uno modo dicitur absolute, quia scilicet secundum se est tale quod possit esse verum vel non possit esse verum, propter habitudinem terminorum ad invicem; alio modo dicitur possibile et impossibile alicui, quod scilicet potest vel secundum potentiam activam vel passivam. Et sic accipitur hic possibile et impossibile, scilicet quod aliquod agens aut patiens potest aut non potest: haec enim significatio maxime congruit rebus naturalibus. 248. Then at [180] he shows how something is described as possible and impossible. And it should be noted that, as the Philosopher says in Metaphysics V, possible and impossible are said in one way absolutely, namely, because in themselves they can be true or cannot be true by reason of the relationship existing between the terms; in another way a thing is said to be possible or impossible to something, namely, what it is able for with respect to its active or passive power. And it is in this sense that "possible" and "impossible" are taken here, namely, as what is, or is not, within the power of an agent or patient — for this is the meaning that is most appropriate to natural things.

Primo ergo ostendit quomodo dicatur aliquid esse possibile vel impossibile;

secundo excludit obiectionem, ibi: nihil autem nos turbet et cetera.

First, then, he shows how something is said to be "possible" or "impossible";

Secondly, he excludes an objection, at 251.

Circa primum duo facit: With respect to the first he does two things:

primo manifestat quomodo dicatur aliquid esse possibile;

secundo ostendit quomodo dicatur aliquid esse impossibile, ibi: et utique si quid et cetera.

First he manifests how some thing is said to be "possible";

Secondly, how something is called "impossible," at 250.

Ad primi autem manifestationem dicit quod, si contingat aliquam rem posse in aliquid magnum, puta quod aliquis homo ambulet per centum stadia, aut possit levare aliquod magnum pondus, semper determinamus sive denominamus eius potentiam per respectum ad plurimum in quod potest; sicut dicimus potentiam huius hominis esse quod potest levare pondus centum talentorum, aut quod potest ire per spatium centum stadiorum, quamvis possit omnes partes infra istam quantitatem contentas, siquidem potest in id quod superabundat. Nec tamen denominatur ab illis partibus, puta quod determinetur eius potentia quia potest ferre quinquaginta talenta, aut ire quinquaginta stadia; sed per id quod est maximum: ita scilicet ut potentia uniuscuiusque denominetur per respectum ad finem, idest per ultimum et per maximum ad quod potest, et per virtutem suae excellentiae; sicut etiam et magnitudo cuiuslibet rei determinatur per id quod est maximum, sicut quantitatem tricubiti notificantes, non dicimus quod sit bicubitum. Et similiter rationem hominis assignamus per rationale, et non per sensibile: quia semper id quod est ultimum et maximum, est completivum et dans speciem rei. 249. To explain the first [180] he says that if a thing is capable of something great, for example, if a man can walk 100 stades or can lift a great weight, we always determine or describe his power in terms of the most he can do. For example, we say that the power of this man is that he can lift a weight of 100 talents or can walk a distance of 100 stades, even though he is capable of all the partial distances included in that quantity, since he can do what exceeds. But his power is not described by these parts — we do not determine his power as being able to carry 50 talents or walk 50 stades, but by the most he can do. Consequently, the power of each thing is described with respect to the end, i.e., with respect to the ultimate, and to the maximum of which it is capable, and with respect to the strength of its excellence. Thus, too, the size of a thing is determined by what is greatest —for example, in describing the size of something that is three cubits, we do not say that it is two cubits. Similarly, we assign as the notion of man that he is rational, not that he is sensible, because what is the ultimate and greatest in a thing is what completes it and puts upon it the stamp of its species.
Sic igitur patet quod ille qui potest in ea quae excellunt, necesse est quod possit etiam in ea quae sunt infra; puta si aliquis potest portare centum talenta, poterit etiam portare duo, et si potest ire per centum stadia, potest ire per duo: sed tamen virtus rei non attribuitur nisi excellentiae, idest, secundum id attenditur virtus rei, quod est excellentissimum omnium eorum in quae potest. Consequently, it is plain that one who can do what exceeds, necessarily can do what is less. For example, if a person can carry 100 talents, he can also carry two, and if he can walk 100 stades, he can also walk two; yet it is to what is excelling that the virtue of a thing is attributed, i.e., the virtue of a thing is gauged in terms of what is most excellent of all the things that can be done.
Et hoc est quod dicitur in alia translatione, virtus est ultimum potentiae, quia scilicet virtus rei determinatur secundum ultimum in quod potest. Et hoc etiam habet locum in virtutibus animae: dicitur enim virtus humana, per quam homo potest in id quod est excellentissimum in operibus humanis, scilicet in opere quod est secundum rationem. This is what is said in another translation, "the virtue is the limit of a power," because, namely, the virtue of a thing is determined according to the ultimate it can do. And this applies also to the virtues of the soul: for a human virtue is that through which a man is capable of what is most excellent in human actions, i.e., in an action which is in accordance with reason.
Deinde cum dicit: et utique si quid etc., ostendit quomodo dicatur aliquid alicui esse impossibile. Et dicit quod, si aliquod tantum est impossibile alicui, si aliquis accipiat ea quae excellunt, manifestum est quod impossibile erit ei portare vel facere plura; sicut ille qui non potest ire per mille stadia, manifestum est quod non potest ire per mille et unum. Unde patet quod, sicut determinatur id quod est possibile alicui per maximum in quod potest, in quo attenditur virtus eius; ita id quod est impossibile alicui determinatur per minimum eorum in quae non potest, in quo consistit eius debilitas. Puta si maximum in quod potest aliquis, est ire viginti stadia, minimum eorum in quae non potest, est viginti et unum; et ab hoc oportet determinare eius debilitatem, non autem ex eo quod non potest ire per centum vel per mille. 250. Then at [181] he tells how something is said to be "impossible" to a thing. And he says that if some amount is impossible to someone if one takes what excels, it is plain that it will be impossible for him to carry or do more. For example, a person who cannot walk 100 stades clearly cannot walk 101. Hence, it is plain that just as the possibility is determined by the greatest that a thing can do — which determines its virtue — so what is impossible is determined by the least that it cannot do, and this determines its weakness. For example, if the most that someone can do is to go 20 stades, the least that he cannot go is 21 — and it is from this that his weakness is to be determined, and not from his inability to walk 100 or 1,000 stades.
Deinde cum dicit: nihil autem nos turbet etc., excludit quandam obiectionem. 251. Then at [182] he excludes an objection.

Et primo movet eam;

secundo solvit, ibi: sed nihil differt et cetera.

First he proposes it;

Secondly, he solves it, at 252.

Dicit ergo primo quod nihil debet nos turbare, quin id quod proprie dicitur possibile, sit determinandum secundum terminum excellentiae. Potest enim aliquis instare, quasi non sit necessarium in omnibus id quod dictum est: videtur enim habere instantiam in visu et in aliis sensibus. Ille enim qui videt aliquam magnam quantitatem, puta unius stadii, non potest propter hoc videre magnitudines minoris quantitatis, quae infra illam quantitatem continentur: sed magis accidit contrarium, quia ille qui potest videre punctum, idest aliquod minimum sensu perceptibile, aut etiam qui potest audire parvum sonum, potest et maiora sentire. He says therefore first [182] that nothing should disturb us in connection with the fact that what is properly called "possible" is determined according to the limit of excellence. Someone could, indeed, object that what has been said is not necessary in all matters — for there seems to be an objection in the case of sight and other senses. For a person who sees some large quantity, for example, something a stadium long, cannot for that reason see magnitudes of smaller size contained below that quantity. Rather it is more the opposite that occurs — for one who can see a "point," i.e., some smallest thing perceptible to sense, or hear a faint sound, can sense what is greater.
Deinde cum dicit: sed nihil differt etc., solvit praedictam obiectionem. Et dicit quod hoc quod dictum est, nihil differt ad rationem qua determinabatur quod possibile determinatur secundum excellentiam: quia huiusmodi excellentia, secundum quam attenditur virtus rei, potest determinari vel secundum virtutem vel secundum rem. Secundum rem quidem, quando in ipsa re est excellentia, sicut dictum est de centum stadiis vel centum talentis: et secundum hanc excellentiam oportet determinari virtutem activam; quia quod potest agere in rem maiorem, potest etiam in rem minorem. Secundum virtutem autem attenditur excellentia, quando aliquid quod non excellit in quantitate, requirit excellentiam virtutis: et hoc maxime videtur accidere circa potentias passivas; quanto enim aliquid est passibilius, tanto a minori potest moveri. Et quia sensus sunt potentiae passivae, ideo in sensibilibus accidit ut qui potest sentire minus, potest sentire maius. 252. Then at [183] he answers this objection and says that what was stated does not affect the argument whereby it is shown that the possible is determined by excellence. For the excellence according to which the virtue of a thing is measured can be determined according to the virtue, or according to the thing. According to the thing, when there is an excellence right in it, as was said about the 100 stades or the 100 talents. According to this excellence the active virtue is determined, because whatever can act on a greater thing can act on a lesser. But with respect to virtue, there is excellence when something which is not outstanding in quantity requires an excellence of virtue. And this is seen to occur especially with passive powers — for the more a thing is passible, the more it can be moved by what is less. And since the senses are passive powers, it happens in sensible things that one who can sense what is less can sense what is greater.
Illud autem quod dictum est, hoc modo manifestat: quia visus qui est sensitivus minoris corporis, excedit in virtute, et sic attenditur hic excellentia in virtute, non in re; sed velocitas est excellentior quae est maioris magnitudinis (illud enim est velocius, quod in eodem tempore per maius spatium movetur), et talis excellentia non solum est in virtute, sed etiam in re. What he has just said he explains as follows: a vision capable of perceiving a smaller body excels in virtue; hence the excellence in this case is an excellence in the virtue and not in the thing. But that speed is more excellent which is of a greater magnitude — for that is speedier which can traverse a greater distance in the same time — and such excellence is an excellence not only in the virtue, but also in the thing.

Lecture 26:
Everything eternal is indestructible and ungenerated
Chapter 12
Διωρισμένων δὲ τούτων λεκτέον τὸ ἐφεξῆς. Εἰ δή ἐστιν ἔνια δυνατὰ καὶ εἶναι καὶ μή, ἀνάγκη χρόνον τινὰ ὡρίσθαι τὸν πλεῖστον καὶ τοῦ εἶναι καὶ τοῦ μή, λέγω δ' ὃν δυνατὸν τὸ πρᾶγμα εἶναι καὶ ὃν δυνατὸν μὴ εἶναι καθ' ὁποιανοῦν κατηγορίαν, οἷον ἄνθρωπον ἢ λευκὸν ἢ τρίπηχυ ἢ ἄλλ' ὁτιοῦν τῶν τοιούτων. Εἰ γὰρ μὴ ἔσται ποσός τις, ἀλλ' ἀεὶ πλείων τοῦ προτεθέντος καὶ οὐκ ἔστιν οὗ ἐλάττων, ἄπειρον (281b.) ἔσται χρόνον δυνατὸν εἶναι, καὶ μὴ εἶναι ἄλλον ἄπειρον ἀλλὰ τοῦτ' ἀδύνατον. 184 Having established these distinctions we car now proceed to the sequel. If there are thing! capable both of being and of not being, there must be some definite maximum time of their being and not being; a time, I mean, during which continued existence is possible to them and a time during which continued nonexistence is possible. And this is true in every category, whether the thing is, for example, 'man', or 'white', or 'three cubits long', or whatever it may be. For if the time is not definite in quantity, but longer than any that can be suggested and shorter than none, then it will be possible for one and the same thing to exist for infinite time and not to exist for another infinity. This, however, is impossible.
Ἀρχὴ δ' ἔστω ἐντεῦθεν τὸ γὰρ ἀδύνατον καὶ τὸ ψεῦδος οὐ ταὐτὸ σημαίνει. Ἔστι δὲ τὸ ἀδύνατον καὶ δυνατὸν καὶ ψεῦδος καὶ ἀληθὲς τὸ μὲν ἐξ ὑποθέσεως (λέγω δ', οἷον τὸ τρίγωνον ἀδύνατον δύο ὀρθὰς ἔχειν, εἰ τάδε, καὶ ἡ διάμετρος σύμμετρος). Ἔστι δ' ἁπλῶς καὶ δυνατὰ καὶ ἀδύνατα καὶ ψευδῆ καὶ ἀληθῆ. 185 Let us take our start from this point. The impossible and the false have not the same significance. One use of 'impossible' and 'possible', and 'false' and 'true', is hypothetical. It is impossible, for instance, on a certain hypothesis that the triangle should have its angles equal to two right angles, and on another the diagonal is commensurable. But there are also things possible and impossible, false and true, absolutely.
Οὐ δὴ ταὐτό ἐστι ψεῦδός τέ τι εἶναι ἁπλῶς καὶ ἀδύνατον ἁπλῶς. Τὸ γάρ σε μὴ ἑστῶτα φάναι ἑστάναι ψεῦδος μέν, οὐκ ἀδύνατον δέ. Ὁμοίως δὲ τὸν κιθαρίζοντα, μὴ ᾄδοντα δέ, ᾄδειν φάναι ψεῦδος, ἀλλ' οὐκ ἀδύνατον. Τὸ δ' ἅμα ἑστάναι καὶ καθῆσθαι, καὶ τὴν διάμετρον σύμμετρον εἶναι, οὐ μόνον ψεῦδος, ἀλλὰ καὶ ἀδύνατον. 186 Now it is one thing to be absolutely false, and another thing to be absolutely impossible. To say that you are standing when you are not standing is to assert a falsehood, but not an impossibility. Similarly to say that a man who is playing the harp, but not singing, is singing, is to say what is false but not impossible. To say, however, that you are at once standing and sitting, or that the diagonal is commensurable, is to say what is not only false but also impossible.
Οὐ δὴ ταὐτόν ἐστιν ὑποθέσθαι ψεῦδος καὶ ἀδύνατον. Συμβαίνει δ' ἀδύνατον ἐξ ἀδυνάτου. 187 Thus it is not the same thing to make a false and to make an impossible hypothesis, and from the impossible hypothesis impossible results follow.
Τοῦ μὲν οὖν καθῆσθαι καὶ ἑστάναι ἅμα ἔχει τὴν δύναμιν, ὅτι ὅτε ἔχει ἐκείνην, καὶ τὴν ἑτέραν ἀλλ' οὐχ ὥστε ἅμα καθῆσθαι καὶ ἑστάναι, ἀλλ' ἐν ἄλλῳ χρόνῳ. 188 A man has, it is true, the capacity at once of sitting and of standing, because when he possesses the one he also possesses the other; but it does not follow that he can at once sit and stand, only that at another time he can do the other also.
Εἰ δέ τι ἄπειρον χρόνον ἔχει πλειόνων δύναμιν, οὐκ ἔστιν ἐν ἄλλῳ χρόνῳ, ἀλλὰ τοῦθ' ἅμα. Ὥστ' εἴ τι ἄπειρον χρόνον ὂν φθαρτόν ἐστι, δύναμιν ἔχοι ἂν τοῦ μὴ εἶναι. Εἰ δὴ ἄπειρον χρόνον, ἔστω ὑπάρχον ὃ δύναται. Ἅμα ἄρ' ἔσται τε καὶ οὐκ ἔσται κατ' ἐνέργειαν. Ψεῦδος μὲν οὖν συμβαίνοι ἄν, ὅτι ψεῦδος ἐτέθη. Ἀλλ' εἰ μὴ ἀδύνατον ἦν, οὐκ ἂν καὶ ἀδύνατον ἦν τὸ συμβαῖνον. Ἅπαν ἄρα τὸ ἀεὶ ὂν ἁπλῶς ἄφθαρτον. 189 But if a thing has for infinite time more than one capacity, another time is impossible and the times must coincide. Thus if a thing which exists for infinite time is destructible, it will have the capacity of not being. Now if it exists for infinite time let this capacity be actualized; and it will be in actuality at once existent and non-existent. Thus a false conclusion would follow because a false assumption was made, but if what was assumed had not been impossible its consequence would not have been impossible. Anything then which always exists is absolutely imperishable.
Ὁμοίως δὲ καὶ ἀγένητον εἰ γὰρ γενητόν, ἔσται δυνατὸν χρόνον τινὰ μὴ εἶναι—φθαρτὸν μὲν γάρ ἐστι τὸ πρότερον μὲν ὄν, νῦν δὲ μὴ ὂν ἢ ἐνδεχόμενόν ποτε ὕστερον μὴ εἶναι γενητὸν δὲ ὃ ἐνδέχεται πρότερον μὴ εἶναι—ἀλλ' οὐκ ἔστιν ἐν ᾧ χρόνῳ δυνατὸν τὸ ἀεὶ ὂν ὥστε μὴ εἶναι, οὔτ' ἄπειρον οὔτε πεπερασμένον καὶ γὰρ τὸν πεπερασμένον χρόνον δύναται εἶναι, εἴπερ καὶ τὸν ἄπειρον. Οὐκ ἄρα ἐνδέχεται τὸ αὐτὸ καὶ ἓν ἀεί τε δύνασθαι εἶναι καὶ ἀεὶ μὴ εἶναι. Ἀλλὰ μὴν οὐδὲ τὴν ἀπόφασιν, οἷον λέγω μὴ ἀεὶ εἶναι. Ἀδύνατον ἄρα καὶ ἀεὶ μέν τι εἶναι, φθαρτὸν (282a.) δ' εἶναι. Ὁμοίως δ' οὐδὲ γενητόν δυοῖν γὰρ ὅροιν εἰ ἀδύνατον τὸ ὕστερον ἄνευ τοῦ προτέρου ὑπάρξαι, ἐκεῖνο δ' ἀδύνατον ὑπάρχειν, καὶ τὸ ὕστερον. Ὥστ' εἰ τὸ ἀεὶ ὂν μὴ ἐνδέχεταί ποτε μὴ εἶναι, ἀδύνατον καὶ γενητὸν εἶναι. 190 It is also ungenerated, since if it was generated it will have the power for some time of not being. For as that which formerly was, but now is not, or is capable at some future time of not being, is destructible, so that which is capable of formerly not having been is generated. But in the case of that which always is, there is no time for such a capacity of not being, whether the supposed time is finite or infinite; for its capacity of being must include the finite time since it covers infinite time. It is therefore impossible that one and the same thing should be capable of always existing and of always not-existing. And 'not always existing', the contradictory, is also excluded. Thus it is impossible for a thing always to exist and yet to be destructible. Nor, similarly, can it be generated. For of two attributes if B cannot be present without A, the impossibility A of proves the impossibility of B. What always is, then, since it is incapable of ever not being, cannot possibly be generated.
Postquam philosophus exposuit significationem nominum quae in quaestione proponuntur, hic incipit argumentari ad quaestionem propositam, utrum scilicet aliquid possit esse genitum et incorruptibile, vel ingenitum et corruptibile. 253. After explaining the meanings of the words proposed in the question, the Philosopher here begins to argue on the question proposed, namely, whether something can be generated and indestructible, or ungenerated and destructible.

Et primo ostendit hoc esse impossibile per rationes communes;

secundo per rationem propriam scientiae naturalis, ibi: et naturaliter et cetera.

First he shows with general arguments that this is impossible;

Secondly, with arguments proper to natural science, at 286 (L. 29).

Circa primum duo facit: About the first he does two things:

primo ostendit quid sequitur ex praemissis circa propositum;

secundo incipit argumentari ad propositum ostendendum, ibi: principium autem sit hinc et cetera.

First he shows what follows from the preceding with respect to the present question;

Secondly, he begins to argue to his proposition, at 255.

Dicit ergo primo quod, determinatis praemissis circa significationem nominum, oportet nunc dicere illud quod consequenter se habet in hac consideratione. Dictum est enim supra quod possibile dicitur secundum aliquod determinatum, puta potens currere dicitur aliquis secundum centum stadia. Sunt autem in rebus quaedam quae possunt esse et non esse. Necesse est ergo ex praemissis quod sit determinatum aliquod plurimum tempus et respectu ipsius esse, ita scilicet quod non possit ampliori tempore esse, et respectu ipsius non esse, ita scilicet quod non possit ampliori tempore non esse. 254. He says therefore first [184] that, the preceding having been determined as to the meanings of certain words, it is now time to state what follows in this treatment. For it has been said above that the "possible" is described in terms of something definite — for example, someone's power to run is described in terms of 100 stades. But there exist in external reality some things that can both exist and not exist. Therefore it is necessary according to the foregoing that there be determined some maximum time both affecting existence, such that it is not possible to exist for a greater time, and affecting non-existence, such that it is not possible not to exist for a greater time.
Et ne hoc intelligatur solum de esse substantiali, subiungit quod, cum dicimus possibile vel non possibile rem esse, vel id quod est possibile non esse, potest intelligi secundum quamcumque praedicationem, idest secundum quodcumque praedicamentum: puta hominem esse vel non esse, quod pertinet ad genus substantiae; aut album esse aut non esse, quod pertinet ad genus qualitatis; aut bicubitum esse vel non esse, quod pertinet ad genus quantitatis; aut de quocumque alio consimili. And lest this be understood as applying only to substantial existence, he adds that, when we say that it is possible or not possible for a thing to exist, or that which is able not to exist, such expressions can be understood with regard to any predication, i.e., with regard to any predicament: for example, that a man exist or not exist, which pertains to the genus of substance; or that white exist or not exist, which pertains to the genus of quality; or that "two cubits" exist or not exist, which pertains to the genus of quantity; or any other similar thing.
Et quod oporteat intelligi secundum aliquod determinatum tempus, cum dicitur aliquid posse esse vel non esse, probat ducendo ad impossibile. Quia, sicut ipse dicit, si non est aliquod tempus determinatae quantitatis, in quo possit esse vel non esse, sed semper accipiatur maius tempore proposito (puta si potest esse in quinquaginta annis, et adhuc plus, et iterum plus), et non sit devenire ad aliquod tempus respectu cuius omne tempus in quo potest esse sit minus; cum idem possit esse et non esse, ut dictum est, sequitur quod idem possit esse in tempore infinito, et non esse in tempore infinito; quia eadem ratio est circa hoc quod est non esse, et circa hoc quod est esse. That when something is said to be able to be or not be, that expression must be understood in terms of some determinate time, he now proves by leading to an impossibility. For, as he says, if there is not a time of determinate quantity in which it could be or not be, but a time greater than a given time is always assumed (for example, if it can be for fifty years and then more and again still more), and no limit is reached with respect to which every time in which it can be is less, then, since it is the same thing that can be and not be, as was said, it follows that the same thing can be for an infinite time, and not be for an infinite time, because the same reasoning applies to the non-existence as applies to the existence.
Non tamen ita quod illud tempus respectu cuius aliquid potest non esse, quod concluditur esse infinitum, sit idem cum illo tempore infinito respectu cuius aliquid dicitur posse esse; quia sic posset esse et non esse in eodem tempore, quod est impossibile, ut infra dicetur: sed quod aliud tempus infinitum sit eius quod est non esse, et aliud eius quod est esse. Quod est impossibile: non enim possunt esse duo tempora infinita, quia sic essent duo tempora simul. Hoc autem impossibile sequitur ex hoc quod dicitur quod possibile esse vel possibile non esse non intelligitur respectu determinati temporis: hoc ergo oportet primo esse manifestum, quod possibile esse dicitur respectu determinati temporis, et similiter possibile non esse: quod etiam consonat his quae sunt praemissa de significatione possibilis. This does not mean that the time in respect to which something is able not to be, and which was concluded to be infinite, is the same as the time in respect to which the thing is able to be — because then the same thing would be able to be and not be during the same time, which is impossible, as will be said below. It means rather that there is one infinite time for the thing as non-existing, and another for it as existing. Now this is impossible: for there cannot be two infinite times, because then two times would be simultaneous. But this impossibility follows from saying that the possibility to be or the possibility not to be are not reckoned with respect to some determinate time. Therefore the first thing that must be clear is that the possibility of being is said with respect to a determinate time, and similarly the possibility of not being. And this agrees with what has been already laid down about the meaning of "possible."
Deinde cum dicit: principium autem sit hinc etc., incipit argumentari ad propositum. Et circa hoc duo facit: 255. Then at [185] he begins to argue to his proposition. About this he does two things:

primo argumentatur ad propositum per communes rationes;

secundo per propriam rationem scientiae naturalis, ibi: et naturaliter et cetera.

First he argues to it with general reasons;

Secondly, with an argument proper to natural science, at 286 (L. 29).

Circa primum duo facit: About the first he does two things:

primo ostendit veritatem, scilicet quod incorruptibile et ingenitum se consequuntur, et similiter corruptibile et genitum;

secundo improbat positionem contrariam, ibi: dicere itaque nihil et cetera.

First he shows the truth, namely, that indestructible and unproduced follow one upon the other; and likewise, destructible and generated;

Secondly, he disproves the contrary of this (L. 29).

Circa primum duo facit: Regarding the first he does two things:

primo ostendit propositum, ostendendo quomodo se habeat sempiternum ad ingenitum et incorruptibile, et ad genitum et corruptibile;

secundo quomodo ista se habeant ad invicem, ibi: palam autem et ex determinatione et cetera.

First he proves his proposition by showing how what is eternal is related to the ungenerated and indestructible, and to the generated and destructible;

Secondly, how they are related to one another (L. 28).

Circa primum tria facit: About the first he does three things:

primo ostendit quod omne sempiternum est incorruptibile et ingenitum;

secundo ostendit quod nullum sempiternum est genitum vel corruptibile, neque e converso, ibi: quoniam autem negatio etc.;

tertio concludit quod omne ingenitum et incorruptibile est sempiternum, ibi: igitur si et ingenitum et cetera.

First he shows that everything eternal is indestructible and ungenerated;

Secondly, that nothing eternal is generated or destructible, nor conversely (L. 27);

Thirdly, he concludes that everything ungenerated and indestructible is eternal, at 265 (L. 27).

Circa primum duo facit: About the first he does two things:

primo praemittit quaedam necessaria;

secundo argumentatur ad propositum, ibi: si itaque aliquid et cetera.

First he presents some needed pre-notes;

Secondly, he argues to his proposition, at 257.

Dicit ergo primo quod oportet hinc sumere principium ad propositum ostendendum, quod impossibile et falsum non significant idem. 256. He says therefore first [185] that in order to prove the proposition, it is necessary to start from the fact that "impossible" and "false" do not mean the same.
Circa quod quatuor ponit. Quorum primum est quod tam impossibile quam possibile, tam verum quam falsum, dicuntur dupliciter. Uno modo ex suppositione, quod scilicet necesse est esse verum vel falsum, possibile vel impossibile, suppositis quibusdam: sicut triangulum secundum rei veritatem necesse est habere tres angulos aequales duobus rectis, sed tamen hoc est impossibile suppositis quibusdam, puta si supponamus quod triangulus sit quadratum, ad quod sequitur triangulum habere quatuor rectos. Similiter etiam diametrum quadrati sequetur esse commensurabilem lateri, si quaedam supposita sint vera, puta si ponamus quod quadratum diametri sit quadruplum quadrati lateris: sic enim sequetur quod proportio diametri ad latus sit sicut proportio numeralis, quae est ratio commensurabilis. Alio modo dicuntur aliqua simpliciter, scilicet absolute et secundum se possibilia et impossibilia, falsa et vera. Regarding this he posits four reflections. The first is that both "possible" and "impossible," as well as "true" and "false," are used in two ways. In one way, conditionally, i.e., in the sense that a thing must be true or false, possible or impossible, if certain things are assumed; for example, a triangle must in fact have three angles equal to two right angles, but nevertheless such a property is impossible if certain things are assumed — thus, if we should suppose a triangle to be a square, it would follow that a triangle would have four right angles. In like manner, it will follow that the diagonal of a square is commensurate to the side if certain assumptions are true — for example, if we should assume that the square of the diagonal is 4 times the square of the side — for then it will always follow that the ratio of the diagonal to the side is a numerical proportion, which is a commensurable ratio. In a second way things are said to be simpliciter, i.e., absolutely and in themselves, "possible" and "impossible," "false" and "true.".
Secundum ponit ibi: non autem idem et cetera. Et dicit quod non est idem aliquid esse falsum simpliciter, idest absolute, et esse impossibile absolute. Si enim dicam te stare, qui non stas sed sedes, falsum erit quod dicitur, non autem impossibile; et similiter falsum erit et non impossibile, si quis dicat cantare eum qui citharizat sed non cantat; sed quod aliquis simul stet et sedeat, vel quod diameter sit commensurabilis lateri, non solum est falsum, sed et impossibile. The second he gives at [186], and he says that to be false simpliciter, i.e. absolutely, and to be impossible absolutely, are not the same. For if I say that you are standing, whereas you are not standing but sitting, then what is said will be false but not impossible; likewise it will be false and not impossible if I say that the person playing the harp is singing, whereas he is not singing. But for someone to be standing and sitting at the same time, or for the diagonal to be commensurable with the side, is not only false, but impossible as well.
Tertium ponit ibi: non itaque etc.: quod concluditur ex praemissis. Cum enim non idem sit falsum et impossibile, sequitur quod non sit idem supponere falsum et impossibile: nam ex falso non sequitur impossibile, sed ex impossibili sequitur impossibile. The third he gives at [187] and he concludes it from the foregoing. For since the false and the impossible are not the same, it follows that it is not the same thing to assume what is false and to assume what is impossible: from the false there does not follow the impossible, but from the impossible there follows the impossible.
Quartum ponit ibi: hoc quidem igitur et cetera. Et quia dictum est quod simul stare et sedere est impossibile, concludit quod, licet aliquid simul habeat virtutem ad opposita (puta ad sedere et stare), tali ratione, quia quandoque una potentia reducitur in actum, quandoque altera; nihil tamen hanc habet potentiam ut simul habeat opposita (puta ut simul sedeat et stet), sed oportet hoc in alio et alio tempore esse. He gives the fourth at [188]. And because it has been said that to stand and sit at the same time is impossible, he concludes that, although something may have at the same time the power to do the opposite things — for example, to sit and to stand — in the sense that now one power is actualized and now the other, nevertheless nothing has the power to have both simultaneously (for example, to stand and sit at the same time) but this must be at different times.
Deinde cum dicit: si itaque aliquid etc., ostendit propositum, scilicet quod omne sempiternum sit incorruptibile et ingenitum. 257. Then at [189] he proves the proposition, namely, that everything eternal is indestructible and ungenerated.

Et primo ostendit quod omne sempiternum sit incorruptibile;

secundo quod omne sempiternum sit ingenitum, ibi: similiter autem et ingenitum et cetera.

First he shows that everything eternal is indestructible;

Secondly, that everything eternal is ungenerated, at 259.

Dicit ergo primo, concludens ex praemissis, in quibus dictum est possibile determinari ad aliquod tempus, quod si aliquid habet virtutem ad plura tempore infinito, non potest dici quod possit aliquid eorum respectu unius temporis, et aliud respectu alterius temporis; sed quidquid potest, potest respectu huius temporis, quia non est aliquod tempus extra tempus infinitum. Si ergo ponamus quod aliquid existens in infinito tempore sit corruptibile, sequitur ex hoc quod est corruptibile, quod habeat virtutem ad hoc quod quandoque non sit; quod quidem oportet intelligi respectu eiusdem temporis infiniti in quo est, vel respectu alicuius partis eius. Quia ergo est in infinito tempore, et tamen ponitur potens non esse, eo quod est corruptibile, sit existens quod potest non esse, idest ponatur non esse ex quo dicis quod potest non esse. Et quia poterat non esse respectu infiniti temporis vel alicuius partis eius, sequitur quod simul secundum actum sit et non sit: quia in infinito tempore ponebatur esse, et postea ponitur non esse respectu eiusdem temporis. He says therefore first [189], as a conclusion from the foregoing (in which it was said that the "possible" is determined for some certain time), that if something is capable of several things during an infinite time, it cannot be said that one of them is possible at one time and another at another time, but whatever it is capable of is possible with respect to this time, because there is no time outside the infinite time. If therefore we should posit something existing in an infinite time to be destructible, from its destructibility it follows that it has the power not to exist at some time; and this must be understood with respect to the same infinite time in which it exists or in respect to some part of that time. Now since it exists in an infinite time and yet is supposed capable of not existing (since it is destructible), then let what is capable of not existing be such, i.e., let it be assumed not to exist, since you say that it is able not to exist. And because it had this capability (of not existing) with respect to infinite time, or some part of it, it follows that it simultaneously actually exists and does not exist —since it is assumed to be existing in infinite time, and later not to exist with respect to the same time.
Manifestum est igitur quod hoc falsum accidit ex falso posito, scilicet ex hoc quod tu ponebas istud existens in infinito tempore non esse quandoque. Sed si hoc falsum non esset impossibile, non sequeretur impossibile; sequitur autem impossibile, scilicet idem simul esse et non esse; ergo impossibile fuit illud non esse. Non ergo poterat non esse; et ita non erat corruptibile. Sic ergo patet quod omne quod est semper ens, non potest esse corruptibile; et ita simpliciter est incorruptibile. It is plain that this falsity occurs on account of the false assumption that the thing existing in infinite time does not exist at some time. But if this falsehood were not something impossible, an impossibility would not follow. However an impossibility does follow, namely, that the same thing exists and does