Si enim tanta gravitas tantam in hoc tempore movet, tanta et adhuc in minori. 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, Et analogiam quam gravitates habent, tempora e converso habebunt: puta si media gravitas in hoc, duplum inmedietate eius. 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. Adhuc, finita gravitas omnem finitam movet in quodam tempore finito. 75 Further, a finite weight traverses any finite distance in a finite time. Necesse igitur ex his, si qua est infinita gravitas, moveri quidem secundum tantum quantum finita et adhuc: non moveri quidem, eo quod proportionaliter oportet secundum excellentias moveri, contrarie autem maior in minori. Proportio autem nulla est infiniti ad finitum, minoris autem temporis ad maius finitum: sed semper in minori, minimum autem non est. 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. Neque si esset, quae utilitas utique esset. Alia enim contra finita sumeretur in eadem proportione in qua infinita ad alteram maiorem. Itaque in aequali tempore aequalem utique moveret infinita finitae. Sed impossibile. 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. Sed adhuc necesse, si quidem in qualicumque tempore finito movet infinita, et aliam in ipso finitam gravitatem movere quandam finitam. Impossibile igitur infinitam esse gravitatem: similiter autem et levitatem. Et corpora ergo infinitam gravitatem habere et levitatem, impossibile. 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. 110. 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. 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. Secundo ibi: contrariae enim lationes etc., ostendit propositum: Secondly, he shows the proposition, at for the upward and downward motions 111, et primo ex parte locorum quae sunt huiusmodi corporibus propria; and this first on the part of the places which are proper to such bodies; secundo ex parte gravitatis et levitatis, per quae huiusmodi corpora in propria loca moventur, ibi: et adhuc si gravitas et cetera. secondly, on the part of heaviness and lightness, through which such bodies are moved to their proper places, at but there is a further point, at but there is a further point 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; 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; secundo quantum ad corpora media, quae sunt aer et aqua, ibi: adhuc si sursum et cetera. secondly, as to the intermediate bodies, which are air and water, at further, if up 112. 111. 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. 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. 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. 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. 112. Deinde cum dicit: adhuc si sursum etc., ostendit idem quantum ad media corpora. 112. Then, at further, if up 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. 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. 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. 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. 113. 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 but the bodies [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. Circa hoc ergo duo facit: Then, at but there is a further point [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. With respect to this, then, he does two things: primo probat conditionalem; first, he proves the conditional proposition, at 114; secundo probat destructionem consequentis, ibi: sed adhuc quoniam infinitam et cetera. secondly, he proves the destruction of the consequent, at but the impossibility of infinite 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 this is proved [66], he proves what he had supposed. et primo ponit probationem; First he presents the proof, at 115; secundo excludit obviationes quasdam, ibi: nihil autem differt gravitates et cetera. secondly, he dismisses some objections, at it does not matter whether the weights 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. 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: Therefore, with these presuppositions, he now argues to three incompatible consequences. 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. 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 again, if the weight of a greater [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 and, further, the weight of unequal masses [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. 116. Deinde cum dicit: nihil autem differt etc., excludit duas obviationes contra praemissam rationem: 116. Then, at it does not matter 14 [69] he dismisses two objections against the foregoing argument: primo primam; first, the first; secundo secundam, ibi: nec utique magnitudinem et cetera. secondly, the second, at nor again does it make any difference 118.