II. To the Principal Arguments

A. To the First Argument

314. To the arguments of the question [nn.285-308].

To the first [n.285] I say it is not unacceptable that every creature, however perfect it be (provided however it not have in essence all perfection), is capable of or has a potential with respect to some perfection, although the perfection is lesser than the creature’s nature - just as an angel has intellection, which is a perfection of his intellective power, and yet intellection is less noble simply than angelic nature; and so can one concede about ‘where’ or corporeal presence with an angel [sc. that ‘where’ or corporeal presence is some perfection or act of an angel], just as the angelic nature is said to be an ‘act’ (though in a far different way) for the angel in whom it is.

B. To the Second Argument

315. As to the second argument [n.286], I deny the assumption it makes, namely that ‘nothing successive is continuous’.

1. Rejection of the First Antecedent

316. The antecedent of the assumption (which antecedent is itself assumed for the proof of the assumption), namely that ‘the successive is composed of indivisibles’, I deny. And I prove the falsity of the antecedent from the Philosopher in Physics 6.2.233b19-32 about sesquialterate proportion [the proportion of one and a half to one] (which is more convincing for the adversary, although perhaps some of Aristotle’s reasons are taken more ‘from the cause’), because he supposes that a motion can be taken quicker than every given motion in any proportion whatever - and consequently, when some motion is given that is measured by three instants [sc. on the assumption that motion is composed of such indivisible instants], one will be able to take a motion twice as quick that will be measured by only an instant and a half [sc. which is impossible, because an instant is indivisible].

317. This point about the successive [sc. that it is not composed of indivisibles, n.316] I prove by the continuity of something persisting; because a persisting thing is continuous, so a successive thing is too.

318. The proof of the consequence is that if there are indivisibles in motion [= a successive thing] which are immediate to each other, I raise a question about the movable [= a persisting thing] and about the ‘wheres’ that the movable has in those immediate instants; if there is nothing in the middle between the ultimate of one ‘where’ and the ultimate of another, then the ultimate of one ‘where’ is immediate with the ultimate of the other ‘where’ [sc. and so the ‘wheres’ are continuous like the movable that persists through them]; but if there is some middle between these two ‘wheres’, I raise a question about the ultimate of the movable when it is in the middle (and not in the second indivisible instant); because when it is in the two indivisibles it is in the ‘wheres’ between which the middle was posited, so when it is in the middle it is in some middle between the two instants; therefore the two instants were not immediate [sc. and so the motion of the movable between these instants is no more made up of instants immediate to each other than the movable itself is]. - And this consequence is made clear by Aristotle in Physics 6 [n.292], namely that “the fact motion and magnitude and time are composed or exist of indivisibles and the fact they are divided into indivisibles mean the same.”

319. The antecedent [sc. ‘a persisting thing is continuous’, n.317] can be proved by Aristotle’s reasons, Physics 6.1.231a21-b18, more manifestly about permanent than successive things, because it is more evident and manifest that permanent indivisibles do not make something larger than that indivisibles succeeding each other do.

320. However the antecedent is more efficaciously proved by two geometrical reasons or propositions, of which the first is as follows:

‘About any center a circle can be drawn, occupying any space’, according to the second postulate of Euclid [Elements 1 postul.3]. So about a give center, which may be called a, let two circles be drawn: a smaller circle, which may be called D, and a larger B. If the circumference of the larger circle is composed of points, let two points immediate to each other be marked, and let them be marked as b and c; and let a straight line be drawn from a to b and a straight line from a to c, according to the postulate of Euclid [Elements 1 postul.1], ‘from a point to a point a straight line may be drawn’.

321. These straight lines, so drawn, will pass straight through the circumference of the smaller circle. I ask then whether they will cut the circumference at the same point or at a different point.

If at a different point, then there are as many points in the smaller circle as in the larger; but it is impossible for two unequal things to be composed of parts equal in size and number; for a point does not exceed a point in size, and the points in the circumference of the smaller circle are as many as the points in the larger circle; so the smaller circumference is equal to the larger, and consequently a part is equal to the whole.

But if the two straight lines ab and ac cut the smaller circumference at the same point (let that point be d), then on the line ab let a straight line be erected cutting it at the point d, and let this line be de, so that this line is also tangent to the smaller circle, from Euclid [Elements 3 prop.17, ‘from a given point draw a straight line tangent to a given circle’]. This line de forms with the line ab two right angles or angles equal to two right angles, from Euclid Elements 1 prop.13 [‘if a straight line erected on a straight line makes angles, it will make two right angles or angles equal to two right angles’]; also from the same prop.13, the line de will make two right angles or angles equal to two right angles with the line ac (which is posited as a straight line); therefore the angle ade and the angle bde will equal two right angles; and by parity of reason, the angle ade and the angle cde will equal two right angles. But any two right angles are equal to any two right angles, from Euclid Elements 1 postul.3 [‘all right angles are equal to each other’]; so take away the common angle (namely ade), and the remaining angles will be equal; so the angle bde will be equal to the angle cde, and so a part will equal the whole.Tr. Since points b and c are, by hypothesis, not the same, the lines from b to d and from c to d must form an angle when they meet at d. Hence, since b is, by hypothesis, to one side of c, the angle bde will be smaller than, or a partial amount of, the angle cde; but by the argument from Euclid, bde must equal cde, so a part will equal the whole. ²⁹

322. But to this conclusion the adversary will say that the lines db and dc do not make an angle, because then on that angle a base could be subtended from point b to point c, which is contrary to what was laid down, that the points b and c are immediate. When therefore the supposition is taken that the angle cde is the whole with respect to the angle bde, the supposition is denied, because nothing is added to the angle bde from the angle cde, for between b and c in their coming together at point d there is no angle.

323. This response may seem at first absurd, because it denies an angle where two lines that cover a surface and are not coincident come together, and in this respect it contradicts the definition of an angle in Euclid Elements 1 [def.8, ‘A plane angle is the inclining of one line to another when two lines touch and do not lie in the same direction’] - and also because, by denying that a line can be drawn between b and c, it denies the first postulate of Euclid [n.320, ‘from a point to a point a straight line may be drawn’] -however because these results may not be reckoned unacceptable (because they follow the opponent’s assumption [n.322]), I argue against the response in a different way:

The angle cde includes the whole angle bde and adds to it at least a point (although you perversely say it does not add an angle), and a point for you is a part; therefore the angle cde adds to angle bde some part; therefore the former is a whole in relation to the latter.

324. The assumption [sc. ‘cde adds to bde at least a point’] is plain because, if an angle is called the space between intercepting lines not including the lines, then the first point of the line db outside the smaller circumference will be nothing of the angle bde and will be something of the angle cde [sc. because the angle bde and the line db are, ex hypothesi, included within the angle cde]; but if an angle include, over and above the included space, also the including lines, then the first point of the line dc outside the smaller circumference will be nothing of the angle bde and will be something of the angle cde [sc. because the line dc is, ex hypothesi, not part of the line db but outside it]. And so in either way the angle cde adds a point to the angle bde.

325. Nor can one in any way oppose the principal demonstration [sc. that the lines begin to diverge at point d on the smaller circumference] by supposing the two lines do not begin to diverge from each other at the circumference of the smaller circle but somewhere else, closer to or further from the center, because wherever you put this I will describe there a smaller circumference [sc. than that of the larger circle, though a circumference larger than that of the original smaller circle].

326. This second part, namely that the smaller circumference is not cut at one point if it is cut by two lines, needs to be proved only because of the perversity of the opponent, because it is sufficiently manifest that the same line, if it is continuously extended straight on, will never, from the same point, end at two points, and if this ‘manifest’ truth is conceded, the intended conclusion is plain from the deduction in the first part [n.325].

327. The second proof [n.320] is from Euclid Elements 10 prop.5, 9. For he says in prop.5 that “the proportion of all commensurable quantities with each other is as that of one number to another number,” and consequently, as he maintains in prop.9, “if certain lines are commensurable, the squares on them will be to each other as some square number is to some square number;” but the square on the diagonal is not related to the square on the side as some square number to some square number; therefore neither is the line, which was the diagonal of the square, commensurable with the side of the square.

328. The minor of this syllogism is plain from Euclid Elements 1 prop.47 [“the squares on straight lines commensurable in length have a proportion to each other that is a square number to a square number”], because the square on the diagonal is double the square on the side, because it is equal to the squares on two sides; but no square number is double some other square number, as is plain from running through all the squares, whatever the roots they are drawn from.

329. Hereby is the following conclusion plain, that the diagonal is asymmetrical, that is incommensurate, with the side. But if these lines were composed of points, they would not be incommensurable (for the points of one would be in some numerical proportion to the points of the other); and not only would it follow that they were commensurable lines, but also that they were equal lines, which is plainly nonsensical.

330. Proof of this consequence [sc. ‘if diagonal and side were composed of points they would be equal’].

Let two points in a side be taken that are immediate to each other, and let another two be taken opposite them in the other side, and let two straight lines, equidistant from the base, be drawn joining the opposite points. These lines will cut the diagonal.

I ask therefore whether they will cut it at immediate points or mediate points.

If at immediate points, then there are no more points in the diagonal than in the side; so the diagonal is not larger than the side.

If at mediate points, I take the point between the two mediate points on the diagonal (this in-between point falls on neither line, from the givens). From this point I draw a line equidistant from each line (from Euclid Elements I prop.31, “Through a given point draw a straight line parallel to a given straight line”); let this line be drawn straight on continuously (from the second part of Euclid Elements 1 postul.2, “A terminated straight line may be drawn straight on continuously”); it will cut the side, and at neither of its given points but between both (otherwise it would coincide with one of the other lines from which it was posited to be equidistant - and this is contrary to the definition of equidistance, which is the definition in Elements 1 def.23, “Parallel lines are those that, drawn in the same plane and produced to infinity in either direction, meet on neither side”). Therefore between the two points, which were posited as immediate in the side, there is an intermediate point; this follows from the fact that it was said [just above] there was a middle point between the points on the diagonal; so from the opposite of the consequent follows the opposite of the antecedent [sc. ‘if there is no intermediate point in the side, there is none in the diagonal; but there is an intermediate in the diagonal, therefore there is one in the side’], therefore etc. [‘therefore since, ex hypothesi, there is no intermediate point in the side, there is none in the diagonal, and side and diagonal are equal’].

331. Nay, in general, the whole of Euclid Elements 10 destroys the composition of lines out of points, because then there would be altogether no irrational lines or surds, although however Euclid there treats principally of irrationals, as is plain about the many species of irrational lines there that he assigns.

2. Rejection of the Second Antecedent

332. From the same discussion [nn.316-331], the rejection of the second antecedent [about minima, nn.286, 290] is also apparent - for either the minimum could precisely end a simply indivisible line, or it could be taken between the ends of two lines.

If in the first way, a minimum is posited as simply an indivisible point; and then it is the same, in this way, as positing a minimum and a simply indivisible as a part.

If in the second way, let two lines then be drawn - extended from the center - to the end points of such a minimum in the larger circumference, such that the lines precisely enclose in the circumference such a minimum. I then ask: do they enclose some minimum in the smaller circumference, or do they precisely include nothing but have altogether the same connecting indivisible? If in the first way, then there are as many minima in the smaller circle as in the larger; so the two circles are equal. If in the second way, it follows that the smaller circumference will be cut at one point by two straight lines (proceeding from the same point), which was rejected in the first member [sc. when arguing against the first antecedent, nn.316-331, esp. 321]. Rather, there follows something more absurd, namely: let these lines in the larger circumference enclose the minimum; and let a straight line be drawn from the end of one these lines to the end of the other, according to the first postulate in Euclid Elements 1 [‘From any point to any point a straight line may be drawn’]; and then this line will be the basis of a triangle of two equal sides, and consequently it will be able to be divided into two equal parts (from Elements 1 prop.10, ‘to divide a given terminated straight line into two equal parts’); and so what was given as a minimum will not be a minimum. Nay further: let some other line be drawn [within the triangle] parallel to the base of the triangle; it will be shorter than the base, and so there will be something less than the minimum.

333. Likewise, this position [sc. about minima] (provided a sort of thing be understood as does not have a part in a whole), involves, whether in one way or the other [n.332], the commensurability of the diagonal with the side (nay, its equality), as was proved before against the first opinion [sc. the first antecedent, n.330].

334. [Instance about minima as to form] - To these arguments [nn.332-333] a response is made that they do not conclude against a minimum as to form, and thus a minimum as to form is posited and not a minimum as to matter.

335. And this distinction is got from the Philosopher On Generation 1.5.321b22-24, ‘On Growth’, where he maintains that “any part as to kind increases but not as to matter.”

336. However this statement can be understood in three ways:

First that ‘a part as to kind’ is called a part as to form, but ‘a part as to matter’ is called a part of an extension insofar as it is an extension, a quantum, because quantity follows matter. And then the statement returns to an old saying, namely that ‘extensions are divisible ad infinitum as they are extensions, but not as they are natural entities’.

337. Or, second, ‘a part as to kind’ can be understood to be what can per se be in act, while ‘a part as to matter’ is called a part as to potency, namely the way a part exists in a whole. And then the statement returns to another old saying, that ‘there exists a minimum that can per se exist, but there is in a whole no minimum than which there is not, existing in it potentially, a lesser’.

338. Or, in a third way (not in harmony with the two old sayings), ‘a part as to kind’ can be understood as what is in something as a minimal part of the form, or of the whole thing as it has the form, and is not any minimal part as to matter, or as to the whole thing in respect of matter. And then it seems manifestly false, because no part of matter in the whole is without form in act, or even without a form of the same nature in the case of homogeneous wholes; rather, just as in this case the whole is divided into homogeneous parts, so the matter and form are per accidens divided into their homogeneous parts - and there is a minimum of each part in the way that there is a minimum of the whole, and conversely.

339. [Response to the instance] - Dismissing, then, this third way of understanding [n.338], I show, by excluding the other two understandings [nn.336-337], that they do not stop the preceding proofs [nn.332-333].

So first I argue against the first way [n.336] using the authority of the Commentator ad loc. on Physics 3.6.206b27-29, on the remark “And we saw Plato etc.;” look there.Aristotle in the Arabic-Latin translation: “And we saw Plato for this reason posit two infinites, because he thought that a thing can pass through and proceed to infinity both by increase and by decrease.” Averroes: “When Aristotle declared that an infinite is found in decrease simply and in addition non-simply (but in that which is converse to division), he began to accuse Plato because Plato equated infinity in one way with infinity in the other (namely both in addition and in decrease), and Aristotle said ‘And we saw Plato etc.’; that is, and Plato, because he thought an infinite proceeds to infinity both by increase and by decrease, posited two species of infinite, by addition and by decrease; and Aristotle introduced the term ‘increase’ in place of the term ‘addition’, so as to distinguish between a proposition of nature and one of geometry.” ³⁰

340. Second using the authority of Aristotle On Sense and Sensibles 6.445b20-27, in the first puzzle when he alleges something to the contrary [n.294]. For although he solves the puzzle obscurely there, yet he does definitely say that ‘sensible qualities are determinate in species’ (which he proves by the fact that ‘when extremes are posited, the intermediates must be finite; but in every kind of sensible quality extremes are posited, because contraries are’). But as to whether any one individual quality is able to have a term in itself, he seems to say no, ‘because they exist along with continuity, and so they have something in act and something in potentiality’, as a continuous thing does; that is, as a continuous thing is one per se actually and many potentially (the many it is per se divisible into), so a sensible quality as it exists in a continuous thing is one actually and many potentially, although per accidens. And then, when the potentiality of the extension or of the quantum is per se reduced to act, the potentiality of the quality is per accidens reduced to act, such that the quantity [sc. of the quality] is by division never divided into mathematical extensions; because, just as he himself argued in response to the puzzle [sc. here above] that ‘a natural thing is not composed of mathematical parts but of natural parts’, so too it [sc. the sensible quality] is divided into such parts, namely natural ones.

But as to how the first way does not make for its intended conclusion, this will be plain from the response [n.344].

341. That for which the authorities of the Commentator and Aristotle have been adduced is also proved by reasons:

Because when some property belongs to something precisely according to some idea, then whatever it belongs to equally according to that idea it belongs to simply equally (just as if ‘to see’ is of a nature to belong to an animal precisely according to its eyes and not according to its hands, then whatever it belongs to equally according to its eyes it will belong to equally simply, even though it does not belong to it according to its hands); but to be divided into such integral and extended parts of the same idea belongs formally to something only through quantity, and to a largest natural thing no more than to a smallest one; therefore since being divided belongs to the smallest according to the idea of quantity, so it will belong to the smallest simply, just as it does to the greatest.

342. But if it be said that the form of a minimum prevents it from coming together from a quantity (as far as concerns itself, on the part of quantity) - on the contrary: if certain consequents are per se incompossible, then what those consequents follow on are also incompossible; and, much more, if what are of the essential idea of certain things are incompossible, then the things too are incompossible; but divisibility into such parts either essentially follows quantity or belongs to the per se idea of it (the sort of idea that the Philosopher assigns to it, Metaphysics 5.13.1020a7-8); therefore, any natural form that divisibility is posited to be incompossible with, quantity is incompossible with too; and so it will not be simply divisible insofar as it is an extension, a quantum, because it is not simply an extension.

343. A proof also of this is that it is not intelligible for something to be an extension without its being made of parts, or for something to be made of parts without a part being less than the whole; and so it is not intelligible for something to be an indivisible extension such that there is not anything in it, less than it, present in it. Nor too can any simply indivisible flesh be posited in a whole of flesh [n.292], because, just as a separate point would not make a separate extension, so neither would a separate point of flesh (if it existed) make any greater thing, either continuous or contiguous, along with another separate point of flesh; hence the reasons of the Philosopher in Physics 6 [n.319] refute the indivisibility of any natural thing just as they refute the indivisibility of any part of an extension insofar as it is an extension.

344. I say therefore that if the response [n.366] about a natural thing insofar as it is an extension and insofar as it is natural can possess any truth, this response should be understood by affirmation and denial of the formal idea of divisibility, such that the formal idea which says that a natural thing is divided insofar as it is an extension says that it is divided insofar as it is a natural extension, and that the formal idea which says that it is not divided insofar as it is natural denies that naturalness is the idea of this division - as if one were to say that an animal sees insofar as it has eyes and not insofar as it has hands; and this understanding is true. But from this it does not follow that that does not belong simply to a natural thing which belongs to it according to quantity; for the concurrent naturalness of the natural thing does not impede that which naturally belongs to quantity, just as neither do the concurrent hands in an animal take away that which simply belongs to the animal according to its eyes. So therefore, absolutely, every natural thing is divisible into divisibles ad infinitum, just as if the quantity, which exists along with the natural form, were to exist by itself, without any natural form. And so all the reasons that proceed of quantity absolutely (according to the idea of quantity) are conclusive about it as it exists in natural things, because divisibility is a natural property of quantity - and so as a result the reasons are conclusive about the natural thing to which this property belongs.

345. The second response [n.337] does not seem to exclude the aforesaid reasons that a whole is not composed of indivisibles or of smallest parts within the whole [nn.332-333]. Nevertheless, it does seem possible to posit a minimum in motion because of the fact that a part of motion per se exists before it is part of something else, of some whole; and thus a part of a form, according to which there is motion, precedes all the parts of that form (not only in nature but also in duration), and so it seems to exist per se and not in the whole. If therefore there may be a minimum in natural things that could exist per se, then this seems to be the smallest part of a form that could be introduced by motion, and so to be a smallest motion [response in nn.350-352].

346. But against this response [nn.337, 345] I argue that just as it is essential to an extension that it can be divided into parts, so it is essential to it that each individual part of the parts it is divided into can be a ‘this something’; therefore existing per se is repugnant to none of them.

347. There is confirmation of this reason and of this consequence:

First because these parts are, as to both matter and form, of the same idea as the whole; therefore they can have per se existence just as the whole also can.

Second because if these parts existed per se, they would be individuals of the species of which the whole is also an individual; but it seems absurd that something has in itself the nature whereby it is, or could be, an individual of some species in such a way that its being able to be an individual of that species is not repugnant to it while yet its being able to exist simply is repugnant to it, and this at any rate as to things that are not accidents (we are speaking now of homogeneous substances which are not essentially inherent in something).

Third too because parts are naturally prior to the whole; so their being able to exist naturally prior to that whole is not repugnant by contradiction to them, because their being prior in time to the whole itself is not naturally repugnant to them (in this way, that it is not repugnant by contradiction to them - on their part - to be prior in duration).

348. It seems, as far as this fact is concerned [nn.346-347], that one should say that, just as a natural form does not take away from a natural whole its being in this way a whole that is always quantitatively divisible, in the way a quantity would be if it existed by itself [n.344], so too it does not take away from it the possibility of any division of it existing per se (as far as concerns it on its own part), in the way that any quantitative part that an extension might be divided into could exist per se.

349. And if you say that it would at once be changed into what is containing it [sc. as water would be changed into air when divided, as per below], the response is that this does not seem to relate to the meaning of the question. For we are looking for a minimum able to exist per se by its intrinsic idea, that is, a minimum that, by nothing intrinsic to it, has any contradictory repugnance to the per se existence of something smaller than it; but, if the whole is corrupted, no intrinsic idea of this sort of incompossibility is imputed. For let us set aside everything containing it or corruptive of it, and let us suppose that water alone exists in the universe; let any given amount of water be divided, because this is possible, as is proved above against the first response [nn.341-344]. The parts into which the division is made will not be nothings, because this is against the idea of division - nor will they, from the idea alone of division, be non-waters, because then water would be composed of non-waters; nor is this smallness, which is now actual, repugnant to the form of water, because this ‘small’ water was there before (although within the whole); nor is the water corrupted through the division, because everything corruptive of it was set aside. So there seems to be no intrinsic reason that the possibility of something less of it per se existing should be repugnant to any per se existing natural thing, although perhaps an extrinsic reason preventive of such per se existence could be assigned in the opposition of some corrupting agent to it [nn.341-344].

350. I also argue against both responses together [nn.336-337], because neither saves a minimum in motion (although it was to reject this charge that the preceding deduction [n.345] was to some extent touched on); for although a medium for local motion cannot be ground for a movable thing unless the medium is natural, yet if per impossibile a mathematical medium could be ground for a mathematical movable, there would truly be succession in such motion, because of the divisibility of the medium; for the movable would pass through a prior part of the space before it passed through a later part. And even now, just as it is per accidens for a thing in place (on the part of the thing as it is in place) that it has natural qualities (as is plain from the Philosopher about a cube in Physics 4.8.216a27-b8 [n.218]), and just as it is per accidens for place (on the part of place as it is place) that it has a natural quality (from q.1 n.235 about place, because although naturalness belongs to what gives a thing place, yet it belongs per accidens to place) - so too it belongs, albeit necessarily in a way that is altogether per accidens, to motion in place or to motion as to ‘where’ (which is per se in a thing in place insofar as it per se regards place) that a natural quality is in the motion, or that it is in it according as it is motion or is in a magnitude over which there is motion. Therefore quantity is per se the reason for succession, whether in a magnitude or in a movable thing or in both.

351. Hereby is the first response [n.336] destroyed, because it does not make for a minimum in motion; because from the fact that - according to this response - one cannot take a minimum in motion according as it is a quantum [n.336], and that succession is per se in local motion by reason of something insofar as it is a quantum, the result follows that in local motion there can in no way be a minimum. And so not in other motions either, because although this may not be as immediately conceded about alteration (if motion or succession be posited according to form), yet it follows by the argument ‘a maiore’ [a fortiori] negatively; for no motion is quicker than passage in place, and thus no motion can have indivisible parts if passage in place necessarily has divisible parts.

352. By the same fact is the second response [n.337] also destroyed, that it does not make for a minimum in motion [n.345]; because in a magnitude over which there is motion one cannot take a minimal part existing in it; therefore neither can one take a minimal passage over the magnitude, because in that minimal passage one should be able to pass through a minimal part of the magnitude.

353. In addition, the second response - as to a minimal motion - is also destroyed by other facts:

First because when a mover is present and is overcoming the movable, one cannot posit the extrinsic reason because of which such a minimum is denied to be capable of existing per se, namely the presence of something corruptive of it [n.349]—because the presence of the cause moving it and producing such a minimum is then overcoming every corruptive contrary.

Likewise [second], ‘for a minimum in successive things to be able to exist in flux is for a minimum there simply to exist in the whole’, because the part of something successive does not have any being in the whole other than that one part flows by before another, and these flowing by parts integrally make up the whole; so just as, in the case of a permanent whole, ‘for a part to be in the whole is for a permanent part to be in the whole’ so, in the case of successive things, ‘for a part to be in the whole is for a flowing by part to be continuous with another part’.

So therefore, now that the two antecedents [nn.286, 290] have been rejected, reply must be made to the proofs of them adduced on their behalf [nn.288-289, 292-300].

3. To the Proofs of the First Antecedent

a. To the First Proof

354. [On the division of the continuous at every mark in it] - To the first argument [n.288] the response is that ‘although it is possible for the continuous to be divided at every point, yet it is not possible for it to exist as so divided, because this division exists in potency and in becoming and can never be complete in a having come to be’. And then as to the proofs adduced for the opposite [n.288], they are conceded as to any single potency for a single making to be, but not as to infinite makings to be, since when one potency has been reduced to act there necessarily remains another not reduced to act; so it is in the issue at hand, that there are infinite potencies for being divided into infinites (since when one potency has been reduced to act, necessarily another remains not reduced to act), and so, although a possibility for being divided is conceded, yet a possibility for having been divided is not.

355. This response is confirmed by the Commentator on Physics 3.7.207b15-18 where he gives the reason for the Philosopher’s proposition that “an [extensive] magnitude happens to be in potency as much as it happens to be in actuality (it is not so in the case of numbers),” namely: “For the reason that all the potencies that there are for parts of a magnitude are potencies of the same potentiality and of the same nature - not so in the case of numbers.”

356. Against this: it follows for you [from the concession made in n.354] that ‘a continuum can be divided at a, therefore it can exist divided at a’ - and so on for b and c and for any individual point (whether determinate or indeterminate), because there cannot be any single division that cannot be carried out. Therefore all the individuals in the antecedent entail all the individuals in the consequent. The antecedent therefore entails the consequent: if a continuum can be divided to infinity, then it will be possible for this division to have been actually done to infinity.

357. But if you say that the singulars in the consequent are repugnant but not the singulars in the antecedent - on the contrary: from something possible no incompossibles follow; but from the singulars of the antecedent the singulars of the consequent follow (as is plain by induction); therefore etc.

358. [On the division of the continuous according to any mark in it] - However, the proposition ‘it is possible for the continuous to be divided at any point whatever’ can be distinguished according to composition and division - so that the sense of composition would be that this proposition ‘it is possible for the continuous to be divided etc.’ is possible, and the sense of division would be that in something continuous there is a potency for it to be at any point divided. The first sense is true and the second false.

359. Or the proposition can be distinguished like this, that it can distribute point divisively or collectively [sc. ‘it is possible for the continuous to be divided at any point singly’ and ‘it is possible for the continuous to be divided at any point together’].

360. It can also be distinguished according as ‘possible’ can precede point or follow it; and if it precedes then the proposition is false, because it would indicate that there is one potency for the attribution of the predicate; if it follows then it is true, because it would indicate that the potency is multiplied on the multiplication of the subject [sc. ‘the continuous is possible to be divided at any point’ and ‘the continuous at any point is possible to be divided’].

361. These responses do not seem very logical; not the third because the mode of putting the proposition together - namely possibility - does not seem it can be distributed to several possibilities (or one possibility to several possible instants), and it would not indicate that the predicate is united to the subject for some one instant; nor is the second response valid, because its distinction has place only when taking ‘any point’ in the plural, as in the proposition ‘all the apostles of God are twelve’; nor is the first response valid, because it still must be that, taking the extremes for the same time (or for a different time), possibility state the mode of composition uniting the extremes [sc. regardless of the distinction between ‘composition’ and ‘division’, ‘possible’ remains the mode by which the proposition combines subject and predicate; see n.362].

362. So passing over long and prolix evasions for these refutations [n.361], I say that this proposition [sc. ‘it is possible for the continuous to be divided at any point whatever’] indicates the union, possibly, of predicate with subject for some one ‘now’ (although the ‘now’ be indeterminate), provided such ampliation of composition can be done by virtue of possibility; for no ampliation can be made for several ‘nows’ such that the possibility of composition for some one ‘now’ not be indicated, whether the extremes are taken for the same ‘now’ or for a different one (to wit, if ‘sitting’ is taken for one instant and ‘standing’ for another). In every sense ‘possibility’ must modify the composition uniting the extremes for some one ‘now’, however indeterminate.

363. So it is in the issue at hand, that the ‘to be divided’ is indicated as being joined to the continuous at a point and at any point of it you like - and this for some indeterminate now. But this is impossible, because whenever the predicate [sc. ‘divided’] is united to it for some singular or singulars [sc. ‘at point a or b’], this predicate is repugnant to it for other singulars; for it is necessary - as the first response says [n.354] -that along with the reduction of a potency (not only to having become but also to becoming) there stands another potency not reduced either to act of having become or even to becoming, because it is necessary that, when division exists ‘in becoming or having become’ at a, something continuous be terminated by a - and thus necessary that the potency which is in that part of the continuous is not reduced to act.

364. But if you argue that any singular is true, therefore the universal is too, one can say that the singulars are true but not compossible, and both are needed for the possibility of a universal.

365. On the contrary: this proposition is true at once ‘a continuum can be divided at a and at b and at c’, and so on about any other singular at once.

366. I reply. I say that singular propositions of possibility, taken absolutely, do not entail formally a universal proposition of possibility, but there is a fallacy of figure of speech ‘from many determinates to one determinate’. For singulars can, from the force of their signification, unite a predicate with a subject for some ‘now’, but a universal unites a predicate with a subject for any now of it universally; and so, by the form of signifying, there is a process ‘from many determinates to one determinate’.Vatican Editors quote from Peter of Spain Logical Summaries tr.7 n.37: “The third mode of fallacy of figure of speech comes from diverse mode of supposition, as ‘an animal is Socrates, an animal is Plato, and so on about each one; therefore an animal is every man’; for a process is made from many determinate suppositions to one determinate supposition... Hence since ‘animal’ supposits in each premise for one supposit and in the conclusion for diverse supposits, its supposition varies.” ³¹ This is the reason why there does not follow from a premise possible for some ‘now’ and a premise possible for another ‘now’ a conclusion about a universal possible as now, because the premises do not - from their form - signify that the extremes are combined with the middle term; and so the union of the extremes to each other does not follow, nor is it even possible for some one and the same now.Tr. That this division is possible at this now and that division possible at that now does not entail that all divisions are possible now, because the particulars do not combine the same now with each division, nor can they. ³²

367. And if you say that the singulars are compossible when taking the potency (but not the act terminating the potency) for the same now, to wit ‘it is at once possible for the continuous to be divided at a and at b etc.’ (but not ‘it is possible for the continuous to be divided at a and at b etc. at once’) - I argue that there is no need for possibility to be divided to the same now in order for the universal to be true, because singular propositions that absolutely assert the predicate of singular subjects, these subjects being sufficiently asserted, entail a universal that absolutely asserts the predicate; if such singular propositions are true, all of them, in themselves, absolutely - then the universal is true as well.

368. And if you ask how singular propositions of possibility are to be taken as sufficiently asserted - I say that they must be taken with specific composition, for the same indeterminate now; to wit, ‘it is possible for the continuous to be divided at a for some now, and possible for it to be divided at c and at b for the same now’, and so on about each of them; and then the universal follows, but otherwise not.

369. And if you argue that these are singulars of a different universal, namely of this universal ‘it is possible for the continuous to be divided at any point whatever according to a single now’, and this universal differs formally from the other [sc. ‘it is possible for the continuous to be divided at any point whatever for the same indeterminate now’ nn.358, 362] - I reply that they differ in words, because that which the former expresses the other by the co-signification of the verb denotes, namely that the extremes are united.

370. And if you say that even in this way, by specification of the predicate to some determinate or indeterminate ‘now’, no singular proposition is repugnant to another, because, just as it is possible for the continuous to be divided at a for some ‘now’, so it is possible for it to be divided at b for the same ‘now’, and so on about c and about any other singular (because if any singular were repugnant, it would be one that took up a point either immediate [sc. to point a] or a point mediate to it; but not one that takes a mediate point, because division at one point does not impede division at another point, even an immediate one; nor one that takes division at an immediate point, because no point is immediate [sc. to point a]; therefore the singular propositions, as they introduce the universal, are true and compossible) - I reply and say that to no singular proposition taken or take-able is any singular proposition repugnant that is determinately taken or take-able with indeterminate composition for the same now, nor are these repugnant among themselves; yet infinite indeterminate propositions are repugnant to any taken singular - and the reason for this repugnance was assigned before, a real one, namely from the incompossibility of the reduction to act of all potentials at once [n.363].

371. An example similar to this in other cases is not easy to get. For one can well posit an example where any singular is possible and yet the universal is not possible, because any one singular is incompossible with any one singular, in the way that the proposition ‘it is possible for every color to be in you’ is impossible, because any determinate singular is repugnant to another determinate singular, as ‘you are white’ is repugnant to ‘you are black’. However, let us posit an example of a man who cannot carry ten stones but only nine (and let the stones be equal), then this proposition ‘it is possible for every stone to be carried by him’ is false; and not because any singular is in itself false, nor because any determinate singular is incompossible with any other determinate singular - but because with some determinate singulars some indeterminate singular is incompossible; for any nine singulars are compossible and the indeterminate tenth is incompossible with them.

372. And in this way must the response of the Commentator at On Generation 1 com.9 be understood where he says that “when a division has been made at one point, a division at another point is prevented from being made,”Averroes: “And it would be possible for a magnitude to be divided at every point at once if the points were in contact with each other, which however is impossible... And so we see that when we divide a magnitude at some point, it is impossible for a division to be made at the point following on that point, although this was possible before the division was made at that point.; but when a division was made at the first point, the possibility of division at the second point was immediately destroyed. When therefore we have taken some point, at any place we wish, it will be possible for the magnitude to be divided at that point; but when the magnitude has been divided at a point and at some place, then it will be impossible for it to be divided at a second point in any place we wish, since it is impossible for it to be divided at a point following on the first point.” ³³ namely not indeed at any indeterminate point (marked or mark-able), but at some determinate one.

373. And then I reply to the argument made above against me, about mediate and immediate points [n.370], namely that it is against the objector. I say therefore that one should not allow a division to be made at some point immediate to another point, but at some mediate one; not however at a determinate mediate one (whether marked or markable), but at an indeterminate one - because let any determinate mediate point be taken, then a division at the initial point could still stand together with a division at this mediate point; yet to the division at the initial point there will be repugnant a division at another mediate point, namely at one that is not an indivisible any longer in the determinate continuum.

374. [On the division of the continuous at any and every mark in it] - If however you ask about this proposition, ‘it is possible for a continuum to be divided at any point whatever’ - this proposition can be conceded, because ‘any whatever’ is not only a distributive particle but also a partitive one, such that for the truth of the universal, whose subject is distributed through the term ‘any whatever’, there suffices a single attribution of the predicate to any singular whatever; so not to every singular at once, but to any singular whatever indifferently (there is no need for it to be attributed to others). But ‘all’ does not signify in this way, but signifies that the subject is taken at once for any respect of the predicate.

375. However about the term ‘any you like’ there is doubt whether it signifies the same as ‘all’ does or the same as ‘any whatever’ does; but whichever of these is posited, one should say the same about it as about what it is equivalent to; for when the meaning is clear, one should not use force about the word.

b. To the Second Proof

376. To the second proof of the antecedent [n.289] it is said that ‘the indivisible is nothing other than lack of the continuous, so that nothing save lack of continuous succession is formally an instant- and so a point is lack of length and states nothing positive’. And in that case the proposition that ‘the successive has precisely being because its indivisible exists’ [n.289] needs to be denied; rather it has precisely successive being because a part of it flows by, and never because an indivisible of it is something positive.

377. Many things seem to make for this opinion:

First, that, when the idea alone of the continuous is posited and everything absolute is removed, the continuous seems to have a term, provided it is not absolute; and it does not seem that God can separate finiteness from line nor - as a consequence - a point from it either, which does not seem likely were a point ‘an absolute essence’ different from line.

378. Likewise, if point and line were two absolute essences, it does not seem possible that some one thing would be made from them unless one of them were an accident of the other; for they are not one by perfect identity since they are posited as two absolute essences; nor possible that a single third thing would be made composed of them, because neither is act or potency with respect to the other. The indivisible then has being and not-being without generation and corruption, because if it is only in the middle of a continuous line it is only one point, but when the line is divided there are two points actually; so there is there some point that was not there before, and there without generation, because it does not seem probable that a generator has generated there some absolute essence.

379. Likewise, it seems, from the author of Six Principles about the figure of an incision,Book of Six Principles 1.4, “In the case of certain things there is doubt whether their beginning is from nature or from act, as in the figure of an incision; for no addition is made but a certain separation of parts.” ³⁴ that this is not something said positively, and yet there is a surface there in actuality that was not in actuality before.

380. But against this [nn.376-379]:

Then the result is that the generation of a substance that is not per se the term of a continuum will be nothing (or at any rate in nothing), because there is no positive measure of it; and so it is in the case of illumination and all sudden changes that are not the per se terms of motion. And although this result could be avoided in the case of changes that are terms of motion and come to be in an instant (as nothing in the case of nothing or privation of continuity in the case of privation of continuity), yet it seems absurd about the former cases, for they are not the per se terms of the continuity of any continuous thing, because they are nothing of anything continuous, whether positively or privatively.

381. Further, according to the Philosopher Posterior Analytics 1.4.7334-37, the idea of line comes from points, that is, point falls into the essential idea of line and is said of line in the first mode of saying per se [sc. the mode of per se when the predicate falls into the definition of the subject]; but no privation pertains per se to the idea of something positive; therefore etc. [sc. point must state something positive, contra n.376].

382. From the same [sc. statement of the Philosopher, n.381] the result also follows that, if a point is only a privation, line too will be only a privation, as well as surface and solid; for a termed thing is defined by what terminates it and something positive does not essentially include a privation.

383. Likewise the same result [n.382] follows (for another reason [sc. from what is said in n.376 and not from Aristotle’s statement in n.381]) that, if a point is only a lack of length, a line will be only a lack of width and a surface only a lack of depth; and then there will only be a single dimension, which solid would be posited to be, although however the dimension which is called ‘depth’ could in another respect be called ‘width’ (for the three dimensions are distinguished by imagining three lines intersecting each other at the same point).

384. And from this further is inferred something unacceptable, that if a surface is only the privation of depth, how will a point be the privation of a privation? For nothing seems to deprive a privation unless it is something formally positive.

385. In addition, there are on a surface many corporeal or sensible qualities, as it seems; therefore a surface is not merely a privation.

The antecedent is proved about colors and figures, each of which is per se visible and consequently something positive. The figure too [sc. of a surface] seems most properly to follow the kind or species, and so seems to be an accident manifestive of the species; but it does not seem probable that there is no positive entity to something that is such as to follow a species naturally and to manifest it.

386. If it be said differently [sc. to the proof, nn.289, 376] that ‘the indivisible by which the successive has being exists only in potency’ - this is no help, because, when the indivisible is gone, what succeeds to it in the way it has being in the whole? If another indivisible does, the argument [n.289] stands; if not, then the successive will not exist.

387. My response to the argument [n.289] is that, when the indivisible is gone, a continuous part flows by and not an indivisible; nor does anything succeed immediately, save as the continuous is immediate to the indivisible.

388. And if it be objected ‘therefore time does not always have being uniformly and equally (because, when the indivisible instant is posited, time exists, for its indivisible exists, but when the indivisible has gone, time immediately does not exist, because another indivisible of it does not exist)’ - I reply that, just as a line does not have being uniformly everywhere insofar as ‘everywhere’ distributes over the parts of a line and the indivisibles of a line (because a line has being in the former as it is in the parts and in the latter as it is in the ultimates), and yet a line exists everywhere uniformly to the extent that ‘everywhere’ distributes precisely over the latter or precisely over the former, so it is in the issue at hand of time; if the ‘always’ [at the beginning here, n.388] distributes precisely for the indivisibles or precisely for the parts, then time does have being uniformly; but if for both at once then it does not have being uniformly.

4. To the Proofs of the Second Antecedent

389. To the proofs of the second antecedent, about minimal parts [nn.292-300], I reply:

To the first [n.292] that the Philosopher has enough against Anaxagoras if the whole is diminished by a taking away from the whole such that an equal amount cannot go on being taken from it forever; for Anaxagoras had to say (as Aristotle imputed to him [Averroes Physics 1 com.37]) that, after separation of anything generable out of flesh has been made from the flesh, there would still remain as much flesh as could have anything generable further separated off from it; and this is impossible, because however much the flesh can be divided and diminished ad infinitum, not as much flesh at any rate would remain as could have anything generable generated from it, because anything generable requires a determinate quantity of that from which it is generated (especially if, as is imputed to Anaxagoras, generation is only separation or local motion, and the flesh is diminished, by continual separation of other parts from it, beyond the total quantity that generation might come from). So one is not required by Aristotle’s intention there [n.292] to posit also a separate minimum in natural things which exists per se and not in the whole.

390. To the statement of the Philosopher On Sense and Sensibles [n.294] I say that properties are divisible as much as may be, so that a quantum cannot be divided without dividing the property; and yet the property is not divided ad infinitum as it is sensible (that is, insofar as it is perceptible by sense), just as Aristotle maintains there that ‘a part, however minimal, can be sensible virtually although not in act’; that is, that such a part can cooperate along with other parts so as to affect the senses - and yet, although division could also be made in it as it is a per se existent, it would not however affect the senses.

And then the response to the argument of Aristotle adduced for the opposite [n.294 ‘the senses could be intensified infinitely’] is plain, that ‘the senses grow ad infinitum in intensity if a property divisible ad infinitum is presented to them’; and this is true if the sensible, insofar as it is actually perceptible by the senses, could be divided ad infinitum - but the same does not follow if the thing that is sensible can be divided ad infinitum.

391. As to the statement from On the Soul [n.293], it is plain that Aristotle is speaking of the quantity of something capable of increase and decrease; and this I concede, because the quantity that is a perfect quantity for any natural thing is determinate as to being greater or smaller, speaking of the quantity in which the natural thing is naturally produced; or at any rate it is determinate as to being smaller in the case of animate things, speaking of the quantity which diminution leads to. However, the Philosopher is only speaking there [sc. in the passage from On the Soul] of the limit of size and increase; and so he is precisely in this place understanding the perfect quantity of any natural thing to be determinate as to being greater - and from this he gets his conclusion, which he intended to prove, namely that ‘fire is not the principle of increase in any generation or in any species’; for the principal agent in any species must be determinate to the perfect quantity of that species, so that it may produce that quantity and not more than it; but fire is not determinate as to determinate quantity in any species, because - as for as concerns itself - it would go on producing a greater amount, for it grows ad infinitum if combustible material is added to it ad infinitum.

392. And when the antecedent about the minimum [n.290] is proved through the premise [n.295] that ‘it is possible to take a first part of motion’, the consequence can be denied [n.295, ‘therefore it is possible to take a smallest part of motion’], because those who asserted a first part in motion asserted that change is this first part of motion; however I deny a first in both ways (both a first motion and a first change), because the

Philosopher in Physics 6.6.236b32-7b22 of express intention shows the opposite, namely that every moving is preceded by a having moved ad infinitum, and conversely [n.297].

393. And he gives proof of this as follows: that if fire were to cause some first in motion, by parity of reason it would cause something equal to that first, simultaneous with it, and immediate. And so one would need to imagine that between the first simultaneous caused thing and the second one - equal to it - the agent would either have to be at rest, and so motion would be composed of motions and intermediate rests, or the agent would, after having introduced the first, need to introduce the attained successive whole, which seems thoroughly irrational, because, since the agent is of equal virtue for, and equally near to, the passive subject, then just as the agent can simultaneously introduce any (first) degree simultaneously caused, so it can, simultaneous with that introduced degree, introduce the whole thing, and so the whole motion would be caused immediately of immediate changes, or composed of changes - whether motions or rests -that are intermediate.

394. So here is the following process. Let there be a form subject to change needing to be corrupted by motion, for instance, in the case of an alteration, under a heat that is at rest. Of this motion, I say, it is possible to take a last, namely the terminating change, because a movable thing is now disposed indivisibly as previously it was disposed divisibly, and this ‘being affected’ - just like ‘being changed’ - is a being now indivisibly disposed otherwise than it was disposed divisibly before [n.181]. Now for this reason it is under the same form - under which it was at rest - in the instant of change, because then the agent that ought to be moving it did nothing before, and is not now doing anything in respect of it. From this instant the movable begins to move, and that successively - either because of the parts of the movable, for no parts of the movable are equally close to the agent but one part is nearer ad infinitum than another (only a point of the movable is with all of itself immediate to the agent, and a point is not movable), or because of the parts of the form according to which there must be motion, each of which parts can be introduced before another by the present mover, since the extrinsic reason why a minimum cannot exist per se in natural things is the presence of a corrupter - but this is removed by the presence of the agent, which corrupts everything corruptive of its own effect [nn.349-353].

395. Therefore, from this instant of change, the heat that was present is continually diminished and coldness takes over. For it is not likely that there is only a movement of diminishing up to some instant and then first some coldness is introduced; for in that case either the heat to be diminished would have an ultimate of its being (which the Philosopher denies in Physics 8.8.263b20-26), or, if not, at least the coldness immediately following it would have a first of its being, and then there would be a first change of the motion of cooling, which is as unacceptable as that there is a first diminishing of the motion of the heat. It also seems unacceptable that an agent should diminish heat save by causing in it something according to some degree incompossible with it, and, as it causes that incompossible something in greater or lesser degree, it corrupts degree after degree of the existing heat; now Aristotle manifestly maintains this in Physics 6 [n.302], that everything moved has something of both extremes - and it seems manifest to sense that there is something of heat in water being successively heated, while the coldness still remains and is not yet wholly corrupted.

396. So, from the instant of change, the motion of remission of heat and the motion of intensifying of coldness run together - and of neither of these is anything first and in some instant in which, by a sudden change, some degree of coldness is introduced that is altogether incompossible with the heat; in the first there is no heat and up to it there was heat - such that heat has no ultimate of its being but did have an ultimate in its being at rest; and coldness has no first simply of its being, although it have a first in being of rest (namely what it receives through the change, although this is not rest).

397. When therefore the proof is given by the Philosopher in Physics 8 [n.297], I say that the intention of the Philosopher is this, namely to prove that not everything is always in motion. And against those who say that ‘everything is always in motion’ he says that they are manifestly refuted if we consider the motions by which they were moved; for the motions - for their positing of this view - were taken from the increase and decrease of animate things, which they saw coming about in some great length of time (as in a year), and yet from this fact they concluded for no reason that these motions were coming about throughout the whole time but not perceptibly in any part of the time. To them Aristotle objects that such a movable can very well be at rest for a certain time and be moved in some small period of time, so that there is no need that it be always moving with that motion; and he proves this with an example about drops of water wearing away a stone, which drops fall in some certain number and take nothing away from the stone - eventually, however, one falling drop (let it be the hundredth) takes away, by virtue of all the drops, some part of the stone, and this part is taken away whole at once and not part before part.The Vatican Editors point out that, in his interpretation of Aristotle here, Scotus is in agreement with the like interpretation of Aquinas in his commentary, ad loc., on the Physics. ³⁵

398. Hereby the Philosopher does not intend that this taking away of a part of the stone happen in an instant and be in this way whole at once, for this taking away belongs to local motion (and so the motion is local), which cannot at all happen unless a part of the movable pass over the space before the whole movable does; but although this one part of the stone - which is taken away by the last drop in virtue of all the preceding drops - is taken away successively, yet the taking away of it is not successive corresponding to the whole succession of the falling of the drops; for it is not the case that there were as many parts of this taking away of a part from the stone as there were falling drops, but this whole small part is taken away by the last drop, albeit successively. The Philosopher, therefore, is denying a succession corresponding to this succession, namely to the whole falling of the succession of drops - and for this reason the moving of the stone was not always being moved, although when it was being moved by the last drop it was then being successively moved.

399. And, in accord with this intention, he subjoins afterwards about alteration that “there is no need, for this reason, that the whole alteration be infinite, for frequently it is swift” [n.297], where the translation of the Commentator has “sudden” for the “swift” in our translation; now the Commentator expounds ‘sudden’ thus, “that is, in an instant,” and infers “not in time.” But this exposition is contrary to Aristotle’s text, as is plain from our translation ‘frequently it is swift’, and from his own translation which has ‘suddenly’ - because in Physics 4.13.222b14-15, where our translation has “at once,” his translation has “suddenly,” and he has a note there, “that is said to happen suddenly which happens in an imperceptible time” - and thus does he himself there expound it. So to expound ‘swiftly’ or ‘suddenly’ as an instant is to expound time as an instant.

400. However, the intention of the Philosopher [sc. in Physics 8, nn.297, 399] is as follows: there is no need that, as the alterable is divisible ad infinitum, so a time ad infinitum should correspond to the alteration of the alterable - or that always, while the alterable exists, part after part of it should alter continuously, the way alteration could be a succession by reason of the parts of the alterable; but ‘frequently alteration is swift or sudden’, that is, when the alterable is at rest, and then the parts are not simultaneous (either according to the first change or according to the first part of motion) but in succession.

401. And this is what is immediately added by the reason that the Philosopher appends for the same conclusion, namely that when someone is healed the healing is in time “and not at the limit of time;” and yet the movable is not always in motion with this motion, because this motion is finite between two contraries. How then would Aristotle, for the purpose of proving that ‘not everything is in motion’, be taking in the preceding reason [n.400] that ‘alteration happens in an instant’ [sc. as Averroes interprets Aristotle, n.399], and in this second reason he is taking the opposite, namely that ‘healing is not at the limit of time but in time’, and still healing is, on this account, ‘not always’ because it is between contraries, and so, when the contrary is acquired, the motion ceases?

402. Therefore the Philosopher subjoins that “to say ‘everything is continually in motion’ is extravagant quibbling” (where ‘continually’ is taken for ‘always’, because he rejects, for all these reasons [nn.397, 399, 401, 402], the second member of the five membered divisionAristotle gives in the passage at Physics 8 five arguments against the thesis that everything is always in motion: from increase and decrease, from the wearing away of a stone, from the freezing of water, from health, and from stones remaining hard. ³⁶). And yet too a further exposition is there posited, because ‘stones remain hard’; so they do not undergo alteration.

403. Aristotle does not then deny his whole opinion in Physics 6 because of anything he says here, in Physics 8 [nn.297, 392]; and granted that here there were some term that seems expressly to carry this meaning (although there is not but only one taken from a false interpretation), yet it would seem to need being expounded according to what is said in Physics 6 rather than to retract somewhere else [sc. Physics 8] the whole of what is chief in Physics 6 because of certain things that somewhere else are not said as chiefly or of as express intention as in Physics 6.

404. To the passage from On Sense and Sensibles [n.299] response will be made in the last argument of this distinction [nn.519-520].

405. To the argument about contradictories [n.300] a response is made that statements are contradictories that are taken to hold for the same time (and according to the other required conditions), and statements are not contradictories that are not taken to hold for the same time - as is proved by the definition of contradiction set down in Sophistical Refutations 1.5.167a23-27 [‘A refutation is a contradiction of a same and single thing in the same respect and in relation to the same thing and in like manner and at the same time’]; and so the non-being of heat as it went before in the last instant of change, and the being of heat taken up in the completed time, are not contradictories with respect to heat.

406. On the contrary: the being and non-being of color, taken absolutely (not as they understood to be in the same instant), are incompossible simply, so that because they are incompossible simply they cannot be in the same instant - not conversely; and the reason for this incompossibility ‘for the same instant’ is not other than that they are formally opposed with no other opposition formally than contradictory opposition.

407. This is confirmed by a likeness in other things, that a contrary succeeding to a contrary is truly contrary to it, although the two are not together in the same instant; likewise, a form as the term ‘to which’ of privation is truly opposed to it privatively - and this motion is formally between opposites. Hence the Philosopher in Physics 1.5.188a30-b26, 5.5.229a7-b22 maintains that every motion is between opposites that are contrary or privative or some intermediate of the two, and yet they are, as terms of change, never simultaneous.

408. It could also be argued that the terms of creation were not contraries, because the non-being that preceded the being of the created thing cannot be a contrary or a privative or an intermediate between them because it is not in any susceptive subject -and thus it would not be contradictory to being. Creation therefore would not be between contradictories or contraries, which seems absurd.

409. But as to what is adduced about the definition of a contradiction [n.405], there is an equivocation because contradiction exists in one way in propositions and in another way in terms. Propositions are not contradictory unless they are taken to be for the same instant, and for this instant both must assert the predicate of the subject; but terms absolutely taken, without determination to any being, are contradictories. About the first contradiction the Philosopher speaks in On Interpretation 6.17b16-26, and about the second in Categories 10.13b27-35.

410. I reply in another way to the argument [n.300], because ‘immediate’ can be taken in two ways: in one way that there is no middle between what is a whole in itself and something else, and in another way that what is a whole in itself is at once with something else or after something else. In the first way the continuous is immediate with its term, because nothing falls in the middle between the indivisible point that terminates and the divisible continuum that is terminated. In the second way there is nothing immediate to the indivisible point terminating a continuum; for nothing that is a whole in itself immediately follows the indivisible but a part of the whole does; and what is an immediate whole in the first way follows an indivisible according to a part before a part ad infinitum.

411. To the issue at hand therefore I say that as the measures are disposed to each other so are the things measured, namely that when one contradictory is measured by an indivisible the other is measured by an indivisible as well. And then the minor is false [sc. in n.300, ‘if there is no first between the being of the form that is to be introduced through motion and the non-being of it, the ‘first’ would be indivisible’]; for there is no middle between a contradictory ‘as it is in its whole measure’ and the other contradictory, just as neither between its whole measure and the measure of the other; a contradictory, however, that is measured by an indivisible is not immediate to anything, such that according to some of its being (namely as it is in its measure) it immediately follow the other contradictory. So I say as to the issue at hand that the non-being was in an indivisible, but the being of the form introduced by motion is in the whole completed time - and so nothing is intermediate between them; and yet what follows in time is not immediate - in the second way [n.410] - to what pre-exists in an instant.

C. To the Third Argument

412. As to the third principal argument of the question, when the argument is made that ‘an angel cannot be moved [continuously] because he is indisivible’ [n.301] -although one could easily reply that an angel occupies a divisible place and so, in respect of place, he is disposed as if he were divisible - or that, if he occupies existing as a point a point-place, he cannot be moved continuously so as always to have point-existence -yet, because there seems no reason to deny that an indivisible is moved (even if it were a per se existing indivisible of quantity), then one can concede that an angel, occupying a ‘where’-point, can, as always existing in a point, be continuously moved.

413. And what is here assumed about an indivisible can be proved in many ways: First that a sphere moved over a plane describes a line on the plane and yet only touches the plane at a point; therefore the point passes through the whole line, and yet not for this reason is the line that the point thus passes through composed of points. Therefore, by similarity, neither would this result follow if the point existed per se.

414. Multiple responses are made here:

That there is no spherical thing in nature but only in the intellect or imagination. -But this reply is nothing, because the heaven is simply spherical; and anyway, given that there were no simply spherical thing in nature, there would still be no contradiction on the part of sphere and plane that this thing move over that thing as a sphere over a plane (but there would be a contradiction if, from an indivisible moved over something, the result was that the thing moved over was indivisible).

415. A response in another way is that a natural sphere touches a plane at a line and not at a point. - But this seems impossible, because what is applied to a circular line (so as to touch the whole of it) is necessarily circular, because any circular part is circular in any part; but of a straight line no part is circular or curved.

416. Another response is that, because the point of the sphere [n.413] is moved per accidens, therefore there is no need that the space over which it moves be commensurate with it; but the sphere itself is moved per se, and it is divisible. - But against this is that, although a part in a whole is moved per accidens, yet it is always in a space equal to it, and it describes - in its passage - the whole space; indeed, if a whiteness (which is moved, when the extension is moved, more per accidens than any part or term of the extension) is compared to space according to the quantity it has per accidens , its accidental quantity would still be measured by space. Hence - as far as commensuration is concerned - it does not seem that ‘being moved per accidens’ takes away anything other than ‘being moved per se’.

417. Second [n.413], the line laid down by the sphere is not commensurate with the sphere (because then it would be a solid), and it is commensurate with something moved over it; therefore only with the point that is moved over it. If too the sphere is posited to be in a vacuum and only the line to be a plenum, and if per impossibile the sphere could be moved in a vacuum and the point could be moved over the line-plenum, the line-plenum would only be precisely described by the point. And so the conclusion intended follows from these considerations.

418. Further, take a solid cube and let it be moved. Its primary surface is always on something equal to it, and so always on a surface; or something corresponds to it in the magnitude placed underneath [sc. the magnitude over which the cube is moving], to wit a line - and thus, by always passing over something of the magnitude before something else of it, the cube passes over the whole magnitude; therefore the whole magnitude underneath is composed of a line, if their reasoning be valid [sc. those who say an indivisible cannot be moved continuously, n.412].The idea seems to be that if a cube is moved over a magnitude not continuously but indivisible by indivisible, then the surface of the cube in contact with the magnitude beneath will move over one line of the magnitude before another. So if we focus on just one line in the magnitude, we can consider the whole surface moving over that one line, which will thus be the magnitude the surface moves over. The magnitude moved over will then be composed of that one line. We can repeat this process for every subsequent line of the magnitude, and consequently the surface will always be moving over a magnitude composed of a line. Since this result is unacceptable, we must suppose that the cube moves continuously and not indivisible by indivisible. ³⁷

419. Further, let a first point be marked on a line over which another line is moving. This point on the line placed underneath describes the whole of the moved line, because just as any point of the moved line is always continuously at different points of the line underneath, so conversely any point of the line underneath is underneath different points of the moved line; and yet along with all of these points there stands a continuity of motion.

420. It can therefore be conceded (since the statement about motion per accidens [n.416] seems nothing but a subterfuge) that an indivisible could be per se moved if it existed per se, and still be moved continuously; nor from this does it follow that the magnitude passed over is composed of indivisibles.

421. However, because of what Aristotle means in the passages quoted [nn.302-304], one needs to understand that in local motion there is succession for two reasons, namely the divisibility of the movable and the divisibility of the space, and each of these causes, if it existed per se and precisely, would be a sufficient reason for succession; for any movable first passes over one part of the space before it passes over another, and so there would be succession on the part of the space when comparing the movable to the diverse parts of space; further any same part of the space goes by a first part of the movable before a second, and so there would be succession on the part of the movable when comparing it to any same part of the space. In like way too can it be said of the motion of alteration and perhaps of the motion of increase.

422. The philosopher denies therefore, and well denies, that an indivisible ‘as far as concerns itself’ can be moved or can move such that a continuity of motion on its part can be taken such that it is a movable possessing in itself the complete idea of continuous movable, because being continuously moved is not something it has in itself; yet moving or being continuously moved is not repugnant to an indivisible when taking the continuity of motion from something else [n.421].

423. And such, and nothing more, is what Aristotle’s reasons prove, as is plain by running through all of them:

For when Aristotle takes the principle that ‘everything that is moved is partly in the term from which and partly in the term to which’ [n.302], this principle is true if the movable is of the sort that, from its own idea, there is succession of motion; for such a movable is in both terms according to different parts of itself. However things are not so here [sc. in the case of an indivisible], but an indivisible is partly in one term and partly in the other according to the same part of itself - that is, it is in some intermediate stage, not by being at rest, but insofar as this intermediate stage is something of both terms, that is, insofar as it is that through which the indivisible tends from one term to the other; this is to say that it is under change and under something lying under change, and in this way the parts of motion are continuous. - But when the principle is taken that ‘the indivisible cannot be partly in one term and partly in the other because it does not have parts’ [n.302], this principle is true of the first sense of partly (and so I conclude and concede that the indivisible is not thus a movable), but it is false of the second sense of partly [sc. first and second in this paragraph: the first is that of a movable from whose own idea there is succession, and the second is that of an indivisible].

424. To the other argument [n.303], when it is said that ‘a movable passes through a space equal or less than itself before it passes through a greater space’, I reply by saying that ‘to pass through’ can be understood of a divisible passage or of an indivisible passage.

If for an indivisible one, the proposition is false if the understanding is that before passing through any greater space the movable universally passes indivisibly through some equal space; for then one would have to concede that there would be a first change in local motion; and not even those perverters (and not expositors), who say that Aristotle retracts [in Physics 8] what he said in Physics 6 [n.297], can reasonably say that he contradicts himself within Physics 6 itself. There is no need, then, that any successive passage, which is greater than the movable, be preceded by an indivisible passage.

But if ‘to pass through’ be understood of a divisible passage, then it can be understood of the whole, not by reason of the whole, but by reason of the part; and this not by comparing the part to a ‘where’ equal to it and the whole to a ‘where’ equal to it, because the continuous is that ‘whose motion is one and indivisible’, Metaphysics 5.6.1016a5-6; and in this way the part passes through a space corresponding to it at the same time as the whole movable passes through a whole space corresponding to it. But when understanding ‘to pass through’ with respect to some definite and determinate point in the space, the whole passes first through that point by reason of some part (and, in having passed through it, the whole has passed through something less than itself, speaking of a ‘where’ different from its own first total ‘where’), before it thus passes through a space equal or greater than itself; and this it does per accidens, insofar as the movable can have a ‘where’ less than its total ‘where’.

425. But if we speak of greater or lesser or equal ‘wheres’, according to which continuity of motion is immediately expected (and an infinite number of which ‘wheres’ are something in the first ‘where’), then simply the whole passes through a space greater than itself before it passes through one equal to itself. As such is the response to the issue at hand, saving what belongs to the per se idea of continuous motion and not what does not belong to the per se idea of it.

426. And if you object that, however it may be with Aristotle’s argument in itself [n.303], this point is always in a space equal to itself and so passes through the whole (so it is commensurate with the whole line underneath, and so this line underneath will be composed of points) - I say that it is ‘always’ in the sense that, in any indivisible, it is in a space equal to itself; but it is not ‘always’ in the sense of any part of time.

The same could be argued about the first surface of the cube solid [n.418], that although in any ‘now’ of time it is lying precisely on the line over which it is moved, yet in the intermediate time between two instants it is flowing over the continuous intermediate between the two extremes.

427. As to the last reason [n.304], I well concede that it is possible to take a time less than any given time, but from this does not follow that in that lesser time a lesser movable can be moved, save when speaking of a continuous movable that was, on its own part, the cause of the continuity of the motion.

D. To the Fourth Argument

428. To the fourth principal argument [nn.305-306], about the cause of succession in motion, I say that, although there can be contention and dispute about Averroes’ intention and about what he contradicts Avempace in (the way it appears in Physics 4 com.71, ‘On the Vacuum’), yet I say briefly that the cause of succession in any motion is the resistance of the movable to the mover; not indeed such that the mover cannot overcome the movable (for then it would not move it), nor indeed such that the movable is inclined back toward the opposite (for then precisely it is in violent motion) - but resistance such that the movable is always under something to which the term intended by the mover cannot immediately succeed. And this resistance of the movable to the mover is because of a defect in the virtue of the mover and thereby because of the resistance of the medium to the mover and the movable, and by the ‘medium’ can be understood all that necessarily precedes the introduction of the term to be introduced. But such a medium is not necessarily a medium save to a limited virtue; for if there were an infinite virtue, it could put the movable at once in the term ‘to which’ - such that neither because of the opposed form in the term ‘from which’ (the form that the movable would already have), nor because of the mediums naturally ordered between the form that the movable has and the term ‘to which’, would there be a necessity that such a mover should move through such mediums before it introduced the term.

429. The possibility, then, of succession comes from the resistance of the movable to the mover, which is from the resistance of the medium to the movable and the mover, such that this is one resistance. For the movable, insofar as it has a form of the sort that between it and the term such mediums are of a nature to exist, can be continuously moved through the mediums to the term - and by these mediums, which resist the movable so that it cannot at once be in the term, can be understood the divisibility of the parts of the movable, or the divisibility of the parts of the form according to which there is motion, or both these two together. However the necessity of succession is never from this resistance, but is precisely by comparing this resistance to the agent, which the movable resists because of the resistance of the medium to the agent - such that, just as the possibility was from the resistance alone of the medium to the movable, so the limited virtue cannot take away this resistance; and therefore this resistance resists the agent so that it does not at once introduce the term.

430. Then to the arguments introduced for the opposite [n.306], namely that ‘there is no resistance of an angel to himself’ - I say that, as an angel does not act from the infinity of active virtue when he is in heaven, between which ‘where’ and his own ‘where’ on earth many mediums are of a nature to exist, which are also mediums for his own motive virtue - so neither can his own motive virtue make all those mediums and the term, nor even can he at once make the term save by first making those mediums; and for this reason there is here the whole resistance that is required for succession in motion.

431. And when argument is made about the saying of Averroes, about a heavy object, that ‘if it were put in a vacuum it would descend immediately because of a defect of resistance on the part of the medium’ [n.306] - I say that if a vacuum is posited then the heavy object would not move (according to the Philosopher, Physics 4.8.214b12-215a24), because a vacuum cannot give way to a heavy object and because separate dimensions cannot be together. However, if it were posited that a vacuum could give way and that there was space in it, and not that the sides of the plenum were together (because then there would not be a vacuum) - I say then that there would be motion successively of the heavy object in the vacuum, because a prior part of the vacuum would be prior and also because the whole heavy object would pass through this part of space before that part; and, as was said before in the second argument [n.350], per se succession is only in local motion and in space insofar as space is a quantum.

432. To the arguments of Aristotle as far as they are adduced for the issue at hand [nn.307-308]:

I say that the proposition ‘what the proportion of medium to medium is in rareness and density, so the proportion of motion to motion is in quickness and slowness’ is true (ceteris paribus), and so it follows that there is no motion in a vacuum - or at least this is true against those [sc. the ancient atomists, Democritus and Leucippus] who posited the vacuum to be the whole cause of motion or of succession in motion; but as to the issue at hand, by arguing similarly here about movables as there about spaces, this proposition ‘what the proportion of movable to movable is in rareness, so the proportion of motion to motion is in quickness’ can be denied. And if you take the proposition ‘what the proportion is of this movable under the idea by which it is movable and of that movable under the idea by which it is movable’, I concede it but then the minor is false [sc. that ‘there is no proportion of angel to body in rareness’, n.307]; for an angel is capable of moving continuously insofar as he has a virtual quantity according to which he can coexist in an extended place, just as a body can, according to its quantity, stand in an extended place.

433. Likewise, as to what the Philosopher infers from his second reason, that ‘motion is in an equal time through a vacuum and a plenum’ [n.308] - if something similar to this is inferred, namely that angel and body would be moved in an equal time, it is not impossible; but there is an impossibility there from the idea of mediums, according to which this sort of reason seems to proceed.

434. But although Aristotle’s reasons would not prove much to the purpose (because movables here are not disposed as spaces are there), yet his reasons are simply valid, such that his major is probative and the other argument leads to an impossibility [sc. the major of the reason in n.307 is probative and is allowed to be true in n.432, and the second reason in n.308 leads to an impossibility by reason of the mediums, n.433].

435. I reply, therefore, that if a vacuum could yield and motion were possible in it, then I say that from the divisibility of the space the motion would have divisibility and succession, just as now from the divisibility of the space of a plenum motion would have per se an essential succession; but over and above this succession can be superadded speed or slowness, by reason of the accidental condition of the medium itself (insofar as it promotes or impedes the motion), or by reason of its rareness (whereby it promotes or at least does not impede motion), or by reason of its opposed density. So in that case there would be motion in a vacuum, and proportionality to the motion in a plenum, and this when speaking of essential succession, but not of the superadded speed or slowness, because a movable in a vacuum would altogether have no superadded speed or slowness

(but it would have some in a plenum, but there is no proportion between ‘nothing’ and something).

436. Therefore Aristotle [n.307] has precisely from this fact [n.435] - against the adversary who says there is motion in a vacuum [n.432] - that there can in a vacuum be no motion having any speed or slowness superadded to essential succession. And this would not be unacceptable if one posited precisely that there was motion in a vacuum -but it would be unacceptable if along with this one were to posit a vacuum as a promotive medium in motion (or a necessary medium in motion), on whose part speed or slowness of motion could be taken.

437. In the same way, what is inferred in the second reason [n.308] is not impossible for an adversary who says precisely that there is motion in a vacuum, because a medium that is a plenum can be equated with a medium that is a vacuum insofar as there is reason or cause for essential succession in the motion; and if some plenum were taken in the sort of proportion to a given motion that Aristotle takes it in, it would be altogether neutral (bestowing no accidental quality), being neither a plenum medium nor a vacuum medium.

438. What then does the Philosopher get against the adversary from this reason [n.308]? - I say that he gets only that a vacuum has no accidental quality over and above essential succession; because if it did, some equal medium could be given and then through the plenum medium and the vacuum medium there would be a motion in as much time as corresponds to the accidental condition of the motion, which is impossible - because if so the mediums would be proportional.