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].