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.