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.