Si autem est continuum, et quod tangit, et consequenter, sicut definitum est prius (continua quidem quorum ultima unum, quae vero tanguntur quorum simul, consequenter autem quorum nihil est medium sui generis), impossibile est ex indivisibilibus esse aliquid continuum, Now, if the terms “continuous,” “in contact,” and “consecutive” are understood as defined above—things being continuous if their extremities are one, in contact if their extremities are together, and consecutive if there is nothing of their own kind intermediate between them—nothing that is continuous can be composed of indivisibles. ut lineam esse ex punctis; si vere linea quidem continuum est, punctum autem indivisible. For instance, a line cannot be composed of points, the line being continuous and the point indivisible. Neque enim unum sunt ultima punctorum: non est enim hoc quidem ultimum, illud autem aliqua pars indivisibilis. Neque simul sunt ultima: non enim est ultimum ullum impartibilis; alterum enim est ultimum et cuius est ultimum. For the extremities of two points can be neither one (since there can be no extremity of an indivisible as distinct from some other part) nor together (since that which has no parts can have no extremity, the extremity and the thing of which it is the extremity being distinct). Amplius, necesse est aut continua esse puncta, aut tangentia se ad invicem, ex quibus est continuum: eadem autem res est et in omnibus indivisibilibus. Moreover, if that which is continuous is composed of points, these points must be either continuous or in contact with one another, and the same reasoning applies in the case of all indivisibles. Continua igitur non erunt propter praedictam rationem. Tangit autem omne, aut totum totum, aut pars partem, aut totum pars. Now, for the reason given above, they cannot be continuous, and one thing can be in contact with another only if whole is in contact with whole or part with part or part with whole. Quoniam autem impartibile est indivisibile, necesse est totum tangere totum. But, since indivisibles have no parts, they must be in contact with one another as whole with whole. Totum autem totum tangens, non est continuum: continuum enim habet hoc quidem aliam, illud vero aliam partem; et dividitur in sic diversas et loco separatas. And, if they are in contact with one another as whole with whole, they will not be continuous, for that which is continuous has distinct parts, and these parts into which it is divisible are different in this way, that is, in distinct places. At vero neque consequenter erit punctum ad punctum, aut ipsum nunc ad ipsum nunc, ut ex his sit longitudo aut tempus. Nor, again, can a point be consecutive to a point or a moment to a moment in such a way that length can be composed of points or time of moments. Consequenter enim sunt, quorum nullum est medium proximum: punctorum autem semper est medium linea, et ipsorum nunc tempus. For things are consecutive if there is nothing of their own kind intermediate between them, but that which is intermediate between points is always a line and that which is intermediate between moments is always a period of time. Amplius, dividerentur in indivisibilia, si ex quibus est utrumque, in ipsa dividitur. Sed nullum est continuorum in impartibilia divisibile. Again, if length and time could thus be composed of indivisibles, they could be divided into indivisibles, since each is divisible into the parts of which it is composed. But, as we saw, no continuous thing is divisible into things without parts. Nullum autem aliud genus potest esse medium punctorum et ipsorum nunc. Si namque erit, manifestum est quod aut divisibile aut indivisibile erit: et si divisibile, aut in indivisibilia aut in semper divisibilia. Hoc autem est continuum. Nor can there be anything of any other kind intermediate between the parts or between the moments, for if there could be any such thing, it is clear that it must be either indivisible or divisible, and if it is divisible, it must be divisible either into indivisibles or into divisibles that are infinitely divisible, in which case it is continuous. Manifestum autem est quod omne continuum est divisibile in semper divisibilia. Si enim in indivisibilia divideretur continuum, esset indivisibile indivisibile tangens: unum enim est ultimum continuorum et quae tanguntur. Moreover, it is plain that everything continuous is divisible into divisibles that are infinitely divisible; if it were divisible into indivisibles, we should have an indivisible in contact with an indivisible, since the extremities of things that are continuous with one another are one and are in contact. 750. Postquam Philosophus determinavit de divisione motus in suas species, et de unitate et contrarietate motuum et quietum, in hoc sexto libro intendit determinare ea quae pertinent ad divisionem motus, secundum quod dividitur in partes quantitativas. 750. After the Philosopher has finished dividing motion into its species and discussing the unity and contrariety of motions and of states of rest, he proposes in book 6 to discuss the things that pertain to the division of motion precisely as it is divisible into quantitative parts. Et dividitur in partes duas. The whole book is divided into two parts. In prima ostendit motum, sicut et omne continuum, esse divisibilem; In the first, he shows that motion, as every continuum, is divisible; in secunda ostendit qualiter motus dividatur, ibi: necesse est autem et ipsum nunc etc. in the second, he shows how motion is divided, at the present also is (233b33; [787]). Prima autem pars dividitur in duas: The first part is subdivided into two sections: in prima ostendit nullum continuum ex indivisibilibus componi; in the first, he shows that no continuum is composed of indivisibles; in secunda ostendit nullum continuum indivisibile esse, ibi: manifestum igitur ex dictis est etc. in the second, that no continuum is indivisible, at it is evident, then (233b15; [785]). Prima autem pars dividitur in duas: The first is further subdivided into two parts: in prima ostendit nullum continuum ex indivisibilibus componi; in the first, he shows that no continuum is composed of indivisibles; in secunda parte (quia probationes praemissae magis ad magnitudinem pertinere videntur) ostendit quod eadem ratio est de magnitudine, motu et tempore, ibi: eiusdem autem rationis est etc. in the second (because the proofs for the first seem to be applicable mainly to magnitudes), he shows that the same proofs apply to magnitudes, to motion, and to time, at the same reasoning applies (231b18; [758]). Circa primum duo facit: In regard to the first part, he does two things: primo resumit quasdam definitiones supra positas, quibus nunc utitur ad propositum demonstrandum; first, he recalls some definitions previously given, with a view to using them in demonstrating his proposition; secundo probat propositum, ibi: neque enim unum sunt etc. second, he proves the proposition, at for the extremities of (231a26; [752]). 751. Dicit ergo primo quod si definitiones prius positae continui, et eius quod tangitur, et eius quod est consequenter, sunt convenientes (scilicet quod continua sint, quorum ultima sunt unum: contacta, quorum ultima sunt simul: consequenter autem sint, quorum nihil est medium sui generis), 751. He says first (231a21) that, if the previously given definitions of continuum, of that which is touched, and of that which is consecutive are correct (namely, that continua are things whose extremities are one; touched things are those whose extremities are together; and consecutive things are those between which nothing of the same type intervenes), ex his sequitur quod impossibile sit aliquod continuum componi ex indivisibilibus, ut lineam ex punctis; si tamen linea dicatur aliquid continuum, et punctum aliquid indivisibile. then it would follow that it is impossible for any continuum to be composed of indivisibles; that is, it is impossible, for example, for a line to be composed of points, provided, of course, that a line is said to be a continuum and that a point is an indivisible. Addit autem hoc, ne aliquis nomine lineae et puncti aliter uteretur. This proviso is added to prevent other meanings being attached to “point” and “line.” 752. Deinde cum dicit: neque enim unum sunt etc., probat propositum. 752. Then, at for the extremities of (231a26), he proves the proposition: Et primo inducit rationes duas ad probandum propositum; first, he gives two proofs of the proposition; secundo manifestat quaedam quae poterant esse dubia in suis probationibus, ibi: nullum autem aliud genus etc. second, he explains things that might be misunderstood in his proofs, at nor can there be (231b12; [756]). Circa primam rationem duo facit: In regard to the first proof, he does two things: primo ostendit quod ex indivisibilibus non componitur aliquod continuum, neque per modum continuationis, neque per modum contactus; first, he shows that no continuum is composed of indivisibles, either after the manner of continuity or after that of contact; secundo quod neque per modum consequenter se habentium, ibi: at vero neque consequenter etc. second, nor after the manner of things that are consecutive, at nor, again, can a point (231b6; [754]). Circa primum ponit duas rationes, In regard to the first, he gives two reasons. quarum prima talis est. Ex quibuscumque componitur aliquid unum, vel per modum continuationis, vel per modum contactus, oportet quod habeant ultima quae sint unum, vel quae sint simul. Sed ultima punctorum non possunt esse unum: quia ultimum dicitur respectu alicuius partis; in indivisibili autem non est accipere aliquid quod sit ultimum, et aliud quod sit aliqua alia pars. The first is that, whatever things a unit is composed of, either after the manner of continuity or after that of contact, the extremities must either be one, or they must be together. But the extremities of points cannot be one, because an extremity is spoken of in relation to a part, but in an indivisible, it is impossible to distinguish that which is an extremity from something else that is a part.