Authors/Thomas Aquinas/physics/L6/lect1

Lecture 1 No continuum is composed of indivisibles

Latin	English
	Lecture 1 No continuum is composed of indivisibles
lib. 6 l. 1 n. 1 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. Et dividitur in partes duas. In prima ostendit motum, sicut et omne continuum, esse divisibilem; in secunda ostendit qualiter motus dividatur, ibi: necesse est autem et ipsum nunc et cetera. Prima autem pars dividitur in duas: in prima ostendit nullum continuum ex indivisibilibus componi; in secunda ostendit nullum continuum indivisibile esse, ibi: manifestum igitur ex dictis est et cetera. Prima autem pars dividitur in duas: in prima ostendit nullum continuum ex indivisibilibus componi; 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 et cetera. Circa primum duo facit: primo resumit quasdam definitiones supra positas, quibus nunc utitur ad propositum demonstrandum; secundo probat propositum, ibi: neque enim unum sunt et cetera.	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 this Sixth Book to discuss the things that pertain to the division of motion precisely as it is divisible into quantitative parts. The whole book is divided into two parts. In the first he shows that motion, as every continuum, is divisible; In the second he shows how motion is divided, at L. 5. The first part is subdivided into two sections: In the first he shows that no continuum is composed solely of indivisibles; at L. 4. In the second that no continuum is indivisible, near the end The first is further subdivided into two parts: In the first he shows that no continuum is composed of indivisibles only; 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 L, 2. In regard to the first part he does two things: First he recalls some definitions previously given, with a view to using them in demonstrating his proposition; Secondly, he proves the proposition, at 752.
lib. 6 l. 1 n. 2 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), 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. Addit autem hoc, ne aliquis nomine lineae et puncti aliter uteretur.	751. He says therefore first (562 231 a21) that if the previously given definitions of continuum, of that which is touched, of that which is consecutive to are correct (namely, that continua are things whose extremities are one; contigua are things whose extremities are together; consecutive things are those between which nothing of the same type intervenes), then it would follow that it is impossible for any continuum to be composed solely of indivisibles; i.e., it is impossible, for example, for a line to be composed of points only, provided, of course, that a line is conceded to be a continuum and that a point is an indivisible. This proviso is added to prevent other meanings being attached to point and line.
lib. 6 l. 1 n. 3 Deinde cum dicit: neque enim unum sunt etc., probat propositum. Et primo inducit rationes duas ad probandum propositum; secundo manifestat quaedam quae poterant esse dubia in suis probationibus, ibi: nullum autem aliud genus et cetera. Circa primam rationem duo facit: primo ostendit quod ex indivisibilibus non componitur aliquod continuum, neque per modum continuationis, neque per modum contactus; secundo quod neque per modum consequenter se habentium, ibi: at vero neque consequenter et cetera. Circa primum ponit duas rationes, 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. Similiter non potest dici quod ultima punctorum sunt simul: quia nihil potest esse ultimum rei impartibilis, cum semper alterum sit ultimum et illud cuius est ultimum; in impartibili autem non est accipere aliud et aliud. Relinquitur ergo quod linea non potest componi ex punctis, neque per modum continuationis, neque per modum contactus.	752. Then at (563 231 a26) he proves the proposition: First he gives two proofs of the proposition; Secondly, he explains things that might be misunderstood in his proofs, at 756. In regard to the first proof he does two things: First he shows that no continuum is composed solely of indivisibles, either after the manner of continuity or of contact; Secondly, or after the manner of things that are consecutive, at 754. In regard to the first he gives two reasons, of which the first is: Whatever things a unit is composed of, either after the manner of continuity or 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, whereas in an indivisible it is impossible to distinguish that which is an extremity and something else that is a part. Similarly, it cannot be said that the extremities are together, because nothing can be the extremity of a thing that cannot be divided into parts, whereas an extremity must always be distinct from that of which it is the extremity. But in a thing that cannot be divided into parts, there is no way of distinguishing one thing and another. It follows therefore that a line cannot be composed of points either after the manner of continuity or after the manner of contact.
lib. 6 l. 1 n. 4 Secundam rationem ponit ibi: amplius necesse est etc. quae talis est. Si ex punctis constituitur aliquod continuum, necesse est quod aut sint continua ad invicem, vel se tangant: et eadem ratio est de omnibus aliis indivisibilibus, quod ex eis non componatur continuum. Ad probandum autem quod indivisibilia non possunt sibi invicem esse continua, sufficiat ratio prima. Sed ad probandum quod non possunt se tangere, inducitur alia ratio, quae talis est. Omne quod tangit alterum, aut totum unum tangit totum aliud, aut pars unius partem alterius, aut pars unius totum aliud. Sed cum indivisibile non habeat partem, non potest dici quod pars unius tangat partem alterius, aut pars totum; et sic necesse est, si duo puncta se tangunt, quod totum tangat totum. Sed ex duobus, quorum unum totum tangit aliud totum, non potest componi continuum; quia omne continuum habet partes seiunctas, ita quod haec sit una pars, et haec alia; et dividitur in partes diversas et distinctas loco, idest positione, in his quae positionem habent: quae autem se secundum totum tangunt, non distinguuntur loco vel positione. Relinquitur ergo quod ex punctis non possit componi linea per modum contactus.	753. The second reason is given at (564 231 a29). If a continuum is composed solely of points, they must be either continuous with one another or touch (and the same is true of all other indivisibles, i.e., that no continuum is composed solely of them). To prove that they are not continuous with one another, the first argument suffices. But to prove that they cannot touch one another, another argument is adduced, which is the following: Everything that touches something else does so either by the whole touching the other wholly, or by a part of one touching a part of the other or the whole of the other. But since an indivisible does not have parts, it cannot be said that part of one touches either a part or the whole of the other. Hence if two points touch, the whole point touches another whole point. But when a whole touches a whole, no continuum can be formed, because every continuum has distinct parts so that one part is here and another there, and is divisible into parts that are different and distinct in regard to place, i.e., position (in things that have-position)—whereas things that touch one another totally are not distinguished as to place or position, It therefore follows that a line cannot be composed of points that are in contact.
lib. 6 l. 1 n. 5 Deinde cum dicit: at vero neque etc., probat quod continuum non componatur ex indivisibilibus per modum eius quod est consequenter. Non enim punctum consequenter se habebit ad aliud punctum, ita quod ex eis constitui possit longitudo, idest linea; aut unum nunc alteri nunc, ita quod ex eis possit componi tempus: quia consequenter est unum alteri, quorum non est aliquid medium eiusdem generis, ut supra expositum est. Sed inter duo puncta semper est linea media: et sic si linea composita est ex punctis, ut tu das, sequitur quod semper inter duo puncta sit aliud punctum medium. Et similiter inter duo nunc est tempus medium. Non ergo linea componitur ex punctis, aut tempus ex nunc, sicut consequenter se habentibus.	754. Then at (565 231 b6) he shows that no continuum is composed of indivisibles after the manner of things that are consecutive. For no point will be consecutive to another so as to form a line; and no “now” is consecutive to another “now” so as to form a period of time, because consecutive things are by definition such that nothing of the same kind intervenes between any two. But between any two points there is always a line, and so, if a line is composed of points only, it would follow that between any two points there is always another, mediate, point. The same is true for the “now’s’”. if a period of time is nothing but a series of “now’s”, then between any two “now’s” there would be another “now”. Therefore, no line is composed solely of points, and no time is composed solely of “now’s”, after the manner of things that are consecutive.
lib. 6 l. 1 n. 6 Secundam rationem principalem ponit ibi: amplius dividerentur etc., quae sumitur ex alia definitione continui, quam supra posuit in principio tertii, scilicet quod continuum sit quod est in infinitum divisibile: et est ratio talis. Ex quibuscumque componitur vel linea vel tempus, in ipsa dividitur: si igitur utrumque istorum componitur ex indivisibilibus, sequitur quod in indivisibilia dividatur. Sed hoc est falsum, cum nullum continuorum sit divisibile in impartibilia: sic enim non esset divisibile in infinitum. Nullum igitur continuum componitur ex indivisibilibus.	755. The second reason is given at (566 231 b10) and is based on a different definition of continuum—the one given at the beginning of Book III—that a continuum is “that which is divisible ad infinitum”. Here is the proof: A line or time can be divided into whatsoever they are composed of. If, therefore, each of them is composed of indivisibles, it follows that each is divided into indivisibles. But this is false, since neither of them is divisible into indivisibles, for that would mean they would not be divisible ad infinitum. No continuum, therefore, is composed of indivisibles.
lib. 6 l. 1 n. 7 Deinde cum dicit: nullum autem aliud etc., manifestat duo quae supra dixerat. Quorum primum fuit, quod inter duo puncta sit linea media, et inter duo nunc, tempus. Et hoc manifestat sic. Si sunt duo puncta, oportet quod differant secundum situm: alias non essent duo sed unum. Non autem possunt se contingere, ut supra ostensum est: unde relinquitur quod distent, et sit aliquod medium inter ea. Sed nullum aliud medium potest esse inter ea quam linea inter puncta, et tempus inter nunc. Quod sic probat: quia si inter puncta esset aliud medium quam linea, manifestum est aut illud medium esse indivisibile aut divisibile. Si autem sit indivisibile, oportet quod sit distinctum ab utroque in situ; et cum non tangat, oportet iterum quod sit aliquod alterum medium inter indivisibile quod ponitur medium et extrema, et sic in infinitum, nisi ponatur medium divisibile. Si autem medium duorum punctorum fuerit divisibile, aut erit divisibile in indivisibilia, aut in semper divisibilia. Sed non potest dici quod dividatur in indivisibilia, quia tunc redibit eadem difficultas, quomodo ex indivisibilibus possit componi divisibile. Relinquitur igitur quod illud medium sit divisibile in semper divisibilia. Sed haec est ratio continui: ergo illud medium erit quoddam continuum. Nullum autem aliud continuum potest esse medium inter duo puncta quam linea: ergo inter qualibet duo puncta est linea media. Et eadem ratione inter qualibet duo nunc, tempus; et similiter in aliis continuis.	756. Then at (567 231 b11) he explains two statements he made in the course of his proofs. The first of these was that between two points there is always a line and that between two “now’s” there is always time. He explains it thus: If two points exist, they must differ in position; otherwise, they would not be two, but one. But they cannot touch one another, as was shown above; hence they are distant, and something is between them. But no other intermediate is possible, except a line between two points, and time between “now’s”: for if the intermediate between two points were other than a line, that intermediate must be either divisible or indivisible. If indivisible, it must be distinct from the two points—at least in position—and, since it touches neither, there must be another intermediate between that indivisible and the original extremities and so on ad infinitum, until a divisible intermediate is found. However, if the intermediate is divisible, it will be divisible into indivisibles or into what are further divisible. But it cannot be divided into indivisibles only, because then the same difficulty returns—how a divisible can be composed solely of indivisibles. It must be granted, then, that the intermediate is divisible into what are further divisible. But that is what a continuum is. Therefore, that intermediate will be a continuum. But the only continuous intermediate between two points is a line. Therefore, between any two points there is an intermediate line. Likewise, between two “now’s” there is time; and the same for other types of continua.
lib. 6 l. 1 n. 8 Deinde cum dicit: manifestum autem etc., manifestat secundum, quod supposuerat, scilicet quod omne continuum sit divisibile in divisibilia. Quia si daretur quod continuum esset divisibile in indivisibilia, sequeretur quod duo indivisibilia se contingerent, ad hoc quod possent constituere continuum. Oportet enim quod continuorum sit unum ultimum, ut ex definitione eius apparet, et quod partes continui se tangant: quia si ultima sunt unum, sequitur quod sint simul, ut in quinto dictum est. Cum igitur sit impossibile duo indivisibilia se contingere, impossibile est quod continuum in indivisibilia dividatur.	757. Then at (568 231 b15) he explains the second statement referred to at the beginning of 756, that every continuum is divisible into divisibles. For on the supposition that a continuum is divisible solely into indivisibles, it would follow that two indivisibles would have to be in contact in order to form the continuum. For continua have an extremity that is one, as appears from the definition thereof; moreover, the parts of a continuum must touch, because if the extremities are one, they are together, as was stated in Book V. Therefore, since it is impossible for two indivisibles to touch, it is impossible for a continuum to be divided into indivisibles.