Authors/Thomas Aquinas/metaphysics/liber1/lect16

Lecture 16

Latin	English
lib. 1 l. 16 n. 1 Hic improbat opinionem Platonis de speciebus inquantum ponebat eas esse numeros. Et circa hoc duo facit. Primo disputat contra ea quae posita sunt ab ipso de numeris. Secundo contra ea quae posita sunt ab ipso de aliis mathematicis, ibi, volentes autem substantias et cetera. Circa primum ponit sex rationes: quarum prima talis est. Eorum quae sunt idem secundum substantiam, unum non est causa alterius: sed sensibilia secundum substantiam sunt numeri secundum Platonicos et Pythagoricos: si igitur species sunt etiam numeri, non poterunt species esse causae sensibilium.	239. Here he destroys Plato’s opinion about the Forms inasmuch as Plato claimed that they are numbers. In regard to this he does two things. First, he argues dialectically against Plato’s opinion about numbers, and second (254), against his opinion about the other objects of mathematics (“Now when we wish”). In regard to the first part he gives six arguments. The first (122) is this: in the case of things which are substantially the same, one thing is not the cause of another. But sensible things are substantially numbers according to the Platonists and Pythagoreans. Therefore, if the Forms themselves are numbers, they cannot be the cause of sensible things.
lib. 1 l. 16 n. 2 Si autem dicatur quod alii numeri sunt species, et alii sunt sensibilia, sicut ad literam Plato ponebat: ut si dicamus quod hic numerus est homo, et ille alius numerus est Socrates et alius numerus est Callias, istud adhuc non videtur sufficere: quia secundum hoc sensibilia et species conveniunt in ratione numeri: et eorum, quae sunt idem secundum rationem, unum non videtur esse causa alterius: ergo species non erunt causae horum sensibilium.	240. But if it is said that some numbers are Forms and others are sensible filings, as Plato literally held (as though We were to say that this number is man and another is Socrates and still another is Callias), even this would not seem to be sufficient; for according to this view the intelligible structure of number will be common both to sensible things and the Forms. But in the case of things which have the same intelligible structure, one does not seem to be the cause of another. Therefore the Forms will not be the causes of sensible things.
lib. 1 l. 16 n. 3 Non iterum potest dici quod sunt causae; quia illi numeri, si sunt species, sunt sempiterni. Illa enim differentia non sufficit ad hoc quod quaedam ponantur causae aliorum; quia aliqua differunt per sempiternum et non sempiternum secundum esse suum absolute consideratum; sed per causam et causatum differunt secundum habitudinem unius ad alterum: ergo diversa numero non differunt per causam et causatum per hoc, quod quaedam sunt sempiterna, et quaedam non sempiterna.	241. Nor again can it be said that they are causes for the reason that, if those numbers are Forms, they are eternal. For this difference, namely, that some things differ from others in virtue of being eternal and non-eternal in their own being considered absolutely, is not sufficient to explain why some things are held to be the causes of others. Indeed, things differ from each other as cause and effect rather because of the relationship which one has to the other. Therefore things that differ numerically do not differ from each other as cause and effect because some are eternal and some are not.
lib. 1 l. 16 n. 4 Si autem dicatur quod haec sensibilia sunt quaedam rationes, idest proportiones numerorum, et per hunc modum numeri sunt causae horum sensibilium, sicut videmus in symphoniis, idest in musicis consonantiis, quia numeri dicuntur esse causae consonantiarum, inquantum proportiones numerales, quae applicantur sonis, consonantias reddunt: palam est quod oportebat praeter ipsos numeros in sensibilibus ponere aliquod unum secundum genus, cui applicantur proportiones numerales: ut scilicet eorum, quae sunt illius generis proportiones, sensibilia constituant; sicut praeter proportiones numerales in consonantiis inveniuntur soni. Si autem illud, cui applicatur illa proportio numeralis in sensibilibus est materia, manifestum est quod oportebat dicere, quod ipsi numeri separati qui sunt species, sint proportiones alicuius unius, scilicet ad aliquod aliud. Oportet enim dicere quod hic homo, qui est Callias vel Socrates, est similis homini ideali qui dicitur autosanthropos idest per se homo. Si igitur Callias non est numerus tantum, sed magis est ratio quaedam vel proportio in numeris elementorum, scilicet ignis, terrae, aquae et aeris; et ipse homo idealis erit quaedam ratio vel proportio in numeris aliquorum; et non erit homo idealis numerus per suam substantiam. Ex quo sequitur, quod nullus numerus erit praeter ea, id est praeter res numeratas. Si enim numerus specierum est maxime separatus, et ille non est separatus a rebus, sed est quaedam proportio rerum numeratarum, nunc nullus alius numerus erit separatus: quod est contra Platonicos.	242. Again, it is said that sensible things are certain “ratios” or proportions of numbers, and that numbers are the causes of these sensible things, as we also observe to be the case “in harmonies,” i.e., in the combinations of musical notes. For numbers are said to be the causes of harmonies insofar as the numerical proportions applied to sounds yield harmonies. Now if the above is true, then just as in harmonies there are found to be sounds in addition to numerical proportions, in a similar way it was obviously necessary to posit in addition to the numbers in sensible things something generically one to which the numerical proportions are applied, so that the proportions of those things which belong to that one genus would constitute sensible things. However, if that to which the numerical proportion in sensible things is applied is matter, evidently those separate numbers, which are Forms, had to be termed proportions of some one thing to something else. For this particular man, called Callias or Socrates, must be said to be similar to the ideal man, called “man-in-himself,” or humanity. Hence, if Callias is not merely a number, but is rather a kind of ratio or numerical proportion of the elements, i.e., of fire, earth, water and air, and if the ideal man-in-himself is a kind of ratio or numerical proportion of certain things, the ideal man will not be a number by reason of its own substance. From this it follows that there will be no number “apart from these,” i.e., apart from the things numbered. For if the number which constitutes the Forms is separate in the highest degree, and if it is not separate from things but is a kind of proportion of numbered things, no other number will now be separate. This is opposed to Plato’s view.
lib. 1 l. 16 n. 5 Sequitur autem, quod homo idealis sit proportio aliquorum numeratorum, sive ponatur esse numerus, sive non: tam enim secundum ponentes substantias esse numeros, quam secundum naturales, qui numeros substantias esse non dicebant, oportet quod in rerum substantiis aliquae proportiones numerales inveniantur: quod patet praecipue ex opinione Empedoclis, qui ponebat unamquamque rerum sensibilium constitui per quamdam harmoniam et proportionem.	243. It also follows that the ideal man is a proportion of certain numbered things, whether it is held to be a number or not. For according to those who held that substances are numbers, and according to the philosophers of nature, who denied that numbers are substances, some numerical proportions must be found in the substances of things. This is most evident in the case of the opinion of Empedocles, who held that each one of these sensible things is composed of a certain harmony or proportion [of the elements].
lib. 1 l. 16 n. 6 Deinde cum dicit amplius ex hic ponit secundam rationem, quae talis est. Ex multis numeris fit unus numerus. Si igitur species sunt numeri, ex multis speciebus fiet una species, quod est impossibile. Nam si ex multis diversarum specierum aliquid unum in specie constituatur, hoc fit per mixtionem, in qua non salvantur species eorum quae miscentur, sicut ex quatuor elementis fit lapis. Et iterum ex huiusmodi diversis secundum speciem non fit aliquod unum ratione specierum, quia ipsae species non coniunguntur ad aliquod unum constituendum, nisi secundum rationem individuorum, qui alterantur, ut possint permisceri: ipsae autem species numerorum binarii et ternarii simul coniunctae numerum constituunt quinarium, ita quod in quinario uterque numerus remanet et salvatur.	244. Again, one number (123). Here he gives the second argument which runs thus: one number is produced from many numbers. Therefore, if the Forms are numbers, one Form is produced from many Forms. But this is impossible. For if from many things which differ specifically something specifically one is produced, this comes about by mixture, in which the natures of the things mixed are not preserved; just as a stone is produced from the four elements. Again, from things of this kind which differ specifically one thing is not produced by reason of the Forms, because the Forms themselves are combined in such a way as to constitute a single thing only in accordance with the intelligible structure of individual things, which are altered in such a way that they can be mixed together. And when the Forms themselves of the numbers two and three are combined, they give rise to the number five, so that each number remains and is retained in the number five.
lib. 1 l. 16 n. 7 Sed quia ad hanc rationem posset responderi ex parte Platonis, quod ex multis numeris non fit unus numerus, sed quilibet numerus immediate constituitur ex unitatibus, ideo consequenter cum dicit sed si nec excludit etiam hanc responsionem. Si enim dicitur quod aliquis numerus maior, ut millenarius, non constituatur ex eis, scilicet ex duobus vel pluribus numeris minoribus, sed constituitur ex unis, idest ex unitatibus, remanebit quaestio quomodo se habent unitates adinvicem, ex quibus numeri constituuntur? Aut enim oportet, quod omnes unitates sint conformes adinvicem, aut quod sint difformes adinvicem.	245. But since someone could answer this argument, in support of Plato, by saying that one number does not come from many numbers, but each number is immediately constituted of units, Aristotle is therefore logical in rejecting this answer (124) (“But if one number”). For if it is said that some greater number, such as ten thousand, is not produced “from them,” namely, from twos or many smaller numbers, but from “units,” i.e., ones, this question will follow: How are the units of which numbers are composed related to each other? For all units must either conform with each other or not.
lib. 1 l. 16 n. 8 Sed primo modo sequuntur multa inconvenientia, et praecipue quantum ad ponentes species esse numeros; quia sequitur quod diversae species non differant secundum substantiam, sed solum secundum excessum unius speciei super aliam. Inconveniens etiam videtur, quod unitates nullo modo differant; et tamen sunt multae, cum diversitas multitudinem consequatur.	246. But many absurd conclusions follow from the first alternative, especially for those who claim that the Forms are numbers. For it will follow that different Forms do not differ substantially but only insofar as one Form surpasses another. It also seems absurd that units should differ in no way and yet be many, since difference is a result of multiplicity.
lib. 1 l. 16 n. 9 Si vero non sint conformes, hoc potest esse dupliciter. Uno modo, quia unitates unius numeri sunt differentes ab unitatibus alterius numeri, sicut unitates binarii ab unitatibus ternarii; et tamen unitates unius et eiusdem numeri sibi invicem sunt conformes. Alio modo ut unitates eiusdem numeri non sibi invicem, nec unitatibus alterius numeri conformes existant. Hanc divisionem significat, cum dicit, nec eaedem sibi invicem, idest quae ad eumdem numerum pertinent, nec aliae omnes etc., scilicet quae pertinent ad diversos numeros. Quocumque autem modo ponatur difformitas inter unitates, videtur inconveniens. Nam omnis difformitas est per aliquam formam vel passionem; sicut videmus quod corpora difformia differunt calido et frigido, albo et nigro, et huiusmodi passionibus: unitates autem huiusmodi passionibus carent, cum sint impassibiles secundum Platonicos; ergo non poterit inter ea poni talis difformitas vel differentia, quae causatur ab aliqua passione. Et sic patet quod ea quae Plato ponit de speciebus et numeris, nec sunt rationabilia, sicut illa quae per certam rationem probantur, nec sunt intelligentiae confessa, sicut ea quae sunt per se nota, et solo intellectu certificantur, ut prima demonstrationis principia.	247. But if they do not conform, this can happen in two ways. First, they can lack conformity because the units of one number differ from those of another number, as the units of the number two differ from those of the number three, although the units of one and the same number will conform with each other. Second, they can lack conformity insofar as the units of one and the same number do not conform with each other or with the units of another number. He indicates this distinction when he says, “For neither will they be the same as one another (125),” i.e., the units which comprise the same number, “nor all the others the same as all,” i.e., those which belong to different numbers. Indeed, in whatever way there is held to be lack of conformity between units an absurdity is apparent. For every instance of non-conformity involves some form or attribute, just as we see that bodies which lack conformity differ insofar as they are hot and cold, white and black, or in terms of similar attributes. Now units lack qualities of this kind, because they have no qualities, according to Plato. Hence it will be impossible to hold that there is any non-conformity or difference between them of the kind caused by a quality. Thus it is evident that Plato’s opinions about the Forms and numbers are neither “reasonable” (for example, those proved by an apodictic argument), nor “in accord with our understanding” (for example, those things which are self-evident and verified by [the habit of] intellect alone, as the first principles of demonstration).
lib. 1 l. 16 n. 10 Deinde cum dicit amplius autem hic ponit tertiam rationem contra Platonem, quae talis est. Omnia mathematica, quae a Platone sunt dicta intermedia sensibilium et specierum, sunt ex numeris, aut simpliciter, sicut ex propriis principiis, aut sicut ex primis. Et hoc ideo dicit, quia secundum unam viam videtur quod numeri sint immediata principia aliorum mathematicorum; nam unum dicebant constituere punctum, binarium lineam, ternarium superficiem, quaternarium corpus. Secundum vero aliam viam videntur resolvi mathematica in numeros, sicut in prima principia et non in proxima. Nam corpora dicebant componi ex superficiebus, superficies ex lineis, lineas ex punctis, puncta autem ex unitatibus, quae constituunt numeros. Utroque autem modo sequebatur numeros esse principia aliorum mathematicorum.	248. Further, [if the Forms] (126). Here he gives the third argument against Plato, which runs thus: all objects of mathematics, which Plato affirmed to be midway between the Forms and sensible substances, are derived unqualifiedly from numbers, either as proper principles, or as first principles. He says this because in one sense numbers seem to be the immediate principles of the other objects of mathematics; for the Platonists said that the number one constitutes the point, the number two the line, the number three surface, and the number four the solid. But in another sense the objects of mathematics seem to be reduced to numbers as first principles and not as proximate ones. For the Platonists said that solids are composed of surfaces, surfaces of lines, lines of points, and points of units, which constitute numbers. But in either way it followed that numbers are the principles of the other objects of mathematics.
lib. 1 l. 16 n. 11 Sicut igitur alia mathematica erant media inter sensibilia et species, ita necessarium est facere aliquod genus numeri, quod sit aliud a numeris qui sunt species, et a numeris qui sunt substantia sensibilium: et quod de huiusmodi numero sit arithmetica, sicut de proprio subiecto, quae est una mathematicarum, sicut geometria de magnitudinibus mathematicis. Hoc autem ponere videtur superfluum esse. Nam nulla ratio poterit assignari quare sunt numeri medii inter praesentia, idest sensibilia et eas scilicet species, cum tam sensibilia quam species sint numeri.	249. Therefore, just as the other objects of mathematics constituted an intermediate class between sensible substances and the Forms, in a similar way it was necessary to devise some class of number which is other than the numbers that constitute the Forms and other than those that constitute the substance of sensible things. And arithmetic, which is one of the mathematical sciences, evidently deals with this kind of number as its proper subject, just as geometry deals with mathematical extensions. However, this position seems to be superfluous; for no reason can be given why number should be midway “between the things at hand,” or sensible things, and “those in the ideal world,” or the Forms, since both sensible things and the Forms are numbers.
lib. 1 l. 16 n. 12 Deinde cum dicit amplius autem hic ponit quartam rationem, quae talis est. Ea quae sunt in sensibilibus et in mathematicis sunt causata ex speciebus: si igitur aliqua dualitas in sensibilibus et in mathematicis invenitur, oportet quod utraque unitas huius posterioris dualitatis sit causata ex priori dualitate, quae est species dualitatis. Et hoc est impossibile, scilicet quod unitas ex dualitate causetur. Hoc enim praecipue oportet dicere, si unitates unius numeri sint alterius speciei ab unitatibus alterius, quia tunc a specie ante illius numeri unitates, species sortientur. Et sic oportet quod unitates posterioris dualitatis sint causatae ex priori dualitate.	250. Again, each of the units (127). Here he gives the fourth argument, which runs thus: those things which exist in the sensible world and those which exist in the realm of mathematical entities are caused by the Forms. Therefore, if some number two is found both in the sensible world and in the realm of the objects of mathematics, each unit of this subsequent two must be caused by a prior two, which is the Form of twoness. But it is “impossible” that unity should be caused by duality. For it would be most necessary to say this if the units of one number were of a different species than those of another number, because then these units would acquire their species from a Form which is prior to the units of that number. And thus the units of a subsequent two would have to be produced from a prior two.
lib. 1 l. 16 n. 13 Deinde cum dicit amplius quare hic ponit quintam rationem, quae talis est. Multa non conveniunt ad unum constituendum, nisi propter aliquam causam, quae potest accipi vel extrinseca, sicut aliquod agens quod coniungit, vel intrinseca, sicut aliquod vinculum uniens. Vel si aliqua uniuntur per seipsa, oportet ut unum sit ut potentia, et aliud ut actus. Nullum autem horum potest dici in unitatibus quare numerus idest ex qua causa numerus erit quoddam comprehensum, idest congregatum ex pluribus unitatibus: quasi dicat: non erit hoc assignare.	251. Further, why is (128). Here he gives the fifth argument, which runs thus: many things combine so as to constitute one thing only by reason of some cause, which can be considered to be either extrinsic, as some agent which unites them, or intrinsic, as some unifying bond. Or if some things are united of themselves, one of them must be potential and another actual. However, in the case of units none of these reasons can be said to be the one “why a number,” i.e., the cause by which a number, will be a certain “combination,” ‘ i.e., collection of many units; as if to say, it will be impossible to give any reason for this.
lib. 1 l. 16 n. 14 Deinde cum dicit amplius autem hic ponit sextam rationem, quae talis est. Si numeri sunt species et substantiae rerum, oportet, sicut praemissum est, dicere vel quod unitates sint differentes, aut convenientes. Si autem differentes, sequitur quod unitas, inquantum unitas, non sit principium. Quod patet per similitudinem sumptam a naturalium positione. Naturales enim aliqui posuerunt quatuor corpora esse principia. Quamvis autem commune sit ipsis hoc quod est esse corpus, non tamen ponebant corpus commune esse principium, sed magis ignem, terram, aquam et aerem, quae sunt corpora differentia. Unde, si unitates sint differentes, quamvis omnes conveniant in ratione unitatis, non tamen erit dicendum, quod ipsa unitas inquantum huiusmodi sit principium; quod est contra positionem Platonicorum. Nam nunc ab eis dicitur, quod unum sit principium, sicut primo de naturalibus dicitur quod ignis aut aqua aut aliquod corpus similium partium principium sit. Sed si hoc est verum quod conclusum est contra positionem Platonicorum, scilicet quod unum inquantum unum non sit principium et substantia rerum, sequeretur quod numeri non sunt rerum substantia. Numerus enim non ponitur esse rerum substantia, nisi inquantum constituitur ex unitatibus, quae dicuntur esse rerum substantiae. Quod iterum est contra positionem Platonicorum, quam nunc prosequitur, qua scilicet ponitur, quod numeri sint species.	252. And, again, in addition (129). Here he gives the sixth argument, which runs thus: if numbers are the Forms and substances of things, it will be necessary to say, as has been stated before (245), either that units are different, or that they conform. But if they are different, it follows that unity as unity will not be a principle. This is clarified by a similar case drawn from the position of the natural philosophers. For some of these thinkers held that the four [elemental] bodies are principles. But even though being a body is common to these elements, these philosophers did not maintain that a common body is a principle, but rather fire, earth, water and air, which are different bodies. Therefore, if units are different, even though all have in common the intelligible constitution of unity, it will not be said that unity itself as such is a principle. This is contrary to the Platonists’ position; for they now say that the unit is the principle of things, just as the natural philosophers say that fire or water or some body with like parts is the principle of things. But if our conclusion against the Platonists’ theory is true-that unity as such is not the principle and substance of things-it will follow that numbers are not the substances of things. For number is held to be the substance of things only insofar as it is constituted of units, which are said to be the substances of things. This is also contrary to the Platonists’ position which is now being examined, i.e., that numbers are Forms.
lib. 1 l. 16 n. 15 Si autem dicas quod omnes unitates sunt indifferentes, sequitur quod omne, idest universum totum sit aliquid unum et idem, ex quo substantia rei cuiuslibet est ipsum unum, quod est commune indifferens. Et ulterius sequitur, quod idem illud sit unum principium omnium: quod est impossibile ratione ipsius rationis, quae de se est inopinabilis, ut scilicet sint omnia unum secundum rationem substantiae; tum quia includit contradictionem ex eo quod ponit unam esse substantiam rerum, et tamen ponit illud unum esse principium. Nam unum et idem non est sui ipsius principium: nisi forte dicatur quod unum multipliciter dicitur, ut distincto uno ponantur omnia esse unum genere, et non specie vel numero.	253. But if you say that all units are undifferentiated, it follows that “the whole,” i.e., the entire universe, is a single entity, since the substance of each thing is the one itself, and this is something common and undifferentiated. Further, it follows that the same entity is the principle of all things. But this is impossible by reason of the notion involved, which is inconceivable in itself, namely, that all things should be one according to the aspect of substance. For this view contains a contradiction, since it claims that the one is the substance of all things, yet maintains that the one is a principle. For one and the same thing is not its own principle, unless, perhaps, it is said that “the one” is used in different senses, so that when the senses of the one are differentiated all things are said to be generically one and not numerically or specifically one.
lib. 1 l. 16 n. 16 Volentes autem substantias hic disputat contra positionem Platonis quantum ad hoc quod posuit de magnitudinibus mathematicis. Et primo ponit eius positionem. Secundo obiicit contra ipsam, ibi, attamen quomodo habebit et cetera. Dicit ergo primo, quod Platonici volentes rerum substantias reducere ad prima principia, cum ipsas magnitudines dicerent esse substantias rerum sensibilium, lineam, superficiem et corpus, istorum principia assignantes, putabant se rerum principia invenisse. Assignando autem magnitudinum principia, dicebant longitudines, idest lineas componi ex producto et brevi, eo quod principia rerum omnium ponebant esse contraria. Et quia linea est prima inter quantitates continuas, ei per prius attribuebant magnum et parvum, ut per hoc quod haec duo sunt principia lineae, sint etiam principia aliarum magnitudinum. Dicit autem ex aliquo parvo et magno, quia parvum et magnum etiam in speciebus ponebantur, ut dictum est, sed secundum quod per situm determinatur et quodammodo particulari ad genus magnitudinum, constituunt primo lineam, et deinde alias magnitudines. Planum autem, idest superficiem eadem ratione dicebant componi ex lato et arcto, et corpus ex profundo et humili.	254. Now when we wish (130). Here he argues against Plato’s position with reference to his views about mathematical extensions. First (130), he gives Plato’s position; and second (255), he advances an argument against it (“Yet how will”). He says, first, that the Platonists, wishing to reduce the substances of things to their first principles, when they say that continuous quantities themselves are the substances of sensible things, thought they had discovered the principles of things when they assigned line, surface and solid as the principles of sensible things. But in giving the principles of continuous quantities they said that “lengths,” i.e., lines, are composed of the long and short, because they held that contraries are the principles of all things. And since the line is the first of continuous quantities, they first attributed to it the great and small; for inasmuch as these two are the principles of the line, they are also the principles of other continuous quantities. He says “from the great and small” because the great and small are also placed among the Forms, as has been stated (217). But insofar as they are limited by position, and are thus particularized in the class of continuous quantities, they constitute first the line and then other continuous quantities. And for the same reason they said that surface is composed of the wide and narrow, and body of the deep and shallow.
lib. 1 l. 16 n. 17 Deinde cum dicit attamen quomodo hic obiicit contra praedictam positionem duabus rationibus: quarum prima talis est. Quorum principia sunt diversa, ipsa etiam sunt diversa; sed principia dictarum magnitudinum secundum praedictam positionem sunt diversa. Latum enim et arctum, quae ponuntur principia superficiei, sunt alterius generis quam profundum et humile, quae ponuntur principia corporis. Et similiter potest dici de longo et brevi quod differunt ab utroque; ergo etiam linea et superficies et corpus erunt adinvicem distincta. Quomodo ergo poterat dici quod superficies haberet in se lineam, et quod corpus habeat lineam et superficiem? Et ad huius rationis confirmationem inducit simile de numero. Multum enim et paucum, quae simili ratione ponuntur principia rerum, sunt alterius generis a longo et brevi, lato et stricto, profundo et humili. Et ideo numerus non continetur in his magnitudinibus, sed est separatus per se. Unde et eadem ratione nec superius inter praedicta erit etiam in inferioribus, sicut linea non in superficie, nec superficies in corpore.	255. Yet how will a surface (130). Here he argues against the foregoing position, by means of two arguments. The first is as follows. Things whose principles are different are themselves different. But the principles of continuous quantities mentioned above are different, according to the foregoing position, for the wide and narrow, which are posited as the principles of surface, belong to a different class than the deep and shallow, which are held to be the principles of body. The same thing can be said of the long and short, which differ from each of the above. Therefore, line, surface and body all differ from each other. How then will one be able to say that a surface contains a line, and a body a line and a surface? In confirmation of this argument he introduces a similar case involving number. For the many and few, which are held to be principles of things for a similar reason, belong to a different class than the long and short, the wide and narrow, and the deep and shallow. Therefore number is not contained in these continuous quantities but is essentially separate. Hence, for the same reason, the higher of the above mentioned things will not be contained in the lower; for example, a line will not be contained in a surface or a surface in a body.
lib. 1 l. 16 n. 18 Sed quia posset dici, quod quaedam praedictorum contrariorum sunt genera aliorum, sicut quod longum esset lati genus, et latum genus profundi; hoc removet tali ratione. Sicut habent se principia adinvicem, et principiata: si igitur latum est genus profundi, et superficies erit genus corporis. Et ita corpus erit aliquod planum, idest aliqua species superficiei: quod patet esse falsum.	256. But because it could be said that certain of the foregoing contraries are the genera of the others, for example, that the long is the genus of the broad, and the broad the genus of the deep, he destroys this [objection] by the following argument: things composed of principles are related to each other in the same way as their principles are. Therefore, if the broad is the genus of the deep, surface will also be the genus of body. Hence a solid will be a kind of plane, i.e., a species of surface. This is clearly false.
lib. 1 l. 16 n. 19 Deinde cum dicit amplius puncta hic ponit secundam rationem, quae sumitur ex punctis; circa quam Plato videtur dupliciter deliquisse. Primo quidem, quia cum punctus sit terminus lineae, sicut linea superficiei, et superficies corporis; sicut posuit aliqua principia, ex quibus componuntur praedicta, ita debuit aliquid ponere ex quo existerent puncta; quod videtur praetermisisse.	257. Further, from what will (132). Here he gives the second argument, which involves points; and in regard to this Plato seems to have made two errors. First, Plato claimed that a point is the limit of a line, just as a line is the limit of a surface and a surface the limit of a body. Therefore, just as he posited certain principles of, which the latter are composed, so too he should have posited some principle from which points derive their being. But he seems to have omitted this.
lib. 1 l. 16 n. 20 Secundo, quia circa puncta videbatur diversimode sentire. Quandoque enim contendebat totam doctrinam geometricam de hoc genere existere, scilicet de punctis, inquantum scilicet puncta ponebat principia et substantiam omnium magnitudinum. Et hoc non solum implicite, sed etiam explicite punctum vocabat principium lineae, sic ipsum definiens. Multoties vero dicebat, quod lineae indivisibiles essent principia linearum, et aliarum magnitudinum; et hoc genus esse, de quo sit geometria, scilicet lineae indivisibiles. Et tamen per hoc quod ponit ex lineis indivisibilibus componi omnes magnitudines, non evadit quin magnitudines componantur ex punctis, et quin puncta sint principia magnitudinum. Linearum enim indivisibilium necessarium esse aliquos terminos, qui non possunt esse nisi puncta. Unde ex qua ratione ponitur linea indivisibilis principium magnitudinum, ex eadem ratione et punctum principium magnitudinis ponitur.	258. The second error is this: Plato seems to have held different opinions about points. For sometimes he maintained that the whole science of geometry treats this class of things, namely, points, inasmuch as he held that points are the principles and substance of all continuous quantities. And he not only implied this but even explicitly stated that a point is the principle of a line, defining it in this way. But many times he said that indivisible lines are the principles of lines and other continuous quantities, and that this is the class of things with which geometry deals, namely, indivisible lines. Yet by reason of the fact that he held that all continuous (quantities are composed of indivisible lines, he did not avoid the consequence that continuous quantities are composed of points, and that points are the principles of continuous quantities. For indivisible lines must have some limits, and these can only be points. Hence, by whatever argument indivisible lines are held to be the principles of continuous quantities, by the same argument too the point is held to be the principle of continuous quantity.