Authors/Thomas Aquinas/metaphysics/liber1/lect8

Lecture 8

Latin	English
lib. 1 l. 8 n. 1 Hic ponit opinionem Pythagoricorum de principiis. Et circa hoc duo facit. Primo ostendit quid circa rerum principia aestimabant. Secundo ad quod genus causae principia ab eis posita reducuntur, ibi, ab his igitur ambobus et cetera. Ponit autem circa primum tres opiniones. Secunda incipit ibi, eorumdem autem alii et cetera. Tertia ibi, quemadmodum videtur. Dicit ergo primo, quod huius gratia venit ad opiniones Pythagoricorum recitandas, ut ostenderet per eorum opiniones, quae sunt rerum principia, et quomodo rerum principia ab eis posita incidunt in causas suprapositas. Videntur enim Pythagorici ponere numerum esse principium entium sicut numerum, et passiones numeri esse sicut passiones entium, et sicut habitus; ut per passiones intelligamus accidentia cito transeuntia, per habitus accidentia permanentia. Sicut ponebant quod passio alicuius numeri secundum quam dicitur aliquis numerus par, erat iustitia propter aequalitatem divisionis, quia talis numerus aequaliter per media dividitur usque ad unitatem, sicut octonarius in duos quaternarios, quaternarius vero in duos binarios, et binarius in duas unitates. Et simili modo alia accidentia rerum assimilabant accidentibus numerorum.	124. Here he states what the Pythagoreans had to say about the principles of things. In regard to this he does two things. First, he expounds their opinions about the principles of things; and second (132), he indicates to what class of cause the principles laid down by them are reduced (“From both of these”). In regard to the first he gives three opinions. The second (127) begins at the words “But other members”; and the third (131), where he says “Alcmaeon of Croton.” He says first (59), then, that the reason he came to examine the opinions of the Pythagoreans is that he might show from their opinions what the principles of things are and how the principles laid down by them fall under the causes given above. For the Pythagoreans seem to hold that number is the principle of existing things as matter,1 and that the attributes of number are the attributes and states of existing things. By “attributes” we mean transient accidents, and by “states,” permanent accidents. They also held that the attribute of any number according to which any number is said to be even is justice, because of the equality of division, since such a number is evenly divided into two parts right down to the unit. For example, the number eight is divided into two fours, the number four into two twos, and the number two into two units. And in a similar way they likened the other accidents of things to the accidents of numbers.
lib. 1 l. 8 n. 2 Principia vero numerorum dicebant esse par et impar, quae sunt primae numerorum differentiae. Paremque numerum dicebant esse principium infinitatis, imparem vero principium finitatis, sicut exponitur in tertio physicorum: quia infinitum in rebus praecipue videtur sequi divisionem continui. Par autem est numerus aptus divisioni. Impar enim sub se numerum parem concludit addita unitate, quae indivisionem causat. Probat etiam hoc, quia numeri impares per ordinem sibi additi semper retinent figuram quadrati, pares autem figuram variant. Ternarius enim unitati quae est principium numerorum additus facit quaternarium, qui primus est quadratus. Nam bis duo quatuor sunt. Rursus quaternario quinarius additus, qui est impar, secundum novenarium constituit, qui est etiam quadratus: et sic de aliis. Si vero binarius qui est primus par, unitati addatur, triangularem numerum constituit, scilicet ternarium. Cui si addatur quaternarius, qui est secundus par, constituit heptangulum numerum, qui est septenarius. Et sic deinceps numeri pares sibiinvicem additi, figuram non eamdem servant. Et hac ratione infinitum attribuebant pari, finitum vero impari. Et quia finitum est ex parte formae, cui competit vis activa, ideo pares numeros dicebant esse feminas, impares vero masculos.	125. in fact, they said that the even and odd, which are the first differences of numbers, are the principles of num hers. And they said that even number is the principle of unlimitedness and odd number the principle of limitation, as is shown in the Physics, Book III, because in reality the unlimited seems to result chiefly from the division of the continuous. But an even number is capable of division; for an odd number includes within itself an even number plus a unit, and this makes it indivisible. He also proves this as followswhen odd numbers are added to each other successively, they always retain the figure of a square, whereas even numbers change their figure. For when the number three is added to the unit, which is the principle of numbers, the number four results, which is the first square [number], because 2 x 2 = 4. Again, when the number five, which is an odd number, is added to the number four, the number nine results, which is also a square number; and so on with the others. But if the number two, which is the first even number, is added to the number one, a triangular number results, i.e., the number three. And if the number four, which is the second even number, is added to the number three, there results a septangular number, i.e., the number seven. And when even numbers are added to each other successively in this way, they do not retain the same figure. This is why they attributed the unlimited to the even and the limited to the odd. And since limitedness pertains to form, to which active power belongs, they therefore said that even numbers are feminine, and odd numbers, masculine.
lib. 1 l. 8 n. 3 Ex his vero duobus, scilicet pari et impari, finito et infinito, non solum numerum constituebant, sed etiam ipsum unum, idest unitatem. Unitas enim et par est virtute et impar. Omnes enim differentiae numeri unitati conveniunt in virtute, quia quaecumque differentiae numeri in unitate resolvuntur. Unde in ordine imparium primum invenitur unitas. Et similiter in ordine parium et quadratorum et perfectorum numerorum, et sic de aliis numeri differentiis: quia unitas licet non sit actu aliquis numerus, est tamen omnis numerus virtute. Et sicut unum dicebat componi ex pari et impari, ita numerum ex unitatibus: caelum vero et omnia sensibilia ex numeris. Et hic erat ordo principiorum quem ponebant.	126. From these two, namely, the even and odd, the limited and unlimited, they produced not only number but also the unit itself, i.e., unity. For unity is virtually both even and odd; because all differences of number are virtually contained in the unit; for all differences of number are reduced to the unit. Hence, in the list of odd numbers the unit is found to be the first. And the same is true in the list of even numbers, square numbers, and perfect numbers. This is also the case with the other differences of number, because even though the unit is not actually a number, it is still virtually all numbers. And just as the unit is said to be composed of the even and odd, in a similar way number is composed of units. In fact, [according to them], the heavens and all sensible things are composed of numbers. This was the sequence of principles which they gave.
lib. 1 l. 8 n. 4 Deinde cum dicit eorumdem autem hic ponit aliam opinionem Pythagoricorum de principiis; dicens, quod de numero eorumdem Pythagoricorum fuerunt aliqui, qui non posuerunt unam tantum contrarietatem in principiis, sicut praedicti; sed posuerunt decem principia secundum coelementationem dicta, idest accipiendo unumquodque illorum cum suo coelemento, idest cum suo contrario. Et huius positionis ratio fuit, quia non solum accipiebant prima principia, sed etiam proxima principia singulis rerum generibus attributa. Ponebant ergo primo finitum et infinitum, sicut et illi qui praedicti sunt; et consequenter par et impar, quibus finitum et infinitum attribuitur. Et quia par et impar sunt prima rerum principia, et primo ex eis causantur numeri, ponebant tertio differentiam numerorum, scilicet unum et plura, quae duo ex pari et impari causabantur. Et quia ex numero constituebantur magnitudines, secundum quod in numeris positionem accipiebant (nam secundum eos punctus nihil aliud erat quam unitas positionem habens, et linea dualitas positionem habens), ideo consequenter ponebant principia positionum dextrum et sinistrum. Dextrum enim invenitur perfectum, sinistrum autem imperfectum. Et ideo dextrum erat ex parte imparis, sinistrum ex parte paris. Quia vero naturalia super magnitudines mathematicas addunt virtutem activam et passivam, ideo ulterius ponebant principia masculum et feminam. Masculum enim ad virtutem activam pertinet, femineum ad passivam: quorum masculum pertinet ad imparem, femineum vero ad parem numerum, ut dictum est.	127. But other members (60). Here he gives another opinion which the Pythagoreans held about the principles of things. He says that among these same Pythagoreans there were some who claimed that there is not just one contrariety in principles, as the foregoing did, but ten principles, which are presented as co-elements, that is, by taking each of these principles with its co-principle, or contrary. The reason for this position was that they took not only the first principles but also the proximate principles attributed to each class of things. Hence, they posited first the limited and the unlimited, as did those who have just been mentioned; and subsequently the even and the odd, to which the limited and unlimited are attributed. And because the even and odd are the first principles of things, and numbers are first produced from them, they posited, third, a difference of numbers, namely, the one and the many, both of which are produced from the even and the odd. Again, because continuous quantities are composed of numbers, inasmuch as they understood numbers to have position (for according to them the point was merely the unit having position, and the line the number two having position), they therefore claimed next that the principles of positions are the right and left; for the right is found to be perfect and the left imperfect. Therefore the right is determined from the aspect of oddness, and the left from the aspect of evenness. But because natural bodies have both active and passive powers in addition to mathematical extensions, they therefore next maintained that masculine and feminine are principles. For masculine pertains to active power, and feminine to passive power; and of these masculine pertains to odd number and feminine to even number, as has been stated (125).
lib. 1 l. 8 n. 5 Ex virtute autem activa et passiva sequitur in rebus motus et quies: quorum motus quia deformitatem habet et alteritatem, in ordine infiniti et paris ponitur, quies vero in ordine finiti et imparis. Differentiae autem motuum primae sunt circulare et rectum. Et ideo consequenter rectum ad parem numerum pertinet; unde et lineam rectam dualitatem esse dicebant. Curvum vero sive circulare ratione uniformitatis pertinet ad imparem, qui indivisionem ex forma unitatis retinet.	128. Now it is from active and passive power that motion and rest originate in the world; and of these motion is placed in the class of the unlimited and even, because it partakes of irregularity and otherness, and rest in the class of the unlimited and odd. Furthermore, the first differences of motions are the circular and straight, so that as a consequence of this the straight pertains to even number. Hence they said that the straight line is the number two; but that the curved or circular line, by reason of its uniformity, pertains to odd number, which retains its undividedness because of the form of unity.
lib. 1 l. 8 n. 6 Nec solum ponebant principia rerum quantum ad actiones naturales et motus, sed etiam quantum ad actiones animales. Et quantum quidem ad cognitionem ponebant lucem et tenebras: quantum vero ad appetitum, bonum et malum. Nam lux est cognitionis principium, tenebra vero ignorantiae ascribitur. Bonum etiam est in quod appetitus tendit, malum vero a quo recedit.	129. And they not only posited principles to account for the natural operations and motions of things, but also to account for the operations of living things. In fact, they held that light and darkness are principles of knowing, but that good and evil are principles of appetite. For light is a principle of knowing, whereas darkness is ascribed to ignorance; and good is that to which appetite tends, whereas evil is that from which it turns away.
lib. 1 l. 8 n. 7 Diversitas autem perfectionis et imperfectionis non solum in naturalibus et voluntariis virtutibus et motibus invenitur, sed etiam in magnitudine et figuris. Quae quidem figurae intelliguntur ut supervenientes substantiis magnitudinum, sicut virtutes motus et actiones substantiis rerum naturalium. Et ideo quantum ad hoc ponebant principium quadrangulare, idest quadratum, et altera parte longius. Dicitur autem quadratum figura constans ex quatuor lateribus aequalibus, cuius quatuor anguli sunt recti; et provenit talis figura ex ductu alicuius lineae in seipsam. Unde cum ex ipsa unitate causetur, ad numerum imparem pertinet. Figura vero altera parte longior dicitur, cuius omnes anguli sunt recti, et latera vicissim sibi opposita sunt aequalia, non tamen omnia latera sunt aequalia omnibus. Unde patet quod sicut quadratum consurgit ex ductu unius lineae in seipsam, ita figura altera parte longior, ex ductu duarum linearum in unam. Et sic pertinet ad numerum parem, qui primus est dualitas.	130. Again, [according to them] the difference of perfection and imperfection is found not only in natural things and in voluntary powers and motions, but also in continuous quantities and figures. These figures are understood to be something over and above the substances of continuous quantities, just as the powers responsible for motions and operations are something over and above the substances of natural bodies. Therefore with reference to this they held that what is quadrangular, i.e., the square and oblong, is a principle. Now a square is said to be a figure of four equal sides, whose four angles are right angles; and such a figure is produced by multiplying a line by itself. Therefore, since it is produced from the unit itself, it belongs to the class of odd number. But an oblong is defined as a figure whose angles are all right angles and whose opposite sides alone, not all sides, are equal to each other. Hence it is clear that, just as a square is produced by multiplying one line by itself, in a similar way an oblong is produced by multiplying one line by another. Hence it pertains to the class of even number, of which the first is the number two.
lib. 1 l. 8 n. 8 Deinde cum dicit quemadmodum hic ponit tertiam opinionem Pythagoricorum, dicens, quod Alcmaeon Crotoniates, sic dictus a civitate unde oriundus fuit, videtur suscipere quantum ad aliquid idem quod praedicti Pythagorici dixerunt, scilicet quod plura contraria sint principia. Aut enim accepit a Pythagoricis, aut illi ab isto. Et quod utrumque esse potuerit, patet per hoc quod fuit contemporaneus Pythagoricorum: ita tamen quod incoepit philosophari Pythagora sene existente. Sed qualitercumque fuerit, multum similiter enunciavit Pythagoricis. Dixit enim multa quae sunt humanorum idest multa rerum sensibilium esse in quadam dualitate constituta, intelligens per dualitatem opposita contrarie. Sed tamen in hoc differt a praedictis, quia Pythagorici dicebant determinatas contrarietates esse rerum principia. Sed ille proiecit quasi inordinate ponens quascumque contrarietates, quae a fortuna ad mentem suam deveniebant, esse rerum principia: sicut album nigrum, dulce amarum, et sic de aliis.	131. Akmaeon of Croton (61). Here he gives the third opinion of the Pythagoreans, saying that Alcmaeon of Croton, so named from the city in which he was raised, seems to maintain somewhat the same view as that expressed by these Pythagoreans, namely, that many contraries are the principles of things. For either he derives the theory from the Pythagoreans, or they from him. That either of these might be true is clear from the fact that he was a contemporary of the Pythagoreans, granted that he began to philosophize when Pythagoras was an old man. But whichever happens to be true, he expressed views similar to those of the Pythagoreans. For he said that many of the things “in the realm of human affairs,” i.e., many of the attributes of sensible things are arranged in pairs, understanding by pairs opposites which are contrary. Yet in this matter he differs from the foregoing philosophers, because the Pythagoreans said that determinate contraries are the principles of things. But he throws them in, as it were, without any order, holding that any of the contraries which he happened to think of are the principles of things, such as white and black, sweet and bitter, and so on.
lib. 1 l. 8 n. 9 Deinde cum dicit ab his igitur hic colligit ex praedictis quid Pythagorici de principiis senserunt, et quomodo principia ab eis posita ad aliquod genus causae reducantur. Dicit ergo quod ex ambobus praedictis, scilicet Alcmaeone et Pythagoricis una communis opinio accipi potest, scilicet quod principia entium sunt contraria; quod non est ab aliis dictum. Quod intelligendum est circa causam materialem. Nam circa causam efficientem posuit Empedocles contrarietatem. Antiqui vero naturales, contraria posuerunt principia, ut rarum et densum; contrarietatem tamen ex parte formae assignantes. Empedocles vero etsi principia materialia posuerit quatuor elementa, non tamen posuit ea principia prima materialia ratione contrarietatis, sed propter eorum naturas et substantiam: isti vero contrarietatem ex parte materiae posuerunt.	132. From both of these (62). Here he gathers together from the above remarks what the Pythagoreans thought about the principles of things, and how the principles which they posited are reduced to some class of cause. He says, then, that from both of those mentioned above, namely, Alcmaeon and the Pythagoreans, it is possible to draw one common opinion, namely, that the principles of existing things are contraries; which was not expressed by the other thinkers. This must be understood with reference to the material cause. For Empedocles posited contrariety in the case of the efficient cause; and the ancient philosophers of nature posited contrary principles, such as rarity and density, although they attributed contrariety to form. But even though Empedocles held that the four elements are material principles, he still did not claim that they are the first material principles by reason of contrariety but because of their natures and substance. These men, however, attributed contrariety to matter.
lib. 1 l. 8 n. 10 Quae etiam sint ista contraria quae isti posuerunt, patet ex dictis. Sed quomodo praedicta principia contraria ab eis posita possunt conduci idest reduci ad praedictas species causarum, non est manifeste articulatum, idest distincte expressum ab eis. Tamen videtur quod huiusmodi principia ordinentur secundum speciem causae materialis. Dicunt enim quod substantia rerum constituitur et plasmatur ab istis principiis, sicut ex his quae insunt: quod est ratio causae materialis. Materia enim est ex qua fit aliquid cum insit. Quod quidem dicitur ad differentiam privationis, ex qua etiam dicitur aliquid fieri, non tamen inest, sicut dicitur musicum fieri ex non musico.	133. The nature of the contraries posited by these men is evident from the foregoing discussion. But how the aforesaid contrary principles posited by them can be “brought together under,” i.e., reduced to, the types of causes described, is not clearly “expressed,” i.e., distinctly stated, by them. Yet it seems that such principles are allotted to the class of material cause; for they say that the substance of things is composed and moulded out of these principles as something inherent, and this is the notion of a material cause. For matter is that from which a thing comes to be as something inherent. This is added to distinguish it from privation, from which something also comes to be but which is not inherent, as the musical is said to come from the non-musical.