- See Johannis Kepleri astronomi opera omnia Frisch Christian Frankfurt am Main 1871 VIII 145 161 together with the other mathematical writings from volume XXII of the Pulkowo Manuscripts, the latter preserved at the Academy of Sciences of Leningrad. De quantitatibus has not yet appeared in the Caspar edition (W. von Dyck, M. Caspar, and F. Hammer, Johannes Kepler Gesammelte Werke (Munich, 1937- ). I have been working on the photographs of the autograph manuscript available at the Kepler Commission of the Bavarian Academy of Sciences (and for this I wish to thank Dr Martha List and Dr Volker Bialas at the Commission) as well as on the Frisch edition. I have indicated in the footnotes the (rare but sometimes not irrelevant) points in which my text does not coincide with Frisch's edition.
- Berneggerus , Or . 1582–1640 . professor of history and oratory at the Gymnase/Academy of Strasbourg, where he also taught mathematics. He had a leadership position at that Academy between 1620 and 1640. He wrote some mathematical treatises, especially on trigonometry, and corresponded intensely with Galileo and Kepler. Incidentally, he is said to be responsible for one of the few honours accorded to Kepler in his lifetime, namely to have exhibited a portrait of Kepler at the library of the Academy.
- Nam demonstrationes addere non est compendii, transferre vero propositiones ad usum, exercitii, non memoriae verborum conceptorum . Johannes Kepler Gesammelte Werke , XVIII 375 – 375 .
- Ursin , Benjamin . 1587–1633 . He was, around 1630, professor of mathematics in Frankfurt/Oder, and wrote a few books on trigonometry.
- Kepler discusses explicitly this part of the Aristotelian doctrine in De quantitatibus, Book I, chapter 3. It would be very interesting to know which edition of Aristotle was used by Kepler. To my knowledge, this is not certain, even though Pacius's edition is probably the best candidate. What we can be sure of, is that he read the original in Greek. Therefore, for all Aristotelian works except Posterior Analytics, I will use throughout the translation by Smith J.A. Ross W.D. Oxford 1908 For Posterior Analytics, instead, I will use the translation by J. Barnes for Clarendon Press (Oxford, 1975).
- ‘figura, quae proprie modus aut positus est quantitatum, ut Circulus, Triangulum, Quadrangulum. Etsi autem hic positus re ipsa non aliud est, nisi talis ordo partium quantitatis, seu positus partium, tamen quia Mathematici sic loquuntur, qui hos modos vocant corporis πoιoτητας, vetustissimam consuetudinem et Aristoteles secutus est. Recentiores contra disputarunt. Sed…honestissimum est, loqui cum ingeniosissimis et doctissimis hominibus veteribus Geometris'. See Erotemata dialectices Philippi Melanthoni opera quae supersunt omnia Breitschneider C.G. Halle 1846 XIII in col. 543.
- See, e.g. von Cues N. Die mathematische Schriften Hamburg 1952 translated by J. Hofmann, introduction and notes by J. E. Hofmann
- See on this interpretation Heath Mathematics in Aristotle Oxford 1949 209 211 Further remarks on quantity and relation are given by Kepler in Chapter 2.
- For an interesting comparison with another Keplerian work: (Apologia Tychonis contra Ursum) in which this doctrine plays a role, as well as for a discussion of its significance in astronomy, see Jardine N. The Birth of History and Philosophy of Science Cambridge 1984 as well as the previous article, ‘The forging of modern realism: Clavius and Kepler against the sceptics’, Studies in the History and Philosophy of Science, 10 (1979), 141-73.
- ‘sicut autem homo conditus est, ut in eo luceat noticia dei, et ut ei Deus communicet suam sapientiam et bonitatem, ita mentem humanam voluit evidentissimum de ipso testimonium esse. Cui et insita est lux, qua esse Deum agnoscimus, et insitae sunt noticiae, discernentes honesta et turpia…Impossibile est noticias numerorum et alias, et discrimen honestorum ac turpium, ortas esse a bruta natura, aut casu sic nasci. Est igitur mens architectatrix sapiens. Et intelligimus numeros, et discrimen honestorum et turpium in mentibus humanis immutabilite lucere,…Talis est igitur Deus, ut hunc ordinem velit, et hae noticiae radii sunt sapientiae divinae’. See Melanthoni opera omnia Breitschneider C.G. Halle 1846 XIII col. 138.
- Esset tamen haec lux in nobis multo clarior, si natura hominum non languefacta est, sed tamen adhuc reliquae sunt scintillae tantae, ut, de numeri nulla est dubitatio'. See Melanthoni opera omnia Breitscheider C.G. Halle 1846 XIII col. 138.
- See Summa theologica Pars prima, quaestio XLV, art. VII.
- See Crapulli Giovanni Mathesis universalis Rome 1969 31 44
- For a detailed historical study of this debate, see Vasoli La dialettica e la retorica nell' Umanesimo Milan 1968
- See Petersen P. Geschichte der Aristotelischen Philosophie im Protestantischen Deutschland Leipzig 1921
- ‘Mathemata prae caeteris sudiis amavit. In philosophia textum Aristotelis ipse legit, quaestiones conscripsit in Physics, Ethica fere neglexit, sic et topicis neglectis analytica posteriora sumsit. Sed Planerus illi hic placuit’. See Johannes Kepler Gesamm. Werke XIX 328 328
- As we can see in his unpublished work Analyseis geometricae sex librorum Euclidis composed around 1566.
- See, on the whole discussion on method, on demonstration and on the propaedeuticity of Euclid for Aristotle's corpus Gilbert N. Renaissance Concepts of Method New York 1960
- See also Met. XI 7 7
- To situate mathematics as intermediate between physics and ‘the perfect science’ is Plato's idea (Rep. 6, 509D-511E). In his context, however, physics was not a science, and the perfect science was dialectic. Aristotle argued the question in detail, and Kepler mentions his main points. Besides the mentioned passage, see Met. I, 9. The ‘great absent’ in this treatise is Proclus, who should however be considered among the ‘Philosophers’ who dispute these matters. Proclus, in fact, begins his main mathematical commentary by a Neoplatonic argument for the intermediate position of mathematics between theology and physics (see Proclus A commentary on the first book of Euclid's Elements Princeton 1970 translated by Glenn R. Morrow
- What follows is a faithful paraphrase from Met. IV 3 3 until ‘much of the divine’.
- For the ‘divine’ as the content of the first philosophy, compare this passage with Met. I 2 2 and also Met. XI, 7.
- The following does not seem to reproduce any specific passage, except some aspects of Met. XI 3 3
- Arguments similar to what follows are found in the quoted Met. VI 1 1 as well as in Met. XI, 7.
- In fact, this does not appear to have been ever said by Aristotle, either in Met VI 1 1 or elsewhere. The assumption that mathematics is superior to physics for the certainty of its proofs, however, underlies the whole discussion in Posterior Analytics (see particularly I, 13, 14, 27). It is interesting to notice, however, that Kepler nowhere in the treatise refers explicitly to this work, which had been the basis for most debates on mathematics in the sixteenth century. It is therefore remarkable that Kepler on one hand corrects Aristotle claiming the superiority of mathematics over all sciences based on the certainty of proofs, and, on the other hand does not acknowledge any auctoritas of the Posterior Analytics on his ideas.
- Here Kepler seems to have interpreted Aristotle in the light of his views on the matter. Aristotle talks about the one as being the measure of all things, in so far as the unit is the measure for each genus. The stronger arithmetical connotation of measure as a key notion for ‘epistemology’ seems to be Kepler's, whereas Aristotle seems to use the term analogically. For, in Met. X 1 1 we read: “to be one…means, especially, ‘to be the first measure of a kind’, and above all of quantity; for it is from this that it has been extended to the other categories… The one is the starting point of the number qua number. And hence in the other classes too ‘measure’ means that by which each is first known and the measure for each is a unit—in length, in breadth… Thus, then, the one is the measure of all things, because we come to know the elements of the substance by dividing the things either in respect of quantity or in respect of kind”. And below: “The measure is always homogeneous with the thing measured”.
- demonstrationes fere oculis subjici et ad vivum depingi possunt 148 – 148 .
- The principles of proofs, or common notions, or common axioms, are discussed extensively in Posterior Analytics. See, in particular, Post, Anal. I 11 11
- Aristotle in Met. IV 3 3 discusses whether the inquiry about axioms belongs to the science of substance. The answer is yes, for ‘since these truths clearly hold good for all things qua being (for this is what is common to them) to him who studies being qua being belongs the inquiry into these as well’. He then proceeds to talk about the best-known and most certain of all principles: ‘it is, that the same attribute cannot at the same time belong and not belong to the same subject’. Notice that the statement of this principle is explicitly attributed to the first philosopher, ‘he whose subject is existing things qua existing must be able to state the most certain principle of all things’. Thus, when Kepler writes ‘It is clear that he is speaking of the logical principles that geometers frequently employ…’ one should not interpret it as if implying that mathematics is the source of such general common principles. Rather, it is the origin of ‘a good more special principles…’ (see below).
- What follows is a paraphrase of Met. IV 3 3 until ‘and with Being’.
- From an Aristotelian point of view, these are the only ones that the subordinate sciences of astronomy, optics, mechanics and harmonics receive from the part of mathematics they depend on (e.g., optics from geometry), and from what Aristotle called ‘universal mathematics’: ‘Nor can the theorems of one science be demonstrated by means of another, unless these theorems are related as subordinate to superior (e.g. optical theorems to geometry or harmonic theorems to arithmetic)’, but in Met. XI 7 7 ‘Each of the mathematical sciences deals with some one determinate class of things, but universal mathematics applies alike to all’, and the same statement appears also in the mentioned Met. VI, 1.
- Here Kepler states the main tenets of the doctrine of lumen naturae. This humanistic idea, which was traced back to Aristotle's ‘common truths’ (see for instance Post. Anal. I 10 10 and Cicero's ‘notiones communes’, as well as to the Augustinian theory of knowledge, was particularly central in Nicholas of Cusa and Melanchthon, both among Kepler's most prominent philosophical sources. See especially N. von Kues, Dialogus de possest, and Philipp Melanchthon, Erotemata dialectices and De anima. Kepler elaborates on this doctrine below, in paragraph 9.
- This is a technical Aristotelian term. ‘To abstract’ is the counterpart of ‘to separate’. In general, what is abstracted is subtracted, removed, omitted; whereas what is separated is retained. But in connection with mathematical things the two notions are correlative. The matter is abstracted and thus the form is separated. Therefore mathematical things (and only them) can be called ‘constituted by abstraction’. I have followed here the argument as presented in Owens The Doctrine of Being in Aristotle's Metaphysics. A study in the Greek background of Mediaeval thought Toronto 1963
- Compare with De Anima III 7 7 ‘It is thus that the mind when it is thinking the objects of mathematics thinks as separate elements which do not exist separately’ (p. 664).
- Aristotle had discussed the ‘mixed sciences’ in Posterior Analytics at first in connection with the question of the ‘demonstration only by genus’: ‘One cannot, therefore, prove by crossing from another kind—e.g. something geometrical by arithmetic…nor can one prove by any other science the theorems of a different one, except such as are so related to one another that the one is under the other—e.g. optics to geometry or harmonics to arithmetic’ (Post. Anal. I, 7) and, below, mechanics is also considered as subordinate to geometry (Post. Anal. I, 9). In other words, the question of subordinate sciences is connected to that of common basic truths, as explicitly in Post. Anal. I, 12, ‘not every question will be geometrical, but only those from which either there is proved one of the things about which geometry is concerned, or something which is proved from the same things as geometry, such as optical theory’. The relationship between mathematics and subordinate sciences should actually be taken as the crucial meaning of the discussion on common truths: this, of course, in Aristotle. Since the late Middle Ages the Arabic developments had given new relevance to the issue of their status, and this slowly transformed into the emphasis on the mathematical sciences and the extension of their scope. However, Kepler, in this treatise, does not so much focus on the topic of subordinate sciences (he seems rather to take for granted the extension of their number and scope), as on the related question of common truths.
- Physics , II 2 – 2 . is stronger: ‘It seems absurd that the physicist should be supposed to know the nature of sun or moon, but not to know any of their essential attributes, particularly as the writers of physics obviously do discuss their shape also and whether the earth and the world are spherical or not’.
- Physics , II 2 – 2 . ‘That is why he separates them; for in thought they are separable from motion’.
- In Cat. I 4 4 Aristotle actually writes: ‘Expressions which are in no way composite signifiy substance, quantity, quality, relation, place, time, position, state, action, or affection’.
- Aristotle addresses the questions posed by the Pythagoreans throughout the Metaphysics, but in particular in Met. XIII 1 1 6, 7, 8, 9.
- Gap not indicated in MS, but by Frisch 150 150
- ‘in definitione lineae inest particula quantum’ Frisch 151 151 Aristotle writes: ‘Some things are called quanta in virtue of their own nature, others incidentally; e.g. the line is a quantum by its own nature, the musical is one incidentally’ (Met. V, 13).
- See Cat. 6 6 also for the next quotations.
- Frisch . 151 – 151 . this time did not correct the MS, which has ‘Ut si quinque (sic) sint pars denarii’. But this is a quotation from Cat. 6, where we read: ‘two fives makes ten, but the two fives have no common boundary, but are separate; the parts three and seven also do not join at any boundary’.
- Kepler writes Phys. II 1 1 but Frisch rightly corrects it.
- Cat. , 6 – 6 .
- A discussion of this topic is given in Phys. IV 4 4 5, though Aristotle does not summarize in these terms.
- See Jacobi Schegkii Schondorffensis in octo Physicorum, sive, De auditione Physica, libros Aristotelis commentaria longe doctissima, nunc primum in lucem edita Basileae per Johannem Hervagium 1546. In particular, pp. 69–71.
- See Cat. 6 6
- It is actually the discussion that Aristotle, summarizing the debates on the infinite (like the Zeno's paradoxes) between Pythagoreans and the Eleatic school, formulated in terms of the existence of actual infinity. A more extended critical analysis of Pythagoreans, Plato, Anaxagoras, see Phys. III 4 4
- This is the conclusion Aristotle arrives at in Phys. III 7 7 ‘the infinity (of number) is not a permanent actuality but consists in a process of coming to be, like time and the number of time’.
- ‘The fourth sort of quality is the figure and the shape that belong to a thing; and besides this, straightness and curvedness and any other qualities of this type; each of these defines a thing as being such and such. Because it is triangular or quadrangular a thing is said to have a specific character, or again because it is straight or curved; in fact a thing's shape in every case gives rise to a qualification of it. … The qualities expressed by the terms ‘triangular’ and ‘quadrangular’ do not appear to admit of variation of degree, nor indeed do any that have to do with figure. For those things to which the definition of the triangle or circle is applicable, cannot be said to differ from one another in degree; the square is no more a circle than the rectangle, for to neither is the definition of the circle appropriate. In short, if the definition of the term proposed is not applicable to both objects, they cannot be compared. Thus it is not all qualities which admit of variation of degree’ Categories VIII 8 8
- The MS has, at this point: ‘numerus ut forma et species est, unitates, ut materia et elementa. Inquit Aristoteles Met. c.’ i.e. ‘number is like form and species, unities like matter and elements, says Aristotle Met. III c.’ (The chapter number is unclear.) Frisch omitted this passage, probably because it interrupts the argument.
- In Cat. 6 a 20, i.e. the only similar passage, Aristotle states something slightly different, like ‘one three is not, so to speak, three in a greater degree than another’.
- Only in Met. IV 2 2
Annals of Science
Volume 43, 1986 - Issue 3
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