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Mathematics in Aristotle PDF

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MATHEMATICS IN • ARISTOTLE By SIR THOMAS HEATH OXFORD A T THE CLARENDON PRESS Oxford University Press, Ely HQUSe. London W. I GLASGOW NEW YORK TORONTO MELBOURNE WElllNGTON CAPE TOWN SALISBURY IBAnAN NAIROBI DAR ES SALAAM LUSAKA ADDIS ABABA BOMBAY CALClJ'ITA MADRAS KARACHI LAHOR! DACCA KUALA LUMPUR SlNGAPOR! HONG KONG TOKYO FIRST PUBLISHED 1949 REPRINTED LITHOGRAPHICALI YIN CREAT BRITAIS AT THE UNIVERSITY PRESS, OXFORD BY VIVIAN RIDLER PRISTER ro"tHE UNI\'ERSIT\' 1970 PREFACE MATHEMATICS IN ARISTOTLE is the result of work done during the • last years of my husband's life, which he devoted to reading all that had been written on the subject and to making his own translations from the Greek. His eagerness to return to this work too soon after a serious illness in 1939 was probably instrumental in hastening his end. After his death in 1940 I found the manuscript (which I took to be unfinished) and consulted Sir David Ross as to the possibility of getting it completed. He most kindly read it through and, to my immense satisfaction, reported that the rough copy was, in fact, complete and advised that the fair copy, much of which was already done, should be finished in typescript. This I undertook to do myself, and it proved not only a welcome distraction from the more disturbing elements of' total war', but, in addition, an unexpectedly enjoyable introduction to the fascinating world of Aristotle, which I had hitherto regarded as far above the head of one not a classical scholar. I had, however, not failed to remark the concomitance of Aristotelian learning with a love of music. During a short visit to Corpus Christi College during the presidency of Thomas Case, I had the unforgettable experience, having returned somewhat early from a Balliol Sunday concert, of discovering the President and my husband playing, by candle-light on the Broadwood grand piano, with unrestrained fortissinro, the four-hand arrangements of Haydn Symphonies. The conventional fiction obtained that my being a professional pianist precluded such indulgence within my hearing! My very grateful thanks are due to Sir David Ross for his invalu able help and encouragement. For the reading of the proofs I am indebted to hinr and to Mr. Ivor Thomas, M.P., whose knowledge of Greek mathematics has been of the greatest assistance. Having no Greek myself, I have to thank Mr. Kerry Downes and Mr. P. R. W. Holmes for their assistance in transcribing the Greek quotations, and also my son Geoffrey T. Heath who, since his demobilization, has helped me in reading proofs and revising Greek quotations, mathematical figures, and formulae. I regret the absence of a bibliography, as my husband left no note of authorities consulted, though some indications are given in £oot-. notes as well as in the text. ADA MARY HEATH SUMMARY OF CONTENTS INTRODUCTION , Aristotle's view of mathematics in relation to the sciences I • (a) General I (b) Physics and mathematics . 9 (c) Mathematics and applied mathematics I I (d) Astronomy and physics 12 II. CATEGORIES '7 (a) Squaring of the circle 17 Cat. 7. 7bz7-33. Knowable prior to known. Squaring of circle not yet discovered. Eclipse prior to knowledge of cause (Thales). (b) Figure '9 Cat. 8. IoRII-23. Figure as quality (triangle, square, straight line, curved, etc.); but figures do not admit of more and less. (c) Gnomon • 20 Cat. 14. 15829-32. History of gnomon. III. PRIOR ANALYTICS 22 (a) Incommensurability of the diagonal 22 An. Pr. I. 23. 4Ia23-7. An. Post. I. 33. 89a29-30. (b) Euclid I. 5-Aristotie's proof 23 An. Pro I. 24. 4IbI3-22. (c) Observational astronomy-phaenomena 25 An. Pr. I. 30. 46817-21. (d) Syllogism: forms of terms. The sum of the angles of a triangle is equal to two right angles 25 An. Pro I. 35. 48:l:z9-39. (e) The geometer's hypotheses 26 An. Pro I. 41. 49b33-7. 5oal-4. An. Posl. I. 10. 76b39-77a3. (f) Parallel straight lines: petieio principii 27 An. Pro II. 16.6Sa4-9. (g) Incommensurability of diagonal and Zeno's bisection. Chrystal's proof 30 An. Pro II. 17. 65br6-21. (h) The squaring of the circle by means of Iunes 33 An. Pt'. II. 25. 6g920-4. 30-4. viii SUMMARY OF CONTENTS IV. POSTERIOR ANALYTICS 37 (a) General 37 An. Post. 1. 1. 7111-9, r-I-2I. (b) Essential attributes: primary and universal 39 An. Post. I. 4. 7312S-hS; 5. 7414-16. (e) Proposition about parallels 41 An. Post. I. 5. 74al5-b4. (d) General: you cannot to prove a thing pass from one genus to another 44 An. Post. I. 6-7. 7Sa35-bI7. (e) Bryson and the squaring of the circle 47' An. Post. I. 9. 7Sb37-76a3. Sopko El. II. 171b12-r8. 34-17217. (f) First principles of mathematics 50 An. Post. I. 10. 76a31-77a4. An. Post. I. 2. 72114-24. (g) 'Ungeometrical' questions 57 . An. Post. I. 12. 77br6-33. (h) • Applied' mathematics 58 An. Post. I. 13. 7sh32-7gaI3. (i) Universal proof versus 'particular' 61 An. Post. I. 23. 84b6--9; 24. 85120-31, h4-IS; 86122-30. (j) Exterior angles of rectilinear figures 62 An. Post. I. 24. 8Sb37-86a3. (k) 'Exactness'. priority. 'fa i~ &,rpa,plaw€ s. 'fa EK 'lrpoaO/aw€ s 64 An. Post. I. 27. 87a31-7. MtJtaph. K. 3. 106la28-b3. Phys. II. 2. 193b31-S. 194al-7. De an. III. 7. 43IbI2-16. (t) No knowledge through sense alone 67 An. Post. I. 31. 87b3S-88a5. 8SaII-17; II. 2. goa24-30. (m) Definition and demonstration 68 An. Post. II. 3. goa31)-b17. gob28--9la6; 7. 92bI2--25. (n) Angle in a semicircle 71 An. Post. II. II. 94124-35. Metaph. 8.9. I05Ia27'"'9. (0) Altct'nando in proportion 74 An. Post. II. 17. 9gal-23. V. TOPICS 76 (a) Pseudograph,mata 76 ToPics I. 10115-17. (b) • Indivisible lines' 78 Topics IV. I. :I2IbIS-23. SUMMARY OF CONTENTS ix (c) Definition of 'same ratio' 80 Topics VIII. 3. I58b29-15~I. (d) Definitions An. Post. II. 13. 96aZ4-bI. Topics VI. 4. 141a24-142a9. • (e) Definitions: failure to define by prior terms and need to specify genus 86 TOPics VI. 4. 14282ZJlI9; 5. 14ZbZ2-g. (f) Definition of 'line' 88 TOPics VI. 6. 143bU-I44a4; 12. 14!?Z9-36. (g) Definition of . straight line' 92 Topics VI. II. 14Sbz3-32. (h) Utility of arguments: multiplication table 93 Topics VIII. 14. r63bI7-z8. VI. PHYSICS 94 (a) Quadrature of the circle: Hippocrates, Bryson, Antiphon 94 SOPko El. It. 17IbI2-I8, 34-172a7. Phys. I. 2. 185&14-17. (b) Things known to us and things prior in the order of nature. Definition of circle 97 PAys. I. t. 184&10-1'12. (c) Mathematics and physics 98 Phys. II. 2. I93b22-194aI5. (d) Necessity in mathematics 100 Pkys. II. 9. 2001015-19. (e) The gnomons lot Phys. III. 4. 203a1O-I5. (f) Infinity r02 Phys. III. 4. Z03bI5-30, 204a2-7; 5. 204a34-b4; 6. z06~b27, 206b33- 207a2; 7. 207a33-b34; S. 20saI4-22. (g) Place II 3 Phys. IV. I. 20S827-2oga30. (h) Void and motion . 115 Phys. IV. 8. 215814-22, a24-hIo, bI3-22, b22-216811, 216alI-ZI. (i) The' now' in time and the point in space. 120 Phys. IV. 10. 21S86-S; II. 219blI-15, 22&4-13,18-21. (j) Some definitions: 'together', 'in contact', 'successive', 'contiguous', 'continuous' 121 Phys. V. 3. ~26b21-2, b23, b34-227a7. 227a1O--32; 4. 22Sbl-6, b15-25• (k) Motion divisible ad infinitum 124 Phys. VI. I. 231a21-232a22; 2. 232a23-b20, 233aI3-31. x SUMMARY OF CONTENTS (l) Zeno's arguments against motion . 133 PAys. VI. 8. 239aZ7-b4; 9. 239b5-9. 9-33. 33-240aI8. (m) Motion on a circle and on a straight line . 140 PAys. VII. 4. 248110-13. 18-b7. bIo-IZ, 24gll8-13. 13-17. (n) Motion caused by different forces 142 Phys. VII. 5. 249bz7-2SoRz8. (0) Circular motion and rectilinear motion; Zeno's dichotomy 146 PAys. VIII. 8. Z6IbZ7-Z6zIS. z6zaI2-b4. z6ZbZl-Z631I. (P) Zeno's 'Dichotomy' (further) 149 Phys. VIII. 8. z6314-b9. (q) Circular motion (,ocVI(Ace) may be one and continuous 150 PAys. VIII. 8. z64b9-28. (1') A finite movent cannot cause motion for an infinite time lSI Phys. VIII. 10. 266aI2-23. aZ4-b6. b6--24. (s) Motion of things thrown . '55 PAys. ,VIII. 10. z66hz7-z67azo. VII. DE CAELO 159 (a) Bodies, dimensions, etc. 159 De caelo I. I. 26811-13. (b) 'Heavy' and 'light' 160 De caela I. 3. 269b2o-32. (e) Agelessness of the universe 160 De caelo I. 3. 270bl..,..20. (d) Two simple motions, circular and rectilinear 161 be caeto I. 3. 27ob26-31. (e) Is the universe finite or infinite? Is there an infinite body? 162 De caelo I. 5 and, in particular, I. 5. 27Ib2-It. (f) A circularly moving body must be finite 163 De caelo I. 5. 271b26-272azo, 27Zbz5-B. (g) Bndies and weight 164 De caeto I. 6. Z73aZl-7. etc.; I. 7. (h) Falling bodies, etc. .67 De caela 1. 8. Z77az7-b8. (i) Mathematical' impossibility' 168 De caelo 1. It. Z81a4-7; 12. Z81b3-7. (j) Priority of circle and sphere among figures 169 De caelo II. 4. 286bz7"':'33. (k) A revolving heaven must be spherical 170 De caela II. 4. z87all-22. (I) Shortest line returning on itself 171 De caela II. 4. 287a27-8. SUMMARY OF CONTENTS xi (m) Surface of water at rest is spherical 172 De caelo II. 4. 287b4-I4. (n) Construction of bodies out of planes 174 De caeZo III. I. zggliZ-II; bz3-31; 4· 303a3I-hI; 5. 304h2-4. 305b28-300a5,~ 306~o-3. 7-8, 306aZ3-bs. (o) Principles should be the fewest possible 178 • De caelo III. 4. 302b26-30. (p) Motion of falling and rising bodies 178 De caelo III. 5. 304bI7-IB; IV. 1. 308a29-33; 2. 3ogbIZ-I5; 4. 3IIbI-I3. 3IIb33-3IZal. VIII. METEOROLOGY 180 (a) A geometrical proposition 180 Meteor. III. 3. 373a3-19. (b) A locus-proposition 181 Meteor. III. 5 and. in particular, 37Sbt6-376blZ. 376bu-22. b28-377an. IX. DE ANIMA 191 (a) A straight line touching a (brazen) sphere 191 De an. 1. I. 403aI2-16. (b) Definition of 'squaring' 191 De an. II. 2. 413aI3-20. Metaph B. 2. 996br8-z2. (c) 'Point', 'division', as • privation , 193 De an. III. 6. 430b2o-1. (d) The abstractions of mathematics 194 De an. III. 7. 431bI2-16. X. METAPHYSICS 195 (a) History of mathematics 195 lvle.taPh. A. I. 981b2o-S. (b) Incommensurability of the diagonal 196 Metaph. A. 2. 9830113-20. (c) Pythagoreans and mathematics 197 Metaph. A. S. 98Sb23--986a3; N. 3. I0g08Zo-S. (d) Plato on • points' and indivisible lines 199 Metaph. A. 9. 992aro-24; a. 2. 994b2Z-5. (e) Beauty in mathematics 201 Metaph. B. 2. gg6azg-bl. (f) . Axioms • 201 Metaph. B. 2. 995bz6-33. 99?lo-n, Ig-21; r. 3. 1005a1~7; r. K. 4. 1061bI7-2S; 3· 1005bII-20; 4. 100&'5-15. (g) Geometry and geodesia 203 Metaph. B. '1. 997b26-34.

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