Mathematics in Aristotle

304 Pages·1949·59.9 MB·English

by  Thomas Heath

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.