Ancient Mathematics. History of Mathematics in Ancient Greece and Hellenism

462 Pages·2022·26.319 MB·English

by  Dietmar Herrmann

Dietmar Herrmann
Ancient
Mathematics
History of Mathematics in Ancient
Greece and Hellenism
Ancient Mathematics
Dietmar Herrmann
Ancient Mathematics
History of Mathematics in Ancient
Greece and Hellenism
Dietmar Herrmann
Anzing, Germany
ISBN 978-3-662-66493-3 ISBN 978-3-662-66494-0 (eBook)
https://doi.org/10.1007/978-3-662-66494-0
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Preface
From whoever wants to deliver the history of any knowledge, we can rightly demand that
he give us news of how the phenomena became known in due course, what has been fanta-
sized about them, what has been meant and thought.1
Even though the history of mathematics is not a relevant lecture course at German
universities, a supplementary book can be of interest. It offers for all mathematics
teachers and -interested people quite a new view on the diverse problems, which
were developed over the course of one millennium (Thales 580 BC until Proklos
420 AD) in ancient Greek mathematics. For reasons of space, only facets of the
different works can be presented, which however assemble to a kaleidoscope of
science. A broad spectrum of tasks, constructions and historical illustrations are
put together to a new overall picture, which provides more insight than conven-
tional summary descriptions.
The versatile methods, which the Greek researchers have invented, also demand
respect and recognition from today’s viewer. These amazing achievements have
been created without any aids like calculators and modern communication. It
was important to describe the whole range of Greek mathematics, especially by
means of literary sources and a well-chosen selection of pictures also to bring in
the context of Pythagorean-Platonic philosophy. There are three possibilities of a
historical processing: strictly chronological, biographical-person related or sub-
ject-related by means of special thematic areas. The present representation chooses
a mixture of the latter two.
A first problem in the representation of ancient mathematics is raised by the
famous article On the Need to Rewrite the History of Greek Mathematics by
Sabetai Unguru. The author expresses the opinion that it is basically inappropri-
ate to present ancient findings with modern formulas. The formula and concept
apparatus of modern mathematics included concepts and abstractions, which
might veil the authentic historical procedure. As an example the binomial for-
mula (a+b)2 = a2+2ab+b2 is chosen. In modern mathematics, it applies to
all abstract elements of a commutative ring; such a concept is completely foreign
1 J.W. von Goethe: From the preface of the Color Theory (1810).
v
vi Preface
to Euclid. A product of two numbers or a square is always associated with an area
in Euclid and can only be linked with quantities of the same dimension. The Greek
word αριθμος (arithmos) must be seen in the Pythagorean-Platonic environment
and cannot be adequately translated with the word number. In order to make the
representations readable and compact, the usual formula language is used.
A second goal is the description of the political and cultural environment in
which the Greek scientist is located. The cultural flourishing of Athens in a phase
of relative peace between the Persian Wars—due to its leading role in the alli-
ance against the Persians—made it possible to build an academy that attracted
educationally motivated people from all over Greece, like Aristotle. Alexander
freed Egypt from Persian occupation and caused a shift of power to the southeast.
The Egyptian-Syrian province that emerged after his death through the division
of the empire became the intellectual and economic center of the Mediterranean
region with its capital Alexandria. The schools founded there, the Museion and
the Serapeion, survived the collapse of the Ptolemaic Empire and flourished even
under Roman occupation. Only the rise of Christianity as the state religion ended
the fate of the scientists still clinging to Platonic doctrine, as seen in the fate of
Hypatia.
Another concern is the inclusion of new, critical perspectives in comparison
to older literature. Stories that the vegetarian Pythagoras sacrificed several bulls
when discovering a theorem or that Archimedes set fire to the sails of the Roman
fleet with mirrors can be dismissed as fairy tales. A modern interpretation of
Diophantus, criticism of the work of Claudius Ptolemy and Heron, as well as new
translations of Nicomachus and Theon of Smyrna provide a new view of Greek
mathematics. The extensive work of Pappus is completely re-evaluated. The meth-
ods used usually require only moderate knowledge.
Geometry is currently taking a back seat in education; but this is not a sufficient
reason to completely abolish Euclidean geometry according to the motto of J.
Dieudonné (member of the Bourbaki group) Euclid must go!
To the 2nd edition The author is indebted to the publisher for the publication of
the book now in its 2nd, revised edition. In this way, a new, unique chapter on
Roman mathematics could be introduced and illustrated with new images. Many
other chapters have been updated and extended, in particular the section dedicated
to the continued influence of Hellenistic mathematics in Byzantium and Baghdad.
Furthermore, I would like to thank Professor Lothar Profke for his helpful com-
ments on the first edition. Special thanks go to the program planner Dr. Annika
Denkert for her support of the project!
In a variation of the poet Horace's quote aut prodesse volunt aut delectare poe-
tae, the author wishes "benefit and/or pleasure" in reading!
Dietmar Herrmann
Contents
1 Introduction .............................................. 1
1.1 To the Content of the Book ............................ 1
1.2 On the State of Mathematical Historical Research .......... 3
References ................................................ 10
2 How Greek Science Began .................................. 11
2.1 The Origins of Mathematics ........................... 15
2.2 The Greek Numeral Symbols ........................... 18
2.3 The Greek School ................................... 21
Further Reading ........................................... 25
3 Thales of Miletus .......................................... 27
3.1 Mathematical Work .................................. 27
3.2 More Reports on Thales ............................... 32
Further References ......................................... 35
4 Pythagoras and the Pythagoreans ............................ 37
4.1 Pythagoras of Samos ................................. 37
4.2 The Pythagoreans .................................... 42
4.3 Mathematical Discoveries of the Pythagoreans ............. 44
4.4 Figured Numbers .................................... 46
4.5 Pythagoras’ Theorem ................................. 51
4.6 Pythagorean Number Triplets .......................... 53
4.7 Heronian Triangles and Applications ..................... 56
4.8 Pythagoras and Music ................................ 57
Further References ......................................... 62
5 Hippocrates of Chios ....................................... 63
5.1 Quadratures According to Alexander of Aphrodisias ........ 65
5.2 Quadratures According to Eudemos ..................... 66
References ................................................ 69
6 Athens and the Academy ................................... 71
6.1 Athens ............................................ 71
6.2 The Academy ....................................... 74
6.3 The Mathematicians of the Academy .................... 77
vii
viii Contents
6.4 Eudoxus of Knidus ................................... 78
6.5 Theodorus of Cyrene ................................. 81
6.6 Theaitetos of Athens ................................. 82
References ................................................ 83
7 Plato .................................................... 85
7.1 Plato’s Most Beautiful Triangles ........................ 88
7.2 From the Book Menon ................................ 90
7.3 Platonic Solids ...................................... 94
7.4 Plato’s Lambda ...................................... 99
7.5 The Role of Mathematics in Plato ....................... 101
Further References ......................................... 103
8 Aristotle and the Lyceum ................................... 105
8.1 Life and Work of Aristotle ............................. 105
8.2 Mathematics with Aristotle ............................ 111
References ................................................ 116
9 Alexandria ............................................... 117
9.1 The Library ........................................ 124
Further References ......................................... 129
10 Euclid of Alexandria ....................................... 131
10.1 From Book I of the Elements ........................... 137
10.2 From Book II of the Elements .......................... 141
10.3 The Circle Theorems in Book III ........................ 146
10.4 Perfect and Friendly Numbers .......................... 148
10.5 The Euclidean Algorithm .............................. 150
10.6 Euclid’s Theorem on Primes ........................... 152
10.7 The Parallel Axiom .................................. 153
10.8 Equivalent Postulates to the Parallel Axiom ............... 154
10.9 Book of Area Divisions ............................... 160
Further References ......................................... 163
11 The Classical Problems of Greek Mathematics ................. 165
11.1 Incommensurability .................................. 165
11.2 The Constructability According to Euclid ................. 166
11.3 Angle Trisection ..................................... 168
11.4 Squaring the Circle .................................. 173
11.5 Doubling the Cube ................................... 173
11.6 Constructability of the Pentagon ........................ 174
11.7 Constructibility of the Heptagon ........................ 175
11.8 Quadratures of Lunes ................................. 176
11.9 The Division in extreme and mean ratio (EMR) ............ 179
Further References ......................................... 185
Contents ix
12 Archimedes of Syracuse .................................... 187
12.1 On the Centers of Gravity ............................. 190
12.2 Problem of the Broken Chord .......................... 191
12.3 The Regular Heptagon ................................ 192
12.4 The Book of Measuring the Circle ....................... 194
12.5 From the Book of Spirals .............................. 196
12.6 The Book of Lemmata ................................ 201
12.7 The Quadrature of the Parabola ......................... 209
12.8 The Palimpsest ...................................... 212
12.9 The Stomachion ..................................... 213
12.10 The Method, Theorem 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
12.11 Archimedes’ Tomb Figure ............................. 217
12.12 More Works by Archimedes ............................ 220
References ................................................ 221
13 Eratosthenes of Cyrene ..................................... 223
13.1 Eratosthenes as a Mathematician ........................ 224
13.2 Eratosthenes as a Geographer .......................... 225
References ................................................ 230
14 Conic Sections ............................................ 231
14.1 The Parabola ....................................... 235
14.2 The Ellipse ......................................... 241
14.3 Hyperbola .......................................... 243
References ................................................ 249
15 Apollonius of Perga ........................................ 251
15.1 From Book III of the Conica ........................... 254
15.2 Apollonius’ Theorem ................................. 258
15.3 The Touching Problem of Apollonius .................... 260
15.4 The Theorem of Apollonius ............................ 261
References ................................................ 263
16 Beginnings of Trigonometry ................................. 265
16.1 Aristarchus of Samos ................................. 267
16.2 Hipparchus of Nicaea ................................. 270
16.3 Menelaus’ Theorem .................................. 271
16.4 Applications in Geography ............................ 273
Further References ......................................... 277
17 Heron of Alexandria ....................................... 279
17.1 From the Definitions ................................. 283
17.2 From Metrica and Geometrica .......................... 285
17.3 From the Stereometrica ............................... 291
17.4 Heron’s Formula .................................... 293
x Contents
17.5 Cube Doubling by Heron .............................. 296
17.6 Area of the Regular Pentagon .......................... 297
17.7 Root Calculation Among the Greeks ..................... 298
17.8 Other Works by Heron ................................ 301
References ................................................ 305
18 Claudius Ptolemy ......................................... 307
18.1 Trigonometry in the Almagest .......................... 309
18.2 Ptolemy’s Theorem .................................. 315
18.3 The Addition Theorem ................................ 316
18.4 Construction of the Pentagon According to Ptolemy ......... 318
18.5 Construction of the 15-gon ............................ 320
18.6 The Geographical Work ............................... 320
References ................................................ 322
19 Nicomachus of Gerasa ..................................... 325
19.1 From the Arithmetica ................................. 327
19.2 Proportions and Averages .............................. 330
19.3 The Nikomachos Theorem ............................. 332
19.4 From the Commentary of Lamblichos .................... 334
References ................................................ 338
20 Theon of Smyrna .......................................... 339
20.1 The Side or Diagonal Numbers ......................... 340
20.2 Geometric Interpretation .............................. 342
20.3 Number Theory ..................................... 345
References ................................................ 346
21 Diophantus of Alexandria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
21.1 From Diophantus’ Book I and II ........................ 353
21.2 From Diophantus’ Book III to V ........................ 357
21.3 From Diophantus’ Book VI ............................ 361
21.4 From Diophantus’ Books in Arabic ...................... 363
21.5 To the Mathematics of Diophantus’ ...................... 366
21.6 Linear Diophantine Equation ........................... 370
21.7 Outlook ........................................... 371
Further References ......................................... 375
22 Pappus of Alexandria ...................................... 377
22.1 From Book VII of the Collectio ......................... 379
22.2 Pappos’ Rule ....................................... 380
22.3 Pappos’ Touching Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
22.4 Pappos’ Theorem .................................... 383
22.5 The Complete Quadrilateral ............................ 385
22.6 Harmonic Division ................................... 386