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On the first Greek mathematical proofAuthor(s): Vassilis KarasmanisSource: Hermathena, No. 169, Essays on the Platonic Tradition: Joint Committee forMediterranean & Near Eastern Studies (Winter 2000), pp. 7-21Published by: Trinity College DublinStable URL: http://www.jstor.org/stable/23041319 .

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On the first Greek mathematical proof

by Vassilis Karasmanis

The main characteristic of early Presocratic thought is rationality. Greek thinkers were rational not so much because they rejected

supernatural and mythological explanations about nature, but

mainly because with them, for the first time in human history,

unargued fables were replaced by argued theory. Or, in other

words, dogma gave way to reason. Their views - independently of

their truth or not - were supported by argument and established

upon evidence. In the area of mathematics, rationality is mani

fested in the introduction, by the early Greek thinkers, of mathe

matical demonstration. In this essay, I am going to attempt a

reconstruction of the earliest Greek geometrical proofs, attributed

to Thales.

1. PRF.-HF.1.1 F.NTC. AND GREEK MATHEMATICS

Pre-Hellenic Babylonian and Egyptian mathematics consisted

mainly in practical and empirical techniques — sometimes very

sophisticated - of solving concrete problems. In the documents of

mathematical content of these peoples no theorems or demon

strations have come down to us. Even more, no general proposi tions are stated. To all appearances, at this early stage in the devel

opment of mathematics, no such fundamental concepts as theo

rem, demonstration, deduction, definition, axiom etc. had been as

yet formed. Mathematics prior to ancient Greek civilization was at

best a useful collection of prescriptions or rules of empirical ori

gin. Even geometrical problems are solved as practical arithmeti

cal ones. So, the Babylonian formula for calculating the area of a

circle was 3r2, and the Egyptians used to multiply together the

averages of the two pairs of opposite sides of a quadrilateral in

order to calculate its area.1

1 For pre-hellenic mathematics see: O. Neugebauer and A. Sachs,

Mathematical Cuneiform Texts (New Haven 1945); J. Hoyrup, Old Babylonian Mathematical Procedure Texts, Max-Planck Institute for the History of

Science,(Berlin 1994); G. Robins & C. Shute, The RhindMathematical Papyrus (London 1987); R. J. Gillings, Mathematics in the Time of Pharaohs (New York 1872); Van der Waerden, Science Awaking (Groningen 1954) chs I, III; J. Gow



8 Vassilis Karasmanis

It is well attested that the ideal of mathematics as a demon

strative discipline is the work of the Greeks. With the Greeks the

notion of proof or demonstration becomes the main characteristic

of any science and especially of mathematics. For them everything in mathematics must be proved. According to the ancient tradi

tion, Greek science and philosophy starts with Thales2 and the

other Milesian thinkers. However, no mathematical documents

from this early period are preserved. Euclid wrote his Elements

around 300 B.C.3 and the only pre-Euclidean surviving mathe

matical text is Eudemus' (late fourth century B.C.) report of four

proofs by Hippocrates of Chios (late fifth century B.C.) concern

ing the squaring of lunes and preserved by Simplicius (sixth cen

tury A.D.) in his commentary on Aristotle's PhysicsA This text

reveals that Hippocrates knew a great deal of the geometrical

propositions found in Euclid and used strict logic and sophisticat ed demonstrative methods.

The difference between pre-hellenic mathematics and Euclid's

Elements, or even Hippocrates' proofs, is great. We cannot imag ine that Greek mathematicians passed automatically from the first

stage of mathematics to the other. We have probably to suppose the existence of an early stage of mathematical demonstration, more primitive and empirical than that found in Hippocrates or in Euclid. However, because of the absence of documents, any

attempt to reconstruct the earliest mathematical demonstration remains a matter of speculation and cannot be conclusive.

Hippocrates' text does not help us to discover what kind of proofs the early Greek mathematicians used. Nevertheless, I find it very attractive and fascinating to attempt a reconstruction of the proofs

A Short History of Greek Mathematics (Cambridge 1884) 123-133; D. Fowler,

'Egyptian Land Measurement as the Origin of Greek Geometry?' in 2-Manifold 1983;T. Heath, A History ofGreek Mathematics, vol 1 (Oxford 1921) 122-8; A. Szabo, 'The transformation of mathematics into deductive science and the

beginnings of its foundation on definitions and axioms', in Scripta Mathematica, vol. XXVII 1964, 28-9.

2 See Plato, Theaetetus 174; Aristotle, Metaphysics 983b20; Eudemus

reported by Proclus In Primum Euclidis Elementorum Librum (ed. Friedlein) 65.

7-11; Plutarch, Solon 2, and De Placitis Philosophorum II 12,24,28, III 10-11; Herodotus I 74; Clemens Alex. Stromateis 114; Diog. Laert. I 23.

3 However, some modern historians believe that Euclid's Elements were

written at least fifty years later: see A. Bowen and B.Goldstein, in Proc. Amer.

Philos. Soc., vol. 135 (1991) 246. 4

Simplicius, Commentary on Aristotle's Physics (ed. Diels) 60.22-68.32.



On the first Greek mathematical proof 9

of the first geometrical theorems in the history of mathematics, attributed by Eudemus to Thales5.

2. The term aeikntmi and visualization in mathematics

Firstly, I think that it is necessary to examine the term apodeixis that the Greeks used for demonstration. The corresponding verb

used is SeiKvupx or dTToSeuKvu^L. In Euclid's Elements the word

8etKvu(jLL is the technical term for logical display or for proof. After

the proof of any theorem, Euclid ends with the following expres sion: hoper edei deixai (quod erat demonstrandum).

From the LSJ (s v)6 we find three groups of meaning of the

word deiknymi: a) to show, to point out, to display, to bring to

light; b) to point out by words, to tell, to explain, to teach; c) to

display by logical argument, to prove, to demonstrate. The first

meaning of the word is not only the primary and original one, but

also the most common in the everyday Greek language7. However, even from the time of Homer we can find the second meaning of

the word deiknymi where something is 'shown' not by pointing the

finger but explained by words8. The third meaning of the word

does not appear in early authors but only later.9

At least from the time of Plato and Aristotle the words Selkv

aTToSeiKvufjiL and dn68ei£is are used as technical terms for

mathematical demonstration.10 But the use of the word §€Ckvu|xi

by the Greek mathematicians as a technical term for demonstra

tion may indicate the existence of an early stage in mathematical

proof where the mere showing of the properties of a figure was

5 For a reconstruction of the early Pythagorean proofs in Arithmetics via

pebble-methods, see A. Szabo, The Beginning of Greek Mathematics (Dordrecht - Boston 1978) 191-5 who also refers to Becker, and W.R. Knorr, The Evolution

of the Euclidean Elements (Dordrecht 1975) ch. V; W. Burkert, Lore and Science

in Ancient Pythagoreanism (Cambridge Mass. 1972) 427-46. 6 H.G. Liddell, R. Scott and H.S. Jones, A Greek-English Lexicon (Oxford

1940); see also Szabo, op. cit. n. 5 above, 187-9. 7 See Plato, Cratylus 430e: 'and when I say 'show' (8ei£ai), 1 mean bring

before the sense of sight.' 8 See Odyssey, M 25; cf. Aeschylus, Prometh. 458, 482. 9 See Plato, Phaedo 66d, Politicus 284d3. However, the words onroSe'iKvu

|xl and diToSeiijis are used with this third meaning even from the 5th century; see e.g. Thucydides, 2.13 , and Aristophanes, Clouds 1334.

™ See Plato, Theaet. 162e4-6, Epinom. 983a4 (referring to astronomy);

Aristotle, Post. An. 90a36ff, Nic. Eth. 1094b26-7.




taken as demonstration. Moreover, it is certain that early Greek

mathematicians employed heuristic or practical methods and were

content with a loose, informal notion of proof. Such a method is

that of inclination or neusis used by Hippocrates of Chios.

According to Iamblichus11 the earlier Pythagoreans regarded

geometry as loropLT], i.e. as a science inseparable from experience.

According to Szabo, 'in the early Greek empirico-illustrative

geometry which had not yet become [xd0T||xa but was only histo

rie, demonstration was probably no more than simple visualiza

tion'.12 Is it possible that in early times the correctness of a math

ematical statement was 'proved' by illustrating or by showing con

cretely the same truth that later was demonstrated by strict logic? Szabo13 finds such a historical example of the demonstration

of a mathematical statement by means of practical visualization in

the famous mathematical passage of Plato's Meno (82b-85e), where Socrates asks an uneducated slave how to double a square that has sides two feet long. Socrates shows the mistake of the

slave's answers by drawing squares and calculating their area.

When, after two unsuccessful attempts, the slave is not able to give another answer about the length of the double square, Socrates

asks: 'Try to tell us exactly; and if you don't want to count it up, just show (8et£ai) us'. This example shows the role of the concrete

visualization and the illustration by drawing in geometrical demonstrations. The solution 'depends entirely on the correct

construction, and once this construction has been carried out the solution to the problem is treated as obvious on direct inspec tion'.14 Aristotle (Metaph., 1051 a21 -31) insists on the relevance of the right diagram and auxiliary constructions in geometry which make obvious the proof of the geometrical proposition.

Another example of mathematical demonstration by means of

1' Vita Pythagorica, 89. 12

Szabo, op.cit. n. 1 above, 35. '3

Op.cit., n. 1 above 35-6; and op.cit. n. 5 above, 190-1. '4 See, G.E.R. Lloyd, Magic, Reason and Experience (Cambridge 1979,)

107-8. Of course visualization and illustration alone could not be regarded as demonstration at the time of Plato. What is shown on the diagram is support ed and completed by a step-by-step logical argument. After all, Plato's intention in this passage is to show the a priori character of mathematical truths; cf. G.

Vlastos, 'Anamnesis in the Meno\ in Dialogue, 1965.




concrete visualization we may find in early Pythagorean arith

metic. According to the existing evidence, the kind of arithmetic

Pythagoreans were concerned with was that in which numbers

were represented by figures made with dots or pebbles (i|n]cj>oi,).15 The basic distinction and the fundamental starting point of

Pythagorean arithmetic is that of the odd and the even numbers.16

Becker showed that the propositions of Euclid's Elements Book IX

21-34 (theory of the odd and the even), which stand isolated in

that Book, have an early Pythagorean origin and can be easily treated by pebble-method.17 However, this kind of arithmetic

does not have any theoretical character. There is no attempt of

general proof. The rule is illustrated by a few examples and this is

considered as verification.

3. Eudemus on Thales' geometry

All ancient sources agree that Greek science, and specifically

geometry, begins with Thales18 who was also regarded as one of

the seven wise men. According to the ancient tradition, early in his

life he engaged in commerce, for which he seems to have had great

aptitude.19 He was at the very centre of the intense Ionian life of

his time. He advised his fellow citizens against an alliance with

Croesus and, later on, attempted to establish a federation of

Ionian cities with Teos as the capital.20 It seems that Thales was a

practical man, ready to apply his knowledge to practical aims. It is

said that he was distinguished as an engineer21, that he measured

15 See Aristotle Metaph. 1092bl0-13, Physics 203al3; Epicharmus, in

Diels-Kranz Die Fragmente der Vorsokratiker, 23B 2; Theophrastus Metaph. 11

(Diels-Kranz, Eurytus 45 B 2). The use of gnomon in creating series of figurate numbers is testified from the above passage of Physics.

'6 See Philolaus, in Diels-Kranz, 44B 5; Epicharmus, in Diels-Kranz 23B

2; According to Aristotle (.Metaph. 986al4-l6, cf 1083b27fF), the Pythagoreans consider that 'number is the principle both as matter of things and as forming their modifications and their permanent states, and hold that the elements of

number are the odd and the even...' '7 See in Knorr, op. cit. n. 5 above, ch. v; Szabo, op. cit. n. 5 above, 192

4; Burkert, op cit n. 5 above, 434-5. *8 See references in n. 2.

Plutarch, Solon 2-3; Aristotle, Politics 1259a. 20 Herodotus I 170. 21 Herodotus I 75; Plato, Republic, 600a.




the height of the pyramids through their shadows,22 and the dis

tance of a ship from the shore23, and also that he foresaw an eclipse of sun.24

The most relevant source about Thales' geometry is Eudemus, a pupil of Aristotle, who wrote the first history of Geometry.

Fragments of this book were preserved by ancient commentators

and mainly by Proclus in his commentary on the first book of

Euclid's Elements. Eudemus attributes the following four general theorems of elementary geometry to Thales: 1) the circle is bisect

ed by its diameter (Proclus, In Eucl. 157); 2) the angles at the base

of an isosceles triangle are equal (Proclus, In Eucl 250.20-251.2);

3) if two straight lines cut one another, the opposite angles are

equal (Proclus, In Eucl. 299); 4) if two triangles have two angles and the side adjoining the angles respectively equal, the triangles are congruent (Proclus, In Eucl. 352).25 Diogenes Laertius reports a fifth theorem the attribution of which to Thales is not, howev

er, certain. According to him: 'Pamphila says that Thales, who

learned geometry from the Egyptians, was the first to inscribe in a

circle a right-angled triangle, and that he sacrificed an ox. Others,

however, including Apollodorus the calculator, say that it was

Pythagoras' (Diog. Laert. I 24-25). Proclus says that Thales proved (diroSet^ai,) the first theorem

and that 'he was the first to have known and to have enunciated'

(emcrrT]o-ai. Kax eliTeLv) the second. According to Proclus the third

theorem 'was first discovered, as Eudemus says, by Thales, though the scientific demonstration was improved by the writer of the

Elements' Regarding the fourth theorem Proclus says that

'Eudemus in his History of Geometry attributes this theorem to

Thales. For he says that the method by which he is reported to

have determined the distance of ships at sea shows that he must

22 Diog. Laert. I 27; Pliny Nat. Hist. XXXVI 12.

2^ Proclus, In. Eucl. 352. 24 Herodotus I 74; Diog. Laert. I 23. 25 Proclus refers to Eudemus as his authority only for the two last theorems.

Regarding the first two, Eudemus is not specifically cited by Proclus, but it is

generally assumed that they derive from Eudemus. Proclus uses the words 'they

say' and 'it is said' and it is well known that he often derives information from

Eudemus' book without citing his name. At the second passage, Proclus records that 'in the more archaic manner he [Thales] described the equal angles as simi

lar'. I suppose, therefore, that Eudemus is the source for all the four theorems.




have used it.' It is worth noting that only for the first theorem

Eudemus says clearly that Thales had demonstrated it. The strange

thing is that Euclid does not prove this theorem in his Elements, but merely states the fact in Book I def. 17. On the other hand we

are told that Thales 'had known and enunciated' the second the

orem and discovered without scientifically proving the third.

From this statement it follows that Thales proved (less scientifi

cally than Euclid) the third theorem, while, regarding the second

theorem, the word emcxTTjcxaL together with enreiv implies some

thing stronger than the simple enunciation of this geometrical

proposition. With regard to the fourth theorem it will be observed

that Eudemus had no information about the attribution of it to

Thales but only inferred that it was known to him from the fact

that it was necessary to Thales' determination of the distance of a

ship from the shore. The passage gives no information on whether

Thales not only knew the theorem but also proved it.

The achievements - and especially the geometrical ones — of

Thales have been strongly disputed by modern scholars.26 Dicks

says that 'because Thales was the most notable name in early Greek history, ... because he also had a reputation for putting his

technical knowledge to practical use, ... and because it soon

became firmly fixed in the tradition that he learnt geometry in

Egypt [Eudemus being responsible for that, according to Dicks], then it seemed obvious to later generations brought up on Euclid

and the logical, analytical method of expressing geometrical

proofs, that Thales must certainly have known the simpler theo

rems in the Elements, which it was supposed he formulated in the

terms familiar to post-Euclidean mathematicians In fact, how

ever, the formal, rigorous method of proof by a process of step-by

step deduction from certain fixed definitions and postulates was

not developed until the time of Eudoxus.'27 Dicks is sceptical about the truth of all the above statements of Proclus. He asks, for

example, how is it possible for Thales to prove the first theorem

when even Euclid did not claim to do this?28

2^ See, for example, J. Burnet, Early Greek Philosophy (London 1892) 45f; P.M. Schuhl, Essai sur la formation de la pensee grecque (Paris 1934) 175ff; D.R. Dicks, 'Thales', in Class. Quart., 1959. All of them show an extreme scepticism.

27 Op.cit. n. 26 above, 303-5.

28 Similarly, Heath, op.cit. n.l above, 131.




I think that the extreme scepticism of Dicks and other schol

ars can easily be refuted. Dicks seems to ignore two references of

Aristophanes to Thales29 (not included in Diels-Kranz) which

show that the name of Thales was closely associated with geome

try, for the Athenian public of the 5th century. Moreover, the geo metrical character of Thales' universe (substituting earlier mytho

logical images by geometrical figures) seems to be evidence that he

did engage in geometry. Dicks observes the cautious manner in which Aristotle speaks

about Thales and he rightly concludes that no written work of

Thales was available to Aristotle.30 He also finds hardly convinc

ing -

although without argumentation - the thesis that Aristotle

was using a pre-Platonic work containing information about

Thales.31 However, he does not disregard the Aristotelian evidence

about Thales. It is, therefore, strange that he entirely rejects the

evidence from Eudemus who was an immediate pupil of Aristotle

and who, very probably, had available all information about

Thales that Aristotle had. The fact that Aristotle says nothing about Thales' mathematical activity indicates nothing because

Aristotle, in his Metaphysics, is interested in metaphysical ques tions regarding the first cosmological principles and not in math

ematics. On the other hand, Eudemus, who wrote a history of

geometry, was interested in the mathematical achievements of

Thales.

Dicks claims that post-Euclidean writers are responsible for

the ascription of the above theorems to Thales. But it is Eudemus

who attributes at least two of them to Thales, and Eudemus writes

before Euclid. Also, the geometrical propositions ascribed to Thales do not presuppose a logical and axiomatic system of geom

29 Clouds 175ff; Birds 995ff. 3° See Dicks, op.cit. n. 26 above, 298. It is not certain whether Thales pro

duced any written work. Diogenes Laertius (1,23) says that 'according to some

he left nothing in writing ... But according to others he wrote nothing but two

treatises, one On the Solstice and one On the Equinox.'

According to B. Snell (Philologus 1944, 170-82; see in Dicks, op.cit., n.26 above, 298, and in Burkert, op.cit. n.5 above, 415) Aristotle derives his

information from Hippias of Elis (cf. Diog. Laert., I, 24). A. Lebedev

(Aristarchus of Samos on Thales' Theory of Eclipses', in Apeiron, 1990, 77-85), based on new evidence, believes that a book of Democritus, containing infor

mation about early science, was available at that time.




etry as Dicks believes. Hippocrates of Chios writes before the

axiomatization of geometry, but this does not prevent him from

stating general theorems and also from using 'rigorous method of

proof by a process of step-by-step deduction.' Moreover, we do

not have to suppose that Thales' proofs had the form of the

Euclidean ones. After all, Proclus speaks about less 'scientific'

proof regarding the third proposition. It is also certain that the

propositions attributed to Thales were formulated quite early in

the history of Greek mathematics because at least two of them (the first and the second) are presupposed in Hippocrates' proofs.

Of course, the above objections to the extreme scepticism of

Dicks do not constitute a strong defence of the acceptance of the

traditional thesis about Thales' geometrical achievements.

Eudemus writes more than two centuries after Thales and even

Aristotle is not quite sure about the credibility of his sources.

However, I believe that we do not have the right to reject the

whole tradition about Thales' geometry without strong evidence

and arguments. On the contrary, I think that we can give credit to

Eudemus' words - but probably not to other later writers - espe

cially if we are able to give a plausible interpretation of Thales'

kind of demonstration, taking into account that he lived almost

three centuries before Euclid and did not have available a system of geometrical propositions at his time. Our problem is, then, to

find out what Thales meant by 'proof'.

4. What kind of demonstration?

We must first note that Thales' theorems are general propositions

stating properties of geometrical figures. This is something new in

the history of mathematics and shows a kind of logical abstraction

compared with the pre-hellenic mathematics.32 The propositions ascribed to Thales are very elementary theorems of plane geometry. However, they are elementary propositions in the sense that they form part of a systematic logical exposition of mathematics.33 The

pre-scientific mind takes for granted such propositions, and asks

questions like these: how do I calculate the area of this quadran

32 We see easily the same tendency in Thales' cosmology. 33

Although not in the axiomatic form and the strictly deductive

of Euclid's Elements.




gle, of this circle, etc? These are questions with which the Egyptian and Babylonian texts are concerned.34 It is only later that the ques tion arises: how do I prove all these? It is probable that Thales was

the first to introduce a kind of proof and to develop a logical struc

ture for geometry. The main difference between the propositions attributed to Thales and those of Babylonian mathematics is that

the former are general propositions, while the latter are particular. Therefore, in the case of Thales, his propositions are theorems, while the Babylonian mathematical propositions relate to practi cal problems regarding specific cases. This means that when we

solve a practical problem of the Babylonian kind, we can solve it

by measurement and calculation and it is solved for this specific case (that is for the specific pyramid or rectangle whose sides have

a specific length). On the other hand, Thales' theorem that the

diameter divides the circle into two equal parts is a general propo sition referring to every circle. Now 'diameter' and 'circle' are uni

versal and not particulars. We cannot prove such propositions by calculation and we need other logical methods.

But what kind of proofs did Thales undertake? Eudemus

(Proclus, In Eucl. 65)35 says that 'Thales who had travelled in

Egypt, was the first to introduce this science into Greece. He made

many discoveries himself and taught the principles of many oth ers to his successors, attacking some problems more generally (i<a0oAiKa>Tepov) and others more empirically (alcr0T]TiKa)Tepov)'. This passage is directly relevant. It says that Thales' proofs (or at least some of his proofs) were more general, that is, proofs of gen eral propositions and not of concrete examples as in pre-hellenic mathematics. On the other hand, these same proofs were more

34 See references in n. 1 above. 35 This passage is part of the famous 'summary of geometers' of Proclus

that is also called 'Eudemian summary' on the assumption that it is extracted

from the lost History of Geometry by Eudemus. Indeed, after speaking about

the geometers of Plato's Academy, Proclus says: 'Those who have compiled his

tories bring the development of this science up to this point. Not much

younger than these is Euclid...' (translation G. Morrow, Proclus: A Commentary on the First Book of Euclid's Elements, Princeton 1970). Knowing that the only one who wrote history of geometry before Euclid was Eudemus, we can safely assume that the 'summary' is either a fragment of his book or - most probably — is based on it. See, Heath, op.cit. n. 1 above, 118-20; Burkert, op.cit. n. 5

above, 409.




empirical, which probably means that Thales' proofs did not have

the strictly logical character of Euclid's ones, but were closer to an

empirical 'showing'.36 Becker37 pointed out that all the theorems attributed to Thales

(except the fifth, about the attribution of which to him Diogenes has reservations) can be derived simply from considerations of

symmetry.38 Commenting on I def. 17 of Euclid's Elements and

after attributing the first of the above theorems to Thales, Proclus

39 proposes a proof of the statement that the diameter bisects the

circle. His proof is according to the empirical method of superpo sition (£<))ap^6I,€Lv), that is, the method of putting one geometri cal figure over the other in order to show their congruence. Kurt

von Fritz believes that Proclus gives Thales' actual proof and holds

that the method of Thales was the empirical method of superpo sition. 'This method must have been applied in earlier times more

frequently than appears in Euclid. It seems to be an earlier stage in

the history of mathematics, and it may not be incidental that of

the five theorems attributed to Thales by tradition, four can be

proved directly by the method of superposition and the fifth indi

rectly'.40 I believe that this interpretation is very plausible and I will try

to give further support to it. The attribution of the method of

superposition to Thales fits very well with Eudemus' words 'more

generally' and 'more empirically'. With the method of superposi

36 There is a problem with the above reading of the words Ka0oXiKu>Tepov and al<70ir]TiKa)Tepov. Thales' proofs were 'more general' in comparison with the

pre-hellenic mathematics, while 'more empirical' in comparison with the later

Greek mathematics. The only other reading I could see is that Thales attacked

some problems more generally and some other problems more empirically. But

even under this second reading, we can conclude that some of Thales' proofs were general and also that some others had an empirical character.

" See in Burkert, op.cit. n.5 above, 417.

38 Indeed Proclus comments on the first theorem attributed to Thales:

'The cause of this bisection is the undeviating course of the straight line

through the centre; for since it moves through the middle and throughout all

parts of its identical movement refrains from swerving to either side, it cuts off

equal lengths of the circumference on both sides' (In EucL, 157. 12-16; trans

lation G. Morrow). 311n EucL. 157.17ff. 4° K. von Fritz 'Die' Apxau in der griechischen Mathematik' in Archiv fur

Begrijfsgeschichte 1955, 77; see in A. Szabo, op.cit., n. 1 above, 38-9. Cf. also

Lloya, op.cit. n.14 above, 104-5; Burkert, op.cit. n.5 above, 417.




tion, the geometrical proof is at the stage of a perceptible 'show

ing' (d-TToSei^is) relying exclusively on the geometrical figure. Nevertheless, this method exhibits general characteristics of geo metrical figures, so geometrical propositions are shown in all their

generality and necessity.41 This technique of demonstration by superposition is intro

duced by Euclid in his Elements as a 'common notion' (the sev

enth, or the fourth if we consider common notions 4-6 as an

interpolation). It serves as an axiom of congruency saying that:

'Things that coincide with one another are equal to one anoth

er'.42 However, he systematically avoids using it, having recourse

to it only twice, in the theorems I prop. 4 and I prop. 8 that refer

to equality of triangles. Heath43 says that 'it is clear that Euclid

disliked the method and avoided it wherever he could, e.g. in I. 26

[similarly in I. 5,1. 15 and other cases], where he proves the equal

ity of two triangles which have two angles respectively equal to

two angles and one side of the one equal to the corresponding side

of the other. It looks as though he found the method handed

down by tradition ... and followed it, in the few cases where he

does so, only because he had not been able to see his way to a sat

isfactory substitute'.44 Euclid, therefore, not being able to avoid this method completely, tried at least to give an axiomatic foun dation to it. That the method of superposition was in practice in earlier times is shown by Aristotle, in De Caelo 299b23ff, where he

says that lines and planes can be superposed45 and seems to blame Plato for not making use of this.

The method of superposition has two features that make it

very useful in an early stage of geometry. The first is that can be

41 Proclus (In Eucl. 243.5-9) had already observed these characteristics of the method of superposition because, although he regarded it as a legitimate method of proof, he observed that 'for congruence, as well as the equality which is inferred from it [the superposition], is completely dependent on the clear

judgement of sense-perception (transl. Morrow). 42 Proclus (In Eucl. 240.24-241.8) observes that this method of proof pre

supposes not only this axiom but also its converse. 43 T. Heath, Euclid: the Thirteen Books of the Elements (Cambridge 1926)

2nd ed., vol. I, 255. Hilbert, in his Grundlangen der Geometrie, avoids this method assuming

Euclid's theorem 1.4 as an axiom. We note that Euclid's theorem I. 4 is the fourth theorem attributed to Thales.

See also Physics 228b24 'or a spiral, or any other figure, parts of which taken at random will not fit upon each other'.




used as a criterion of geometrical congruency in general, so over

coming the earlier stage of measuring in order to show the equal

ity of geometrical magnitudes. Secondly, because it depends almost exclusively on sense experience, we can prove with its help theorems without appeal to other known or 'prior' geometrical

propositions. I find it plausible, therefore, that this method marks

the passage from the pre-hellenic practical mathematics, relying on calculations and measurement, to the more theoretical Greek

mathematics that state and prove general propositions. From this

point of view it seems to me very likely that proof by superposi tion goes back to Thales.

Another very relevant characteristic of this method is that it is

easily combined and supplemented by the method of reductio ad

absurdum (diraytD'yr] eis o/tottov). Assume that we have to prove one of the propositions attributed to Thales (say the first) and that

we do it by the method of superposition. The diameter divides the

circle into two equal parts. We fold the one part over the other and

we can easily see that the two parts coincide because every point of the circumference of the first part is the extreme of a radius and

so will fall over the other part of the circumference. But imagine someone skeptical who objects to this procedure of 'seeing' that

every point of the first part of the circumference will fall over the

second. A clever geometer could proceed indirectly by saying: 'All

right, say that the two parts of the circle do not coincide but the

one falls out of the other. Now, bring a line from the centre O that

cuts the one part of the circumference at point A and the other at

point B. We have OAcOB. But OA and OB are both radii of the

circle and, therefore, equal. Therefore, OA=OB and our hypothe sis that the two parts of the circle do not coincide when we fold

the one over the other is false and consequently the diameter

divides the circle into two equal parts'. Indeed, Proclus uses super

position together with reductio in order to prove 'mathematically' this theorem.46

It has often been said that the indirect proof in the form of

reductio ad absurdum was the most common kind of proof in early

46 In Eucl, 157.17-158.2. Euclid does not use reductio ad absurdum when

he proves propositions I. 4 and I. 8 by the method of superposition. But both

of these theorems can be easily proved by using reductio together with super

position.




Greek mathematics and led to antiempiricism in mathematics.47

This is because this kind of proof employs strict logical argument without resort to sense experience. Szabo observes that six out of

the seventeen theorems of the old theory of the odd and the even

(found in Book IX of Euclid's Elements) are demonstrated by this

indirect method. Similarly with the first thirty-six theorems of

Book VII of the Elements which are considered that belong to an

early period. In fifteen of them reductio ad absurdum is used.48 But

why is this method of proof so common at this early stage of

Greek mathematics? I think that in pre-axiomatic mathematics an

indirect method of proof is more effective than a direct one. This

is because in the method of reductio we start from the negation of

the proposition to be proved and going back we arrive at some

thing which is impossible. In this way, we exploit the structure of the proposition to be disproved. It is possible, then for the geome ter to begin the proof without the need of other premisses (prior known theorems). So, the indirect proof does not presuppose a

complete system of anterior propositions.49 Szabo has maintained that the indirect demonstration in

Greek mathematics does not have its origin in 'historically more

primitive forms of mathematical thinking'. According to him 'the earliest Greek mathematicians, the Pythagoreans, borrowed the

method of indirect demonstration from Eleatic philosophy; con

sequently, the creation of deductive mathematical science can be attributed to the influence of Eleatic philosophy.'50

Szabo's thesis does not seem to me very plausible. I am not

going to argue at length against it. I cannot, however, understand

47 See Szabo, op.cit., n.l above, 43-47; Szabo, op.cit. n. 5 above, 216-220;

Lloyd, op.cit. n. 14 above, 11 Off. 48 See Szabo, op.cit., n. 1 above, 42-3. Aristotle {An.Pr. 41a23ff) takes as

an example of reasoning per impossibile the proof of the incommensurability between the side and the diameter of a square, which goes back to the

Pythagoreans of the mid-fifth century. However, some modern historians believe that the discovery of the incommensurability arose from application of

anthyphairesis; see W. Knorr, op.cit., n.5 above, chh. 2 and 6, who also explores two other possibilities; D. Fowler, Mathematics of Plato's Academy (Oxford 1987) 302-8.

49 This is also valid for the method of analysis and synthesis: see, V.

Karasmanis, The Hypothetical Method in Plato's Middle Dialogues, (D.Phil, thesis, Oxford 1987) 53.

5° See Szabo, op.cit. n.l above, 46-7; cf. op.cit. n.5 above, 216-220.




why mathematicians could not arrive alone at this form of proof. Moreover, it seems that arguments by reductio are prior to Eleatic

philosophy According to Aristotle (De Caelo, 295bl0-296a23) Anaximander used such an argument in order to support his the

sis that the earth stays in the middle of the world.51 Similarly,

Xenophanes' (late sixth century) argument against the anthropo

morphism of the gods is an attempt to reduce that view to absurd

ity.52 Instances of this kind of argument we can discern also in

Heraclitus and even in Homer.53

In conclusion, I find very plausible the thesis that Thales did

engage in geometry, was the first to state general propositions and

used, for the proof of his theorems, the method of superposition. We have no evidence to support that he supplemented his empir ical but general method with reductio ad absurdum arguments, but

it is quite probable that his immediate successors in geometry

(perhaps Anaximander?) carried out this step.

Vassilis Karasmanis

National Technical University of Athens

51 For the arguments of Anaximander see J. Barnes, The Presocratic

Philosophers (London 1979) 23ff; Lloyd, op.cit. n. 14 above, 67-8. 5^

Fragment 15 in Diels-Kranz; cf. also Fragments 14 and 16. 53 For Heraclitus see Fr. 7 ('if everything became smoke, noses would dis

criminate them') and Fr. 23 ('they would not know the name of justice, if these

things did not exist). For Homer see Iliad 563ff and Odyssey 194ff. In the for mer, Achilles says he knows that some god guided Priam to the Achaians' ships: 'for no mortal, not even one in the prime of life, would dare to come to the

camp'. In the latter, Telemachus, faced with a transformed Odysseus, infers

that a god is deceiving him: 'for no mortal man could devise these things with

his own mind, at least... for just now you were an old man'. For a presentation and analysis of early Greek argumentation, see Lloyd, op.cit. n.14 above, 65

79.



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