Thales and Geometry Thales is the founder of Greek geometry. 41 This statement implies not only that he introduced geometrical studies into Greek usage, but also that he founded Greek geometry with its striking specificity.
According to the evidence of Proclus (who used Eudemus's book systematically), Thales proved that the circle is halved by its diameter; he found out and stated that the angles at the base of an isosceles triangle are equal; and he discovered the theorem of the equality of vertical angles and that of the equality of triangles with equal bases and adjoining angles (11A20 DK). In the two latter cases there are direct references to Eudemus (frag. 135, 134 Wehrli). [End Page 404]
There are no grounds for believing that tradition has preserved all of Thales' achievements. Thus, according to Pamphila's evidence quoted by Diogenes Laertius (D.L.1.24), but naturally absent from Proclus's commentary on the first book of Euclid's Elements, Thales also inscribed a right-angled triangle in a circle.
A. I. Zaitsev defines Thales' achievements as a "real revolution in the forms of human cognition," to wit: "first, Thales realized the necessity or at least the desirability of proving geometrical statements that seemed self-evident, and second, he gave those proofs." Now we may ask, why did Thales (unlike the Egyptians and Babylonians) begin to prove theorems? "The first mathematical proofs," Zaitsev writes, "were the natural fruit of a social climate where the discovery of a new truth not only gave an immediate satisfaction but could also bring fame. For it is clear that in these conditions, mathematical truths confirmed with proof became a particularly attractive object of search; one who found a faultless proof could as a rule count on public recognition, while the achievements in any other field of knowledge could as a rule be disputed." 42
This ingenious and of course heuristically valuable explanation does not, however, agree with the neighboring statement that the revolution accomplished by Thales consisted in proving statements that seemed self-evident. Who would have disputed something self-evident? Who would have admired the proof of something that was clear without it? It is significant that in the conclusion of Zaitsev's book we meet the following words: Thales "was the first to feel the need of proving geometrical statements that seemed self-evident." But a reference to intuition, as we have already seen, does not solve the problem, for we will then have to postulate similar intuition on the part of all his followers in order to explain why Thales's initiative was recognized and taken up.
B. L. van der Waerden tried to explain the origin of "proving geometry" by the fact that Thales was acquainted with both Egyptian and Babylonian mathematical traditions; having found some discrepancy between them, Thales wondered who was right and how to ascertain it. 43 Such a train of thought may well be appropriate when explaining the origin of the water thesis, but in the given case it does not work because the geometrical problems considered by Thales did not concern such matters as the formula for the area of a circle, in which the data of the two Near Eastern traditions [End Page 405] were at variance. Besides, if we admit that Thales proved self-evident theses, the discrepancies are extraneous. I assert that proof comes from the demand for proof.The demand for proof arises, on the one hand, in connection with solving a problem the answer to which cannot be obtained by means of immediate observation or measurement, and, on the other hand, in connection with the need to submit an answer in the form of a substantiated and not an arbitrary statement.
The information preserved by the textual tradition allows us to suggest how the first geometrical proofs could have been given. Eudemus, when attributing the theorem of the equality of triangles with equal bases and adjoining angles to Thales, remarks in this connection: "And now to estimate the distance to a ship in the sea it is necessary to use the method which, they say, has been shown by Thales." (DK 11A20) Thales' method was convincingly reconstructed by Paul Tannery; this reconstruction has been supported by many others, among them Thomas Heath, who adduced some additional arguments in its favor. 44 It consists of constructing two equal right-angled triangles, one of which is situated on land and therefore can be measured. Heath shows how it is possible to simplify the technical aspect of the solution without changing its geometrical essence, by means of a simple device fixing the angle.45
One cannot agree with Burnet and many others who consider the estimation of the distance to a ship to be the solution of a practical problem by means of a practical rule borrowed from the Egyptians. As for the borrowing, the Egyptians did not deal with the question of the triangles' equality. Moreover, it is difficult to find any substantial practical import in the solution of such a problem. It is, rather, an interesting problem, a puzzle, something like the estimation of the height of a pyramid or of the ratio between the size of the sun and the length of the circle traced by it--these being other problems whose solution is traditionally ascribed to Thales. In all these cases it was impossible to make measurements, and if [End Page 406] Thales nevertheless gave an answer he needed to prove its correct- ness.
It should be noted that the solving of problems concerning the estimation of the distance to an inaccessible object had been developed also in Chinese mathematics, but this did not become a deductive system analogous to the Greeks'--even though in Chinese mathematics one encounters sporadic proofs. 46 This leads us to the idea that it is not only the proofs in themselves that really matters. And if we still choose some proof as a decisive event, we should perhaps prefer the discovery of incommensurability (or irrationality) by Hippasus of Metapontum, who lived a century later than Thales, 47 to the first proof given by Thales. "Properly speaking, we may date the very beginnings of "theoretical" mathematics to the first proof of irrationality, for in "practical" (or applied) mathematics there can exist no irrational numbers." 48 Here a problem arose that is analogous to the one whose solution initiated theoretical natural science: it was necessary to ascertain something that it was absolutely impossible to observe (in this case, the incommensurability of a square's diagonal with its side). This was a more fundamental case of the demand for proof, in comparison with the solution of the problem concerning the distance to an inaccessible object, though we should not disregard the similarity between these two issues. It is not the case that "deductive mathematics begins just that very moment when the knowledge obtained only from practice ceases to be considered quite convincing: when the need...of proof appears even in cases where everyday practice gives, one would think, a full explanation."49 Rather to the contrary: what is called deductive mathematics begins when knowledge that can by no means be obtained empirically starts to be recognized as convincing. It is not the proof of something self-evident, but the proof of something that cannot be seen, that produces a substantiated [End Page 407] knowledge, one that is theoretical and at the same time organized into a system, consecutively brought into correlation with what is already ascertained.
The discovery of incommensurability was attended by the introduction of indirect proof and, apparently in this connection, by the development of the definitional system of mathematics. 50 In general, the proof of irrationality promoted a stricter approach to geometry, for it showed that the evident and the trustworthy do not necessarily coincide. And, of course, when choosing between trustworthiness and evidence, the mathematicians preferred the principle of noncontradictory reasoning--not because they adopted it from Eleatic philosophy, but because otherwise they would be unable to answer the question: "Where did you get it from?" 51
As for the proving of evident statements, that is a secondary occurrence, probably connected with both philosophical and professional exigency. In the course of demonstrative reasoning it was necessary to come to premises that the opponent could agree with, and then to show the inevitability of the correlation between these premises and the consequence being upheld; 52 since such premises belong to the sphere of the evident, there would generally be no need for further persuasion--though if the listener happened to be a follower of Zeno, one had to expect some cavils. On the other hand, a person arranging the material of his proof in a chain so that the transition from one link to another would leave no room for doubt and would win common recognition 53 would certainly want every link to be solid; after all, the boundary between the evident and the not-quite-evident is not always unequivocal.
Geometrical truths are not deduced consecutively from one single fundamental fact, but from a complex of these. The complexforming properties are not deduced one from another, but should be ascertained in their interdependence. The totality of visually evident truths tends to coincide with such a complex. What an indirect proof can do is to show the indispensable concomitance of some fundamental properties. Thus, strict proofs of the evident [End Page 408] correlations and, above all, the formation of a punctiliously arranged deductive system are hardly conceivable without indirect proof. 54
But still, what is to be done with Proclus's assertion that Thales proved (apodeixai) that a circle is halved by its diameter? Heath suggests we should not take Proclus's words too literally, for even Euclid does not prove this statement but takes it for a fact. 55 On the other hand, according to Zaitsev, it is out of the question that Thales advanced just the geometrical statements ascribed to him, while the posterior tradition attributed to him their proof, since we are speaking of visually evident truths. 56 It seems that a compromise can be found here. "Undoubtedly, many ratios were first discovered by means of drawing figures of different kinds and lines inside them along with the concomitant observation of the evident ratios of equality and so on between the parts" 57 --and a reconstruction of the "basic Thales figure," which would be sufficient for the proof of all the theorems ascribed to him, has been suggested: it is only necessary to inscribe a rectangle inside a circle and link its apexes with diagonals.58 In the course of various constructions of this kind it was possible to observe the ratios of different degrees of evidence. Since within the limits of one draft they appeared interdependent, the study of the problematical ratios could entail the formulation and demonstration of all the ratios. Moreover, it is only within the limits of a special draft that a lot of obvious geometrical truths (like the equality of vertical angles, or the equality of two triangles with equal bases and adjoining angles) could be realized and formulated as facts. Besides, ratios that were quite obvious within the limits of the "basic Thales figure" and similar ones were far less obvious in a system of intricate construction with a number of intermediate links--like the one necessary for the construction of the water supply in Samos.59
Fortunately, we have a text at our disposition that may throw some light on how Thales proved evident things. Strangely, this [End Page 409] important evidence has received little attention from those who have discussed the problem. "They say it was Thales himself who first proved that a circle is halved by its diameter"--these words of Proclus are quoted in Diels-Kranz (11A20), but the continuation is omitted: "And the reason of halving is that the straight line passes through the center unflinchingly. For passing the center and always preserving the same movement, being indifferent to both sides, it separates the equal towards the circumference of the circle on both sides and in all its parts" (Proclus Comm. in Eucl., p. 157 Friedlein). The passage goes on: "And if you want to prove the same by scientific means [dia mathematikes ephodou]"--and an indirect proof follows.
Thus Proclus comments on the seventeenth Euclidean definition ("The diameter of a circle is any straight line drawn through the center and bounded at both ends by the circumference of the circle, the same line cutting the circle in halves"). First he gives a general commentary, then states that the correctness of the second part of the definition was first proved by Thales, gives a proof, and finally opposes to it another proof, a scientific one. It appears that Proclus is attributing the first proof to Thales. It is hard to say how the detailed knowledge of Thales' geometrical studies was preserved, but it is nevertheless a fact that Eudemus of Rhodes, a pupil of Aristotle, was informed about different methods of proof used by Thales and the peculiarities of his mathematical phraseology.60
What Thales deals with here is a kind of etiology, not deduction. He refers to properties that are not deduced one from another, but ascertained in their interdependence: the equality of the parts of a chord dissected by a diameter when the angles of dissection are equal (with the concomitant equality of the segments into which the circle is divided, two by two). 61 At the same time we have before us a reasoning, and not a primitive demonstration by means of folding and superimposing the figures (which, in spite of the ancients' habit of drawing on the ground, is generally supposed to have been used by Thales).62 What we have here is a study, and not a crafty answer. [End Page 410]
It seems possible to say that the path that later proved to be so fruitful was outlined already in Thales' geometrical studies. In these an interest in the properties of figures expressed in ratios (contrary to the operational potentialities connected with them) comes to the foreground. Thales is interested in what the diameter does with the circle, which triangles are equal, what happens to the angles when straight lines cross. It was this way of thinking that led to the formation of geometry as a system of interdependencies. Later on this development was extremely favored by the fact that after having discovered the correlation between the pitch of a sound and the length of the string producing that sound, Pythagoras and his followers considered the problem of ratios, and above all quantitative ones, to be among the most important and significant problems. The study of ratios became a more or less deliberate program, Pythagoras's theorem and the proof of incommensurability being its fruits.
This path was fruitful also in two other interdependent aspects. First, it gave the opportunity for cumulative development, for it was possible to construct a new figure out of a given one, and to advance from the solution of one problem to the solution of another. Second, in the course of studying geometrical (and then, naturally, arithmetical) ratios and their quantitative expression there was an apparatus worked out whose application to the description of nature proved to be effective in physics, astronomy, and geography. Eudoxus, Eratosthenes, and Archimedes made considerable progress in this work in ancient times, while the great scientists of the seventeenth century brought it up to a new level.
No wonder that at the beginning of this path there is a Greek. Nobody hindered him from solving fascinating problems: on the contrary, solving them gave him a chance to demonstrate the brilliance of his intellect. And if it proved necessary to pass from just fascinating problems to the realm of high significance in order to make a complete revolution in mathematics, the Pythagorean mathematicians were people who chose their way of life voluntarily and therefore knew why they were racking their brains over questions that were far from being of practical use, knowing at the same time that they could count for understanding at least on the part of their colleagues. But the essence of the colossal efficiency of Greek culture lay in the fact that the way to the amazed admiration for which a man could rightfully hope led--as he addressed his [End Page 411] equals--through recognition. In the sphere of thought, this led to objectivity.
This is page one
These pages were developed through GirlTECH teacher training program sponsored by the Center for Research on Parallel Computation (CRPC), a National Science Foundation Science and Technology Center. Pages copyright July 1999 by Ricardo Jesus.