THE GEOMETRY OF THALES BY PERCY J. HARDING. In this paper there will, of course, be found no new facts ; but merely a review of well-known historical statements and an attempt to connect them. It has often been stated that Thaïes introduced an abstract Geometry of lines and angles; whereas the Egyptians dealt with a practical Geometry of areas and volumes, and possessed only slight beginnings of any more advanced views. This seems to be true when we compare the papyrus of Ahmes with the theorems attributed to Thaïes. But the latter, like men of all ages, must have made his advances gradually and with difficulty; and to understand his position we ought to avoid ascribing to him too great or too sudden a bound forward into abstract science. The gradual nature of his advances is clearly pointed out by the practical measurements that are related of him, forming, as these do, a natural link between the geometry of the Egyptians and his own more abstract work. One of the most remarkable of the theorems attributed to Thaïes is that the angle in a semicircle is a right angle ; and it is in connection with this theorem that I wish to enquire with what amount of geometrical knowledge he should be credited. Allman and Cantor consider that Thaïes knew not only the theorem just mentioned, but also that the sum of the angles of a triangle equals two right angles ; and further that he succeeded in deducing one of these theorems from the other with the help of his knowledge that the angles at the base of an isosceles triangle are equal. They differ from one another as to which of these two theorems he deduced from the other; but by each, in his conjectured proof, Thaïes is made to treat angles very freely as magnitudes. In each case he adds angles together, regards one angle of a triangle as equal to the sum of the other two, or to half the sum of all three. In fact a veritable algebraical equation is worked with angles as the magnitudes concerned ; and Allman even goes so far as to say that in this he laid the foundations of algebra. The question for this paper is whether Thaïes had such a complete view of angles as magnitudes that he could deal with them in the way just mentioned. Before answering this question, I propose to consider the different stages that can be traced in the development of the complete notion of a plane angle stages
534 PERCY J. HARDING that every individual has to go through, and that every day present difficulties to beginners. (i) An angle originally is simply a corner. In this stage there is no clear idea of an angle as a magnitude. Corners may be of different shapes ; but one is hardly regarded as greater than another. The corner of a square or rectangle is one of a special shape rather than of special size. (ii) In the next stage an angle is a slope or inclination. Here there is some notion of magnitude attached to an angle, but not a very complete one. Two lines may have the same or different slopes ; at most the inclination of one of them to a base line may be regarded as greater than that of another. But in this stage angles are not thought of as pure magnitudes that may be the subjects of addition or subtraction, or of any form of equation. (iii) In the third stage we have the fully developed view of an angle as a magnitude, generally -explained with the help of the notion of revolution. It is generally allowed that, in an early stage of thought, an angle may be regarded only from the first of the preceding points of view. But surely there may be a period in the development of thought in which an angle is looked at from the point of view of (ii); whilst (iii) is not understood at all. I think the Egyptians were in this stage, and my reason for thinking so is to be found in their calculations of seqt. In the seqt we find a cosine in embryo. The seqt was not a true ratio, but was sufficient to determine the slope of a slanting edge of a pyramid on a square base. The seqt seems to have been used to enable the builders of a pyramid to give the required slope to its edges; from which it follows that the designers felt that a pyramid might be built with an edge either too steep or not steep enough. That is, they felt that the idea of greater and less could be applied to the inclination of an edge to the base, though they may not verbally have reached quite so far. That stages corresponding to the three just mentioned occur generally in the development of the idea of magnitude, and that it is possible for this development to be arrested at the end of stage (ii), may be verified by watching the thoughts of children. But a very striking case of this arrested development is to be seen in Euclid's Fifth Book. It is often discussed* whether, in the Fifth Book, Euclid regarded a ratio as a magnitude; and the different answers given suggest that he hesitated in the matter. The correct answer seems to be that he left this thought where he found it, i.e. in stage (ii). In the fifth and sixth definitions he speaks of the sameness of two ratios : in the seventh definition one ratio may be greater than another. But ratios are never said to be equal, nor are two ratios added together. The avoidance of the use of the word " equal " with regard to two quantities of the same kind, although some notion of magnitude is clearly present, seems to be characteristic of stage (ii). If the word " equal " is found in any edition of a work * See Prof. Hill on the Theory of Proportion in the Mathematical Gazette for July 1912, Arts. 12 and 13.
THE GEOMETRY OF THALES 535 that is really in this stage, it is probably merely an alteration made almost unconsciously by an editor who is himself in a more advanced stage of thought. An illustration of this is that some editors of Euclid's Fifth Book, though apparently intending to keep pretty close to his words, use the word "equal," or let it slip in, without any note or qualifying remark*. Having now seen the level of the Egyptian view- of an angle, the view that we may suppose came before Thaïes through his Egyptian teachers, we may ask whether, in considering an angle as a magnitude, he himself ever reached stage (iii). I submit that historical evidence answers this question in the negative; and that whoever, in reconstructing proofs for Thaïes, represents him as regarding an angle from the point of view of stage (iii), is reading his own more advanced thoughts into Thaïes' work. In the first place Proclus says that in stating the theorem that the angles at the base of every isosceles triangle are equal he did not use the Greek word for equal ; but "in archaic fashion he phrased it, like (o/j,ocaù)f." Next, if, disregarding the angle in a semicircle and the sum of the three angles of a triangle, we look at the other cases, either of measurement or theory, in which Thaïes deals with angles, we can see that, although he looked at angles in a more abstract way than the Egyptians and had a better notion of their use and importance, yet there is no suggestion of the addition of angles, but only a better view of slope and seqt. But if Thaïes knew the theorem of the sum of the angles of a triangle his thought must have been in stage (iii). And if he was in stage (iii), he might easily have seen a connection between this theorem and that of the angle in a semicircle. But, as other things suggest that he was only in stage (ii), direct historical evidence is certainly required to satisfy us that he knew the theorem of the sum of the angles of a triangle. There is no such direct evidence ; but only an inference from a statement of Geminus. All the rest is pure conjecture. Now, the statement of Geminus is, that the ancient geometers proved the theorem about the sum of the angles of a triangle for three species of triangles, viz. equilateral, isosceles and scalene, separately; and as further Proclus states that this theorem was first proved in a general manner by the Pythagoreans, it is inferred that the older geometers mentioned were none other than Thaïes and his successors in the Ionic school. To this I say that it is not certain, and not even probable, that Thaies was one of the older geometers referred to. First Proclus ascribes the general proof to the Pythagoreans rather than to Pythagoras. Hence its date may be after the death of the latter; and if so the older geometers mentioned might have been decidedly younger than Thaïes. Next, all the attempts that have been made to reconstruct the proof of these older geometers have required a perpendicular to be drawn from the vertex of a * See Hill's quotation from Simson in the same article. f See Allman's Greek Geometry from Thaïes to Euclid, Chap. i.
536 PERCY J. HARDING triangle to its base : a construction that ought not to be supplied so lightly. For this construction is conspicuous by its absence from the Rhind papyrus ; and that this is not a mere accidental omission may be inferred from the fact that the priests of the temple of Edfu, and even later land measurers, were no further advanced in this particular. It is this absence of the perpendicular that spoils the Egyptian measurements. What could cause the area of an isosceles triangle to be found by taking the product of the lengths of one side and half the base, except unfamiliarity with the perpendicular mentioned? The statement that Oinopides was the first to draw a perpendicular from an external point on to a straight line is generally received with some hesitation or qualification. It is supposed that the passage means that he was the first to give the pure ruler and compass method. This may be the meaning. But the very fact that the perpendicular was in any way coupled with the name of Oinopides seems to suggest that it was at least not common before his time. But the Egyptians were familiar with the carpenter's square and could draw right angles and rectangles. True, but to draw a right angle as a corner between the bounding lines of a figure under construction, or as part of an ornamental drawing, is a very different thing from introducing an internal perpendicular into a triangle for the sake of discovering or proving some further property. In fact, after the bounding lines of a figure had been drawn, the introduction of further lines preliminary to a proof a proceeding common enough in Euclid would not suggest itself very readily to people who had little or no experience of a proof in any form. Further, as the ancient historians have given us a fairly detailed list of constructions and theorems due to Thaïes, is it possible that, if he had discovered the theorem of the sum of the angles of a triangle, no one of the ancients should have recollected to attribute to so celebrated a man a discovery that would certainly have been his most remarkable accomplishment? I conclude that it is not even probable that Thaies was one of the older geometers who knew, through some archaic proof, that the sum of the angles of any triangle is equal to two right angles; and hence, to render it probable that he did not advance beyond stage (ii) in the notion of an angle as a magnitude, it only remains to deal with the theorem of an angle in a semicircle. The historical foundation for the opinion that Thaïes discovered this theorem is a passage from Diogenes Laertius in which, on the authority of Pamphila, it is stated that Thaïes was the first to describe a right-angled triangle in a circle. The objections to the acceptance of this passage as evidence of Thaïes' knowledge of the theorem in question are, first, either Diogenes Laertius, or Pamphila (perhaps both) did not quite understand their own tale and confused a theorem with a problem ; secondly, it is strange that, if, in ancient times, this theorem was generally attributed to Thaïes, Proclus should have omitted all mention of it in his account of Thaïes' work.
THE GEOMETRY OF THALES 537 As far as this paper is concerned, it may be said that if this theorem is not due to Thaïes the question as to his view of an angle is at an end. There is nothing left in his reported work that lifts him out of stage (ii). But if it is considered that Pamphila's statement is evidence that Thaïes knew that the angle in a semicircle is a right angle, we have to consider in what way Thaïes could have arrived at this theorem, and whether it was necessary, or even likely, that he should take a stage (iii) view of an angle. The following method seems to me probable. The Egyptians were experts in ornamental Geometrical drawing. They were familiar with squares and with surfaces mapped out into squares by horizontal and vertical lines. They must also have been familiar with rectangles, for a row of these squares would produce a rectangle. And also, though the author of the papyrus of Ahmes does not express himself very clearly, he plainly, in measuring areas, had squares and rectangles in his mind. Further, squares with their inscribed circles are found in Egyptian decorations. The most natural way for them to obtain these would be to draw the diagonals of a square* and so obtain the centre from which they could draw the inscribed circle. Though I do not know that figures of squares and rectangles with their circumscribed circles have been discovered in Egyptian drawings, it seems difficult to suppose that the Egyptians stopped short of these ; but even if they did, that is if no such wall decoration met the gaze of Thaïes, it would have been no great step in advance for him to have drawn such a figure himself. And, bearing in mind that Proclus says that Thaïes attempted some things in a general or more abstract manner, we may regard him as lifting the whole subject away from wall decoration. He might thus have before him the abstract figure of a rectangle, its two diagonals, and the circumscribing circle with the intersection of the diagonals as its centre. Proclus further says that Thaïes first demonstrated that the circle was bisected by its diameter. We may, of course, erase the word " demonstrated," as belonging to a later time and substitute " observed " or " stated." Then applying this to the figure of a rectangle and its circumscribed circle, and omitting from consideration one of the diagonals of the rectangle, Thaïes might easily have observed that the other diagonal was a diameter of the circle, and that the figure on one side of it was a semicircle in which was an angle that was first drawn as an angle of a rectangle. If next, the figure were to be drawn in a different order, the circle first, then a diameter and then an angle in the semicircle on one side of the diameter, I think that the statement of Pamphila is quite justified and fairly applicable to the thought of a time before Pythagoras. The criticism of Proclus comes from a mind of a later date. My final conclusion is that, though Thaïes dealt with angles in a far more advanced way than the Egyptians, there is not sufficient reason for supposing him to have reached stage (iii) or to have known the theorem about the sum of the angles * The introduction of the diagonals of a square for the purpose of further construction is very different from the introduction of a perpendicular into a triangle for the purpose of proof.
538 PERCY J. HARDING of a triangle. The different statements made about him form a much more consistent whole without these suppositions. Moreover, the statement of Apollodorus that it was Pythagoras who first drew a right angle in a circle is not, if the preceding view is adopted, decidedly contradictory of Pamphila. For the Pythagoreans, with their deductive method, and after the general proof of the theorem of the sum of the angles of a triangle, could easily advance to a deductive proof of the theorem of the angle in a semicircle, and might hence be reported to be its discoverers. Finally, I may add that since writing this paper I have seen in Zeuthen's History of Mathematics a suggestion that Thaïes might have reached the theorem of the angle in a semicircle in the way just given; but as what is here written was sketched out in my own mind long before the publication of Zeuthen's book, I have thought it better to leave this paper as it is.