'This is unquestionably the meaning of the statement of Pamphila (temp. Nero), ap. Diog. Laer. i. 24, that he was the first person to describe a right-angled triangle in a circle.
12) .
theory of the scalene triangle and the theory of lines'. Proclus in his summary of the history of geometry before Euclid, says that Thales introduced geometry into Greece from Egypt, and com municated the beginnings of many propositions to his successors.
From these indications it is difficult to determine what Thales brought from Egypt and what he himself discovered. This diffi culty has, however, been lessened since the translation and publica tion of the papyrus Rhind by and we can deduce cer tain facts from it. [I] Thales must have known the theorem that the sum of the three angles of a triangle are equal to two right angles. This inference is made from theorems (4) and (2). We know from Proclus, on the authority of Eudemus, that Euclid i. 32 was first proved in a general way by the Pythagoreans; but, on the other hand, we learn from Geminus that the ancient geometers discovered the equality to two right angles in each kind of triangle —equilateral isosceles, and scalene (Apoll. Conica, ed. Halleius, p. 9) ; and the geometers older than the Pythagoreans can only have been Thales and his school. The theorem was probably arrived at by induction, and may have been suggested by the con templation of floors or walls covered with regular triangular, square or hexagonal tiles. [2] We see also in the theorem (4) the first trace of the conception of geometrical loci, which we, there fore, attribute to Thales. It was in this manner that this remark able property of the circle, with which, in fact, abstract geometry was inaugurated, presented itself to the imagination of Dante:— "0 se del mezzo cerchio far si puote Triangol si, ch'un retto non avesse."—Par. c. xiii. loi.
[3] Thales discovered the theorem that the sides of equiangular triangles are proportional. This theorem was probably made use of also in his determination of the distance of a ship at sea.
Let us now consider the importance of the work of Thales. I. In a scientific point of view: (a) by his two theorems he founded the geometry of lines, which has ever since remained the principal part of geometry; (b) he may be considered to have laid the foundation of algebra, for his first theorem establishes an equation in the true sense of the word, while the second institutes a II. In a philosophic point of view; we see that in these two theorems of Thales the first type of a natural law, i.e., the expression of a fixed dependence between different quan tities, or, in another form, the disentanglement of constancy in the midst of variety—has decisively III. Lastly, in a practical point of view: Thales furnished the first example of an application of theoretical geometry to practice', and laid the foundation of the methods of measurement of heights and distances.
As to the astronomical knowledge of Thales we have the follow ing notices :—(i) besides the prediction of the solar eclipse, Eudemus attributes to him the discovery that the circuit of the sun between the solstices is not always uniforms; (2) he called the last day of the month the thirtieth (Diog. Laer. i. ; (3) he divided the year into 365 days (Id. i. 27) ; (4) he determined the diameter of the sun to be the 72oth part of the he remarked on the constellation of the Lesser Bear and instructed his countrymen to steer by it [as nearer the pole] instead of the Great Bear (Callimachus ap. Diog. Laer. i. 23 ; cf. Aratus, Phae nomena, v. 36 seq.). Other discoveries in astronomy are attributed to Thales but on doubtful authorities. He did not know, for example, that "the earth is spherical," as is erroneously stated by Plutarch (Placita, iii. io) ; on the contrary, he conceived it to be a flat disk. The doctrine of the sphericity of the earth, for which the researches of Anaximander had prepared the way, was in fact one of the great discoveries of Pythagoras, was taught by Parmenides, who was connected with the Pythagoreans, and re Laertius (i. 25).