# Greek Mathematics

# V. Thales

# The circle is bisected by its diameter

# Procl. in Eucl. i., ed. Friedlein 157. 10–13

Τὸ μὲν οὖν διχοτομεῖσθαι τὸν κύκλον ὑπὸ τῆς διαμέτρου πρῶτον Θαλῆν ἐκεῖνον ἀποδεῖξαί φασιν, αἰτία δὲ τῆς διχοτομίας ἡ τῆς εὐθείας ἀπαρέγκλιτος διὰ τοῦ κέντρου χώρησις.

The angles at the base of an isosceles triangle are equalIbid. 250. 22–251. 2

Λέγεται γὰρ δὴ πρῶτος ἐκεῖνος ἐπιστῆσαι καὶ εἰπεῖν, ὡς ἄρα παντὸς ἰσοσκελοῦς αἱ πρὸς τῇ βάσει γωνίαι ἴσαι εἰσίν, ἀρχαϊκώτερον δὲ τὰς ἴσας ὁμοίας προσειρηκέναι.

# Thales

# V. Thales

# The circle is bisected by its diameter

# Proclus, on Euclid i., ed. Friedlein 157. 10–13

They say that Thales was the first to demonstrate^{a} that the circle is bisected by the diameter, the cause of the bisection being the unimpeded passage of the straight line through the centre.

[Thales] is said to have been the first to have known and to have enunciated [the theorem] that the angles at the base of any isosceles triangle are equal, though in the more archaic manner he described the equal angles as similar.^{b}

^{a}The word “demonstrate” (ἀποδεῖξαι) must not be taken too literally. Even Euclid did not demonstrate this property of the circle, but stated it as the 17th definition of his first book. Thales probably was the first to point out this property. Cantor (Gesch. d. Math. i^{3}., pp. 109, 140) and Heath (H.G.M. i. 131) suggest that his attention may have been drawn to it by figures of circles divided into equal sectors by a number of diameters. Such figures are found on Egyptian monuments and vessels brought by Asiatic tributary kings in the time of the eighteenth dynasty.^{b}This theorem is Eucl. i. 5, the famous pons asinorum. Heath notes (H.G.M. i. 131): “It has been suggested that the use of the word ‘similar’ to describe the equal angles of an isosceles triangle indicates that Thales did not yet conceive of an angle as a magnitude, but as a figure having a certain shape, a view which would agree closely with the idea of the Egyptian se-qet, ‘that which makes the nature,’ in the sense of determining a similar or the same inclination in the faces of pyramids.”