THE ANGLE IN A SEMICIRCLE
135
that the sum of the angles of any triangle is equal to two
right angles; but if Thales was aware of the truth of the
latter general proposition and proved the proposition about
the semicircle in this way, by means of it, how did Eudemus
come to credit the Pythagoreans, not only with the general
proof, but with the discovery, of the theorem that the angles
of any triangle are together equal to two right angles 1 1
Cantor, who supposes that Thales proved his proposition
after the manner of Euclid III. 31, i.e. by means of the general
theorem of I. 32, suggests that Thales arrived at the truth of
the latter, not by a general proof like that attributed by
Eudemus to the Pythagoreans, but by an argument following
the steps indicated by Geminus. Geminus says that
‘ the ancients investigated the theorem of the two right
angles in each individual species of triangle, first in the equi
lateral, then in the isosceles, and afterwards in the scalene
triangle, but later geometers demonstrated the general theorem
that in any triangle the three interior angles are equal to two
right angles ’. 2
The ‘ later geometers ’ being the Pythagoreans, it is assumed
that the ‘ancients’ may be Thales and his contemporaries.
As regards the equilateral triangle, the fact might be suggested
by the observation that six such triangles arranged round one
point as common vertex would fill up the space round that
point; whence it follows that each angle is one-sixth of four
right angles, and three such angles make up two right angles.
Again, suppose that in either an equilateral or an isosceles
triangle the vertical angle is bisected by a straight line meet
ing the base, and that the rectangle of which the bisector and
one half of the base are adjacent sides is completed; the
rectangle is double of the half of the original triangle, and the
angles of the half-triangle are together equal to half the sum
1 Proclus on Euel. I, p. 879. 2-5.
2 See Eutocius, Comm, on Conics of Apollonius (vol. ii, p. 170, Heib.).
