PYTHAGOREAN GEOMETRY 
143 
Discoveries attributed to the Pythagoreans. 
(a) Equality of the sum of the three angles of a triangle 
to two right angles. 
We have seen that Thales, if he really discovered that the 
angle in a semicircle is a right, angle, was in a position, first, 
to show that in any right-angled triangle the sum of the three 
angles is equal to two right angles, and then, by drawing the 
perpendicular from a vertex of any triangle to the opposite 
side and so dividing the triangle into two right-angled 
triangles, to prove that the sum of the three angles of any 
triangle whatever is equal to two right angles. If this method 
of passing from the particular case of a right-angled triangle to 
that of any triangle did not occur to Thales, it is at any rate 
hardly likely to have escaped Pythagoras. But all that we know 
for certain is that Eudemus referred to the Pythagoreans 
the discovery of the general theorem that in any triangle 
the sum of the interior angles is equal to two right angles. 1 
Eudemus goes on to tell us how they proved it. The method 
differs slightly from that of Euclid, but depends, equally with 
Euclid’s proof, on the properties of parallels; it can therefore 
only have been evolved at a time when those properties were 
already known. 
Let ABC be any triangle; through A draw DE parallel 
to BO. DAE 
Then, since BC, DE are parallel, the 
alternate angles DAB, ABC are equal. 
Similarly the alternate angles EAC, 
ACB are equal. 
Therefore the sum of the angles ABC, 8 c 
ACB is equal to the sum of the angles DAB, EAC. 
Add to each sum the angle BAG-, therefore the sum of the 
three angles ABC, ACB, BAG, i.e. the three angles of the 
triangle, is equal to the sum of the angles DAB, BAG, CAE, 
i.e. to two right angles. 
We need not hesitate to credit the Pythagoreans with the 
more general propositions about the angles of any polygon, 
1 Proclus on End. I, p. 397. 2.