134 THE EARLIEST GREEK GEOMETRY. THALES 
Diogenes Laertius was mistaken in bringing Apollodorus into 
the story now in question at all; the mere mention of the 
sacrifice in Pamphile’s account would naturally recall Apollo- 
dorus’s lines about Pythagoras, and Diogenes may have 
forgotten that they referred to a different proposition. 
But, even if the story of Pamphile is accepted, there are 
difficulties of substance. As Allman pointed out, if Thales 
knew that the angle in a semicircle 
is a right angle, he was in a position 
at once to infer that the sum of the 
angles of any right-angled triangle is 
equal to two right angles. For suppose 
that BC is the diameter of the semi 
circle, 0 the centre, and A a point on 
the semicircle; we are then supposed 
to know that the angle BAG is a right angle. Joining OA, 
we form two isosceles triangles OAB, OAC; and Thales 
knows that the base angles in each of these triangles are 
equal. Consequently the sum of the angles OAB, OAC is 
equal to the sum of the angles OB A, OCA. The former sum 
is known to be a right angle; therefore the second sum is 
also a right angle, and the three angles of the triangle ABC 
are together equal to twice the said sum, i.e. to two right 
angles. 
Next it would easily be seen that any triangle can be 
divided into two right-angled triangles by drawing a perpen 
dicular AD from a vertex A to the 
opposite side BC. Then the three 
angles of each of the right-angled 
triangles ABD, ADC are together equal 
to two right angles. By adding together 
the three angles of both triangles we 
find that the sum of the three angles of the triangle ABC 
together with the angles ADB, ADC is equal to four right 
angles; and, the sum of the latter two angles being two 
right angles, it follows that the sum of the remaining angles, 
the angles at A, B, C, is equal to two right angles. And ABC 
is any triangle. 
Now Euclid in III. 31 proves that the angle in a semicircle 
is a right angle by means of the general theorem of I. 32 
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