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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