136 THE EARLIEST GREEK GEOMETRY. THALES
of the angles of the rectangle, i. e, are equal to two right
angles; and it immediately follows that the sum of the angles
of the original equilateral or isosceles triangle is equal to two
right angles. The same thing is easily proved of any triangle
by dividing it into two right-angled
triangles and completing the rectangles
which are their doubles respectively, as
in the figure. But the fact that a proof
on these lines is just as easy in the case
of the general triangle as it is for the
equilateral and isosceles triangles throws doubt on the whole
procedure; and we are led to question whether there is any
foundation for Gerninus’s account at all. Aristotle has a re
mark that
‘even if one should prove, with reference to each (sort of)
triangle, the equilateral, scalene, and isosceles, separately, that
each has its angles equal to two right angles, either by one
proof or by different proofs, he does not yet know that the
triangle, i.e. the triangle in general, has its angles equal to
two right angles, except in a sophistical sense, even though
there exists no triangle other than triangles of the kinds
mentioned. For he knows it not qua triangle, nor of every
triangle, except in a numerical sense; he does not know it
nationally of every triangle, even though there be actually no
triangle which he does not know A
It may well be that Geminus was misled into taking for
a historical fact what Aristotle gives only as a hypothetical
illustration, and that the exact stages by which the proposi
tion was first proved were not those indicated by Geminus.
Could Thales have arrived at his proposition about the
semicircle without assuming, or even knowing, that the sum
of the angles of any triangle is equal to two right angles ? It
seems possible, and in the following way.
Many propositions were doubtless first
discovered by drawing all sortsof figures
and lines i n them, and observing apparent
relations of equality, &c., between parts.
It would, for example, be very natural
to draw a rectangle, a figure with four right angles (which, it
1 Arist. Anal. Post. i. 5, 74 a 25 sq.
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