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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