Note 282 — Chapter XX · (continued 3)

The demonstration of the thirty-second proposition of Euclid could leave no one in doubt whether this property were common to all triangles, after it had been proved in a single instance. It is said, however, to be recorded by an ancient writer, that this discovery was first made as to equilateral, afterwards as to isosceles, and lastly as to other triangles. Stewart’s philosophy of Human Mind, vol. ii., chap., iv., sect. 2 The mode of proof must have been different from that of Euclid. And this might possibly lead us to suspect the truth of the tradition. For if the equality of the angles of a triangle to two right angles admitted of any elementary demonstration, such as might occur in the infancy of geometry, without making use of the property of parallel lines, assumed in the twelfth axiom of Euclid, the difficulties consequent on that assumption would readily be evaded. See the Note on Euclid, i. 29. in Playfair, who has given a demonstration of his own, but one which involves the idea of motion rather more than was usual with the Greeks in their elementary propositions.