Page:Carroll - Euclid and His Modern Rivals.djvu/238

200 request, to 'write down its converse theorem,' it is so powerfully borne in upon me that the writer of the Question is probably himself a biped, that I feel I must, however reluctantly, decline the task.

Th. 5 (p. 18) is Euc. I. 4, proved as in Euclid.

Th. 6 (p. 20) is Euc. I. 5, proved by supposing the vertical angle to be bisected, thus introducing a 'hypothetical construction' (see p. 20).

Th. 7 (p. 21) is Euc. I. 26 (1st part), proved by superposition. Euclid's proof, by making a new Triangle, is quite as good, I think. The areas are here proved to be equal, a point omitted by Euclid: I think it a desirable addition to the Theorem.

Th. 8 (p. 22) is Euc. I. 5, proved by reversing the Triangle and then placing it on itself (or on the trace it has left behind), a most objectionable method (see p. 48).

Theorems 9 to 13 (pp. 22 to 26) are Euc. I. 16, 18, 19, 20, 21, with Euclid's proofs.

Th. 14 (p. 27) is Euc. I. 24, proved by supposing an angle to be bisected: another 'hypothetical construction.'

Th. 15 (p. 28) is Euc. I. 8, for which two proofs are offered:—one by Euc. I. 24 (which seems to be reversing the natural order)—the other by an application of Euc. I. 5, a method involving three cases, of which only one is given. All this is to save the introduction of Euc. I. 7, a Theorem which I think should by no means be omitted. (See p. 220.) Here, as in Th. 7, the equality of the areas is, I think, a desirable addition to Euclid's Theorem.

Th. 16 (p. 29) is Euc. I. 25, with old proof.