Page:Carroll - Euclid and His Modern Rivals.djvu/176

138 words 'the bisector of the angle FAC,' the words 'the perpendicular to FC drawn through A.'

(5) Vertical angles together greater than two right angles.

This also requires a new proof, as we must insert, after the words 'the bisector of the angle FAC,' the words 'produced through A,' and must then prove (by your Th. 1) that the angles OAC, OAF, are equal.

On the whole, I take this to be the most cumbrous proof yet suggested for this Theorem.

We now come to what is probably the most extraordinary Corollary ever yet propounded in a geometrical treatise. Turn to pages 30 and 31.

Th. 20. 'If two triangles have two sides of the one equal to two sides of the other, and the angle opposite that which is not the less of the two sides of the one equal to the corresponding angle of the other, the triangles shall be equal in all respects.

'Cor. 1. If the side opposite the given angle were less than the side adjacent, there would be two triangles, as in the figure; and the proof given above is inapplicable.

'This is called the ambiguous case.'

The whole Proposition is a grand specimen of obscure writing and bad English, 'is' and 'are,' 'could,' and 'would,' alternating throughout with the most charming impartiality: but what impresses me most is the probable effect of this wondrous Corollary on the brain of a simple reader, coming breathless and exhausted from a death-struggle with the preceding theorem. I can imagine him saying wildly to himself 'If two Triangles fulfil such and