Page:Carroll - Euclid and His Modern Rivals.djvu/211

. § 4.] it is luminous when compared with the note which follows it. 'If the linear unit and angular unit are left arbitrary, any angle has for measure the ratio of the numbers of linear units contained in the arcs which the angle in question and the irregular unit intercept in any circumference described from their summit as common centres.' Is not that a useful note? 'The irregular unit'! Linear, or angular, I wonder? And then 'common centres'! How many centres does a circumference usually require? I will only trouble you with one more extract, as a bonne bouche to wind up with.

'Th. 9. (P. 126.) Every convex closed Line ABCD enveloped by any other closed Line PQRST is less than it.



'All the infinite Lines ABCD, PQRST, &c.'by the way, these are curious instances of 'infinite Lines'?

Nie. (hastily) We mean 'infinite' in number, not in length.

Min. Well, you express yourself oddly, at any rate 'which enclose the plane surface ABCD, cannot be equal. For drawing the straight Line MD, which does not cut ABCD, MD will be less than MPQD; and adding