uniform motions in C H and H P will be described uniformly the strait line P C. Seeing therefore the two strait lines A P and P C are described in the time A E with the same increase of impetus, wherewith the crooked line A B O C is described in the same time A E, that is, seeing the line A P C and the line A B O C are transmitted by the same body in the same time and with equal velocities, the lines themselves are equal; which was to be demonstrated. By the same method (if any of the semiparabolasters in the table of art. 3 of the precedent chapter be exhibited) may be found a strait line equal to the crooked line thereof, namely, by dividing the diameter into two equal parts, and proceeding as before. Yet no man hitherto hath compared any crooked with any strait line, though many geometricians of every age have endeavoured it. But the cause, why they have not done it, may be this, that there being in Euclid no definition of equality, nor any mark by which to judge of it besides congruity (which is the 8th axiom of the first Book of his Elements) a thing of no use at all in the comparing of strait and crooked; and others after Euclid (except Archimedes and Apollonius, and in our time Bonaventura) thinking the industry of the ancients had reached to all that was to be done in geometry, thought also, that all that could be propounded was either to be deduced from what they had written, or else that it was not at all to be done: it was therefore disputed by some of those ancients themselves, whether there might be any equality at all between crooked and strait lines; which question Archimedes, who assumed that some strait line was equal to the circumference of a circle, seems to have despised, as he had reason. And there is a late writer that granteth that between a strait and a crooked line there is equality; but now, says he, since the fall of Adam, without the special assistance of Divine Grace it is not to be found. ------------------------------------------------------------------------ [Illustration: _Vol. 1. Lat. & Eng._ C. XVIII. _Fig. 1-2_ ] ------------------------------------------------------------------------ ------- CHAPTER XIX. OF ANGLES OF INCIDENCE AND REFLECTION, EQUAL BY SUPPOSITION.