To the fourteenth article you object, that (in the sixth figure) I assume gratis, that F G, D E, B C, are proportional to A F, A D, A B; and you refer it to be judged by the reader: and to the reader I refer it also. The not exact drawing of the figure (which is now amended) is it that deceived you. For A F, F D, D B, are equal by construction. Also, A G, G E, E C, are equal by construction. And F G, D K, B H, K E, H I, I C, are equal by parallelism. And because A F, F G, are as the velocities wherewith they are described; also 2 A F (that is A D) and 2 F G (that is D E) are as the same velocities. And finally, 3 A F (that is A B) and 3 F G (that is B C) are as the same velocities. It is not therefore assumed gratis, that F G, D E, B C are proportional to A F, A D, A B, but grounded upon the sixth article of the thirteenth chapter; and consequently your objection is nothing worth. You might better have excepted to the placing of D E, first at adventure, and then making A D two-thirds of A B; for that was a fault, though not great enough to trouble a candid reader; yet great enough to be a ground, to a malicious reader, of a cavil. That which you object to the third _corollary_ of art. 15, was certainly a dream. There is no assuming of an angle C D E, for an angle H D E, or B D E, neither in the demonstration, nor in any of the corollaries. It may be you dreamt of somewhat in the twentieth article of chapter XVI. But because that article, though once printed, was afterwards left out, as not serving to the use I had designed it for, I cannot guess what it is: for I have no copy of that article, neither printed nor written; but am very sure, though it were not useful, it was true.