For in the first place we have this A B. A G :: B G. G E, by analogism article 18. Then by composition we have this A B + A G. A B :: B G + G E that is, B E. B G. And by taking the halves of the ½A B + ½A G. A B :: ½B G + ½G E, antecedents this third that is, B H. B G. And by conversion a fourth A B. ½A B + ½A G :: B G. B H. And by division this fifth ½A B - ½A G. ½A B + ½A G :: H G. B H. And by doubling the first antecedent and the first consequent A B - A G. A B + A G :: H G. B H. Also by the same method may be found out this analogism A B - A I. A B + AI :: K I. B K. Now seeing the proportion of A B to A E is greater than that of A B to A F, the proportion of A B to A G, which is half the greater proportion, is greater than the proportion of A B to A I the half of the less proportion; and therefore A I is greater than A G. Wherefore the proportion of A B - A G to A B + A G, by the precedent lemma, will be greater than the proportion of A B - A I to A B + A I; and therefore also the proportion of H G to B H will be greater than that of K I to B K, and much greater than the proportion of K I to B H, which is greater than B K; for B H is the half of B E, as B K is the half of B F, which, by supposition, is less than B E. Wherefore H G is greater than K I; which was to be proved. Coroll. It is manifest from hence, that if any quantity be supposed to be divided into equal parts infinite in number, the difference between the arithmetical and geometrical means will be infinitely little, that is, none at all. And upon this foundation, chiefly, the art of making those numbers, which are called Logarithms, seems to have been built. 29. If any number of quantities be propounded, whether they be unequal, or equal to one another; and there be another quantity, which multiplied by the number of the propounded quantities, is equal to them all; that other quantity is a mean in arithmetical proportion to all those propounded quantities. ------- CHAP. XIV. OF STRAIT AND CROOKED, ANGLE AND FIGURE.