12. Let such ordinate dh, or (equal to it in the Asymptote) AF, be so divided in L, M, N, &c. (by Perpendiculars cutting the Hyperbola in l, m, n, &c.) as that FL, LM, MN, be as 1⁄m, 1⁄mm, 1⁄m3, &c. That is, so continually decreasing as that each Antecedent be to its Consequent, as 1 to 1⁄m, or as m to 1. See Fig. 5. Tab. 5.
13. This is done by taking AF, AL, AN, &c. in such proportion. For, of continual Proportionals, the Differences are also continually proportional, and in the same proportion. For let A, B, C, D, &c. be such Proportionals, and their Differences a, b, c, &c. That is, A - B = a, B - C = b, C - D = c, &c.
Then, because A, B, C, D, &c. are in continual proportion,
That is, A. B :: B. C :: C. D :: &c.
And dividing (A - B). B :: (B - C). C :: (C - D). D :: &c.
That is, a. B :: b. C :: d. D :: &c.
And alternly a. b. c. &c. :: B. C. D. &c. :: A. B. C. &c.
That is, in continual proportion as A to B, or as m to 1.
14. This being done; the Hyperbolick Spaces Fl, Lm, Mn, &c. are equal. As is demonstrated by Gregory San-Vincent; and as such is commonly admitted.
15. So that Fl, Lm, Mn, &c. may fitly represent equal Times, in which are dispatched unequal Lengths, represented by FL, LM, MN, &c.