Or if _Bd_ be taken beyond _B_; then as _Ad_ = 1 + 1/_m_ to _AB_ = 1, or as _m_ + 1 to _m_, so is (1/_m_)_DH_ to _dh_, which is therefore equal to 1/(_m_ + 1)DH; that is (as will appear by like dividing of 1 by _m_ + 1;) = to 1/_m_ - 1/_mm_ + 1/_m_³ - _&c._ of _DH_. 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/_m_³, _&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._ 16. And because they are in Number infinite (though equal to a finite Magnitude) the Duration is infinite: And consequently the impressed Force, and Motion thence arising, never to be wholly extinguished (without some further Impediment) but perpetually approaching to _A_, in the Nature of Asymptotes. 17. The Spaces _Fl_, _Fm_, _Fn_, &c. are therefore as Logarithms (in Arithmetical Progression increasing) answering to the Lines _AF_, _AL_, _AM_, &c. or to _FL_, _LM_, _MN_, &c. in Geometrical Progression decreasing. 18. Because _FL_, _LM_, _MN_, &c. are as 1/_m_, 1/_mm_, 1/_m_³, &c. (infinitely) terminated at _A_; therefore (by ¶ 10) their Aggregate _FA_ or _dh_, is to _DH_, (so much Length as would have been dispatched, in the same time, by such impressed Force undiminished) as 1 to _m_ - 1 = _n_. 19. If therefore we take, as 1 to _n_, so _AF_ to _DH_; this will represent the Length to be dispatched, in the same time, by such undiminished Force. 20. And if such _DH_ be supposed to be divided into equal Parts innumerable (and therefore infinitely small;) these answer to those (as many) Parts unequal in _FA_, or _hd_. 21. But, what is the Proportion of _r_ to 1, or (which depends on it) of 1 - _r_ to 1, or 1 to _m_; remains to be inquired by Experiment?