S53- PHILOSOPHICAL TRANSACTIONS. [aNNO 1687. 



minished ; or the line to be described in such time, by a celerity answerable to 



such undiminished force: also let B S, a like ordinate, be — of it. In yi B, 



which is put = 1, let Bdhe such a part of it, as B S is o( D H. Now because 

 all the inscribed parallelograms, in the exterior hyperbola; AS, AH, &c. are 

 equal; and therefore their sides reciprocally proportional ; consequently as 



Ad = 1 isto^fissl, orasm— lto?w, so is 65= — DH, to dh. 



which is therefore equal to r of D H\ that is, by division, 1 + 



^ m — \ 'J ' m ' mm ' 



— , &c. of DH. Or, if B d he taken beyond B; then as ^ d = 1 H — , is 

 to A B ss I, or as w + 1 to m, so is — D H to d h, which is therefore 



kM'^'—-^ DH; that is, by division, = — + —, — &cc. o( D H. 



' m -\- I ' m mm m' 



10. Let such ordinate dh, or its equal in the asymptote A F, fig. 3, be so 

 divided in L, M, N, &c. by perpendiculars cutting the hyperbola, in /, m, n, &c. 



as that FL, LM, MN, be as — ,— ^, — r, &c. ; that is, so continually 

 •' ' ' m' mm ' m^ ' / 



, 1 '' 



decreasing, as that each antecedent be to tts consequent, as 1 to — ; or as m 



to 1. This is done by taking A F, A L, AM, &c. in such proportion. For 

 the differences of continual proportionals are also continually proportional, and 

 in the same proportion. 



11. This being done; the hyperbolic spaces Fl, Lm, Mn, &c. are equal. 

 So that Fl, Lm, Mn, &c. may fitly represent equal times, in which are dis- 

 patched unequal lengths, represented by F L, LM, M N, &c. — 12. And be- 

 cause they are infinite in number, 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. — 13. The spaces 

 Fl, Fm, Fn, &c. are therefore as logarithms, in arithemetical progression in- 

 creasing, answering to the lines ^F, A L, AM, &c.; or to FX, LM, M N, 

 &c. in geometrical progression decreasing. 



14, Because F L, LM, MN, &c. are as — , -i-, A, &c. infinitely 



' ' ' m^ mm' m> ■' 



terminated at A; therefore their aggregate FA or dh, is to D H, as 1 to 

 TO — 1 = n.— 15. If therefore we take, as 1 to «, so A F to D H ; this 

 will represent the length to be dispatched, in the same time, by such undimi- 

 nished force. — 16. And if such DH he supposed to be divided into innumer- 

 able equal parts, and therefore infinitely small, these answer to those (as many) 



