1^73 J 
proportion. For let J^B^D, 8^c. be fuch proportionals; 
and their differences ay h^i^^ kc. That is J — B B — 
Then, becaufe A, B, C, D, &c. are in continual proport: 
That is A. B :: B. C :: C. D :: &:c. 
And dividing A-B.B:: B~C. C:: C -D. D ::&Co 
That is a.B::kC::lD:i^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 i. 
14. Tliis being done; tlie Hyperbolick fj^aces Fl^Lm^ 
Mn^ &c. are equal. As is demonftrated hy Gregory San* 
Vincenp\ and as fuch is commonly admitted. 
15. So thatf'/, Lm^ Huy &c.may fitly reprefent e- 
qual time§, in which are difpatched unequal lengths, re- 
prefented by FL, LA/, MN^&cc. 
16. And becauie they are in number infinite ( though 
equal to a finite Magnitude j the duration is infinite / And 
conlequenlty the impreffed force, and motion thence ari*- 
fing, never to be wholly extinguiflied ( without fome fur- 
ther impediment ) but perpetually approaching to Ay in 
the nature of Afymptotes. 
17. The fpaces jF/-F;», F^, Sec. are therefore as Lo« 
garithms f in Arithmetical progrefTion increafing j an- 
1 wering to the lines J FyJLj JMj &c. ^ or to P L, L M, 
MAV&c. in Geometrical progreffion decreafing, 
i i J. 
1 8. Becaufe FL^LM, MN^ &c. are as mm^ m\hc 
{ infinitely ) terminated at A ; therefore by ^ 10 j their 
Aggregate FA or dhj h to DH, ( much length as 
would have been difpatched, in the fame time, by fuch 
impreffed force undiminiflied j as i to^^ — 1 = 
19. If therefore we take, as i to {bAFtoDH'y this 
will reprefent the length to be difpatched,in the fame time, 
by fuch undiminifhed force. 
20. And if fuch DHbefuppofed to be divided into e- 
qual parts innumerable ( and therefore infinitely frnall ; J 
thefe anfwer to thofe ( as many ) parts unequal in FAy 
ov h d. 2 1 
