1

to A B = r,

s 575 )

Or is B d be. taken beyond B ; then as A d = i -i-

m

or as m *+ i to m, (b is- D H, to d h, which is thcresore Equal to

m

--—- D H ; that is, (as will appear by Uke Dividing os i by m -+■ i

m -*• i

l

I 1 I

= to -->-:-, dc. os D H.

m mm m*

io. Let such Ordinate d or (Equal to it ia the Asmptote) A F, be & vig.iji.

Divided in L, M, N, dc. (by Perpendicuhrs cutting the Hyperbata in /, m, n,

dc.) as that FL, LM, MN, beas,—7, —-, dc. .That is,. Io

" M' m ni m 5

contirtually Decreasing, as that each Antecedent be to its Conlequant, as 1 to

I:

--, ..oras m to 1.

m

n. Xhis is done by taking AF, A L, ANI, dc. in stich Proportion.

For, os Continual Proportionais the Differences are also ContinuaJly Pro-portional, and in the tarne Proportjon. For let A, B, C,' D, dc. be sichProportionais ; and their Difterences, /?, b, c, dc. That is, A — B = a,

B — C — b, C — D c, dc.

Then bccaule, A, B, C, D, dc. are in Continua! Proportion ;

That is, A : B : : B: C :: C : D : : , dc.-

And Dividing A — B:B::B — C : C ; : C — D : D : dc.

That is, * : B : : b : C : : d : D : :, dc.

And Alternly, a. b. c. dc. : : B. C. D.* dc, : : A. B. C . .dc..

That is, In Continual Proportion, as A to B, or as mto 1.

iz. This being done ; the !hpcrbo!ick_ Spdccs F /, L m, M n, dc. are"

Equal, as is Demonstrated by Grcgory Sait-Vinccn! j and as sicli is commonlyadmitted.

iz. So that F/, L M «, £>4. rruy iqtly .represnt Equal Times, .in ■

which are dispatehed Unequal . I.engths, reprefented by F E, •> L M, ; ;

M N, dc. ... '

14. And because they are in Numbcr Infinite; (though Eqiiml to a Fi-nite Magnitude) the Duration is Infinite : and conlequently the Imprdfed.

Force, and Motion thence Arising, never to be .wholly Extinguiflicd (with-out forne further Impedimenti but perpetually Approachiijg to A, in tfic -Nature os Ajymptstcs. ...

x y. The Spaces F /, F m, F», dc."' are therefore di Loga r >tl>ns (in.Arithnictic/tl Progression Increafing) arilWcring to the Lines, A F) A L,

A M, dc. or to F L, L NI, M N, dc. in Gcomcirical Progression De-cre.ifing,

:6. B-