C i]r ) 
the Fluxion of the Hy* 
/ s — 1 1 _ r r . _ . . 
• m— — m — 7 r nh 
2 r zr 
perbolic Sector A R H yt whofe flowingQuantiry is there-** 
fore equal to - r m = ~ A R Jg.E.D. 
iy, This Ihews another Property of the Catenaria, viz. 
that it fquares the Hyperbola ; for R M is equal to NT 
the Ordinate of theCatenaria. 
1 6. Jn Fig. 4. Let A R be Radius, A C B the Equitangen- 
dal Curve ; M R N its Afymptote, in which let M , N, be 
any two Points equally diftant from R* Upon M draw< 
M L. parallel to A R and equal to the Difference of the 
Secant and Tangent of that Latitude whofe Meridional 
Part is R:M iby § 4,) Upon N' draw N O parallel to 
AR , and equal to the$»ff«*:of the forefaid Secant and 
Tangent. Do thus from as many Points in the Afymptote 
as is convenient, and a Curve drawn equably through the 
Points L - - ~ A - &c. will be a logarithmic Curve, 
whofe Suhtangent (being conftantj is equal, to Radius 
AR. '1 / uB v'l] ^ f ; 'r./.’ij'j t ■ * io 'ih | • ; 
17. Let no be an Ordinate infinitely near and patal— 
lal to NO. Opcz Nn the Fluxion of the Afymptote ; 
07 ”. the Tangent, and T N the Subtangent , to the Loga« 
rich.. Curve in 0 . Then- op.: pO ON:' NT. But. 
ON— s Ar't ; ; therefore op — l-f- f. pO — m-. (the 
Fluxion of the Meridian or Afymptote.) So- the Analo-. 
gy is s H“ t : m 1: s -j~ t : NT By Seff. 1 3 _■ 14, s : m< 
: : t : r. alfo, ,t : m s '■ r ; and thence s 
t \ s : r. wherefore is NT fthe Subtangent to L AO) e? 
qual to Radius A R a conftant Line, and consequently 
the Curve L AO is the Logarithmic Curve, and its Sub-i 
tangent known. 
iS. The fame Demonftration ferves for , L M, (any 
Ordinate on the other Side of A R) only chaining the 
Sine -f- into — 3 and then it agrees with Mr. James < 
Gregory’s Prop. l>pag. 17. of his Exercitations, viz.. That 
G §g„?. ik 
