188 PHILOSOPHICAL TRANSACTIONS. [anNO 17 J 5. 



triangle rnh is4^*/+-Lij. Thefluxionof anhIs^^. So the fluxion of the sector arh 

 is ist-\'^ts— ts = l^st—^ts. It is found before, sect. 13, that r:s{s:- :: m : t; 



whence st= -rti. And because of the like triangles cde, Efe, cd : de :: sf : fe. 

 But Ef := Mm = 7W, because both Ef and Mm are to Md in the same ratio viz. 

 as s to r ; therefore r •.t{t'.-) y.m'.s: whence ts =i-m, and ^ ~ ^ = ^* ~ *^ 



2r 



— m=. ±rm = the fluxion of the hyperbolic sector arh, whose flowing quan - 

 tity is therefore equal to \rm = -I-armgi. a. e. d. 



15. This shows another property of the Catenaria, viz. that it squares the 

 hyperbola; for rm is equal to nt, the ordinate of the Catenaria. 



16. In fig. 10, let AR be radius, acb the equitangential curve, mrn its asymp- 

 tote, in which let m, n, be any two points equally distant from r. On m draw 

 ML parallel to ar, and equal to the diflPerence of the secant and tangent of that 

 latitude whose meridional part is rm (by sect. 3, 4). On n draw no parallel to 

 ar, and equal to the sum of the aforesaid secant and tangent. Do thus from 

 as many points in the asymptote as is convenient. And a curve drawn equably 

 through the points l. . .a. . .0, &c. will be a logarithmic curve, whose subtan- 

 gent, being constant, is equal to the radius ar. 



17. Let no be an ordinate infinitely near and parallel to no, op = Nn the 

 fluxion of the asymptote, ot the tangent, and tn the subtangent to the loga- 

 rithmic curve in o. Then op :po :: on : nt. But on = * -}- /; therefore op = 

 s -\- t, po = m, the fluxion of the meridian or asymptote. So the analogy is 

 s -{- t : m :: s -{■ t: nt. By sect. 13, 14, s : m :: t : r, also, i : m :: s : r, and 

 thence s -[■ i:m::t -\- s :r; therefore is nt (the subtangent to lao) equal to 

 radius ar a constant line, and consequently the curve lao is the logarithmic 

 curve, and its subtangent known. 



18. The same demonstration serves for lm, any ordinate on the other side of 

 AR, only changing the sine -|- into — ; and then it agrees with Mr. James 

 Gregory's prop. 3, p. 17, of his Exercitations, viz. " That the nautical meri- 

 dian is a scale of logarithms of the differences whereby the secants of latitude 

 exceed their respective tangents, radius being unity." So here rm is the loga- 

 rithm of ML, the difference of the secant and tangent of the latitude whose 

 meridional part is rm. 



19. Supposing the preceding construction, if through any point c of the 

 curve ACB be drawn a right line gcw, parallel to mr, terminated by the loga- 

 rithmic curve in w, and the radius ar in g ; then the same right line wo is equal 

 to the intercepted part of the curve line AC. 



