( m ) 
9. So that fuppofing AT t a Curve Line in which arc 
all the Centers of Curvature of the Particles of A CB, 
any point as T being found as before, the Length A T 
(by the nature of Evolution of Curves, ) is every where 
equal to the Tangent of its correfpondent Circular Arc DM. 
The Point 7 ~is alfo found by making M T perpendicular 
to R M, and equal ro the Secant C E : for fo is the An- 
gle C MT— MC D i and the Triangle McT equal to 
the Triangle CDE. 
10. Let A El h bean Equilater Hyperbola whofc Semi- 
axis is A fland Center R. In the Vieridian let R P be e- 
qual rothe Tangent D R. Join A P, and draw P H— AP 
and parallel to A R. Compleat the Parallelogram H NR P, 
fo will the Point// be in the Hyperbola, and its ordi- 
nate H N(=R P — DE — C T) be equal te the Curve 
AT t. From whence, and from Prop 3 Cor oil 2. of 
Dr. Gregory's Catenaria(PhiU Tranj. N. 23 it) it appears that 
the Curve A Tt is that call’d the Catenaria or Funicularia , 
viz. the Curve into whofe Figure a flack Cord or Chain na-- 
turally difpofes its felf by the Gravity of its Particles. 
“ 1 1. Hence we have another Property of tire Catenaria 
“ not hitherto taken notice of (that I know of,) viz,, that 
“ fuppofing A R(— a, the conftant Line in Dr. Gregory) 
" eqaal to rhe Radius of the Nautical Proje&ion, and 
** RN the Secant of a given Latitude, then is NT the 
“ Cat en arias Ordinate at N, equal to R M the Meridio- 
“ nal Parc anfwering to the Latitude whofe Secant is 
“ RN 
1 1. That r^is the Catenaria is alio demonftrable from 
Dr. Gregorys firft Prop Let Tu be the the Fluxion of 
the Ordinate NT ; and tu (= N n) the Fluxion of the 
Axe A N. Then becaufe of like Triangles TQM, Tut, 
C M : CT (— T A) :: Tu : u t, that is } as C M a con- 
ftant Line to T A the Curve : : fo is the Fluxion of the 
G gg z Ordi- 
