( ) 
Ordinate, to that of the Axe ( ) : x ) according to Prof . 
i, CatemrU. 
13. From the Premifes the Conftrudion and feveral 
Properties of the Catenaria are eafily deducible ; one or 
two of which I’ll fet down- 
1 , The Area AT M R is equal to A OP R a Redangle 
contained by Radius A R and R P the Tangent anfwe- 
ring to Secant HP —T M. For becaufe of the like Tri^ 
angles C Mm, C Ee\ C M : C E :: Mm : E e, that is, 
putting r, s , t 9 m for Radius* Secant, Tangent and Me- 
ridional part RM.) r : s :: m : t whence r sm, and 
all the r t — all the s in, that is A O P R =. A X MR* 
which agrees with Dr. Gregory’s Cor. $. of Prof. 7. 
14. Suppofing the former Conftrudion, let be added 
the Line R H, including the Hyperbolic Seffor ARH, 
I fay the fame Sedor. is equal to half the Redangle , 
A R MQc ontained by Radius A R and theMeridional Part 
RM, (— rr m), For the Sedor A R Triangle/? NH 
wanting the Semifegment AN H. The Fluxion of the 
Triangle RNH is — The Fluxion of AN His 
■ ' 2 
t s . 
So the Fluxion of the Sedor A RH is 
its 
’Tis found before (Sett, 13.) that 
J 1 , # 1/ ^ • 
r : s (s -) m : whence s t=: — m» And becaufe 
r r / 1 -■ 
of the like Triangles C D E, Efe,C D : D £ : E f : 
fe. But Ef— Mm— in, becaufe both Ef and Mm are 
to Md in the fame Reafon, m, as s to r; therefore r : t 
, r K - , t* - 7 » t- 
\.t : s : whence t s — — m, and — 
r r 
t s 
3c 
SS 
X 
