VOL. XXIX.] PHILOSOPHICAL TRANSACTIONS. 187 



These angles being equal, their radii are proportional to their arcs; therefore 

 Md : cc :: Mc : ct. But cc =; dm (because of cm = cm) so that Md : dm (:: cd 

 : de) :: cm : ct. But cd = cm; therefore ct = de = tangent of the arc dm. 

 Q. So that supposing ATt a curve line, in which are all the centres of curva- 

 ture of the particles of acb, any point as t being found as before, the length 

 AT, by the nature of evolution of curves, is every where equal to the tangent 

 of its correspondent circular arc dm. The point t is also found by making mt 

 perpendicular to RM, and equal to the secant ce; for so is the angle cmt = 

 MCD, and the triangle mct equal to the triangle cde. 



10. Let AHh be an equilateral hyperbola, whose semi-axis is ar and centre r. 

 In the meridian let rp be equal to the tangent de. Join ap, and draw ph = ap 

 and parallel to ar. Complete the parallelogram hnrp ; so will the point h be 

 in the hyperbola, and its ordinate hn (= rp = de = ct) be equal to the curve 

 ATt. From whence, and from prop. 3, corol. 2, of Dr. Gregory's Catenaria, 

 Phil. Trans. N° 23 1 , it appears that the curve ATt is that called the catanaria 

 or funicularia, viz. the curve into whose figure a slack cord or chain naturally 

 disposes itself by the gravity of its particles. 



" J I . Hence we have another property of the catenaria not hitherto taken 

 notice of, that I know of, viz. that supposing ar (= «, the constant line in 

 Dr. Gregory) equal to the radius of the nautical projection, and rn the secant 

 of a given latitude, then is nt, the catenaria's ordinate at n, equal to rm the 

 meridional part answering to the latitude whose secant is rn." 



12. That TA is the catenaria, is also demonstrable from Dr. Gregory's first 

 prop. Let Tu be the fluxion of the ordinate nt; and tu (:= Nn) the fluxion 

 of the axe an. Then because of like triangles tcm, Tut, cm : ct (= ta) :; tu 

 : ut, that is, as cm a constant line to ta the curve :: so is the fluxion of the 

 ordinate, to that of the axe (i/ : i) according to prop. 1, Catenaria. 



13. From the premises, the construction and several properties of the cate* 

 naria are easily deducible, one or two of which I will set down. 



1. The area atmr is equal to aopr, a rectangle contained by radius ar and 

 RP the tangent answering to secant hp = tm. For because of the like triangles 

 CMm, CEc; CM : CE :: Mm : EC, that is, (putting r, s, t, m, for radius, secant, 

 tangent, and meridional part rm) r : s :: m : i, whence rt = sm^ and all the rt 

 = all the srhj that is, aopr = atmr, which agrees with Dr. Gregory's cor. 5, 

 of prop. 7- 



14. Supposing the former construction, let be added the line rh, including 

 the hyperbolic sector arh. I say the same sector is equal to half the rectangle 

 armq contained by radius ar and the meridional part rm, (= ^rm). For the 

 sector ARH = triangle rnh wanting the semisegment anh. The fluxion of the 



B B 2 



