178 • PHILOSOPHICAL TRANSACTIONS. [aNNO 1738. 



upper focus, for the orbit of a planet; or that of a motion in a parabola for 

 the perihelion part of the orbit of a comet, or some other such) it would be 

 impossible to proceed one step in it. But as no general rule has ever yet been 

 laid down, to assist this method, so as to make it always operate, it is the 

 same in effect as if there were no method at all. And accordingly in experi- 

 ence it is found, that there is no rule now subsisting but what is absolutely 

 useless in the elliptic orbits of comets ; for in such cases there is no other way 

 to proceed, but that which was used by Kepler : to compute a table for some 

 part of the orbit, and therein examine if the time to which the place is required, 

 will fall out anywhere in that part. So that, on the whole, it appears evident, 

 that this problem, contrary to the received opinion, has never yet been ad- 

 vanced one step towards its true solution : a consideration which will furnish a 

 sufficient plea for meddling with a subject so frequently handled ; especially if 

 what is offered shall at the same time appear, as he trusts it will, to contribute 

 towards supplying the main defect. 



Lemma I.— The Tangent of an jirch being given, to find the Tangent of its 

 Multiple. — Let r be the radius of the circle, t the tangent of a given arch a, 

 and n a given number. And let t be the tangent of the multiple arch nX a, 

 to be found. 



Then if f j be put for — rr, and tt for — tt ; 



~~, — iB In 



The tangent t will be ' \p : 



r+r\ +r—r\ 



Which binomials being raised according to Sir Isaac Newton's rule, the ficti- 

 tious quantities t and j will disappear, and the tangent t will become equal to 



», n— t. «— 2. ^' , B. n— 1. «— 2. n— 3. n— 4. ^5 o 



" 1 ~2~ 3 j^"^! ~2 3 4~ 5 r*~ 



». »— 1 . ^ _i » . n^l.n— 2. n— 3. *■* » - 



r 2 rr ' 1 2 3~ ~4~ H 



This theorem, which he formerly found for the quadrature of the circle, at 

 a time when it was not known here to have been invented before, has now been 

 common for many years ; for which reason it is premised at present, without 

 any proof; only for the sake of some uses that have not yet been made of it. 



Carol. 1. From this theorem for the tangent, the sine, suppose y, and 

 cosine z of the multiple arch n X a, may be readily found. 



For if y be the sine, and z the cosine of the given arch a, then putting w 



TT 



for — WW, and substituting -^ for Y, and — for t, and - , , - for y: 



The sine y will be !± F I -'-^I f. 

 2r«_ 



mu • -11 u 2 + '^l" + 2 - "I" 



The cosme z will be — ' 



2r»-' 



