VOL. XXVIII.] PHILOSOPHICAL TRANSACTIONS. * 3 



tion is from the aphelion towards the periheHon. But when the planet tends 

 from the perihelion towards the aphelion, as in fig. 4, the area Bsa = sector 

 BCQ — triangle csq, and therefore it will be proportional to the arch Ba — 

 right line sf. Hence if there be taken the arch an, in fig. 2, 3, and bn in 

 fig. 4, proportional to the times, it will be aq + sp = an, and Ba — sp = 

 BN. Whence sp will be equal to qn, if an or bn are proportional to the times 

 in which the areas asq or bsq are described. Now that the measure of the arcs 

 in the periphery aqb, which arc is equal to the right line sf, may be found in 

 degrees and parts of a degree : let it be made, as cq to cs, so is the arch of 

 57.29578 degrees, (which is equal to the radius ca) to a fourth arc, which will 

 be equal to cs. Let that arc be b. But it is cs to sp, so is radius to the sine 

 of the angle scp or acq. Therefore let it be made, as radius to the sine of the 

 angle acq, or the arc aq, so is the arc b to another d; that arc d will be equal 

 to the right line sf ; therefore if, at a given time, the area asq were propor- 

 tional to the time, the arc d would be equal to nq ; and taking the arc np = d, 

 the point p would fall on q. But if the area asq should not exactly answer to 

 the time, the point p will fall above or below q, according as the area asq is 

 greater or less than the true area which answers to the time. Let it be as^, 

 and on cq let fall the perpendicular sh ; then, by what has been demonstrated, 

 it will be SH = n^. But it is sf = np, whence it will be sh — sp or sp — sh, 

 that is nearly he = 9P = qp — aq or q^ — qp. And if the angle qc^' be 

 small, it will be ch : cq :: he : q^ :: qp — q^ : q^; whence cq -j- ch : cq :: qp : 

 Q^, when the arc aq is less than a quadrant. But when it is greater than a 

 quadrant, it will be cq — ch : CQ :: qp : (kq. And in like manner, when the 

 arc BQ is less than a quadrant, it will be cq — ch : cq :: qp : Q^'. 



When the angle acq or bcq is small, that is, when the planet is near the 

 apses, it will be ca + cs : ca :: qp : <Aq. 



Make as cs to cq so radius n to a certain length l, then will cq = . 



But radius is to the cosine of the angle acq, as sc : cf or ch (for ch and cp 

 are nearly equal) ; 



, r sc X CM.ACQ J . 1 c CS X L + CS XCOS : ACQ CS X L 



therefore ch = , and therefore qp : qo :: : 



:: L + cos.kcql : l, when the arc aq is less than a quadrant. But if aq be 

 greater than a quadrant, then will qp : a^ :: l — co^. acq : l. 



And in this manner, if the arc aq be any how taken, which is somewhat less 

 or greater than the truth, there will thence be found an arc aq, to be added to 

 it, or taken from it, which will make the area as^ very nearly proportional to 

 the time. And if, instead of aq, there be taken an arc xq, and a process like 



B 2 



