VOL. v.] PHILOSOPHICAL TRANSACTIONS. 426 



In the triangle S FT, are given, SF the distance of the foci; FT = FE 

 4- ES = AP; and AFT, the external angle or mean anomaly, equal to the 

 sum of the angles FST and T: Therefore the angle FSE, or the true anoma- 

 ly, = the difference of the angles FST andT, may be found thus: viz. As half 

 the sum of the sides FT and FS is to half their difference, so the tangent of 

 half the sum of the angles FST and T, to the tangent of half their difference. 

 But half the sum of the sides FT and FS is found, by substituting for FT its 

 equal AP, whose half is AC, which added to CS, the half of FS, makes the 

 half sum AS, the greatest distance of the planet. If then we subtract from 

 AS the lesser side FS, there remains the half difference of the sides FA = PS, 

 the least distance of the plant. Then the rule for finding the true anomaly 

 from the mean will be; As AS the greatest distance of the planet, is to PS 

 its least distance; so is the tangent of half the mean anomaly, to the tangent of 

 half the true anomaly. 



Corol. 1. If SE be produced to V, so as that EV be = EF, and the whole 

 SV = AP the axis; then the angle V of the triangle FSV will be half of the 

 prosthaphaeresis FES, consequently = to the half difference of the angles of 

 the true and mean anomaly, that is, of the angles AFE and ASE; and the 

 external angle AFV is = their half sum, after subtracting the half difference 

 VFE from the greater AF E; and hence arise these two analogies : 



1. As the sine of half the sum of the mean and true anomalies, AFV, is to 

 the sine of half their difference V; so is SV = AP the transverse axis to SF, 

 the distance of the foci. 



2. As the sine of half the sum of the mean and true anomalies AFV, is to 

 the sine of the true anomaly, FSV ; so is SV, or the axis AP, to FV, the sub- 

 tense of the true anomaly ; and so likewise is the semiaxis AC, to the semi- 

 subtense VX or FX. 



Corol 2. If, in the same triangle FSV, be erected on half the subtense FV, 

 the perpendicular XE; it will cut SV into two parts, of which the one VE is = 

 FE the line of the mean motion, and the other SE is the line of the true 

 motion. 



cdh is the angle of half the true 



II. In fig. 4, let a be the centre of the 

 concentric chfi. 



cad the diam. and line of the apses. 



ch the arc of true anomaly, to which 

 answers 



di the arc of mean anomaly ; there- 

 fore 



anomaly ; and 



dci the angle of half the mean ano- 

 maly. 



ci and dh are right lines, which inter- 

 sect in g. 



gb is perp. to cd. — Then, 



VOL. I. 3 H 



