66 AVAl.OGY FOR DEDUCING THE 



Ferdinartdia, he mentions a fimple method of deducing the 

 greateft equation of a planet's centre from the execntricity, 

 which he discovered a few years ago ; and he invites your 

 mathematical correspondents to give a demonftration of this 

 method, refpeciing the accuracy of which he feems to have 

 little doubt. In confequence of Mr. Pearfon's invitation, I 

 take the liberty of addreiling you on the fubject. It will (ave 

 Dcmonftration much circumlocution to refer to a diagram : let, therefore, 

 accuracy?^ ^ the elli P fls A P G Q in the annexed figure ] . PI. VII. repre- 

 fent the orbit of a planet moving about the fun S in one of the 

 foci ; and let A N QE be a circle defcribed upon the major axis 

 of the orbit as a diameter. Then, if P be the place of the planet 

 in its orbit, A N will meafure the excentric anomaly ; alfo, if the 

 arc A D be taken proportional to the time from the aphelion, 

 it will reprefent the mean anomaly ; and, letting fall the per- 

 pendicular S T from S upon N C continued, it is (hewn, by 

 Ktill and many others, that AN + ST = AD. It has alfo 

 been ihewn that, when the equation of the centre is a maxi- 

 mum in any orbit, the diftance S P of the planet from the fun 



is = *J S A x S Q, and when this is the cafe, in all orbits of 

 fmall eccentricity the point P nearly coincides with G, or the 

 place of the planet will be near an extremity of its orbit's mi- 

 nor axis. In this fituation of S P, it is manifeft that ST will 

 be almoft coincident with S C, and nearly equal to it ; S D 

 and C N will be nearly parallel ; and D C N -f- S N C nearly 

 equal to 2 D C N ; that is, the greateft equation will be nearly 

 equal to 2ND, or 2 S T, or 2SC. But, when GS the 

 mean diftance reprefents the radius of a circle, as the angle 

 G C S is 90°, S C will manifeftly reprefent the fine of an an- 

 gle : therefore, as G S, the mean diftance, to S C, the eccen- 

 tricity, fo is 1, or radius, to the fine of \ the greateft equa- 

 tion, which is the fame as IVJr. Pearfon's analogy. 

 It is merely an It is obvious from this investigation, that the (imple method 

 approximation. p i n ted out by Mr. P. is merely an approximation. It fup- 

 pofes that the orbits are nearly circular, and will therefore 

 produce a refult deviating moil widely from the truth when 

 the eccentricity is the greaieft. Thus in Mr. P's table (p. 49, 

 No. V. N. S.) the greateft error is in the orbit of mercury, 

 where the eccentricity is '20551 in terms of its own mean 

 diftance. The next greateft error is in the orbit of Mars, 

 where the eccentricity is -09308. The next error in order is 



that 



