74 PHiLOsbfHfdAt TRANSACTIONS. [anno 1763. 



be drawn, ad will be to ap as the sine of the angle apd to the sine of the angle 

 PDA ; this angle therefore is given, and the angles apd, pda being given, the an- 

 gle pad is given. 



Coroll. Here where the orbit of the earth is supposed a circle, the ratio of i 

 to AB, that is, of the rectangle under ab, i to the square of ab, will . be com- 

 ])ounded of the subduplicate ratio of am, the semidiameter of the earth's orbit, 

 to half the latus rectum to the greater axis of the planet's orbit, and of the ratio 

 of radius to the co-sine of the inclination of the planet's orbit to the plane of the 

 ecliptic ; and, adding on both sides the ratio of the square of ab to the square of 

 AM, the ratio of the rectangle under ab, i, to the square of am will be compounded 

 of the ratio of the square of ab to the rectangle under am and the mean propor- 

 tional between am and the half of this latus rectum of the planet's orbit, and of 

 the ratio of the radius to the co-sine of the inclination of the planet's orbit. 



In the next place, though the earth's orbit is not a circle concentric to the sun ; 

 yet if the projection of the planet falls on the line perpendicular to the axis of 

 the earth's orbit, the point a will still bisect lm. In this case draw, to the points 

 L and m, tangents to the ellipses meeting in p, whence through d draw pd meeting 

 the ellipses again in a, and intersecting lm in o. Here if a tangent be drawn to 

 the ellipses in a, it will meet the tangent at d on the line lm in the point g, fig. 

 26. Now LG will be to gm as lo to om, and the point a bisecting lm, the rect- 

 angle under gao will be equal to the square of am. But bg is to bk as ag to i. 

 Therefore bn being taken equal to i, ab will be to kn as ag to i, and the rect- 

 angle under ab, i equal to that under ag, kn; whence ao being to kn as the 

 rectangle under gao to that under ag and kn, ao will be to kn as the given 

 square of am to the rectangle under ab and i also given. 



Draw RP parallel to cb, and take PS to ap, also nt to ar, in this given ratio 

 inverted. Then will the points t and s be both given, also ao will be to kn, 

 and Ro to kt, as ar to nt, that is, as ap to ps. Therefore if tv be drawn par- 

 allel to cb, that is, to kd, and vs parallel to lm, these lines will be both given in 

 position ; and wdxy being also drawn parallel to lm, wd will be equal to kt, and 

 RO being to kt as ap to ps, dy will be to wd as xp to ps, and by composition 

 YW to WD as xs to PS, and the given rectangle under yw, or sv, and ps equal to 

 that under wd and xs. Whence sv being parallel to lm, the point d will be 

 in an hyperbola through p, and having for asymptotes the lines vs, vt, given in 

 position. 



But if the projection of the planet fall on the axis of the earth's orbit, or the 

 same continued, ab extended to the earth's orbit in l and m will be the axis of 

 that orbit. If also cb should be perpendicular to ab, kd would be ordinately ap- 

 plied to lm, fig. 27 ; and the point r being taken, so that a being the centre of the 

 orbit, the rectangle under aqr be equal to the square of om, the same will be 



