'6 PHILOSOPHICAL TRANSACTIONS. [aNNO 1763. 



Thus in this problem in all cases solved either by a right line, or an hyperbola 

 given in position, which shall intersect the projected orbit in the point sought. 

 For though in each case the projection of the planet has here been considered as 

 within the orbit of the earth, the form of argumentation will be altogether 

 similar, were the projection of the planet without. And this is agreeable to the 

 method pursued throughout this discourse, where I have always accommodated the 

 expression to one situation only of the terms given and sought in each article ; 

 the variation necessary for the other cases, when one has been duly explained, 

 being sufficiently obvious. 



In the 5th vol. of the Commentaries of the Royal Academy at Petersburg, 

 is given an algebraical computation for a general solution of this problem, in the 

 orbits of any two planets projected on the plane of the ecliptic ; but with this 

 oversight of applying to the projected orbits a proposition from Dr. Keil's Astro- 

 nomical Lectures, which relates to the real orbits.* 



However, from the geometrical solution now given, a calculation for assigning 

 the point d may be formed without difficulty, ldm being the orbit of the earth, 

 A is the focus, and hp perpendicular to the axis. Let this axis be ab meeting rp 

 in c, rz in d, pt in e and wv iny. Then the angle a\u is given, being the 

 distance between the heliocentric place of the planet in the ecliptic from the 

 earth's aphelion. Also pt being parallel to cb , the angle kTe, and conse- 

 quently the angle act, will in like manner be given, whence the points r, b, t, 

 V being given, as in the solution above, the points (/, c, e, andy will be given, the 

 triangles arc, ATe, being given in species, and similar respectively to the trian- 

 gles xYd, and Av/i Also the rectangle under wdz being equal to that under rF 

 vt, if DK be continued to the axis in g, and dA be drawn parallel to pr, the rect- 

 angle under y^, hd, is equal to that under fe, dc, and both being deducted from 

 the rectangle under fhd, the excess of the rectangle under /hd above that under 

 fe, dc, will be equal to that under ghd, so that this difference will be a mean pro- 

 portional between the square of /id and the square oUig, which is in a given ratio 

 to the square of hi), and therefore is a given ratio to the rectangle under ahb, dA 

 being ordinately ap])lied to the axis ab. 



Thus a biquadratic equation may be formed, by which the point k shall be found, 

 and thence the point D, whose distance from A is to he as the excentricity of the 

 earth's orbit to half its axis. Therefore I shall only observe further, that here 

 occurs an obvious question, what, in so extended a search for principles leading to 

 the solution of any problem, as the ancient analysis admits of, can conduct to 

 the most genuine on each several occasion. But for this end, where commodi- 

 ous principles do not offer themselves, the most general means is to consider first 



* The demonstration of Dr. Keil's proposition proceeds on the known property in the planets of 

 having their periodic tiroes in the sesquiplicate ratio of tlie axis of their orbits, which confines the pro- 

 portion to the real orbits j for in each planet the periodic time through the projected orbit, is the 

 same as through the real, though the axis in one be not equal to the axis of the other. — Orig. 



