VOL. LIII.] PHILOSOPHICAL TRANSACTIONS. 75 



equal also to the rectangle under oaK ; whence as go to Aa so rq to qk, and ag 

 to AQ as KR to aK. But, as above, bg being to bk as ag to i, and bn taken equal to 

 I, BG will be to BK, as ag to bn, and ab to kn also as ag to bn or i. There- 

 fore if NS be taken to ab as i to Aa, by equality ns will be to nk as aq tP aq, 

 that is, as kr to qk ; and in the last place ns to ks as kr to qr, that-iS, the rect- 

 angle under skr equal to the given rectangle ns, qr, whence the point k, the 

 position of kd, and thence the point d will be given. 



But if DK be not ordinately applied to lm, let do be ordinately applied to lm. 

 Then here the rectangle under aqr, equal to the squareof qm, will be equal to that 

 under oqg, and gq to aq as qr to oq, fig. 28 ; whence by composition ag to aq 

 as OR to OQ. But BN being now also taken equal to i, and ns to ab as i to aq, 

 AB will be here in like manner to kn as ag to i, and ns to kn as ag to aq ; 

 therefore ns will be to kn as or to oq, and by conversion ns to ks as or to qr. 

 But NS and qr being both given in magnitude, if sp be taken to ns as qr to pr, 

 the point p will be given, and also by equality sp will be to ks as or to pr : 

 whence if rv be drawii parallel to no, and st to kd, both rv and sx will be given 

 in position, one passing through the given point r, parallel to the ordinates applied 

 to the axis lm, and the other through the point s also given, and parallel to kd or 

 CB : also DTV being drawn parallel to ml, dt will be equal to Ks,and dv equal to 

 OR ; therefore as sp to dt so dv to pr, and the rectangle under spr equal to that 

 under tdv, consequently the point d in an hyperbola passing through p, and 

 having for asymptotes the lines st, rv, given in position. 



In the last place, when the line lm drawn through the sun in a, and the pro- 

 jected place of the planet in b, is neither the axis of the earth's orbit, nor bi- 

 sected in a, fig. 29, the tangents to the points l, m, being drawn to meet in p, let 

 LM be bisected in q, and the point r taken, so that the rectangle under aqr be 

 equal to the square of qm, which by pdo being drawn, the rectangle under aqr 

 shall be equal to that under oqg, and qg to aq as qr to qo, or by composition 

 AG to aq as OR to QO. Therefore if nb be here also taken equal to i, and ns to 

 AB as I to AQ, ab being, as before, to nk as ag to i : by equality ns will be to 

 NK as AG to AQ, that is, as or to qo. Whence by conversion ns will be to ks as 

 OR to QR ; and if pt be drawn parallel to cb, and sv be here taken to ns as qr 

 to TR, by equality sv will be to ks as or to tr, and also by conversion sv to kv as 

 OR to ot. Also sv will be given in magnitude, and the point v given ; therefore 

 vw drawn parallel to cb, or kd, will here be given in position. But wdxy being 

 also drawn parallel to rv, sv will be to kv, or dw, as yd to xd , and yz being 

 taken equal to the given line sv, yz will be to dw as zd to xw, equal to tv, and 

 the given rectangle under yz, tv equal to that under wdz. Therefore rz being 

 drawn parallel to rp, rF, and its equal yz, being given, the line rz is given in 

 position, and the point d in an hyperbola having for asymptotes vvv, rz, and pass- 

 ing through p. 



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