VOL. LIII.3 PHILOSOPHICAL TRANSACTIONS. Qj 



the ratio of mt to at is given, and thence the point t. But also the diameter 

 Rs is given in magnitude, the square of rt or of st being equal to the rectangle 

 under mta ; whence in the last place the transverse diameter conjugate to this is 

 also given ; for the square of this diameter is to the square of rt, as the given 

 square of bm to the rectangle under amt now also given, - 



But a more simple case may also be proposed in 3 lines, when the ratio of the 

 rectangle under edf should be equal to the rectangle under a given line, and 

 that drawn from D to bc in a given angle, %. J 5 . This line will bear, both to 

 BE and PC, a given ratio, and the rectangle under edf will be in a given ratio 

 to the rectangle under the given line and eb or cf. — Let the line given be h, 

 and take mb and nc, so that the rectangle under mbc, and that under bcn be to 

 that under ba and h, in the given ratio of the rectangle under edf to that under 

 be and h, bm and cn being equal. Then draw from m and n lines parallel to 

 BA, CA, which shall intersect ef in k and l, by which, mk cutting ca in i, the 

 rectangle under mbc will be to that under ba and h, as the rectangle under bmc 

 to that under mi and h, and also as the rectangle under ekf to that under ki 

 and H, that is, as the rectangle under edf to that under h and be or mk ; 

 whence, by adding the antecedents and consequents, the rectangle under kdl 

 will be to the rectangle under h and mi, in the same given ratio, which is also 

 that of the rectangle under bmc to the same rectangle under h and mi : the 

 ppint d therefore is in an hyperbola passing through b and c having for asymp- 

 totes the lines mk and nl given in position, the rectangle under kdl being equal 

 to that under bmc, or that under mbn. 



If the two lines ab and ac are parallel, the locus may be known to be a para- 

 bola by the last proposition of the 4 th book of Pappus. But if bc were parallel 

 to one of the other, the locus will be an hyperbola, as the preceding, but as- 

 signed by a shorter process. Suppose the given lines to be ae, af, fig. l6, and 

 bc parallel to af. And let the rectangle under edf be equal to that under do 

 and the given line h, the line eg making given angles with ae, af. Here take 

 Ei equal to h, and deduct from both the rectangles that under ei or h, and dp, 

 by which will be left the rectangle under idf equal to that under h and pg, both 

 whose sides are given. Draw therefore ik parallel to ae, and the rectangle 

 under idf will be equal to this given rectangle, the given lines ki, af being the 

 asymptotes to the hyperbola passing through d. 



Curoll. If LM be drawn through b parallel to ef, lb shall be equal to fg, and 

 bm equal to ei or h, by which the hyperbola opposite to that passing through d 

 will pass through b. 



Scholium. — The propositions of Pappus, which have been here referred to, 

 are given by him, among others, for Lemmas subservient to the lost treatise of 

 ApoUonius De sectione determinata ; and the four here cited respect and com- 



k2 



