304 PHILOSOPHICAL TRANSACTIONS. [aNNO 1706. 



+ KG : id-^- J)d :: Aa : nd. But Bb -{- Aa = Sd -\- dc?, as shown above; there- 

 fore Aa = ad. Aa is the fluxional particle of the curve fa, and j)d the fluxional 

 particle of the line hd ; these fluxions or augments, being equal, and their 

 flowing cfuantities beginning together, are themselves therefore equal, viz. 



PA ^ HD. 



Let FG = ^, GA = HB = 2/, AD = tf BB = s. So is the curve pa = hd = 

 y + SI that is, the curve from the vertex to any given point in it, is equal to 

 the sum of its ordinate and subtangent, to the same point ; which is its second 

 property. 



III. The next property, and from which I call it the hyperbolic quadratrix, 

 is this, in fig. Q, let fae be a part of the curve, &c. as before ; fikh a square 

 on the line ph ; ail an equilateral hyperbola, whose vertex is i, its asymp- 

 totes ho, hr, and its axis hi/x. From a given point l in the hyperbola, below 

 its vertex i, draw la parallel to the asymptote eh, intersecting the diagonal ih 

 in M, FH in g, and touching the quadratrix in a. I say, that the hyperbolic 

 area ilm, is equal to a rectangle, whose sides are the ordinate ga and twice fh, 

 the axis to the quadratrix, that is, the trilineal ilm = 2ph X ga. 



Let FH = a, PG = jCj ga = y. Because of the hyperbola gl X gh (ls) = 



WR^ ; therefore gl = — ; and lm = gh (mg), that is, lm = a 



' GH GH \ /' ■> a — X 



-f. a: =: *" "" — , and consequently the fluxion of the area ilm = — -^ — ,f. 



In the rectangle triangle adb, ab =: a — x,bd = s, at> = t = a — s: then 

 is ad" = ab" -}- BD* : or aa — 2as -\- ss = aa — 2ax -\- xx -\- ss, which being 



2ax — XX 



reduced, gives s = — . 



Let la be a right line supposed infinitely near and parallel to da, and inter- 

 secting AB in c. Because of the similar triangles Aca and abd, ab : bd :; ac : 



ca, that \s, a — x:s (= — — — ) ::x:y, therefore y = - — -^ i; multiply each 

 by 2a, then it is2a^ = — — — .f. The flowing quantity of 2a^ is 2ay; and the 



flowing quantity of — — — i is the hyperbolic area ilm, as above. These two 



areas beginning together at p and i, and having every where equal fluxions, or 

 augments, are therefore themselves every where equal. 



Note. — ^The quadrature of the trilinear figure ilm being thus found, any 

 other area bounded by the curve line il, and any other right lines, is also 



given. - .' wf- ,^;, --T >sk 



IV. Supposing the same things as in the precedent proposition, I say, that 

 the area of the quadratrix po^hf, is equal to half the square of fg, wanting the 



cube of F^ divided by 6fh, or fc^hf = ^ — ^, The fluxion of this area 



