304 PHILOSOPHICAL TRANSACTIONS. [aNNO 1706. 



+ Ka : ^^4- D^ :: Aa : dc^. But i^h -\- Ka =. H -\- od, as shown above; there- 

 fore Aa = T)d. Aa is the fluxional particle of the curve fa, and Dd 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 = y, AD = t, BJ) = s. So is the curve pa = hd = 

 7/ -\- s', 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/a. 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 = 'Zfh X ga. 



Let FH = a, FG = ar, ga = y. Because of the hyperbola gl x gh (ls) = 



fh'* ; therefore gl = — ; and lm = gh (mg), that is, lm = a 



' GH GH V /' -> a — X 



-|- a: =: °^ "" '^ , and consequently the fluxion of the area ilm = — — — .r. 



In the rectangle triangle adb, ab = a — a:, bd = *, ad = i = a — j: then 

 is ad'^ = ab^ + bd* : or aa — 2as -{- ss = aa — 2ax -^ xx -\- ss, which being 



2flX — 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 is, a — a: : 5 (= — — — ) ::i:y, therefore ^r = - — — — i; multiply each 

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



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



areas beginning together at f 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. 



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

 the area of the quadratrix FabHF, is equal to half the square of fg, wanting the 



XX XXX 



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



