VOL. II.J PHILOSOPHICAL TRANSACTIONS. 273 



With this book, my Lord, which is just come out, I was so much pleased, 

 that I could not quit it before I read it quite through. The doctrine on which 

 it is founded, and by which the logarithms may be expeditiously and ingeni- 

 ously constructed, is perspicuous, and ingeniously treated. The quadrature of 

 the hyperbola, subjoined to it, is very elegant and ingenious ; and is to this 

 effect. , 



After the author had demonstrated, prop. 14, that in the hyperbola MB F, 

 (fig. 8, pi. 7.) having its asymptotes, AH, AN, meeting at right-angles^ and 

 drawing B I, F H, s p, &c. parallel to the asymptote A N, then the rectangles 

 A I B, A H F, A p s, &c. are all equal among themselves, and that therefore 

 their sides are reciprocally proportional, being the known property of the hy- 

 perbola : putting then A I = B I = 1, and H I =a; he shows, prop. 15, that 

 F H = , namely from this analogy H A : A I : *. B I : F H, that is 1 -f a : 



1 : : 1 : =FH=1 — a + a^ — G^-f-a* &c. by dividing the numerator 



1 by the denominator 1 + «> continued by the powers of a, alternately nega- 

 tive and affirmative. And since this holds equally true for every point H be- 

 yond I, putting A I, as before, = 1, and making any continuation of it, as Ir, 

 = A, which is conceived as divided into innumerable equal parts, each of 

 which, as I p, p q, &c. is called a ; therefore I p, I q, &c. will be a, 2 a, 3 a, 

 &c. till the last term be A. Then the right lines p s, q t, &c. corresponding 

 to these, comprehending the space B I r u, are, 



1 — a + c^ — c^ -\- a^ &c. Since then it is, 



1+1 + 1 &c. (to the last) = A 



a + 2 a -i- 3 a &c. (till A) = -lA^ 



a" -\- Aa' + g a" &c. (till A') =z i-A* 



a3 _^ 8 a' + 27 ef &c. (till A') =^A* 



and so on, as he Fhows in prop. 16, and which I have elsewhere demonstrated. 

 Hence he properly infers, that the hyperbolic space BIruis = A ^ 4.A^ 

 + ^A^ — ^A* -\- ^A^ &c. So that, assigning to A=lr, any value in 

 numbers, and distributing the series into two classes, viz. the affirmative 

 powers A, -^A^ ^A\ &c. and the negative powers ^A\^A\ &c. the ag- 

 gregate of the latter being deducted from that of the former, the remainder 

 will be the value of the hyperbolic space B I r u. 



Then putting ^=0.1, or = 0.21, 01; any other decimal fraction, and con- 

 sequently less than 1, that is, making I r less than A or 1, the last powers of 

 A become so small, that they may be neglected. For example, putting A I = 1 

 and I r = 0.21, then the terms will be as follow : 



VOL. I. Mm 



1 — 2 a + 4 a'^ — 8a^ -j- 16 a'' &c. 

 1 — 3g!+ Qa^— 27a^+ Sla'^&c. 



and so on to 

 I - A -\- A' - A^ + A* &c. 



