274 



PHILOSOPHICAL TRANSACTIONS. 



[anno 1668. 



A — 0.21 

 ^^^ = 0.003087 

 ■^u4' =z 0.000081682 

 -\.A'' = 0.000002572 

 .1-^^ = 0.000000088 

 T-V^^^= 0.000000003 



4-^^ = 0.02205 

 4. ^* = 0.000486202 

 ^A^ = 0.000014294 

 ^A^ = 0.000000472 

 ^'-A^°z=i 0.000000016 



+ 0.213171345 

 — 0.022550984 



— 0.022550984 



Gives 0.190620361 = B I r u the hyperbolic space. 

 But if the quadrature of the whole space B I H F be required, when the side 

 I H is longer than A I, this method would not succeed so well ; for in that 

 case ^ being greater than 1, it is manifest that the higher powers of it would 

 be too considerable to be neglected. To remedy this inconvenience, proceed 

 thus ; Suppose H F u r the space to be squared, A H being of any length what- 

 ever, either greater or less than A I, or equal to it : taking the point r any- 

 where between A and H, let AH = 1, and Hr = ^, which is to be con- 

 ceived as divided into innumerable equal parts, each =: a; then, after AH = 1, 

 the other parts continually decreasing will be, 1 — a, 1 — 2 a, 1 — 3 a, &c. 

 to A r = 1 —A. Hence, because of the equal rectangles F H A, u r A, B I A, 



&c. each of which suppose = b^, it will be H F = —-, and the rest m order 



■I 2 \\7, V»S Vt.2 



• , 1 — T-, - — —, &c. till ru = -; -, completing: the space H Fur, as is 



shown in the Arithm. Infinit. prop. 88, 94, 95. Then dividing b^ by 1 — «, 

 the quotient will be b^ -|- b^ a -|- b* a^ -f b^ a^ &c, that is b^ into \-\-a-\-a^ +«'' 

 &c, and all the right lines between 



into 



H F and r u will be, 



1 -h a + a^ -f- a^hc. ' 

 - 1 H- 2 a -I- 4 a^ -h Qa^ he. 

 i -i- 3 « + 9 a^ -I- 27a^ &c. y 



and so on till 

 1 +^-h^' + ^'&c. 

 Then the aggregate of all is A-\- 

 ^A'' + i-A^ + ^A* he. Xb^ = 

 F H r u, by Arith. Inf. pr. 64. 



For example, let A H = 1 , H r =: 

 A = 0.21, A I = b = 0.1, and there- 

 fore b^ = 0.01 ; then. 



A = 0.21 

 i-A^ = 0.02205 

 0-^^ = 0.003087 

 4-^^ = 0.000486203 — 

 i^^ = 0.000081682 -I- 

 ^A^ = 0.000014294 -h 

 1-A^ = 0.000002573 - 

 i-A^ = 0.000000473 — 



■^A^ = 0.000000088 -t- 



vv^'°= 0.000000017 4- 



.tV^" = 0.000000003 -|- 



Their sum 0.235722333 

 Which drawn into b^ or 0.01, 

 Gives 0.00235722333 = FH r u; 



