VOL. LXVII.] PHILOSOPHICAL TRANSACTIONS. 185 



epual to LR, KN has to no a ratio compounded of the ratios of ln to no and nr 

 to op; that is, of the ratios a to b, c to d, and e to f. Therefore a magnitude 

 of the same kind with a and b, which has to b the ratio compounded of these 



ratios, is expressed by a + a. - -|- a. 1- a. . , 



Again, if nr, op, be supposed to represent g, h, respectively, and kv a 4th 

 proportional to op, kn, and qr; vq will be equal to kr (14 e 6) and consequently 

 VN will have to no a ratio compounded of the ratios of kn to no and nr to op; 

 that is, of the ratios a to B, c to d, e to p, g to h. But vk is by construction 



, G — H, C— DG— H, E— FG — II, C — DE— FG— H 



equal to a. \- a. — . \- a. . \- a. . . . 



T II 1> H F H D F H 



And this added to kn, above found, gives a + a. 1- a. -^ 1- a. — ^^- 



C — D E— F, C— D G — H, E — !■ G — H, C — D E — F G — H 



+ A. . h A. . h A. . \- A. . . , 



' D F D H F 11 D F H 



for the magnitude of the same kind with a and b, which has to b, the ratio 

 compounded of the ratios a to b, c to d, e to p, g to h; whence the law of 

 continuation is manifest. The same conclusions may be derived from (e. 3); so 

 that no principle can be simpler or more geometrical than that here made use of. 

 Thus then these magnitudes will stand: 



1. A + A. ~ , when two ratios are compounded. 



1. A + A. h A. 1- A. . , when three are compounded. 



C— D, E— F, G— II, C— DG — F, C — DG — H 



3. A + A. h A. ^A. ^ A. . \- K. . \- 



' D F h' D f' D h' 



E — FG— H, C— DE— FG— II , f ^. ,. 



A. . h A. . . — , when lour ratios are compounded. 



F H D F H r ' 



&c. &c. 



By continuing this operation much further, I found on examination that the 

 number of terms in which a is connected with the differences c — d, e — p, 

 G — H, &c. taken 1 by 1, 2 by 2, 3 by 3, &c. if/> denote the number of ratios 

 compounded, is expressed respectively by ^—r~, ^—r— • ^—w~i ^~r~ ■ ^— • ^— — ,&c. 

 Thus if the ratio of a to b be supposed equal to the ratios of c to d, k to f, 

 G to H, &c. respectively, these expressions will give the following ones : 



,2 — 1 A - B 



1. A H ; — .A. . 



' 1 B 



,3-1 A — B, 3— 13-2 (a-b)« 



2. A 4 ; .A. ;— .— -— .A.^ -. 



' 1 B ' 1 2 B 



„ ,4-1 A-B 4-1 4-2 (a-b)«,4-1 4-2 4-3 (a-b)» 



o. A H ; .A. h ; . — ;- — .A. 1 . — — — . .A.i :. • 



'l b'I 2 b'123 b* 



for magnitudes of the same kind with a and b, which have to b respectively, the 

 duplicate, triplicate, and quadruplicate ratio of A to b ; where p is successively 

 equal to 2, 3, and 4. And universally, by the same geometrical reasoning, it is 



r Ji-U.. iP— 1 A — B,p— Ip— 2 (a— b)*,. (a— b)/-! 



found, that a + '—-— .a. \- <--— . ~—.a.- 4- &c. a. ^^ — 



'1 B ' I 2 B ' B ' 



VOL. XIV. Bb 



