Magnitudes of any Kind together. 45 5 
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 
fame kind with a and b, which has to b the ratio com- 
pounded of thefe ratios is expreffed by a+ a. + a. 
C — D E — F v 
+ A ’ ~T~ ' T -0 
Again, if nr, op, be fuppofed to reprefent g, h, re- 
fpedtively, and kv a fourth proportional to op, kn, and 
qr; vQ.will be equal to kr (14. e. 6.) and confequently 
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 F, G to H. But vk is by conhruchion equal to 
G H C — D G — H E — F G — H C — D E — F G — H 
Ai T3 A® _ • 77 A* “ ” • ~ “I” A» ~ • ' ~ 
And this added to kn above found gives a + a. 
H 
C — - D 
D 
> E — F G — H C D E- 
+ A. — - + A. — h A. 
G— H 
+ A.— 
H D 
F H D 
C— — D E — F G — H 
F H * 
•F C— D G H E — F 
- + A.— .— + A.— 
for the magnitude of the fame 
kind with A and b, Vvhich has to b the ratio compounded 
of the ratios a to b, c to d, e to f, g to h ; whence the 
law of continuation is manifefh 
The fame conclufions may be derived from (e. 5.); 
fo that no principle can be Ampler or more geometrical 
than that here made ufe of. 
Thus then thefe magnitudes will hand. 
a- a+ a. when two ratios are compounded. 
a. A 
