400 
EUCLID 
proving that, according as a 2 does or does not measure b 2 , 
a does or does not measure h and vice versa; and similarly, 
according as d 3 does or does not measure b 3 , a does or does not 
measure b and vice versa. 13 proves that, if a, b, c ... are in 
G. P., so are a 2 , b 2 , c 2 ... and a 3 , b 3 , c 3 ... respectively. 
Proposition 4 is the problem, Given as many ratios as we 
please, a:b, c:d... to find a series p, q, r, ... in the least 
possible terms such that r p\q — a\b, q:r = c:d, This is 
done by finding the L. C. M., first of b, c, and then of other 
pairs of numbers as required. The proposition gives the 
means of compounding two or more ratios between numbers 
in the same way that ratios between pairs of straight lines 
are compounded in VI. 23 ; the corresponding proposition to 
VI. 23 then follows (5), namely, that plane numbers have 
to one another the ratio compounded of the ratios of their 
sides. 
Propositions 8-10 deal with the interpolation of geometric 
means between numbers. If a:b = e:/, and there are n 
geometric means between a and h, there are n geometric 
means between e and/also (8). If a n , a n ~ 1 b ... ab n-1 , b n is a 
G. P. of n+ 1 terms, so that there are (n — 1 ) means between 
a n , b n , there are the same number of geometric means between 
1 and a n and between 1 and b n respectively (9); and con 
versely, if l, a, a 2 ... a n and 1, b, b 2 ... b n are terms in G. P., 
there are the same number (n — 1) of means between a n , b n (10). 
In particular, there is one mean proportional number between 
square numbers (11) and between similar plane numbers (18), 
and conversely, if there is one mean between two numbers, the 
numbers are similar plane numbers (20) ; there are two means 
between cube numbers (12) and between similar solid numbers 
( 19), and conversely, if there are two means between two num 
bers, the numbers are similar solid numbers (21). So far as 
squares and cubes are concerned, these propositions are stated by 
Plato in theTimaeus, and Nicomachus, doubtless for this reason, 
calls them : Platonic ’. Connected with them are the proposi 
tions that similar plane numbers have the same ratio as a square 
has to a square (26), and similar solid numbers have the same 
ratio as a cube has to a cube (27). A few other subsidiary 
propositions need no particular mention. 
Book IX begins with seven simple propositions such as that