THE ELEMENTS. BOOK IX 
401 
the product of two similar plane numbers is a square (1) and, 
if the product of two numbers is a square number, the num 
bers are similar plane numbers (2) ; if a cube multiplies itself 
or another cube, the product is a cube (3, 4); if a 3 B is a 
cube, B is a cube (5) ; if A 2 is a cube, A is a cube (6). Then 
follow six propositions (8-13) about a series of terms in geo 
metrical progression beginning with 1. If 1, a, b, c ... k are 
n terms in geometrical progression, then (9), if a is a square 
(or a cube), all the other terms b, c, ... k are squares (or 
cubes) ; if a is not a square, then the only squares in the series 
are the term after a, i. e. h, and all alternate terms after h ; if 
a is not a cube, the only cubes in the series are the fourth 
term (c), the seventh, tenth, &c., terms, being terms separated 
by two throughout ; the seventh, thirteenth, &c., terms (leaving 
out live in each case) will be both square and cube (8, 10). 
These propositions are followed by the interesting theorem 
that, if 1, Oj, a 2 ... a n ... are terms in geometrical progression, 
and if a r , a n are any two terms where r < n, a r measures a n , 
and a n = a r .a n _ r (11 and For.) ; this is, of course, equivalent 
to the formula a m+n = a m . a n . Next it is proved that, if the 
last term k in a series 1, a, b, c ... k in geometrical progression 
is measured by any primes, a is measured by the same (12) ; 
and, if a is prime, k will not be measured by any numbers 
except those which have a place in the series (13). Proposi 
tion 14 is the equivalent of the important theorem that a 
number can only be resolved into 'prime factors in one way. 
Propositions follow to the effect that, if a, b be prime to one 
another, there can be no integral third proportional to them 
(16) and, if a, b, c ... k be in G. P. and a,k are prime to one 
another, then there is no integral fourth proportional to a, b, k 
(17) . The conditions for the possibility of an integral third 
proportional to two numbers and of an integral fourth propor 
tional to three are then investigated (18, 19). Proposition 20 
is the important proposition that the number of prime numbers 
is infinite, and the proof is the same as that usually given in 
our algebraical text-books. After a number of easy proposi 
tions about odd, even, ‘ even-times-odd ’, ‘ even-times-even ’ 
numbers respectively (Propositions 21-34), we have two im 
portant propositions which conclude the Book. Proposition 3 5 
gives the summation of a G. P. of n terras, and a very elegant 
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