THE ELEMENTS. BOOKS YII-VIII 
399 
measures a the same number of* times that n measures b, and 
that numbers prime to one another are the least of those which 
have the same ratio with them. These propositions lead up to 
Propositions 22-32 about numbers prime to one another, prime 
numbers, and composite numbers. This group includes funda 
mental theorems such as the following. If two numbers be 
prime to any number, their product will be prime to the same 
(24). If two numbers be prime to one another, so will their 
squares, their cubes, and so on generally (27). If two numbers 
be prime to one another, their sum will be prime to each 
of them ; and, if the sum be prime to either, the original 
numbers will be prime to one another (28). Any prime number 
is prime to any number which it does not measure (29). If two 
numbers are multiplied, and any prime number measures the 
product, it will measure one of the original numbers (30). 
Any composite number is measured by some prime number 
(31). Any number either is prime or is measured by some 
prime number (32). 
Propositions 33 to the end (39) are directed to the problem 
of finding the least common multiple of two or three numbers ; 
33 is preliminary, using the G. C. M. for the purpose of solving 
the problem, ‘ Given as many numbers as we please, to find the 
least of those which have the same ratio with them.’ 
It seems clear that in Book VII Euclid was following 
earlier models, while no doubt making improvements in the 
exposition. This is, as we have seen (pp. 215-16), partly con 
firmed by the fact that in the proof by Archytas of the 
proposition that ‘no number can b(f a mean between two 
consecutive numbers ’ propositions are presupposed correspond 
ing to VII. 20, 22, 33. 
Book VIII deals largely with series of numbers ‘in con 
tinued proportion ’, i. e. in geometrical progression (Propositions 
1-3, 6-7, 13). If the series in G. P. be 
a n , a n ~ l b, a n ~ 2 b 2 ,... a 2 b n ~ 2 , ab n ~ l , b n , 
Propositions 1-3 deal with the case where the terms are the 
smallest that are in the ratio a:b, in which case a n , b n are 
prime to one another. 6-7 prove that, if a n does not measure 
a n ~ x b, no term measures any other, but if a n measures b n , 
it measures a n ~ x b. Connected with these are Propositions 14-17