403 
EUCLID 
solution it is. Suppose that a v a 2 , a. s , ... a n+l are n+ 1 terms 
in G. P.; Euclid proceeds thus : 
®n+1 <,J n 
We have 
and, separando. 
Adding antecedents and consequents, we have (VII. 12) 
a 2 — a, 
a, 
a n -\-a n _\ J r ... + a l 
which gives a n + a n _ x + ... + a x or S n . 
The last proposition (36) gives the criterion for perfect 
numbers, namely that, if the sura of any number of terms of 
the series 1, 2, 2 2 ... 2 n is prime, the product of the said sum 
and of the last term, viz. (1 + 2 + 2 2 + ... + 2 n ) 2 n , is a perfect 
number, i. e. is equal to the sum of all its factors. 
It should be added, as regards all the arithmetical Books, 
that all numbers are represented in the diagrams as simple 
straight lines, whether they are linear, plane, solid, or any 
other kinds of numbers; thus a product of two or more factors 
is represented as a new straight line, not as a rectangle or a 
solid. 
Book X is perhaps the most remarkable, as it is the most 
perfect in form, of all the Books of the Elements. It deals 
with irrationals, that is to say, irrational straight lines in rela 
tion to any particular straight line assumed as rational, and 
it investigates every possible variety of straight lines which 
can be represented by V{ Va+ Vb), where a, b are two com 
mensurable lines. The theory was, of course, not invented by 
Euclid himself. On the contrary, we know that not only the 
fundamental proposition X. 9 (in which it is proved that 
squares which have not to one another the ratio of a square 
number to a square number have their sides incommen 
surable in length, and conversely), but also a large part of 
the further development of the subject, was due to Theae 
tetus. Our authorities for this are a scholium to X. 9 and a 
passage from Pappus’s commentary on Book X preserved 
in the Arabic (see pp. 154-5, 209-10, above). The passage