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