THE ELEMENTS. BOOKS I-II 
379 
I make no apology for having dealt at some length with 
Book I and, in particular, with the preliminary matter, in 
view of the unique position and authority of the Elements 
as an exposition of the fundamental principles of Greek 
geometry, and the necessity for the historian of mathematics 
of a clear understanding of their nature and full import. 
It will now be possible to deal more summarily with the 
other Books. 
Book II is a continuation of the third section of Book I, 
relating to the transformation of areas, but is specialized in 
that it deals, not with parallelograms in general, but with 
rectangles and squares, and makes great use of the figure 
called the gnomon. The rectangle is introduced (Def. 1) as 
a ‘ rectangular parallelogram ’, which is said to be ‘ contained 
by the two straight lines containing the right angle ’. The 
gnomon is defined (Def. 2) with reference to any parallelo 
gram, but the only gnomon actually used is naturally that 
which belongs to a square. The whole Book constitutes an 
essential part of the geometrical algebra which really, in 
Greek geometry, took the place of our algebra. The first ten 
propositions give the equivalent of the following algebraical 
identities. 
1. a (b + c + d +...) = ab + ac + ad + ..., 
2. {a + b) a + (a + b) b = {a + b) 2 , 
3. (a + b) a = ab + a 2 , 
4. (a + b) 2 = a 2 + b 2 + 2 ab, 
5. ab+{i{a + b) — b] 2 = {%{a + b)} 2 , 
or (a + p) (a — /3) + /3 2 = a 2 , 
6. (2 a + b)b + a 2 = (a + b) 2 , 
or (oc + p) (¡3 — a) + a 2 = P 2 , 
7. (a + b) 2 + a 2 = 2 (a + b)a + b 2 , 
or oc 2 + ft 2 = 2a.p + (a —(3) 2 , 
4 (a + b)a + b 2 = {(a + b) + a} 2 , 
or 4a/3 + (a — P) 2 = (cx + P) 2 , 
8.