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16 THE NATURE OF MATHEMATICS 

We shall fiud that algebra was hardly touched by those 
Greeks who made of geometry such an important science, partly, 
perhaps, because the almost universal use of the abacus l 
rendered it easy for them to add and subtract without any 
knowledge of theoretical arithmetic. And here we must 
remember that the principal reason why Ahmes' arithmetical 
problems seem so easy to us, is because of our use from childhood 
of the system of notation introduced into Europe by the Arabs, 
who originally obtained it from the Hindoos. In this system 
an integral number is denoted by a succession of digits, each 
digit representing the product of that digit and a power of ten, 
and the number being equal to the sum of these products. 
Thus, by means of the local value attached to nine symbols and 
a symbol for zero, any number in the decimal scale of notation 
can be expressed. It is important to realise that the long and 
strenuous work of the most gifted minds was necessary to 
provide us with simple and expressive notation which, in nearly 
all parts of mathematics, enables even the less gifted of us to 
reproduce theorems which needed the greatest genius to dis- 
cover. Each improvement in notation seems, to the uninitiated, 
but a small thing ; and yet, in a calculation, the pen sometimes 
seems to be more intelligent than the user. Our notation is an 
instance of that great spirit of economy which spares waste of 
labour on what is already systematised, so that all our strength 
can be concentrated either upon what is known but unsys- 
tematised, or upon what is unknown. 

Let us now consider the transformation of Egyptian geometry 
in Greek hands. Thales of Miletus (about 640-546 B.C.), 
who, during the early part of his life, was engaged partly in 
commerce and partly in public affairs, visited Egypt and first 
brought this knowledge into Greece. He discovered many 
things himself, and communicated the beginnings of many to 
his successors. We cannot form any exact idea as to how 

1 The principle of the abacus is that a number is represented by 
counters in a series of grooves, or beads strung on parallel wires ; 
as many counters being put on the first groove as there are units, as 
many on the second as there are tens, and so on. The rules to be 
followed in addition, subtraction, multiplication, and division are 
given in various old works on arithmetic.