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MODERN MATHEMATICS ALGEBRA S3 

the less exactly 8 times. Now this property is possessed by 
any number, and not 8 alone. In fact, if we denote the 
number we start with by "a," we have, by the rules of algebra, 

a+ 1 

y- y = a This is an instance of a general property of 

numbers proved by algebra. 

Algebra contains many rules by which a complicated alge- 
braical expression can be reduced to its simplest terms. Owing 
to the suggestive and compact notation, we can easily acquire 
an almost mechanical dexterity in dealing with algebraical 
symbols. This is what Descartes means when he speaks of 
algebra as not being a science fitted to cultivate the mind. On 
the other hand, this art is due to the principle of the economy 
of thought, and the mechanical aspect becomes, as Descartes 
foresaw, very valuable if we could use it to solve geometrical 
problems without the necessity of fatiguing our imaginations 
by long reasonings on geometrical figures. 

I have already mentioned that the valuable notation " of 1 " 
was due to Descartes. This was published, along with all his 
other improvements in algebra, in the third part of his Geometry 
of 1637. I shall speak in the next chapter of the great dis- 
covery contained in the first two parts of this work ; here I 
will resume the improvements in notation and method made 
by Descartes and his predecessors, which make the algebraical 
part of the Geometry very like a modern book on algebra. 

It is still the custom in arithmetic to indicate addition by 
juxtaposition : thus " 2 " means " 2 -f ." In algebra, we 
always, nowadays, indicate addition by the sign " + " and 
multiplication by juxtaposition or, more rarely, by putting a 
dot or the sign " x " between the signs of the numbers to be 
multiplied. Subtraction is indicated by " ". 

Here we must digress to point out what is often, owing to 
confusion of thought, denied in text-books that, where "a" 
and "b" denote numbers, "a b" can only denote a number 
if a is equal to or greater than b. If a is equal to b, the 
number denoted is zero ; there is really no good reason for 
denying, say, that the numbers of Charles II. 's foolish sayings 
and wise deeds are equal, if a well-known epitaph be true. 
Here again we meet the strange way in which mathematics