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32 THE NATURE OF MATHEMATICS 

multiplication, and division 1 are interpretable in much the 
same way as new points whether the "a" and "6" in "a + b," 
"a b," "ab," and so on, stand for numbers or factors, and 
we have, for example, 

a + b = b + a, ab = ba, a(b + c)=ab+ac, 

always. Because of this very strong analogy, mathema- 
ticians have called the fractions "numbers" too, and they 
often speak and write of " generalisations of numbers," of 
which this is the first example, as if the conception of 
number were generalised, and not merely the name "num- 
ber," in virtue of a great and close and important 
analogy. 

When once the points of a line were made to represent 
numbers, there seemed to be no further difficulty in admitting 
certain " irrational numbers " to correspond to the end-points 
of the incommensurable lines which had been discovered by 
the Greeks. This question will come up again at a later 
stage : there are necessary discussions of principle involved, 
but mathematicians did not go at all deeply into questions 
of principle until fairly modern times. Thus it has happened 
that, until the last sixty years or so, mathematicians were 
nearly all bad reasoners, as Swift remarked of the mathe- 
maticians of Laputa in Ghtlliver's Travels, and were unpar- 
donably hazy about first principles. Often they appealed to 
a sort of faith. To an intelligent and therefore doubting 
beginner, an eminent French mathematician of the eighteenth 
century said : " Go on, and faith will come to you." It is a 
curious fact that mathematicians have so often arrived at truth 
by a sort of instinct. 

Let us now return to our numerical algebra. Take, say, 
the number 8, and the fraction, which we will now call a 
" number " also ^. Add 1 to both ; the greater contains 

1 The operation of what is called, for the sake of analogy, 
" multiplication " of fractions is denned in the manner indicated 
in the following example. If of a yard costs lOd, how much 

10 x 4 x 7 
does of a yard cost ? The answer is - pence, and we define 

O X o 

4x7 1 

5 5 as " multiplied by " f , by analogy with what would happen 

o x o J- 

if | were 1 and % were, say, 3.