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6 THE NATURE OF MATHEMATICS 

grumbling at the lateness of his dinner. Now, we will admit 
at once that higher mathematics is of no more use in the 
grocery trade than the grocery trade is in the navigation of a 
ship ; but that is no reason why we should condemn mathe- 
matics as entirely useless. I remember reading a speech made 
by an eminent surgeon, who wished, laudably enough, to spread 
the cause of elementary surgical instruction. "The higher 
mathematics," said he with great satisfaction to himself, " do 
not help you to bind up a broken leg ! " Obviously they do 
not ; but it is equally obvious that surgery does not help us 
to add up accounts ; ... or even to think logically, or to 
accomplish the closely allied feat of seeing a joke. 

To the question about the use of mathematics we may reply 
by pointing out two obvious consequences of one of the applica- 
tions of mathematics : mathematics prevents much loss of life 
at sea, and increases the commercial prosperity of nations. 
Only a few men a few intelligent philosophers and more 
amateur philosophers who are not highly intelligent would 
doubt if these two things were indeed benefits. Still, probably, 
all of us act as if we thought that they were. Now, I do not 
mean that mathematicians go about with life-belts or serve 
behind counters ; they do not usually do so. What I mean I 
will now try to explain. 

Natural science is occupied very largely with the prevention 
of waste of the labour of thought and muscle when we want to 
call up, for some purpose or other, certain facts of experience. 
Facts are sometimes quite useful. For instance, it is useful for 
a sailor to know the positions of the stars and sun on the nights 
and days when he is out of sight of land. Otherwise, he cannot 
find his whereabouts. Now, some people connected with a 
national institution publish periodically a Nautical Almanac 
which contains the positions of stars and other celestial things 
you see through telescopes, for every day and night years and 
years ahead. This Almanac, then, obviously increases the 
possibilities of trade beyond coasting-trade, and makes travel 
by ship, when land cannot be sighted, much safer ; and there 
would be no Nautical Almanac if it were not for the science of 
astronomy ; and there would be no practicable science of 
astronomy if we could not organise the observations we make 
of sun and moon and stars, and put hundreds of observations