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INTRODUCTION 7 

in a convenient form and in a little space in short, if we could 
not economise our mental or bodily activity by remembering or 
carrying about two or three little formulse instead of fat books 
full of details ; and, lastly, we could not economise this activity 
if it were not for mathematics. 

Just as it is with astronomy, so it is with all other sciences 
both that of Nature and mathematical science : the very 
essence of them is the prevention of waste of the energies of 
muscle and memory. There are plenty of things in the un- 
known parts of science to work our brains at, and we can only 
do so efficiently if we organise our thinking properly, and con- 
sequently do not waste our energies. 

The purpose of this little volume is not to give like a text- 
book a collection of mathematical methods and examples, but 
to do, firstly, what text-books do not do : to show how and 
why these methods grew up. All these methods are simply 
means, contrived with the conscious or unconscious end of 
economy of thought-labour, for the convenient handling of long 
"and complicated chains of reasoning. This reasoning, when 
applied to foretell natural events, on the basis of the applica- 
tions of mathematics, as sketched in the fourth chapter, often 
gives striking results. But the methods of mathematics, 
though often suggested by natural events, are purely logical. 
Here the word "logical" means something more than the 
traditional doctrine consisting of a series of extracts from the 
science of reasoning, made by the genius of Aristotle and frozen 
into a hard body of doctrine by the lack of genius of his school. 
Modern logic is a science which has grown up with mathematics, 
and, after a period in which it moulded itself on the model of 
mathematics, has shown not only that the reasonings but also 
conceptions of mathematics are logical in their nature. 

In this book I shall not pay very much attention to the 
details of the elementary arithmetic, geometry, and algebra of 
the many text-books, but shall be concerned with the dis- 
cussion of those conceptions such as that of negative number 
which are used and not sufficiently discussed in these 
books. Then, too, I shall give a somewhat full account of the 
development of analytical methods and certain examinations of 
principles.