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ANALYTICAL GEOMETRY 53 

the next chapter in an important question of mechanics is 
the same in principle as "the integral calculus." The 
integral calculus grew out of the work of Cavalieri and his 
successors, among whom the greatest are Eoberval (1602- 
1675), Blaise Pascal (1623-1662), and John Wallis (1616- 
1703), and mainly consists in the provision of a convenient 
and suggestive notation for this method. The discovery of the 
infinitesimal calculus was completed by the discovery that the 
inverse of the problem of finding the areas of figures enclosed 
by curves was the problem of drawing tangents to these curves, 
and the provision of a convenient and suggestive notation 
for this inverse and simpler method, which was, for certain 
historical reasons, called " the differential calculus." 

Both analytical geometry and the infinitesimal calculus are 
enormously powerful instruments for solving geometrical and 
physical problems. The secret of their power is that long and 
complicated reasonings can be written down and used to solve 
problems almost mechanically. It is the merest superficiality 
to despise mathematicians for busying themselves, sometimes 
even consciously, with the problem of economising thought. 
The powers of even the most god-like intelligences amongst us 
are extremely limited, and none of us could get very far in 
discovering any part whatever of the Truth if we could not 
make trains of reasoning which we have thought through and 
verified, very ready for and easy in future application by 
being made as nearly mechanical as possible. In both 
analytical geometry and the infinitesimal calculus, all the 
essential properties of very many of the objects dealt with in 
mathematics, and the essential features of very many of the 
methods which had previously been devised for dealing with 
them are, so to speak, packed away in a well-arranged (and 
therefore readily got at) form, and in an easily usable way. 

CHAPTER IV 

THE BEGINNINGS OF THE APPLICATION OF MATHEMATICS 
TO NATURAL SCIENCE THE SCIENCE OF DYNAMICS 

THE end of very much mathematics and of the work of 
many eminent men is the simple and, as far as may be,