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THE INFINITESIMAL CALCULUS 75 

useful. Let us now try to resume what we have seen of the 
development of mathematics along what seem to be its main 
lines. 

In the earliest times men were occupied with particular 
questions the properties of particular numbers and the 
geometrical properties of particular figures, together with 
simple mechanical questions. With the Greeks, a more general 
study of classes of geometrical figures began. But traces of an 
earlier exception to this study of particulars are afforded by 
'' algebra." In it and its later form symbols like our present 
x and y took the place of numbers, so that, what is a great 
advance in economy of thought and other labour, a part of 
calculation could be done with symbols instead of numbers, so 
that the one result stated, in a manner analogous to that of 
Greek geometry, a proposition valid for a whole infinite class of 
different numbers. 

The great revolution in mathematical thought brought about 
by Descartes in 1637 grew out of the application of this 
general algebra to geometry by the very natural thought of 
substituting the numbers expressing the lengths of straight 
lines for those lines. Thus a point in a plane for instance 
is determined in position by two numbers x and y, or co- 
ordinates. Now, as the point in question varies in position, 
x and y both vary ; to every x belongs, in general, one or more 
?/'s, and we arrive at the most beautiful idea of a single 
algebraical equation between x and y representing the whole of 
a curve the one " equation of the curve " expressing the 
general law by which, given any particular x out of an infinity 
of them, the corresponding y or y's can be found. 

The problem of drawing a tangent the limiting position of 
a secant, when the two meeting points approach indefinitely 
close to one another at any point of a curve came into 
prominence as a result of Descartes' work, and this, together 
with the allied conceptions of velocity and acceleration " at an 
instant," whicli appeared in Galileo's classical investigation, 
published in 1638, of the law according to which freely falling 
bodies move, gave rise at length to the powerful and convenient 
"infinitesimal calculus" of Liebniz and the "method of 
fluxions" of Newton. Mathematically, the finding of the