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ANALYTICAL GEOMETRY 49 

second chapter. The ideas of function and variable were not 
prominent until the time of Descartes, and names for these 
ideas were not introduced until much later. 

The conventions of analytical geometry as to the signs of 
co-ordinates in different quadrants of the plane had an important 
influence in the transformation of trigonometry from being a 
mere adjunct to a practical science. In the same notation as 
that used at the end of the first chapter, we may conveniently 

AP 

call the number - -, which is the same for all lengths of OP, 

by the name "w," for short, and define and as the 

"sine of u," and the "cosine of u" respectively. Thus 
" sin u " and " cos u," as we write them for short, stand for 
numerical functions of u. Considering as the origin of a 
system of rectangular co-ordinates of which OA is the ce-axis, so 

that u measures the angle POA and - and - are cos u and sin u 

r r 

respectively. Now, even if u becomes so great that POA is 
successively obtuse, more than two right angles . . ., these 
definitions can be preserved, if we pay attention to the signs 
of x and y in the various quadrants. Thus sin u and cos u 
become separated from geometry, and appear as numerical 
functions of the variable u, whose values, as we see on re- 
flection, repeat themselves at regular intervals as u becomes 
larger and larger. Thus, suppose that OP turns about O in a 
direction opposite to that in which the hands of a clock move. 

In the first quadrant, sin u and cos u are ^ and - : in the 

r r 

second they are - and ; in the third they are -^ and ? : 

r r r r 

Al SYl 

in the fourth they are - and - ; in the fifth they are 

At fV 

- and - again ; and so on. Trigonometry was separated from 

geometry mainly by John Bernoulli and Euler, whom we shall 
mention later. 

We will now turn to a different development of mathematics. 



The ancient Greeks seem to have had something approaching 
a general method for finding areas of curvilinear figures. In- 
deed, infinitesimal methods, which allow indefinitely close 
approximation, naturally suggest themselves. The determina- 
tion of the area of any rectilinear figure can be reduced to that 
of a rectangle, and can thus be completely effected. But this 
process of finding areas this " method of quadratures "- 
failed for areas or volumes bounded by curved lines or surfaces, 
respectively. Then the following considerations were applied. 
"When it is impossible to find the exact solution of a question, 
it is natural to endeavour to approach to it as nearly as possible 
by neglecting quantities which embarrass the combinations, if 
it be foreseen that these quantities which have been neglected 
cannot, by reason of their small value, produce more than a 
trifling error in the result of the calculation. For example, as 
the properties of curves are with difficulty discovered, it is 
natural to consider them as polygons of a great number of 
sides. If a regular polygon be supposed to be inscribed in a 
circle, it is evident that these two figures, although always 
different, are nevertheless more and more alike according as 
the number of the sides of the polygon increases. Their peri- 
meters, their areas, the solids formed by their revolving round 
a given axis, the angles formed by these lines, and so on, are, 
if not respectively equal, at any rate so much the nearer ap- 
proaching to equality as the number of sides becomes increased. 
Whence, by supposing the number of these sides very great, it 
will be possible, without any perceptible error, to assign to the 
circumscribed circle the properties that have been found be- 
longing to the inscribed polygon. Thus, if it is proposed to 
find the area of a given circle, let us suppose this curve to be 
a regular polygon of a great number of sides : the area of any 
regular polygon whatever is equal to the product of its peri- 
meter into the half of the perpendicular drawn from the centre 
upon one of its sides ; hence, the circle being considered as a 
polygon of a great number of sides, its area ought to equal the 
product of the circumference into half the radius. Now, this 
result is exactly true. However, the Greeks, with their taste 
for strictly correct reasoning, could not allow themselves to 
consider curves as polygons of an "infinity" of sides. They 
were also influenced by the arguments of Zeno, and thus re- 
garded the use of " infinitesimals " with suspiciou.