148 Bibliography. 
subtlety of analysis. Descartes was a truly great man. If he wasto0 
clumsy, it will be the natural and fitting vestibule of analytical geom® 
try. The clear geometrical conceptions which it would then g'® 
should precede the analysis of lines and surfaces. Without this pre 
aration there is danger of analytical geometry degenerating into * 
algebra, and thus becoming nearly barren of interest and value: 
tween two variables who does not keep the geometry of the on 
clearly in mind. It is a fault of most of our analytical researches 
their i that 
and Surfaces are discussed by the aid of algebraic symbols. In treat 
ing plane curves two axes are assumed, and the codrdinates of a" 
— of the plane are the lines from the point to the axes, draw® 
alle 
w 
to the axes. e length of these codrdinates for the differen! 
nts curve are called variables, and are represented by 7 @ 
The relation between these variables for an plane curve js € of 
y an equation other words, the codrdinates of all the points 
any particular curve, as a circle, a parabola, or a cycloid, bear 
stant and particular relation to each other, which relation would be # 
Proposition in common language, but in analysis is an equation. 
discussing this equation, all the properties of a particular curve may 
be determined in a rigorous and beautiful manner, singularly in at 
trast with the verboseness and indirectness of pure geometry: | 
using three axes and codrdinates, curves of double curvature like th? 
inat 
helix, and all geometrical surfaces like the sphere, the ellipsoid oF the 
helicoid, can be discussed. The method is absolutely general for al 
lines and surfaces of a regular geometrical character, though it 18. 
customary in elementary treatises to introduce those whose equatio™ 
involve logarithms, sines, &e. The curves of conic se 
