CHAPTER XVII 
GEOMETRIC NOTIONS 
Plane Curves 
562. In this chapter we propose to examine the notions of 
curve and surface together with other allied geometric concepts. 
Like most of our notions, we shall see that they are vague and 
uncertain as soon as we pass the confines of our daily experience. 
In studying some of their complexities and even paradoxical 
properties, the reader will see how impossible it is to rely on his 
unschooled intuition. He will also learn that the demonstration 
of a theorem in analysis which rests on the evidence of our 
geometric intuition cannot be regarded as binding until the 
geometric notions employed have been clarified and placed on a 
sound basis. 
Let us begin by investigating our ideas of a plane curve. 
563. Without attempting to define a curve we would say on 
looking over those curves most familiar to us that a plane curve 
has the following properties : 
1° It can be generated by the motion of a point. 
2° It is formed by the intersection of two surfaces. 
3° It is continuous. 
4° It has a tangent at each point. 
5° The arc between any two of its points has a length. 
6° A curve is not superficial. 
7° Its equations can be written in any one of the forms 
V =f O), 
s = <KO > y = 'KOi 
P& y') = 0 ; 
(1 
(2 
(3 
and conversely such equations define curves. 
678