We now have a definition of area which is in accordance with the 
promptings of our geometric intuition. It must be remembered, 
however, that this definition has been only recently discovered, 
and that the definition which for centuries has been accepted leads 
to results which flatly contradict our intuition, which leads us to 
say that a figure bounded by a continuous closed curve has an 
area. 
583. At this point we will break off our discussion of the 
relation between our intuitional notion of a curve, and the con 
figuration determined by the equations 
* = <K0 i V = 'KO 
where <£, yjr are one-valued continuous functions of t in an interval 
T. Let us look back at the list of properties of an intuitional 
curve drawn up in 563. We have seen that the analytic curve 
1) does not need to have tangents at a pantactic set of points on 
it; no arc on it needs have a finite length; it may completely fill 
the interior of a square; its equations cannot always be brought 
in the forms y=f(x) or F(xy') = 0, if we restrict ourselves to 
functions/or F employed in analysis up to the present; it does 
not need to have an area as that term is ordinarily understood. 
On the other hand, it is continuous, and when closed and with 
out double point it forms the complete boundary of a region. 
Enough in any case has been said to justify the thesis that 
geometric reasoning in analysis must be used with the greatest 
circumspection. 
Detached and Connected Sets 
584. In the foregoing sections we have studied in detail some 
of the properties of curves defined by the equations 
* = <K 0 , y = fC 0- 
Now the notion of a curve, like many other geometric notions, is 
independent of an analytic representation. We wish in the fol 
lowing sections to consider some of these notions from this point 
of view.