372 NATURE OF CAPITAL AND INCOME 
installments of $1 each, they are represented by¢,¢,¢, and so 
on indefinitely, in each case the lines becoming shorter but more 
numerous. If this process is continued indefinitely, it is clear 
that continuous income would simply be represented by an in- 
finite number of infinitesimally small lines, —a representation 
which would be unintelligible. It is for this reason that the area 
method becomes necessary. To show how it may be used, even 
for discontinuous income, let a series of annual payments, a, be 
represented in Figure 36 by the rectangles whose bases are equal 
to unity and whose altitudes, therefore, are equal toa. The point 
of time to which each rectangle is referred is taken, for conven- 
ience, as the end of each year in which it occurs. Thus the rec- 
tangle OV refers to the point of time P, and PWto Q. Ifthe 
payments are semi-annual, we represent them by the areas of 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
RS T.U V Ww 
i Q 
; Fia. 36. 
the rectangles OT, YV, etc., in the same manner. But as the 
rectangles are each equal to one half, the altitudes will no 
longer represent the individual payments, but double those semi- 
annual payments, i.e. the per annum rate. Thus, if the annu- 
ity is $4 per annum payable semi-annually, the rectangle OT 
means $2, its base is one half, and its altitude, YT, will not be 
2, but 4, the rate per annum. 
Similarly, quarterly payments are represented by rectangles 
08, XT, YU, etc., whose altitudes will again represent the rate 
per annum of each quarterly payment. 
Finally, for continuous payments, we shall have an infinite 
number of infinitesimal rectangles, forming in the aggregate 
the whole figure represented, the altitude of which at any point 
will be the rate per annum at which income is flowing at that 
point.