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 infinite
 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 convenience,
 as the end of each year in which it occurs. Thus the rectangle
 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rectangles are each equal to one half, the altitudes will no
longer represent the individual payments, but double those semiannual
 payments, i.e. the per annum rate. Thus, if the annuity
 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.