APPENDIX TO CHAPTER XIII 395

year will bring the radius vector to the position OD, at which
time the same interest, DFE, may be detached, and so on indefinitely,
 the result being a toothed wheel. Each tooth being
+5 of the radius 04, the sum of the twenty-five teeth will be
exactly equal to the radius. In case the interest is reckoned
semi-annually, the teeth will be fifty in number instead
of twenty-five, but each will be half as large; and so on
until, for “continuous” reckoning, we have an infinite number
of teeth each of infinitesimal size; but the sum of the whole
number in the complete revolution will still be exactly equal
to 0A.
In case no income is received, the accumulation of capital is
represented by increasing the length of the radius vector, the
end of which thus traces a spiral. The radius vector revolves
 around the spiral once every twenty-five years,
and the ratio between the “amount,” 04', after a complete
 revolution, and the original “principal” 04, will be
e, that is, 2.718, provided the rate of interest is reckoned
continuously.
The same spiral represents the accumulation of capital,
whatever may be the rate of interest. For a complete revolution
does not represent a definite length of time, but the purchase
period ; and it is clear that a more rapid rate of interest is represented
 by a more rapid turning of the radius vector. Thus, if
the rate is not 49, but 89, the radius vector swings around
through the same spiral once in twelve and a half years instead
of once in twenty-five years. Again, if the rate is 29, the
revolution is fifty years. The spiral is what is known as the
“equiangular spiral,” and has the property that the tangent at
any point is inclined at a constant angle to the radius vector.

The angle is in this case such that its tangent is Zr. This

: SP
angle is 80° 57". The equation of this spiral is =e
0
which p represents the radius vector, 8 the angle of revolution,
and p, the initial radius vector; e and = are, of course, the
magnitudes ordinarily represented by these letters, namely, the
base of the Napierian system of logarithms and the ratio of

the circumference of a circle to its diameter.