362 NATURE OF CAPITAL AND INCOME

AC, the rate reckoned semi-annually, and so on until the tangent
 AT", or AT, is again reached, when the distinction between
interest and discount disappears. It is easy to prove (by similar
 triangles) that the reciprocal of the rate of interest continuously
 reckoned, in other words the “year’s purchase,” is
represented by aS the subtangent.

§ 3 (ro Cm. XII, § 6)

A Premium Rate of 49 one Year and 3% Each Year after means a Price
Rate of 3.089 the First Year.

We have given 4% as the rate of interest (as premium) in
the first year, and 3% as the rate for each year thereafter.
Let us suppose that $100 to-day is invested for $104 next
year, and that at the end of the year, of this $104, $100 is reinvested.
 Since thereafter the rate of interest is always 3%
in the premium sense, it must be, by proof in the text, also 3%
in the price sense. Consequently, the $100 reinvested next
year may be used to buy $3 a year forever. The result is that
for the $100 to-day the return is $4 next year and $3 each
year thereafter. This series of payments is the same as $3 a
year forever, together with one extra dollar at the end of the
first year. This extra dollar has a present worth (since the
interest premium between this year and next year is 4%) of
ory OF 962. If we deduct this present value of the extra
dollar from the $100 (which is the present value of the entire
series of $4, $3, $3, $3, etc, ad inf.), we have the present
value of the series without the extra dollar, i.e. of $3 a year
forever. The remainder is $99.0311. Since, then, this
$99.0314 will buy $3 a year forever, the rate of interest in the
price sense is $3 + $99.031, or 3.039 approximately. This
is the rate at the outset. After the first year it will evidently

always be 3%.
§4 (ro Cu. XII, § 6)

A Price Rate of 49 this Year and 39% Each Year after means a Premium
Rate of 37} 9 the First Year.

We have given 4 % as the rate (in the price sense) for the
first year, and 3% as the rate (in the same sense) for each year
after.