is the allowance given for the use of money by the borrower to the lender, and is either simple or compound. The method of computing both interests is explained in the article ALGEBRA, (Encycl.) page 427, &c.; and the subject of simple interest is again referred in ARITHMETIC, (Encycl.) p. 20. The application of the canons for the computation of compound interest, to the value of annuities, the only case in which that interest is allowed by the laws of this country, may be seen in the articles ANNUITY and SURVIVORSHIP, (Encycl.) where various tables are given to facilitate the different computations. Some of our readers, however, have expressed a wish to have the rule for computing compound interest stated, as to be understood by those who are unacquainted with algebraic symbols. Their wish may be easily gratified.
The general formula $S = pR^t$ answers for the amount of any sum, whether the interest be payable yearly, half-yearly, quarterly, or daily. Let $R$ denote the amount of one pound for the first payment, and $t$ the number of payments, the unit being from the commencement till the first payment is due; also, let $l$ denote the logarithm of any quantity before which it is wrote; then, from the known property of logarithms, the theorem may be expressed thus, $l.S = l.p + l.R \times t$.
Required the amount of L. 250 at 5 per cent. compound interest, for 12 years, reckoning the interest payable yearly, half-yearly, quarterly, and daily?
**Yearly**, $p = 250$, $R = 1.05$, $t = 12$.
\[ \begin{align*} 0.0211893 &= l.R \\ \frac{2542716}{12} &= l.R \times t \\ 2.3979400 &= l.p. \end{align*} \]
\[ l.S = 2.6522116 - L.448 : 19 : 3\frac{1}{2} = \text{Amount}. \]
\[ \begin{align*} 250 \\ 198 : 19 : 3\frac{1}{2} = \text{Comp. interest}. \end{align*} \]
**Half yearly**, $p = 250$, $R = 1.025$, $t = 24$.
\[ \begin{align*} 0.0107239 &= l.R \\ \frac{428956}{24} &= l.R \times t \\ 214478 &= l.p. \end{align*} \]
\[ l.S = 2.6553136 - L.452 : 3 : 7\frac{1}{2} = \text{Amount}. \]
\[ \begin{align*} 250 \\ 202 : 3 : 7\frac{1}{2} = \text{Interest}. \end{align*} \] Quarterly, \( p = 250, R = 10125, t = 48 \).
\[ \begin{align*} 0.0053950 &= l.R. \\ 431600 & \\ 215800 & \\ 2589600 &= l.R \times t. \\ 23979400 &= l.p. \\ \end{align*} \]
\( l.S = 26569000 - L.453 : 16 : 8\frac{1}{2} = \text{Amount}. \)
\[ \begin{align*} 250 & \\ 203 : 16 : 8\frac{1}{2} &= \text{Interest}. \\ \end{align*} \]
Daily, \( p = 250, R = 1 + \frac{0.05}{365} = \frac{365.05}{365}, t = 365 \times 12. \)
\[ \begin{align*} 25623524 & \\ 25622929 & \\ 0.000595 &= l.R. \\ 4380 & \\ 47600 & \\ 1785 & \\ 2380 & \\ 2606100 &= l.R \times t. \\ 23979400 &= l.p. \\ \end{align*} \]
\( l.S = 26585500 - L.455 : 11 : 3\frac{1}{2} = \text{Amount}. \)
\[ \begin{align*} 250 & \\ 205 : 11 : 3\frac{1}{2} &= \text{Interest}. \\ \end{align*} \]