Chapter XIII – Of the Calculation of Interest
Summary: Applies geometrical progressions and logarithms to compound interest, including annual additions or withdrawals, present value, and annuities.
Sources: chapter-1.3.13
Last updated: 2026-05-02
Simple interest
Simple interest is computed by the Rule of Three: interest . For principal at 5%: interest (§540).
Compound interest
With compound interest the interest is added to the principal each year, so the base for the next year’s interest grows. At 5% interest, a principal becomes after one year, after two years, and in general (§544):
More generally, at rate %:
This is a geometrical progression in .
Computing compound interest with logarithms
For large , compute via logarithms (§546):
where .
Example (§547): , , rate 5%:
Example (§548): , , rate 6%:
For very large more decimal places of are needed (§549); at 5% for 500 years gives
Principal with annual additions
If a fixed sum is added each year in addition to compound interest, the principal after years is (§550–551):
This is derived by summing the geometric series of contributions. Computed via logarithms: then subtract .
Example (§552): , , : amount
Finding the time: given the same parameters, when does the principal reach ? (§553)
Principal with annual withdrawals
If is withdrawn each year, the principal after years is (§554–555):
This is obtained from the addition formula by replacing with . If the principal decreases to zero.
Example (§557): , (interest = 5000l., so overspending by 1000l./yr):
Interest for fractional years
The exponent need not be an integer (§558–559). For time (days): Computed by logarithms as before.
Example (§559): , 8 days at 5%:
Present value
Since at present grows to in one year, a sum due in one year is currently worth only . More generally, due in years has present value (§560):
Present value of an annuity
An annual rent lasting years (starting now) is worth the geometric sum (§562):
Example (§561): annual rent for 5 years at 5%: