Compound interest is the bedrock of modern wealth building, yet Roman law and common law once condemned it as the worst kind of usury. Its old name was anatocism: interest charged on the principal plus all the interest already accumulated. Simple interest on $1,000 at 5 percent is a steady $50 a year. Compounding earns $52.50 in the second year, a difference that looks trivial until the curve bends upward over decades. Ancient borrowers had no way to model that curve, which is why lawmakers tried to ban it outright.

This episode traces how the forbidden math became standard practice. The trail runs from a Babylonian clay tablet dated to roughly 2000 to 1700 BC, through the merchant tables of Renaissance Florence and the Rule of 72, to Jacob Bernoulli’s accidental discovery of the constant e in 1683. Along the way it explains why banks’ compounding schedules confuse savers, what the annual equivalent rate reveals, why US mortgages do not actually compound, and how continuous compounding prices derivatives today.

• Around 1340 the Florentine merchant Francesco Balducci Pegolotti published a table showing interest on 100 lire at rates from 1 to 8 percent for up to 20 years.
• The Rule of 72 first appeared in Luca Pacioli’s 1494 Summa de arithmetica. Seventy-two is used because it divides cleanly, though 69.3 would be slightly more accurate.
• Richard Witt’s 1613 book, the first devoted entirely to compound interest, included 124 worked examples and based its tables on London’s 10 percent legal cap on loans.
• Persian merchants in the 19th century worked out monthly compound payments in their heads using a slightly modified linear Taylor approximation.
• US home mortgages follow an amortizing schedule in which each payment covers that month’s interest, while Canadian mortgages are generally compounded semi-annually.