Mathematical Treasure: Bachet’s Mathematical Recreations

Frank J. Swetz (The Pennsylvania State University)

Claude Gaspar Bachet (1581-1638) was a French aristocrat, poet, and writer. In 1612 he published Problèmes plaisans et delectable, qui se font par les nombres, a book of arithmetical problems and puzzles. The book became very popular, going through five editions over three centuries. In style and content, it became the forerunner of modern books of mathematical recreations. A modern reader will recognize many of the book’s problems.

Much of Bachet’s Problèmes plaisans et delectables concerns number theory set forth as theorems, demonstrations, and exercises. Theorem I, on page 1, states:

If a given number is multiplied by another, the product [then] divided again by another [number], there will be a like proportion of the given number to the quotient of the division, [as] there exists a factor to the multiplier.

Bachet then supplied an example, using letters of the alphabet A, B, C, … to indicate numbers in the computation: 8 x 3 = 24, 24 ÷ 4 = 6 → 8:6 = 4:3. As reinforcement of this statement, Bachet referred to Euclid VII.19:

If four numbers are proportional, then the number produced from the first and fourth equals the number produced from the second and third; and, if the number produced from the first and fourth equals that produced from the second and third, then the four numbers are proportional.

Let A, B, C, and D be four numbers in proportion, so that A is to B as C is to D, and let A multiplied by D make E, and let B multiplied by C make F.

Another problem and its solution are shown on pages 62 and 63 above.

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