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Mathematical Treasure: Arithmetic Perfected for Commerce

Frank J. Swetz (The Pennsylvania State University)

L’Arithmetique en sa Perfection was first published in 1657 and became a very popular book. It went through many editions, the last published in 1781. Material from this Arithmetic was frequently plagiarized by other writers. Little is known of its author, François Le Gendre, other than that he was a well-respected arithmetician. Arithmetic Perfected with Practical Implementations for the Use of Bankers, Financers and Merchants promises much. These images are from the ninth edition of 1690.

Title page for 1690 edition of The Arithmetic in its Perfection.

On page 27, the “casting out of nines” is demonstrated as a check for the correctness of a multiplication result: 706 x 57 = 40242. Nines are “cast out” of the multiplicand, 706, leaving 4, which is then placed at the apex of a cross. The same is done to the multiplier, 57, giving 3, placed at the base of the cross. These results are multiplied together, 4 x 3 = 12, and nine cast out, providing a 3 which is placed in the left apex of the cross. Finally, nines are cast out of the product, 40242, producing a 3, placed in the right apex of the cross. The left and right entries, 3, are compared and seen to be the same, affirming the correctness of the operation.

Page 27 of 1690 edition of The Arithmetic in its Perfection.

On pages 42 and 43, Le Gendre described the operation of multi-digit division. His heading on page 42 introduced three Italian methods of division. At this period of history, the Italian merchant community was the leader in arithmetic calculation employing “Hindu-Arabic” numerals. However, on page 43, the author provided the example, 898108 ÷ 999, and obtained the correct quotient using the “popular” French, Spanish, and Italian algorithms.

Page 42 of 1690 edition of The Arithmetic in its Perfection.Page 43 from 1690 edition of The Arithmetic in its Perfection.

Sections on geometry are included in the Arithmetic. A practical example of surveying and land measurement is considered on pages 316 and 317. An irregular polygonal region has been measured and its area is required. The region is partitioned into several smaller regions whose areas can be easily computed. These areas are combined to obtain the required measure.

Page 316 from 1690 edition of The Arithmetic in its Perfection.Page 317 from 1690 edition of The Arithmetic in its Perfection.

In a 1781 edition of the Arithmetic, a section on the use of jetons has been added. While it would seem that, at this late period of history, the use of the counting board and jetons would have become obsolete, a justification, supposedly written by the author, is included:

This arithmetic is quite as useful as that which is done with a pen since with jetons, we can perform all the rules we need in all the computation used in business. This method of calculation is more practiced by women than by men; however, many people employed in the Exchequer and in the courts make use of it with great success.

This material is obtained through the courtesy of the University of California Libraries. A complete digital copy can be read on the UC Libraries’ Internet Archive.

Index to Mathematical Treasures

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Arithmetic Perfected for Commerce," Convergence (May 2017)