# Mathematical Treasure: Arithmetic of Pierre de Savonne

Author(s):
Frank J. Swetz (The Pennsylvania State University)

L’Arithmetique de Pierre de Savonne d’Avignon is a typical arithmetic of its time written for the needs of merchants and businessmen. The claim on the title page promises all good rules necessary for business. Little is known of the author other than that he was probably a French reckoning master who wrote several books on arithmetic and bookkeeping. He published the first French treatise on accounting in 1567. This book went through several editions as did his popular L’Arithmetique, which was released in Paris in 1565 and in Lyon in 1571 (WorldCat). Images from the tenth edition of 1630 are shown here.

A place value chart for the writing of numbers up to hundreds of millions is given on page 23.

On page 3 is a table of multiplication facts. Note the error in the first fact presented: “1 fois 2 = 4.” This should read “2 fois 2 = 4” or “2 times 2 = 4.”

On page 40, the author discussed an algorithm for division. The technique appears to be an abbreviation of the “Galley Method.” The example on the left is: 30519 ÷ 19 = 1605 remainder 12; a check for correctness is then performed using multiplication. The problem on the right is similar: 190011 ÷ 99 = 1919 remainder 30; once again, the inverse operation, multiplication, is used to check the result.

On page 44, as another algorithm for division, “Galley Division” is presented using a full array of partial factors. The quotient on the left, 530763 ÷ 679, is found to be 781 remainder 464. The quotient on the right, 19999100007 ÷ 99999, gives an exact answer of 199993. In both instances, the results are checked by multiplication.

On page 55, a discussion and demonstration of the subtraction of fractions takes place. In the French of this period, the term for “fraction” was nombre rompu, or broken number. The example 7/8 – 2/3 is given; the least common denominator of the two fractions is obtained, 24, and the process proceeds from there. A “proof” of the process is given. The interested reader may attempt to discern the process of this proof.

On page 126, de Savonne discussed the “Inverse Rule of Three” with an example.

On pages 272 and 273, the “Rule of Company [Partnership]” is discussed. The rule usually involves the distribution of funds among business partners, but here the situation involves two partners and a factor or business agent.

You can also view three images of the 1571 version of this work.

The images above are presented courtesy of the Thomas Fisher Rare Book Library, University of Toronto, and are available via Internet Archive.

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Arithmetic of Pierre de Savonne," Convergence (July 2016)