Diophanti Alexandrini Arithmeticorum libri sex, et De numeris multangulis liber vnus. Cum commentariis C. G. Bacheti V. C. & obseruationibus D. P. de Fermat senatoris Tolosani

Claude Gaspard Bachet de Méziriac was a French prolific reader of poetry, history, commentary, scholarship, and mathematics. He is regarded as a poet and early mathematician of the French Academy, best known for his translation in 1621 of Diophantus's Arithmetica.

Background

Claude Gaspard Bachet de Méziriac was born on October 9, 1581, in Bourg-en-Bresse, France. This was a region which frequently allied itself to France, Spain or Italy. Bachet's grandfather, on his father's side, was Pierre Bachet de Meyzétiat who was a counselor to King Henry II of France. Bachet's father Jean Bachet, however, was a counselor to the Duke of Savoy while his mother Marie de Chavanes was a noblewoman. Savoy had been occupied a number of times by French troops but the dukes of Savoy were able to recover the territory. Twenty years before Bachet was born, Emmanuel Philibert duke of Savoy moved the capital of Savoy to Turin and made Italian the official language (before this it had been Latin).

Education

Orphaned at the age of six, the precocious Claude-Gaspar received his early education in a house of the Jesuit order that belonged to the duchy of Savoy. Presumably, he also studied in Padua as a young man.

Career

After completing his studies in Padua, Claude Gaspar Bachet may have taught in a Jesuit school in Milan or Como. Bachet also spent a few years in Paris and in Rome, where, with his friend Claude Vaugelas, he composed a great deal of Italian verse.

By the time when the Académie Française was founded in 1634, he was too ill to attend the inaugural ceremony, however, he was made a member of the Académie in the following year. His early literary works, humanist in outlook, consisted of poems in Latin, French, and Italian. Between 1614 and 1628 he published a Latin epistle from the Virgin Mary to her Son, canticles, brief sacred and profane Latin poems, translations of psalms, and a metrical translation of seven of Ovid’s Epistulae heroium. He also published an anthology of French poetry, Délices, and the Epistles of Ovid (1626); the latter assured his reputation as a mythologist.

Prior to Bachet’s translation, only a few scholars had written on the work of Diophantus; Maximus Planudes, who gave an incomplete commentary on the first two books of the Arithmetica (ca. 1300); Raphael Bombelli, who embodied all of the problems of the first four books in his Algebra Wilhelm Holzmann, better known as Xylander, who gave a complete Latin translation (1575); and Simon Stevin, who gave a French translation of the first four books (1585). Bachet’s translation, Diophanti Alexandrini Arithmeticorum libri sex, was based largely on the writings of Bombeli and Xylander, particularly the latter, although he admitted this with reluctance. Indeed, it is the opinion of T.L. Heath that although Bachet generally has been regarded as the only writer to interpret the contributions of Diophantus effectively, perhaps as much - if not more - of the credit is due to Xylander.

He asserted that he corrected many errors in Xylander’s version; filled in numerous omissions, such as proofs of porisms and abstruse theorems that Diophantus merely mentioned; and clarified much of the exposition. He apparently added few original contributions to number theory or Diophantine analysis, however, except for a generalization of the solution of the system ax + v = u2, cx + d = w2. Yet despite its imperfections, Bachet’s work is commendable for being the first and only edition of the Greek text Diophantus. It was subsequently reprinted, with the addition of Fermat’s notes, in 1670; and while Fermat’s notes are significant, the Greek text is inferior to that of the first edition. Bachet’s penchant for arithmetical rather than geometric problems is also obvious in the contents of his Problemes plaisans et delectables.

Of greater mathematical significance was Bachet’s problem of the weights: to determine the least number of weights that would make possible the weighing of any integral number of pounds from one pound to forty pounds, inclusive. Bachet gave two solutions: the series of weights 1, 2, 4, 8, 16, 32; and the series 1, 3, 9, 27 - depending upon whether the weights may be placed in only one scale pan or in each of the two scale pans. Last, there is the celebrated prototype of ferrying problems or difficult crossings, the problem of the three jealous husbands and their wives who wish to cross a river in a boat that can hold no more than two persons, in such a manner as never to leave a woman in the company of a man unless her husband is also present. Eleven crossings are required, but Bachet gave a solution that asserts “II faut qu’ils passent en six fois en cette sorte.” The analogous problem with four married couples cannot be solved; Bachet stated this fact without proof. It should also be noted that Bachet gave a method for constructing magic squares which is essentially that of Moschopulous (ca. 1300), although Bachet appears to have discovered it independently.

He died on February 26, 1638 at the age of 56 in Bourg-en-Bresse and was interred in Eglise Notre Dame Bourg-en-Bresse, Departement de l'Ain, Rhone-Alpes, France.

In his religious affiliation Bachet was a Roman Catholic and for a year in 1601 he was a member of the Jesuit Order.

Views

Bachet was a writer of books on mathematical puzzles and tricks which formed the basis for almost all later books on mathematical recreations. Bachet wrote Problèmes plaisans et delectables qui se font par les nombres (1612), of which there were at least five editions, the last as late as 1959. The emphasis was placed on arithmetical rather than geometrical puzzles. The Problèmes plaisans contains: questions involving number bases other than 10; card tricks; watch-dial puzzles depending on numbering schemes; think-of-a-number problems; river crossing or ferry problems which were similar to ones which Alcuin had given; problems concerning magic squares containing constructions for the squares which had been given by Moschopoulos; the Josephus problem in a similar form to that given by Tartaglia; various weighing problems; and liquid pouring problems.

Personality

Physical Characteristics:
By the 1630s Bachet was suffering rather severe health problems, particularly with rheumatism and gout. Later in life Bachet struggled with this work while suffering from a server fever.

Connections

On the year of 1612 Bachet married Philiberte de Chabeu, by whom he had seven children.