Aug. 18. 1855.] 



NOTES AND QUERIES. 



117 



LONDON. SATURDAY, AUGUST 18, 1856. 



ARITHMETICAL NOTES, NO. III. 



I find among my books the fourth edition of 

 Van Etten's Recreations (Vol. xi., pp. 459. 504. 

 516.), Paris, 1627, 8vo. At this rate the work 

 started with yearly editions, so that it is odd that 

 the edition of 1660 (Vol. xi., p. 459.) should only . 

 be called the fifth. By an old note I find that 

 Brunet also attributes the authorship to Leurichon 

 ^ol. xi., p. 516.). It appears that Henrion, said 

 to have been the first French translator of Euclid^ 

 very soon took up these Recreations. This fourth 

 edition is marked D. H. P. E. M., meaning Denis 

 Henrion, Prof esseur es Mathematiques, or Philo- 

 sophe et Mathematicien. The earlier editions seem 

 to contain some foolish things which do not appear 

 in the English translations, and were probably 

 struck out of later editions. For example, what 

 would happen if the stars should fall ? You will 

 tell us, says the author, that we should catch 

 plenty of larks, and the ancient Gauls used to say 

 that this was the only thing of which they stood 

 jin awe. But If very great heat, or other adverse 

 influence, should not interfere, a mathematician 

 might venture to say that since the stars and the 

 earth are round, a star would touch the earth only 

 in a point, and then those who were not near that 

 point would be in no danger, &c. 



I had intended in the next number of these 

 Notes, to give some little account of the work 

 which really suggested these Recreations, a work 

 of some importance in the history of mathematics. 

 Claude Gaspar Bachet de Mezlriac, the author 

 (died 1638), an account of whom is given in the 

 supplement to Moreri, and in Bayle, published 

 several literary works, and two of a mathematical 

 character. His edition of Diophantus, Paris, 

 1621, folio (Gr. Lat.), is the first print of the 

 Greek text, and is beautifully printed, but loaded 

 with those unfortunate contractions which in print- 

 ing are no contractions at all. Bachet had ac- 

 cordingly been a reader of the manuscripts of 

 Diophantus ; and there is one account, if not 

 more, of some of the manuscripts containing com- 

 mentator's allusions to the Indian algebra, though 

 it must also be said that these manuscripts have not 

 since been found. I mention this because we 

 .shall presently see that Bachet produced and 

 .printed one of the most remarkable points of the 

 Indian algebra, get it how he might. 



The other work is the Prohlemes plaisans et 

 delectables qui se font par les nombres. This work 

 .'was first published in 1612, when the author, ac- 

 cording to the usual accounts, was only twenty 

 years old. The same accounts state that he joined 

 the Jesuits, intending to become a member of 

 No. 303.] 



their order, at twenty years old. Bayle, however, 

 gives authority for his being the son of a first 

 marriage, the second marriage being made in 

 1586 ; and this is no doubt a more correct state- 

 ment. The first edition of this work is not the 

 remarkable one ; there is a copy in the British 

 Museum ; and both editions are rare. 



The second edition (Lyons, 1624, Svo.) has ad- 

 ditions by the author. One of them is the re- 

 markable piece of Indian algebra of which I 

 have spoken. Algebraists call it the solution of 

 indeterminate equations of the first degree. It is 

 a method of answering such questions as the fol- 

 lowing : — In how many ways can a thousand 

 pounds be paid in five-shilling pieces and seven- 

 shilling pieces ? How may all the ways be de- 

 tected by which one man may pay another thirteen 

 shillings when the first has nothing but five-shilling 

 pieces, and the second nothing but seven-shilling 

 pieces? The mode in which Bachet proceeds is 

 that which the Hindus call the Kuttaka, or pul- 

 verizer, and which the European algebraists now 

 connect with continued fractions. Hence this 

 work Is, for Europe at least, an incunabulum of the 

 theory of numbers. Whether Bachet was an 

 original inventor cannot be directly ascertained. 

 His title-page tells us that the work is partly de- 

 rived and partly original. His method was an- 

 nounced, though not fully given. In the first edition, 

 so that he possessed it before 1612. It is his only 

 claim to great power of original discovery. The 

 case then stands thus : A method is known iu 

 India, where it Is at least as old as the Christian 

 £era. In the sixteenth century Bombelli, whose 

 sufficiency as evidence Is well known, found in 

 the Vatican library a manuscript of a certain 

 Diophantus, with which he and another were so 

 struck that they actually translated five books, 

 intending to publish the whole. In [the notes to] 

 this manuscript he and his comrade found frequent 

 citations from Hindu writers, by which they learnt 

 that algebra was in India before it was In Arabia. 

 But this manuscript has never been found, though, 

 on the other hand, the Vatican library contains a 

 great deal which we do not know to have been 

 closely examined. Add to this that of all the 

 Hindu algebra, the method in question is the part 

 which a commentator on Diophantus would have 

 cited if he had known it. On the other hand, it 

 would be very strange (though by no means without 

 parallel) that Bombelli should have omitted to 

 bring away and publish so remarkable a thing, if 

 he had ever seen it. In the next century Bachet, 

 who had resided at Rome, with the intention of 

 editing Diophantus, which Intention he fulfilled, 

 and who was acquainted with the assertion of Bom- 

 belli, published this Indian method in a work which, 

 a(!Cording to himself. Is partly derived from other 

 writers ; and did nothing else of the same note. 

 This is the case as it now stands ; possibly farther 



