Sept. 29. 1855.] 



NOTES AND QUERIES. 



237 



LOKTDON. SATURDAY, SEPTEMBER 29. 1855. 



ARITHMETICAL NOTES. NO. XT. 



The arithmetical puzzles given by Bachet, in 

 his work described in my last, consist mostly of 

 ways of finding out a number thought of, of which 

 there are plenty of examples in Hutton's Recrea- 

 tions. Tliere is also the well-known puzzle of the 

 three jealous husbands, their wives, and the boat 

 which will only take two. Passing over these, I 

 mention the equally well-known problem of the 

 fifteen Christians and fifteen Turks, which de- 

 serves notice as the earliest of this class of riddles. 



The original story is that Josephus, after the 

 siege of Jotapata in Galilee, saved himself with 

 certain companions in a cave. Despairing of final 

 safety, the refugees had recourse to the proceed- 

 ing afterwards adopted by the snakes in- Ireland, 

 according to the ballad of St. Patrick, of saving 

 themselves from slaughter by suicide. Josephus 

 contrived that the lot should fall last upon him- 

 self, and persuaded the last but one to join him in 

 living on. Hegesippus (or the author of the book 

 called by his name) coined or received the story 

 that the artifice was of the kind afterwards em- 

 ployed in the problem of the Turks and Christians. 

 The change of characters was necessary to spice 

 the question : the idea of Christians jockeying 

 Turks out of their lives by an unfair manoeuvre 

 edified our ancestors greatly. 



Bachet gives the question as he says it had 

 commonly been given. He seems to hint at some 

 previous works of the same kind as his own ; but 

 I have never met with them. If any of your 

 readers should find out works of arithmetical re- 

 creation of the sixteenth century — that is, works 

 expressly devoted to puzzles and difficulties — I 

 hope they will communicate them to you. 



The problem is as follows : — Fifteen Christians 

 and as many Turks, being in a storm at sea, find 

 it necessary to lighten the vessel by throwing half 

 the crew overboard. It is agreed that they shall 

 stand in a row, and that every ninth, beginning 

 again when the row is ended, shall be thrown over. 

 The question is how to manaire their position, so 

 that the lot shall fall only on Turks. The arrange- 

 ment is as follows: four Christians, five Turks, 

 two Christians, &c., as thus abbreviated : 

 4C 5T 2C T 3C T C 2T 2C 3T C 2T 20 T. 

 Allowing the vowels a, e, i, o, u, to stand for 1, 2, 

 3, 4, 5, the arrangement was indicated by Bachet 

 in the vowels of the following lines : 



" Mort, tu ne falliras pas 

 En me livrant le trespas." 



It is worth notice that, though Bachet's printer 

 uses u for v in the word livrant, the author does 

 No. 309.] 



not make any remark : he seems to think nobody 

 will be led into confusion. 



Subsequently, the vowels were fitted with con- 

 sonants in the following way : 



" Populeam virgam mater regina ferebat." 

 When the Turks ceased to be the bugbear of 

 Christendom, and the lesson ceased to be practical, 

 another version of the story was adopted, suggested 

 perhaps by the Latin line. I take it from the 

 Arithmetique Demontree of C. F. Gaignat Del'- 

 auln'ais, Paris, 1770, 8vo. An English ship has 

 been sunk by a French one, and fifteen of the 

 crew saved : the victors are also reduced to fifteen. 

 Provisions failing, the French captain says : 



" Mes enfans, afin que la moiti^ de nous vive quelque 

 tems, apr^s avoir consult^ Dieu et la Sainte Vierge, j'ai 

 juge a propos que I'autre moiti^ soit jete k la mer." 



The rest as before, and the heretics disposed of. 



Another problem of Bachet is that of magic 

 squares. Your readers are all aware that a square 

 of numbers is so called, when it gives the same sum 

 from any row, any column, or from the diagonals. 

 Thus, one of the ways of forming a magic square 

 from the first twenty-five numbers, is as follows : 

 11 24 7 20 3 



12 25 



8 16 



17 

 10 



6 



18 



13 

 1 



21 

 14 



9 



22 



23 6 19 2 15 



which gives 65 in every row, in every column, 

 and in the diagonals, or diameters, as they were 

 called in this subject. Bachet rejects the mystical 

 meaning of these squares ; but as a word on this 

 subject may be interesting to the readers of 

 Walter Scott, I give it. 



Writers of fiction are very often inaccurate in 

 their technicalities, and most frequently perhaps 

 in those of arithmetic or other mathematics. The 

 most amusing instance I ever met with, is in Mr. 

 Warren's story of the martyr-philosopher, lately 

 alluded to by one of your correspondents. The 

 philosopher tells his guests that he had suspected 

 an error in Laplace, and then goes on thus : 



" Only look at the quantity of evidence that was neces- 

 sary to convince that I was a simpleton by the side of 

 Laplace — pointing to two or three sheets of paper cram- 

 med with small algebraical characters in pencil — a fear- 

 ful array of symbols : 



V— 3«2, n?C+9— n=9, nxlog. e, 



and sines, cosines, series, &c., without end." 

 Certainly a philosopher who needed pages of these 

 symbols to convince himself that he was a simple- 

 ton by the side of Laplace, would really be a 

 simpleton by the side of any junior optime. For 

 a parody without caricature, on the supposition 



