The vowels in these lines, taken in order, indicate alternately the number of Christians and Turks to be placed together; i.e., take o 4 Christians, then u = 5 Turks, then e 2 Christians, etc. Bachet de Méziriac, Tartaglia, and Cardan give each different verses to represent the rule. According to a story related by Hegesippus,1 the famous historian Josephus, the Jew, while in a cave with 40 of his countrymen, who had fled from the conquering Romans at the siege of Jotapata, preserved his life by an artifice like the above. Rather than be taken prisoners, his countrymen resolved to kill one another. Josephus prevailed upon them to proceed by lot and managed it so that he and one companion remained. Both agreed to live.

The problem of the 15 Christians and 15 Turks has been called by Cardan Ludus Joseph, or Joseph's Play. It has been found in a French work of 1484 written by Nicolas Chuquet and in MSS. of the twelfth, eleventh, and tenth centuries. Daniel Adams gives in his arithmetic the following stanza:

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"As I was going to St. Ives,

I met seven wives.

Every wife had seven sacks;
Every sack had seven cats;

Every cat had seven kits :

Kits, cats, sacks, and wives,

How many were going to St. Ives?"

Compare this with Fibonaci's "Seven old women go to Rome," etc., and with the problem in the Ahmes papyrus, and we perceive that of all problems in "mathematical recreations" this is the oldest.

Pleasant and diverting questions were introduced into some English arithmetics of the latter part of the seventeenth and

1 De Bello Judaico, etc., III., Ch. 15. 2 CANTOR, Vol. II., p. 332. 8 M. CURTZE, in Biblio. Mathem., 1894, p. 116 and 1895, pp. 34–36.

of the eighteenth centuries. In Germany this subject found entrance into arithmetic during the sixteenth century. Its aim was to make arithmetic more attractive. In the seventeenth century a considerable number of German books were wholly devoted to this subject.1

1 WILDERMUTH,

ALGEBRA

The Renaissance

ONE of the great steps in the development of algebra during the sixteenth century was the algebraic solution of cubic equations. The honour of this remarkable feat belongs to the Italians.1 The first successful attack upon cubic equations was made by Scipio Ferro (died in 1526), professor of mathematics at Bologna. He solved cubics of the form 23+mx = n, but nothing more is known of his solution than that he taught it to his pupil Floridus in 1505. It was the practice in those days and during centuries afterwards for teachers to keep secret their discoveries or their new methods of treatment, in order that pupils might not acquire this knowledge, except at their own schools, or in order to secure an advantage over rival mathematicians by proposing problems beyond their reach. This practice gave rise to many disputes on the priority of inventions. One of the most famous of these quarrels arose in connection with the discovery of cubics, between Tartaglia and Cardan. In 1530 one Colla proposed to Tartaglia several problems, one leading to the equation a+paq. The latter found an imperfect method of resolving this, made known his success, but kept

1 The geometric solution had been given previously by the Arabs.

his solution secret. This led Ferro's pupil Floridus to proclaim his knowledge of how to solve x3+mx=n. Tartaglia challenged him to a public contest to take place Feb. 22, 1535. Meanwhile he worked hard, attempting to solve other cases of cubic equations, and finally succeeded, ten days before the appointed date, in mastering the case ama+n. At the contest each man proposed 30 problems. The one who should be able to solve the greater number within fifty days was to be the victor. Tartaglia solved his rival's problems in two hours; Floridus could not solve any of Tartaglia's. Thenceforth Tartaglia studied cubic equations with a will, and in 1541 he was in possession of a general solution. His fame began to spread throughout Italy. It is curious to see what interest the enlightened public took in contests of this sort. A mathematician was honoured and admired for his ability. Tartaglia declined to make known his method, for it was his aim to write a large work on algebra, of which the solution of cubics should be the crowning feature. But a scholar of Milan, named Hieronimo Cardano (1501-1576), after many solicitations and the most solemn promises of secrecy, ceeded in obtaining from Tartaglia the method. Cardan thereupon inserted it in a mathematical work, the Ars Magna, then in preparation, which he published in 1545. This breach of promise almost drove Tartaglia mad. His first step was to write a history of his invention, but to completely annihilate Cardan, he challenged him and his pupil Ferrari to a contest. Tartaglia excelled in his power of solving problems, but was treated unfairly. The final outcome of all this was that the man to whom we owe the chief contribution to algebra made in the sixteenth century was forgotten, and the discovery in question went by the name of Cardan's solution. Cardan was a good mathematician, but the association of his name with the discovery of the solution of cubics is a

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gross historical error and a great injustice to the genius of Tartaglia.

The success in resolving cubics incited mathematicians to extraordinary efforts toward the solution of equations of higher degrees. The solution of equations of the fourth degree was effected by Cardan's pupil, Lodovico Ferrari. Cardan had the pleasure of publishing this brilliant discovery in the Ars Magna of 1545. Ferrari's solution is sometimes ascribed to Bombelli, who is no more the discoverer of it than Cardan is of the solution called by his name. For the next three centuries algebraists made innumerable attempts to discover algebraic solutions of equations of higher degree than the fourth. It is probably no great exaggeration to say that every ambitious young mathematician sooner or later tried his skill in this direction. At last the suspicion arose that this problem, like the ancient ones of the quadrature of the circle, duplication of the cube, and trisection of an angle, did not admit of the kind of solution sought. To be sure, particular forms of equations of higher degrees could be solved satisfactorily. For instance, if the coefficients are all numbers, some method like that of Vieta, Newton, or Horner, always enables the computor to approximate to the numerical values of the roots. But suppose the coefficients are letters which may stand for any rational quantity, and that no relation is assumed to exist between these coefficients, then the problem assumes more formidable aspects. Finally it occurred to a few mathematicians that it might be worth while to try to prove the impossibility of solving the quintic algebraically; that is, by radicals. Thus, an Italian physician, Paolo Ruffini (1765-1822), printed proofs of their insolvability,' but these proofs were declared inconclusive by his countryman Malfatti.

1 See H. BURKHARDT, "Die Anfänge der Gruppentheorie und Paolo Ruffini" in Zeitschr. f. Math. u. Physik, Suppl., 1892,