His Propositiones ?lrithmeticx ad acuendos juvenes, con tain various ways of discovering a number that may have been thought of by another person, after it has been subjected to certain arithmetical operations, but of which only the result is given. The questions, of trans porting the three jealous husbands with their wives across a river, in a boat which can only carry two persons at a time, so that no woman should be in the company of any of the men unless her own husband was present ; of di viding equally among three persons twenty-one casks, of which seven are full, seven half full, and seven empty ; and others of a similar nature, were first proposed and solved by Alcuin. That work probably suggested to Bachet the first idea of his treatise, entitled, Proble'mes plaisans et dclectables qui se font par les nombres. A superb edition of the works of Alcuin was given in 1777, by the learned abbot of St Emeran.
Alcuin may be said to have given a new xra to lite rature and science. His royal pupil founded, by his persuasion, the universities of Paris and Pavia, the first institutions of this kind in Europe. Whatever effect universities have upon learning at present, it must be acknowledged, that they afforded an asylum to science during the dark ages, and were the means of transmit ting it to more enlightened times.
Though we are indebted to the Arabs for our present system of arithmetic, that people do not pretend to have been its inventors ; but, on the contrary, confess with candour, that they received it originally from India. Accordingly, manuscript copies of Arabian treatises on arithmetic are to be found in several libraries, entitled, The art of calculating according to the method of the In dians ; Of the Indian Calculation, Scc.; and one of these, in the library of Leyden, is said to employ symbols of computation, which have a striking resemblance to the arithmetical characters of Planudes. Alsephadi, in his commentary on a poem of Tograi, a work that is held in high estimation by the Arabs, mentions three things, which, in his opinion, are highly creditable to the In dians: Golaila -oe damma, a kind of moral tales ; their method of calculation ; and the Game of Chess. The authority of Aben-Ragel, an Arabian writer of the thir teenth century, ought to be of great weight in determin ing this point: He affirms, in the preface to his treatise on Astronomy, a copy of which has been preserved in the library of Leyden, that the invention of the notation in question is clue to the philosophers of India. Plaint des, a Greek writer, who lived about the same time, confirms the testimony of Aben-Ragel. Manuscripts of a work of Planudes, entitled, A ..6y,G-Tor.n '40:4, or NI-Eppo va xa7ct IveNs, still exist. The system of numeration which he explains is essentially the same with our own ; and though the characters are somewhat different, they are very like those of Alsephadi. (See Plate XXXI. Fig. 1.) After giving the nine characters, which he observes are all Indian, he adds, there is still another called Te,pece,* which is expressed by 0, and denotes nothing.
If it were necessary to bring forward an additional proof, that the Arabian arithmetic is of Indian origin, it would be sufficient to state, that it was generally be lieved to be so, when it was introduced into Europe. Joannes de Sacro Bosco, a native of Halifax, who com posed a treatise on arithmetic in Latin verse, about the beginning of the thirteenth century, mentions it as an opinion of which no doubt was entertained : Hoc algorithmus ars przesens dicitur, in qua Talibus Indorum fruimur bis quiuque figuris.
Notwithstanding this abundant evidence of the Indian origin of our arithmetic, and for which we are chiefly indebted to Montucla, that opinion is by no means uni versally admitted, but has been controverted by writers of the greatest respectability. Vossius, among others,
pretends, that the Indians borrowed this notation from the Arabs ; and that the Arabs, in like manner, borrow ed it at a more remote period from the Greeks. This assertion, however, is neither supported by the testimony of history, nor entitled to much credit as a critical con jecture. His sole argument is drawn from the form of the figures, which are said to be imitations of the first nine letters of the Greek alphabet ; but, as Montucla well observes, they have so little resemblance to their supposed originals, that they must have suffered greatly in the tansformation. The most distant likeness, indeed, can with difficulty be traced between any two of them, and none whatever, if we take them in order, except between el and 8. A fanciful imagination, by adding one part to a letter, taking away another from it, and distor ting the position of the whole, may certainly force out something like a general resemblance ; and, in the same manner, may resemblances be traced between objects the most unlike, provided we are equally regardless of probability, and merely anxious about establishing some imagined similarity. But no unprejudiced mind can, for a moment, admit, that such a likeness exists between the Arabian figures, and the first nine letters of the Greek alphabet, as to warrant even a conjecture, that the former were copied from the latter. Bossut dismisses the controversy in a very summary manner :---" With out entering upon this frivolous discussion," says he, " I shall satisfy myself with observing, that we are indebted immediately to the Arabs for our present system of arithmetic," (Essai sur l'Hise. de Mathem. Periode IT. chap. 1.) This remark of Bossut must be considered as one of those modes of speech into which authors are sometimes betrayed, when they wish to assume airs of superior discernment, and get rid, at the same time, of a painful and laborious investigation. To us, at least, we confess the proofs which Montucla has adduced are quite decisive, and demonstrate, beyond all doubt, that the Arabs obtained their arithmetic from the East. The Indians have a tradition, that numbers and geometry were transmitted to them by a people who inhabited the northern parts of Tartary. M. Bailly supposes, that this people was none other than the Atlantides of Plato ; but no vestige of the arts has been found in that part of Asia which can support this conjecture, or lead us to believe that that quarter of the globe was ever inhabited by a very enlightened race of men.
The first writer who is known to have employed the Arabian algorithm, is Jordanus Nemorarius. He wrote ten books on arithmetic, about the year 1230. Bossut mentions the work in his History of Mathematics ; and Wallis says, that there is a manuscript copy of it in the Savilian library at Oxford ; but we believe it has never been printed. Joan. Ls de S.icro Bosco, whom we alrea dy mentioned, was a contemporar) of Jordanus.
After the introduction of the Arabian notation, the Sexagesimal arithmetic gradually MI into disuse; the Sexagesimal fractions, however, were retained till the middle of the fifteenth century.
About that time, a kind of arithmetical amusement, called magic squares, was invented by Emanuel Mos copulius, which we shall mention, not so much on ac count of their own importance, as of the attention that has been bestowed upon them by mathematicians of the greatest eminence. These magic squares consist of a certain number of square cells, formed by the intersec tion of vertical and horizontal lines, into each of which is inserted a number belonging to the series of an arithme tical progression, and so disposed, that the sum of each of the vrical and horizontal columns, as well as of the iwo diagonal columns, is equal to the same number. See