[261] Daventria Illustrata, p. 35.
[262] Lambinet.
Physical sciences in middle ages. 29. We have in the last chapter made no mention of the physical sciences, because little was to be said, and it seemed expedient to avoid breaking the subject into unnecessary divisions. It is well known that Europe had more obligations to the Saracens in this, than in any other province of research. They indeed had borrowed much from Greece, and much from India; but it was through their language that it came into use among the nations of the west. Gerbert, near the end of the tenth century, was the first who, by travelling into Spain, learned something of Arabian science. A common literary tradition ascribes to him the introduction of their numerals, and of the arithmetic founded on them, into Europe. |Arabian numerals and method.| This has been disputed, and again re-asserted, in modern times.[263] It is sufficient to say here, that only a very unreasonable scepticism has questioned the use of Arabic numerals in calculation during the thirteenth century; the positive evidence on this side cannot be affected by the notorious fact, that they were not employed in legal instruments, or in ordinary accounts; such an argument, indeed, would be equally good in comparatively modern times. These numerals are found, according to Andrès, in Spanish manuscripts of the twelfth century; and, according both to him and Cossali, who speak from actual inspection, in the treatise of arithmetic and algebra by Leonard Fibonacci Pisa, written in 1202.[264] This has never been printed. It is by far our earliest testimony to the knowledge of algebra in Europe; but Leonard owns that he learned it among the Saracens. “This author appears,” says Hutton, or rather Cossali, from whom he borrows, “to be well skilled in the various ways of reducing equations to their final simple state by all the usual methods.” His algebra includes the solution of quadratics.
[263] See Andrès, the Archæologia, vol. viii., and the Encyclopædias, Britannic and Metropolitan, on one side, against Gerbert; Montucla, i. 502, and Kästner, Geschichte der Mathematik, i. 35, and ii. 695, in his favour. The latter relies on a well-known passage in William of Malmsbury concerning Gerbert: Abacum certe primus a Saracenis rapiens, regulas dedit, quæ a sudantibus abacistis vix intelliguntur; upon several expressions in his writings, and upon a manuscript of his geometry, seen and mentioned by Pez, who refers it to the twelfth century, in which Arabic numerals are introduced. It is answered, that the language of Malmsbury is indefinite, that Gerbert’s own expressions are equally so, and that the copyist of the manuscript may have inserted the cyphers.
It is evident that the use of the numeral signs does not of itself imply an acquaintance with the Arabic calculation, though it was a necessary step to it. Signs bearing some resemblance to these (too great for accident) are found in MSS. of Boethius, and are published by Montucla, (vol. i. planch. ii.) In one MS. they appear with names written over each of them, not Greek, or Latin, or Arabic, or in any known language. These singular names, and nearly the same forms, are found also in a manuscript well deserving of notice,—No. 343 of the Arundel MSS., in the British Museum, and which is said to have belonged to a convent at Mentz. This has been referred by some competent judges to the twelfth, and by others to the very beginning of the thirteenth century. It purports to be an introduction to the art of multiplying and dividing numbers; quicquid ab abacistis excerpere potui, compendiose collegi. The author uses nine digits, but none for ten, or zero, as is also the case in the MS. of Boethius. Sunt vero integri novem sufficientes ad infinitam multiplicationem, quorum nomina singulis sunt superjecta. A gentleman of the British Museum, who had the kindness, at my request, to give his attention to this hitherto unknown evidence in the controversy, is of opinion that the rudiments, at the very least, of our numeration are indicated in it, and that the author comes within one step of our present system, which is no other than supplying an additional character for zero. His ignorance of this character renders his process circuitous, as it does not contain the principle of juxtaposition for the purpose of summing; but it does contain the still more essential principle, a decuple increase of value for the same sign, in a progressive series of location from right to left. I shall be gratified if this slight notice should cause the treatise, which is very short, to be published, or more fully explained.
[264] Montucla, whom several other writers have followed, erroneously places this work in the beginning of the fifteenth century.
Proofs of them in thirteenth century. 30. In the thirteenth century, we find Arabian numerals employed in the tables of Alfonso X., king of Castile, published about 1252. They are said to appear also in the Treatise of the Sphere, by John de Sacro Bosco, probably about twenty years earlier; and there is an unpublished treatise, De Algorismo, ascribed to him, which treats expressly of this subject.[265] Algorismus was the proper name for the Arabic notation and method of reckoning. Matthew Paris, after informing us that John Basing first made Greek numeral figures known in England, observes, that in these any number may be represented by a single figure, which is not the case “in Latin nor in Algorism.”[266] It is obvious that in some few numbers only this is true of the Greek; but the passage certainly implies an acquaintance with that notation, which had obtained the name of Algorism. It cannot, therefore, be questioned that Roger Bacon knew these figures; yet he has, I apprehend, never mentioned them in his writings: for a calendar, bearing the date 1292, which has been blunderingly ascribed to him, is expressly declared to have been framed at Toledo. In the year 1282, we find a single Arabic figure 3 inserted in a public record; not only the first indisputable instance of their employment in England, but the only one of their appearance in so solemn an instrument.[267] But I have been informed that they have been found in some private documents before the end of the century. In the following age, though they were still by no means in common use among accountants, nor did they begin to be so till much later, there can be no doubt that mathematicians were thoroughly conversant with them, and instances of their employment in other writings may be adduced.[268]
[265] Several copies of this treatise are in the British Museum. Montucla has erroneously said that this arithmetic of Sacro Bosco is written in verse. Wallis, his authority, informs us only that some verses, two of which he quotes, are subjoined to the treatise. This is not the case in the manuscripts I have seen. I should add, that only one of them bears the name of Sacro Bosco, and that in a later handwriting.
[266] Hic insuper magister Joannes figuras Græcorum numerales, et earum notitiam et significationes in Angliam portavit, et familiaribus suia declaravit. Per quas figuras etiam literæ repræsentantur. De quibus figuris hoc maxime admirandum, quod unica figura quilibet numerus representatur; quod non est in Latino, vel in Algorismo. Matt. Paris, A.D. 1252, p. 721.
[267] Parliamentary Writs, i. 232, edited under the Record Commission by Sir Francis Palgrave. It was probably inserted for want of room, not enough having been left for the word IIIum. It will not be detected with ease, even by the help of this reference.