This concurrent history of Arabic writers is also further confirmed by the fact that the Arabians wrote their figures from left to right, after the manner of the Hindus, but contrary to the order of their writing, which was from right to left. Under the reign of Almamun, Ptolemy's great work on astronomy and arithmetic was translated into Arabic, with the title of Almagest, a word formed from the Arabic article, and one of the words of oúvražis μeyiorn, the title of Ptolemy's work. The arithmetics of Diophantus were translated into Arabic by Buzjani, in the fourth century of the Hegira, nearly two centuries after the Arabians had become acquainted with the arithmetic and the astronomy of the Hindus. Among the writers subsequent to Mohammed Ben Musa was Abulfaraj, the author of a treatise on computation. He lived in the twelfth century, and notices a work on numerical computation which Mohammed Ben Musa amplified, and is described as a most expeditious and concise method, and testifies the ingenuity and acuteness of the Hindus." 66 Before the end of the eleventh century the Saracens had extended their conquests along the northern parts of Africa, and at an earlier period had established a flourishing kingdom in the southern provinces of Spain, which existed for upwards of seven centuries, until the reign of Ferdinand and Isabella in 1491, when Granada was taken, and the Saracen power in Spain came to an end. During the rule of the Saracens the arts and sciences of the East were cultivated and promoted, and the schools of the learned in Spain were in high repute in those early times. In the latter part of the tenth century, Gerbert, a Benedictine monk, of Aurillac in Auvergne, is reported to have travelled into Spain, and there to have acquired a knowledge of the sciences of the Saracens, and also of the Arabic numerals. He was without doubt one of the remarkable men of his age, but how far he promoted or assisted in the extension of the knowledge of the Arabic arithmetic is not made out satisfactorily. William of Malmesbury writes (De Gestis Anglorum) of Gerbert: "Abacum certe primus a Saracenis rapiens, regulas dedit, quæ a sudantibus abacistis vix intelliguntur.' Gerbert afterwards became Archbishop of Rheims and of Ravenna, and in A.D. 999 was raised to the Popedom under the title of Sylvester II. He died in the year 1003. It is not improbable that other persons in their travels, both in Spain and in the East, acquired a knowledge of the Arabic numerals long before the use of them became general in the west of Europe. The intercourse of merchants in traffic, and of the hosts which, from the west of Europe, joined in the expeditions to the Holy Land during the Crusades, may also have afforded opportunities of gaining some acquaintance with the arts and knowledge of the Saracens. The Arabic numerals are said to have been found in manuscripts of Spain of the eleventh and twelfth centuries. It is probable that much interesting information might be brought to light from any remains of the manuscript literature of the Saracens in Spain during these centuries. The Arabic numerals were certainly employed in the astronomical tables made by Alphonsus X., King of Castile, about A.D. 1252. Leonardo of Pisa first made known from the Arabians the Hindu arithmetic and algebra in Italy. A manuscript of Leonardo's treatise bearing the title of "Liber Abbaci compositus a Leonardo filio Bonacci Pisano in anno 1202;" and a transcript of another treatise entitled "Leonardi Pisani de filiis, Bonacci . . . Practica Geometria composita anno 1220, were found about 1750 by Targioni Tozzetti in the Magliabecchian Library at Florence, of which he had the care. In his preface to the Liber Abbaci, Leonardo relates that he had travelled into Egypt, Barbary, Syria, Greece, and Sicily. During his youth, at Bugia, in Barbary, where his father was scribe at the custom house for the merchants of Pisa who resorted thither, he there learned the Indian method of counting by nine figures. He states it to be more commodious than the methods used in other countries which he had visited; he therefore prosecuted the study, and, with some additions of his own, and some things taken from Euclid's Elements, he undertook the composition of his treatise that "the Latin race might no longer be found deficient in the complete knowledge of that method of computation." In the epistle prefixed to the revision of his Liber Abbaci in 1228, he professes to have taught the complete doctrine of numbers according to the Indian method. The study of the Indian method of computation through the medium of Arabic in an African city having been introduced into Italy, the Italians were the first European people who cultivated this and its kindred sciences. After the introduction of this new knowledge into Italy, numerous treatises were composed, and manuscript copies of the works of that age are found in the libraries of Italy and other parts of Europe. Villani, the earliest Florentine historian, writes of Paoli di Dagomari, who died about 1350, as a great geometer and most skilful arithmetician, and who surpassed both the ancients and moderns in the knowledge of equations. And Raffaelo Caracci, a Florentine arithmetician of the same century, wrote a work entitled "Ragionamento di Algebra," in which he speaks of Guglielmo di Lunis, who before his time had translated a treatise on Algebra from Arabic into Italian. This was most probably a translation of the Algebra of Mohammed Ben Musa, which Bombelli some years after spoke of as if it were well known in Italy. Matthew Paris writes of John de Basingstoke:-"This Master John, moreover, brought into England the Greek numerical characters, and explained to his friends the knowledge and meaning of them. And this is chiefly remarkable of them, that every number is represented by a single character, which is not the case in the Roman numerals or in Algorithm."1 It may be remarked that the word algorithm, as well as the word algebra, were, at that time, both new words from the Arabic, the former word being applied to the science of arithmetic with the Arabic numerals, and the latter to the generalised science of number. John de Basingstoke was advanced to the Archdeaconry of Leicester by Robert Grosstete, Bishop of Lincoln, who was at that time a zealous promoter of Greek learning and of the sciences. He was 1 Hic insuper magister Joannes figuras Græcorum numerales, et earum notitian et significationes in Angliam portavit, et familiaribus suis declaravit. Per quas figuras etiam literæ repræsentantur. De quibus figuris hoc maxime admirandum, quod unica figura quilibet numerus repræsentatur; quod non est in Latino, vel in Algorismo.-Matthew Paris. the author of a work, "De Computo Ecclesiastico," and of the "Kalendarium Lincolniense," which latter was long held in high estimation. Copies of it still exist, of which only the Latin manuscript copies contain the Arabic numerals. After he was made bishop, Grosstete's firmness in resisting the encroachments of the papacy drew down upon him the censures of the Pope, but the Pope's censures had not the effect of inducing him to alter the line of conduct he deemed it his duty to his sovereign to adopt in the administration of his diocese. He died in the year 1253, at Buckden, the year before the death of John of Basingstoke. Johannes de Sacro Bosco, or John of Holywood or of Halifax as he is sometimes called, studied at Oxford. He was the author of a treatise on the Sphere, and of a tract, "De Arte Numerandi,"1 both of which were celebrated works. This tract of Sacro Bosco gives the Arabic numerals, explains the local value, and gives the rules for arithmetical operations, including the rules for the square and cube roots. He was also the author of a work entitled "De Computo Ecclesiastico." His death took place at Paris, A.D. 1256. Contemporary with Matthew Paris, Sacro Bosco, and Robert Grosstete, was Roger Bacon, a native of Ilchester, who was born about 1214, and died in 1292. He was one of the great men that held forth the light of truth in a dark age, only a few years after the kingdom of England had endured the degradation of a Papal Interdict of the third Pope Innocent. In his work entitled "Opus Majus" he highly commends the sciences, and that of number among them. He employs the word algoristicus several times, and repeats the names of the same set of rules as are given in the treatise of Sacro Bosco. The following extract from the first chapter of the "Opus Majus" can scarcely fail of being interesting to the student. "There are four principal stumbling-blocks in the way of arriving at truth-authority, confirmed habit, appearances as they present themselves to the vulgar, and concealment of ignorance under the ostentation of 1 In 1839 a small volume (pp. 120) was printed by Mr. Halliwell, with the title of "Rara Mathematica. In the collection will be found five tracts on the Arabic numerals. Of these the most important is the treatise of Sacro Bosco, the text of which he states is taken from a manuscript he purchased at the sale of the library of the Abbate Canonici at Venice. There is another treatise entitled "Carmen de Algorismo," in hexameter verse, containing 255 lines. It appears that Alexander de Villa Dei was the author, and that he lived in the fourteenth century. The manuscript copies existing of this poem are very numerous, from which it may be inferred that it was highly valued and extensively read. There are manuscript copies of this treatise preserved both in the University Library and in the Library of Trinity College, Cambridge. Mr. Halliwell quotes the following lines, which he has appended as a note, to the twenty-sixth line of the Carmen de Algorismo : En argorisme devon prendre Vii especes Doubloison mediacion Monteploie et division Et de radix enstracion A chez vii especes savoir Doit chascun en memoire avoir These lines are probably as old as the time of Roger Bacon. 2 Shakespeare saw the same causes at work in his day, and has left the record of his opinion of them in one of his sonnets in these words : "And Art made tongue-tied by Authority, knowledge. The authority I mean here is that which many have violently usurped of their own self-will and with the lust of power, and to which the ignorant vulgar have yielded, to their own ruin, by the just judgment of God (in pernicionem propriam judicio Dei iusto). Now, where these obstructions exist, no reason can move, no judge decide, no law bind; right has no place, the dictates of nature no force; vice flourishes, virtue fades; truth expires, and falsehood rules supreme. Even if the first three can be got over by some great effort of reason, the fourth remains. Men presume to teach before they have learned, and fall into so many errors both in science and common life, that we see a thousand falsehoods for one truth. And this being the case, we must examine most strictly the opinions of our predecessors, that we may add what is lacking in them, and correct what is erroneous, but with all modesty and allowance. We must with all our strength prefer reason to custom, and the judgments of the wise and good to the opinions of the vulgar; and we must not use the triple argument-it is established-it is customary-it is common,,—and therefore it is to be retained, whether in opposition to, or in accordance with the dictates of truth and reason." who held and expressed such opinions was a very dangerous person; and accordingly he was imprisoned, his works forbidden to be read, and his lectures prohibited in the University of Oxford. The learned monk, while engaged in his inquiries into the works of nature, and in his experiments in alchemy, was seriously believed to have practised magic, and to have had converse with evil spirits. So gross in that age was the ignorance of the clergy that even Anthony à Wood, the Oxford antiquary, who had no prejudices against the clergy, has stated with respect to their knowledge of geometry, that "they knew no property of the circle but that of keeping out the devil, and thought that the angles of a triangle would wound religion." The man Thomas Bradwardine was born at Hatfield, in Sussex, about the close of the thirteenth century, and received his education at Merton College, Oxford. He was distinguished both as a divine and as a mathematician. He constantly attended Edward the Third during his wars in France, and was most probably present at the battle of Cressy in 1346. His works, "De Arithmetica Practica," and "De Proportionibus," were printed at Paris, the former in 1502, and the latter in 1495. He died in 1349, forty days after his consecration to the see of Canterbury. Mabillon, in his work entitled "De Re Diplomatica," after the examination of above 6000 documents, writes that he found no authentic date in Arabic figures earlier than that of 1355, and that date in the handwriting of Petrarch.1 Geoffrey Chaucer, the poet, who died A.D. 1400, calls, in one of his poems, the Arabic numerals "the figures newe." He had visited Italy, where he would have learned that the science of number was 1 In the "Journal of the Archæological Institute," vol. vii., p. 85, is a fac-simile of a date in a public document of 19 Edw. II., 1325 A.D., in which the date of the year is expressed in one part in Roman numerals, and in another in Arabic. The document is a warrant from Hugh le Despenser to Bonefez de Peruche and his partners, merchants of a company, to pay forty pounds, dated February 4, 19 Edw. II. In a different hand on the dorse, is a memorandum of the payment, written by one of the Italian merchants to whom the warrant was addressed. there more cultivated than in England. At Florence, Venice, and other cities of Italy, both literature and the arts flourished in the thirteenth and fourteenth centuries, and the science of the arithmetic of commerce was both cultivated and improved by their extensive trade with other countries. He wrote a treatise on the Astrolabe in English for the use of his son. This is the earliest treatise in the English language written on any scientific subject. It has lately been edited by Mr. Skeat, of Christ's College, to which he has prefixed a very interesting preface. In pages xliii.-xlvi. will be found a table' of the fixed stars, copied from a manuscript which bears the date of 1223 in the Arabian characters, a date prior to that of Petrarch by more than a century. The character of the writing, however, is very much like that of manuscripts written in the fourteenth century. If these characters are really copied as written in the original manuscript, they constitute the earliest date as yet discovered in these characters. They do not appear in the dates of the works of Caxton. In "The Myrrour or Ymage of the World," however, printed in 1480, where, treating of Arsmetryke, or Algorithm, among other sciences, he has given a woodcut of an arithmetician sitting before a desk, on which are tablets or papers marked with the nine figures. At St. Albans "The Myrrour of the World" was reprinted in 1506, in which the Arabic figures appear under the forms now in use. The ancient calendars of the fourteenth and the early part of the fifteenth centuries, written before the invention of printing, supply some evidence of the manner in which a knowledge of the Arabic notation was generally made known both in England and in the countries of Europe. Copies of these calendars are found in almost all the 1 "Tabula stellarum fixarum que ponuntur in Astrolabio certificata ad civitatem parisius cuius latitudo est 48 gradus et 30 minuta. In anno domini nostri Iesu Christi 1223."-Pref. p. xliii.—[MS. Camb. Univ. Lib. Hh., 6. 8, fol. 236.] : 2 In Archbishop Parker's manuscript library, preserved in Corpus Christi College, Cambridge, there is a table of eclipses from 1330 to 1340, to which is subjoined a table, in three columns, containing the Roman and Arabic numerals, and another nearly the same as the Roman, but the characters different. The following explanation is subjoined :—" Omnis numerus vel omnis figura in algorismo primo loco se ipsum significat; secundo loco, decies se ipsum significat; tertio loco, centies se; quarto loco, milesies se; quinto loco, decies milesies se; sexto loco, centies milesies se; septimo loco, mille milesies se; octavo loco, decies mille milesies se; nono loco, centies mille milesies se ; decimo loco, mille milesies milesies se. Et sic multiplicando per decem centum et mille usque in infinitum computando versus sinistram." The calendar of John Somers, of Oxford, written in 1380, was one of the most popular of the time, and the copies in general have this addition :-" -"Tabula docens algorismum legere, cujus utilitas est in brevi satis spatio numerum magnum comprehendere. Et quin numeri in Kalendario positi vix excedunt sexaginta, ultra illam summam non est protensa." There is a copy of this calendar in the British Museum, and several English translations among the manuscripts in the Ashmolean Library at Oxford. Mr. Halliwell states in his "Rara Mathematica," that, in the year 1812, a small octavo volume was published at Hackney, containing an account of an almanack for the year 1386, probably one of the oldest in English. At this time the Indian notation appears to have been imperfectly understood, if one may judge from the mixture of Roman and Indian notations in numbers consisting of more than two figures; as 52,220 is written thus-52 MCC 20. There is a calendar preserved in the British Museum of about the date of 1403, which contains the numerals in the form they usually appear before the end of the fifteenth century. There is another described in the "Archæologia," vol. xiii., p. 153, which |