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important was an Encyclopædia, which, with other subjects, treated of the liberal arts of geometry, arithmetic, music, and astronomy. Bede, styled the Venerable, was the most learned man of his time, as his extant writings clearly prove. He lived in the eighth century, and among his numerous writings is one entitled, “De Temporum Ratione." It begins with an explanation of the mode of numerical computation by means of the hands, the fingers, &c., and gives an account of the Greek and Roman notations for recording numbers. This treatise, as its name imports, contains an account of ecclesiastical chronology, and in some manuscripts it includes the tract, “De Rerum Natura."1

The treatise “ De Numerorum Divisione,” attributed to Bode, clearly belongs to a later period. It will be found in pp. 159-163, book i., of the folio edition of his works printed in 1529. The treatise is one of the works of Gerbert, on the Abacus, and was dedicated by him to his friend Constantinus.

Alcuin, a disciple of the Venerable Bede, was the author of numerous works, chiefly theological. Among his miscellaneous writings will be found an elementary introduction to the different sciences, and a tract entitled, “Propositiones Arithmeticæ (LIII.] ad acuendos juvenes," which has been, without evidence, included in the works of Bede.

In the middle ages, the Councils of the Roman Church, in their decrees, directed bishops in their pastorals to recommend to their clergy the study of sacred chronology, and that in every monastery there should be one person at least who understood the order of the festivals of the Church as determined by the Calendar of the current years. In this way computation always maintained a small place in

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1 “Bede's work, entitled, 'De Rerum Natura,' has two great merits; it assembles into one focus the wisest opinions of the ancients on the subjects he discusses, and it continually refers the phenomena of nature to natural causes. The imperfect state of knowledge prevented him from discerning the true natural cause of many things, but the principle of referring the events and appearances of nature to its own laws and agencies, displays a mind of sound philosophical tendency, which was calculated to lead his countrymen to a just mode of thinking on these subjects. Although to teach that thunder and lightning were the collisions of the clouds, and that earthquakes were the effect of winds rushing through the spongy caverns of the earth, were erroneous deductions, yet they were light itself compared with the superstitions which other nations have attached to these phenomena. Such theories directed the mind into the right path of reasoning, though the correct series of the connected events and the operating laws had not then become known. The works of Bede supply evidence that the establishment of the Teutonic nations in the Roman empire did not barbarise knowledge. He collected and taught more natural truths, with fewer errors, than any Roman book on the same subjects had accomplished. Thus. his work displays an advance, not a retrogradation, of human knowledge ; and from its judicious selection and concentration of the best natural philosophy of the Roman empire, it does high credit to the Anglo-Saxon good sense.”-Sharon Turner's, History of the Anglo-Sacons, vol. iii., p. 403.

the education of the clerical order. In a manuscript of the date A.D. 944, we are told that reckoning was seldom done on paper, but more frequently on the fingers, according to the ancient custom. There is no mention even of the abacus nor of the counters.

In the eyes of Alcuin, and many others of the priestly order, theology gave to the science of number its real importance, in that it was supposed to throw light on the symbolical numbers found in Holy Writ. He considers the number 6 the number of God, who created all things, because it is a perfect number, or the sum of its aliquot parts; but the number 8 is an imperfect number, the sum of the aliquot parts of it, 1, 2, 4, being equal to 7, a number less than 8. From this number 8 sprung the second origin of mankind after the deluge; as we read that in Noah's ark 8 persons were saved, from whom descended the whole human race, which shows that the second origin of the race was as imperfect as the first that were made after the number 6. A similar use was made of astronomy. By such speculations the learned

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· Nichomachus of Gerasa, who lived about the time of Augustus Cæsar, the first Roman Emperor, was the author of a treatise entitled “Isagoge Arithmetica.” Iam. blicus wrote a commentary on this work, and surpassed, in his visionary specula. tions on numbers, the niost extravagant of his predecessors. Porphyry also wrote on the mysteries of numbers.

The propensity for finding mysteries in numbers, and in their relations and analogies, did not end with the latest of the commentators on the Platonic philosophy, but has descended even to modern times. Several works have appeared since the introduction of the Indian science of numbers, for the express purpose of ex. plaining their mysterious properties.

Meursius published in 1631 his work, which he called “Denarius Pythagoricus.” It contains a collection of the names and properties of the first ten numbers, from the writings of the Pythagorean philosophers.

In the year 1502, a work was published at Leipsic for the use of the students of that University, under the title of “ Heptalogium Virgilii Saltzburgensis,” on the properties of the number seven. The book is divided into seven parts, and each part into seven chapters, and forms a collection of quaint conceits and absurdities respecting the mystical and other properties of the number seven.

But the most extraordinary of these productions is the work of Peter Bungus, published in 1618, under the title of “Numerorum Mysteria.” The work is written in Latin, and contains about 700 quarto pages. It professes to illustrate all numbers and their properties, mathematical, metaphysical, and theological, embracing all the opinions of the Pythagoreans and Platonists, with the addition of speculations on the numbers which incidentally occur in the writings of the Old and New Testaments.

In England, the same propensity to treat on the mysteries of numbers and the advantages of the study, has been exhibited in a work of which the following is a copy of the title-page : "The Secrets of Numbers according to Theological, Arith. metical, Geometrical, and Harmonical Computation. Drawn, for the better part, out of those Ancients, as well as Neoteriques. Pleasing to read, profitable to understande, opening themselves to the capacities of both learned and unlearned; being no other than a key to lead men to any doctrinal knowledge whatsoever. By William Ingpen, Gent. London, 1624."

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doctor sought to make it intelligible, "How pleasant and useful was the knowledge of arithmetical discipline!”

Gerbert was the most remarkable man of his age, and at an early period was devoted to the cloister, where he was brought up as a monk. He left his home to undertake a student's journey into Spain. On his return he became pupil in the convent school of Rheims. Endowed with great abilities and with extensive knowledge, both in the mathematical sciences as well as in classical learning, and being of an ardent character, thirsting for knowledge, he soon found himself distinguished as a scholar and a teacher. In the year A.D. 980 he was appointed Abbot of the Monastery of Bobbio in Lombardy; he was promoted Bishop of Rheims, next of Ravenna, and at length closed his learned and useful life in A.D. 1003, at Rome, where for four years he filled the Papal chair, under the name of Sylvester II. Gerbert, distinguished by the title of “Reparator Studiorum,” reached the highest eminence that was pos. sible in his time. He had entirely made his own the learning of his time in its widest extent, and he knew how to employ it in a manner scarcely known in that age. Gerbert's knowledge of astronomy, his construction of the globe, and his sun-dials, appeared so wonderful to the unlearned and superstitious clergy of his time, that it was reported of him after his death that he had made a compact with the devil.?

Leonardo Bonacci, of Pisa, first made known to his countrymen the Hindu Arithmetic and Algebra from the Arabians. A manuscript of his 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 discovered in the Magliabecchian Library at Florence by Targioni Tozzetti, about the middle of the eighteenth century. In the preface to the Liber Abbaci, Leonardo relates that he had travelled in Egypt and other countries. In 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 Arithmetic. He describes their method 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 composed the treatise which he named Liber Abbaci. In the index to the fifteenth chapter

1 The student is referred for more ample information on the early history of arithmetic and algebra to the learned works which bear the following titles :

Die Algebra der Griechen. Nach den Quellen bearbeitet von Dr. C. H. F. Nesselmann, Privat-docenten an der Universitat zu Konigsberg. Berlin, 1842.

Zur Geschichte der Mathematik, in Alterthum und Mittelalter, von Dr. Hermann Hankel, Weil. Ord. Professor der Math. an der Universitat zu Tubingen, Leipzig, 1874.

of his work he writes :-"Incipit Pars Tertia de solutione quarundam quæstionum secundum modum Algebræ et Almucabalæ, scilicet oppositionis et restaurationis." This work appears to contain the earliest notice of the introduction of the Hindu Arithmetic and Algebra into Italy. Leonardo, the earliest scholar of the Arabian algebraists, is reported by Targioni Tozzetti to have employed the small letters of the alphabet to denote quantities. But it would appear that he only did so because he represented quantities by straight lines, and designated these lines by letters in the elucidation of his Algebraic Solution of Problems. Leonardo and his disciples translated the Arabic word shai by the Latin res and the Italian cosa. The Arabic word mal was rendered by the Latin census and the Italian censo, as terms of the same import; for it was in the acceptation of amount of property or estate (census, quicquid fortunarum quis habet) that census was used by Leonardo. And it appears that the name of the science was derived from the first and second powers of the unknown quantity, “Ars Rei et Census," and sometimes “Regula Rei et Census," the rule of the thing and the product, or the rule of the root and the square, in allusion to equations of the second degree.

Villani, the historian of Florence, writes of Paoli di Dagomari, who died about 1350, as a great geometer and most skilful arithmetician, and who surpassed both the ancients and the moderns in the knowledge of equations, which at that time constituted the most advanced branch of the science. Also, Raffaelo Caracci, of the same century, composed a work entitled “Ragionamento di Algebra," in which he writes of Guglielmo di Lunis, who had translated a treatise on Algebra from Arabic into Italian. This was the work of Mohammed Ben Musa, and was executed not long after the introduction of that science into Italy by Leonardo of Pisa. All these combine to show that the Italians were in possession of the knowledge of the Arabic numerals and the science of Algebra long before the rest of the nations of Europe.

Paciolus, and all the Italian writers on arithmetic and algebra, declare that Leonardo of Pisa was the first who introduced the knowledge of algorithm and algebra to his countrymen. Paciolus appears to have taught these sciences at Venice about 1460, and he specially makes mention of his three predecessors, who had in succession filled the office expressly dedicated to the exposition of these sciences. Their names were Paolo della Pergola, Demetrio Bragadini, and Antonio Cornaro, the last of whom had been his fellow-disciple.

The first Italian writer on the science of Algebra whose treatise was printed in Italy was Lucas Paciolus, or Lucas de Burgo. This work was published in 1494, with the title, “L'Arte Maggiore ditta dal Volgo la Regola de la Cosa over Alghebra e Almucabala.” He adopted the name of L'Arte Maggioreto distinguish it from the science of number, which was then called L'Arte Minore." He generally employs both the terms Algebra and Almucabala, and from the former word devised the adjective Algebratico. He informs his readers that the science is named Algebra wa Almucabala in the Arabic language, and those words are thus explained in the Roman language : Algebra, id est, restauratio; Almucabala, id est, oppositio vel comparatio et solidatio." These two words being only the names of two preliminary operations in the preparation of equations for solution.

The words Algebra and Algorism were introduced at the same time into Europe, and the latter term was always applied to treatises on Arithmetic with the Indian figures. Algorism, or Algorithm, was corrupted into Augrym, and the counters which were used in calculation were called "

augrym stones," as may be read in Chaucer's Miller's Tale.

After the appearance in print of the works of Lucas de Burgo on arithmetic and algebra, the knowledge of these sciences in the following century, not only in Italy, but also in other countries of Europe, engaged the attention of learned men, who promoted the advancement of them and introduced numerous improvements.

It appears from the treatises of the first Italian writers on Algebra that they did not understand how to turn the extension of the science of arithmetic to the purposes for which it has subsequently been found most useful. The application of the different methods of reckoning in common life appeared to them, as a contrast with algebra, an entirely theoretic knowledge. They appear to have held ideas as inadequate as their teachers of the real meaning of the word “equation." Neither they nor the Arabians knew equality to be any other than two or more absolutely positive quantities, and the Italians follow the special rules of their Arabian teachers.

As it has been very aptly described—" they worked with an instrument the use of which they did not fully comprehend, and employed a language which expressed more than they were prepared to understand; a language, under the notion first of negative, and then of imaginary quantities, seemed to involve such mysteries as the accuracy of mathematical science must necessarily refuse to admit.”

The early Italian writers on Algebra introduced some abbreviations of the terms they learned from their Arabian instructors. They employed the initial letters p and m of the words plus and minus to denote a positive and a negative quantity, and used Co and qa for the first and second unknown quantities. They also adopted the characters C, C", C", to denote the first, second, and third powers respectively, and denominated the biquadrate and the higher powers as relato primo, secondo, tertio, &c. They assumed R, the initial letter of radix, de te the square root.

It appears that the Italian mathematicians adopted the same classification of cubic equations as the Arabic writers; and that in the

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