Another contributor to arithmetic during this interesting period was a prominent Spanish Jew called variously John of Luna, John of Seville, Johannes Hispalensis, Johannes Toleta.n.u.s, and Johannes Hispanensis de Luna.[499] {125} His date is rather closely fixed by the fact that he dedicated a work to Raimund who was archbishop of Toledo between 1130 and 1150.[500] His interests were chiefly in the translation of Arabic works, especially such as bore upon the Aristotelian philosophy. From the standpoint of arithmetic, however, the chief interest centers about a ma.n.u.script ent.i.tled _Joannis Hispalensis liber Algorismi de Practica Arismetrice_ which Boncompagni found in what is now the _Bibliotheque nationale_ at Paris. Although this distinctly lays claim to being Al-Khow[=a]razm[=i]'s work,[501] the evidence is altogether against the statement,[502] but the book is quite as valuable, since it represents the knowledge of the time in which it was written. It relates to the operations with integers and s.e.xagesimal fractions, including roots, and contains no applications.[503]

Contemporary with John of Luna, and also living in Toledo, was Gherard of Cremona,[504] who has sometimes been identified, but erroneously, with Gernardus,[505] the {126} author of a work on algorism. He was a physician, an astronomer, and a mathematician, translating from the Arabic both in Italy and in Spain. In arithmetic he was influential in spreading the ideas of algorism.

Four Englishmen--Adelhard of Bath (c. 1130), Robert of Chester (Robertus Cestrensis, c. 1143), William Sh.e.l.ley, and Daniel Morley (1180)--are known[506] to have journeyed to Spain in the twelfth century for the purpose of studying mathematics and Arabic. Adelhard of Bath made translations from Arabic into Latin of Al-Khow[=a]razm[=i]'s astronomical tables[507] and of Euclid's Elements,[508] while Robert of Chester is known as the translator of Al-Khow[=a]razm[=i]'s algebra.[509] There is no reason to doubt that all of these men, and others, were familiar with the numerals which the Arabs were using.

The earliest trace we have of computation with Hindu numerals in Germany is in an Algorismus of 1143, now in the Hofbibliothek in Vienna.[510] It is bound in with a {127} _Computus_ by the same author and bearing the date given. It contains chapters "De additione," "De diminutione," "De mediatione," "De divisione," and part of a chapter on multiplication. The numerals are in the usual medieval forms except the 2 which, as will be seen from the ill.u.s.tration,[511] is somewhat different, and the 3, which takes the peculiar shape [Symbol], a form characteristic of the twelfth century.

It was about the same time that the _Sefer ha-Mispar_,[512] the Book of Number, appeared in the Hebrew language. The author, Rabbi Abraham ibn Mer ibn Ezra,[513] was born in Toledo (c. 1092). In 1139 he went to Egypt, Palestine, and the Orient, spending also some years in Italy. Later he lived in southern France and in England. He died in 1167. The probability is that he acquired his knowledge of the Hindu arithmetic[514] in his native town of Toledo, but it is also likely that the knowledge of other systems which he acquired on travels increased his appreciation of this one. We have mentioned the fact that he used the first letters of the Hebrew alphabet, [Hebrew: A B G D H W Z CH T'], for the numerals 9 8 7 6 5 4 3 2 1, and a circle for the zero. The quotation in the note given below shows that he knew of the Hindu origin; but in his ma.n.u.script, although he set down the Hindu forms, he used the above nine Hebrew letters with place value for all computations.

{128}

CHAPTER VIII

THE SPREAD OF THE NUMERALS IN EUROPE

Of all the medieval writers, probably the one most influential in introducing the new numerals to the scholars of Europe was Leonardo Fibonacci, of Pisa.[515] This remarkable man, the most noteworthy mathematical genius of the Middle Ages, was born at Pisa about 1175.[516]

The traveler of to-day may cross the Via Fibonacci on his way to the Campo Santo, and there he may see at the end of the long corridor, across the quadrangle, the statue of Leonardo in scholars garb. Few towns have honored a mathematician more, and few mathematicians have so distinctly honored their birthplace. Leonardo was born in the golden age of this city, the period of its commercial, religious, and intellectual prosperity.[517]

{129} Situated practically at the mouth of the Arno, Pisa formed with Genoa and Venice the trio of the greatest commercial centers of Italy at the opening of the thirteenth century. Even before Venice had captured the Levantine trade, Pisa had close relations with the East. An old Latin chronicle relates that in 1005 "Pisa was captured by the Saracens," that in the following year "the Pisans overthrew the Saracens at Reggio," and that in 1012 "the Saracens came to Pisa and destroyed it." The city soon recovered, however, sending no fewer than a hundred and twenty ships to Syria in 1099,[518] founding a merchant colony in Constantinople a few years later,[519] and meanwhile carrying on an interurban warfare in Italy that seemed to stimulate it to great activity.[520] A writer of 1114 tells us that at that time there were many heathen people--Turks, Libyans, Parthians, and Chaldeans--to be found in Pisa. It was in the midst of such wars, in a cosmopolitan and commercial town, in a center where literary work was not appreciated,[521] that the genius of Leonardo appears as one of the surprises of history, warning us again that "we should draw no horoscope; that we should expect little, for what we expect will not come to pa.s.s."[522]

Leonardo's father was one William,[523] and he had a brother named Bonaccingus,[524] but nothing further is {130} known of his family. As to Fibonacci, most writers[525] have a.s.sumed that his father's name was Bonaccio,[526] whence _filius Bonaccii_, or Fibonacci. Others[527] believe that the name, even in the Latin form of _filius Bonaccii_ as used in Leonardo's work, was simply a general one, like our Johnson or Bronson (Brown's son); and the only contemporary evidence that we have bears out this view. As to the name Bigollo, used by Leonardo, some have thought it a self-a.s.sumed one meaning blockhead, a term that had been applied to him by the commercial world or possibly by the university circle, and taken by him that he might prove what a blockhead could do. Milanesi,[528] however, has shown that the word Bigollo (or Pigollo) was used in Tuscany to mean a traveler, and was naturally a.s.sumed by one who had studied, as Leonardo had, in foreign lands.

Leonardo's father was a commercial agent at Bugia, the modern Bougie,[529]

the ancient Saldae on the coast of Barbary,[530] a royal capital under the Vandals and again, a century before Leonardo, under the Beni Hammad. It had one of the best harbors on the coast, sheltered as it is by Mt. Lalla Guraia,[531] and at the close of the twelfth century it was a center of African commerce. It was here that Leonardo was taken as a child, and here he went to school to a Moorish master. When he reached the years of young manhood he started on a tour of the Mediterranean Sea, and visited Egypt, Syria, Greece, Sicily, and Provence, meeting with scholars as well as with {131} merchants, and imbibing a knowledge of the various systems of numbers in use in the centers of trade. All these systems, however, he says he counted almost as errors compared with that of the Hindus.[532] Returning to Pisa, he wrote his _Liber Abaci_[533] in 1202, rewriting it in 1228.[534] In this work the numerals are explained and are used in the usual computations of business. Such a treatise was not destined to be popular, however, because it was too advanced for the mercantile cla.s.s, and too novel for the conservative university circles. Indeed, at this time mathematics had only slight place in the newly established universities, as witness the oldest known statute of the Sorbonne at Paris, dated 1215, where the subject is referred to only in an incidental way.[535] The period was one of great commercial activity, and on this very {132} account such a book would attract even less attention than usual.[536]

It would now be thought that the western world would at once adopt the new numerals which Leonardo had made known, and which were so much superior to anything that had been in use in Christian Europe. The antagonism of the universities would avail but little, it would seem, against such an improvement. It must be remembered, however, that there was great difficulty in spreading knowledge at this time, some two hundred and fifty years before printing was invented. "Popes and princes and even great religious inst.i.tutions possessed far fewer books than many farmers of the present age. The library belonging to the Cathedral Church of San Martino at Lucca in the ninth century contained only nineteen volumes of abridgments from ecclesiastical commentaries."[537] Indeed, it was not until the early part of the fifteenth century that Palla degli Strozzi took steps to carry out the project that had been in the mind of Petrarch, the founding of a public library. It was largely by word of mouth, therefore, that this early knowledge had to be transmitted. Fortunately the presence of foreign students in Italy at this time made this transmission feasible.

(If human nature was the same then as now, it is not impossible that the very opposition of the faculties to the works of Leonardo led the students to investigate {133} them the more zealously.) At Vicenza in 1209, for example, there were Bohemians, Poles, Frenchmen, Burgundians, Germans, and Spaniards, not to speak of representatives of divers towns of Italy; and what was true there was also true of other intellectual centers. The knowledge could not fail to spread, therefore, and as a matter of fact we find numerous bits of evidence that this was the case. Although the bankers of Florence were forbidden to use these numerals in 1299, and the statutes of the university of Padua required stationers to keep the price lists of books "non per cifras, sed per literas claros,"[538] the numerals really made much headway from about 1275 on.

It was, however, rather exceptional for the common people of Germany to use the Arabic numerals before the sixteenth century, a good witness to this fact being the popular almanacs. Calendars of 1457-1496[539] have generally the Roman numerals, while Kobel's calendar of 1518 gives the Arabic forms as subordinate to the Roman. In the register of the Kreuzschule at Dresden the Roman forms were used even until 1539.

While not minimizing the importance of the scientific work of Leonardo of Pisa, we may note that the more popular treatises by Alexander de Villa Dei (c. 1240 A.D.) and John of Halifax (Sacrobosco, c. 1250 A.D.) were much more widely used, and doubtless contributed more to the spread of the numerals among the common people.

{134}

The _Carmen de Algorismo_[540] of Alexander de Villa Dei was written in verse, as indeed were many other textbooks of that time. That it was widely used is evidenced by the large number of ma.n.u.scripts[541] extant in European libraries. Sacrobosco's _Algorismus_,[542] in which some lines from the Carmen are quoted, enjoyed a wide popularity as a textbook for university instruction.[543] The work was evidently written with this end in view, as numerous commentaries by university lecturers are found.

Probably the most widely used of these was that of Petrus de Dacia[544]

written in 1291. These works throw an interesting light upon the method of instruction in mathematics in use in the universities from the thirteenth even to the sixteenth century. Evidently the text was first read and copied by students.[545] Following this came line by line an exposition of the text, such as is given in Petrus de Dacia's commentary.

Sacrobosco's work is of interest also because it was probably due to the extended use of this work that the {135} term _Arabic numerals_ became common. In two places there is mention of the inventors of this system. In the introduction it is stated that this science of reckoning was due to a philosopher named Algus, whence the name _algorismus_,[546] and in the section on numeration reference is made to the Arabs as the inventors of this science.[547] While some of the commentators, Petrus de Dacia[548]

among them, knew of the Hindu origin, most of them undoubtedly took the text as it stood; and so the Arabs were credited with the invention of the system.

The first definite trace that we have of an algorism in the French language is found in a ma.n.u.script written about 1275.[549] This interesting leaf, for the part on algorism consists of a single folio, was noticed by the Abbe Leboeuf as early as 1741,[550] and by Daunou in 1824.[551] It then seems to have been lost in the mult.i.tude of Paris ma.n.u.scripts; for although Chasles[552] relates his vain search for it, it was not rediscovered until 1882. In that year M. Ch. Henry found it, and to his care we owe our knowledge of the interesting ma.n.u.script. The work is anonymous and is devoted almost entirely to geometry, only {136} two pages (one folio) relating to arithmetic. In these the forms of the numerals are given, and a very brief statement as to the operations, it being evident that the writer himself had only the slightest understanding of the subject.

Once the new system was known in France, even thus superficially, it would be pa.s.sed across the Channel to England. Higden,[553] writing soon after the opening of the fourteenth century, speaks of the French influence at that time and for some generations preceding:[554] "For two hundred years children in scole, agenst the usage and manir of all other nations beeth compelled for to leave hire own language, and for to construe hir lessons and hire thynges in Frensche.... Gentilmen children beeth taught to speke Frensche from the tyme that they bith rokked in hir cradell; and uplondissche men will likne himself to gentylmen, and fondeth with greet besynesse for to speke Frensche."

The question is often asked, why did not these new numerals attract more immediate attention? Why did they have to wait until the sixteenth century to be generally used in business and in the schools? In reply it may be said that in their elementary work the schools always wait upon the demands of trade. That work which pretends to touch the life of the people must come reasonably near doing so. Now the computations of business until about 1500 did not demand the new figures, for two reasons: First, cheap paper was not known. Paper-making of any kind was not introduced into Europe until {137} the twelfth century, and cheap paper is a product of the nineteenth. Pencils, too, of the modern type, date only from the sixteenth century. In the second place, modern methods of operating, particularly of multiplying and dividing (operations of relatively greater importance when all measures were in compound numbers requiring reductions at every step), were not yet invented. The old plan required the erasing of figures after they had served their purpose, an operation very simple with counters, since they could be removed. The new plan did not as easily permit this.

Hence we find the new numerals very tardily admitted to the counting-house, and not welcomed with any enthusiasm by teachers.[555]

Aside from their use in the early treatises on the new art of reckoning, the numerals appeared from time to time in the dating of ma.n.u.scripts and upon monuments. The oldest definitely dated European doc.u.ment known {138} to contain the numerals is a Latin ma.n.u.script,[556] the Codex Vigila.n.u.s, written in the Albelda Cloister not far from Logrono in Spain, in 976 A.D.

The nine characters (of [.g]ob[=a]r type), without the zero, are given as an addition to the first chapters of the third book of the _Origines_ by Isidorus of Seville, in which the Roman numerals are under discussion.

Another Spanish copy of the same work, of 992 A.D., contains the numerals in the corresponding section. The writer ascribes an Indian origin to them in the following words: "Item de figuris arithmetic[e,]. Scire debemus in Indos subtilissimum ingenium habere et ceteras gentes eis in arithmetica et geometria et ceteris liberalibus disciplinis concedere. Et hoc manifestum est in n.o.bem figuris, quibus designant unumquemque gradum cuiuslibet gradus. Quarum hec sunt forma." The nine [.g]ob[=a]r characters follow.

Some of the abacus forms[557] previously given are doubtless also of the tenth century. The earliest Arabic doc.u.ments containing the numerals are two ma.n.u.scripts of 874 and 888 A.D.[558] They appear about a century later in a work[559] written at Shiraz in 970 A.D. There is also an early trace of their use on a pillar recently discovered in a church apparently destroyed as early as the tenth century, not far from the Jeremias Monastery, in Egypt. {139} A graffito in Arabic on this pillar has the date 349 A.H., which corresponds to 961 A.D.[560] For the dating of Latin doc.u.ments the Arabic forms were used as early as the thirteenth century.[561]

On the early use of these numerals in Europe the only scientific study worthy the name is that made by Mr. G. F. Hill of the British Museum.[562]

From his investigations it appears that the earliest occurrence of a date in these numerals on a coin is found in the reign of Roger of Sicily in 1138.[563] Until recently it was thought that the earliest such date was 1217 A.D. for an Arabic piece and 1388 for a Turkish one.[564] Most of the seals and medals containing dates that were at one time thought to be very early have been shown by Mr. Hill to be of relatively late workmanship.

There are, however, in European ma.n.u.scripts, numerous instances of the use of these numerals before the twelfth century. Besides the example in the Codex Vigila.n.u.s, another of the tenth century has been found in the St.

Gall MS. now in the University Library at Zurich, the forms differing materially from those in the Spanish codex.

The third specimen in point of time in Mr. Hill's list is from a Vatican MS. of 1077. The fourth and fifth specimens are from the Erlangen MS. of Boethius, of the same {140} (eleventh) century, and the sixth and seventh are also from an eleventh-century MS. of Boethius at Chartres. These and other early forms are given by Mr. Hill in this table, which is reproduced with his kind permission.

EARLIEST Ma.n.u.sCRIPT FORMS

[Ill.u.s.tration]

This is one of more than fifty tables given in Mr. Hill's valuable paper, and to this monograph students {141} are referred for details as to the development of number-forms in Europe from the tenth to the sixteenth century. It is of interest to add that he has found that among the earliest dates of European coins or medals in these numerals, after the Sicilian one already mentioned, are the following: Austria, 1484; Germany, 1489 (Cologne); Switzerland, 1424 (St. Gall); Netherlands, 1474; France, 1485; Italy, 1390.[565]

The earliest English coin dated in these numerals was struck in 1551,[566]

although there is a Scotch piece of 1539.[567] In numbering pages of a printed book these numerals were first used in a work of Petrarch's published at Cologne in 1471.[568] The date is given in the following form in the _Biblia Pauperum_,[569] a block-book of 1470,

[Ill.u.s.tration]

while in another block-book which possibly goes back to c. 1430[570] the numerals appear in several ill.u.s.trations, with forms as follows:

[Ill.u.s.tration]

Many printed works anterior to 1471 have pages or chapters numbered by hand, but many of these numerals are {142} of date much later than the printing of the work. Other works were probably numbered directly after printing. Thus the chapters 2, 3, 4, 5, 6 in a book of 1470[571] are numbered as follows: Capitulem [Symbol 2]m.,... [Symbol 3]m.,... 4m.,...

v,... vi, and followed by Roman numerals. This appears in the body of the text, in s.p.a.ces left by the printer to be filled in by hand. Another book[572] of 1470 has pages numbered by hand with a mixture of Roman and Hindu numerals, thus,

[Ill.u.s.tration] for 125 [Ill.u.s.tration] for 150 [Ill.u.s.tration] for 147 [Ill.u.s.tration] for 202

As to monumental inscriptions,[573] there was once thought to be a gravestone at Katharein, near Troppau, with the date 1007, and one at Biebrich of 1299. There is no doubt, however, of one at Pforzheim of 1371 and one at Ulm of 1388.[574] Certain numerals on Wells Cathedral have been a.s.signed to the thirteenth century, but they are undoubtedly considerably later.[575]

The table on page 143 will serve to supplement that from Mr. Hill's work.[576]

{143}

EARLY Ma.n.u.sCRIPT FORMS

[577] [Ill.u.s.tration] Twelfth century A.D.

[578] [Ill.u.s.tration] 1197 A.D.

[579] [Ill.u.s.tration] 1275 A.D.

[580] [Ill.u.s.tration] c. 1294 A.D.

[581] [Ill.u.s.tration] c. 1303 A.D.

[582] [Ill.u.s.tration] c. 1360 A.D.

[583] [Ill.u.s.tration] c. 1442 A.D.