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 1217A.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 manuscripts, numerous instances of the use of these numerals before the twelfth century. Besides the example in the Codex Vigilanus, another of the tenth century has been found in the St. Gall MS. now in the University Library at Zürich, 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(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 Manuscript Forms
This is one of more than fifty tables given in Mr. Hill's valuable paper, and to this monograph studentsare 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 theBiblia Pauperum,[569]a block-book of 1470,
Numerals 1470
while in another block-book which possibly goes back to c. 1430[570]the numerals appear in several illustrations, with forms as follows:
Numerals
Many printed works anterior to 1471 have pages or chapters numbered by hand, but many of these numerals areof 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: CapitulemSymbol 2m.,...Symbol 3m.,... 4m.,... v,... vi, and followed by Roman numerals. This appears in the body of the text, in spaces 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,
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 assigned 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]
Powers of 2.
For the sake of further comparison, three illustrations from works in Mr. Plimpton's library, reproduced from theRara Arithmetica, may be considered. The first is from a Latin manuscript on arithmetic,[584]of which the original was written at Paris in 1424 by Rollandus, a Portuguese physician, who prepared the work at the command of John of Lancaster, Duke of Bedford, at one time Protector of England and Regent of France, to whom the work is dedicated. The figures show the successive powers of 2. The second illustration is from Luca da Firenze'sInprencipio darte dabacho,[585]c. 1475, and the third is from an anonymous manuscript[586]of about 1500.
Numerals.
As to the forms of the numerals, fashion played a leading part until printing was invented. This tended to fix these forms, although in writing there is still a great variation, as witness the French 5 and the German 7 and 9. Even in printing there is not complete uniformity,and it is often difficult for a foreigner to distinguish between the 3 and 5 of the French types.
Numerals.
As to the particular numerals, the following are some of the forms to be found in the later manuscripts and in the early printed books.
1. In the early printed books "one" was often i, perhaps to save types, just as some modern typewriters use the same character for l and 1.[587]In the manuscripts the "one" appears in such forms as[588]
Variations of 1.
2. "Two" often appears as z in the early printed books, 12 appearing as iz.[589]In the medieval manuscripts the following forms are common:[590]
Variations of 2.
It is evident, from the early traces, that it is merely a cursive form for the primitive2 horizontal strokes, just as 3 comes from3 horizontal strokes, as in the Nānā Ghāt inscriptions.
3. "Three" usually had a special type in the first printed books, although occasionally it appears asSymbol.[591]In the medieval manuscripts it varied rather less than most of the others. The following are common forms:[592]
Variations of 3.
4. "Four" has changed greatly; and one of the first tests as to the age of a manuscript on arithmetic, and the place where it was written, is the examination of this numeral. Until the time of printing the most common form wasSymbol, although the Florentine manuscript of Leonard of Pisa's work has the formSymbol;[593]but the manuscripts show that the Florentine arithmeticians and astronomers rather early began to straighten the first of these forms up to forms likeSymbol[594]andSymbol[594]orSymbol,[595]more closely resembling our own. The first printed books generally used our present form[596]with the closed topSymbol, the open top used in writing (Symbol) beingpurely modern. The following are other forms of the four, from various manuscripts:[597]
Variations of 4.
5. "Five" also varied greatly before the time of printing. The following are some of the forms:[598]
Variations of 5.
6. "Six" has changed rather less than most of the others. The chief variation has been in the slope of the top, as will be seen in the following:[599]
Variations of 6.
7. "Seven," like "four," has assumed its present erect form only since the fifteenth century. In medieval times it appeared as follows:[600]
Variations of 7.
8. "Eight," like "six," has changed but little. In medieval times there are a few variants of interest as follows:[601]
Variations of 8.
In the sixteenth century, however, there was manifested a tendency to write itSymbol.[602]
9. "Nine" has not varied as much as most of the others. Among the medieval forms are the following:[603]
Variations of 9.
0. The shape of the zero also had a varied history. The following are common medieval forms:[604]
Variations of 0.
The explanation of the place value was a serious matter to most of the early writers. If they had been using an abacus constructed like the Russian chotü, and had placed this before all learners of the positional system, there would have been little trouble. But the medievalline-reckoning, where the lines stood for powers of 10 and the spaces for half of such powers, did not lend itself to this comparison. Accordingly we find such labored explanations as the following, fromThe Crafte of Nombrynge:
"Euery of these figuris bitokens hym selfe & no more, yf he stonde in the first place of the rewele....
"If it stonde in the secunde place of the rewle, he betokens ten tymes hym selfe, as this figure 2 here 20 tokens ten tyme hym selfe, that is twenty, for he hym selfe betokens tweyne, & ten tymes twene is twenty. And for he stondis on the lyft side & in the secunde place, he betokens ten tyme hym selfe. And so go forth....
"Nil cifra significat sed dat signare sequenti. Expone this verse. A cifre tokens noȝt, bot he makes the figure to betoken that comes after hym more than he shuld & he were away, as thus 10. here the figure of one tokens ten, & yf the cifre were away & no figure byfore hym he schuld token bot one, for than he schuld stonde in the first place...."[605]
It would seem that a system that was thus used for dating documents, coins, and monuments, would have been generally adopted much earlier than it was, particularly in those countries north of Italy where it did not come into general use until the sixteenth century. This, however, has been the fate of many inventions, as witness our neglect of logarithms and of contracted processes to-day.
As to Germany, the fifteenth century saw the rise of the new symbolism; the sixteenth century saw it slowlygain the mastery; the seventeenth century saw it finally conquer the system that for two thousand years had dominated the arithmetic of business. Not a little of the success of the new plan was due to Luther's demand that all learning should go into the vernacular.[606]
During the transition period from the Roman to the Arabic numerals, various anomalous forms found place. For example, we have in the fourteenth century cαfor 104;[607]1000. 300. 80 et 4 for 1384;[608]and in a manuscript of the fifteenth century 12901 for 1291.[609]In the same century m. cccc. 8II appears for 1482,[610]while MoCCCCo50 (1450) and MCCCCXL6 (1446) are used by Theodoricus Ruffi about the same time.[611]To the next century belongs the form 1vojj for 1502. Even in Sfortunati'sNuovo lume[612]the use of ordinals is quite confused, the propositions on a single page being numbered "tertia," "4," and "V."
Although not connected with the Arabic numerals in any direct way, the medieval astrological numerals may here be mentioned. These are given by several early writers, but notably by Noviomagus (1539),[613]as follows[614]:
Astrological numerals.
Thus we find the numerals gradually replacing the Roman forms all over Europe, from the time of Leonardo of Pisa until the seventeenth century. But in the Far East to-day they are quite unknown in many countries, and they still have their way to make. In many parts of India, among the common people of Japan and China, in Siam and generally about the Malay Peninsula, in Tibet, and among the East India islands, the natives still adhere to their own numeral forms. Only as Western civilization is making its way into the commercial life of the East do the numerals as used by us find place, save as the Sanskrit forms appear in parts of India. It is therefore with surprise that the student of mathematics comes to realize how modern are these forms so common in the West, how limited is their use even at the present time, and how slow the world has been and is in adopting such a simple device as the Hindu-Arabic numerals.
Transcriber's note: many of the entries refer to footnotes linked from the page numbers given.