Popular Science Monthly/Volume 81/December 1912/The Hindu-Arabic Numerals
|THE HINDU-ARABIC NUMERALS|
PROFESSOR OF EUROPEAN HISTORY, UNIVERSITY OF MICHIGAN
AT present the Hindu-Arabic numerals hold nearly unlimited sway in the realm of number. In China, in Japan, in southeast Asia, and in parts of India, it is true, they are employed only by the upper classes and by foreign traders; but all over Europe, in Australia, and in North America, they are supreme; while in South America and in Africa they are used wherever civilized men make arithmetical computation.
Scarcely ever has a more wonderful device been perfected. By means of these characters prodigious calculations can be made. Through tables, logarithms, and counting-machines amazing swiftness and accuracy are obtained. Their power and their scope seem almost limitless.
And yet there was a time when they were confined to a few districts in India, and not heard of by the rest of mankind; when they were cumbersome and inert and difficult to use; when they were no better than number signs which had been developed elsewhere, and not nearly so well known. Even when their superiority was manifest, their progress into other lands was uncertain, difficult, and slow. The story of the development of the Hindu numerals and of their conquest of the world make an interesting but oft-forgotten chapter in the history of civilization.
The idea of number originates in sense experience. The conception of two and three as different from one rests fundamentally upon experience in dealing with one thing and with more than one thing. The ideas of an object and of several objects which the mind forms through the eye or through the sense of touch constitute the basis upon which all knowledge of number rests. Thus gradually in infancy or during the childhood of the race are obtained the conceptions of what we now call in English "one," "two," and "three."
In the lower stages of culture only dim ideas of higher number exist, and the lowest, basic numbers are used in combination to express what is more numerous. Thus, three and one or three and two take the place of four and five. This method was actually incorporated into the systems of the and the ancient Hindus. Indeed, the habit persists in the minds of people later on, when they possess many number symbols and the ability to use them in calculation. While the more acute nowadays can without great difficulty comprehend four and five, yet five will frequently be apprehended better as a combination of two and three, while almost inevitably six will be thought of as a combination of three and three, and seven as made of three and four.
Nevertheless as the mind of man becomes more powerful, and the need for calculation becomes more frequent, larger numbers are made use of, even though they can only be comprehended as combinations of smaller ones. The designation of these number ideas soon becomes necessary, and it must be made both for the eye and for the ear. For the eye this may be done by symbols; for the ear by words. Thus, the Greeks may write Ὡ (90) and call it ὲνενῄκοντα; the Romans XL, and call it quadraginta.
The designation of these number ideas either by symbol or by sound is exceedingly difficult for the reason that number ideas are necessarily abstract. It is true that the lower, which represent a few objects, can be designated by pictures. So, man may originate the symbols for his lower numbers in the same manner that he first makes the symbols for his words, by ideographs or picture-drawings. In old Chinese the rude representation of a man designates man. Similarly the Chinaman writes a simple stroke to represent a single object; two strokes to represent two objects; and three, to represent three:
The Hindu once employed the same device, except that his strokes were usually perpendicular:
These symbols were used wherever men began to write their numbers. They were employed by the Latin peoples, and as the Roman numerals are still in common use. In cursive form they are the numerals which we use to-day:
whether this is a combination of four strokes, and so a true ideograph, can not be known. For six the Chinaman writes
Thus a series of signs may be accumulated.
In many cases the need for a series of number symbols arises when considerable progress has been made in the construction of an alphabet. The alphabet then furnishes a series of signs which follow each other in definite sequence, and as signs are fairly well understood and familiar. The result is that the letters of the alphabet are employed to designate the number ideas. This was done by the Hebrews, who make use of their twenty-two letters, and by the Greeks who had the twenty-four letters of their alphabet with three archaic signs interspersed.
As an example of this alphabetic designation the Greek system may be taken. The letters accented were the numerals: α', 1; β', 2; γ', 3; δ', 4; ε', 5; ς', 6; ζ', 7; η', 8; θ', 9; ι', 10; κ', 20; λ', 30; μ', 40; ν', 50; ξ', 60; ο', 70; π', 80; ρ', 90; ρ', 100; σ', 200; τ', 300; υ', 400 φ', 500; χ', 600; ψ', 700; ω', 800. The intervening numbers were expressed by combination; thus, γ' = 3 and ι' = 10, therefore ιγ' = 13; while the numbers for the thousands were expressed by sub-accenting the lower symbols; thus β' = 2, ͵β = 2000.
Here is a system comprehensive and excellent for the mere writing of numbers. It was, however, because of the numerous signs employed, cumbersome and complex. For example, in multiplication, where our nine numerals now in use require a knowledge of forty-five combinations—one times one, one times two, one times three, and so on—the Greek system with its twenty-seven characters required the memorizing of three hundred and seventy-eight—α' times α', α' times β', α times γ', and so forth. Other arithmetical processes were correspondingly difficult.
Another scheme, apparently much simpler, consists in using only a few letters or signs. As an example, the Roman system may be taken. For one a single stroke was employed, I; while groups of strokes were used for the numbers following, II, III, IIII. Five was designated by a symbol of its own, V, which was once thought to be a representation of the thumb and four fingers held up, but this theory has been abandoned. For ten, X was employed, the origin of which is not entirely clear. A study of the inscriptions, however, affords ground for the belief that the Romans in counting from one to ten used one, two, three, four, five, six, seven, eight, nine strokes; then, to avoid confusion, denoted ten by drawing a tenth stroke over the nine parallel ones,
and that this was abbreviated to two strokes crossed, X. The upholders of this theory assert that five, half of ten, was then denoted by V, half of X. For fifty, L was used; for one hundred, C; for five hundred, D; for one thousand, M. The numbers in between were expressed by combination: LX = 60, DCXV = 615; or by addition to or subtraction from the nearest one of the seven symbols: XII = 12, IX = 9.
The advantage of this system lay in the fewness of the symbols employed. Where the Hebrew had twenty-two characters and the Greek twenty-seven, the Roman made use only of seven. Because of this fewness the value of the characters could be easily remembered. On the other hand the smaller number of characters employed made necessary the greater use of combination. The Greeks had a symbol for sixty or for ninety, but the Latins must place together several numeral signs of smaller value so that the combination would equal the total required In Greek 60 might be expressed by ξ', one character; in Latin by two, LX. The more complex the number the greater became the relative cumbersomeness of the method. In Greek 1863 could be expressed by ,αωξγ'; in Latin it would be MDCCCLXIII. The result of these cumbersome combinations was that with the Roman numerals it was virtually impossible to make calculations of any intricacy, and exceedingly difficult to make even simple ones. They might be employed for mere designation, as they are to this day used to express dates and to distinguish the pages of a preface; but for addition, subtraction, multiplication, division, and intricate arithmetical work, the Roman mathematicians were driven to use the Greek symbols and methods.
Meanwhile in the east other systems of numeral notation had been developed, some of which, in modified form, are in use there at the present time. The Babylonians were expert calculators; and the Chinese invented a notation which they still have. It was in India, however, that a system arose destined to supersede all others among civilized people.
There were probably some numerals in use among the Hindus a thousand or more years before our era, but no records exist earlier than the time of Asoka, in the third century B.C. From this time on occur inscriptions which contain some of the native number symbols. Two systems may be discerned, which possess respectively the characteristics of the Roman and the Greek.
In the Kharosthi inscriptions of the third century B.C. four numerals occur, the origin and meaning of which are evident:
In the Saka inscriptions of the first century before Christ more characters appear, and the resemblance to the Roman becomes striking:
This system is constructed of the symbols for 1, 4, 10, 20, 100, and so forth, as the Roman is built upon the I, V, X, L, C, D, and M.
In the same period another system was invented in which greater flexibility and power were obtained by using an increased number of signs. In the third century B.C. certain of King Asoka's inscriptions in the Brahmi writing contain these characters:
In the following century an inscription in the Nana Ghat cave near Poona in central India has even more interesting ones:
Here the 1, 6, and 7, which we now use, appear plainly; while the 2 and 9 are in rudimentary form. About two hundred years later inscriptions in the Nasik cave contain all of the important Hindu numerals:Here the characters for 1, 4, 6, and 7 are easily recognizable, while the 2, 3, 5, and 9 can be developed without difficulty.
This system is neither different from that of the Greeks nor superior to it. Neither has a zero, and in neither have the characters a value of place. That is, all of the important numbers must be designated by signs of their own. In our system 2 moved one place to the left becomes 20, but in Greek β′ = 2 and κ′ = 20. So, in the Nasik inscription 2 and 20 are designated by characters separate and distinct.
In this far there is similarity; but while the Greeks used the letters of their alphabet, the Hindus did not. The meaning of some of these signs, such as the strokes for 1, 2, and 3, is apparent, but the origin and meaning of others are not known. They may have been made from alphabetic characters, but as yet no theory has been substantiated.
It may be seen that all of the different systems of numerical notation which had been developed, whether in China, in India, or in Greece, had the same general characteristics: there was no zero, and the symbols had no place value. Because of this it was necessary to employ a great many different symbols. Such a system might be used for mathematical calculation, but it was bound to be complicated and intricate. What was needed was a system with fewer signs; but when this was constructed, as it was among the Romans, and the Greeks of Solon's time, it was so rigid and inflexible as to admit of no progress in mathematics. Something altogether different was needed.
Gradually, by processes of which we know little, a revolution was wrought in all mathematical work and new numeral systems were developed. This revolution was effected by the use of the counting-board, or, as we now call it, the abacus, from a Greek word the meaning of which is in dispute. At first all calculation was probably mental or performed upon the fingers, but as time went by, a mechanical device was perfected wherever men strove to make readier and more elaborate computations. This device is said to have been invented by the Chinese, though of such tradition there is no certain proof At all events it was used by the Chinese, the Babylonians, and the Hindus in immemorial times. The Greeks and the Romans had it; and it continued to be used in Europe throughout the middle ages. According to the "Dialogus de Scaccario" of the twelfth century, the officials of the exchequer reckoned the king of England's revenue by means of it. To this day it is employed generally in Russia, and in schools wherever children are learning to count.
Fundamentally the abacus consists of a board or table marked off in parallel columns within which counters can be placed. The principle is the same if the counters are strung along parallel wires. The important thing is that on the abacus each column has a value of its own, a value of place. Thus, several numerals may be employed with reference to the counters in the first column, for example, our 1, 2, 3, 4, 5, 6, 7, 8, 9. Then these same numerals may be employed for the second, but they will now have in their new place a new value, let us say, ten times as great, so that a 2 in the second column will denote 2 tens, or 20. And so in other columns, which give the value of 100, or 1000. By this means the entire number of numeral signs may at once be reduced to the number of signs used in the first column. Without the abacus the Greeks must have nineteen signs in counting from one to a hundred: on the abacus there is need only of nine.
The place value assigned on such an instrument would depend upon the practical system of counting which had been developed. Just as nowadays the workman, or the boy playing a game, will tally five, and then begin another five, and then another, thus making a group of fives which he can handle easily, rather than one longer series, so the practical calculators of bygone days worked with fives, or sixes, or tens, or twenties. Various systems have been used. The Babylonians employed the sexagesimal, reckoning by sixties. Some of the African tribes count by sixes, and some of the New Zealanders are said to use elevens The duodecimal or twelve system has passed away, but in the dozen we still preserve traces of it ourselves.
As a rule, however, the system has been none of these, but counting has been done by fives, by tens, or twenties, and this simply because it has been based upon the antique but persistent habit which man has of counting upon his fingers. To this day there is a widespread custom of reckoning roughly by fives. The Mayas of Yucatan used the vigesimal or twenty system; so did the men of Palmyra in Zenobia's time, and the Syrians before the days of Mohammed. The same is said to have been true of the Celts, and the French seem to preserve traces of it now when they say quatre-vingts, four twenties, for 80. But after all that system which has been most widely adopted is the decimal, based upon all the fingers of the two hands.
The Hindus came to employ the decimal system, as did the Greeks and the Chinese. It found place in the written language as well as in the numeral notation; but, as has been said, it existed only in complicated form. Thus, the Greeks and the Romans had words to express numbers from one to ten, as they had signs. In Greek εἲς and d denoted one; δέκα and ί, ten; after which there were words and signs for twenty, thirty, and so on, at intervals of ten up to one hundred; followed by words and signs at intervals of a hundred. In between, the numbers were expressed by combination: ἕνδεκα (one-ten), eleven; δίο καῚ τριάκοτα (two and thirty), thirty-two. The Roman system was entirely similar, except that it employed fewer signs. The Hindus used the decimal system even more consistently, since they preserved it in counting beyond thousands indefinitely.
Thus, the decimal system was developed in language and in numeral notation at the same time that it was being employed in the construction of the abacus. In each case its origin was due to the habit or practise among people of counting upon the ten fingers until they came by custom to reckon in tens. In each case, however, the decimal system was unwieldy in that it was built up upon a large number of words and signs. It was the function of the abacus to make it simple by reducing the number of signs.
Since the Hindus employed the decimal system, so, on an Indian counting device the counters in the second column had ten times the value of those in the first; the ones in the third ten times the value of those in the second, and one hundred times the value of the counters in the first. It was necessary now for the Hindus to use only the signs which in their present form are 1, 2, 3, 4, 5, 6, 7, 8, 9. Thus, 591 would be represented by 5 | 9 | 1. In like manner 501 would be 5 | | 1, the middle space being vacant since there were only five hundreds and one unit, but no tens. Among the Greeks it is probable that the same method was worked out. In this manner the number signs could now be attached to a definite place, and so had a definite place value. This is the most important step which has ever been made in mathematical science.
But a difficulty arose when the calculation was transferred from the abacus-board and became a written operation. 591 could be transferred without difficulty, since the digits by mere juxtaposition would preserve their place value; but 5(0)1 taken from the abacus might be 51, since the vacant place was no longer indicated. Accordingly mathematicians were led to invent a character to stand for the vacant space. By so doing they perfected the system of place value, since they could now show that even when there was no one of the nine numerals in a particular place, the value of the place remained, and the values of the adjoining places could be maintained. The invention of a symbol for nothing is the crowning, transcendent achievement in the perfection of the decimal system, and lies at the base of all subsequent arithmetical progress. It is the peculiar triumph of the Hindu mathematicians to have made this contribution to the science of number.
A symbol for nothing was employed among the Chaldeans, but merely for notation, and apparently never in calculating. In the cuneiformit occurs as
Among the Hindus it was at first a dot (. ). In this guise it was borrowed by the Arabs, who still use it. Very soon, however, the Hindus began to employ a circle, 0. The earliest known use is in an inscription at Gwalior, 876 A.D. Fifty and two hundred and seventy are written respectively
So at last the decimal system was complete, and it had been worked cut in the Hindu numerals. It remained now to carry them from India into the countries nearby.
The introduction of the Hindu numerals into Europe is one of the obscurest matters in history. Most probably the truth can never be completely discovered. This is as it should be, since the adoption of these numerals was natural and slow, and not premeditated or artificial. After a while they were in use among the merchants of the east, who carried them along the highways of the world's commerce It may be that they entered into countries upon bales of goods or in the accounts of traders, and so would be unnoticed by the scholars. Perhaps for a long time the local mathematicians would know nothing of them, and would continue to use the symbols and the systems of their forefathers. It may be said, however, that the fame of the Hindu characters was spreading into the countries near by even before the addition of the zero. In 662 Sebocht, a Syrian ecclesiastic, writing in a monastery on the Euphrates, refers to the "science of the Hindus . . . and of the easy method of their calculations, and of their computation, which surpasses words. I mean that made with nine symbols." No doubt there were others in the neighboring lands who came to know of this wondrous art.
Of one thing there is no doubt: the Arabs soon adopted the Hindu numerals, and when they spread their conquest across the world they carried these numerals with them. In the ninth century a group of brilliant mathematicians were employing them at Bagdad, while a long line of scholars used them in a slightly different form, the Gobar numerals, in Spain. From the Arabs these numbers were taken by Christian Europe, and for this reason came to be known for a long time as Arabic numbers.
There will always be the question whether in some form these numerals were known and used in southern Europe before the coming of the Arabs. It is not likely that this matter can ever be entirely determined. In an eleventh century manuscript of the "Geometry" of Boethius there is a passage where certain numerals are introduced, curiously like the Hindu symbols:
As Boethius wrote at the beginning of the sixth century this was at one time thought to indicate an early introduction of the symbols, or even an independent origin; but it is now as certain as such things can be, that the passage is a medieval interpolation, and was not written by Boethius. The subject is exceedingly obscure, but there is reason for thinking that these characters, apices, as they were called, were used in Europe some time before the interpolation was made.
However this may be, there was apparently among the Christian peoples of Europe no widespread use of the ten symbols as they were used by the Hindus, until the Christians borrowed them from the Saracens of Spain. The date of their introduction from Spain cannot be determined, but it is fairly certain that Gerbert, who as Sylvester II. was Pope from 999 to 1003, brought them back from the Saracens among whom he had studied. He seems to have described the nine Gobar numerals without the zero:
After a while some of the monkish mathematicians learned of the symbol for nothing. O'Creat in the twelfth century employed it in three forms, o, ō, τ. At this time when the new numerals are used the whole subject is confused. Sometimes the Hindu symbols are used without the zero; sometimes the Roman characters with it, the Roman characters then acquiring a place value. Thus, when O'Creat writes 1200, he puts it I.II.τ.τ.; for 1089 he uses I. O. VIII. IX. At the beginning of the twelfth century Radulph of Laon used a mixture of Gobar and Roman characters. About the same time an unknown German scribe wrote them in a manuscript now in the Hof-Bibliothek in Vienna.
The mathematician who had most to do with spreading the Hindu numerals in Europe was Leonardo Fibonacci, a merchant of Pisa, who was born in 1175. In 1202 he completed his "Liber Abaci," or arithmetic, rewriting it in 1228. He it was who, when employing the Hindu numerals, first clearly explained their use. The progress was furthered when Alexander de Villa Dei about 1240, and Johannes de Sacrobosco about 1250, wrote popular treatises. Of Sacrobosco's "Algorismus" there remain now nearly one hundred manuscripts. It was due to him particularly that the Hindu signs came to be known in Europe as Arabic numerals.
The Hindu characters were now known, but they came into general use very slowly, for the same reason that the metric system is not adopted immediately nowadays. The Roman characters, computation on the abacus, and various devices held their own obstinately through force of custom. Moreover, before the era of printing no knowledge could be spread rapidly among the mass of the people, and for the most part the counting of the ordinary man could be done upon his fingers.
A coin of Roger of Sicily is dated with the Hindu numerals in 1138. In 1390 an Italian coin is so dated; in France, one of 1485. There is a Scotch piece of 1538, but apparently none for England until 1551. A French manuscript of 1275 contains a treatise upon these numerals. They are used in the inscription upon a grave at Pforzheim in 1371, and upon one at Ulm in 1388. In 1170 they are used to date a book:
In 1471 the pages are so numbered in a work of Petrarch printed at Cologne.
They were still used in curious and confused combinations. In a work of 1470, 147 and 150 are written respectively
CC2 stands for 202. About the same time 12901 means 1291, m.cccc.8II stands for 1482, and MCCCCXL6 for 1446. Somewhat earlier ca is written for 104; somewhat later 1vojj for 1502.
The forms of the numeral signs varied as much as they do now upon the slates of children learning in school. These forms were not well fixed until after the general use of printing.
One is frequently written i. Other forms are
Two is often z, so that iz = 12.
Three is usually recognizable.
Five has changed a great deal, though it can usually be recognized.
Six has changed but little.
Up to the fifteenth century seven was usually recumbent in posture.
Eight has preserved its shape.
So has nine for the most part.
The symbol for nothing has varied greatly, and in an arbitrary manner.
The name of this last symbol is interesting. The Hindus called it sunya, void. In Arabic this became sifr or as-sifr. In 1202 Leonardo Fibonacci translated it zephirum; in 1330 Maximus Planudes called it τζιφρα, tziphra. During the fourteenth century Italian writers shortened it to zeuero and ceuro, which became zero, now in general use. Meanwhile it had passed more nearly in Arabic form into French as chiffre, and into English as cipher, taking on new significations. Perhaps the schoolboy of to-day, who speaks of getting "zip" for an answer, is himself reverting to the Arabic.
After the Renaissance the Hindu numerals gradually supplanted other forms, and by the seventeenth century the process was nearly complete. From Europe in turn they have spread over the world, until they are now in general use wherever civilized man is living, except in some countries of the far east, near where they had their origin.
Such is the story of the numerals, and such is one phase of the development of the mind of man. In the childhood of the race we see him counting upon his lips, his fingers, or his toes. Then he assigns to his ideas symbols, and finally with prodigious difficulty gives to them a value of place. But when at last this has been done, and when he understands what he has done, his mind can travel upon the wings of light. In a twinkling he is able to make calculations which in physical terms express aeons of time or stupendous distance along the pathway of the infinite. And all this he can do with ten little symbols, which can be written upon a shred of paper, or told off upon the fingers of his hands.
- In preparing this paper I have had the assistance of Dr. Louis Charles Karpinski, who has put at my disposal the results of research in a field which he is making peculiarly his own. By permission of Messrs. Ginn several illustrations are reproduced from Smith and Karpinski, "The Hindu-Arabic Numerals." Ginn and Company, 1911, the best work upon the subject, and a work of which I have made free use.
- From Smith and Karpinski, "The Hindu-Arabic Numerals."
- From Smith and Karpinski, "The Hindu-Arabic Numerals."
- From Smith and Karpinski, "The Hindu-Arabic Numerals."
- From Smith and Karpinski, "The Hindu-Arabic Numerals."