Read Ebook: The Hindu-Arabic Numerals by Karpinski Louis Charles Smith David Eugene

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More absurd is the hypothesis of a Greek origin, supposedly supported by derivation of the current symbols from the first nine letters of the Greek alphabet. This difficult feat is accomplished by twisting some of the letters, cutting off, adding on, and effecting other changes to make the letters fit the theory. This peculiar theory was first set up by Dasypodius , and was later elaborated by Huet.

A bizarre derivation based upon early Arabic sources is given by Kircher in his work on number mysticism. He quotes from Abenragel, giving the Arabic and a Latin translation and stating that the ordinary Arabic forms are derived from sectors of a circle, .

Of absolute nonsense about the origin of the symbols which we use much has been written. Conjectures, however, without any historical evidence for support, have no place in a serious discussion of the gradual evolution of the present numeral forms.

TABLE OF CERTAIN EASTERN SYSTEMS Siam Burma Malabar Tibet Ceylon Malayalam

LATER HINDU FORMS, WITH A PLACE VALUE

Before speaking of the perfected Hindu numerals with the zero and the place value, it is necessary to consider the third system mentioned on page 19,--the word and letter forms. The use of words with place value began at least as early as the 6th century of the Christian era. In many of the manuals of astronomy and mathematics, and often in other works in mentioning dates, numbers are represented by the names of certain objects or ideas. For example, zero is represented by "the void" , or "heaven-space" ; one by "stick" , "moon" , "earth" , "beginning" , "Brahma," or, in general, by anything markedly unique; two by "the twins" , "hands" , "eyes" , etc.; four by "oceans," five by "senses" or "arrows" ; six by "seasons" or "flavors"; seven by "mountain" , and so on. These names, accommodating themselves to the verse in which scientific works were written, had the additional advantage of not admitting, as did the figures, easy alteration, since any change would tend to disturb the meter.

Mention should also be made, in this connection, of a curious system of alphabetic numerals that sprang up in southern India. In this we have the numerals represented by the letters as given in the following table:

has the value 1,565,132, reading from right to left. This, the oldest specimen known of this notation, is given in a commentary on the Rigveda, representing the number of days that had elapsed from the beginning of the Kaliyuga. Burnell states that this system is even yet in use for remembering rules to calculate horoscopes, and for astronomical tables.

As already stated, however, the Hindu system as thus far described was no improvement upon many others of the ancients, such as those used by the Greeks and the Hebrews. Having no zero, it was impracticable to designate the tens, hundreds, and other units of higher order by the same symbols used for the units from one to nine. In other words, there was no possibility of place value without some further improvement. So the Nn Ght symbols required the writing of "thousand seven twenty-four" about like T 7, tw, 4 in modern symbols, instead of 7024, in which the seven of the thousands, the two of the tens , and the four of the units are given as spoken and the order of the unit is given by the place. To complete the system only the zero was needed; but it was probably eight centuries after the Nn Ght inscriptions were cut, before this important symbol appeared; and not until a considerably later period did it become well known. Who it was to whom the invention is due, or where he lived, or even in what century, will probably always remain a mystery. It is possible that one of the forms of ancient abacus suggested to some Hindu astronomer or mathematician the use of a symbol to stand for the vacant line when the counters were removed. It is well established that in different parts of India the names of the higher powers took different forms, even the order being interchanged. Nevertheless, as the significance of the name of the unit was given by the order in reading, these variations did not lead to error. Indeed the variation itself may have necessitated the introduction of a word to signify a vacant place or lacking unit, with the ultimate introduction of a zero symbol for this word.

To enable us to appreciate the force of this argument a large number, 8,443,682,155, may be considered as the Hindus wrote and read it, and then, by way of contrast, as the Greeks and Arabs would have read it.

As Woepcke pointed out, the reading of numbers of this kind shows that the notation adopted by the Hindus tended to bring out the place idea. No other language than the Sanskrit has made such consistent application, in numeration, of the decimal system of numbers. The introduction of myriads as in the Greek, and thousands as in Arabic and in modern numeration, is really a step away from a decimal scheme. So in the numbers below one hundred, in English, eleven and twelve are out of harmony with the rest of the -teens, while the naming of all the numbers between ten and twenty is not analogous to the naming of the numbers above twenty. To conform to our written system we should have ten-one, ten-two, ten-three, and so on, as we have twenty-one, twenty-two, and the like. The Sanskrit is consistent, the units, however, preceding the tens and hundreds. Nor did any other ancient people carry the numeration as far as did the Hindus.

It should also be mentioned as a matter of interest, and somewhat related to the question at issue, that Varha-Mihira used the word-system with place value as explained above.

The first kind of alphabetic numerals and also the word-system are plays upon, or variations of, position arithmetic, which would be most likely to occur in the country of its origin.

Many early writers remarked upon the diversity of Indian numeral forms. Al-Brn was probably the first; noteworthy is also Johannes Hispalensis, who gives the variant forms for seven and four. We insert on p. 49 a table of numerals used with place value. While the chief authority for this is B?hler, several specimens are given which are not found in his work and which are of unusual interest.

For purposes of comparison the modern Sanskrit and Arabic numeral forms are added.

Sanskrit, Arabic,

NUMERALS USED WITH PLACE VALUE

THE SYMBOL ZERO

What has been said of the improved Hindu system with a place value does not touch directly the origin of a symbol for zero, although it assumes that such a symbol exists. The importance of such a sign, the fact that it is a prerequisite to a place-value system, and the further fact that without it the Hindu-Arabic numerals would never have dominated the computation system of the western world, make it proper to devote a chapter to its origin and history.

"The earliest undoubted occurrence of a zero in India is an inscription at Gwalior, dated Samvat 933 . Where 50 garlands are mentioned , 50 is written . 270 is written ." The Bakhl Manuscript probably antedates this, using the point or dot as a zero symbol. Bayley mentions a grant of Jaika Rashtrak?ta of Bharuj, found at Okamandel, of date 738 A.D., which contains a zero, and also a coin with indistinct Gupta date 707 , but the reliability of Bayley's work is questioned. As has been noted, the appearance of the numerals in inscriptions and on coins would be of much later occurrence than the origin and written exposition of the system. From the period mentioned the spread was rapid over all of India, save the southern part, where the Tamil and Malayalam people retain the old system even to the present day.

Aside from its appearance in early inscriptions, there is still another indication of the Hindu origin of the symbol in the special treatment of the concept zero in the early works on arithmetic. Brahmagupta, who lived in Ujjain, the center of Indian astronomy, in the early part of the seventh century, gives in his arithmetic a distinct treatment of the properties of zero. He does not discuss a symbol, but he shows by his treatment that in some way zero had acquired a special significance not found in the Greek or other ancient arithmetics. A still more scientific treatment is given by Bhskara, although in one place he permits himself an unallowed liberty in dividing by zero. The most recently discovered work of ancient Indian mathematical lore, the Ganita-Sra-Sagraha of Mahvrcrya , while it does not use the numerals with place value, has a similar discussion of the calculation with zero.

Although the dot was used at first in India, as noted above, the small circle later replaced it and continues in use to this day. The Arabs, however, did not adopt the circle, since it bore some resemblance to the letter which expressed the number five in the alphabet system. The earliest Arabic zero known is the dot, used in a manuscript of 873 A.D. Sometimes both the dot and the circle are used in the same work, having the same meaning, which is the case in an Arabic MS., an abridged arithmetic of Jamshid, 982 A.H. . As given in this work the numerals are . The form for 5 varies, in some works becoming or ; is found in Egypt and appears in some fonts of type. To-day the Arabs use the 0 only when, under European influence, they adopt the ordinary system. Among the Chinese the first definite trace of zero is in the work of Tsin of 1247 A.D. The form is the circular one of the Hindus, and undoubtedly was brought to China by some traveler.

THE QUESTION OF THE INTRODUCTION OF THE NUMERALS INTO EUROPE BY BOETHIUS

Just as we were quite uncertain as to the origin of the numeral forms, so too are we uncertain as to the time and place of their introduction into Europe. There are two general theories as to this introduction. The first is that they were carried by the Moors to Spain in the eighth or ninth century, and thence were transmitted to Christian Europe, a theory which will be considered later. The second, advanced by Woepcke, is that they were not brought to Spain by the Moors, but that they were already in Spain when the Arabs arrived there, having reached the West through the Neo-Pythagoreans. There are two facts to support this second theory: the forms of these numerals are characteristic, differing materially from those which were brought by Leonardo of Pisa from Northern Africa early in the thirteenth century ; they are essentially those which tradition has so persistently assigned to Boethius , and which he would naturally have received, if at all, from these same Neo-Pythagoreans or from the sources from which they derived them. Furthermore, Woepcke points out that the Arabs on entering Spain would naturally have followed their custom of adopting for the computation of taxes the numerical systems of the countries they conquered, so that the numerals brought from Spain to Italy, not having undergone the same modifications as those of the Eastern Arab empire, would have differed, as they certainly did, from those that came through Bagdad. The theory is that the Hindu system, without the zero, early reached Alexandria , and that the Neo-Pythagorean love for the mysterious and especially for the Oriental led to its use as something bizarre and cabalistic; that it was then passed along the Mediterranean, reaching Boethius in Athens or in Rome, and to the schools of Spain, being discovered in Africa and Spain by the Arabs even before they themselves knew the improved system with the place value.

A recent theory set forth by Bubnov also deserves mention, chiefly because of the seriousness of purpose shown by this well-known writer. Bubnov holds that the forms first found in Europe are derived from ancient symbols used on the abacus, but that the zero is of Hindu origin. This theory does not seem tenable, however, in the light of the evidence already set forth.

Two questions are presented by Woepcke's theory: What was the nature of these Spanish numerals, and how were they made known to Italy? Did Boethius know them?

In form these obr numerals resemble our own much more closely than the Arab numerals do. They varied more or less, but were substantially as follows:

The question of the possible influence of the Egyptian demotic and hieratic ordinal forms has been so often suggested that it seems well to introduce them at this point, for comparison with the obr forms. They would as appropriately be used in connection with the Hindu forms, and the evidence of a relation of the first three with all these systems is apparent. The only further resemblance is in the Demotic 4 and in the 9, so that the statement that the Hindu forms in general came from this source has no foundation. The first four Egyptian cardinal numerals resemble more the modern Arabic.

This theory of the very early introduction of the numerals into Europe fails in several points. In the first place the early Western forms are not known; in the second place some early Eastern forms are like the obr, as is seen in the third line on p. 69, where the forms are from a manuscript written at Shiraz about 970 A.D., and in which some western Arabic forms, e.g. for 2, are also used. Probably most significant of all is the fact that the obr numerals as given by Sacy are all, with the exception of the symbol for eight, either single Arabic letters or combinations of letters. So much for the Woepcke theory and the meaning of the obr numerals. We now have to consider the question as to whether Boethius knew these obr forms, or forms akin to them.

This large question suggests several minor ones: Who was Boethius? Could he have known these numerals? Is there any positive or strong circumstantial evidence that he did know them? What are the probabilities in the case?

First, who was Boethius,--Divus Boethius as he was called in the Middle Ages? Anicius Manlius Severinus Boethius was born at Rome c. 475. He was a member of the distinguished family of the Anicii, which had for some time before his birth been Christian. Early left an orphan, the tradition is that he was taken to Athens at about the age of ten, and that he remained there eighteen years. He married Rusticiana, daughter of the senator Symmachus, and this union of two such powerful families allowed him to move in the highest circles. Standing strictly for the right, and against all iniquity at court, he became the object of hatred on the part of all the unscrupulous element near the throne, and his bold defense of the ex-consul Albinus, unjustly accused of treason, led to his imprisonment at Pavia and his execution in 524. Not many generations after his death, the period being one in which historical criticism was at its lowest ebb, the church found it profitable to look upon his execution as a martyrdom. He was accordingly looked upon as a saint, his bones were enshrined, and as a natural consequence his books were among the classics in the church schools for a thousand years. It is pathetic, however, to think of the medieval student trying to extract mental nourishment from a work so abstract, so meaningless, so unnecessarily complicated, as the arithmetic of Boethius.

He was looked upon by his contemporaries and immediate successors as a master, for Cassiodorus says to him: "Through your translations the music of Pythagoras and the astronomy of Ptolemy are read by those of Italy, and the arithmetic of Nicomachus and the geometry of Euclid are known to those of the West." Founder of the medieval scholasticism, distinguishing the trivium and quadrivium, writing the only classics of his time, Gibbon well called him "the last of the Romans whom Cato or Tully could have acknowledged for their countryman."

The second question relating to Boethius is this: Could he possibly have known the Hindu numerals? In view of the relations that will be shown to have existed between the East and the West, there can only be an affirmative answer to this question. The numerals had existed, without the zero, for several centuries; they had been well known in India; there had been a continued interchange of thought between the East and West; and warriors, ambassadors, scholars, and the restless trader, all had gone back and forth, by land or more frequently by sea, between the Mediterranean lands and the centers of Indian commerce and culture. Boethius could very well have learned one or more forms of Hindu numerals from some traveler or merchant.

To justify this statement it is necessary to speak more fully of these relations between the Far East and Europe. It is true that we have no records of the interchange of learning, in any large way, between eastern Asia and central Europe in the century preceding the time of Boethius. But it is one of the mistakes of scholars to believe that they are the sole transmitters of knowledge. As a matter of fact there is abundant reason for believing that Hindu numerals would naturally have been known to the Arabs, and even along every trade route to the remote west, long before the zero entered to make their place-value possible, and that the characters, the methods of calculating, the improvements that took place from time to time, the zero when it appeared, and the customs as to solving business problems, would all have been made known from generation to generation along these same trade routes from the Orient to the Occident. It must always be kept in mind that it was to the tradesman and the wandering scholar that the spread of such learning was due, rather than to the school man. Indeed, Avicenna in a short biography of himself relates that when his people were living at Bokhra his father sent him to the house of a grocer to learn the Hindu art of reckoning, in which this grocer was expert. Leonardo of Pisa, too, had a similar training.

The whole question of this spread of mercantile knowledge along the trade routes is so connected with the obr numerals, the Boethius question, Gerbert, Leonardo of Pisa, and other names and events, that a digression for its consideration now becomes necessary.

Even in very remote times, before the Hindu numerals were sculptured in the cave of Nn Ght, there were trade relations between Arabia and India. Indeed, long before the Aryans went to India the great Turanian race had spread its civilization from the Mediterranean to the Indus. At a much later period the Arabs were the intermediaries between Egypt and Syria on the west, and the farther Orient. In the sixth century B.C., Hecataeus, the father of geography, was acquainted not only with the Mediterranean lands but with the countries as far as the Indus, and in Biblical times there were regular triennial voyages to India. Indeed, the story of Joseph bears witness to the caravan trade from India, across Arabia, and on to the banks of the Nile. About the same time as Hecataeus, Scylax, a Persian admiral under Darius, from Caryanda on the coast of Asia Minor, traveled to northwest India and wrote upon his ventures. He induced the nations along the Indus to acknowledge the Persian supremacy, and such number systems as there were in these lands would naturally have been known to a man of his attainments.

A century after Scylax, Herodotus showed considerable knowledge of India, speaking of its cotton and its gold, telling how Sesostris fitted out ships to sail to that country, and mentioning the routes to the east. These routes were generally by the Red Sea, and had been followed by the Phoenicians and the Sabaeans, and later were taken by the Greeks and Romans.

In the fourth century B.C. the West and East came into very close relations. As early as 330, Pytheas of Massilia had explored as far north as the northern end of the British Isles and the coasts of the German Sea, while Macedon, in close touch with southern France, was also sending her armies under Alexander through Afghanistan as far east as the Punjab. Pliny tells us that Alexander the Great employed surveyors to measure the roads of India; and one of the great highways is described by Megasthenes, who in 295 B.C., as the ambassador of Seleucus, resided at Ptalpura, the present Patna.

The Hindus also learned the art of coining from the Greeks, or possibly from the Chinese, and the stores of Greco-Hindu coins still found in northern India are a constant source of historical information. The Rmyana speaks of merchants traveling in great caravans and embarking by sea for foreign lands. Ceylon traded with Malacca and Siam, and Java was colonized by Hindu traders, so that mercantile knowledge was being spread about the Indies during all the formative period of the numerals.

Greece must also have had early relations with China, for there is a notable similarity between the Greek and Chinese life, as is shown in their houses, their domestic customs, their marriage ceremonies, the public story-tellers, the puppet shows which Herodotus says were introduced from Egypt, the street jugglers, the games of dice, the game of finger-guessing, the water clock, the music system, the use of the myriad, the calendars, and in many other ways. In passing through the suburbs of Peking to-day, on the way to the Great Bell temple, one is constantly reminded of the semi-Greek architecture of Pompeii, so closely does modern China touch the old classical civilization of the Mediterranean. The Chinese historians tell us that about 200 B.C. their arms were successful in the far west, and that in 180 B.C. an ambassador went to Bactria, then a Greek city, and reported that Chinese products were on sale in the markets there. There is also a noteworthy resemblance between certain Greek and Chinese words, showing that in remote times there must have been more or less interchange of thought.

The Romans also exchanged products with the East. Horace says, "A busy trader, you hasten to the farthest Indies, flying from poverty over sea, over crags, over fires." The products of the Orient, spices and jewels from India, frankincense from Persia, and silks from China, being more in demand than the exports from the Mediterranean lands, the balance of trade was against the West, and thus Roman coin found its way eastward. In 1898, for example, a number of Roman coins dating from 114 B.C. to Hadrian's time were found at Pakl, a part of the Hazra district, sixteen miles north of Abbottbd, and numerous similar discoveries have been made from time to time.

Augustus speaks of envoys received by him from India, a thing never before known, and it is not improbable that he also received an embassy from China. Suetonius speaks in his history of these relations, as do several of his contemporaries, and Vergil tells of Augustus doing battle in Persia. In Pliny's time the trade of the Roman Empire with Asia amounted to a million and a quarter dollars a year, a sum far greater relatively then than now, while by the time of Constantine Europe was in direct communication with the Far East.

In view of these relations it is not beyond the range of possibility that proof may sometime come to light to show that the Greeks and Romans knew something of the number system of India, as several writers have maintained.

Returning to the East, there are many evidences of the spread of knowledge in and about India itself. In the third century B.C. Buddhism began to be a connecting medium of thought. It had already permeated the Himalaya territory, had reached eastern Turkestan, and had probably gone thence to China. Some centuries later the Chinese emperor sent an ambassador to India, and in 67 A.D. a Buddhist monk was invited to China. Then, too, in India itself Aoka, whose name has already been mentioned in this work, extended the boundaries of his domains even into Afghanistan, so that it was entirely possible for the numerals of the Punjab to have worked their way north even at that early date.

Furthermore, the influence of Persia must not be forgotten in considering this transmission of knowledge. In the fifth century the Persian medical school at Jondi-Sapur admitted both the Hindu and the Greek doctrines, and Firdus tells us that during the brilliant reign of Khosr I, the golden age of Pahlav literature, the Hindu game of chess was introduced into Persia, at a time when wars with the Greeks were bringing prestige to the Sassanid dynasty.

As to the Arabs, it is a mistake to feel that their activities began with Mohammed. Commerce had always been held in honor by them, and the Qoreish had annually for many generations sent caravans bearing the spices and textiles of Yemen to the shores of the Mediterranean. In the fifth century they traded by sea with India and even with China, and ira was an emporium for the wares of the East, so that any numeral system of any part of the trading world could hardly have remained isolated.

Long before the warlike activity of the Arabs, Alexandria had become the great market-place of the world. From this center caravans traversed Arabia to Hadramaut, where they met ships from India. Others went north to Damascus, while still others made their way along the southern shores of the Mediterranean. Ships sailed from the isthmus of Suez to all the commercial ports of Southern Europe and up into the Black Sea. Hindus were found among the merchants who frequented the bazaars of Alexandria, and Brahmins were reported even in Byzantium.

Such is a very brief r?sum? of the evidence showing that the numerals of the Punjab and of other parts of India as well, and indeed those of China and farther Persia, of Ceylon and the Malay peninsula, might well have been known to the merchants of Alexandria, and even to those of any other seaport of the Mediterranean, in the time of Boethius. The Brhm numerals would not have attracted the attention of scholars, for they had no zero so far as we know, and therefore they were no better and no worse than those of dozens of other systems. If Boethius was attracted to them it was probably exactly as any one is naturally attracted to the bizarre or the mystic, and he would have mentioned them in his works only incidentally, as indeed they are mentioned in the manuscripts in which they occur.

In answer therefore to the second question, Could Boethius have known the Hindu numerals? the reply must be, without the slightest doubt, that he could easily have known them, and that it would have been strange if a man of his inquiring mind did not pick up many curious bits of information of this kind even though he never thought of making use of them.

Let us now consider the third question, Is there any positive or strong circumstantial evidence that Boethius did know these numerals? The question is not new, nor is it much nearer being answered than it was over two centuries ago when Wallis expressed his doubts about it soon after Vossius had called attention to the matter. Stated briefly, there are three works on mathematics attributed to Boethius: the arithmetic, a work on music, and the geometry.

The genuineness of the arithmetic and the treatise on music is generally recognized, but the geometry, which contains the Hindu numerals with the zero, is under suspicion. There are plenty of supporters of the idea that Boethius knew the numerals and included them in this book, and on the other hand there are as many who feel that the geometry, or at least the part mentioning the numerals, is spurious. The argument of those who deny the authenticity of the particular passage in question may briefly be stated thus:

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