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CHAPTER

PRONUNCIATION OF ORIENTAL NAMES vi

INDEX 153

PRONUNCIATION OF ORIENTAL NAMES

B, D, F, G, H, J, L, M, N, P, SH , T, TH , V, W, X, Z, as in English.

, see .

, see .

THE HINDU-ARABIC NUMERALS

EARLY IDEAS OF THEIR ORIGIN

It has long been recognized that the common numerals used in daily life are of comparatively recent origin. The number of systems of notation employed before the Christian era was about the same as the number of written languages, and in some cases a single language had several systems. The Egyptians, for example, had three systems of writing, with a numerical notation for each; the Greeks had two well-defined sets of numerals, and the Roman symbols for number changed more or less from century to century. Even to-day the number of methods of expressing numerical concepts is much greater than one would believe before making a study of the subject, for the idea that our common numerals are universal is far from being correct. It will be well, then, to think of the numerals that we still commonly call Arabic, as only one of many systems in use just before the Christian era. As it then existed the system was no better than many others, it was of late origin, it contained no zero, it was cumbersome and little used, and it had no particular promise. Not until centuries later did the system have any standing in the world of business and science; and had the place value which now characterizes it, and which requires a zero, been worked out in Greece, we might have been using Greek numerals to-day instead of the ones with which we are familiar.

Of the first number forms that the world used this is not the place to speak. Many of them are interesting, but none had much scientific value. In Europe the invention of notation was generally assigned to the eastern shores of the Mediterranean until the critical period of about a century ago,--sometimes to the Hebrews, sometimes to the Egyptians, but more often to the early trading Phoenicians.

The first of the Arabic writers mentioned is Al-Kind , who wrote five books on arithmetic and four books on the use of the Indian method of reckoning. Sened ibn `Al, the Jew, who was converted to Islam under the caliph Al-Mmn, is also given as the author of a work on the Hindu method of reckoning. Nevertheless, there is a possibility that some of the works ascribed to Sened ibn `Al are really works of Al-Khowrazm, whose name immediately precedes his. However, it is to be noted in this connection that Casiri also mentions the same writer as the author of a most celebrated work on arithmetic.

To Al-f, who died in 986 A.D., is also credited a large work on the same subject, and similar treatises by other writers are mentioned. We are therefore forced to the conclusion that the Arabs from the early ninth century on fully recognized the Hindu origin of the new numerals.

"Hec algorism' ars p'sens dicit' in qua Talib; indor fruim bis quinq; figuris.

"This boke is called the boke of algorim or augrym after lewder use. And this boke tretys of the Craft of Nombryng, the quych crafte is called also Algorym. Ther was a kyng of Inde the quich heyth Algor & he made this craft.... Algorisms, in the quych we use teen figurys of Inde."

EARLY HINDU FORMS WITH NO PLACE VALUE

While it is generally conceded that the scientific development of astronomy among the Hindus towards the beginning of the Christian era rested upon Greek or Chinese sources, yet their ancient literature testifies to a high state of civilization, and to a considerable advance in sciences, in philosophy, and along literary lines, long before the golden age of Greece. From the earliest times even up to the present day the Hindu has been wont to put his thought into rhythmic form. The first of this poetry--it well deserves this name, being also worthy from a metaphysical point of view--consists of the Vedas, hymns of praise and poems of worship, collected during the Vedic period which dates from approximately 2000 B.C. to 1400 B.C. Following this work, or possibly contemporary with it, is the Brahmanic literature, which is partly ritualistic , and partly philosophical . Our especial interest is in the Stras, versified abridgments of the ritual and of ceremonial rules, which contain considerable geometric material used in connection with altar construction, and also numerous examples of rational numbers the sum of whose squares is also a square, i.e. "Pythagorean numbers," although this was long before Pythagoras lived. Whitney places the whole of the Veda literature, including the Vedas, the Brhmaas, and the Stras, between 1500 B.C. and 800 B.C., thus agreeing with B?rk who holds that the knowledge of the Pythagorean theorem revealed in the Stras goes back to the eighth century B.C.

The importance of the Stras as showing an independent origin of Hindu geometry, contrary to the opinion long held by Cantor of a Greek origin, has been repeatedly emphasized in recent literature, especially since the appearance of the important work of Von Schroeder. Further fundamental mathematical notions such as the conception of irrationals and the use of gnomons, as well as the philosophical doctrine of the transmigration of souls,--all of these having long been attributed to the Greeks,--are shown in these works to be native to India. Although this discussion does not bear directly upon the origin of our numerals, yet it is highly pertinent as showing the aptitude of the Hindu for mathematical and mental work, a fact further attested by the independent development of the drama and of epic and lyric poetry.

It should be stated definitely at the outset, however, that we are not at all sure that the most ancient forms of the numerals commonly known as Arabic had their origin in India. As will presently be seen, their forms may have been suggested by those used in Egypt, or in Eastern Persia, or in China, or on the plains of Mesopotamia. We are quite in the dark as to these early steps; but as to their development in India, the approximate period of the rise of their essential feature of place value, their introduction into the Arab civilization, and their spread to the West, we have more or less definite information. When, therefore, we consider the rise of the numerals in the land of the Sindhu, it must be understood that it is only the large movement that is meant, and that there must further be considered the numerous possible sources outside of India itself and long anterior to the first prominent appearance of the number symbols.

Thereupon Vivamitra crya expresses his approval of the task, and asks to hear the "measure of the line" as far as yjana, the longest measure bearing name. This given, Buddha adds:

... "'And master! if it please, I shall recite how many sun-motes lie From end to end within a yjana.' Thereat, with instant skill, the little prince Pronounced the total of the atoms true. But Viswamitra heard it on his face Prostrate before the boy; 'For thou,' he cried, 'Art Teacher of thy teachers--thou, not I, Art Gr.'"

As to authentic histories, however, there exist in India none relating to the period before the Mohammedan era . About all that we know of the earlier civilization is what we glean from the two great epics, the Mahbhrata and the Rmyana, from coins, and from a few inscriptions.

It is with this unsatisfactory material, then, that we have to deal in searching for the early history of the Hindu-Arabic numerals, and the fact that many unsolved problems exist and will continue to exist is no longer strange when we consider the conditions. It is rather surprising that so much has been discovered within a century, than that we are so uncertain as to origins and dates and the early spread of the system. The probability being that writing was not introduced into India before the close of the fourth century B.C., and literature existing only in spoken form prior to that period, the number work was doubtless that of all primitive peoples, palpable, merely a matter of placing sticks or cowries or pebbles on the ground, of marking a sand-covered board, or of cutting notches or tying cords as is still done in parts of Southern India to-day.

The early Hindu numerals may be classified into three great groups, the Kharoh, the Brhm, and the word and letter forms; and these will be considered in order.

The Kharoh numerals are found in inscriptions formerly known as Bactrian, Indo-Bactrian, and Aryan, and appearing in ancient Gandhra, now eastern Afghanistan and northern Punjab. The alphabet of the language is found in inscriptions dating from the fourth century B.C. to the third century A.D., and from the fact that the words are written from right to left it is assumed to be of Semitic origin. No numerals, however, have been found in the earliest of these inscriptions, number-names probably having been written out in words as was the custom with many ancient peoples. Not until the time of the powerful King Aoka, in the third century B.C., do numerals appear in any inscriptions thus far discovered; and then only in the primitive form of marks, quite as they would be found in Egypt, Greece, Rome, or in various other parts of the world. These Aoka inscriptions, some thirty in all, are found in widely separated parts of India, often on columns, and are in the various vernaculars that were familiar to the people. Two are in the Kharoh characters, and the rest in some form of Brhm. In the Kharoh inscriptions only four numerals have been found, and these are merely vertical marks for one, two, four, and five, thus:

| || ||| ||||

In the so-called aka inscriptions, possibly of the first century B.C., more numerals are found, and in more highly developed form, the right-to-left system appearing, together with evidences of three different scales of counting,--four, ten, and twenty. The numerals of this period are as follows:

This system has many points of similarity with the Nabatean numerals in use in the first centuries of the Christian era. The cross is here used for four, and the Kharoh form is employed for twenty. In addition to this there is a trace of an analogous use of a scale of twenty. While the symbol for 100 is quite different, the method of forming the other hundreds is the same. The correspondence seems to be too marked to be wholly accidental.

It is not in the Kharoh numerals, therefore, that we can hope to find the origin of those used by us, and we turn to the second of the Indian types, the Brhm characters. The alphabet attributed to Brahm is the oldest of the several known in India, and was used from the earliest historic times. There are various theories of its origin, none of which has as yet any wide acceptance, although the problem offers hope of solution in due time. The numerals are not as old as the alphabet, or at least they have not as yet been found in inscriptions earlier than those in which the edicts of Aoka appear, some of these having been incised in Brhm as well as Kharoh. As already stated, the older writers probably wrote the numbers in words, as seems to have been the case in the earliest Pali writings of Ceylon.

The following numerals are, as far as known, the only ones to appear in the Aoka edicts:

These fragments from the third century B.C., crude and unsatisfactory as they are, are the undoubted early forms from which our present system developed. They next appear in the second century B.C. in some inscriptions in the cave on the top of the Nn Ght hill, about seventy-five miles from Poona in central India. These inscriptions may be memorials of the early Andhra dynasty of southern India, but their chief interest lies in the numerals which they contain.

There is considerable dispute as to what numerals are really found in these inscriptions, owing to the difficulty of deciphering them; but the following, which have been copied from a rubbing, are probably number forms:

The inscription itself, so important as containing the earliest considerable Hindu numeral system connected with our own, is of sufficient interest to warrant reproducing part of it in facsimile, as is done on page 24.

The next very noteworthy evidence of the numerals, and this quite complete as will be seen, is found in certain other cave inscriptions dating back to the first or second century A.D. In these, the Nasik cave inscriptions, the forms are as follows:

From this time on, until the decimal system finally adopted the first nine characters and replaced the rest of the Brhm notation by adding the zero, the progress of these forms is well marked. It is therefore well to present synoptically the best-known specimens that have come down to us, and this is done in the table on page 25.

TABLE SHOWING THE PROGRESS OF NUMBER FORMS IN INDIA

NUMERALS 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 1000 Aoka aka Aoka Ngar Nasik Katrapa Kuana Gupta Valhab Nepal Kaliga Vkaka

With respect to these numerals it should first be noted that no zero appears in the table, and as a matter of fact none existed in any of the cases cited. It was therefore impossible to have any place value, and the numbers like twenty, thirty, and other multiples of ten, one hundred, and so on, required separate symbols except where they were written out in words. The ancient Hindus had no less than twenty of these symbols, a number that was afterward greatly increased. The following are examples of their method of indicating certain numbers between one hundred and one thousand:

for 174 for 191 for 269 for 252 for 400 for 356

To these may be added the following numerals below one hundred, similar to those in the table:

for 90 for 70

We have thus far spoken of the Kharoh and Brhm numerals, and it remains to mention the third type, the word and letter forms. These are, however, so closely connected with the perfecting of the system by the invention of the zero that they are more appropriately considered in the next chapter, particularly as they have little relation to the problem of the origin of the forms known as the Arabic.

Having now examined types of the early forms it is appropriate to turn our attention to the question of their origin. As to the first three there is no question. The or is simply one stroke, or one stick laid down by the computer. The or represents two strokes or two sticks, and so for the and . From some primitive came the two of Egypt, of Rome, of early Greece, and of various other civilizations. It appears in the three Egyptian numeral systems in the following forms:

Hieroglyphic Hieratic Demotic

The last of these is merely a cursive form as in the Arabic , which becomes our 2 if tipped through a right angle. From some primitive came the Chinese symbol, which is practically identical with the symbols found commonly in India from 150 B.C. to 700 A.D. In the cursive form it becomes , and this was frequently used for two in Germany until the 18th century. It finally went into the modern form 2, and the in the same way became our 3.

There is, however, considerable ground for interesting speculation with respect to these first three numerals. The earliest Hindu forms were perpendicular. In the Nn Ght inscriptions they are vertical. But long before either the Aoka or the Nn Ght inscriptions the Chinese were using the horizontal forms for the first three numerals, but a vertical arrangement for four. Now where did China get these forms? Surely not from India, for she had them, as her monuments and literature show, long before the Hindus knew them. The tradition is that China brought her civilization around the north of Tibet, from Mongolia, the primitive habitat being Mesopotamia, or possibly the oases of Turkestan. Now what numerals did Mesopotamia use? The Babylonian system, simple in its general principles but very complicated in many of its details, is now well known. In particular, one, two, and three were represented by vertical arrow-heads. Why, then, did the Chinese write theirs horizontally? The problem now takes a new interest when we find that these Babylonian forms were not the primitive ones of this region, but that the early Sumerian forms were horizontal.

What interpretation shall be given to these facts? Shall we say that it was mere accident that one people wrote "one" vertically and that another wrote it horizontally? This may be the case; but it may also be the case that the tribal migrations that ended in the Mongol invasion of China started from the Euphrates while yet the Sumerian civilization was prominent, or from some common source in Turkestan, and that they carried to the East the primitive numerals of their ancient home, the first three, these being all that the people as a whole knew or needed. It is equally possible that these three horizontal forms represent primitive stick-laying, the most natural position of a stick placed in front of a calculator being the horizontal one. When, however, the cuneiform writing developed more fully, the vertical form may have been proved the easier to make, so that by the time the migrations to the West began these were in use, and from them came the upright forms of Egypt, Greece, Rome, and other Mediterranean lands, and those of Aoka's time in India. After Aoka, and perhaps among the merchants of earlier centuries, the horizontal forms may have come down into India from China, thus giving those of the Nn Ght cave and of later inscriptions. This is in the realm of speculation, but it is not improbable that further epigraphical studies may confirm the hypothesis.

As to the numerals above three there have been very many conjectures. The figure one of the Demotic looks like the one of the Sanskrit, the two like that of the Arabic, the four has some resemblance to that in the Nasik caves, the five to that on the Katrapa coins, the nine to that of the Kuana inscriptions, and other points of similarity have been imagined. Some have traced resemblance between the Hieratic five and seven and those of the Indian inscriptions. There have not, therefore, been wanting those who asserted an Egyptian origin for these numerals. There has already been mentioned the fact that the Kharoh numerals were formerly known as Bactrian, Indo-Bactrian, and Aryan. Cunningham was the first to suggest that these numerals were derived from the alphabet of the Bactrian civilization of Eastern Persia, perhaps a thousand years before our era, and in this he was supported by the scholarly work of Sir E. Clive Bayley, who in turn was followed by Canon Taylor. The resemblance has not proved convincing, however, and Bayley's drawings have been criticized as being affected by his theory. The following is part of the hypothesis:

B?hler rejects this hypothesis, stating that in four cases the facts are absolutely against it.

There is also some slight possibility that Chinese influence is to be seen in certain of the early forms of Hindu numerals.

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.

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