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The Hindu-Arabic Numerals c6: Chapter II. Early Hindu Forms with No Place Value

The Hindu-Arabic Numerals c6

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Chapter II. Early Hindu Forms with No Place Value

c6.2

While it is generally conceded that the scientific development of astronomy among the Hindus towards the beginning of the Christian era rested upon Greek¹ or Chinese² 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 view³ — 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.⁴ Following this work, or possibly contemporary with it, is the Brahmanic literature, which is partly ritualistic (the Brāhmaṇas), and partly philosophical (the Upanishads). Our especial interest is in the Sūtras, 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. Whitney¹ places the whole of the Veda literature, including the Vedas, the Brāhmaṇas, and the Sūtras, between 1500 b.c. and 800 b.c., thus agreeing with Bürk² who holds that the knowledge of the Pythagorean theorem revealed in the Sūtras goes back to the eighth century b.c.

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¹ Thibaut, *Astronomie, Astrologie und Mathematik*, Strassburg, 1899.

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² Gustave Schlegel, *Uranographie chinoise ou preuves directes que l'astronomie primitive est originaire de la Chine, et qu'elle a été empruntée par les anciens peuples occidentaux à la sphère chinoise; ouvrage accompagné d'un atlas céleste chinois et grec*, The Hague and Leyden, 1875.

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³ E. W. Hopkins, *The Religions of India*, Boston, 1898, p. 7.

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⁴ R. C. Dutt, *History of India*, London, 1906.

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¹ W. D. Whitney, *Sanskrit Grammar*, 3d ed., Leipzig, 1896.

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² “Das Āpastamba-Śulba-Sūtra,” *Zeitschrift der deutschen Morgenländischen Gesellschaft*, Vol. LV, p. 543, and Vol. LVI, p. 327.

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The importance of the Sūtras as showing an independent origin of Hindu geometry, contrary to the opinion long held by Cantor³ of a Greek origin, has been repeatedly emphasized in recent literature,⁴ especially since the appearance of the important work of Von Schroeder.⁵ 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.

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³ *Geschichte der Math.*, Vol. I, 2d ed., p. 595.

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⁴ L. von Schroeder, *Pythagoras und die Inder*, Leipzig, 1884; H. Vogt, “Haben die alten Inder den Pythagoreischen Lehrsatz und das Irrationale gekannt?” *Bibliotheca Mathematica*, Vol. VII (3), pp. 6–20; A. Bürk, loc. cit.; Max Simon, *Geschichte der Mathematik im Altertum*, Berlin, 1909, pp. 137–165; three Sūtras are translated in part by Thibaut, *Journal of the Asiatic Society of Bengal*, 1875, and one appeared in *The Pandit*, 1875; Beppo Levi, “Osservazioni e congetture sopra la geometria degli indiani,” *Bibliotheca Mathematica*, Vol. IX (3), 1908, pp. 97–105.

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⁵ Loc. cit.; also *Indiens Literatur und Cultur*, Leipzig, 1887.

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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,¹ 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.

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¹ It is generally agreed that the name of the river Sindhu, corrupted by western peoples to Hindhu, Indos, Indus, is the root of Hindustan and of India. Reclus, *Asia*, English ed., Vol. III, p. 14.

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No one attempts to examine any detail in the history of ancient India without being struck with the great dearth of reliable material.² So little sympathy have the people with any save those of their own caste that a general literature is wholly lacking, and it is only in the observations of strangers that any all-round view of scientific progress is to be found. There is evidence that primary schools existed in earliest times, and of the seventy-two recognized sciences writing and arithmetic were the most prized.¹ In the Vedic period, say from 2000 to 1400 b.c., there was the same attention to astronomy that was found in the earlier civilizations of Babylon, China, and Egypt, a fact attested by the Vedas themselves.² Such advance in science presupposes a fair knowledge of calculation, but of the manner of calculating we are quite ignorant and probably always shall be. One of the Buddhist sacred books, the *Lalitavistara*, relates that when the Bōdhisattva³ was of age to marry, the father of Gopa, his intended bride, demanded an examination of the five hundred suitors, the subjects including arithmetic, writing, the lute, and archery. Having vanquished his rivals in all else, he is matched against Arjuna the great arithmetician and is asked to express numbers greater than 100 koṭis.⁴ In reply he gave a scheme of number names as high as 10⁵³, adding that he could proceed as far as 10⁴²¹,⁵ all of which suggests the system of Archimedes and the unsettled question of the indebtedness of the West to the East in the realm of ancient mathematics.⁶ Sir Edwin Arnold, in *The Light of Asia*, does not mention this part of the contest, but he speaks of Buddha's training at the hands of the learned Viṣvamitra:

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² See the comments of Oppert, *On the Original Inhabitants of Bharatavarṣa or India*, London, 1893, p. 1.

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¹ A. Hillebrandt, *Alt-Indien*, Breslau, 1899, p. 111. Fragmentary records relate that Khāravela, king of Kaliṅga, learned as a boy *lekhā* (writing), *gaṇanā* (reckoning), and *rūpa* (arithmetic applied to monetary affairs and mensuration), probably in the 5th century b.c. [Bühler, *Indische Palaeographie*, Strassburg, 1896, p. 5.]

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² R. C. Dutt, *A History of Civilization in Ancient India*, London, 1893, Vol. I, p. 174.

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³ The Buddha. The date of his birth is uncertain. Sir Edwin Arnold put it c. 620 b.c.

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⁴ I.e. 100·10⁷.

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⁵ There is some uncertainty about this limit.

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⁶ This problem deserves more study than has yet been given it. A beginning may be made with Comte Goblet d'Alviella, *Ce que l'Inde doit à la Grèce*, Paris, 1897, and H. G. Keene's review, "The Greeks in India," in the *Calcutta Review*, Vol, CXIV, 1902, p. 1. See also F.

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"And Viswamitra said, 'It is enough,
Let us to numbers. After me repeat
Your numeration till we reach the lakh,¹
One, two, three, four, to ten, and then by tens
To hundreds, thousands.' After him the child
Named digits, decads, centuries, nor paused,
The round lakh reached, but softly murmured on,
Then comes the kōti, nahut, ninnahut,
Khamba, viskhamba, abab, attata,
To kumuds, gundhikas, and utpalas,
By pundarīkas into padumas,
Which last is how you count the utmost grains
Of Hastagiri ground to finest dust;²
But beyond that a numeration is,
The Kātha, used to count the stars of night,
The Kōti-Kātha, for the ocean drops;
Ingga, the calculus of circulars;
Sarvanikchepa, by the which you deal
With all the sands of Gunga, till we come
To Antah-Kalpas, where the unit is
The sands of the ten crore Gungas. If one seeks
More comprehensive scale, th' arithmic mounts
By the Asankya, which is the tale
Of all the drops that in ten thousand years
Would fall on all the worlds by daily rain;
Thence unto Maha Kalpas, by the which
The gods compute their future and their past.'"

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¹ I.e. to 100,000. The lakh is still the common large unit in India, like the myriad in ancient Greece and the million in the West.

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² This again suggests the *Psammites*, or *De harenae numero* as it is called in the 1544 edition of the *Opera* of Archimedes, a work in which the great Syracusan proposes to show to the king "by geometric proofs which you can follow, that the numbers which have been named by us . . . are sufficient to exceed not only the number of a sand-heap as large as the whole earth, but one as large as the universe." For a list of early editions of this work see D. E. Smith, *Rara Arithmetica*, Boston, 1909, p. 227.

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Woepcke, *Propagation*, p. 253; G. R. Kaye, loc. cit., p. 475 seq., and "The Source of Hindu Mathematics," *Journal of the Royal Asiatic Society*, July, 1910, pp. 749–760; G. Thibaut, *Astronomie, Astrologie und Mathematik*, pp. 43–50 and 76–79. It will be discussed more fully in Chapter VI.

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Thereupon Viṣvamitra Ācārya¹ expresses his approval of the task, and asks to hear the "measure of the line" as far as yōjana, the longest measure bearing name. This given, Buddha adds:

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¹ *I.e.* the Wise.

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. . . "'And master! if it please,
I shall recite how many sun-motes lie
From end to end within a yōjana.'
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 Gūrū.'"

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It is needless to say that this is far from being history. And yet it puts in charming rhythm only what the ancient *Lalitavistara* relates of the number-series of the Buddha's time. While it extends beyond all reason, nevertheless it reveals a condition that would have been impossible unless arithmetic had attained a considerable degree of advancement.

c6.31

To this pre-Christian period belong also the *Vedāṅgas*, or "limbs for supporting the Veda," part of that great branch of Hindu literature known as *Smṛiti* (recollection), that which was to be handed down by tradition. Of these the sixth is known as *Jyotiṣa* (astronomy), a short treatise of only thirty-six verses, written not earlier than 300 B.C., and affording us some knowledge of the extent of number work in that period.² The Hindus also speak of eighteen ancient Siddhāntas or astronomical works, which, though mostly lost, confirm this evidence.¹

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² Sir Monier Monier-Williams, *Indian Wisdom*, 4th ed., London, 1893, pp. 144, 177. See also J. C. Marshman, *Abridgment of the History of India*, London, 1893, p. 2.

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¹ For a list and for some description of these works see R. C. Dutt, *A History of Civilization in Ancient India*, Vol. II, p. 121.

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As to authentic histories, however, there exist in India none relating to the period before the Mohammedan era (622 A.D.). About all that we know of the earlier civilization is what we glean from the two great epics, the Mahābhārata ² and the Rāmāyana, from coins, and from a few inscriptions.³

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² Professor Ramkrishna Gopal Bhandarkar fixes the date as the fifth century B.C. ["Consideration of the Date of the Mahābhārata," in the *Journal of the Bombay Branch of the R. A. Soc.*, Bombay, 1873, Vol. X, p. 2.]

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³ Marshman, loc. cit., p. 2.

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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,⁴ 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.⁵

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⁴ A. C. Burnell, *South Indian Palæography*, 2d ed., London, 1878, p. 1, seq.

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⁵ This extensive subject of palpable arithmetic, essentially the history of the abacus, deserves to be treated in a work by itself.

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The early Hindu numerals¹ may be classified into three great groups, (1) the Kharoṣṭhī, (2) the Brāhmī, and (3) the word and letter forms; and these will be considered in order.

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¹ The following are the leading sources of information upon this subject: G. Bühler, *Indische Palaeographie*, particularly chap. vi; A. C. Burnell, *South Indian Palæography*, 2d ed., London, 1878, where tables of the various Indian numerals are given in Plate XXIII; E. C. Bayley, "On the Genealogy of Modern Numerals," *Journal of the Royal Asiatic Society*, Vol. XIV, part 3, and Vol. XV, part 1, and reprint, London, 1882; I. Taylor, in *The Academy*, January 28, 1882, with a repetition of his argument in his work *The Alphabet*, London, 1883, Vol. II, p. 265, based on Bayley; G. R. Kaye, loc. cit., in some respects one of the most critical articles thus far published; J. C. Fleet, *Corpus inscriptionum Indicarum*, London, 1888, Vol. III, with facsimiles of many Indian inscriptions, and *Indian Epigraphy*, Oxford, 1907, reprinted from the *Imperial Gazetteer of India*, Vol. II, pp. 1–88, 1907; G. Thibaut, loc. cit., *Astronomie* etc.; R. Caldwell, *Comparative Grammar of the Dravidian Languages*, London, 1856, p. 262 seq.; and *Epigraphia Indica* (official publication of the government of India), Vols. I–IX. Another work of Bühler's, *On the Origin of the Indian Brāhma Alphabet*, is also of value.

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The Kharoṣṭhī numerals are found in inscriptions formerly known as Bactrian, Indo-Bactrian, and Aryan, and appearing in ancient Gandhāra, 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 Aśoka, 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 Aśoka¹ 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 Kharoṣṭhī characters, and the rest in some form of Brāhmī. In the Kharoṣṭhī inscriptions only four numerals have been found, and these are merely vertical marks for one, two, four, and five, thus:

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¹ The earliest work on the subject was by James Prinsep, "On the Inscriptions of Piyadasi or Aśoka," etc., *Journal of the Asiatic Society of Bengal*, 1838, following a preliminary suggestion in the same journal in 1837. See also "Aśoka Notes," by V. A. Smith, *The Indian Antiquary*, Vol. XXXVII, 1908, p. 24 seq., Vol. XXXVIII, pp. 151–159, June, 1909; *The Early History of India*, 2d ed., Oxford, 1908, p. 154; J. F. Fleet, "The Last Words of Aśoka," *Journal of the Royal Asiatic Society*, October, 1909, pp. 981–1016; E. Senart, *Les inscriptions de Piyadasi*, 2 vols., Paris, 1887.

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In the so-called Śaka 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:

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There are several noteworthy points to be observed in studying this system. In the first place, it is probably not as early as that shown in the Nānā Ghāt forms hereafter given, although the inscriptions themselves at Nānā Ghāt are later than those of the Aśoka period. The four is to this system what the X was to the Roman, probably a canceling of three marks as a workman does to-day for five, or a laying of one stick across three others. The ten has never been satisfactorily explained. It is similar to the A of the Kharoṣṭhī alphabet, but we have no knowledge as to why it was chosen. The twenty is evidently a ligature of two tens, and this in turn suggested a kind of radix, so that ninety was probably written in a way reminding one of the quatre-vingt-dix of the French. The hundred is unexplained, although it resembles the letter *ta* or *tra* of the Brāhmī alphabet with 1 before (to the right of) it. The two hundred is only a variant of the symbol for hundred, with two vertical marks.¹

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¹ For a discussion of the minor details of this system, see Bühler, loc. cit., p. 73.

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This system has many points of similarity with the Nabatean numerals² in use in the first centuries of the Christian era. The cross is here used for four, and the Kharoṣṭhī 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.

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² Julius Euting, *Nabatäische Inschriften aus Arabien*, Berlin, 1885, pp. 96–97, with a table of numerals.

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It is not in the Kharoṣṭhī 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 Brāhmī characters. The alphabet attributed to Brahmā 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,¹ 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 Aśoka appear, some of these having been incised in Brāhmī as well as Kharoṣṭhī. 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.²

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¹ For the five principal theories see Bühler, loc. cit., p. 10.

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² Bayley, loc. cit., reprint p. 3.

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The following numerals are, as far as known, the only ones to appear in the Aśoka edicts:³

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³ Bühler, loc. cit.; *Epigraphia Indica*, Vol. III, p. 134; *Indian Antiquary*, Vol. VI, p. 155 seq., and Vol. X, p. 107.

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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 Nānā Ghāt 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.

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The cave was made as a resting-place for travelers ascending the hill, which lies on the road from Kalyāna to Junar. It seems to have been cut out by a descendant of King Śātavāhana,¹ for inside the wall opposite the entrance are representations of the members of his family, much defaced, but with the names still legible. It would seem that the excavation was made by order of a king named Vedisiri, and “the inscription contains a list of gifts made on the occasion of the performance of several *yagnas* or religious sacrifices,” and numerals are to be seen in no less than thirty places.²

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¹ Pandit Bhagavānlāl Indrājī, “On Ancient Nāgārī Numeration; from an Inscription at Nānēghāt,” *Journal of the Bombay Branch of the Royal Asiatic Society*, 1876, Vol. XII, p. 404.

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² Ib., p. 405. He gives also a plate and an interpretation of each numeral.

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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:³

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³ These may be compared with Bühler’s drawings, loc. cit.; with Bayley, loc. cit., p. 337 and plates; and with Bayley’s article in the *Encyclopædia Britannica*, 9th ed., art. “Numerals.”

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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.

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*Nânâghât Inscriptions*

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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 Nasik¹ cave inscriptions, the forms are as follows:

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¹ E. Senart, "The Inscriptions in the Caves at Nasik," *Epigraphia Indica*, Vol. VIII, pp. 59-96; "The Inscriptions in the Cave at Karle," *Epigraphia Indica*, Vol. VII, pp. 47-74; Bühler, *Palaeographie*, Tafel IX.

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From this time on, until the decimal system finally adopted the first nine characters and replaced the rest of the Brāhmī notation by adding the zero, the progress of these forms is well marked. It is therefore well to present

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c Brāhmī numerals, Aśoka inscriptions, c. 250 B.C. *Indian Antiquary*, Vol. VI, p. 155 seq.

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j Bendall's Table of Numerals, in *Cat. Sansk. Budd. MSS.*, British Museum.

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k *Indian Antiquary*, Vol. XIII, 120; *Epigraphia Indica*, Vol. III, 127 ff.

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Table Showing the Progress of Number Forms in India

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a Kharoṣṭhī numerals, Aśoka inscriptions, c. 250 B.C. Senart, *Notes d'épigraphie indienne*. Given by Bühler, loc. cit., Tafel I.

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b Same, Śaka inscriptions, probably of the first century B.C. Senart, loc. cit.; Bühler, loc. cit.

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d Same, Nānā Ghāt inscriptions, c. 150 B.C. Bhagavanlāl Indrājī, *On Ancient Nāgarī Numeration*, loc. cit. Copied from a squeeze of the original.

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e Same, Nasik inscription, c. 100 B.C. Burgess, *Archeological Survey Report, Western India*; Senart, *Epigraphia Indica*, Vol. VII, pp. 47–79, and Vol. VIII, pp. 59–96.

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f Kṣatrapa coins, c. 200 A.D. *Journal of the Royal Asiatic Society*, 1890, p. 639.

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g Kuṣana inscriptions, c. 150 A.D. *Epigraphia Indica*, Vol. I, p. 381, and Vol. II, p. 201.

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h Gupta Inscriptions, c. 300 A.D. to 450 A.D. Fleet, loc. cit., Vol. III.

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i Valhabī, c. 600 A.D. *Corpus*, Vol. III.

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l Fleet, loc. cit.

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[Most of these numerals are given by Bühler, loc. cit., Tafel IX.]

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synoptically the best-known specimens that have come down to us, and this is done in the table on page 25.¹

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¹ See Fleet, loc. cit. See also T. Benfey, *Sanskrit Grammar*, London, 1863, p. 217; M. R. Kále, *Higher Sanskrit Grammar*, 2d ed., Bombay, 1898, p. 110, and other authorities as cited.

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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,² a number that was afterward greatly increased. The following are examples of their method of indicating certain numbers between one hundred and one thousand:

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² Bayley, loc. cit., p. 335.

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³ From a copper plate of 493 A.D., found at Kārītalāī, Central India. [Fleet, loc. cit., Plate XVI.] It should be stated, however, that many of these copper plates, being deeds of property, have forged dates so as to give the appearance of antiquity of title. On the other hand, as Colebrooke long ago pointed out, a successful forgery has to imitate the writing of the period in question, so that it becomes evidence well worth considering, as shown in Chapter III.

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⁴ From a copper plate of 510 A.D., found at Majhgawāin, Central India. [Fleet, loc. cit., Plate XIV.]

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⁵ From an inscription of 588 A.D., found at Bōdh-Gayā, Bengal Presidency. [Fleet, loc. cit., Plate XXIV.]

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⁶ From a copper plate of 571 A.D., found at Māliyā, Bombay Presidency. [Fleet, loc. cit., Plate XXIV.]

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⁷ From a Bijayagadh pillar inscription of 372 A.D. [Fleet, loc. cit., Plate XXXVI, C.]

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⁸ From a copper plate of 434 A.D. [*Indian Antiquary*, Vol. I, p. 60.]

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

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[Figure:] 𑁦¹ for 90 [Figure:] 𑁥² for 70

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¹ Gadhwa inscription, *c.* 417 A.D. [Fleet, loc. cit., Plate IV, D.]

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² Kārītalāī plate of 493 A.D., referred to above.

c6.100

We have thus far spoken of the Kharoṣṭhī and Brāhmī 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.

c6.101

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:

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Hieroglyphic || Hieratic [Figure:] Demotic [Figure:] [Figure:]

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The last of these is merely a cursive form as in the Arabic ۲, 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 Z, 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.

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There is, however, considerable ground for interesting speculation with respect to these first three numerals. The earliest Hindu forms were perpendicular. In the Nānā Ghāt inscriptions they are vertical. But long before either the Aśoka or the Nānā Ghāt inscriptions the Chinese were using the horizontal forms for the first three numerals, but a vertical arrangement for four.¹ Now where did China get these forms? Surely not from India, for she had them, as her monuments and literature² 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.³ 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.¹

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¹ It seems evident that the Chinese four, curiously enough called "eight in the mouth," is only a cursive ||||.

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² Chalfont, F. H., *Memoirs of the Carnegie Museum*, Vol. IV, no. 1; J. Hager, *An Explanation of the Elementary Characters of the Chinese*, London, 1801.

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³ H. V. Hilprecht, *Mathematical, Metrological and Chronological Tablets from the Temple Library at Nippur*, Vol. XX, part I, of Series A, *Cuneiform Texts Published by the Babylonian Expedition of the University of Pennsylvania*, 1906; A. Eisenlohr, *Ein altbabylonischer Felderplan*, Leipzig, 1906; Maspero, *Dawn of Civilization*, p. 773.

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¹ Sir H. H. Howard, "On the Earliest Inscriptions from Chaldea," *Proceedings of the Society of Biblical Archæology*, XXI, p. 301, London, 1899.

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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 Aśoka's time in India. After Aśoka, and perhaps among the merchants of earlier centuries, the horizontal forms may have come down into India from China, thus giving those of the Nānā Ghāt 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.

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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 (reversed) like that of the Arabic, the four has some resemblance to that in the Nasik caves, the five (reversed) to that on the Kṣatrapa coins, the nine to that of the Kuṣana 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.¹ There has already been mentioned the fact that the Kharoṣṭhī numerals were formerly known as Bactrian, Indo-Bactrian, and Aryan. Cunningham² 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,³ who in turn was followed by Canon Taylor.⁴ 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:¹

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¹ For a bibliography of the principal hypotheses of this nature see Bühler, loc. cit., p. 77. Bühler (p. 78) feels that of all these hypotheses that which connects the Brāhmī with the Egyptian numerals is the most plausible, although he does not adduce any convincing proof. Th. Henri Martin, “Les signes numéraux et l’arithmétique chez les peuples de l’antiquité et du moyen âge” (being an examination of Cantor’s *Mathematische Beiträge zum Culturleben der Völker*), *Annali di matematica pura ed applicata*, Vol. V, Rome, 1864, pp. 8, 70. Also, same author, “Recherches nouvelles sur l’origine de notre système de numération écrite,” *Revue Archéologique*, 1857, pp. 36, 55. See also the tables given later in this work.

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² *Journal of the Royal Asiatic Society, Bombay Branch*, Vol. XXIII.

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³ Loc. cit., reprint, Part I, pp. 12, 17. Bayley’s deductions are generally regarded as unwarranted.

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⁴ *The Alphabet*, London, 1883, Vol. II, pp. 265, 266, and *The Academy* of Jan. 28, 1882.

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¹ Taylor, *The Alphabet*, loc. cit., table on p. 266.

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| *Numeral* | *Hindu* | *Bactrian* | *Sanskrit* |
| 4 | [Figure:] | [Figure:] =ch | chatur, Lat. quattuor |
| 5 | [Figure:] | [Figure:] =p | pancha, Gk. πε′ντε |
| 6 | [Figure:] | [Figure:] =s | .ṣaṣ / sapta &The s and ṣ are interchanged as / &occasionally in N.W. India |
| 7 | [Figure:] | [Figure:] =ṣ | |

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Bühler² rejects this hypothesis, stating that in four cases (four, six, seven, and ten) the facts are absolutely against it.

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² Bühler, *On the Origin of the Indian Brāhma Alphabet*, Strassburg, 1898, footnote, pp. 52, 53.

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While the relation to ancient Bactrian forms has been generally doubted, it is agreed that most of the numerals resemble Brāhmī letters, and we would naturally expect them to be initials.³ But, knowing the ancient pronunciation of most of the number names,⁴ we find this not to be the case. We next fall back upon the hypothesis that they represent the order of letters¹ in the ancient alphabet. From what we know of this order, however, there seems also no basis for this assumption. We have, therefore, to confess that we are not certain that the numerals were alphabetic at all, and if they were alphabetic we have no evidence at present as to the basis of selection. The later forms may possibly have been alphabetical expressions of certain syllables called *akṣaras*, which possessed in Sanskrit fixed numerical values,² but this is equally uncertain with the rest. Bayley also thought³ that some of the forms were Phœnician, as notably the use of a circle for twenty, but the resemblance is in general too remote to be convincing.

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³ Albrecht Weber, *History of Indian Literature*, English ed., Boston, 1878, p. 256: "The Indian figures from 1–9 are abbreviated forms of the initial letters of the numerals themselves . . .: the zero, too, has arisen out of the first letter of the word ṣunya (empty) (it occurs even in Piṅgala). It is the decimal place value of these figures which gives them significance." C. Henry, "Sur l'origine de quelques notations mathématiques," *Revue Archéologique*, June and July, 1879, attempts to derive the Boethian forms from the initials of Latin words. See also J. Prinsep, "Examination of the Inscriptions from Girnar in Gujerat, and Dhauli in Cuttach," *Journal of the Asiatic Society of Bengal*, 1838, especially Plate XX, p. 348; this was the first work on the subject.

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⁴ Bühler, *Palaeographie*, p. 75, gives the list, with the list of letters (p. 76) corresponding to the number symbols.

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¹ For a general discussion of the connection between the numerals and the different kinds of alphabets, see the articles by U. Ceretti, "Sulla origine delle cifre numerali moderne," *Rivista di fisica, matematica e scienze naturali*, Pisa and Pavia, 1909, anno X, numbers 114, 118, 119, and 120, and continuation in 1910.

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² This is one of Bühler's hypotheses. See Bayley, loc. cit., reprint p. 4; a good bibliography of original sources is given in this work, p. 38.

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³ Loc. cit., reprint, part I, pp. 12, 17. See also Burnell, loc. cit., p. 64, and tables in plate XXIII.

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There is also some slight possibility that Chinese influence is to be seen in certain of the early forms of Hindu numerals.⁴

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⁴ This was asserted by G. Hager (*Memoria sulle cifre arabiche*, Milan, 1813, also published in *Fundgruben des Orients*, Vienna, 1811, and in *Bibliothèque Britannique*, Geneva, 1812). See also the recent article by Major Charles E. Woodruff, "The Evolution of Modern Numerals from Tally Marks," *American Mathematical Monthly*, August-September, 1909. Biernatzki, "Die Arithmetik der Chinesen," *Crelle's Journal für die reine und angewandte Mathematik*, Vol. LII, 1857, pp. 59-96, also asserts the priority of the Chinese claim for a place system and the zero, but upon the flimsiest authority. Ch. de Paravey, *Essai sur l'origine unique et hiéroglyphique des chiffres et des lettres de tous les peuples*, Paris, 1826; G. Kleinwächter, "The Origin of the Arabic Numerals," *China Review*, Vol. XI, 1882-1883, pp. 379-381, Vol. XII, pp. 28-30; Biot, "Note sur la connaissance que les Chinois ont eue de la valeur de position des chiffres," *Journal Asiatique*, 1839, pp. 497–502. A. Terrien de Lacouperie, "The Old Numerals, the Counting-Rods and the Swan-Pan in China," *Numismatic Chronicle*, Vol. III (3), pp. 297–340, and Crowder B. Moseley, "Numeral Characters: Theory of Origin and Development," *American Antiquarian*, Vol. XXII, pp. 279–284, both propose to derive our numerals from Chinese characters, in much the same way as is done by Major Woodruff, in the article above cited.

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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.¹ 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 Dasypodius² (Conrad Rauhfuss), and was later elaborated by Huet.³

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¹ The Greeks, probably following the Semitic custom, used nine letters of the alphabet for the numerals from 1 to 9, then nine others for 10 to 90, and further letters to represent 100 to 900. As the ordinary Greek alphabet was insufficient, containing only twenty-four letters, an alphabet of twenty-seven letters was used.

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² *Institutiones mathematicae*, 2 vols., Strassburg, 1593–1596, a somewhat rare work from which the following quotation is taken:

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³ P. D. Huet, *Demonstratio evangelica*, Paris, 1769, note to p. 139 on p. 647: "Ab Arabibus vel ab Indis inventas esse, non vulgus eruditorum modo, sed doctissimi quique ad hanc diem arbitrati sunt. Ego vero falsum id esse, merosque esse Graecorum characteres aio; à librariis Graecae linguae ignaris interpolatos, et diuturna scribendi consuetudine corruptos. Nam primum 1 apex fuit, seu virgula, nota *μονάδος*. 2, est ipsum *β* extremis suis truncatum. *γ*, si in sinistram partem inclinaveris & cauda mutilaveris & sinistrum cornu sinistrorsum flexeris, fiet 3. Res ipsa loquitur 4 ipsissimum esse *Δ*, cujus crus sinistrum erigitur *κατὰ κάθετον*, & infra basim descendit; basis vero ipsa ultra crus producta eminet. Vides quam 5 simile sit *τῷ ε*; infimo tantum semicirculo, qui sinistrorsum patebat, dextrorsum converso. *ἐπίσημον βαῦ* quod ita notabatur *ϛ*, rotundato ventre, pede detracto, peperit *τὸ* 6. Ex Z basi sua mutilato, ortum est *τὸ* 7. Si H inflexis introrsum apicibus in rotundiorem & commodiorem formam mutaveris, exurget *τὸ* 8. At 9 ipsissimum est *ϑ*." I. Weidler, *Spicilegium observationum ad historiam notarum numeralium*, Wittenberg, 1755, derives them from the Hebrew letters; Dom Augustin Calmet, "Recherches sur l'origine des chiffres d'arithmétique," *Mémoires pour l'histoire des sciences et des beaux arts*, Trévoux, 1707 (pp. 1620–1635, with two plates), derives the current symbols from the Romans, stating that they are relics of the ancient "Notae Tironianae." These "notes" were part of a system of shorthand invented, or at least perfected, by Tiro, a slave who was freed by Cicero. L. A. Sedillot, "Sur l'origine de nos chiffres," *Atti dell' Accademia pontificia dei nuovi Lincei*, Vol. XVIII, 1864–1865, pp. 316–322, derives the Arabic forms from the Roman numerals.

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"*Quis est harum Cyphrarum autor?*
"A quibus hae usitatae syphrarum notae sint inventae: hactenus incertum fuit: meo tamen iudicio, quod exiguum esse fateor: a graecis librarijs (quorum olim magna fuit copia) literae Graecorum quibus veteres Graeci tamquam numerorum notis sunt usi: fuerunt corruptae. vt ex his licet videre.

"*Graecorum Literae corruptae.*

[Figure:]

"*Sed qua ratione graecorum literae ita fuerunt corruptae?*

"Finxerunt has corruptas Graecorum literarum notas: vel abiectione vt in nota binarij numeri, vel additione vt in ternarij, vel inuersione vt in septenarij, numeri nota, nostrae notae, quibus hodie utimur: ab his sola differunt elegantia, vt apparet."

See also Bayer, *Historia regni Graecorum Bactriani*, St. Petersburg, 1738, pp. 129–130, quoted by Martin, *Recherches nouvelles*, etc., loc. cit.

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A bizarre derivation based upon early Arabic (c. 1040 A.D.) sources is given by Kircher in his work¹ on number mysticism. He quotes from Abenragel,² giving the Arabic and a Latin translation³ and stating that the ordinary Arabic forms are derived from sectors of a circle, ⊕.

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¹ Athanasius Kircher, *Arithmologia sive De abditis Numerorum mysterijs qua origo, antiquitas & fabrica Numerorum exponitur*, Rome, 1665.

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² See Suter, *Die Mathematiker und Astronomen der Araber*, p. 100.

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³ "Et hi numeri sunt numeri Indiani, a Brachmanis Indiae Sapientibus ex figura circuli secti inuenti,"

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Out of all these conflicting theories, and from all the resemblances seen or imagined between the numerals of the West and those of the East, what conclusions are we prepared to draw as the evidence now stands? Probably none that is satisfactory. Indeed, upon the evidence at hand we might properly feel that everything points to the numerals as being substantially indigenous to India. And why should this not be the case? If the king Srong-tsan-Gampo (639 A.D.), the founder of Lhāsa,¹ could have set about to devise a new alphabet for Tibet, and if the Siamese, and the Singhalese, and the Burmese, and other peoples in the East, could have created alphabets of their own, why should not the numerals also have been fashioned by some temple school, or some king, or some merchant guild? By way of illustration, there are shown in the table on page 36 certain systems of the East, and while a few resemblances are evident, it is also evident that the creators of each system endeavored to find original forms that should not be found in other systems. This, then, would seem to be a fair interpretation of the evidence. A human mind cannot readily create simple forms that are absolutely new; what it fashions will naturally resemble what other minds have fashioned, or what it has known through hearsay or through sight. A circle is one of the world's common stock of figures, and that it should mean twenty in Phœnicia and in India is hardly more surprising than that it signified ten at one time in Babylon.² It is therefore quite probable that an extraneous origin cannot be found for the very sufficient reason that none exists.

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¹ V. A. Smith, *The Early History of India*, Oxford, 2d ed., 1908, p. 333.

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² C. J. Ball, "An Inscribed Limestone Tablet from Sippara," *Proceedings of the Society of Biblical Archæology*, Vol. XX, p. 25 (London, 1898). Terrien de Lacouperie states that the Chinese used the circle for 10 before the beginning of the Christian era. [*Catalogue of Chinese Coins*, London, 1892, p. xl.]

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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.¹

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¹ For a purely fanciful derivation from the corresponding number of strokes, see W. W. R. Ball, *A Short Account of the History of Mathematics*, 1st ed., London, 1888, p. 147; similarly J. B. Reveillaud, *Essai sur les chiffres arabes*, Paris, 1883; P. Voizot, "Les chiffres arabes et leur origine," *La Nature*, 1899, p. 222; G. Dumesnil, "De la forme des chiffres usuels," *Annales de l'université de Grenoble*, 1907, Vol. XIX, pp. 657-674, also a note in *Revue Archéologique*, 1890, Vol. XVI (3), pp. 342-348; one of the earliest references to a possible derivation from points is in a work by Bettino entitled *Apiaria universae philosophiae mathematicae in quibus paradoxa et noua machinamenta ad usus eximios traducta, et facillimis demonstrationibus confirmata*, Bologna, 1545, Vol. II, Apiarium XI, p. 5.

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² *Alphabetum Barmanum*, Romae, mdccclxxvi, p. 50. The 1 is evidently Sanskrit, and the 4, 7, and possibly 9 are from India.

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³ *Alphabetum Grandonico-Malabaricum*, Romae, mdcclxxii, p. 90. The zero is not used, but the symbols for 10, 100, and so on, are joined to the units to make the higher numbers.

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⁴ *Alphabetum Tangutanum*, Romae, mdcclxxiii, p. 107. In a Tibetan MS. in the library of Professor Smith, probably of the eighteenth century, substantially these forms are given.

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⁵ Bayley, loc. cit., plate II. Similar forms to these here shown, and numerous other forms found in India, as well as those of other oriental countries, are given by A. P. Pihan, *Exposé des signes de numération usités chez les peuples orientaux anciens et modernes*, Paris, 1860.

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Table of Certain Eastern Systems

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We may summarize this chapter by saying that no one knows what suggested certain of the early numeral forms used in India. The origin of some is evident, but the origin of others will probably never be known. There is no reason why they should not have been invented by some priest or teacher or guild, by the order of some king, or as part of the mysticism of some temple. Whatever the origin, they were no better than scores of other ancient systems and no better than the present Chinese system when written without the zero, and there would never have been any chance of their triumphal progress westward had it not been for this relatively late symbol. There could hardly be demanded a stronger proof of the Hindu origin of the character for zero than this, and to it further reference will be made in Chapter IV.

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