History of Hindu Mathematics: A Source Book v1.c7.s6: 6. Kharoṣṭhī Numerals
History of Hindu Mathematics: A Source Book v1.c7.s6
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6. Kharoṣṭhī Numerals
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**Early Occurrence.** The *Kharoṣṭhī lipi* is a script written from right to left. The majority of the Kharoṣṭhī inscriptions have been found in the ancient province of Gāndhāra, the modern eastern Afghanistan and the northern Punjab. It was a popular script meant for clerks and men of business. The period during which it seems to have been used in India extends from the fourth century B.C. to the third century A.D. In the Kharoṣṭhī inscriptions of Aśoka only four numerals have been found. These are the primitive vertical marks for one, two, four, and five, thus:
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More developed forms of these numerals are found in the inscriptions of the Śakas, of the Pārthians and of the Kuṣânas, of the 1st century B.C. and the 1st and 2nd centuries A.D., as well as in other probably later documents. The following are some of the numerals of this period:
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**Forms and their Origin.** It cannot be satisfactorily explained why the number four, which was previously represented by four vertical lines came to be represented by a cross later on. The representation of the numbers five to eight follows the additive principle, with four as the base. This method of writing the numbers 4 to 8 is not met with in the early records of the Semites. We do not know how the number nine was written. It is very probable that it was written as [Figure:] , *i.e.*, 4+4+1 (reading from right to left, the order being the same as that of the script). The number 10 has an entirely new sign. The question why it was not written as [Figure:] , or why the base [Figure:] (4) was abandoned cannot be satisfactorily answered.
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It is accepted by all that the Kharoṣṭhî is a foreign script brought into India from the west. The exact period at which it was imported is unknown. It might have been introduced at the time of the conquest of the Punjab by Darius (*c.* 500 B.C.) or earlier.¹ The numerals given above undoubtedly belong to this script as they proceed from right to left.
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¹ The theory of the foreign origin of the script has to be revised in the light of the discoveries at Mohenjo-daro and Harappa, especially in view of the fact that the Mohenjo-daro alphabet ran from right to left.
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The old symbols of the inscriptions of Aśoka, however, seem to have undergone modification in India, especially the numbers from 4 to 19. The symbols for four and ten seem to have been coined in India, in order to introduce simplification and also to bring the Kharoṣṭhî numeral system in line with the Brâhmî notation already in extensive use. The symbol [Figure:] seems to have been derived by turning the Brâhmî symbol [Figure:] which represents 4 in the inscriptions of Aśoka. The inclined cross to represent 4 is found in the Nabatean numerals in use in the earlier centuries of the Christian Era.² The Nabatean numerals resemble the Kharoṣṭhî also in the use of the scale of twenty and in the method of formation of the hundreds. It is possible that the Semites might have borrowed the Kharoṣṭhî symbol for 4, although it is not unlikely, as Bühler thinks, that the symbol might have been invented independently by both nations.
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² J. Euting, *Nabatäische Inschriften aus Arabien*, Berlin, 1885, pp. 96-97.
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The numeral [Figure:] (10) closely resembles the letter *a* of the Brâhmî alphabet. The symbol for twenty [Figure:] appears to be a cursive combination of two tens. It resembles one of the early Phoenician forms found in the papyrus Blacas¹ (5th century B.C.). The mode of expressing the numbers 30, 40, etc., by the help of the symbols for 10 and 20, is the same as amongst the early Phoenicians and Aramaeans.
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¹ Bühler, *Palaeography*, p. 77; Ojha, *l.c.*, p. 128; see Table II(*b*).
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The symbol for 100 resembles the letter *ta* or *tra* of the Brâhmî script, to the right of which stands a vertical stroke.
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The symbols for 200, 300, etc., are formed by writing the symbols for 2, 3, etc., respectively to the right of the symbol for 100. This evidently is the use of the multiplicative principle, as is found amongst the early Phoenicians.²
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² See Table II(*c*).
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The formation of other numbers may be illustrated by the number 274 which is written with the help of the symbols for 2, 100, 20, 10 and 4 arranged as
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in the right to left order. The 2 on the right of 100 multiplies 100, whilst the numbers written to the left are added, thus giving 274.
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The ancient Kharoṣṭhî numerals are given in Table I.