B 1. Rule (sūtra) Bakhsha¯lı¯ Manuscript 2. Example (udāharan. a) . Statement (nyāsa/sthāpanā) TAKAO HAYASHI . Computation (karan. a) . Verification(s) (pratyaya/pratyānayana) The Bakhshālī Manuscript is the name given to the oldest extant manuscript in Indian mathematics. It is so A decimal place-value notation of numerals with zero called because it was discovered by a peasant in 1881 at (expressed by a dot) is employed in the Bakhshālī a small village called Bakhshālī, about 80 km northeast Manuscript. The terms for mathematical operations are of Peshawar (now in Pakistan). It is preserved in the often abbreviated, especially in tabular presentations of Bodleian Library at Oxford University. computations. Thus we have yu for yuta (increased), gu The extant portion of the manuscript consists of 70 for gun. a or gun. ita (multiplied), bhā for bhājita (divided) fragmentary leaves of birchbark. The original size of a or bhāgahāra (divisor or division), che for cheda leaf is estimated to be about 17 cm wide and 13.5 cm (divisor), and mū for mūla (square root). For subtraction, high. The original order of the leaves can only be the Bakhshālī Manuscript puts the symbol, + (similar to conjectured on the bases of rather unsound criteria, the modern symbol for addition), next (right) to the such as the logical sequence of contents, the order of number to be affected. It was originally the initial letter the leaves in which they reached A. F. R. Hoernle, who of the word .rn. a, meaning a debt or a negative quantity did the first research on the manuscript, physical in the Kus.ān.a or the Gupta script (employed in the appearance such as the size, shape, degree of damage, second to the sixth centuries). The same symbol is also and knots, and the partially preserved serial numbers of used in an old anonymous commentary on Śrīdhara’s mathematical rules (9–11, 13–29, and 50–58). Pātīgan. ita, which is uniquely written in the later type of The script is the earlier type of the Śāradā script, the Śāradā script (after the thirteenth century). Most which was in use in the northwestern part of India, works on mathematics, on the other hand, put a dot namely in Kashmir and the neighboring districts, from above a negative number. the eighth to the twelfth centuries. G. R. Kaye, who The problems treated in the extant portion of the succeeded Hoernle, has shown that the writing of the Bakhshālī work involve five kinds of equations, manuscript can be classified into at least two styles, one namely (1) simple equations with one unknown (15 of which covers about one-fifth of the work. There is, types of problems), (2) systems of linear equations with however, no definitive reason to think that the present more than one unknown (14 types), (3) quadratic manuscript consists of two different works. equations (two types, both of which involve an The information contained in the manuscript, the arithmetical progression), (4) indeterminate systems title of which is not known, is a loose compilation of of linear equations (three types, including the so-called mathematical rules and examples collected from “Hundred Fowls Problem,” in which somebody is to different works. It consists of versified rules, examples, buy 100 fowls for 100 monetary units of several kinds), most of which are versified, and prose commentaries on and (5) indeterminatepffiffiffiffiffiffiffiffiffiffiffi equationspffiffiffiffiffiffiffiffiffiffiffi of the second degree the examples. A rule is followed by an example or (two types: x þ a ¼ u and x À b ¼ v, where u and v examples, and under each one the commentary gives a are rational numbers; and xy ¼ ax þ by). “statement,”“computation,” and a “verification” or The rules of the Bakhshālī work may be classified as verifications. The statement is a tabular presentation of follows: the numerical information given in the example, and the computation works out the problem by following, 1. Fundamental operations, such as addition and and often citing, the rule step by step. subtraction of negative quantities, addition, multi- Thus, the most typical pattern of exposition in the plication, and division of fractions, reduction of Bakhshālī Manuscript is: measures, and a root-approximation formula, 2 Bakhshālī manuscript pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 A style of exposition similar to that of the Bakhshālī r ðÞr=2a 2 þ þ À : work (“statement,” etc.) is found in Bhāskara I’s a r a ðÞþ = 2a 2 a r 2a commentary (AD 629) on the second chapter called gan. ita (mathematics) of the Āryabhat.īya (AD 499). 2. General rules applicable to different kinds of Both Bhāskara I’s commentary and the Bakhshālī work problems: regula falsi, rule of inversion, rule of attach much importance to the verification; it became three, proportional distribution, and partial addition obsolete in later times. The unusual word yāva and subtraction. (yāvakaran. a in Bhāskara I’s commentary) meaning 3. Rules for purely numerical problems: simple equations the square power, and the apparently contradictory with one unknown, systems of linear equations with meanings of the word karan. ī, the square number and more than one unknown, indeterminate equations, the square root, occur in both works. and period of an arithmetical progression. Bhāskara I does not use the symbol yā (the initial 4. Rules for problems of money: equations of proper- letter of yāvattāvat or “as much as”) for unknown ties, wages, earnings, donations, etc., consumption numbers in algebraic equations even when it is of income and savings, buying and selling, purchase naturally expected, while he employs the original word in proportion, purchase of the same number of yāvattāvat itself in the sense of unknown quantities (in articles, price of a jewel, prices of living creatures, his commentary on Āryabhat.īya 2.30). This probably mutual exchange of commodities, installments, a implies that he did not know the symbol. The symbol sales tax paid both in cash and in kind, and a bill of is, on the other hand, utilized once in the Bakhshālī exchange. work in order to reduce the conditions given in 5. Rules for problems of travelers: equations of an example to a form to which the prescribed rule is journeys, meeting of two travelers, and a chariot easily applicable; after the reduction, the symbol and horses. is discarded and the rule is, so to speak, applied 6. Rules for problems of impurities of gold. mechanically (fol. 54v). This restricted usage of the 7. Rules for geometrical problems: volume of an symbol seems to indicate that the work belongs to a irregular solid and proportionate division of a triangle. period when the symbol was already invented, but not very popular yet. All the rules of the first category, namely the There has been quite a bit of dispute over the dates of fundamental operations, occur only as quotations in the manuscript. Hoernle assigned the work to the third the computations of examples. Many of the other rules or the fourth century AD, Kaye to the twelfth century, could belong to either miśraka-vyavahāra (on mixture) Datta to “the early centuries of the Christian era,” and or średhī (of series) in a book of pātī (algorithms) such . Ayyangar and Pingree to the eighth or the ninth century. as Śrīdhara’s Pātīganita and Triśatikā (eighth century), . The above points suggest that the Bakhshālī work etc., but they have not been arranged according to the (commentary) was composed not much later than ordinary categories of vyavahāra. Bhāskara I (the seventh century). We apparently owe the present manuscript to four types of persons: the authors of the original rules and ▶ ▶Ś ī ▶ ā examples, the compiler, the commentator, and the scribe. See also: Zero, r dhara, Bh skara I Possibly, however, the commentator was the compiler himself, and “the son of Chajaka” (his name is unknown), References ā ī by whom the Bakhsh l Manuscript, or at least part of it, ā ī “ ” Ayyangar, A. A. K. The Bakhsh l Manuscript. Mathematics was written, was the commentator, or one of the Student 7 (1939): 1–16. commentators. The colophon to the section that deals Channabasappa, M. N. On the Square Root Formula in the exclusively with the trairāśika (rule of three) reads: Bakhshālī Manuscript. Indian Journal of History of Science 2 (1976): 112–24. This has been written by the son of Chajaka, a ---. The Bakhshālī Square-Root Formula and High Speed brāhman.a and king of mathematicians, for the Computation. Gan. ita Bhāratī 1 (1979): 25–7. sake of Hasika, son of Vasis.t.ha, in order that it ---. Mathematical Terminology Peculiar to the Bakhshālī may be used (also) by his descendents. Manuscript. Gan. ita Bhāratī 6 (1984): 13–8. Datta, B. The Bakhshālī Mathematics. Bulletin of the Immediately before this statement occurs a fragmentary Calcutta Mathematical Society 21 (1929): 1–60. word rtikāvati, which is probably the same as the Gupta, R. C. Centenary of Bakhshālī Manuscript’s Discov- ā ī – country of Mārtikāvata mentioned by Varāhamihira (ca. ery. Gan. ita Bh rat 3 (1981): 103 5. ---. Some Equalization Problems from the Bakhshālī AD 550) among other localities of northwestern India ś ā ā ā Manuscript. Indian Journal of History of Science 21 such as Taks.a il (Taxila), Gandh ra, etc. (Br. hatsam. hit (1986): 51–61. 16.25). It may be the place where the Bakhshālī work Hayashi, T. The Bakhshālī Manuscript: An Ancient Indian was composed. Mathematical Treatise. Groningen: Egbert Forsten, 1995. Balkhī school of arab geographers 3 Hoernle, A. F. R. On the Bakhshālī Manuscript. Verhandlun- It is obvious that the maps are conceived as a set gen des VII Internationalen Orientalisten Congresses covering the Muslim Empire with reasonable detail, and – (Vienna 1886).