Algebra (Burrow, 1790)

Excerpts from Dharampalji's book Indian Science and Technology in the 18th Century (1971)

I mean very shortly to publish translations of the Leelvatty and Beej Ganeta, or the Arithmetic and Algebra of the Hindoos. [As noted by Burton (1985), the Lilavati and the Vijagnita form the first two parts of Bhaskara's book Siddhanta Siromani. The book was written in 1150 A.D. and translated into Arabic in 1587 A.D.] With respect to the binomial theorem, the application of it to fractional indices will perhaps remain forever the exclusive property of Newton; but the following question and its solution shows that the Hindoos understood it in whole numbers to the full as well as Briggs, and much better than Pascal.
[Briggs (1561-1630) was an English mathematician who invented the logarithm to the base 10. Pascal (1623-1662) was a French mathematician, physicist, and philosopher. He developed 'Pascal's triangle', which displays the binomial coefficients.]

A Raja's palace had eight doors; now these doors may either be opened by one at a time; or by two at a time; or by three at a time; and so on through the whole, till at last all are opened together. It is required to tell the numbers of times that this can be done.

Set down the number of the doors, and proceed in order to gradually decreasing by one to unity and then in a contrary order as follows:

8 7 6 5 4 3 2 1
1 2 3 4 5 6 7 8

Divide the first eight by the unit beneath it, and the quotient eight shows the number of times that the doors can be opened by one at a time: multiply this eight by the next term seven, and divide the product by the two beneath it, and the result twenty-eight is the number of times that two doors may be opened; multiply the last found twenty-eight by the next figure six, and divide the product by the number three beneath it, and the quotient of fifty-six shows the number of times that three different doors may be opened: again this fifty-six multiplied by the next number five and divided by the four beneath it, is seventy, the number of times that four different doors may be opened: in the same manner fifty-six is the number of fives that can be opened: twenty-eight the number of times that six can be opened: eight the number of times that seven can be opened and lastly, one is the number of times the whole may be opened together, and the sum of all the different times is 255.

By K. Kesava Rao
Department of Chemical Engineering
Indian Institute of Science
Bengaluru 560012, India
kesava@iisc.ac.in

...To be continued ...

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