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 thinly liquid paint which made marks that could be easily erased, or upon a white tablet, less than a foot square, strewn with red flour, on which they wrote the figures with a small stick, so that the figures appeared white on a red ground."[7] Since the digits had to be quite large to be distinctly legible, and since the boards were small, it was desirable to have a method which would not require much space. Such a one was the above method of multiplication. Figures could be easily erased and replaced by others without sacrificing neatness. But the Hindoos had also other ways of multiplying, of which we mention the following: The tablet was divided into squares like a chess-board. Diagonals were also drawn, as seen in the figure. The multiplication of $$\scriptstyle{12 \times 735=8820}$$ is exhibited in the adjoining diagram.[3] The manuscripts extant give no information of how divisions were executed. The correctness of their additions, subtractions, and multiplications was tested "by excess of 9's." In writing fractions, the numerator was placed above the denominator, but no line was drawn between them.

We shall now proceed to the consideration of some arithmetical problems and the Indian modes of solution. A favourite method was that of inversion. With laconic brevity, Aryabhatta describes it thus: "Multiplication becomes division, division becomes multiplication; what was gain becomes loss, what loss, gain; inversion." Quite different from this quotation in style is the following problem from Aryabhatta, which illustrates the method:[3] "Beautiful maiden with beaming eyes, tell me, as thou understandst the right method of inversion, which is the number which multiplied by 3, then increased by $$\scriptstyle{\frac{3}{4}}$$ of the product, divided by 7, diminished by $$\scriptstyle{\frac{1}{3}}$$ of the quotient, multiplied by itself, diminished by 52, the square