Page:A History of Mathematics (1893).djvu/109

 7 of these a grain of dust whirled up by the wind, and so on. Thus he proceeded, step by step, until he finally reached the length of a mile. The multiplication of all the factors gave for the multitude of primary atoms in a mile a number consisting of 15 digits. This problem reminds one of the 'Sand-Counter' of Archimedes.

After the numerical symbolism had been perfected, figuring was made much easier. Many of the Indian modes of operation differ from ours. The Hindoos were generally inclined to follow the motion from left to right, as in writing. Thus, they added the left-hand columns first, and made the necessary corrections as they proceeded. For instance, they would have added 254 and 663 thus: $$\scriptstyle{2+6=8}$$, $$\scriptstyle{5+6=11}$$, which changes 8 into 9, $$\scriptstyle{4+3=7}$$. Hence the sum 917. In subtraction they had two methods. Thus in $$\scriptstyle{821-348}$$ they would say, $$\scriptstyle{8\text{ from }11=3}$$, $$\scriptstyle{4\text{ from }11=7}$$, $$\scriptstyle{3\text{ from }7=4}$$. Or they would say, $$\scriptstyle{8\text{ from }11=3}$$, $$\scriptstyle{5\text{ from }12=7}$$, $$\scriptstyle{4\text{ from }8=4}$$. In multiplication of a number by another of only one digit, say 569 by 5, they generally said, $$\scriptstyle{5 \cdot 5=25}$$, $$\scriptstyle{5 \cdot 6=30}$$, which changes 25 into 28, $$\scriptstyle{5 \cdot 9=45}$$, hence the 0 must be increased by 4. The product is 2845. In the multiplication with each other of many-figured numbers, they first multiplied, in the manner just indicated, with the left-hand digit of the multiplier, which was written above the multiplicand, and placed the product above the multiplier. On multiplying with the next digit of the multiplier, the product was not placed in a new row, as with us, but the first product obtained was corrected, as the process continued, by erasing, whenever necessary, the old digits, and replacing them by new ones, until finally the whole product was obtained. We who possess the modern luxuries of pencil and paper, would not be likely to fall in love with this Hindoo method. But the Indians wrote "with a cane-pen upon a small blackboard with a white,