rangement seems to have come the fundamental law of the decimal notation in which its superior utility consists, and upon which quite recently has been based the metric system of weights and measures. By placing any of the digits in the place of the zero to make the numbers between ten and twenty, we have the law established. The science of arithmetic, like all other sciences, was very limited and imperfect at the beginning, and the successive steps by which it has reached its present extension and perfection have been taken at long intervals, and among different nations. It has been developed by the necessities of business, by the strong love for mathematical science, and by the call for its higher offices by other sciences, especially that of astronomy. In its progress, we find that the Arabians discovered the method of proof by casting out the 9's, and that the Italians early adopted the practice of separating numbers into periods of six figures, for the purpose of enumerating them. The property of the number 9 affords an ingenious method of proving each of the fundamental operations in arithmetic, and it seems to be an incidental attribute of this number. It arises from the law of increase in the decimal notation. It universally belongs to the number that is one less than the radix of the system of notation. And in this connection it may not be irrelevant to state some facts or curiosities with regard to this number 9. It cannot be multiplied away, or got rid of in any manner. Whatever we do, it is sure to turn up again, as was the body of Eugene Aram's victim. One remarkable property of this figure (said to have been discovered by W. Green, who died in 1794) is, that all through the multiplication-table the product of 9 comes to 9. Multiply any number by 9, as 9×2=18, add the digits together, 1+8=9. So it goes on until we reach 9×11=99. Very well add the digits 9+9=18, and 1+8=9. Going on to any extent it is impossible to get rid of the figure 9. Take any number of examples at random, and we have the same result. For instance, 339×9=3,051. Add the digits 3+0+5+1=9. Take one more, 5,071×9=45,639, and the sum of the digits, 4+5+6+3+9=27, and 2+7=9.
The French mathematicians found out another queer thing about this number, namely: if we take any row of figures, and, reversing their order, make a subtraction, and add the digits, the final sum is sure to be 9. For example, 5,071-1,705=3,366; add these digits 3+3+6+6=18, and 1+8=9. The same result is obtained if we raise the numbers so changed to their squares or cubes. Starting with 62, and reversing the digits, we have 26, then 62-26=36, and 3+6=9. The squares of 26 and 62 are respectively 676 and 3,844, and 3,844-676=3,168; add 3+1+6+8=18, and 1+8=9. This may be exemplified in another way. Write down any number, as, for example, 7,549,132, subtract the sum of its digits 7+5+4+9+1+3+2=31, and 7,549,132-31=
