73 8 THE POPULAR SCIENCE MONTHLY. 



rano-ement seems to have come the fundamental law of tbe 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, espe- 

 cially 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 connec- 

 tion 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 1*794) is, that all through the multiplication-table the product of 9 

 comes to 9. Multiply any number by 9, as 9 x 2 = 18, add the digits 

 together, 1+8 = 9. So it goes on until we reach 9 x 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 num- 

 ber of examples at random, and we have the same result. For in- 

 stance, 339 x 9 = 3,051. Add the digits 3 + + 5 + 1=9. Take 

 one more, 5,071 x 9 = 45,639, and the sum of the digits, 4 + 5 + 

 + 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 ob- 

 tained 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, 

 '""I 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 = 



