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PROPOSITION 1.—Take a number containing wo figures, say 83, reverse the figures, which will make 38, then subtract them from the original number, 83, and the difference will be 45nine times the difference between the two figures 8 and 3, which is 5. Example : 83 38 45=9 X (83) =5 The fallowing formula shows the fact anc reason, taking the value of x equal to 8, y equal to 5, and = equal to 3 : 10x+z) (10: +X)=9(x-:). 2nd.Take a number containing three fi gurus reverse and subtract as above, and the difference will be equal to 99 times the differ ence of the first and last figures. Example 853 358 495=99X (83) =99X5 Formula (IOOX+IOIJ+Z) (lOOz+lOy+x) =99 (xz) 3rd.But if, instead of reversing the thre figures as in the second proposition, you plac the centre one first, and the last in the centre then the difference will be 9 times 11 time the first figure less the two last. Example : 853 538 315=9+ (11X853) =9 X (8853) Formula: (lOOi+lOy+s) (100y+10H i) =9(111 (10+=) from which subtract 9 times the first figure and the difference will be equal to the sum o amount of the two figures added togethe Example: 83 added together make 11 72, nine times first figure, 8, subtract Formula : l0x+z9x=x+z. 5thReverse this, and from the two figure take their sum or amount added together, an you will have 9 times the first figure. Ex. 53 8, the sum of 5 and 3 45, 9 times the first figure, f> Formula10y+: (y+) 9y. A curious result is obtained on the principl of Prop. 5 : take, for example, a number con taining two figures (a number containing an amount of figures will do as well) say 86 se parate each figure into two others containin together the same number of digits, say 5 an 3 for the 8, and 4 and 2 for the six, then yo will have 5342 ; now you may change thei places so as to destroy the connection of th 5 and 3 and the 4 and 2 ; for example, plac the 3 first, then the 2, then the 5, and lastl the 4 (or any other way you may desire) the you have 3254; now take the original figure from any part of the number, and you will in variably find the difference to be a multipl of 9. Example : 3254 3254 3425 8 6 or 68 or 6 8 9-2448 2574 2817 If, after this is done, a number is left out c the difference, it can be detected withou knowing the figures used in the calculation for example, if a 4 is left out of the first c the above three examples, you will have 24 which, divided by 9 (or added up until yo have only one figure, as 2 plus 4 plus 8 mak 14, and 1 plus 4 make 5) will have 5, and yo see immediately that there is a 4 wanting t make up the 9. JAMES SWAIM. Philadelphia, Pa., 1853.
