The Masonic Properties Of Numbers.

is incessantly composed before our eyes , after having undergone a thousand decompositions . 6 th . —I propose to conclude my observations on the number 9 , by showing some curious properties which it possesses , the singularity of which has gained for it so great a celebrity . And firstlet us write downin their natural order

, , , the simple sequence of numbers from one to ten inclusive , thus : 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 10 . Under these , respectively , let us place the product of each by nine , thus : 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 10 .

9 . IS . 27 . 36 . 45 . 54 . 63 . 72 . 81 . 90 . It is remarkable that the lower line of figures reproduces tbe simple sequence of the upper line in a twofold sense ; first , from the left hand to the right , taking the left hand digits , from the left hand digit of 18 up " to that of 90 ; secondly , from the right hand backwards to the left , taking the right hand digits , from the right hand digit

of 81 to the initial figure of the line , 9 . Furthermore , it is singular that by adding up each column of figures in the last-mentioned double row , we have the natural decimal series of numbers , or their simple progression , by tens , in the decimal scale ; viz ., ten , twenty , thirty , forty , fifty , < S . c , up to one hundred , thus : 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 10 . 9 . 18 . 27 . 36 . 45 . 54 . 63 . 72 . 81 . 90 .

10 . 20 . 30 . 40 . 50 . 60 . 70 . 80 . 90 . 100 . And here again , in the third or lowest row of figures , we have the original natural sequence of numbers reproduced , from one to ten , by the left hand digits taken from left to right . But further ; the number 9 has the property of constantly reproducing itself in any multiple , thus :

2 x 9 = 18 , and , 1 + 8 = 9 . 3 x 9 = 27 , and , 2 + 7 = 9 . 4 x 9 = 36 , and , 3 + 6 = 9 . 5 x 9 = 45 , and , 4 + 5 = 9 . & c ., & c , & o . Again , in higher multiples : 81 x 81 = 6561 , and , 6 + 5 + 6 + 1 = 18 , and , 1 + 8 = 9 . 27 x 27 = 729 , and , 7 + 2 + 9 = 18 , and , 1 + 8 = 9 . 18 18 324 and 3 2

x = , , + + 4 = 9 . 36 x 36 = 1290 , and 1 + 2 + 9 + 6 = 18 , and , 1 + 8 --= 9 . Another peculiar property of the number 9 , is its power of testing the accuracy of any arithmetical multiplication or division . The operation by which this test is applied , may be called " casting out the nines . " It may be thus illustrated .

Let us take any number and multiply it by any other ; say the number 83246 , by the number 2765—thus .-multiplicand 83246 5 multiplier 2765 2 416230 10 499476 9

582722 — 166492 1 product 230175190 = 10 : 9 j 1 :

Now , in order to test the accuracy of this calculation , it would be most tedious to go through exactly the same operation a second time ; letting alone the obvious probability of repeating any error which may have crept into the first computation . But , by calling in the aid of the number 9 , we have at once a most curious and accurate test of our workba so distinct in its

, y process method from the original multiplication , that the detection of any error amounts well nigh to a certainty . I proceed , without further comment , to show the method of " casting out nines . "

Operation 1 st . —Take the digits in the multiplicand , and add them up together thus : 8 + 3 = 11 ; cast out 9 , leaves 2 ; 2 + 2 ( the next digit to 3 ) = 4 + 4 ( next digit to 2 ) = 8 + 6 ( last digit ) = 14 ; cast ont 9 ( from the 14 ) = 5 . Operation 2 nd . —Perform the same operation with the multiplier thus : 2 + 7 = 9 ; cast out 9 , leaves 0 ; 0 + 6

= 6 + 5 ( last digit ) = 11 ; cast out 9 , leaves 2 . Operation 3 rd . —Multiply the 5 found from operation 1 st , by the 2 found from operation 2 nd , the product is 10 ; cast out 9 , leaves 1 . If the operation of multiplication in the first instance ( of 83 , 246 by 2 , 765 ) has been correctly performed , the casting out of the nines in product ( by a similar method ) will result in leaving unity , the number found by operation 3 rd . Let us try this : — Operation 4 th . —Add up the digits in the product , casting out the nines thus : —

2 + 3 + 0 + 1 + 7 = 13 ; cast out 9 , leaves 4 ; 4 + 5 = 9 ; cast out 9 , leaves 0 . 0 + 1 + 9 = 10 ; oast out 9 , leaves 1 . which , having been found by operation 3 rd , has proved the sum originally set to have been correctly performed . This method of investigation is evidently applicable to division as well as to multiplication , because every division may be reducedfor a checkto a multilication

, , p of the quotient by the divisor , the product being the dividend . Finally , by the help of the number 9 , a curious arithmetical trick may be performed , thus : You tell a person that he may write down any number ; add up the digits , subtract their sum from the original number , and strike out any digit in the new number ( just found as a

remainder ) . He must then tell you the digits in this last number that are not expunged , and you may , by the assistance of the number 9 , tell him what figure he has struck out , though you have no idea of any of the numbers originally thought of by him . This is done as follows : — Let the number originally written down be 14 , 368 .

The digits added together = 1 + 4 + 3 + 6 + 8 = 22 . Subtract 22 from 14 , 368 , leaves 14 , 346 . These operations have all been formed under your instructions , without your having any knowledge , up to the present stage , of any of the figures employed . You now tell tbe persou who has been hitherto operating on the number 14 , 368 , to strike out any figure he pleases from the number ( last found ) 14 , 346 ; we will suppose he strikes out the

6—thus—14 , 3 4 . He is then to tell you the remaining figures , or digits , which are , of course , 1 4 , 3 4 .

You are then enabled to tell him that the number erased was a 6 by tbe following method : —Directly you are told that the remaining figures are 1 , 434 , you add them up together in your own mind , which gives 1 + 4 + 3 + 4 = 12 ) . This number 12 you subtract from the next higher multiple of nine , which is 18 . The difference being 6 ( 18 — 12 = 6 ) . the number struck out . And the same

method holds good for all possible cases ; the sum of tbe digits last told you being always subtracted from tbe next higher multiple of nine . Thus again .- —Supposing that , out of the number 14 , 346 the number 4 had been expunged instead of the number 6 , then we should have had 1 4 , 3 6 , and the sum of the remaining digits would have been

1 + 4 + 3 + 6 = 14 ; and 18 ( next higher multiple of nine)— 14 = 4 , the number expunged . From these and other properties of the number 9 , brethren , you may conceive that it became of no small importance in the eyes of the ancient Cabalists ; and even to the present day many traces of traditional superstition , in connection with this cipher , may be observed in the world . But having already too long detained you on this part of my subject , I pass ou to tbe number 10 , or tbe denary .