Arithmetical Recreations. 353 



nately takes place at the bright limb, and the observer's success 

 will therefore depend upon the very uncertain contingency of 

 his having the point of reappearance at the dark limb just 

 under his eye. This may indeed be secured by a wire in the 

 field with an equator eal mounting ; but it will demand very 

 pertinacious watching, as the actual times, and the consequent 

 amount of duration, will vary from those here given, in propor- 

 tion to the distance of the station from Greenwich. 30th, 55 

 Leonis, 6 mag., will be hidden from lOh. 13m. to lOh. 31m. 



ARITHMETICAL RECREATIONS. 



The following paper, communicated byM. Plateau to the Belgian 

 Academy, is translated from Cosmos, and will not fail to interest 

 a large circle of our readers by its exhibition of amusing and 

 unexpected properties of numbers : — 



M. Plateau states that a long while ago he heard the 

 following recreative problem proposed : — Given as a multipli- 

 cand the curious number 12,345,679, it is required to find a 

 multiplier which shall furnish a product entirely composed of 

 a repetition of any one number, chosen at pleasure. If the 

 figure required to be repeated in the product be 1, the multi- 

 plier required will be 9. If the repeated figure is 2, 3, 4, etc., 

 the multiplier must be the product of 9 multiplied by such 

 figure. Thus, to have a product consisting of 2 repeated in 

 succession, the multiplier must be 9 x 2 = 18 ; and to obtain 

 a repetition of threes, it must be 9 x 3=27, etc. 



Mr. Plateau supplies the following general rule applied to 

 this class of proposition, and he believes it to be his own 

 discovery : — 



Any unequal number being given as a multiplicand, pro- 

 vided it does not terminate in 5, it is always possible to find 

 another number as a multiplier which shall give, with the 

 former, a product composed entirely of the iteration of any 

 figure that may be named. 



To do this, divide unity by the number given as a multipli- 

 cand, which will give a periodic decimal fraction, in which }.he 

 period will commence immediately after the point, it being 

 understood that the ciphers preceding the significant figures 

 belong to the period. 



If the number given be not divisible by 3, the period will 

 be divisible by 9 ; make this division, and the quotient will 

 be the factor sought, taken singly, or multiplied by 2, 3, 4, 

 according as the product is required to be composed of ones, 

 twos, threes, fours, etc. If the number given be divisible by 



