CARDAN, 3 



seventeen can each be thrown in three ways ; and so on. Cardan 

 also gives the following list of the number of cases in Fritillo : 



12 34 5 6789 10 11 12 



108 111 115 120 12G 133 33 36 37 36 33 26 



Here we have corrected two misprints by the aid of Cardan's 

 verbal statements. It is not obvious what the table means. It 

 might be supposed, in analogy with what has already been said, 

 that if a person throws three dice he is to be considered to have 

 thrown a given number when one of the dice alone exhibits that 

 number, or when two dice together exhibit it as their sum, as 

 well as when all the three dice exhibit it as their sum : and this 

 would agree wdth Cardans remark, that for numbers higher than 

 twelve the favourable cases are the same as those already given by 

 him for three dice. But this meaning does not agree with Cardan's 

 table ; for with this meaning we should proceed thus to find the 

 cases favourable for an ace : there are 5^ cases in which no ace 

 appears, and there are 6' cases in all, hence there are 6^ — 5^ cases 

 in which we have an ace or aces, that is 91 cases, and not 108 as 

 Cardan gives. 



The connexion between the numbers in the ordinary mode of 

 using dice and the numbers which Cardan gives appears to 

 be the following. Let n be the number of cases which are favour- 

 able to a given throw in the ordinary mode of using three dice, 

 and N the number of cases favourable to the same throw in 

 Cardan's mode ; let m be the number of cases favourable to the 

 given throw in the ordinary mode of using two dice. Then for any 

 throw not less than thirteen, N=n ; for any throw between seven and 

 twelve, both inclusive, N = Sni + n ; for any throw not greater than 

 six, i\^= 108 + 3?/i + n. There is only one deviation from this law ; 

 Cardan gives 26 favourable cases for the throw twelve, and our 

 proposed law would give 3 + 25, that is 28. 



We do not, however, see what simple mode of playing with 

 three dice can be suo'o-ested which shall oive favourable cases 

 agreeing in number with those determined by the above law. 



6. Some further account of Cardan's treatise will be found 



1—2 



