2 CARDAN. 



Montmort says, " Jerome Cardan a donne un Traits De Ludo 

 Alese ; mais on n'y trouve que de I'erudition et des reflexions 

 morales." Essai d'Analyse.-.ip. XL. Libri says, "Cardan a ecrit 

 un traite special de Ludo Alece, ou se trouvent resolues plusieurs 

 questions d'analyse combinatoire." Histoire, Vol. ill. p. 176. The 

 former notice ascribes too little and the latter too much to 

 Cardan. 



4. Cardan's treatise occupies fifteen folio pages, each containing 

 two columns; it is so badly printed as to be scarcely intelligible. 

 Cardan himself was an inveterate gambler ; and his treatise may 

 be best described as a gambler's manual. It contains much mis- 

 cellaneous matter connected with gambling, such as descriptions of 

 games and an account of the precautions necessary to be employed 

 in order to guard against adversaries disposed to cheat : the 

 discussions relating to chances form but a small portion of the 

 treatise. 



5, As a specimen of Cardan's treatise we will indicate the 

 contents of his thirteenth Chapter. He shews the number of 

 cases which are favourable for each throw that can be made with 

 two dice. Thus two and twelve can each be thrown in only one 

 way. Eleven can be thrown in two ways, namely, by six appear- 

 ing on either of the two dice and five on the other. Ten can be 

 thrown in three ways, namely, by five a23pearing on each of the 

 dice, or by six appearing on either and four on the other. And 

 so on. 



Cardan proceeds, *'Sed in Ludo fritilli undecim puncta adjicere 

 decet, quia una Alea potest ostendi."...The meaning apparently is, 

 that the person who throws the two dice is to be considered to 

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

 that number, as well as when the number is made up by the sum 

 of the numbers on the two dice. Hence, for six or any smaller 

 number eleven more ftivourablc cases arise besides those already 

 considered. 



Cardan next exhibits correctly the number of cases which are 

 favourable for each throw that can be made with three dice. Thus 

 three and eighteen can each be thrown in only one way ; four and 



