Vi PREFACE. 



The third Chapter analyses the treatise in which Huygens in 

 1659 exhibited what was then known of the subject. Works such 

 as this, which present to students the opportunity of becoming 

 acquainted with the speculations of the foremost men of the 

 time, cannot be too highly commended ; in this respect our sub- 

 ject has been fortunate, for the example which was afforded by 

 Huygens has been imitated by James Bernoulli, De Moivre and 

 Laplace — and the same course might with great advantage be 

 pursued in connexion with other subjects by mathematicians in 

 the present day. 



The fourth Chapter contains a sketch of the early history of 

 the theory of Permutations and Combinations ; and the fifth Chap- 

 ter a sketch of the early history of the researches on Mortality 

 and Life Insurance. Neither of these Chapters claims to be ex- 

 haustive ; but they contain so much as may suffice to trace the 

 connexion of the branches to which they relate with the main sub- 

 ject of our history. 



The sixth Chapter gives an account of some miscellaneous in- 

 vestigations between the years 1670 and 1700. Our attention is 

 directed in succession to Caramuel, Sauveur, James Bernoulli, 

 Leibnitz, a translator of Huygens's treatise whom I take to be 

 Arbuthnot, Roberts, and Craig — the last of whom is notorious for 

 an absurd abuse of mathematics in connexion with the probability 

 of testimony. 



The seventh Chapter analyses the Ars Conjectandi of James 

 Bernoulli. This is an elaborate treatise by one of the greatest 

 mathematicians of the age, and although it was unfortunately 

 left incomplete, it affords abundant evidence of its author's ability 

 and of his interest in the subject. Especially we may notice the 

 famous theorem which justly bears the name of James Bernoulli, 

 and which places the Theory of Probability in a more commanding 

 position than it had hitherto occupied. 



The eighth Chapter is devoted to Montmort. He is not to be 

 compared for mathematical power with James Bernoulli or De 

 Moivre; nor does he seem to have formed a very exalted idea of 

 the true dignity and importance of the subject. But he was en- 

 thusiastically devoted to it; he spai^ed no labour himself, and his 

 influence direct or indirect stimulated the exertions of Nicolas 

 Bernoulli and of De Moivre. 



The ninth Chapter relates to De Moivre, containing a full 

 analysis of his Doctrine of Chances, De Moivre brought to bear 

 on the subject mathematical powers of the highest order ; these 

 powers are especially manifested in the results which he enun- 

 ciated respecting the great problem of the Duration of Play. 

 Unfortunately he did not publish demonstrations, and Lagrange 



