32 LEIBNITZ. 



sertation similar to Pascal's Arithmetical Triangle, and applies it 

 to find the number of the combinations of an assigned set of things 

 taken two, three, four,... together. In the latter part of his disser- 

 tation Leibnitz shews how to obtain the number of permutations 

 of a set of things taken all together ; and he forms the product of 

 the first 24* natural numbers. He brings forward several Latin 

 lines, including that which we have already quoted in Art. 39, 

 and notices the great number of arrangements which can be 

 formed of them. 



The greater part of the dissertation however is of such a 

 character as to confirm the correctness of Erdmann's judgment in 

 including it among the philosophical works of Leibnitz. Thus, 

 for example, there is a long discussion as to the number of moods 

 in a syllogism. There is also a demonstration of the existence of 

 the Deity, which is founded on three definitions, one postulate, 

 four axioms, and one result of observation, namely, aliquod corpus 

 movetur. 



4iD. We will notice some points of interest in the dissertation. 



(1) Leibnitz proposes a curious mode of expression. When 

 a set of things is to be taken two at a time he uses the S3rmbol 

 com2natio (combinatio) ; when three at a time he uses conSnatio 

 (conternatio) ; when four at a time, con4natio, and so on. 



(2) The mathematical treatment of the subject of combina- 

 tions is far inferior to that given by Pascal ; probably Leibnitz 

 had not seen the work of Pascal. Leibnitz seems to intimate 

 that his predecessors had confined themselves to the combina- 

 tions of things two at a time, and that he had himself extended 

 the subject so far as to shew how to obtain from his table the 

 combinations of things taken together more than two at a time ; 

 generaliorem modum nos deteximus, specialis est vidgatus. He 

 gives the rule for the combination of things two at a time, namely, 



that which we now express by the formula ^ — -^ ; but he does 



not give the similar rule for combinations three, four,... at a time, 

 which is contained in Pascal's work. 



(3) After giving his table, which is analogous to the Arith- 



