LEIBNITZ. 3 1 



Similarly if the proposed letters are a, a, a, b, h, it is found 

 that 11 elections can be made. 



In his following sections Schooten proceeds to apply these 

 results to questions relating to the number of divisors in a number. 

 Thus, for example, supposing a, h, c, d, to be different prime 

 factors, numbers of the following forms all have 16 divisors, 

 ahcd, a^hc, a^b^, a^b, a)^. Hence the question may be asked, what is 

 the least number which has 10 divisors? This question must 

 be answered by trial ; we must take the smallest prime numbers 

 2, 8,. . . and substitute them in the above forms and pick out the least 

 number. It will be found on trial that the least number is 2^. 3. 5, 

 that is 120. Similarly, suppose we require the least number which 

 has 24 divisors. The suitable forms of numbers for 24 divisors 

 are ci^bcd, a^¥c, oJ'bc, a^¥, a'b'^, o}^h and a^^. It will be found on 

 trial that the least number is 2^ 3^. 5, that is 360. 



Schooten has given two tables connected with this kind of 

 question. (1) A table of the algebraical forms of numbers which 

 have any given number of divisors not exceeding a hundred ; and 

 in this table, when more than one form is given in any case, the 

 first form is that which he has found by trial will give the least 

 number with the corresponding number of divisors. (2) A table 

 of the least numbers which have any assigned number of divisors 

 not exceeding a hundred. Schooten devotes ten pages to a list of 

 all the prime numbers under 10,000. 



43. A dissertation was pubHshed by Leibnitz in 1666, entitled 

 Dissertatio de Arte Combinatoma; part of it had been previously 

 published in the same year under the title of Disputatio arith- 

 metica de comjilexionihus. The dissertation is interesting as the 

 earliest work of Leibnitz connected with mathematics ; the con- 

 nexion however is very slight. The dissertation is contained in 

 the second volume of the edition of the works of Leibnitz by 

 Dutens ; and in the first volume of the second section of the 

 mathematical works of Leibnitz edited by Gerhardt, Halle, 1858. 

 The dissertation is also included in the collection of the philoso- 

 phical writings of Leibnitz edited by Erdmann, Berlin, 1840. 



44. Leibnitz constructs a table at the beginning of his dis- 



