.so SniOOTEN. 



tionem ipse mihi ostendit, ac etiam demonstranJam proposiiit, ipsam 

 ego san^ miratus sum, sed difficultate territus vix opus suscepi, 

 et ipsi authori relinquendum existimavi; attamen trianguli arith- 

 metici auxilio, sic proclivis facta est via. Pascal then establishes 

 the correctness of the rule by the aid of his Arithmetical Triangle; 

 after which he concludes thus, Hac demonstratione assecuta, jam 

 reliqua quae invitus supprimebam libenter omitto, adeo dulce est 

 amicorum memorari. 



42. In the work of Schooten to which w^e have already re- 

 ferred in Art. 28 we find some very slight remarks on combinations 

 and their applications; see pages 873 — 403. Schooten's first sec- 

 tion is entitled, Ratio inveniendi electiones omnes, qu^ fieri pos- 

 sunt, data multitudine rerum. He takes four letters a, h, c, d, 

 and arranges them thus, 



a. 



h. ah. 



c. ac. he. ahc. 



d. ad. hd. abd. cd. acd. bed. abed. 



Thus he finds that 15 elections can be made out of these four 

 letters. So he adds, Hinc si per a designatur unum malum, jDer b 

 unum pirum, per c unum prunum, et per d unum cerasum, et ipsa 

 alitor atque alitor, ut supra, eligantur, electio eorum fieri poterit 15 

 diversis modis, ut sequitur 



Schooten next takes five letters ; and thus he infers the result 

 which we should now express by saying that, if there are n letters 

 the whole number of elections is 2"— 1. 



Hence if a, b, c, d are prime factors of a number, and all dif- 

 ferent, Schooten infers that the number has 15 divisors excludinsf 

 unity but including the number itself, or 1 6 including also unity. 



Next suppose some of the letters are repeated; as for example 

 suppose we have a, a, b, and c ; it is required to determine how 

 many elections can be made. Schooten arranges the letters thus, 



a. 



a. aa. 



h. ah. aab. 



c. ac. aac. be. ahc. aabc. 

 We have thus 2 + 3 + 6 elections. 



