28 PASCAL. 



by James Bernoulli in his Ars Coiijectandi, where he thus details 

 the history of the problem. 



. , . Quemadmodum cernere est in hexametro a Bernli. Bauhusio Jesuita 

 Lovaniensi in laudem Virginis Deiparse constructo : 



Tot tihi sunt Dotes, Virgo, quot sidera ccdo ; 

 qiiem dignnm peculiari opera duxerunt plures Viri celebres. Erycius 

 Puteanus in libello, quern. Tliaumata Pietatis inscripsit, variationes ejus 

 utiles integris 48 paginis enumerat, easque numero stellarum, quarum 

 vulgb 1022 recensentur, accommodat, omissis scrupulosius illis, quse di- 

 cere videntur, tot sidera cselo esse, quot Marine dotes; nam Mariae 

 dotes esse multo plures. Eundem numerum 1022 ex Puteano repetifc 

 Gerh. Yossius, cap. 7, de Scient. Matliemat. Prestetus Gallus in prima 

 editione Element. Matliemat. pag. 358. Proteo huic 2196 variationes 

 attribuit, sed facta revisione in altera edit. torn. pr. pag. 133. numerum 

 earum dimidio fere auctum ad 3276 extendit. Industrii Actorum Lips. 

 Collectores m. Jun. 1686, in recensione Tractatus Wallisiani de Algebra, 

 numerum in qusestione (quem Auctor ipse definire non fuit ausus) ad 

 2580 determinant. Et ipse postmodum Wallisius in edit, latina operis 

 sui Oxon. anno 1693. impressa, pagin. 494. eundem ad 3096 profert. 

 Sed omnes adliuc a vero deficientes, ut delusam tot Yirorum post 

 adhibitas quoque secundas curas in re levi perspicaciam meritb mireris. 

 Ars Conjectandi, page 78. 



James Bernoulli seems to imply that the two editions of 

 Wallis's Algebra differ in their enumeration of the arrangements 

 of the line due to Bauhusius ; but this is not the case : the two 

 editions agree in investigation and in result. 



James Bernoulli proceeds to say that he had found that there 

 could be 3312 arrangements without breaking the law of metre; 

 this excludes spondaic lines but includes those which have no 

 caesura. The analysis which produces this number is given. 



41. The earliest treatise on combinations which we have ob- 

 served is due to Pascal. It is contained in the work on the 

 Arithmetical Triangle which we have noticed in Art. 22; it will 

 also be found in the fifth volume of Pascal's works, Paris 1819, 

 pages 86—107. 



The investigations of Pascal on combinations depend on his 

 Arithmetical Triangle. The following is his principal result; we 

 express it in modern notation. 



