ON POLYGONAL NUMBERS. Igl 



to allow that, if e can be resolved into m — / polygons, On Fermat's 

 e+/ may be resolved into m polygons; but if e cannot be po°ygona°" 

 resolved into m — / polygons, what proof have we, that numbers. 

 e-\-f can be resolved into m polygons? And that there are 

 many such cases is evident : thus, 14 cannot be resolved into 

 less than three or m triangular numbers, nor 23 into less 

 than four or m squares. 



This ought to be explained, the importance of the pro- 

 position demands it, the last labours of Euler, Lagrange, 

 and le Gendre demand it also, and a few more pages may 

 very well be afforded to complete the demonstration. 



The remaining part of the essay goes on to show, how 

 any given number may be resolved into polygonal numbers 

 of any given denomination ; but, from the examples there 

 given, it does not appear to possess any advantage over the 

 usual method of trials ; and even if it did, it is of no use 

 in demonstrating the proposition, for showing how a thing 

 is -to be done is very different from showing it may always 

 be done. 



Upon the whole therefore we may conclude, that for the 

 present, the celebrated theorem of Fermat is without a de- 

 monstration, and that its importance, as containing one of 

 the most curious properties of numbei-s, renders it worthy 

 the attention of mathematicians. 



VII. 



Sotne farther Remarks on the Doctrines of Chance, in a Let- 

 ter from a Correspondent. 



SIR, 



H- 



AVING observed a letter in the last number of your Certain doc- 

 valuable publication, from a correspondent who has assumed t""esof chance 

 the signature of Opsimath, in which some doubts are express- a former con-o* 

 ed respecting the elementary Doctrines of Chance, and a spondent. 

 request to yourself, or any of your correspondents, either to 

 confute or to confirm his objections, I have ventured to offer 

 the following remarks; though certainly with some diffidence, 



being 



