A 



JOURNAJL 



OF 



NATURAL PHILOSOPHY, CHEMISTRY, 



AND 



THE ARTS. 



JULY, 1808. 



ARTICLE I. 



An Essay on Polygonal Numbers, containing the Demonstra- 

 tion of a Proposition respecting Whole Numbers 171 general. 

 In a Letter from John Gough, Esq. 



SIR, Middleshaw, May 26, 1808. 



JL HE design of the present essay is, to demonstrate the Proposition to 

 following general propositions : every whole number is either ^ ^'^^^ 

 a polygonal number of a given denomination tn; or it may 

 be divided into polygons of that denomination, the number 

 of which does not exceed m. This singular property of num- 

 bers was discovered by Mr. de Fermat, who I believe gave 

 it to the world without a demonstration. Should the pre- 

 sent attempt to supply the defect obtain your approbation, 

 the publication of it will oblige 



Yours, &c. 



JOHN GOUGH. 



Definitions. 1 st. The sum of any arithmetical progres- Definition?, 

 sion, beginning with unity, is called a polygonal number, 

 and a polygon at times in the present essay. 



Vol. XX. No. 83— July, 1308, M 2d, 



