241 



Comparing the terms above with the terms below, it is manifest 

 the terms of the series are divisible into 4 squares whose roots con- 

 form to the law of the theorem. It is then shown that the odd squares, 

 and also all the numbers mentioned in the beginning of the paper, 

 can be made terms in an arithmetic series, and will therefore have 

 the property stated. It is then suggested that the properties of 

 numbers stated in the paper may have been in some form a portion 

 of the mysterious properties of numbers by which Fermat announced 

 he could prove his celebrated theorem of the polygonal numbers. 



A Postscript was added, dated 20th May, which is here given 

 entire. 



Since this paper was sent to the Society, some other theorems of 

 a similar kind have occurred to me, in which the terms of a series 

 (not arithmetical of the 1st order) have a similar relation with regard 

 to the roots of the 4 squares of which they may be composed, that 

 is, those which are equidistant from the middle, or the middle term 

 (according as the number of terms is even or odd), have the middle 

 roots the same, and the exterior roots have an arithmetical relation 

 to each other (varying with the distance from the centre), viz. the 

 one being less and the other greater by the same quantity. 



Thus, if any number of terms (exceeding 3) of either of the 2 

 series above-mentioned (viz. 1, 3, 9, 19, &c., or 1, 5, 13, 25, &c.), 

 and, beginning with the first term, the differences be added " inverso 

 or dine" a new series will be obtained possessing the property in 



0246 8 



question ; thus the first 7 terms of the 1st series are, 1, 3, 9, 19, 33, 

 10 12 



51, 73; the differences are, 2, 6, 10, 14, 18, 22 ; if the differences 

 be added "inverso ordine" the series becomes 1, 23, 41, 55, 65, 



