544 On Fermafs Theorem of the PolygonaLNumbers. [Dec. 15, 



of the roots, two roots equal, and also in some form two roots differing by 

 1 ; also that in some form of the roots the algebraic sum of the roots will 

 be equal to 1 . 



If + (m^-f m-fw^) be equal to any number whatever, odd or 



even, it must equal a number represented by — (c^-f c) ; and as 

 «^ + « + 5^ — (m^ -r m + w^) — (c^ + c), 

 .-. c5^ + «-f 5^4-c^ + c=»^^^-»^ + ?^^+^^ 

 These two expressions are equivalent to each other, and any number which 

 is of the one form is also of the other form ; and if they be doubled and 

 1 be added to each, they will become 



+ 2« + 1 + 2c" + 2c, 2mH 2m + 1 + 271" + 2p\ 

 and either of them will represent any odd number whatever. For a^-^a + 

 b^—(m^-{-m-\-n'^) not only equals (c^ + c), but it also equals |3" — (c" 

 -\-c) + q; and if both be doubled and 1 be added, 



2m" + 2m+l4-2?i"4-2/+22=2a2+2«+l + 262 + 2c2 + 2c; 

 if therefore to either form any even number (2^') be added, it is still of the 

 form of the other, and therefore still of its own form, that is, 



2m'+2m+l-\-2n^+ 2f + H 

 is still of the form 2m^ + 2m+ 1 + 27i^ + 2pV and that form therefore repre- 

 sents any odd number. 



It is shown in the paper that when 2g!^ + 2a+ 1, 2/^^, 2c^ + 2c is expanded, 

 2a^-\-2a+\ becomes a series (by the addition of 4, 8, 12, &c.) whose 

 terms will be 0, 0, 0, 1 ; 0, 0, 1, 2 ; 0, 0, 2, 3, &c., and may be considered as 

 a line whose general expression is 0, 0, a, («+ 1)- 



When 2b^ is added to each term by the addition of 2, 6, 10, 14, &c. it 

 becomes a square whose general term is b, b, «, «+l (these being roots 

 whose squares added together form the term in the square). Lastly, when 

 2c^+2c is added (by decreasing a and increasing «+l, 1 each time) and 

 the square becomes a cube, every term has two roots equal, but is composed 

 of not exceeding 4 square numbers ; and as on the addition to any term of 

 any even number (2q) the term so increased will still be within the cube 

 (extended indefinitely), the cube will contain every odd number; but if 

 2m.^+2m + l-{-2n^ be formed into a square, and then by the addition of 2j>^ 

 be raised into a cube (the terms n, n in each term being one increased, 

 and the other diminished hj \ ), every term in the cube will have two 

 roots differing by 1, and will be composed of not exceeding 4 square num- 

 bers ; and this cube also will contain every odd number for the same 

 reason that the other will. 



Supplement. 



Lastly, «^ + a 4-^2— (m^ + m + ?i^) will (as it equals' any number) equal 



either p^-^-~ or _|.^^ ^nd therefore 2a^+2«+l, +2b^-]-c'-\-c 

 2 2 



will equal 2^^ + 2m + 1 + 2n^ + 2p^ with or without 2q. 



In raising a,a-{-\,b,b to a cube by adding c^ + c, it must be by the 



