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XXI. On pEEiiAi’s Theorem of the Polygonal ITuwhers. 
By Sir Feedekick Pollock, F.B.S., Lord Chief Baron, &c. &c. 
Eeceived in abstract July 11, 1860, in full April 25, 1861, — Bead May 2, 1861. 
Fekmat’s theorem of the polygonal numbers has engaged the attention of some of the 
most eminent mathematicians. It was first aimoimced (about the year 1670) in his 
edition of Diophantus, published after his death (it occm’s in a note on the 31st question, 
p. 180). It is to be found stated at length in Legekdee’s ‘Theorie des Nombres’ (in 
p. 187 of the 2nd edition, &c.). For above a century after it appeared, no proof was 
discovered of any part of it; but in 1770 Lagea^ge (in the Transactions of the Eoyal 
Academy of Sciences at Berlin) gave a proof of the second branch of the theorem (the 
case of the square numbers), from the paper containing which it may be collected that 
Eulee had endeavoured in vain to establish a proof, but had suggested the clue by 
which liAGE-AN'GE succeeded in discovering one. 
In the second volume of Eulee’s ‘ Opuscula Analy tica,’ there is an article on this sub- 
ject, of some length, lamenting the loss of Feemat’s investigations, and pointing out 
that Lageakge’s proof as to the square numbers affords (from its nature) no assistance 
to the discovery of a proof of the other cases ; he adds, “ sine dubio plerique Geometrae 
in his demonstrationibus investigandis frustra desudaverint.” 
About twenty-five years after the death of Eulee (who died in 1783), Legekdee, in 
his ‘Theorie des Nombres,’ published a proof of the first branch of the theorem (the 
case of the triangular numbers), which proof is in part inductive, and not founded on 
pure demonstration ; and subsequently M. Cauchy discovered a proof of all the cases 
(assuming tlie first and second cases to be proved) ; this w^as published about the year 
1816, in a Supplement to Legendee’s ‘Theoiie des Nombres.’ 
Feemat, after stating the proposition, alludes to the proof of it as arising out of 
“ many various and abstruse mysteries of numbers ; ” and he states his intention to 
“ write an entire book on the subject, and very much to advance the bounds of arithmetic^ 
No such work has appeared ; and it is understood that among his papers no trace has 
been found of any materials for such a pubhcation. It becomes a matter of more than 
mere curiosity to consider what could have been the properties of numbers alluded to ; 
obnously they must have been connected, more or less, with the division of numbers 
into squares or other polygonal numbers. 
The general object I have in view is to investigate the properties of numbers on 
which Feemat’s theorem depend. In this paper I wish to call attention to some pro- 
