Sir F. Pollock on the Mysteries of Numbers J* 



115 



"On the Mysteries of Numbers alluded to by Fermat." By 

 the Rt. Hon. Sir Frederick Pollock, Lord Chief Baron, 

 F.U.S., &c* , 



(Abstract.) 



This paper is presented as a continuation of one which appeared in the 

 Philosophical Transactions of the B.oyal Society for 1861, vol. cli. p. 409, 

 and the object of it is to call attention to certain properties of odd numbers 

 when placed in a square, according to an arrangement to be explained 

 below. 



It appears to me probable that these properties are connected with (if, 

 indeed, they be not actually some form of) the mysterious properties of 

 numbers, to which Fermat alludes in the announcement of his theorem (as 

 furnishhig the proof of it) ; for in point of fact these properties give a 

 method by which every odd number can be divided into four square num- 

 berSj and every number (odd or even) can be divided into not exceeding 

 three triangular numbers. 



I am not prepared to say whether or not the method affords a demon- 

 stratioTiy and proves that it can always be done ; but it always does it, and 

 the cause of its success may be distinctly shown. The properties I allude 

 to are scarcely less interesting and curious than the theorem itself, and pre- 

 sent results for which I can find no name more appropriate than the 

 geometry ofnumberSy for relations appear to be established between various 

 numbers in the square, which relations are not founded on any arithmetical 

 connexion between them, but on the positions they respectively occupy in 

 the square of which they form a part. 



The arrangement of the numbers is as shown in diagram No. 1 . Any 

 odd number (which may be the subject of inquiry) is made the first term 

 of a series, increasing from left to right by the numbers 4, 8, 1 2, 1 6, ... (4 ri) 

 This series forms the horizontal line at the top ; each term of the series so 

 formed is made the first term of a series, increasing downwards by the 

 numbers 2, 6, 10, 14 ... . (4w— 2). A square of indefinite magnitude is 

 thus formed, consisting of two sets of series, one set all horizontal, the 

 other set all vertical. 161 is the first number in diagram No. 1. 



The result of the arrangement of the two series in the manner above 

 mentioned is the formation of a third set of series, which may be found in 

 the diagonal lines of the square. 



If from the first number in the square ^ diagonal line be drawn 



towards the opposite corner, it will pass through the first terms of one por- 

 tion of the third set of series ; the second, third, and following terms are 

 taken alternately from each side of the diagonal line. These series increase 



VOL, XV. 



Bead April 19. (See p. lOG.) 



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