[ 304 ] 

 XLIV. Proceedings of Learned Societies. 



ROYAL SOCIETY. 



[Continued from p. 235.] 



Dec. 18, HPHE following papers were read : — 



1851. -■• "A Proof (by means of a series) that every Number 

 is composed of 4 Square Numbers, or less, without reference to the 

 properties of Prime Numbers." Bv Sir Frederick Pollock, Lord 

 Chief Baron, F.R.S. &c. 



The paper contains a proof, that if every number of the form 

 Srt + 4 is composed of 4 odd squares, then every number whatever 

 must be composed of 4 square numbers or less ; also a proof of the 

 converse of this, viz. that if every number is composed of 4 square 

 numbers or less, then every number of the form 8» + 4 must be com- 

 posed of 4 odd squares. 



It is then proposed to show that every number of the form 8m + 4 

 is composed of 4 odd squares, by taking a number of the form 8» + 4, 

 viz. an odd square +3, and showing that Sn + 4 in that case is 

 divisible into 4 odd squares (other than the odd square and 1, 1, 1); 

 thus I6n"-+8n+l is a form that includes every odd square, and 

 16n 9 +8« + 4 is divisible into 



4w~ + 4»+l, 



4» s ±4n-f-l, 



4w-+4n+l, 



4n"- + 4n+ 1. 



8 is then added, and the sum is shown to be still divisible into 4 odd 

 squares ; and again 8, and so on, until by successive additions of 

 8 + 8 + 8, &c, the quantity added to 16?i 2 + 8?« becomes equal to the 

 original term with which the operation commenced. The odd 

 squares +3 form the series 4, 12, 28, 52, 84, &c. ; and if the suc- 

 cessive additions reach the third or some higher term, and also if 

 the sum added to 16» 2 +8m be equal to the term with which the 

 operation commenced, it is contended the following term may be 

 attained, and so on, and ever}' number of the form 8« + 4 will be 

 composed of 4 odd squares. The paper concludes by a suggestion 

 that the method is applicable to several other similar problems. 



2. '* Observations on the Structure and Connexion of the Valves 

 of the Human Heart." By W. Savory, Esq. Communicated by 

 Edward Stanley, Esq., F.R.S. 



The paper contains observations upon the structure and connections 

 of the auriculo- ventricular and arterial valves of the human heart, which 

 the author thinks will assist in explaining their nature and functions. 



The relation of the " four orifices " in the base of the heart is ex- 

 amined, and it is shown that the aortic and left auriculo-ventricular 

 apertures are not separated as the others are ; that no muscular 

 tissue of the ventricle intervenes between them, but that when the 

 auricles and great vessels are separated from the ventricles (which 



