922 



the tail. According to these admeasurements the Pelorosaurus would 

 be 81 feet long, and its body 20 feet in circumference. But if v, e 

 assume the length and number of the vertebree as the scale, we 

 should have a reptile of relatively abbreviated proportions; even in 

 this case, however, the original creature would far surpass in mag- 

 nitude the most colossal of reptilian forms. 



In conclusion, Dr. Mantell comments on the probable physical 

 conditions of the countries inhabited by the terrestrial reptiles of the 

 secondary ages of geology. These highly-organized colossal land 

 saurians appear to have occupied the same position in those ancient 

 faunas as the large mammalia in those of modern times. The trees 

 and plants whose remains are associated with the fossil bones, mani- 

 fest, by their close affinity to living species, that the islands or con- 

 tinents on which they grew possessed as pure an atmosphere, as 

 high a temperature, and as unclouded skies as those of our tropical 

 climes. There are therefore no legitimate grounds for the hypo- 

 thesis in which some physiologists have indulged, that during the 

 Age of Beptiles" the earth was in the state of a half-finished 

 planet, and its atmosphere too heavy, from an excess of carbon, for 

 the respiration of warm-blooded animals. Such an opinion can only 

 have originated from a partial view of all the phenomena which 

 these problems embrace, for there is as great a discrepancy between 

 the existing faunas of different regions, as in the extinct groups of 

 animals and plants which geological researches have revealed. 



The memoir was illustrated by numerous drawings, and the gi- 

 gantic humerus of the Pelorosaurus and otlier bones were placed 

 before the Society. 



February 21, 1850. 



GEORGE RENNIE, Esq., Treasurer, in the Chair. 



Robert Alfred Cloyne Austen, Esq. was admitted into the Society. 

 The following papers were read : — 



1. "On the Extension of the Principle of Fermat's Theorem of 

 the Polygonal Numbers to the higher orders of series whose ultimate 

 differences are constant. With a new Theorem proposed, applicable 

 to all the Orders." By Sir Frederick Pollock, Lord Chief Baron, 

 F.R.S. 



The object of this paper professes to be to ascertain whether the 

 principle of Fermat's theorem of the polygonal numbers may not be 

 extended to all orders of series whose ultimate differences are con- 

 stant. The polygonal numbers are all of the quadratic form, and 

 they have (according to Fermat's theorem) this property, that every 

 number is the sum of not exceeding, 3 terms of the triangular num- 

 bers, 4 of the square numbers, 5 of the pentagonal numbers, &c. 



It is stated in this paper that the series of the odd squares 1,9, 25, 

 49, Sea. has a similar property, and that every number is the sum of 



