36 Mr. C. J. Hargreave's Analytical Besearches 



of animal life in the palaeozoic ages. The coal-fields of the 

 Asturias (of which there are seventy workable seams) seem 

 therefore to be subordinate to the mountain limestone, like 

 those of the north of Northumberland, the south of Scotland, 

 &c. &c. 



The Devonian system has been found to abound in the 

 north of Spain, chiefly through the researches of M. Paillette, 

 who has transmitted many of its fossils to France, where they 

 have been described by M. de Verneuil. 



The Triassic and Jurassic systems are also stated to be con- 

 siderably developed in Spain, and like the palaeozoic rocks they 

 are highly dislocated. 



In conclusion, the author remarks, that the interesting 

 region of the Asturias will soon be better known, first through 

 a very exact geographical map prepared by M. Paillette, 

 particularly in reference to its coal-fields ; and next by a gene- 

 ral geological map of the province by Don. G. Schultz, on 

 which that gentleman has been occupied during four years, 

 and which is spoken of as a work of great merit. 



V. Analytical Researches concerning Numbers. By Chaules 

 James Hargreave, Esq.^ LL.B., FJi.S., Professor of 

 Juris'prudence^ University College^ London^. 



THE theory of numbers is a branch of mathematics which 

 has either been regarded as not falling under the domi- 

 nion of analysis, or has eluded the researches of those who 

 may have attempted its application; so that, notwithstanding 

 the great powers and singular industry of the eminent men who 

 have devoted themselves to this science, it remains, so far as its 

 processes are concerned, nearly in the condition in which the 

 theory of probabilities would have stood, if the higher analysis 

 had not l)een applied to it. It abounds with theorems of 

 remarkable elegance; but is destitute of the processes, which 

 alone, in case of need, could render it a practical science. 



Legendre, in his treatises on this subject, communicated as 

 the result of an observation of numbers in the first million, a 

 formula for determining the number of primes up to any given 

 limit; and adopting this formula as a law, he was conducted 

 to some theorems involving the application of analysis. ( Theorie 

 des Nomhres, vol. ii. § VIII.) That the formula thus given 

 by him is not really the natural law of primes, is apparent 

 from the circumstance that it involves a constant, or rather 



* Communicated by the Author. 



