72 Proceedings of Philosophical Societies. [July", 



instrument is reversed. This would not happen if the tuhe were 

 wider. Tin's construction is so simple, that it is conceived the 

 preceding description will be sufficient to give an exact idea of it 

 without a figure. 



Count Rumford has read to the Class the description of a ther- 

 mometer destined to measure the specific heat of solids and liquids. 

 But this memoir has not been communicated to us. 



Books presented to the Class by Members and Correspondents 

 during the course of 1813. 



Exercises on the Integral Calculus', supplement to the first part 

 by M. le Chevalier Lcgendre. 



The author in this supplement proposes to make known a new 

 and extensive class of definite integrals, which may be expressed in 

 part by eliptic functions, in part by the arcs of circles and by 

 logarithms. These applications, joined to all those that he has given 

 in the first part, demonstrate more and more the necessity of ad- 

 mitting eliptical functions into the integral calculus to the same 

 rank as the arcs of circles and logarithms. This will require pretty 

 extensive tables of the functions of the first and second species. 

 The author has already shown shortly, how such tables may be 

 constructed ; he now promises to give the formulas proper for 

 shortening the labour, and furnishing new means of executing it. 



It is obvious that a work of this kind, which is in fact merelv a 

 collection of formulas, cannot be analysed. We shall therefore 

 merely say, that the author for the sake oi" greater clearness, has 

 arranged all the integrals under 16 cases, which form as many 

 particular tables, in which we may observe the whole results by a 

 glance of the eye, and find those for which we have occasion : that 

 he has often varied the solutions by showing the different ways by 

 which we may arrive at the same expression ; and that he has 

 marked out some particular cases, which from their singularity and 

 difficulty seem to him to deserve all the attention of mathema- 

 ticians. 



Reflections on the Metaphysics of the Infinitesimal Calculus, by 

 M. Carnot. Second Edition. 



In speaking of the first edition of this work M. Lacroix gave this 

 testimony in its favour, that the metaphysics if the calculus was 

 presented in a new, ingenious, and concise manner. Nothing more 

 clear, or more satisfactory has been written in favour of the algo- 

 rithm of Leibnitz. But notwithstanding the preference which he 

 gives to tins system, the author passes in review the different 

 points of view under which this theory may be examined. He 

 answers the different objections which have been started against it. 

 It is always the met hot! of exhaustion of the ancients mote or less 

 simplified) more or less happily appropriated to the requisites of the 

 calculus and reduced to a regular algorithm. The same principles 

 enable him to explain the principles of the calculus of variations of 

 the immortal Lagrange. This new edition is enriched with an ad- 



