42 Proceedings of the American Philosophical Society. 



a careful examination by Mr. James Sowerby, to assimilate very closely 

 to those of the oolitic beds," &c. 



Mr. Lea's paper contains descriptions of the following species : 



Orthocera Humboldtiana, Ammonites Tocaimaensis, Ammonites occi- 

 dentalis, Ammonites Gibboniana, Ammonites Vanuxemensis, Ammonites 

 Americana, Trigonia Gibboniana, Trigonia Tocaimana, Trigonia Hon- 

 daana, Natica Gibboniana, Spatangus Colombianus, Terebratula Taylo- 

 riana, Terebratula Poeyana, Tellina [?] Humboldtiana. 



The committee, consisting Dr. Patterson, Prof Bache, and Mr. Walker, 

 to whom was referred a paper, entitled "On the Insufficiency of Taylor's 

 Theorem as commonly investigated, with Objections to the Demonstra- 

 tions of Poisson and Cauchy, and the assumed Generalization of Mr. 

 Peacock ; to which is added, a New Investigation and Remarks on the 

 Development and Continuity of Functions, by Charles Bonnycastle, Pro- 

 fessor of Mathematics in the University of Virginia," reported in favur of 

 its publication in the Transactions of the Society, which was ordered 

 accordingly. 



The paper of Prof Bonnycastle is composed of three sections. In the 

 first, which is on the " Development of Functions," he points out and 

 discusses what he considers to be " the errors and conflicting views result- 

 ing from the vague manner in which mathematical writers have usually 

 conceived the ultimate object of their peculiar logic." The second sec- 

 tion is on the " Continuity of Functions," and the division of this con- 

 tinuity into classes ; a subject heretofore touched upon only incidentally 

 by other writers. The principal object of the paper is presented in the 

 third section, which treats of" Functions considered in the order of their 

 magnitude," and particularly of " Taylor's Theorem ;" and the author 

 discusses this subject with the care demanded by a theorem which forms 

 the basis of the differential and integral calculus, and which acts so im- 

 portant a part in all the higher mathematics. 



Mr. Walker, from the committee on making and collecting observations 

 of celestial phenomena, reported in part, that they had received observa- 

 tions of Lunar Occultations of the fixed stars, which are given in the 

 mean time of the respective places of observation, being a continuation of 

 the list published in No. 6, pp. 71, 72, of the Society's Proceedings, (Vol. 

 XXXVIII, p. 177, of this Journal ;) and, on motion, the report was ac- 

 cepted. 



The longitudes and latitudes of the American places of observation, as 

 far as they can be determined from a reduction of these and former Amer- 

 ican observations, have been furnished by Messrs. Walker and Kendall, 

 as follows : 



