Ill 



Monday, 18M January 1847- 

 Sir GEORGE S. MACKENZIE, Bart., V.P., in the Chair. 



The following Communications were read : — 



1. An Attempt to Ehicidate and Apply Mr Warren's Doc- 



trine respecting the Square Root of Negative Quantities. 

 By the Right Rev. Bishop Terrot. 



The author shewed, in the firet place, that the successive radii of 

 a circle forming equal angles with one another, are properly symbo- 

 lized by the successive expressions, a, a . ,~^ a .,-— &c., where 



a is the arithmetical length of the radius, and ^ the angle which the 

 several radii make with one another. He proceeded to shew how this 

 theory of symbols, first advanced by Mr Warren, is applicable to the 

 problems of plane trigonometry. This was shewn in several ele- 

 mentary propositions, in a new demonstration of Cotes' properties of 

 the circle, and in the demonstration of a property closely connected 

 with these, namely, that if A be the extremity of a diameter, and 

 P, Pi_ P.2, P3, &c., be the extremities of a series of equal arcs mea- 

 sured in the same direction from it, then A Pi x A P2 . . . x A Pn - 1 

 = nR''-^ The author finally shewed that though this sym- 

 bolism is applicable to the most elementary propositions of geometry, 

 it is applicable under forms which are unsuitable to the purposes of 

 elementary instruction. 



2. Notes on the Superficial Strata of the Neighbourhood of 



Edinburgh, concluded. By Dr Fleming. 



The author began by adverting to the position of Edinburgh as 

 favourable to the study of geology, from the proximity of the transi- 

 tion rocks, the old red sandstone, and the coal measures ; and then 

 stated, that the deposits belonging to the modern epoch were equally 

 accessible and interesting. 



In order to illustrate the character of the superficial strata, the 

 author considered, in the first instance, the condition of the surfaces 

 of the rocks, on which these loose deposits rest. He had found many 

 examples, as at Craigleith, Granton, and Sampson's Ribs, where, on 

 the removal of the boulder-clay, the surfaces of the rocks were dressed 

 and scratched in a direction nearly from W. to E. ; while in other 



