217 



4. On the Real Nature of Symbolical Algebra. By D. F. 

 Gregory, B. A. 



The object of this paper is to determine in what consists the dif- 

 ference between general Symbolical Algebra and the sciences sub- 

 ordinate to it, particularly Arithmetical Algebra. The view which 

 the author takes is, that Symbolical Algebra takes cognizance only 

 of the laws by which the symbols are combined, without consider- 

 ing the nature of the operations represented. The greater part of 

 the paper is occupied in applying this definition, by shewing what 

 are the laws to which are subject the various symbols of operations 

 we are in the habit of using ; and one or two examples are given 

 of the advantages derivable from this way of considering the sub- 

 ject — particularly with respect to the connection between the arith- 

 metical and geometrical meanings of -|- and Tlie chief appli- 

 cation of the theory may be said to be the elucidation of the causes 

 of analogies between operations by no means similar in their na- 

 ture. 



2. On a New Method in the Conic Sections. By J. Scott 

 Russell, Esq. 



In this paper the author institutes an examination of the various 

 methods of conceiving and constructing the conic sections, the dif- 

 ferent standards of comparison to which they have been referred, 

 and the means by which their properties have been developed and 

 expressed in the ancient and modern schools of mathematics. He 

 finds that the constitution of the curves in piano by means of their 

 remarkable properties, and the demonstration of their other pro- 

 perties from the constitution so obtained ; have been noticed by 

 Eutocius in the 6th century, and by him referred to Apollonius, 

 although the formal adoption of the plane method is generally re- 

 ferred to a recent date. 



The author agrees with modern writers in considering the plane 

 method in the conic sections as more philosophical, as well as more 

 useful, than the solid method ; but he diflFers from them in the me- 

 thods of constituting these curves, without reference to what he con- 

 siders their proper origin and standards of reference. Apollonius 

 employed the straight line and circle to constitute his cone ; or, in 

 other words, he derived the conic sections from the straight line and 

 the circle, using the cone as the mechanism of derivation. Mr Russell 

 has succeeded in deriving the conic sections from the straight line 



