XXXVII. Contributions towards a System of Symbolical Geometry and Mechanics. 

 By the Rev. M. O'Brien, Professor of Natural Philosophy and Astronomy 

 in King's College, London, and late Fellow of Caius College, Cambridge. 



[Read March 15, 1847.] 



1. The important distinction which has been made by an eminent Authority in Mathematics 

 between Arithmetical and Symbolical Algebra, may be extended to most of the Sciences which 

 call in the aid of Algebra. Thus we may distinguish between Symbolical Geometry and Arith- 

 metical Geometry, Symbolical Mechanics and Arithmetical Mechanics. This distinction does not 

 imply, that in one division numbers only are used, and in the other symbols, for symbols are 

 equally used in both, but it relates to the degree of generality of the symbolization. In the 

 Arithmetical Science the symbols have a purely numerical signification, but in the Symbolical they 

 represent, not only abstract quantity, but all the circumstances which, as it is usually expressed, 

 aject quantity. The Arithmetical Science is, in fact, the first step of generalization, and the 

 Symbolical the complete generalization. 



In this view of the case, I have ventured to entitle the following Paper "Contributions towards 

 a System o{ Symbolical Geometry and Mechanics." The Geometrical System about to be proposed 

 consists, first, in representing curves and surfaces by symbolical formulas, and secondly, in using 

 the Differential Notation to denote Perpendicularity, according to the principles explained in a Paper 

 read a few months since at a Meeting of the Society. The proposed Mechanical System is analogous in 

 many respects to the Geometrical : examples of it have already been given in the Paper just quoted. 



2. The following well-known principles are those upon which the 

 Geometrical System is based. 



1st. If ABCD be any polygon, then AD = AB + BC + CD. 



This may be regarded as the definition of +. 

 2ndly. Giving the usual definition of - it follows, that, in the triangle ° 



ABC, 



AC - AB = BC. * 



.3rdly. Where it follows that, if a denote any right line, - a denotes an o(iual right line 

 mea.sured in an opposite direction. 



ithly. If m denote any number, ma denotes a line m times the length of a drawn in the same 

 direction as a. This follows immediately from the first principle. 



These principles, with some others which we need not specify here, form the basis of llie 

 Geometrical System about to be proposed. 



3. It will be convenient to consider that every line is traced by the tiiolion of a point, .ind 

 this will lead us to distinguish between the beginning and end of a line, the beginning being the 

 extremity from which the tracing point starts, and the end the otiier extremity. 



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