REPORT ON CERTAIN BRANCHES OF ANALYSIS. 233 



nate states of quantity, are equally inconceivable. We are ac- 

 customed, however, to speak of quantities as infinitely great 

 and infinitely small, as distinguished from finite quantities, 

 whether great or small, and to represent them by the symbols 

 00 and 0. It is this practice of designating such inconceivable 

 states of quantity by symbols, which brings them, in some de- 



of the circle and the rectangular hyperbola. It is well known that the circle 

 and rectangular hyperbola are included in the same equation y = \/{\ — x-), 

 if we suppose x to have 

 any value between + oo 

 and — 00 : let a circle be 

 described with centre C 

 and radius C A = 1, and 

 upon the production of 

 this radius,; let a rectan- 

 gular hyperbola be de- 

 scribed whose semiaxis is 

 1, in a plane at right an- 

 gles to that of the circle: 

 if 6 denote the angle A C P, 

 then the circular cosine and sine (C M and P M) are expressed by 



giVITi _ e- ^ -/^ e^ V^ — e- ^ V^ 



and 



2 2 V— 1 



respectively ; whilst the hyperbolic cosine and sine (to adopt the terms pro- 

 posed by Lambert) corresponding to the angle 6 \f — 1 (in a plane at right 

 angles to the former) are expressed bv 



I j, or by — X_ — and 



2 " ^ \ 2 / '2 2 



if they be considered as determined by the following conditions ; namely, 

 that (hyp. cosine)" — (hyp. sine)^ = 1, and that hyp. cos & = hyp. cos — tf, 

 and hyp. sine ^ = — hyp. sine — ^. A comparison of these processes in the 

 circle and hyperbola would show, says Mr. Playfair, that investigations which 

 are conducted by real symbols, and therefore by real operations, in the hy- 

 perbola, would present analogovs imaginary symbols, and therefore analogous 

 imaginary operations in the circle, and conversely ; and that the same species 

 of analogy which connects the geometrical properties of the circle and hyper- 

 bola, connects the conclusions, of the same symbolical forms, when conducted 

 by real and imaginary symbols. 



This attempt to convert an extremely limited into a very general analogy, 

 and to make the conclusions of symbolical algebra dependent upon an insu- 

 lated case of geometrical interpretation, would certainly not justify us in 

 drawing any general conclusions from processes involving imaginary symbols, 

 unless they could be confirmed by other considerations. The late Professor 

 Woodhouse, who was a very acute and able scrutinizer of the logic of ana- 

 lysis, has criticised this principle of Mr. Playfair with just severity, in a paper 

 in the Philosophical Transactions for 1802, "On the necessary truth of certain 

 conclusions obtained by means of imaginary expressions." The view which he 

 has taken of algebraical equivalence, in cases where the connexion between 

 the expressions which were treated as equivalent could not be shown to be 

 the result of a defined operation, makes a very near approach to the principle 



