264 



Mr DE morgan, ON A PROOF OF THE EXISTENCE OF 



commensurables is proof that a demonstration can be found as to incommensurables. If a 

 step suggested by interpretation, and seen to be a true step by perception of the necessary 

 consequences of interpretation, be allowed to stand part of the proof, without anything further, 

 this question then arises, Can the step of interpretation be supplied by an algebraical substi- 

 tute? If yes, then the substitution ought in strictness to be made, and it must be made on 

 demand : if no, then the proof cannot be called either actually or potentially algebraical. 



The proof which I have given above is not, in the very strictest sense, algebraical. All 

 its geometrical interpretations might very easily be replaced by algebraical ones ; not so its 

 arithmetical interpretations. It hinges on the use o{ greater and less, when we come to apply 

 the preliminary theorem to (pz. Let all the letters denote operations, how are we to prove 

 that X^ + AX + B performed on (px is the result of five successive operations of the form 

 X- a I do not believe that any proof* exists except that which is derived from our know- 

 ledge that transformations deduced from quantitative interpretations, upon no assumptions as 

 to the specific magnitude of the quantities, are symbolically valid. 



I now proceed to supply the algebraical substitute for a geometrical step which occurs 

 in Sturm's proof of Cauchy's theorem, and in Mourey's proof of the fundamental theorem. 

 When a closed circuit is described, say in the positive direction of revolution (that is, in the 

 direction which, on the whole balance of positivej- and negative revolution, makes the radius 

 drawn from soine one point inside it gain four right angles), then the radius drawn from any 

 one point whatsoever inside the circuit gains four right angles ; the radius from any point 

 outside neither gains nor loses, performing as much positive revolution as negative; the radius 

 from any point on the circuit gains two right angles during continuous revolution, and a 

 second pair of right angles per saltum, in passing through its vanishing position. This is as 

 evident as can be when the figure is looked at. 



Let one point, within the contour, be taken as the origin : let the radius from this point to 



{x,y) be r, and its angle with the axis of w be 9. Let there be another point within, on, or 



without, the circuit, at a radius m and angle /m with respect to the origin. Let the radius 



from the point just named to (.r, y) be s, and its angle a. Remember that r, m, s, are posi- 



„, , , „ . « . . rsinO-msin/i 



tive. \Ve have then rcost) = Tn cos m + scoso", rsind=m sm/x + ssma, tana- = . 



r COS0 — mcos^ 



Now it is the algebraical property of this last formula, independent of all geometrical interpre- 

 tation to those who algebraize the sine and cosine, that while 6 changes from to Stt, o- gains 

 Stt, or gains tt, or gains nothing ; and never loses. Let cr = ft. + -^ : we then deduce 



r sin {9 - fi.) 



tan \j/ = 



rcos{9 — fi) — m 



* The celebrated proof of Laplace, or rather his improve- 

 ment of the ])roof j^iven by Foiicenex {Lemons de I'Ecole Nor- 

 male, vol. ii. p. 315), has often been cited as a proof that every 

 equation has roots. The iirst words of Laplace are " Soient 

 a, b,c, (Sic. les diverges racines de cette Equation..." and the 

 proposition proved is that these roots are of the form m + n^ — l. 

 Dr Peacock's form of this proof {Report on Analysis, p. 298), 

 begins by shewing that the possibility of roots stands or falls 

 with the possibility of symbols, all whose symmetrical products 



are given symbols. But the assumption of this possibility is a 

 difficulty of the same kind. 



■]■ The circuit must not be autolomic. Subject to this con- 

 dition it makes any undulations. With respect to an internal 

 point, any point which describes the circuit revolves in one way, 

 positively or negatively, while it is hidden from the internal 

 point by an even number of intervening parts of the circuit, and 

 in the other way, negatively or positively, while it is hidden 

 I by an odd number of intervening parts. 



