1 74 On the Symbol V — 1 in Geometry. 



x \/-l is a real quantity involving no incongruous opera- 

 tions. This, however, is more immediately and more fre- 

 quently the case in the resolution of problems; the cases of 

 this possible transformation of the conditions under which 

 a theorem involving incongruous operations (whether in pure 

 analysis or its applications to geometry or physics) can be con- 

 gruously expressed being comparatively rare. Be it remem- 

 bered, however, that whether the preceding views be admitted 

 or not, they do not actually bear upon the direct argument 

 which I am about to urge, although according to my view 

 they strongly bear upon its illustration. 



My proposition is this, — 



The symbol V — 1 does not express perpendicularity, but 

 only incongruity amongst the geometrical conditions from which 

 the expression was derived. 



1 . When we attach the signs + and — to the symbols of 

 two lines, it expresses that those lines have a contrary direc~ 

 Hon estimated from a given point in a given indefinitely pro- 

 longed line. Their positions become then fixed and incapable 

 of an altered position without cancelling the hypothesis of 

 their existence. 



2. When we define a rectangle, we do it as a parallelogram 

 which has these two lines placed at right angles to each other 

 for adjacent sides. These two lines then become fixed and 

 incapable of an altered position, without cancelling the hypo- 

 thesis of their existence. 



3. Let A B be the same iu both 

 cases ; then A B' is the second line of 

 the first hypothesis, and A C is the 

 second line of the second hypothesis. 

 Whence AB' and AC cannot coincide 



without cancelling one hypothesis or B' A B 



the other. Let AB, A B', AC be of equal magnitude. 



4. In saying that the rectangle + a x -a = AC 2 = a 2 , we 

 allege that —a can have the position AB' and AC at the same 

 time; and in this way we get AC= faV'-l. 



5. In this we have manifestly only prescribed incongruous 

 operations, those of giving to the same line two different posi- 

 tions at the same time. 



6. Perpendicularity is a real property, or the result of a 

 possible and performable series of operations, whilst this al- 

 leged expression of it results from incongruous operations. 

 The expression V — 1 cannot, therefore, be that of perpen- 

 dicularity ; and it has been proved to be that of incongruous 

 operations. 



