Mr. J. Cockle on a nem Imaginary in Algebra. 43 



future course, rather than actually to commence it. But I 

 will not disguise the end which I am desirous of attaining — 

 that of vindicating, for ordinary algebra, a claim to the power 

 of representing space of three dimensions, as completely as, by 

 the aid of unreal quantities, it can denote any conditions 

 whatever in piano, and any modification of such conditions. 

 By way of illustration, let there be given two points, A and 

 B, and let it be required to find a third point, C, such, that the 

 rectangle AC x CB may be equal to half the square described 

 upon AB. Now, if we proceed on the supposition that C is 

 somewhere in the straight line which passes through A and 

 B, we must suppose that C lies between A and B, for other- 

 wise the rectangle would obviously be greater than the square. 

 Bearing this in mind, let AB = 2a, and ACs=:r; then the 

 qucBsitum of the problem gives the equation 



x{2a—x)=2a% (5.) 



or 



x^—2ax=—'^a^, 

 whence 



x-=a + a V' — 1. 

 Hence, the problem is an impossible one, if we regard C as 

 lying in AB. But, if we interpret the symbol v' — 1 as mean- 

 ing perpendicularity* — in which case we must regard (5.) as 

 the representation of the problem in its most general form — 

 we have the following construction. Along AB take AD = iAB; 

 from D draw DC perpendicular to AB, and equal to DA; 

 then, C is the point required. The point C, so obtained, evi- 

 dently fulfils the condition of the question. In fact, any line 

 in a plane being given, as an axis, we may represent any point 

 whatever, in the plane, by the formula jo + Q'V' — 1, p and q 

 being real. It is to be remarked, however, that, a line being 

 given in space, when the symbol \^ —\ occurs in our researches, 

 we may (as in the above problem) draw our perpendicular in 

 whatever direction we please, provided only it be in a plane 

 perpendicular to the primitive axis. But, the perpendicular 

 once drawn, we have a determined plane to which, I apprehend, 

 all our interpretations of V'— 1 are to be confined ; for I con- 

 ceive that, consistently with ordinary algebra, we cannot, on 

 V — I occurring a second time in our investigations, take that 

 symbol to denote a line perpendicular to, or making any angle 

 with, the former perpendicular. Perhaps I have said enough 

 to show what my own views are, as to the applications of the 

 symbol W — 1, and the limitations to be imposed on its inter- 



* Tills has been done by Hamilton, Warren, and others. 



