248 Mr. DE MORGAN, ON TRIPLE ALGEBRA. 



The three axes on which a, 6, c, are laid down, ought not to be rectangular axes, but those 

 of y and «• should be each inclined at 60" to the axis of w, so that units laid down on them may 

 be cube roots of - 1. The planes of acy and xz being at right angles, and A being the diagonal 

 of the parallelepiped on a, b, c, we have /■- = A^ - f 6c. 



Should a simple interpretation be obtained, the ancient difficulty of the imaginary quantity will 

 immediately occur; for -y/m must take the place of m in -\/[/, m, O], and m may be negative. 

 This system therefore will never be completely explained until it is interpreted on the supposition 

 that a, b, &c. have the forms a + a ^— 1, 6 +6,-\/- 1, and also Q, I, kc. By analogy we 

 might have expected this, in the following manner. As soon as pure arithmetic is converted into 

 single Algebra by the extended definitions of + and — , and the new symbol \/- 1 occurs, it occurs 

 in conjunction with both the forms + 1 and — 1 ; and at tlie same time the vehicle of explanation 

 takes two dimensions. If new distinct symbols be added, such as will require space of three 

 dimensions, it is therefore natural to suppose that each of those new symbols will combine with the 

 complete system of the double Algebra. By this, since a + «,-y/- I may mean any line in the 

 plane of ivy, it is reasonable to suppose that two new symbols will be required, to express removal 

 into the planes of yz and «x, and that 



(o + o-v/- 1) + (6 + 6 \/- 1)>; + (c + c,v^- 1)^, 

 will signify some line in space, determined by three lines in the three co-ordinates planes. 



& 4. Redundant biquadratic form. The last remark suggests an examination of the method 

 by which systems have hitherto proceeded, with a view to ascertain whether the hints which analogy 

 might give are exhausted. If we look at the series +1, — 1, \/ — 1, we see that one new 

 unit-symbol is introduced at each step, represented by a square root of the preceding. What then is 

 the system in which one more unit-symbol is introduced, whose action resembles that of ^y — 1, 

 the combination with preceding symbols being of the complete character just described. 



Let the fundamental symbol be 



{a, p, b, q] --= a + J) ^y - \ + (b + q y/ - 1)^, 



where "Q means y/ - \- Accordingly, the product of [o, j), 6, </] and [a', -p , b', 5'] is [_A, P, B, Q] 

 where 



J = aa' — pp - bq — b' q, B = ah' + a' b — pq' — p'q, 



P = ap' + a'p + bb' - qq, Q = aq' + a q + bp + pb' . 



The modulus of multiplication is found to be 



Now it is evident that, a line in space being determined by three data, we have here one to spare,. 



since a, 6, p and g must all be given before the fundamental symbol is completely determined. It 



Would be in our power for instance, to consider the symbol as meaning a line of given length drawn 



from the origin in a given direction at a given time ; or as determining a point which has a given 



position at a given instant. Let a + p ^y - i represent in the usual manner a line in the plane 



of wy, and let "t represent a unit somewhere in the plane of xz ; we may easily see that it must 



be at 45" to the positive axis of x, if the rule of angles in multiplication is to be preserved. To 



satisfy this last condition, let [a, p, b, g] represent a length I making an angle with the axis of 1 



determined bv 



b - q . n b + q 



lcosd = a + — -pr- , Ismd = p + — — ■ 



