and its appUcatioji to the Geometry of three Dimensions. 121 



part of almost all modern systems of symbolic geometry. If 

 we next proceed to inquire how we may represent the product 

 of two lines {xy) and {x^i/)i we shall find the following rule for 

 constructing it : " The projections of the real unit line, the fac- 

 tor lines, and the product line, either upon the axis OA, or upon 

 OB, a right line perpendicular to it, foi-m an algebraic pro- 

 portion.^* This is the geometrical interpretation of the equa- 

 tions (1.). 



5. So far we have been geometrizing only in piano ; but we 

 can pass readily into the geometry of three dimensions, since 

 in the course of its rotation round OA the real unit quits the 

 plane of jry. 



As s denotes the rotation of 180° round OA, 5* may be taken 

 to denote half that rotation. 



Again, the unit of length in the direction of OA is 



l+s{l) , 



and the unit of length in the perpendicular direction is evi- 

 dently 



i-sjl ) 



± v/2' 



Now if this latter unit be operated upon by s4, it will be brought 

 into the position of the axis of z, perpendicular to the axes 

 both of x and y; that is to say, 



i/2 



is the positive x-unit, which we shall henceforth denote by w(l). 

 For the square of this ^-unit we shall find a simpler expres- 

 sion. Squaring the equation 



we have 



5(l)-l=n2(l); 



and operating upon the preceding equation with s, we find 



5n(l)=-«(l). 



6. Having now ascertained the laws of the combination of 

 5 and n, we may proceed to deal with trinomials of the form 

 x + s{y) + n{z), which we shall take to represent the line drawn 

 from the origin of rectangular co-ordinates to the point (03/2). 



As before, the sum of two lines is their resultant. If 



a" + s{y") +»(»") = (a? + s{y) + «(z)) (a;' + 5(2^) + n{z% (2.) 



