434 PROCEEDINGS OF THE AMERICAN ACADEMY. 



31. We have seen that rotation in a (7)-plane about the perpendicu- 

 lar (5)-Une is Euclidean, whereas rotation in a (5)-plane about the 

 normal (7)4ine is non-Euclidean. In this latter case not only do the 

 (5)-planes normal to the axis remain fixed during the rotation, but 

 the hvo singular planes through the axis and tangent to the cone also 

 are fixed ; for the axis remains fixed and the lines in which the planes 

 are tangent to the cone are respectively the two fixed lines in the (5)- 

 plane. As every point in the axis of rotation is fixed, the whole set 

 of lines parallel to the elements of tangency is fixed. The effect in 

 the two singular planes of a rotation is therefore to leave one line, the 

 axis, fixed point for point, to leave a set of lines fixed, and to move 

 the points on these lines either toward the axis or awa^^ from it by 

 an amount which is proportional to the interval from the point to 

 the axis. 



Since a rotation in a (5)-plane multiplies all intervals along one of 

 the fixed directions in a certain ratio, and divides all intervals along 

 the other fixed direction in the same ratio, the effect upon areas in 

 the two singular planes is to multiply all areas in one of the planes 

 in that same ratio, and to divide areas in the other in that ratio. 

 This however is not inconsistent with our condition that areas should 

 remain invariant; for it is evident that, when compared with areas 

 in other planes, areas in singular planes are all of zero magnitude. 

 This is also shown by the fact that the inner product of any singular 

 vector by itself vanishes. That areas in a singular plane have a zero 

 magnitude does not prevent our comparing two areas in the same 

 singular plane or in parallel singular planes, just as the fact that 

 intervals along singular lines had zero magnitude did not prevent our 

 comparing intervals along any such line. 



A limiting case of rotation occurs when the axis of rotation is itself 

 an element of the cone, that is, a singular line. Here the infinity of 

 fixed planes perpendicular to the axis, and the two singular planes 

 through it, have all coalesced into the one singular plane through this 

 line and tangent to the cone. In this plane the rotation consists in a 

 sort of shear. Every point moves along a straight line parallel to the 

 axis. In this case areas are rotated into areas which are from every 

 point of view equal. For if a parallelogram whose base is on the axis, 

 which is fixed point for point, is subjected to this rotation, its base 

 remains fixed and the parallelogram remains enclosed between the 

 same two parallel lines (Theorem IX). 



The geometry in this plane, depending upon translation and upon 

 such a rotation as has just been described, is interesting as affording a 



