The six independent quantities, or "co-ordinates," which serve to specify the position of a rigid body in space may of course be chosen in an endless variety of ways. We may, for instance, employ the three Cartesian co-ordinates of a particular point O of the body, and three angular co-ordinates which express the orientation of the body with respect to O. Thus in fig. 36, if OA, OB, OC be three mutually perpendicular lines in the solid, we may denote by [theta] the angle which OC makes with a fixed direction OZ, by [psi] the azimuth of the plane ZOC measured from some fixed plane through OZ, and by [phi] the inclination of the plane COA to the plane ZOC. In fig. 36 these various lines and planes are represented by their intersections with a unit sphere having O as centre. This very useful, although unsymmetrical, system of angular co-ordinates was introduced by L. Euler. It is exemplified in "Cardan's suspension," as used in connexion with a compass-bowl or a gyroscope. Thus in the gyroscope the "flywheel" (represented by the globe in fig. 37) can turn about a diameter OC of a ring which is itself free to turn about a diametral axis OX at right angles to the former; this axis is carried by a second ring which is free to turn about a fixed diameter OZ, which is at right angles to OX. [Illustration: FIG. 10.] We proceed to sketch the theory of the finite displacements of a rigid body. It was shown by Euler (1776) that any displacement in which one point O of the body is fixed is equivalent to a pure _rotation_ about some axis through O. Imagine two spheres of equal radius with O as their common centre, one fixed in the body and moving with it, the other fixed in space. In any displacement about O as a fixed point, the former sphere slides over the latter, as in a "ball-and-socket" joint. Suppose that as the result of the displacement a point of the moving sphere is brought from A to B, whilst the point which was at B is brought to C (cf. fig. 10). Let J be the pole of the circle ABC (usually a "small circle" of the fixed sphere), and join JA, JB, JC, AB, BC by great-circle arcs. The spherical isosceles triangles AJB, BJC are congruent, and we see that AB can be brought into the position BC by a rotation about the axis OJ through an angle AJB. [Illustration: FIG. 38.] [Illustration: FIG. 39.]