  Can metric properties be fixed in this way? Traditionally one defines the distance between two points (x1, … ,xn) and (y1, … ,yn) of a numerical manifold as the positive square root of (x1 − y1) 2 + … + (xn − y n)2. The group of isometries consists of the transformations that preserve this function. However, this is just a convention, adopted to ensure that the geometry is Euclidean. Using projective geometry, Klein thought of something better. No real-valued function of point pairs, defined on all projective space, is an invariant of the projective group, but there is a function of collinear point quadruples, called the cross-ratio, which is such an invariant. Drawing on work by Arthur Cayley (b. 1821, d. 1895), Klein (1871, 1873) considered the cross-ratio of point quadruples &lt;P1,P2,P3,P4&gt;. such that P3 and P4 belong to a given conic κ on a projective plane, while P1 and P2 range over a region  R  that is bounded by or otherwise fixed by κ. Since P3 and P4 must be the points where the straight line through P1 and P2 meets κ, the said cross-ratio may be regarded as a function of the point pair &lt;P1,P2&gt;. The collineations that map a given conic onto itself form a group, and the said function is clearly an invariant of this group. Klein showed that a certain function of this function behaves like an ordinary distance function on  R.  According to the nature of the conic κ, the structure determined by this function satisfies either (i) all the theorems of Euclidian plane geometry, or (ii) all those of Lobachevskian plane geometry, or (iii) those of a third geometry which Klein himself discovered and dubbed ‘elliptic’. (In elliptic geometry every straight line meets every other, and the three internal angles of a triangle always add up to more than two right angles. Klein's names for the geometries of Euclid and Lobachevsky were ‘parabolic’ and ‘hyperbolic’, respectively.)