  For a philosopher, the most satisfying feature of the tremendous complication attained by 19th-century mathematics was perhaps the promptness with which the newly created (or discovered?) mathematical structures found their way into empirical science, enabling the intellectual grasp and handling of actual phenomena. We shall wind up this survey of 19th-century geometry with a few light remarks on a particularly rich and fruitful structure that has pride of place in current physics, namely, Lie groups, so called after Sophus Lie (1842–1899), the Norwegian mathematician who studied them in depth after 1870. A Lie group is, of course, a group in the algebraic sense we met in §3, that is, a set G such that (i) every ordered pair &lt;x,y&gt; ∈ G is associated to a unique element x · y ∈ G (known as the product or the sum of x and y); (ii) the product operation is associative, i.e., (x · y) · z = x · (y · z), for every x, y, z ∈ G; (iii) there is a one and only one element 0 ∈ G such that for every x ∈ G, x · 0 = 0 · x = x (0 is the identity or neutral element of G); (iv) for every x ∈ G, there is one and only one element x−1 ∈ G such that x · x−1 = 0 (x−1 is known as the inverse of x). But a Lie group is also a smooth manifold, as described in the supplement  A Modern Formulation of Riemann's Theory:   the set G can be represented patchwise by systems of real-valued (or alternatively complex-valued) coordinates, mutually linked by well-defined, differentiable coordinate transformations wherever their respective patches overlap. The group and manifold structures of G are meshed together by the condition that the product operation is a differentiable mapping of G × G into G.