Sophus Lie, the mathematician

line geometry and comes up with the number 16 but here he is talkingabout spheres whose coordinates are complex.Since the line-sphere transformation necessarily passes through thecomplex numbers it has not really found its rightful place in moderndifferential geometry. Of the major classic works on the subject it seemsthat only the books by Darboux and Blaschke treat it, in fact muchof Blaschke's third volume from 1929 is devoted to it. For a moderntreatment see Cecil's book [10].Lie in his thesis, and in the above-mentioned expansion thereof, embarksalso on a general theory of contact-transformations, transformationsof the cotangent bundle T*(C n ) into itself preserving the canonical1-form (the line-sphere transformation being an example) In the usualcoordinates (l,... ,zpnl,..., pn) on T*(Cn), a contact transformationis an isomorphism (i,pi) --* (Xi, Pi) such that Epii = E PiXi.Such transformations appear in the integration of first order partialdifferential equationsF axl'"" xl,...,z,~, ax, = o (1)In this context consider the Poisson bracket of functionsand vector fieldsand the Lie bracket(f g)= (aiapi a ap (2)X= aj Y= E bk (3)jkbk .oak a[X Y = (aj bj aXx a (4)jk - a k -XLie views the Poisson bracket as the effect on f of a vector field Xgassociated with g and interprets the Jacobi identity for (f, g) as the factthat the bracket of the operators corresponding to g and h is associatedto (g, h):[X9, Xh] = X(g,h)Thus the first order differential equation for g