574 GEOMETRY 



C". Not every space curve is obtained in this way, but only those 

 whose tangents belong to a certain linear complex. If C is algebraic 

 so is C", and then an infinite number of algebraic surfaces may be 

 passed through the latter. If C is transcendental, so is C', and 

 usually no algebraic surface can be passed through it. Sometimes, 

 however, one such algebraic surface F exists. (If there were two, 

 C' and C would be algebraic.) It is precisely in this case that the 

 curve C is panalgebraic in the sense of Loria's theory. That such a 

 curve belongs to a definite system is seen from the fact that while the 

 surface F is unique, it contains a singly infinite number of curves 

 whose tangents belong to the linear complex mentioned, and the 

 orthogonal projections of these curves constitute the required system. 

 The principal problems in this field which require treatment are: 

 first, the exhaustive discussion of the simplest systems, correspond- 

 ing to small values of the characteristics n and v ; second, the study of 

 the general case in connection with (1) algebraic differential equa- 

 tions. (2) connexes, and (3) algebraic surfaces and linear complexes. 



Natural or Intrinsic Geometry 



In spite of the immediate triumph of the Cartesian system at the 

 time of its introduction into mathematics, rebellion against what 

 may be termed the tyranny of extraneous coordinates, first expressed 

 in the Characteristica geometrica of Leibnitz, has been an ever-present 

 though often subdued influence in the development of geometry. 

 Why should the properties of a curve be expressed in terms of x's 

 and y's which are defined not by the curve itself, but by its relation 

 to certain arbitrary elements of reference? The same curve in differ- 

 ent positions may have unlike equations, so that it is not a simple 

 matter to decide whether given equations represent really distinct or 

 merely congruent curves. The idea of the so-called natural or in- 

 trinsic coordinates had its birth during the early years of the nine- 

 teenth century, but it is only the systematic treatment of recent 

 years which has created a new field of geometry. 



For a plane curve there is at each point the arc s measured from 

 some fixed point on the curve, and the radius of curvature p; these 

 intrinsic coordinates are connected by a relation p=f(s) which is 

 precisely characteristic of the curve, that is, the curves corresponding 

 to the equation differ only in position. There is, however, still 

 something arbitrary in the point taken as origin. This is eliminated 

 by taking as coordinates p and its derivative 8 taken with respect 

 to the arc; so that the final intrinsic equation is of the form 8 =F(p). 

 There is no difficulty in extending the method to space curves. The 

 two natural equations necessary are here r=<j)(p), S=t/;(p), where 

 p and T are the radii of first and second curvature and 8 is the arc 

 derivative of p. 



