PKINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 60S 



traced by the terminal G is a pedal curve. If the line hk be a 

 continuous function of its own distance or of the angle through 

 which it turns, and the line KG, or kg, make an angle therewith, 

 which is either constant, or a continuous function of the distance 

 kg, or of the angle through which that line turns, the curve traced 

 by the point G- may be said to be pedally generated. 1 These 

 three generative schemes are substantially identical in principle, 

 and may be all characterised as pedal generation. 2 



Three-dimensional figures may similarly be generated, either 

 from three-dimensional surfaces, or by the rotation of two-dimen- 

 sional figures. In generating by rotation the radius of curvature 

 may also be a function of the angle through which it turns. A 

 surface of revolution would be continuous, but for the generated 

 surface to be continuous in other cases it must necessarily be 

 defined by the involutional, evolutional or pedal surface, defined 

 as the locus of the generating points, the centres of curvature, or 

 the pedal points. 



We pass now to examples of more purely functional schemes of 

 generation, e.g. modular and umbilical generation. 



When the locus of points is determined by the constant ratio 3 

 which their distance from some fixed point 4 bears to their distance 

 — measured parallel to a fixed plane 5 — from some fixed straight 

 line, 6 the surface in which the points lie, or which is thus represented, 

 is said to be modularly generated. 7 The ratio, or modulus, may 

 be continuously varied in any specified way, that is it may be 



1 GgJ is a simple type of pedal curve. 



2 E volutes and involutes were discussed by Huygens as far back as 1672 

 in his Horologium Oscillatorium, by Tschirnhausen (Acta Eruditorum 

 1682) by Leibniz, ibid., 1686, and by Newton (Principia). 



3 The modulus. 4 The modular focus. s The directing plane. 6 The 

 directrix. 



7 The ratio being constant the locus may be an elliptic paraboloid, an 

 elliptic or parabolic cylinder, a hyperbolic paraboloid or cylinder, an 

 ellipsoid, or hyperboloid of one or two sheets, the oblate spheroid and 

 hyperboloid af revolution of one sheet. The prolate spheroid, and hyper- 

 boloid of revolution of two sheets cannot be modularly generated. The 

 subject has been treated by MacCullagh, Salmon, Townsend, Frost and 

 Wolstenholme, and others. 



