PKINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 271 



that which is absolutely homaloidal, or at least homaloidal to an 

 infinite order of infinity. We may call a surface that is plane for 

 finite geometrical figures, but not for their infinite enlargement, 

 'finitely homaloidal, and since such a surface must be infinite, it is 

 a finitely-homaloidal infinite surface. 



Similarly a surface homaloidal for geometrical figures of infinite 

 magnitude may be described as homaloidal to infinity of the first 

 order. These may be defined as an homaloid of the 1st order and 

 of 2 dimensions; and of the 2nd order, of 2 dimensions. 



1 3. Resolution of discontinuities in 2-dimensional curves, through 

 infinite paths in 3-dimensional space. — In absolutely homaloidal 

 (conceptual) space every point on a straight line divides it, as we 

 have seen, into absolutely infinite branches, which are consequently 

 absolutely discontinuous. 



If at a point A, on the straight line, a tangent circle of infinite 

 radius 1 be drawn, the deflection 8y say, of a point B on the line, 

 distant x from A is 



S 2/ = 10V + 10 3 ^+ etc (11) 



which is zero 2 for all finite values of x, it follows that every finite 

 straight line may be regarded as the circumference of a circle of 

 infinite radius, the centre of which circle, however, lies indifferently 

 in any point of a circle of equal infinite radius, the locus of the 

 possible centres being in a plane perpendicular to the line. The 

 distinction consequently between + oo and - go may either be 

 regarded as generally evanescent, or at least as evanescent at the 

 opposite or antipodal point on an infinite circle or infinite spherical 

 surface, that is at least for figures of finite dimensions. Schemati- 

 cally, therefore, we can move a point along a line AB in two ways, 

 either through the finite path, or the infinite one, the directions 

 cf motions being opposite 3 ; or algebraically + x= - 2oo + x, that 



1 First order infinity. 2 A first order zero. 



3 See Henrici, Article Geometry, Encyc. Brit. X., 389; also Luigi 

 Cremona, Elements of Projective Geometry (trans. Leuesdorf) Chap. viii. 

 § 52, p. 44, 1885. Infinity is thus the total circle, hence its radius is 

 symbolically co/27r- 



