628 
[797 
797. 
SURFACE, CONGRUENCE, COMPLEX. 
[From the Encyclopaedia Britannica, Ninth Edition, voi. xxn. (1887), pp. 668—672.] 
In the article Curve [785], the subject was treated from an historical point of view 
for the purpose of showing how the leading ideas of the theory were successively 
arrived at. These leading ideas apply to sui'faces, but the ideas peculiar to surfaces 
are scarcely of the like fundamental nature, being rather developments of the former 
set in their application to a more advanced portion of geometry; there is consequently 
less occasion for the historical mode of treatment. Curves in space were briefly con 
sidered in the same article, and they will not be discussed here ; but it is proper to 
refer to them in connexion with the other notions of solid geometry. In plane 
geometry the elementary figures are the point and the line ; and we then have the 
curve, which may be regarded as a singly infinite system of points, and also as a 
singly infinite system of lines. In solid geometry the elementary figures are the point, 
the line, and the plane; we have, moreover, first, that which under one aspect is the 
curve and under another aspect the developable (or torse), and which may be regarded 
as a singly infinite system of points, of lines, or of planes; and secondly, the surface, 
which may be regarded as a doubly infinite system of points or of planes, and also 
as a special triply infinite system of lines. (The tangent lines of a surface are a 
special complex.) As distinct particular cases of the first figure, we have the plane 
curve and the cone : and as a particular case of the second figure, the ruled surface, 
regulus, or singly infinite system of lines ; we have, besides, the congruence or doubly 
infinite system of lines, and the complex or triply infinite system of lines. And thus 
crowds of theories arise which have hardly any analogues in plane geometry ; the 
relation of a curve to the various surfaces which can be drawn through it, and that 
of a surface to the various curves which can be drawn upon it, are different in kind 
from those which in plane geometry most nearly correspond to them,—the relation of 
a system of points to the different curves through them and that of a curve to the 
systems of points upon it. In particular, there is nothing in plane geometry to corre 
spond to the theory of the curves of curvature of a surface. Again, to the single