An algebraic surface is one of the form f(x,y,z) = 0 where f(x,y,z)
is a polynomial in x, y and z. The order of the surface is the degree
of the polynomial. A surface of order one is a plane. A surface
of order two is called a quadric surface and consists of surfaces
such as ellipsiods and hyperboloids. These include
cones, cylinders and paraboloids. The surface whose history
we are interested in for this short article is a surface
of order three which is called a cubic surface.
In 1849 Salmon and Cayley published the results of their correspondence
on the number of straight lines on a cubic surface.
It was Cayley who, in a letter to Salmon, first showed
that there could be only a finite number of straight lines
on a cubic surface while it was Salmon who then proved
that there were exactly 27 such straight lines in general.
Steiner already knew of Cayley-Salmon theorem about 27 straight
lines when he started his own work on cubic surfaces. He wrote an important
article which gave results that allowed a purely geometrical treatment
of cubic surfaces. He proved in 1856 that:-
The nine straight lines in which the surfaces of two arbitrarily
given trihedra intersect each other determine, together
with one given point, a cubic surface.
Klein's synthesis of geometry as the study of the properties
of a space that are invariant under a given group of transformations,
known as the Erlanger Programm,
profoundly influenced mathematical development.