The twenty-seven lines upon the cubic surface ... by Archibald Henderson.

4

HISTORICAL SUMMARY

first constructed a model of the diagonal surface with twenty-seven
real lines. "Instigated by this investigation of Clebsch," says Klein,
" I turned to the general problem of determining all possible forms of
cubic surfaces. I established the fact that by the principle of continuity all forms of real surfaces of the third order can be derived
from the particular surface having four conical points*." Klein's
method established completeness of enumeration-the consideration of
fundamental importance. Klein exhibited a complete set of models of
cubic surfaces at the World's Exposition in Chicago in 1894, including
Clebsch's symmetrical model of the diagonal surface and Klein's
model of the cubic surface having four real conical points. Models of
the typical cases of all the principal forms of cubic surfaces have been
constructed by Rodenbergt for Brill's collection; and these plaster
models may now be purchased. Blythe has constructed models of
certain types of cubic surfaces, and illustrated in some detail the
character of the changes that take place under certain conditions.
The list of those who have written on the mechanical construction of
the configurations of the lines upon a cubic surface and the general
collocation of the lines upon the surface includes the names of Cayley,,
Frost, Zeuthen, De Vries, Taylor and Blythe~.
The configuration of the twenty-seven lines is not only of great.
interest per se, but also because of its close association with, and
relation to, other remarkable configurations. It was also in the year
1869-the year over which Sylvester waxed dithyrambic-that
Geiserll showed that the projection of a cubic surface from a point.
upon it, on a plane of projection parallel to the tangent plane at that
point, is a quartic curve; and that every quartic curve can be
generated in this way. He showed the mutual interdependence of the
configurations of the twenty-eight bitangents to a plane quartic curve
and the twenty-seven lines upon a cubic surface, and the method of
derivation of either configuration from the other. By making use of
* Lectures on Mathematics, Evanston Colloquium, 1894, Macmillan and Co..
Cf. Klein's paper, "Ueber Flachen drifter Ordnung," Math. Ann. Bd. VI. (1873),
pp. 551-581, where are to be found figures and sketches of surfaces having ore
conic node, symmetrical in form.
t "Zur Classification der Flachen dritter Ordnung," Math. Ann. Bd. xiv...
pp. 46-110.
+ On Models of Cubic Surfaces. Cambridge University Press, 1905.
~ Cf. infra, ~~ 18, 21.
1] Math. Ann. Bd. I. (1869), pp. 129-138. Cf. also Crelle's Journ. Vol. LXI1..
p. 377.