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

HISTORICAL SUMMARY

5

Geiser's results, Zeuthen* obtained a new demonstration of the
theorems of Schlifli t upon the reality of the straight lines and triple
tangent planes of a cubic surface. He proves the reality of all the
twenty-eight bitangents to the quartic in the case when the curve
consists of four separate closed portions. In the next year, he points
out the important connection between Klein's researches on cubic
surfaces (1.c.) and his own researches on plane quartic curves. If
a surface with four conic nodes be chosen, the resulting quartic has
four double points. By the principle of continuity, the four ovals of
the quartic are readily obtained; and this, as Zeuthen showed,
corresponds to Klein's derivation of the diagonal surface from the
cubic surface having four conic nodes:. Timerding has shown that
it is feasible to derive the properties of the plane quartic curve and
its bitangents from the known properties of the cubic surface and its
straight lines, and vice versa ~.
In 1877, Cremonall was first to show that the Pascalian configuration might be derived from the configuration of the twenty-one
lines upon the surface of the third degree with one conical point
(Species II in Cayley's enumeration) by projection from the conical
point. Mention should also be made here of the elaborate paper of
Bertini T.
Among recent investigations on the theory of the cubic surface,
the allied problems of the twenty-seven lines, and the bitangents
to the plane quartic curve, with generalizations to higher dimensions,
are the papers, here given in chronological order, of: Richmond
(Camb. Phil. Proc. Vol. xiv. 1908, pp. 475-477), Dixon (Quart.
Journ. Vol. XL. 1909, pp. 381-384; ibid. Vol. XLI. 1910, pp. 203-209),
Burnside (Camb. Phil. Proc. Vol. xv. 1910, pp. 428-430), Miss M. Long
(Proc. London Mlath. Soc. Ser. 2, Vol. Ix. 1910, pp. 205-230), Baker
(Proc. London Mlath. Soc. Ser. 2, Vol. ix. 1910, pp. 145-199; Proc.
R.oyal Soc. A, Vol. LXXXIV. 1911, pp. 597-602), and Bennett (Proc.
London Math. Soc. Ser. 2, Vol. Ix. 1911, pp. 336-351). In the first
of his two papers above mentioned, Baker gives a proof of the theorem
* Math. Ann. Bd. vII. (1874), pp. 410-432.
t Quart. Journ. Vol. II. (1858); Philos. Trans. Royal Soc. Vol. CLIII. (1863).
+ Math. Ann. Bd. vIII. (1875), pp. 1-30.
~ Crelle's Journ. Vol. cxxII. (1900), pp. 209-226.
I1 Reale Accademia dei Lincei, Anno CCLXXIV. (1876-77), Rome. Also cf.
infra, ~~ 45-6.
IT "Contribuzione alla teoria delle 27 rette e dei 45 piani tritangenti di una
superficie di 3~ ordine," Annali di Matematica (1883-4), II. 12, pp. 301-346.