Log24

From pp. 322 ff. of The Development of Mathematics, 
by Eric Temple Bell, Second Edition, McGraw-Hill, 1945, at
https://archive.org/stream/in.ernet.dli.2015.133966/2015.133966.
The-Development-Of-Mathematics-Second-Edition_djvu.txt —

Rising to a considerably higher level of difficulty, we may 
instance what the physicist Maxwell called “Solomon’s seal in 
space of three dimensions,” the twenty-seven real or imaginary 
straight lines which lie wholly on the general cubic surface, 
and the forty-five triple tangent planes to the surface, all so 
curiously related to the twenty-eight bitangents of the general 
plane quartic curve. If ever there was a fascinating snarl of 
interlaced theories, Solomon’s seal is one. Synthetic and analytic 
geometry, the Galois theory of equations, the trisection of 
hyperelliptic functions, the algebra of invariants and covariants, 
geometric-algebraic algorithms specially devised to render the 
tangled configurations of Solomon’s seal more intuitive, the 
theory of finite groups — all were applied during the second half 
of the nineteenth century by scores of geometers who sought to 
break the seal. 

Some of the most ingenious geometers and algebraists in 
history returned again and again to this highly special topic. 
The result of their labors is a theory even richer and more 
elaborately developed than Klein’s (1884) of the icosahedron. 
Yet it was said by competent geometers in 1945 that a serious 
student need never have heard of the twenty-seven lines, the 
forty-five triple tangent planes, and the twenty-eight bitangents 
in order to be an accomplished and productive geometer; and 
it was a fact that few in the younger generation of creative 

CONTRIBUTIONS FROM GEOMETRY 323 

geometers had more than a hazy notion that such a thing as 
tiie Solomon’s seal of the nineteenth century ever existed. 

Those rvho could recall from personal experience the last 
glow of living appreciation that lighted this obsolescent master- 
piece of geometry and others in the same fading tradition looked 
back with regret on the dying past, and wished that mathe- 
matical progress were not always so ruthless as it is. They also 
sympathized with those who still found the modern geometry 
of the triangle and the circle worth cultivating. For the differ- 
ence between the geometry of the twenty-seven lines and that of, 
say, Tucker, Lemoine, and Brocard circles, is one of degree, 
not of kind. The geometers of the twentieth century long since 
piously removed all these treasures to the museum of geometry, 
where the dust of history quickly dimmed their luster. 

For those who may be interested in the unstable esthetics 
rather than the vitality of geometry, we cite a concise modern 
account1 (exclusive of the connection with hyperclliptic func- 
tions) of Solomon’s seal. The twenty-seven lines were discovered 
in 1849 by Cayley and G. Salmon2 (1819-1904, Ireland); the 
application of transcendental methods originated in Jordan’s 
work (1869-70) on groups and algebraic equations. Finally, 
in the 1870’s L. Cremona (1830-1903), founder of the Italian 
school of geometers, observed a simple connection between 
the twenty-one distinct straight lines which lie on a cubic 
surface with a node and the ‘cat’s cradle’ configuration of 
fifteen straight lines obtained by joining six points on a conic 
in all possible ways. The ‘mystic hexagram’ of Pascal and its 
dual (1806) in C. J. Brianchon’s (1783-1864, French) theorem 
were thus related to Solomon’s seal; and the seventeenth 
century met the nineteenth in the simple, uniform deduc- 
tion of the geometry of the plane configuration from that of 
a corresponding configuration in space by the method of 
projection. 

The technique here had an element of generality that was to 
prove extremely powerful in the discovery and proof of cor- 
related theorems by projection from space of a given number of 
dimensions onto a space of lower dimensions. Before Cremona 
applied this technique to the complete Pascal hexagon, his 
countryman G. Veronese had investigated the Pascal configura- 
tion at great length by the methods of plane geometry, as had 
also several others, including Steiner, Cayley, Salmon, and 
Kirkman. All of these men were geometers of great talent; 

324 THE DEVELOPMENT OF MATHEMATICS 

Cremona’s flash of intuition illuminated the massed details of 
all his predecessors and disclosed their simple connections. 

That enthusiasm for this highly polished masterwork of 
classical geometry is by no means extinct is evident from the 
appearance as late as 1942 of an exhaustive monograph (xi + 180 
pages) by B. Segre (Italian, England) on The nonsingular cubic 
surface. Solomon’s seal is here displayed in all its “complicated 
and many-sided symmetry” — in Cayley’s phrase — as never 
before. The exhaustive enumeration of special configurations 
provides an unsurpassed training ground or ‘boot camp’ for 
any who may wish to strengthen their intuition in space of three 
dimensions. The principle of continuity, ably seconded by the 
method of degeneration, consistently applied, unifies the multi- 
tude of details inherent in the twenty-seven lines, giving the 
luxuriant confusion an elusive coherence which was lacking 
in earlier attempts to “bind the sweet influences” of the thirty- 
six possible double sixes (or ‘double sixers,’ as they were once 
called) into five types of possible real cubic surfaces, containing 
respectively 27, 15, 7, 3, 3 real lines. A double six is two sextuples 
of skew lines such that each line of one is skew to precisely one 
corresponding line of the other. A more modern touch appears 
in the topology of these five species. Except for one of the 
three-line surfaces, all are closed, connected manifolds, while 
the other three-line is two connected pieces, of which only one 
is ovoid, and the real lines of the surface are on this second 
piece. The decompositions of the nonovoid piece into generalized 
polyhedra by the real lines of the surface are painstakingly 
classified with respect to their number of faces and other char- 
acteristics suggested by the lines. The nonovoid piece of one 
three-line surface is homeomorphic to the real projective plane, 
as also is the other three-line surface. The topological interlude 
gives way to a more classical theme in space of three dimensions, 
which analyzes the group in the complex domain of the twenty- 
seven lines geometrically, either through the intricacies of the 
thirty-six double sixes, or through the forty triads of com- 
plementary Steiner sets. A Steiner set of nine lines is three sets 
of three such that each line of one set is incident with precisely 
two lines of each other set. The geometrical significance of 
permutability of operations in the group is rather more com- 
plicated than its algebraic equivalent. The group is of order 
51840. There is an involutorial transformation in the group for 
each double six; the transformation permutes corresponding 

CONTRIBUTIONS FROM GEOMETRY 325 

lines of the complementary sets of six of the double six, and 
leaves each of the remaining fifteen lines invariant. If the double 
sixes corresponding to two such transformations have four 
common lines, the transformations are permutable. If the 
transformations are not permutable, the corresponding double 
sixes have six common lines, and the remaining twelve lines 
form a third double six. Although the geometry of the situation 
may be perspicuous to those gifted with visual imagination, 
others find the underlying algebraic identities, among even so 
impressive a number of group operations as 51840, somewhat 
easier to see through. But this difference is merely one of ac- 
quired taste or natural capacity, and there is no arguing about 
it. However, it may be remembered that some of this scintillating 
pure geometry was subsequent, not antecedent, to many a 
dreary page of laborious algebra. The group of the twenty- 
seven lines alone has a somewhat forbidding literature in the 
tradition of the late nineteenth and early twentieth centuries 
which but few longer read, much less appreciate. So long as 
geometry — of a rather antiquated kind, it may be — can clothe 
the outcome of intricate calculations in visualizable form, the 
Solomon’s seal of the nineteenth century will attract its de- 
votees, and so with other famous classics of the geometric 
imagination. But in the meantime, the continually advancing 
front of creative geometry will have moved on to unexplored 
territory of fresher and perhaps wider interest. The world some- 
times has sufficient reason to be weary of the past in mathe- 
matics as in everything else.