Men of Mathematics (76 page)

Before taking up the thread of Cayley's life where it crossed Sylvester's again, we shall let the author of
Rosalind
describe how he made one of his most beautiful discoveries, that of what are called “canonical forms.” [This means merely the reduction of a given “quantic” to a “standard” form. For example
ax
²
+
2bxy
+
cy
²
can be expressed as the sum of two squares, say X
²
+
Y
²
; ax
⁵
+ 5bx
⁴
y + 10cx
³
y
²
+
10dx
²
y
³
+
5exy
⁴
+ fy
⁵
can be expressed as a sum of three fifth powers, X
⁵
+
Y
⁵
+ Z
⁵
.]

“I discovered and developed the whole theory of canonical binary forms for odd degrees, and, so far as yet made out, for even degrees
^I
too, at one sitting, with a decanter of port wine to sustain nature's flagging energies, in a back office in Lincoln's Inn Fields. The work was done, and well done, but at the usual cost of racking thought—a brain on fire, and feet feeling, or feelingless, as if plunged in an ice-pail.
That night we slept no more.”
Experts agree that the symptoms are unmistakable. But it must have been ripe port, to judge by what Sylvester got out of the decanter.

*  *  *

Cayley and Sylvester came together again professionally when Cayley accepted an invitation to lecture at Johns Hopkins for half a year in 1881-82. He chose Abelian functions, in which he was researching at the time, as his topic, and the 67-year-old Sylvester faithfully attended every lecture of his famous friend. Sylvester had still several prolific years ahead of him, Cayley not quite so many.

We shall now briefly describe three of Cayley's outstanding contributions to mathematics in addition to his work on the theory of algebraic invariants. It has already been mentioned that he invented the theory of matrices, the geometry of space of
n
dimensions, and that one of his ideas in geometry threw a new light (in Klein's hands) on non-Euclidean geometry. We shall begin with the last because it is the hardest.

Desargues, Pascal, Poncelet, and others had created
projective
geometry (see chapters 5, 13) in which the object is to discover those properties of figures which are invariant under projection. Measurements—sizes of angles, lengths of lines—and theorems which depend upon measurement, as for example the Pythagorean proposition that the
square on the longest side of a right triangle is equal to the sum of the squares on the other two sides, are not projective but
metrical,
and are not handled by
ordinary
projective geometry. It was one of Cayley's greatest achievements in geometry to transcend the barrier which, before he leapt it, had separated projective from metrical properties of figures. From his higher point of view metrical geometry also became projective, and the great power and flexibility of projective methods were shown to be applicable, by the introduction of “imaginary” elements (for instance points whose coordinates involve
) to metrical properties. Anyone who has done any analytic geometry will recall that two circles intersect in four points, two of which are always “imaginary.” (There are cases of apparent exception, for example concentric circles, but this is close enough for our purpose.) The fundamental notions in metrical geometry are the distance between two points and the angle between two lines. Replacing the concept of distance by another, also involving “imaginary” elements, Cayley provided the means for unifying Euclidean geometry and the common non-Euclidean geometries into one comprehensive theory. Without the use of some algebra it is not feasible to give an intelligible account of how this may be done; it is sufficient for our purpose to have noted Cayley's main advance of uniting projective and metrical geometry with its cognate unification of the other geometries just mentioned.

The matter of
n
-dimensional geometry when Cayley first put it out was much more mysterious than it seems to us today, accustomed as we are to the special case of four dimensions (space-time) in relativity. It is still sometimes said that a four-dimensional geometry is inconceivable to human beings. This is a superstition which was exploded long ago by Plücker; it is easy to put four-dimensional figures on a flat sheet of paper, and so far as
geometry
is concerned the
whole
of a four-dimensional “space” can be easily imagined. Consider first a rather unconventional three-dimensional space:
all
the
circles
that may be drawn in
a plane.
This “all” is a three-dimensional “space” for the simple reason that it takes
precisely three numbers,
or
three coordinates,
to individualize any one of the swarm of circles, namely
two
to fix the position of the center with reference to any arbitrarily given pair of axes, and
one
to give the length of the radius.

If the reader now wishes to visualize a four-dimensional space he may think of
straight lines,
instead of
points,
as the
elements
out of which
our common “solid” space is built. Instead of our familiar solid space looking like an agglomeration of infinitely fine birdshot it now resembles a cosmic haystack of infinitely thin, infinitely long straight straws. That it is indeed four-dimensional in
straight lines
can be seen easily if we convince ourselves (as we may do) that
precisely four numbers
are necessary and sufficient to individualize a particular straw in our haystack. The “dimensionality” of a “space” can be anything we choose to make it, provided we suitably select the elements (points, circles, lines, etc.) out of which we build it. Of course if we take
points
as the elements out of which our space is to be constructed, nobody outside of a lunatic asylum has yet succeeded in visualizing a space of more than three dimensions.

Modern physics is fast teaching some to shed their belief in a mysterious “absolute space” over and above the mathematical “spaces”—like Euclid's, for example—that were
constructed
by geometers to correlate their physical experiences. Geometry today is largely a matter of analysis, but the old terminology of “points,” “lines,” “distances,” and so on, is helpful in suggesting interesting things to do with our sets of coordinates. But it does not follow that these particular things are the most useful that might be done in analysis; it may turn out some day that all of them are comparative trivialities by more significant things which we, hidebound in outworn traditions, continue to do merely because we lack imagination.

If there is any mysterious virtue in talking about situations which arise in analysis as if we were back with Archimedes drawing diagrams in the dust, it has yet to be revealed. Pictures after all may be suitable only for very young children; Lagrange dispensed entirely with such infantile aids when he composed his analytical mechanics. Our propensity to “geometrize” our analysis may only be evidence that we have not yet grown up. Newton himself, it is known, first got his marvellous results analytically and re-clothed them in the demonstrations of an Apollonius partly because he knew that the multitude—mathematicians less gifted than himself—would believe a theorem true only if it were accompanied by a pretty picture and a stilted Euclidean demonstration, partly because he himself still lingered by preference in the pre-Cartesian twilight of geometry.

The last of Cayley's great inventions which we have selected for mention is that of matrices and their algebra in its broad outline. The subject originated in a memoir of
1858
and grew directly out of simple
observations on the way in which the transformations (linear) of the theory of algebraic invariants are combined. Glancing back at what was said on discriminants and their invariance we note the transformation (the arrow is here read “is replaced by”)
Suppose we have two such transformations,

the second of which is to be applied to the
x
in the first. We get

Attending only to the coefficients in the three transformations we write them in square arrays, thus

and see that the result of performing the first two transformations successively could have been written down by the following rule of “multiplication,”