Men of Mathematics (42 page)

Later we shall refer to Felix Klein's unification of Euclidean geometry and the common non-Euclidean geometries into one comprehensive geometry. This unification was made possible by Cayley's revision of the usual notions of
distance
and
angle
on which
metrical
geometry is founded. In this revision, cross-ratio played the leading part, and through it, by the introduction of “ideal” elements of his own devising, Cayley was enabled to reduce
metrical
geometry to a species of
projective
geometry.

To close this inadequate description of the kind of weapons that Poncelet used we shall mention the extremely fruitful “principle of duality.” For simplicity we consider only how the principle operates in plane geometry.

Note first that any continuous curve may be regarded in either of two ways: either as being generated by the motion of a point, or as being swept out by the turning motion of a straight line. To see the latter, imagine the tangent line drawn at each point of the curve. Thus
points
and
lines
are intimately and reciprocally associated with respect to the curve:
through
every
point
of the curve there is a
line
of the curve;
on
every line of the curve there is a point of the curve. Instead of “through” in the preceding sentence, write “on.” Then the two assertions separated by “;” after the “:” are identical except that the words “point” and “line” are interchanged.

As a matter of terminology we say that a line (straight or curved) is
on
a point if the line passes through the point, and we note that if a line is
on
a point, then the point is
on
the line, and conversely. To make this correspondence universal we “adjoin” to the usual plane in which Euclidean geometry (common school geometry) is valid, a so-called
metric plane,
“ideal elements” of the kind already described. The result of this adjunction is a
projective plane:
a projective plane consists of all the ordinary points and straight lines of a metric plane and, in addition, of a set of ideal points all of which are assumed to lie on one ideal line and such that one such ideal point lies on every ordinary line.
^II

In Euclidean language we would say that two parallel lines have the same direction; in projective phraseology this becomes “two parallel lines have the same ideal point.” Again, in the old, if two or more lines have the same direction, they are parallel; in the new, if two or more lines have the same ideal point they are parallel. Every
straight line in the projective plane is conceived of as having on it
one ideal point
(“at infinity”);
all
the ideal points are thought of as making up
one ideal line,
“the line at infinity.”

The purpose of these conceptions is to avoid the exceptional statements of Euclidean geometry necessitated by the postulated existence of parallels. This has already been commented on in connection with Poncelet's formulation of the principle of continuity.

With these preliminaries the
principle of duality
in plane geometry can now be stated: All the propositions of plane projective geometry occur in dual pairs which are such that, from either proposition of a particular pair another can be immediately inferred by interchanging the parts played by the words
point
and
line.

In his projective geometry Poncelet exploited this principle to the limit. Opening almost any book on projective geometry at random we note pages of propositions printed in double columns, a device introduced by Poncelet. Corresponding propositions in the two columns are duals of one another; if either has been proved, a proof of the other is superfluous, as implied by the principle of duality. Thus geometry at one stroke is doubled in extent with no expenditure of extra labor. As a specimen of dual propositions we give the following pair.

Two distinct points are on one, and only one, line.

Two distinct lines are on one, and only one, point.

It may be granted that this is not very exciting. The mountain has labored and brought forth a mouse. Can it do any better?

The proposition in the left-hand column (
page 217
) is Pascal's concerning his
Hexagrammum Mysticum
which we have already seen; that on the right is Brianchon's theorem, which was
discovered
by means of the principle of duality. Brianchon (1785-1864) discovered his theorem while he was a student at the École Polytechnique; it was printed in the
Journal
of that school in 1806. The figures for the two propositions
look nothing alike. This may indicate the power of the methods used by Poncelet.

Brianchon's discovery was the one which put the principle of duality on the map of geometry. Far more spectacular examples of the power of the principle will be found in any textbook on projective geometry, particularly in the extension of the principle to ordinary three-dimensional space. In this extension the parts played by the words
point and plane
are interchangeable;
straight line
stays as it was.

If
A, B, C, D, E, F
are any points on a conic section, the points of intersection of the pairs of lines
AB
and
DE, BC
and
EF, CD
and
FA
are on a straight line; and conversely.

If
A, B, C, D, E, F
are tangent straight lines on a conic section, the lines joining the pairs of intersections of
A
with
B
and
D
with
E, B
with
C
and
E
with
F, C
with
D
and
F
with
A,
meet in one point; and conversely.

*  *  *

The conspicuous beauty of projective geometry and the supple elegance of its demonstrations made it a favorite study with the geometers of the nineteenth century. Able men swarmed into the new goldfield and quickly stripped it of its more accessible treasures. Today the majority of experts seem to agree that the subject is worked out so far as it is of interest to professional mathematicians. However, it is conceivable that there may yet be something in it as obvious as the principle of duality which has been overlooked. In any event it is an easy subject to acquire and one of fascinating delight to amateurs and even to professionals at some stage of their careers. Unlike some other fields of mathematics, projective geometry has been blessed with many excellent textbooks and treatises, some of them by master geometers, including Poncelet himself.

I
. In what precedes the tangents are
real
(visible) if the point
P
lies
outside
the circles; if
P
is
inside,
the tangents are. “
imaginary.”

II
. This definition, and others of a similar character given presently, is taken from
Projective Geometry
(Chicago, 19S0) by the late John Wesley Young. This little book is comprehensible to anyone who has had an ordinary school course in common geometry.

The further elaboration and development of systematic arithmetic, like nearly everything else which the mathematics of our [nineteenth] century has produced in the way of original scientific ideas, is knit to Gauss.
—L
EOPOLD
K
RONECKER

A
RCHIMEDES, NEWTON, AND GAUSS
, these three, are in a class by themselves among the great mathematicians, and it is not for ordinary mortals to attempt to range them in order of merit. All three started tidal waves in both pure and applied mathematics: Archimedes esteemed his pure mathematics more highly than its applications; Newton appears to have found the chief justification for his mathematical inventions in the scientific uses to which he put them, while Gauss declared that it was all one to him whether he worked on the pure or the applied side. Nevertheless Gauss crowned the higher arithmetic, in his day the least practical of mathematical studies, the Queen of all.