Men of Mathematics (41 page)

For the next decade (1815–25) Poncelet's duties as a military
engineer left him only odd moments for his real ambition—the exploitation of his new methods in geometry. Relief was not to come for many years. His high sense of duty and his fatal efficiency made Poncelet an easy prey for short-sighted superiors. Some of the tasks he was set could have been done only by a man of his calibre, for example the creation of the school of practical mechanics at Metz and the reform of mathematical education at the Polytechnique. But the reports on fortifications, his work on the Committee of Defense, and his presidency of the mechanical sections at the international expositions of London and Paris (1851-58), to mention only a few of his numerous routine jobs, could all have been done by lesser men. His high scientific merits, however, were not unappreciated. The Academy of Sciences elected him (1831) as successor to Laplace. For political reasons Poncelet declined the honor till three years later.

Poncelet's whole mature life was one long internal conflict between that half of him which was born to do lasting work and the other half which accepted all the odd or dirty jobs shortsighted politicians and obtuse militarists shoved in its way. Poncelet himself longed to escape, but a mistaken sense of duty, drilled into his very bones in Napoleon's armies, impelled him to serve the shadow and turn his back on the substance. That he did not suffer an early and permanent nervous breakdown is a remarkable testimonial to the ruggedness of his physique. And that he retained his creative abilities almost to his death at the age of seventy nine is a shining proof of his unquenchable genius. When they could think of nothing better for this splendidly endowed man to do with his time they sent him traipsing about France to inspect cotton mills, silk mills, and linen mills. They did not need a Poncelet to do that sort of thing, and he knew it. He would have been the last man in France to object had his unique talents been indispensable in such affairs, for he was anything but the sort of intellectual prude who holds that science loses her perennial virginity every time she shakes hands with industry. But he was not the only man available for the work, as possibly Pasteur was in the equally important matters of the respective diseases of beer, silkworms, and human beings.

*  *  *

We now glance at one or two of the weapons either devised or remodelled by Poncelet for the conquest of projective geometry. First there is his “principle of continuity,” which refers to the permanence
of geometrical properties as one figure shades, by projection or otherwise, into another. This no doubt is rather vague, but Poncelet's own statement of the principle was never very exact and, as a matter of fact, embroiled him in endless controversies with more conservative geometers whom he politely designated as old fossils—always in the dignified diction suitable to an officer and a gentleman, of course. With the caution that the principle is of great heuristic value but does not always of itself provide proofs of the theorems which it suggests, we may see something of its spirit from a few simple examples.

Imagine two intersecting circles. Say they intersect in the points
A
and
B.
Join
A
and
B
by a straight line. The figure presents ocular evidence of two
real
points
A, B
and the common chord
AB
of the two circles. Now imagine the two circles pulled gradually apart. The common chord presently becomes a common tangent to the two circles at their point of contact. At any stage so far the following theorem (usually set as an exercise in school geometry) is true: if
any
point
P
be taken on the common chord,
four
tangent lines may be drawn from it to the two circles, and if the points in which these tangent lines touch the circles are
T
₁
T
₂
, T
₃
, T
₄
, then the segments
PT
₁
, PT
₂
, PT
₃
, PT
₄
, are all equal in length. Conversely, if it is asked where do
all
the points
P
lie such that the four tangent-segments to the two circles shall all be equal, the answer is
on the common chord.
Stating all this briefly in the usual language, we say that the
locus
(which merely means
place)
of a point
P
which moves so that the lengths of the tangent-segments from it to two
intersecting
circles are equal, is the common chord of the two circles.
^I
All this is familiar and straightforward; there is no element of mystery or incomprehensibility
as some may say there is in the next where the “principle of continuity” enters.

Pull the circles completely apart. Their two intersections (or in the last moment their one point of contact) are no longer visible on the paper and the “common chord” is left suspended between the two circles, cutting neither visibly. But it is known that there is still a
locus
of equal tangent-segments, and it is easily proved that this locus is a straight line perpendicular to the line joining the centres of the two circles, just as the original locus (the common chord) was. Merely as a manner of speaking, if we object to “imaginaries,” we continue to
say
that the two circles intersect in two points in the infinite part of the plane, even when they have been pulled apart, and we
say
also that the new straight-line locus is still the common chord of the circles: the points of intersection are “imaginary” or “ideal,” but the straight line joining them (the new “common chord”) is “real”—we actually draw it on the paper.

If we write the equations of the circles and lines algebraically in the manner of Descartes, all that we do in the algebra of solving the equations for the intersections has its unique correlate in the enlarged geometry, whereas if we do not first expand our geometry—or at least increase its vocabulary, to take account of “ideal” elements—much of the meaningful algebra is geometrically meaningless.

All this of course requires logical justification. Such justification has been given so far as is necessary, that is, up to the stage which includes the applications of the “principle of continuity” useful in geometry.

A more important instance of the principle is furnished by parallel straight lines. Before describing this we may repeat the remark a venerable and distinguished judge relieved himself of a few days ago when the matter was revealed to him. The judge had been under the weather; an amateur mathematician, thinking to cheer the old fellow up, told him something of the geometrical concept of infinity. They were strolling through the judge's garden at the time. On being informed that “parallel lines meet at infinity,” the judge stopped dead. “Mr. Blank,” he said with great emphasis, “any man who says parallel lines meet at infinity, or anywhere else, simply hasn't got good sense.” To obviate an argument we may say as before that it is all a way of speaking to avoid irritating exceptions and separations into exasperating distinct cases. But once the language has been agreed upon,
logical consistency demands that it be followed to the end without traversing the rules of logical grammar and syntax, and this is what is done.

To see the reasonableness of the language, imagine a fixed straight line
l
and fixed point P not on
l
. Through
P
draw any straight line
l
' intersecting
l
in
P'
₉
and imagine
l
' to rotate about
P,
so that
P
^f
recedes along
l
. When does
P'
stop receding? We say it stops when
l
,
l
' become parallel or, if we prefer, when the point of intersection
P'
is at infinity. For reasons already indicated this language is convenient and suggestive—not of a lunatic asylum, as the judge might think, but of interesting and sometimes highly practical things to do in geometry.

In a similar manner the visualizable
finite
parts of lines, planes and three-dimensional space (also of higher space) are enriched by the adjunction of “ideal” points, lines, planes, or “regions”
at infinity.
If the judge happens to see this he may enjoy the following shocking example of the behavior of the infinite in geometry:
any two circles in a plane intersect in four points, two of which are imaginary and at infinity.
If the circles are concentric, they touch one another in two points lying on the line at infinity. Further,
all
circles in a plane go through
the same
two points at infinity—they are usually denoted by
I
and
J
, and are sometimes called Isaac and Jacob by irreverent students.

In the chapter on Pascal we described what is meant by projective properties in distinction to metrical properties in geometry. At this point we may glance back at Hadamard's remarks on Descartes' analytic geometry. Hadamard observed among other things that
modern synthetic geometry repaid the debt of geometry in general to algebra by suggesting important researches in algebra and analysis. This modern synthetic geometry was the object of Poncelet's researches. Although all this may seem rather involved at the moment, we shall close the chain by taking a link from the 1840's, as the matter really is important, not only for the history of pure mathematics but for that of recent mathematical physics as well.

The link from the 1840's is the creation by Boole, Cayley, Sylvester and others, of the algebraic theory of invariance which (as will be explained in a later chapter) is of fundamental importance in current theoretical physics. The projective geometry of Poncelet and his school played a very important part in the development of the theory of invariance: the geometers had discovered a whole continent of properties of figures
invariant
under projection; the algebraists of the 1840's, notably Cayley, translated the geometrical
operations of projection
into analytical language, applied this translation to the
algebraic,
Cartesian mode of expressing geometric relationships, and were thus enabled to make phenomenally rapid progress in the elaboration of the theory of algebraic invariants. If Desargues, the daring pioneer of the seventeenth century, could have foreseen what his ingenious method of projection was to lead to, he might well have been astonished. He knew that he had done something good, but he probably had no conception of just how good it was to prove.

Isaac Newton was a young man of twenty when Desargues died. There is no evidence that Newton ever heard the name of Desargues. If he had, he also might have been astonished could he have foreseen that the humble link forged by his elderly contemporary was to form part of the strong chain which, in the twentieth century, was to pull his law of universal gravitation from its supposedly immortal pedestal. For without the mathematical machinery of the tensor calculus which developed naturally (as we shall see) from the algebraic work of Cayley and Sylvester, it is improbable that Einstein or anyone else could ever have budged the Newtonian theory of gravitation.

*  *  *

One of the useful ideas in projective geometry is that of
cross-ratio
or
anharmonic ratio.
Through a point
O
draw any four straight lines
l, m, n, p.
Across these four draw any straight line
x,
and label the points in which
x
cuts the others
L, M, N, P
respectively. We thus have on
x
the line segments
LM, MN, LP, PN.
From these form
the ratios
and
Finally we take the ratio of these two ratios, and get the
cross-ratio
The remarkable thing about this cross-ratio is that it has the same numerical magnitude for
all
positions of the line
x.