Men of Mathematics (59 page)

Now, in navigation at least, an ocean is not thought of as a flat surface (Euclidean plane) if even moderate distances are concerned; it is taken for what it very approximately is, namely a part of the surface of a sphere, and the geometry of great circle sailing is not Euclid's. Thus Euclid's is not the only geometry of human utility. On the plane two geodesics intersect in exactly
one
point
unless
they are parallel, when they do not intersect (in Euclidean geometry); but on the sphere
any
two geodesics always intersect in precisely
two
points. Again, on a plane, no two geodesics can enclose a space—as Euclid
assumed in one of the postulates for his geometry; on a sphere, any two geodesics always enclose a space.

Imagine now the equator on the sphere and two geodesics drawn through the north pole perpendicular to the equator. In the northern hemisphere this gives a triangle with curved sides, two of which are equal. Each side of this triangle is an arc of a geodesic. Draw any other geodesic cutting the two equal sides so that the intercepted parts between the equator and the cutting line are equal. We now have,
on the sphere,
the four-sided figure corresponding to the
AXTB
we had a few moments ago in the plane. The two angles at the base of this figure are right angles and the corresponding sides are equal, as before,
but each of the equal angles at X, T is now greater than a right angle.
So, in the highly practical geometry of great circle sailing, which is closer to real human experience than the idealized diagrams of elementary geometry ever get, it is not Euclid's postulate which is true—or its equivalent in the hypothesis of the right angle—but the geometry which follows from the hypothesis of the obtuse angle.

In a similar manner, inspecting a less familiar surface, we can make reasonable the hypothesis of the acute angle. The surface looks like two infinitely long trumpets soldered together at their largest ends. To describe it more accurately we must introduce the plane curve called the
tractrix,
which is generated as follows. Let two lines
XOX′ TOT'
be drawn in a horizontal plane intersecting at right angles in O, as in Cartesian geometry. Imagine an inextensible fiber lying along
TOT',
to one end of which is attached a small heavy pellet; the other end of the fiber is at
O
. Pull this end out along the line
OX.

As the pellet follows, it traces out one half of the tractrix; the other half is traced out by drawing the end of the fiber along
OX',
and of course is merely the reflection or image in
OT
of the first half. The drawing out is supposed to continue indefinitely—“to infinity”—in each instance. Now imagine the tractrix to be revolved about the line
XOX'.
The double-trumpet surface is generated; for reasons we need
not go into (it has constant negative curvature) it is called
a. pseudosphere.
If on this surface we draw the four-sided figure with two equal sides and two right angles as before, using geodesics, we find that the hypothesis of the acute angle is realized.

Thus the hypotheses of the right angle, the obtuse angle, and the acute angle respectively are true on a Euclidean plane, a sphere, and a pseudosphere respectively, and in all cases “straight lines” are
geodesics
or
extrema.
Euclidean geometry is a limiting, or degenerate, case of geometry on a sphere, being attained when the radius of the sphere becomes infinite.

Instead of constructing a geometry to fit the Earth as human beings now know it, Euclid apparently proceeded on the assumption that the Earth is flat. If Euclid did not, his predecessors did, and by the time the theory of “space,” or geometry, reached him the bald
assumptions
which he embodied in his postulates had already taken on the aspect of hoary and immutable necessary truths, revealed to mankind by a higher intelligence as the veritable essence of all material things. It took over two thousand years to knock the eternal truth out of geometry, and Lobatchewsky did it.

To use Einstein's phrase, Lobatchewsky
challenged an axiom.
Anyone who challenges an “accepted truth” that has seemed necessary or reasonable to the great majority of sane men for 2000 years or more takes his scientific reputation, if not his life, in his hands. Einstein himself challenged the axiom that two events can happen in
different places
at the
same time,
and by analyzing this hoary assumption was led to the invention of the special theory of relativity. Lobatchewsky challenged the assumption that Euclid's parallel postulate or, what is equivalent, the hypothesis of the right angle, is necessary to a consistent geometry, and he backed his challenge by producing a system of geometry based on the hypothesis of the acute angle in which there is not
one
parallel through a fixed point to a given straight line but
two.
Neither of Lobatchewsky's parallels meets the line to which both are parallel, nor does any straight line drawn through the fixed point and lying within the angle formed by the two parallels. This apparently bizarre situation is “realized” by the geodesics on a pseudosphere.

For any everyday purpose (measurements of distances, etc.), the differences between the geometries of Euclid and Lobatchewsky are too small to count, but this is not the point of importance: each is self-consistent and each is adequate for human experience. Lobatchewsky abolished the
necessary
“truth” of Euclidean geometry. His geometry was but the first of several constructed by his successors. Some of these substitutes for Euclid's geometry—for instance the Riemannian geometry of general relativity—are today at least as important in the still living and growing parts of physical science as Euclid's was, and is, in the comparatively static and classical parts. For some purposes Euclid's geometry is best or at least sufficient, for others it is inadequate and a non-Euclidean geometry is demanded.

Euclid in some sense was believed for 2200 years to have discovered an absolute truth or a necessary mode of human perception in his system of geometry. Lobatchewsky's creation was a pragmatic demonstration of the error of this belief. The boldness of his challenge and its successful outcome have inspired mathematicians and scientists in general to challenge other “axioms” or accepted “truths,” for example the “law” of causality, which, for centuries, have seemed as necessary to straight thinking as Euclid's postulate appeared till Lobatchewsky discarded it.

The full impact of the Lobatchewskian method of challenging axioms has probably yet to be felt. It is no exaggeration to call Lobatchewsky the Copernicus of Geometry, for geometry is only a part of the vaster domain which he renovated; it might even be just to designate him as a Copernicus of all thought.

I have finished a monument more lasting than bronze and loftier than the pyramids reared by kings, that neither corroding rain nor the uncontrolled north wind can dash apart, nor the countless succession of years and the flight of ages. I shall not wholly die; that greater part of me shall escape Death and ever shall I grow, still fresh in the praise of posterity.
—H
ORACE
(Odes,
3, xxx)

A
N ASTROLOGER IN THE YEAR
1801 might have read in the stars that a new galaxy of mathematical genius was about to blaze forth inaugurating the greatest century of mathematical history. In all that galaxy of talent there was no brighter star than Niels Henrik Abel, the man of whom Hermite said, “He has left mathematicians something to keep them busy for five hundred years.”

Abel's father was the pastor of the little village of Findö, in the diocese of Kristiansand, Norway, where his second son, Niels Henrik, was born on August
5, 1802.
On the father's side several ancestors had been prominent in the work of the church and all, including Abel's father, were men of culture. Anne Marie Simonsen, Abel's mother, was chiefly remarkable for her great beauty, love of pleasure, and general flightiness—quite an exciting combination for a pastor's helpmeet. From her Abel inherited his striking good looks and a very human desire to get something more than everlasting hard work out of life, a desire he was seldom able to gratify.

The pastor was blessed with seven children in all at a time when Norway was desperately poor as the result of wars with England and Sweden, to say nothing of a famine thrown in for good measure between wars. Nevertheless the family was a happy one. In spite of pinching poverty and occasional empty stomachs they kept their chins up. There is a charming picture of Abel after his mathematical genius had seized him sitting by the fireside with the others chattering and laughing in the room while he researched with one eye on his mathematics
and the other on his brothers and sisters. The noise never distracted him and he joined in the badinage as he wrote.

Like several of the first-rank mathematicians Abel discovered his talent early. A brutal schoolmaster unwittingly threw opportunity Abel's way. Education in the first decades of the nineteenth century was virile, at least in Norway. Corporal punishment, as the simplest method of toughening the pupils' characters and gratifying the sadistic inclinations of the masterful pedagogues, was generously administered for every trivial offense. Abel was not awakened through his own skin, as Newton is said to have been by that thundering kick donated by a playmate, but by the sacrifice of a fellow student who had been flogged so unmercifully that he died. This was a bit too thick even for the rugged schoolboard and they deprived the teacher of his job. A competent but by no means brilliant mathematician filled the vacancy, Bernt Michael Holmboë
(1795-1850),
who was later to edit the first edition of Abel's collected works in
1839.