Men of Mathematics (58 page)

We recall that 1842, the year of the fire, was also the year in which, thanks to the good offices of Gauss, Lobatchewsky was elected a foreign correspondent of the Royal Society of Göttingen for his creation of non-Euclidean geometry. Although it seems incredible that any man so excessively burdened with teaching and administration as Lobatchewsky was, could find the time to do even one piece of mediocre scientific work, he had actually, somehow or another, made the opportunity to create one of the great masterpieces of all mathematics and a landmark in human thought. He had worked at it off and on for twenty years or more. His first public communication on the subject, to the Physical-Mathematical Society of Kazan, was made in 1826. He might have been speaking in the middle of the Sahara Desert for all the echo he got. Gauss did not hear of the work till about 1840.

Another episode in Lobatchewsky's busy life shows that it was not only in mathematics that he was far ahead of his time. The Russia of
1830 was probably no more sanitary than that of a century later, and it may be assumed that the same disregard of personal hygiene which filled the German soldiers in the World War with an amazed disgust for their unfortunate Russian prisoners, and-which today causes the industrious proletariat to use the public parks and playgrounds of Moscow as vast and convenient latrines, distinguished the luckless inhabitants of Kazan in Lobatchewsky's day when the cholera epidemic found them richly prepared for a prolonged visitation. The germ theory of disease was still in the future in 1830, although progressive minds had long suspected that filthy habits had more to do with the scourge of the pestilence than the anger of the Lord.

On the arrival of the cholera in Kazan the priests did what they could for their smitten people, herding them into the churches for united supplication, absolving the dying and burying the dead, but never once suggesting that a shovel might be useful for any purpose other than digging graves. Realizing that the situation in the town was hopeless, Lobatchewsky induced his faculty to bring their families to the University and prevailed upon—practically ordered—some of the students to join him in a rational, human fight against the cholera. The windows were kept closed, strict sanitary regulations were enforced, and only the most necessary forays for replenishing the food supply were permitted. Of the 660 men, women and children thus sanely protected, only sixteen died, a mortality of less than 2.5 per cent. Compared to the losses under the traditional remedies practised in the town this was negligible.

It might be imagined that after all his distinguished services to the state and his European recognition as a mathematician, Lobatchewsky would be in line for substantial honors from his Government. To imagine anything of the kind would not only be extremely naïve but would also traverse the scriptural injunction “Put not your trust in princes.” As a reward for all his sacrifices and his unswerving loyalty to the best in Russia, Lobatchewsky was brusquely relieved in 1846 of his Professorship and his Rectorship of the university. No explanation of this singular and unmerited double insult was made public. Lobatchewsky was in his fifty fourth year, vigorous of body and mind as ever, and more eager than he had ever been to continue with his mathematical researches. His colleagues to a man protested against the outrage, jeopardizing their own security, but were curtly informed
that they as mere professors were constitutionally incapable of comprehending the higher mysteries of the science of government.

The ill-disguised disgrace broke Lobatchewsky. He was still permitted to retain his study at the University. But when his successor, hand-picked by the Government to discipline the disaffected faculty, arrived in
1847
to take up his ungracious task, Lobatchewsky abandoned all hope of ever being anybody ag£Ín in the University which owed its intellectual eminence almost entirely to his efforts, and he appeared thereafter only occasionally to assist at examinations. Although his eyesight was failing rapidly he was still capable of intense mathematical thinking.

He still loved the University. His health broke when his son died, but he lingered on, hoping that he might still be of some use. In
1855
the University celebrated its semicentennial anniversary. To do honor to the occasion, Lobatchewsky attended the exercises in person to present a copy of his
Pangeometry,
the completed work of his scientific life. This work (in French and Russian) was not written by his own hand, but was dictated, as Lobatchewsky was now blind. A few months later he died, on February
24, 1856,
at the age of sixty two.

*  *  *

To see what Lobatchewsky did we must first glance at Euclid's outstanding achievement. The name Euclid until quite recently was practically synonymous with elementary school geometry. Of the man himself very little is known beyond his doubtful dates,
330-275. B.C.
In addition to a systematic account of elementary geometry his
Elements
contain all that was known in his time of the theory of numbers. Geometrical teaching was dominated by Euclid for over
2200
years. His part in the
Elements
appears to have been principally that of a coordinator and logical arranger of the scattered results of his predecessors and contemporaries, and his aim was to give a connected, reasoned account of elementary geometry such that every statement in the whole long book could be referred back to the postulates. Euclid did not attain this ideal or anything even distantly approaching it, although it was assumed for centuries that he had.

Euclid's title to immortality is based on something quite other than the supposed logical perfection which is still sometimes erroneously ascribed to him. This is his recognition that the fifth of his postulates (his Axiom XI) is a pure assumption. The fifth postulate can be stated in many equivalent forms, each of which is deducible from any
one of the others by means of the remaining postulates of Euclid's geometry. Possibly the simplest of these equivalent statements is the following: Given any straight line
l
and a point
P
not on
l
, then in the plane determined by
l
and
P
it is possible to draw
precisely one
straight line
V
through
P
such that
V
never meets
l
no matter how far
l
′ and
l
are extended (in either direction). Merely as a nominal definition we say that two straight lines lying in one plane which never meet are
parallel.
Thus the fifth postulate of Euclid asserts that through
P
there is precisely one straight line parallel to /. Euclid's penetrating insight into the nature of geometry convinced him that this postulate had not, in his time, been deduced from the others, although there had been many attempts to
prove
the postulate. Being unable to deduce the postulate himself from his other assumptions, and wishing to use it in the proofs of many of his theorems, Euclid honestly set it out with his other postulates.

There are one or two simple matters to be disposed of before we come to Lobatchewsky's Copernican part in the extension of geometry. We have alluded to “equivalents” of the parallel postulate. One of these, “the hypothesis of the right angle,” as it is called, will suggest two possibilities, neither equivalent to Euclid's assumption, one of which introduces Lobatchewsky's geometry, the other, Riemann's.

Consider a figure
AXTB
which “looks like” a rectangle, consisting of four straight lines
AX, XT, TB, BA,
in which
BA
(or
AB)is
the base,
AX
and
TB
(or
BT)
are drawn equal and perpendicular to
AB,
and on the same side of
AB.
The essential things to be remembered about this figure are that each of the angles
XA B, TBA
(at the base) is a right angle, and that the sides
AX, BY
are equal in length.
Without using the parallel postulate,
it can be proved that the angles
AXT, BTX,
are
equal,
but,
without
using this postulate,
it is impossible to prove that AXT, BTX are right angles,
although they look it. If we
assume
the
parallel postulate
we can
prove
that
AXT, BTX
are
right angles
and, conversely, if we
assume
that
AXT, BTX
are
right
angles,
we can
prove
the parallel postulate. Thus
the assumption that AXT, BTX are right angles
is equivalent
to the parallel postulate.
This assumption is today called
the hypothesis of the right angle
(since both angles are right angles the singular instead of the plural “angles” is used).

It is known that the hypothesis of the right angle leads to a consistent, practically useful geometry, in fact to Euclid's geometry refurbished to meet modern standards of logical rigor. But the figure suggests two other possibilities: each of the equal angles
AXY, BYX
is
less
than a right angle—
the hypothesis of the acute angle;
each of the equal angles
AXY, BYX
is
greater
than a right angle—
the hypothesis of the obtuse angle.
Since any angle can satisfy one, and only one, of the requirements that it be
equal to, less than,
or
greater than
a right angle, the three hypotheses—of the right angle, acute angle, and obtuse angle respectively—exhaust the possibilities.

Common experience predisposes us in favor of the first hypothesis. To see that each of the others is not as unreasonable as might at first appear we shall consider something closer to actual human experience than the highly idealized “plane” in which Euclid imagined his figures drawn. But first we observe that neither the hypothesis of the acute angle nor that of the obtuse angle will enable us to prove Euclid's parallel postulate, because, as has been stated above, Euclid's postulate is
equivalent
to the hypothesis of the
right angle
(in the sense of interdeducibility; the hypothesis of the right angle is both necessary and sufficient for the deduction of the parallel postulate). Hence if we succeed in constructing geometries on either of the two new hypotheses, we shall not find in them parallels in Euclid's sense.

To make the other hypotheses less unreasonable than they may
seem at first sight, suppose the Earth were a perfect sphere (without irregularities due to mountains, etc.). A plane drawn through the center of this ideal Earth cuts the surface in a
great circle.
Suppose we wish to go from one point
A
to another
B
on the surface of the Earth, keeping always
on
the surface in passing from
A
to
B,
and suppose further that we wish to make the journey by the shortest way possible. This is the problem of “great circle sailing.” Imagine a plane passed through
A, B,
and the center of the Earth (there is one, and only one, such plane). This plane cuts the surface in a great circle. To make our shortest journey we go from
A
to
B
along the shorter of the two arcs of this great circle joining them. If
A, B
happen to lie at the extremities of a diameter, we may go by either arc.

The preceding example introduces an important definition, that of a
geodesic on a surface,
which will now be explained. It has just been seen that the
shortest
distance joining two points on a sphere, the distance itself being measured
on the surface,
is
an
arc of the great circle joining them. We have also seen that the
longest
distance joining the two points is the
other
arc of the same great circle, except in the case when the points are ends of a diameter, when shortest and longest are equal. In the chapter on Fermat “greatest” and “least” were subsumed under the common name “extreme,” or “extremum.” We recall now one usual definition of a straight-line segment joining two points in a plane—“the
shortest distance
between two points.” Transferring this to the sphere, we say that to
straight line
in
the plane
corresponds
great circle
on the
sphere.
Since the Greek word for the Earth is the first syllable ge
of geodesic we call
all extrema joining any two points on any surface the geodesics of that surface.
Thus in a plane the geodesics are Euclid's straight lines; on a sphere they are great circles. A geodesic can be visualized as the position taken by a string stretched as tight as possible between two points on a surface.