by E. T. Bell
Lobatchewsky was a strong believer in the philosophy that in order to get a thing done to your own liking you must either do it yourself or understand enough about its execution to be able to criticize the work of another intelligently and constructively. As has been said, the University was his life. When the Government decided to modernize the buildings and add new ones, Lobatchewsky made it his business to see that the work was done properly and the appropriation not squandered. To fit himself for this task he learned architecture. So practical was his mastery of the subject that the buildings were not only handsome and suited for their purposes but, what must be almost unique in the history of governmental building, were constructed for less money than had been appropriated. Some years later (in 1842) a disastrous fire destroyed half Kazan and took with it Lobatchewsky’s finest buildings, including the barely completed observatory—the pride of his heart. But due to his energetic cool-headedness the instruments and the library were saved. After the fire he set to work immediately to rebuild. Two years later not a trace of the disaster remained.
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.
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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 st
raight 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.
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.