Popular Science Monthly/Volume 67/November 1905/The Value of Non-Euclidean Geometry

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Among conditions to a more profound understanding of even very elementary parts of the Euclidean geometry, the knowledge of the non-Euclidean geometry can not be dispensed with.—E. Study.

ELEMENTARY geometry has been the most stable part of all science. This was due to one book, of which Philip Kelland says:

It is certain, that from its completeness, uniformity and faultlessness, from its arrangement and progressive character, and from the universal adoption of the completest and best line of argument, Euclid's Elements stand preeminently at the head of all human productions. In no science, in no department of knowledge, has anything appeared like this work: for upwards of 2,000 years it has commanded the admiration of mankind, and that period has suggested little towards its improvement.

In all lands and languages, in all the world, there was but one geometry. For the abstractest philosophy, for the most utilitarian technology, geometry is of fundamental importance. For education it is the before and after, the oldest medium and the newest; older, more classic than the classics, as new as the automobile. The first of the sciences, it is ever the newest requisite for their ongo. Says H. J. S. Smith:

I often find the conviction forced upon me that the increase of mathematical knowledge is a necessary condition for the advancement of science, and, if so, a no less necessary condition for the improvement of mankind. I could not augur well for the enduring intellectual strength of any nation of men, whose education was not based on a solid foundation of mathematical learning, and whose scientific conceptions, or, in other words, whose notions of the world and of the things in it, were not braced and girt together with a strong framework of mathematical reasoning.

Of what startling interest then must it be that at length this century-plant has flowered, a new epoch has unfolded. How did this happen? Euclid deduced his geometry from just five axioms and five postulates. These were all very, very short and simple, except the last postulate, which was in such striking contrast to the others that not its truth, but the necessity of assuming or postulating it, was doubted from remotest antiquity. The great astronomer Ptolemæos (Ptolemy) wrote a treatise purporting to prove it, and hundreds after him spent their brains in like attempts. What vast effort has been wasted in this chimeric hope, says Poincaré, is truly unimaginable!

This most celebrated, most notorious of all postulates, Euclid's parallel-postulate, is not used for his first 28 propositions. When at length used, it is seen to be the inverse of a proposition already demonstrated, the seventeenth, as Proklos remarked, therefrom, according to Lambert, arguing its demonstrability. Moreover, its one and only use is in proving the inverse of another proposition already demonstrated, the twenty-seventh. No one had a doubt of the necessary external reality and exact applicability of the postulate. The Euclidean geometry was supposed to be the only possible form of space-science; that is, the space analyzed in Euclid's axioms and postulates was supposed to be the only non-contradictory sort of space. Even Gauss never doubted the actual reality of the parallel-postulate for our space, the space of our external world, according to Dr. Max Simon, who says in his 'Euclid,' 1901, p. 36:

Nur darf man nicht glauben, dass Gauss je an der thatsächlichen Richtigkeit des Satzes für unsern Raum gezweifelt habe, so wenig, wie an der der Dreidimensionalität des Raumes, obwohl er audi hier das logisch Hypothetische erkannte.

But could not this postulate be deduced from the other assumptions and the 28 propositions already proved by Euclid without it? Euclid had among these very propositions demonstrated things more axiomatic by far. His twentieth, 'Any two sides of a triangle are together greater than the third side,' the Sophists said, even donkeys knew. Yet, after he has finished his demonstration, that straight lines making with a transversal equal alternate angles are parallel, in order to prove the inverse, that parallels cut by a transversal make equal alternate angles, he brings in the unwieldy assumption thus translated by Williamson (Oxford, 1781):

11. And if a straight line meeting two straight lines makes those angles which are inward and upon the same side of it less than two right angles, the two straight lines being produced indefinitely will meet each other on the side where the angles are less than two right angles.

This ponderous assertion is neither so axiomatic nor so simple as the theorem it is used to prove. As Staeckel says:

It requires a certain courage to declare such a requirement, alongside the other exceedingly simple axioms and postulates.

Says Baden Powell in his 'History of Natural Philosophy,' p. 34: The primary defect in the theory of parallel lines still remains.

This supposed defect an ever renewing stream of mathematicians tried in vain to remedy. Some of these merely exhibit their profound ignorance, like Ferdinand Hoefer, who in his 'Histoire des Mathematiques,' Paris, 1874, p. 176, says:

Certain defects with which Euclid is reproached may be explained by simple transpositions. Such is the case of the Postulatum V.
He then proceeds to misquote it as follows:

Si une droite, en coupant deux autres droites, fait les angles internes inégaux, ou moindres que deux angles droits, ces deux droits, prolongées à rinfini, se rencontreront du côté ou les angles sont plus petits que deux droits;

and continues,

It is certain that, placed after the definitions, this Postulatum is incomprehensible. But, placed after Proposition XXVI. of the first book, where the author demonstrates that 'if the interior angles together equal two right angles, the lines will not meet,' it acquires almost the evidence of an axiom.

The XXVI. is, of course, a mistake for XXVIII.

Other mathematicians have tried to turn the flank of the difficulty by substituting a new definition of parallels for Euclid's.

Eu. I., Def. 35, is: 'Parallel straight lines are such as are in the same plane, and which being produced ever so far both ways do not meet.'

On this Hobbs petulantly remarks:

How shall a man know that there be straight lines which shall never meet, though both ways infinitely produced?

The answer is simple: Read Eu. I., 27, where if the straight line be infinite, is proven that those making equal alternate angles nowhere meet.

Wolf, Boscovich and T. Simpson substitute for Euclid's the definition: 'Straight lines are parallel which preserve always the same distance from each other.' But this is begging the question, since it assumes the definition, 'two straight lines are parallel when there are two points in the one on the same side of the other from which the perpendiculars to it are equal,' and at the same time assumes the theorem, 'all perpendiculars from one of these lines to the other are equal.'

Just so the assumption that there are straights having the same direction is a petitio principii, since it assumes the definition of Varignon and Bézout, that 'parallel lines are those which make equal angles with a third line,' and at the same time assumes the theorem that 'straight lines which make equal angles with one given transversal make equal angles with all transversals.'

Other and more penetrating geometers have proposed substitutes for the parallel-postulate. Of these the simplest are Ludlam's: 'Two straight lines which cut one another can not both be parallel to the same straight line,' and W. Bolyai's 'Any three points are costraight or concyclic.'

But the largest and most desperate class of attempts to remove this supposed blemish from geometry consists of those who strive to deduce the theory of parallels from reasonings about the nature of the straight line and plane angle, helped out by Euclid's nine other assumptions and first twenty-eight propositions. Hundreds of geometers tried at this. All failed. That eminent man, Legendre, was continually trying at this, and continually failing at it, throughout his very long life.

Naturally, some very respectable mathematicians were deceived.

The acute logician, De Morgan, accepted and reproduced a wholly fallacious proof of Euclid's parallel postulate, recently republished as sound by the Open Court Publishing Company, Chicago, 1898. A like pseudo-proof published in Crelle's Journal (1834) trapped even our well-known Professor W. W. Johnson, head mathematician of the U. S. Naval Academy, who translated and published it in the Analyst (Vol. III., 1876, p. 103), saying:

This demonstration seems to have been generally overlooked by writers of geometrical text-books, though apparently exactly what was needed to put the theory upon a perfectly sound basis.

But a more recent, a veritably shocking, example is at hand. On April 29, 1901, a Mr. Israel Euclid Rabinovitch submitted to the Board of University Studies of the Johns Hopkins University, in conformity with the requirements for the degree of doctor of philosophy, a dissertation in which, after an introduction full of the most palpable blunders, he proceeds to persuade himself that he proves Euclid's parallel postulate by using the worn-out device of attacking it from space of three dimensions, a device already squeezed dry and discarded by the very creator of non-Euclidean geometry, John Bolyai. And his dissertation was accepted by the referees. And since then Dr. (J. H. U.) Israel Euclid Rabinovitch has written, March 25, 1904:

As to Poincaré's assertion about the impossibility to [sic] prove the Euclidean postulate, it is no more than a belief—though an enthusiastic one [sic]—never proved mathematically, and in its very nature incapable of mathematical proof.

Poincaré is undoubtedly a great mathematician, perhaps the greatest now living; but his assertion of his inmost conviction, no matter how strongly put, can not pass for mathematical truth, unless mathematically proved.

His conclusion—shared also by many another noted mathematician as well as by the founders of the non-Euclidean geometries—can only be based on the fact of the existence of these last geometries, self-consistent and perfectly logical. But this is a poor proof of the impossibility to [sic] establish the Euclidean postulate.

If space is regarded as a point-manifold, it is Euclidean, and the postulate can be proved as soon as we are allowed to look for its establishment in three-dimensional geometry.

The two-dimensional elliptic geometry described by Klein, Lindemann and Killing, according to my opinion, is an absurdity for a point-space in the ordinary sense of the term.

Poincaré says that all depends upon convention. But still he deduces from this the perfectly gratuitous conclusion that therefore the parallel-postulate can not be proved.

Alongside this modern instance, too pathetic for comment, we may, however, be allowed to quote what one of the two greatest living mathematicians, Poincare, says in reviewing the work of the other, Hilbert's transcendently beautiful 'Grundlagen der Geometrie,' itself an outcome of non-Euclidean geometry:

What are the fundamental principles of geometry? What is its origin; its nature; its scope?

These are questions which have at all times engaged the attention of mathematicians and thinkers, but which took on an entirely new aspect, thanks to the ideas of Lobachevski and of Bolyai.

For a long time we attempted to demonstrate the proposition known as the postulate of Euclid; we constantly failed; we know now the reason for these failures.

Lobachevski succeeded in building a logical edifice as coherent as the geometry of Euclid, but in which the famous postulate is assumed false, and in which the sum of the angles of a triangle is always less than two right angles. Riemann devised another logical system, equally free from contradiction, in which the sum is, on the other hand, always greater than two right angles. These two geometries, that of Lobachevski and that of Riemann, are what are called the non-Euclidean geometries. The postulate of Euclid then can not be demonstrated; and this impossibility is as absolutely certain as any mathematical truth whatsoever.

It was the attainment of this very perception which in fact led to the creation of the non-Euclidean geometry. Says Lobachevski in the introduction to his 'New Elements of Geometry':

The futility of the efforts which have been made since Euclid's time during the lapse of two thousand years awoke in me the suspicion that the ideas employed might not contain the truth sought to be demonstrated. When finally I had convinced myself of the correctness of my supposition I wrote a paper on it [assuming the infinity of the straight].

It is easy to show that two straights making equal angles with a third never meet.

Euclid assumed inversely, that two straights unequally inclined to a third always meet.

To demonstrate this latter assumption, recourse has been had to many different procedures.

All these demonstrations, some ingenious, are without exception false, defective in their foundations and without the necessary rigor of deduction.

John Bolyai calls his immortal two dozen pages (the most extraordinary two dozen pages in the whole history of thought), 'The Science Absolute of Space, independent of the truth or falsity of Euclid's Axiom XI. (which can never be decided a priori).'

Later we read on the title page of W. Bolyai's 'Kurzer Grundriss': 'the question, whether two straights cut by a third, if the sum of the interior angles does not equal two right angles, intersect or not? no one on the earth can answer without assuming an axiom (as Euclid the eleventh)' [the parallel postulate].

With the ordinary continuity assumptions or the Archimedes postulate, it suffices to know the angle-sum in a single rectilineal triangle in order to determine whether space be Euclidean or non-Euclidean.

How peculiarly prophetic or mystic then that the clairvoyant inspiration of the genius of Dante, the voice of ten silent centuries, should have connected with the wisdom of Solomon and the special opportunity vouchsafed him by God a question whose answer would have established the case of Euclidean geometry seven hundred years before it was born, or that of non-Euclidean geometry three thousand years before its creation.

I. Kings 3 : 5 is:

In Gibeon the Lord appeared to Solomon in a dream by night: and God said, Ask what I shall give thee.
Then says Dante of his asking,
'Twas not to know the number in which are

Or if in semicircle can be made
Triangle so that it have no right angle.
se del mezzo cerchio far si puote
Triangol si, ch'un retto non avesse.
Par. C. XIII. 101-102.

How startling this! How strangely reinforced hy the fact that in the fourth canto of the 'Divina Commedia,' with Caesar greatest of the sons of men, Dante ranks, among exalted personages

. . . who slow their eyes around
Majestically moved, and in their port
Bore eminent authority:

Hippocrates of Chios, who found the quadrature of the lune (nearest that ever man came to the quadrature of the circle until finally John Bolyai squared it in non-Euclidean geometry and Lindemann proved no man could square it in Euclidean geometry); Euclid, the geometer, the elementist, preemptor, by his postulate, of the common universe, Euclidean space; and then Ptolemy, first of the long line of those who have tried by proof to answer the question Dante says Solomon might have asked God and did not, a question crucial as to whether Euclid or Bolyai owns the real world.

Anyhow, the shock currents to scientific somnolence and complacency breaking in to the entrenched thought camp from over the ramparts on the far frontiers almost simultaneously at Kazan on the Volga and at Maros-Vásárhely in far Erdély, started an ever-deepening movement to sift, to revise the foundations of geometry, then of all mathematics, then of all science, a movement of which the latest, as the most charming and weightiest, outcome is that pair of wonderbooks, Poincaré's 'Science and Hypothesis' and 'The Value of Science.'

It was formerly customary to consider the assumption that space is a triply extended continuous number-manifold as a self-evident outcome of the continuity relations of space and the curves and surfaces in it. But since the rise of non-Euclidean geometry, it has come more and more to be seen that such presupposition about space is only admissible when one has already established and developed elementary geometry synthetically. Lobachevski stressed this in 1836 in his introduction to the 'New Elements,' where he says:

One must necessarily make the beginning with synthesis, in order, finally, after one has found the equations, therewith likewise to reach the limit beyond which now all goes over into the science of numbers.

For example, one demonstrates in geometry that two straights perpendicular to a third never meet; that the equality of triangles follows from that of certain of their parts.

In vain would one seek to treat analytically propositions of this species, even as all the theory of parallels. One would never succeed, just as one would not be able to do without synthesis for measuring plane rectilineal figures, or solids terminated by plane surfaces. It is incontestable that in the beginnings of geometry or mechanics, analysis can not serve as sole method.

One may compress the circle of synthesis; but it is impossible completely to suppress it.

From this, however, it follows that all investigations, such as those of Sophus Lie which start with the idea of number-manifold, involve a petitio principii, if interpreted directly as researches on the foundations of geometry. In the same way, the non-Euclidean geometry stops the old wrangle as to whether the axioms of geometry are a priori or empirical by showing that they are neither, but are conventions, disguised definitions, or unprovable assumptions pre-created by auto-active animal and human minds.

As Lambert insisted, for the space problem the mathematical treatment is in essence the treatment by logic. The start is from a system of axioms, assumptions. We postulate that between the elements of a system of entities certain relations shall hold, e. g., two points determine a straight, three a plane. There is to be shown that these axioms are independent and not contradictory, presupposing pure logic and the applicability to the entities of an arithmetic founded by and made of pure logic. That the assumptions considered should be axioms of geometry, they must satisfy a further condition, which Hilbert formulates thus:

A system of assumptions is called a system of axioms of geometry if it gives the necessary and sufficient and independent conditions to which a system of things must be subjected in order that every property of these things should correspond to a geometric fact, and inversely; so that therefore in Hertz's sense these things should be a complete and simple picture of geometric reality.

The physiologic-psychologic investigation of the space problem must give the meaning of the words geometric fact, geometric reality.

It is the set of assumptions which makes the geometry what it is, which determines it. Thus, in my 'Rational Geometry,' one system of assumptions about the elements, points and straights on a plane, makes Euclidean planimetry. Another set makes Riemannean planimetry, in which when we picture it as in Euclidean space, we may call the straights straightests (great circles), and the plane sphere.

In the light of all this we see how the importance of non-Euclidean geometry for the teacher is still emphasized by the text-books of France, which have never recovered from Clairaut and Legendre. Even the latest and best French geometry, that of Hadamard, published under the editorship of Gaston Darboux, never presents nor consciously considers the question of its own foundations. It seems childishly unconscious of what is now requisite for any geometry pretending to be scientific or rigorous. This lack of foundation is allowable in a preliminary course of intuitive geometry which does not attempt to be rigorously demonstrative, which emphasizes the sensuous rather than the rational. But in a serious work it is now no longer permissible to have nothing to start from. Wherever rigorous mathematics, there pure logic.

It may be a relief to many that the non-Euclidean geometry has shown the limitations to the arithmetization of mathematics. The opinion that only the concepts of analysis or arithmetic are susceptible of perfectly rigorous treatment Hilbert considers entirely erroneous.

On the contrary, he says, I think that wherever, from the side of the theory of knowledge, or in geometry, or from the theories of natural or physical science, mathematical ideas come up, the problem arises for mathematical science to investigate the principles underlying these ideas and so to establish them upon a simple and complete system of axioms, that the exactness of the new ideas and their applicability to deduction shall be in no respect inferior to those of the old arithmetical concepts.

The arithmetical symbols are written diagrams and the geometrical figures are graphic formulas.

The use of geometrical signs as a means of strict proof presupposes the exact knowledge and complete mastery of the axioms which underlie those figures; and in order that these geometrical figures may be incorporated in the general treasure of mathematical signs, there is necessary a rigorous axiomatic investigation of their conceptual content.

In other words, the world has outgrown Wentworth's geometry. More than this, as Frankland puts it, the possibility of explaining 'mass' (the fundamental property of matter) as a function of 'electric charge' is on the point of banishing both ordinary gross matter and also ether, since the principle of parsimony forbids needless hypothetical entities. Now the relation between the two opposite electricities so closely resembles that between Bolyaian and Riemannean space that, as Clifford adumbrated, we may expect to see matter, ether and electricity banished in favor of space, the various and changing geometries of which will be found adequate to account for all the phenomena of the material world.

Furthermore, these geometries of physical space will be found not to be 'continuous,' but to be the varied and changing 'tactical' arrangements of a discrete, a discontinuous manifold consisting of indivisible units. The notion of continuous extension, so long considered ultimate, will, by this simplification, be subsumed under the finally ultimate notion of juxtaposition, with which Lobachevski begins his great treatise 'Noviya nachala,' in whose very first article he says of it: "This simple idea derives from no other, and so is subject to no further explanation."