SINCE we have seen that the fifth postulate of Euclid is capable of strict proof, and that his definition of parallels can be solidly established, we are in a position to answer the third of the questions as between Euclidean and non-Euclidean geometry, viz., can one make assumptions contradictory to these postulates and still have a logical body of geometric doctrine consistent with the other assumptions of Euclid, and his deductions from them. It is now evident that we can answer this question in the negative. We have only to accept the criterion of Lobatschewsky himself and judge his system out of his own mouth. Lobatschewsky has shown that, if what he calls the angle of parallelism is a right angle, Euclidean geometry will follow. But if the angle of parallelism is an acute angle then his own would be the true geometry. The angle of parallelism is the angle formed by the second parallel with a perpendicular to the first parallel. Now we have clearly demonstrated that this angle of parallelism is necessarily a right angle. For since the interior angles formed by a transversal on the two parallels equal two right angles, if the transversal is perpendicular to one, or forms a right angle with it, it must also form a right angle with the other to make the two right angles. All non-Euclidean metageometry is then nothing more than pseudo-geometry in so far as it departs from Euclid’s principles. Whatever there is in it that is built on assumptions that do not contradict Euclid’s principles, is Euclidean and not non-Euclidean. Such developments are then by no means specific of or belonging to metageometry. Whatever is built on assumptions contradictory to those of Euclid is pure falsehood. And this part is the only part that is proper to metageometry and strictly belongs to it. Metageometry is thereby judged and condemned.

But since the challenge has been issued by the non-Euclidean Geometers themselves, and they claim to construct as scientifically accurate a geometry as Euclid, and as closely logical, it may be well to investigate the claim at least a little.

In the first place, there is very little body to non-Euclidean Geometry, nothing at all to compare with the various and finely interlaced structure of Euclid. We are speaking now of the non-Euclidean Geometry of its real founders Lobatschewsky and Bolyai. As a geometry of practical utility it is worse than nothing. All that it consists of in its own right is a revamping of the propositions of Saccheri whereby in working out the theory of a triangle having its interior angles less than two right angles, he endeavored to show its absurdity. Nearly all that the non-Euclideans have added has been assurance enough to deny the absurdity. We are, of course, now speaking of geometry properly so called; not the analytic mathematics based on the so-called non-Euclidean principles. The body of this is much greater. The only non-Euclidean Geometry properly so called, or formal geometry, is that based on the theory of the acute angle, i.e., the Hyperbolic Geometry of Bolyai and Lobatschewsky and their followers, who took over Saccheri’s. The Riemann Geometry has never been developed as a formal geometry at all; its principles were merely used as a basis for differential calculations. But what there is of formal Hyperbolic Geometry is a very meagre output. Even its authors found it impossible to develop it to any great extent.

But can it be said that this Geometry is a logical geometry? This depends on what interpretation we give to the word logical. We can start with any principles, even the most absurd, and if we are not afraid of continued absurdity, we can maintain a logical consistency; that is, we can at least adhere to the rules of logic in developing our absurdity; we can be consistent to our false principles. But consistency is not in any sense a criterion of truth. If there is any consistency in non-Euclidean Geometry it is of this kind. But such consistency is rather easy. All that is required is simply to deny whatever would make our system illogical, and we can all be happy.

But is there even such logic in the so-called non-Euclidean Geometry? Let us see. Here is a proposition on what is known as Saccheri’s Quadrilateral. It is taken from the Elements of Non-Euclidean Plane Geometry and Trigonometry, by H. S. Carslaw, London, 1916. It uses a figure Saccheri made frequent use of in his Theory of Parallels.

Here is the theorem.

“In Saccheri’s Quadrilateral, when the right angles are adjacent to the base, the vertical angles are equal acute angles, and the line which bisects the base at right angles also bisect the opposite side at right angles.¹

And here is the proof.

Let AC and BD be the equal sides, and the angles A and B right angles. Let E, F be the middle points of AB and CD respectively.
Join EF, CE, and DE.
Then the triangles ACE and EDB are congruent, and the congruence of CFE and EFD follows:

Thus the angles at C and D are equal, and EF is perpendicular both to AB and CD.

Further, the angles at C and D are acute.

We need not go any further in describing the proof of the latter statement, a thing which is attempted by means of a new figure. What we have amply suffices for our purpose. Here in this so-called quadrilateral of Saccheri, we actually have three perpendiculars to the line AB each forming right angles with it, and the same three perpendiculars falling on the line CD make two acute angles and two right angles. In other words, of three perpendiculars two are not perpendicular, and the third is perpendicular to a second line. Could any one imagine Euclid or any Euclidean giving out such stuff? What intelligible difference is there between the three perpendiculars? Either two of them have gone their own sweet way, or the one exception has. There is apparently no law for them to follow. Easy to be non-contradictory.Only agree to disagree as do the perpendiculars above. Maybe it is not contradictory, but it is funny.

Here is another gem. It follows the other almost immediately.

“If ABCD is a quadrilateral in which the angles A, B, and C are right angles, then the angle D must be acute.”²

To avoid making a new figure, we shall merely change the letters to correspond to the figure given above. Carslaw employs the same form of figure, only his lettering in the second instance is different. Let us then state his proposition in terms of the figure given above.

“If EBFD is a quadrilateral in which the angles E, B, and F are right angles, then the angle D must be acute.”

Here the perpendiculars act in a new way. There has been a political revolution among perpendiculars, or an election and another party is now in control. Before, we had two perpendiculars making two acute angles and two right angles with the two lines falling on them. Now, we have two perpendiculars producing three right angles and one acute angle by the same kind of operation. We see the revolution worked before our very eyes; the election ballots were in an open box and perfectly visible. The whole subversion took place in the quadrilateral of Saccheri. By the simple introduction of an extra perpendicular, a perfectly regular quadrilateral formed by two happy, harmonious and united perpendiculars, producing two right angles and two acute angles, is turned into a disrupted menage of two quadrilaterals each with two perpendiculars, each having three right angles and one acute angle. (I do not think the quadrilateral can be named after Saccheri any more; he could scarcely countenance the revolutionary bolshevism of such a household.) It is a case where two is company and three is a crowd. This again may be logical consistency, but if any one can explain by what logic two perpendiculars between the same two lines can now form quadrilaterals having three right angles and one acute, now two right angles and two acute, he should receive the palm for logic, or something else. He certainly could build without the principle of sufficient reason.

If it is any benefit to shift the contradictoriness from logic to the physical and metaphysical behavior of geometric elements, it is certainly no benefit either to correct thought or to actual practice. Apparently, according to the new Geometry, anything goes, with the sole exception of what common sense demands. A quadrilateral may have two right angles or three right angles, all formed by the same kind of perpendiculars; but never four, which experience, common sense, and even strict logic demand, for that would be to have part with Euclid, and that would never do. Though, to tell the truth, I have never seen a quadrilateral with only one right angle; I imagine the main difficulty is not logical contradiction, but that the new Geometers would scarcely know how to draw one. Some one has called this Astral Geometry. It surely should be banished to astral regions, to the moon or elsewhere, for it would sometimes seem that our carpenters are being corrupted by it, to judge by the ill-fitting doors and windows with which they occasionally supply us. They might find full justification for work not properly squared, since they might easily prove by non-Euclidean Geometry, that even a square must have one or two acute angles as the occasion demands.

The strange and sad thing about this new Geometry is not that it has found advocates. There is nothing so absurd, or opposed to the common sense of mankind that it will not find supporters, for novelty is oftentimes a greater attraction than truth. The thing to be really deprecated is that such trash could attract the erudite and scholarly, and even impose itself on whole realms of science, or approach dangerously near to doing so. The dictum of P. T. Barnum can be universalized far beyond circus matters and the American public. Aristotle tells us about a little error in the beginning leading to a great one in the end. Facilis descensus Averno. Saccheri and those whom he led off the track have a great deal to answer for scientifically, though, of course, with the best of intentions.

We may seem to be rather hard on the new Geometers, but it is not at them we are aiming, but at the system. Nevertheless, we can scarcely be expected to be extremely complimentary to its authors. At least we are not ready to concur in the encomiums heaped upon them. A certain ability we need not deny; but it is scarcely of a kind to make itself felt by its own weight in the world of thought in such an extraordinary way. We would not even attribute originality to them. Even here they are lacking. There is no real novelty in what they have put forward; the only surprising thing is that there was found someone to put it forward. What made the system possible was not originality, but assurance. The two non-Euclidean Geometries, as we have remarked before, were launched by young men. Now, as far as abstract mathematics is concerned, young men with requisite training, and natural ability, are as capable of handling what is within the range of their ability as anyone. But with matters that touch the fundamentals of experience it is different. They cannot be expected to have the philosophical balance, or that concrete grasp of reality that comes only from experience and a complete all around symmetrical intellectual development. They are more inclined to be led by notions alone. This is what has occurred in this case. Yet it was the older men who gave weight to the cause by countenancing it.

The fundamental idea with which the new Geometry started, if we omit the negative one which so influenced Gauss and others after him, viz., their inability to prove the fifth postulate, was the idea introduced into the proof even in ancient times of the relation of parallels and asymptotic lines. We have already seen this idea clearly introduced in a quotation by Proclus from Geminus. Here we find at least the correct distinction between asymptotes and parallels, viz., that both have this in common that they do not meet, but parallels have this beyond the others that they remain the same distance apart. This is of course only remarkable in that it shows that the consideration of asymptotes had been introduced into the discussion of the parallel question, and the first seed of non-Euclidean Geometry was already planted.

The same idea was introduced into the discussion in modern times by Clavius. He urges the conchoid of Nicomedes against the argument of Proclus by which the latter seeks to establish the fifth postulate of Euclid. Proclus supposes that two lines parting from a point and making an angle will get further and further apart as the lines increase in length, until the distance between them will exceed any finite magnitude. Clavius well points out that the first part is correct, except that the distance will not exceed any finite magnitude. But the point is that he urges against this assumption the example of the conchoid of Nicomedes, which continually approaches its asymptote, and therefore gets continually farther away from the tangent at the vertex; yet the perpendicular from any point on the curve to the tangent will always be less than the distance between the tangent and its asymptote. Saccheri concurs with Clavius in this objection.

In this idea we have the germ of that worked out by Saccheri of asymptotic lines that do not meet, brought forward by him with the notion of reducing them to a contradiction, but frankly adopted by Bolyai and Lobatschewsky as the basis of their system, in other words, non-secant lines other than those we consider parallel in Euclidean geometry. Even this very idea was broached by Proclus, who in his proof of the fifth postulate says: “For one might say that, the lessening of the two right angles being subject to no limitation, with such and such an amount of lessening the straight lines remain non-secant, but with an amount of lessening in excess of this they meet.”³

Here we have the exact position of Lobatschewsky and Bolyai clearly indicated. All they had to do was to change the “one might say” for “I do say” and the new Geometry exists. Not much invention in that. For the working out of the Geometry, Saccheri displays the model, and does most of the actual work. His propositions developing the two hypotheses of the acute and the obtuse angle in a series of theorems which were intended to show, as he supposed they would, that such assumptions were self contradictory, but more especially the part wherein he developed his theory of the acute angle, were taken over bodily by Lobatschewsky and Bolyai. In fact Saccheri is the inventor of non-Euclidean Geometry in all but the necessary blindness and rashness to make the final leap. There was not then very much intellectual originality necessary on the part of the actual founders.

With regard to this idea, in which we might say the germ of the new Geometry lodged and started its nefarious work in the organism of geometry, there was as usual a misplacing of the question and an example of “ignorantia elenchi,” or a complete misunderstanding of the point at issue. It starts out from the conception, as we see it in the quotation from Geminus, that the straight and curved are mere species of the generic line. This division is upheld in the classification of lines that is found in the treatment of the subject by Proclus. But this is false, as we expect later to show when we intend to treat the question of the straight line. Proclus attributes the same division to both Aristotle and Plato. But if Aristotle held any such division, which we extremely doubt, at least as a real specific division, his usually keen perception failed him, and he stumbled. And though even Homer may nod, it is certainly allowed to doubt, till proof be forthcoming, that the Philosopher would trip and fall over so simple a thing as a line.

This division has been followed at any rate from that time to this, and has brought, as we shall see, no little confusion into the discussion of the subject. But this would not be sufficient to complicate the question of parallels. The ignorantia elenchi consisted in con-founding straight lines with curved lines. The straight line has one set of properties, the curved line another.

There can be no conclusion to a specific property of a straight line from a specific property of a curved line, since both are specifically distinct. The straight line, to speak in the ordinary terms of common sense and observation, has one direction; the curved line has really two. To recur to the explanation used in physics, the straight line has only one acceleration, and that a direct one. The curved line has two different accelerations and in different directions. It is this double acceleration of a curved path that precisely makes the peculiar relation of a curve to its asymptote possible. It is nothing then but the most patent sophistry to urge a difficulty taken from a twofold acceleration against a path that represents a single direct acceleration. There is no parity.

Yet it is on this sophism that the whole of non-Euclidean Geometry rests. There was then no intellectual keenness shown in adopting such a fallacy to urge against the assumption in the fifth postulate. There was certainly neither genius nor talent, nor mathematical insight, nor logical thinking in the building of a whole theory on such a paralogism. It is not even a subtle sophism. The sophistry in an argument such as follows ought to be apparent to any one. Because one kind of line which has a property which requires it and explains it, can approach another kind of line and yet not meet it, a different kind of line, which has not this property at all, must do the same thing. Here the above effect is evidently due to the curvature in the first line, and it is poor logic to deduce the same effect with regard to straightness. Straightness and curvature are specific differences of the respective lines, and there is not much logic in concluding from one difference to the other; it is even worse logic than concluding from one species to the other. Yet this is the logic that is supposed to be the equal of Euclid’s.

But were we not ourselves guilty of the same paralogism in our exposition of the vicious circle of Euclid’s twenty-seventh proposition? There we objected to Euclid that his disjunction as between parallel lines and lines that meet was not complete. But the two cases are different. We did not deny the real completeness of the disjunction but only the logical completeness. In a word, it was a question of logical assumption, of assuming the disjunction as complete, instead of proving it. Nothing was assumed as existing but only the definitions. This is quite different from arguing the sameness of really distinct things; and it is entirely different from building a new theory on it. We simply pointed out the absence of a middle term in Euclid’s argument, thus showing the argument inconclusive. The neo-Geometers assumed an impossible condition for true, and built a new Geometry on it.

The objection drawn from asymptotes, being a pure sophism and not a reason at all, cannot produce a positive doubt. In fact to any one with a sense of logic, and a proper appreciation of the essential rules of reasoning, it could not produce any kind of a doubt, positive or negative. The objection urged against the fifth postulate only shows how nearly self-evident the postulate of Euclid is, since in all the reasoning of two thousand years no positive doubt could be found against it, that is, a doubt founded on a legitimate reason. All that could ever be really said was, non probatur. In other words, the proposition was not proved, but nothing further could be shown against it. This is all that is valid in the criticism of two thousand years and more, a mere looking for the well known Q. E. D. at the end of a demonstration. Pure negative doubt; and it was on this pure negative doubt, which certain individuals tried to turn into a positive or reasoned doubt, and then into positive certitude by a palpable sophism, that the whole of the so-called non-Euclidean Geometry, or non-Euclidean bosh, has been founded.

But let us suppose for the sake of argument that it is possible to bring the parallel construction into connection with the asymptotic, so that we could reason from the properties of the one to the other. It is true the curved line approaches but does not meet its asymptote theoretically. But the distance between them is a constantly diminishing quantity, an infinitesimal in the language of the calculus. Now an infinitesimal is defined as a variable whose limit is zero. The distance dx can be made as small as we please until there is only a theoretical separation; it can be made so small that the paths of the smallest real things having the smallest extension would meet.

Practically, then, the distinction between asymptotes and the ordinary intersecting straight lines of Euclidean geometry is between lines that meet and intersect and lines that meet only. For geometry as the science of measurement, as opposed to pure mathematics, asymptotes fall in with intersecting lines and not with parallels as far as meeting is concerned. The second thing that throws them practically out of all comparison with parallels as an objection to the fifth postulate, is that in asymptotes the approach or divergence from the curve to which they are asymptotic is independent of distance. The argument used for non-secant straights other than parallels is that of immense distance. We know that converging straight lines in small areas intersect; the very existence of triangles shows it. Some straight lines at least intersect as was shown by Proclus in attempting to answer the ancient sophism we referred to, which attempted to show that converging lines cannot intersect. He objects to the argument used in the sophism that it proves too much, “since we have only to join AG in the figure above⁴ in order to see that straight lines making some angles which are together less than two right angles, do in fact meet.” And he rightly concludes: “Therefore, it is not possible to assert, without some definite limitation, that the straight line produced from angles less than two right angles do not meet.” But the non-Euclideans never worried about this in their arguing from non-secants. That secants and asymptotes are rather in the same class did not seem to strike them, since asymptotes may be made to approach curves and practically coincide with them by the use of a pencil. Distance has no effect on the asymptotic relation; asymptotes always come infinitesimally near theoretically, and practically meet. If we wished to reason by correct analogy from asymptotes to so-called converging straight lines, the conclusion should be drawn that these latter meet, and that distance has no effect on them. The analogy really points in the opposite direction to that taken by the metageometers.

The position of non-Euclidean Geometry before the bar of reason is then as follows. There is not even the wraith of a positive reason for rejecting the postulate of Euclid, for no reason capable of creating positive doubt has ever been brought against it. The arguments that have been adduced are sophistic, and are invented to bolster up a negative doubt that arose purely from inability to prove a proposition that was very nearly self-evident; but this want of proof was more than offset by the complete lack of competency to show even any manner of plausibility for the contradictory proposition. The foundation of non-Euclidean Geometry was not then an intellectual movement at all, but simply a jump in the dark which required no more than pure intellectual suicidal boldness. This is what we stated in the introduction, and we think we have given sufficient reason to show that we were not exaggerating or gratuitously assaulting non-Euclidean mathematicians to rob them of their hardly won honors.

In the next place, besides having no evidence or argument in favor of their own principles and assumptions on which they base their supposed science, nor again the slightest positive suspicion of an argument against the traditional principles and position, we have to weigh the arguments and evidences for the theory of Euclid. All the authority of common sense, all the weight of the accumulated and personal experience of the whole human race, the pragmatical sanction of a wonderfully coordinated and thoroughly consistent body of science that has satisfied all the practical and theoretical needs of the human spirit in those things for which the science may be used,⁵ all are tossed aside for a mere nothing, for a contradiction out of a spirit of petulance at not being able to prove apodictically what all men have always held for true. Such evidently is not the sign spiritual of intellectual benefactors of the human race.

But that is not all. The numerous arguments that have been worked out and brought forward, in a few cases at least by outstanding mathematicians, certainly are entitled to some weight and consideration. It is true none of them were apodictic or necessary in the sense in which geometry demands demonstration. But we must distinguish apodictic demonstration from proof that merely produces conviction. Many of these arguments were, to say the least, extremely plausible, so plausible that one can not find any positive objection other than that the necessity of the conclusion is not positively proved. Take the argument of Wallis; it is extremely plausible and persuasive, and very cleverly worked out. The only objection that we can find is in the extension of his argument. He proves his case for the realm of experience, but he cannot extend it beyond to similar conditions in unmeasured space because, again a pure negative doubt, maybe conditions there are different. The result of his argument is this: as far as we can judge, the matter is so; beyond the confines of our experience, who knows, agnosticism.

We must distinguish between the three kinds of certitude known as metaphysical, physical and moral. Metaphysical certitude is the certitude of necessary truth; it cannot be otherwise. Physical certitude is the certitude of things that occur through natural causes. I am physically certain the sun will rise tomorrow; but it is possible that it may be otherwise. Moral certitude is the certitude that is concerned with free human conduct. It is precisely the metaphysical certitude that is wanting to the postulate of Euclid, as long as it does not receive a strict geometric proof. For instance, that is what is lacking in the proof of Wallis. We are certain it is so, if our intellects have not been turned topsy-turvy by artificial consideration of non-Euclidean Geometry, but it might be otherwise. We feel that the thing demands metaphysical certitude; that the note of necessity should be attached to the proof; that as long as this is absent we are not completely satisfied. But it certainly does not mean that we are not physically certain of it; that the proofs have no persuasive force at all; that the principle which thus lacks metaphysical certainty is false; that we can reasonably assume the contrary proposition as true; in a word, that we can and ought to throw overboard everything connected with it, and assume something else without even a particle of the plausibility that at the very least attaches to this principle.

On the other hand, the assumptions and deductions of the newfangled Geometry contradict all experience, such that even non-Euclidean Geometers admit that Euclidean geometry is the only practical geometry, but that their new Geometry may find itself fulfilled in some larger space with which we are unacquainted, and they seek its justification in this sphere. We have seen some of the implications of these assumptions, when, a few paragraphs before, we discussed some of the theorems of non-Euclidean Geometry. and these are the same people who are so hypercritical of Euclid’s assumptions that, because an almost self-evident intuitive proposition had not been proved to be metaphysically necessary, they not only reject it, but assume for that reason alone the truth of its contradictory. This is certainly a case of straining out the gnat with grimaces, but swallowing the camel whole—sans broncher.

So far we have criticized mainly the non-Euclidean Geometry which owns Lobatschewsky and Bolyai as its authors, and which is ordinarily known as Hyperbolic Geometry. The same criticism applies to the Geometry of Riemann. That we did not deal with his Geometry directly is due to the fact that his treatment of the question is not geometrical, but analytical. Only his postulates are non-Euclidean, and they are at the opposite pole to the assumptions of Lobatschewsky and Bolyai. This is what usually happens whenever the human mind errs from the truth; it is not satisfied with any special direction, but wanders in all directions. The path that leads to truth is one, but error is everywhere else in the wilderness. So in this case; the new Geometers were not satisfied with one extreme; they had to go to both and opposite extremes of the golden mean; in medio stat virtus. Horace’s favorite ethical principle is as true in things of the intellect as it is in manners and conduct:

“Est modus in rebus, sunt certi denique fines,
Quos ultra citraque nequit consistere virtus.”⁶

Riemann was also a young man of twenty-eight when he read the now well-known Habilitationsschrift to the Philosophical Faculty of Göttingen. It is rather a jejune effort written in ponderous language, and is not remarkable for any striking originality either in questions of geometry or on the metaphysics of space. It simply had the good fortune to come at a time when a wave of scientific opinion stirred from the depths of scepticism was forming, and it rode into publicity on the crest of this wave. It consisted in nothing more than the adoption of the only remaining hypothesis of Saccheri, that of the obtuse angle. Lobatschewsky and Bolyai had rejected this, because of the scientific attitude of their time. It was agreed on all hands that Saccheri had been successful in proving the second hypothesis, that of the obtuse angle, to be contradictory; it was likewise agreed that he had failed in the third or that of the acute angle. It was this phase of the problem that was seized upon by the young authors of hyperbolic Geometry to exploit as a new discovery and for the foundation and framework of a new Geometry.

But as always happens, scepticism did not stop there. If the third hypothesis was a fitting base for Geometry, and Euclid’s principle not absolute, there was nothing to hinder constructing a Geometry on the still rejected hypothesis, that of the obtuse angle. All that was required was simply to reject a further postulate of Euclid, and that was easily done. Riemann did it. It no longer required such courage as in the beginning, for it was now becoming the mode, and the brunt of the attack had already been borne by Bolyai and Lobatschewsky.

To do away with Saccheri’s criticism of the second hypothesis, it was necessary to do away with Euclid’s straight line. If that was suppressed, of course the proof of the impossibility of the second hypothesis fell with it. What could be simpler? It was not even a discovery. Lobatschewsky and Bolyai had already done it, for their line was shown to be a curve with negative curvature.⁷

Riemann may have been a young man of certain mathematical attainments, beyond the average in a man of his age, but he did not have either a keen logical perception or a clear grasp of ideas. His strength really was in his mistiness. He not only clouded his own vision, but that of others that came within the cloud. There was nothing clear-cut in Riemann’s geometric ideas, and nobody has been able to cut them clearly to date. But we shall see what the knife of logic can do. It is a pity that the days of commentaries have passed, when disciples were wont to study, develop, and explain every sentence of the founder of their school. One would like to see such a commentary on the Habilitationsschrift by some fervent disciple of Riemann. But perhaps disciples are wiser in this generation than to attempt such a detailed study. Latet dolus in generalibus. Generalities at least pass more easily.

What might be considered special in Riemann may be included under three heads, for his work is really based on the following three conceptions: (l) That space is a manifold. (2) That it is finite. (3) That its measure is a measure of curvature. We shall examine each of these (a) with regard to the reasons for establishing them, (b) for the truth contained in them, (c) for their effects on geometrical theory. We cannot, it is true, enter into a full discussion of the entire theory of Riemann, or even examine critically his entire lecture, for the simple reason that such procedure would carry us entirely too far, and besides that it is not objectively worth the labor. We shall be obliged, then, to restrict ourselves to certain points and paragraphs. Nevertheless, we shall endeavor to justify the strictures we made on Riemann’s matter and method.

In the first place, let us examine the question of manifolds, or the theory by which Riemann reduces space to a multiplicity, for this is the mathematical basis of the whole theory. The fundamental error of Riemann lies precisely here, that he attempts to infer the nature of space from algebraic and analytic representations of multiplicity. But before we come to the development of manifolds, let us for a moment turn our attention to Riemann’s attitude towards this geometric problem, as expressed in his Introduction, or on the plan of the investigation. The whole disquisition bears as title “On the Hypotheses which lie at the Bases of Geometry,” and in the Introduction he makes the doctrine of multiplicities, or, as he says, multiply-extended-magnitudes, the solution of the whole question. He therefore considers it fundamental in his system. Whether or not Riemann was the first to apply the analytic notion of multiplicity to the explanation of space, we cannot say, but at least he makes the claim. Let us hear his own words.

“It is known that Geometry assumes, as things given, both the notion of space, and the first principles of construction in space. She gives definitions of them which are merely nominal, while the true determinations appear in the form of axioms. The relation of these assumptions remains consequently in darkness; we neither perceive whether and how far their connection is necessary, nor a priori, whether it is possible.

“From Euclid to Legendre (to name the most famous of modern reforming geometers) this darkness was cleared up neither by mathematicians nor by such philosophers as concerned themselves with it. The reason of this is doubtless that the general notion of multiply-extended-magnitudes (in which space magnitudes are included) remained entirely unworked. I have in the first place, therefore, set myself the task of constructing the notion of a multiply-extended-magnitude out of general notions of magnitude. It will follow from this that a multiply-extended-magnitude is capable of different measure relations, and consequently that space is only a particular case of a triply extended magnitude. But hence flows as a necessary consequence that the propositions of Geometry cannot be derived from general notions of magnitude, but that the properties which distinguish space from other conceivable triply extended magnitudes are only to be deduced from experience. Thus arises the problem to discover the simplest matters of fact from which the measure-relations of space may be determined; a problem which from the nature of the case is not completely determinate, since there may be several systems of matters of fact which suffice to determine the measure-relations of space,—the most important system for our present purpose being that which Euclid has laid down as a foundation. These matters of fact are—like all matters of fact—not necessary, but only of empirical certainty; they are hypotheses. We may therefore investigate their probability, which within the limits of observation is of course very great, and inquire about the justice of their extension beyond the limits of observation on the side both of the infinitely great and of the infinitely small.”

We now come to Riemann’s notion of a multiply-extended-magnitude, or an n-ply extended magnitude. On this he tells us:

“Magnitude notions are only possible where there is an antecedent general notion which admits of different specializations. According as there exists among these specializations a continuous path from one to another or not, they form a continuous or discrete manifoldness; the individual specializations are called in the first case points, in the second case elements, of the manifoldness. Notions whose specializations form a discrete manifoldness are so common, that at least in the cultivated languages, any things being given, it is always possible to find a notion in which they are included. (Hence mathematicians might found the theory of discrete magnitude upon the postulate that certain given things are to be regarded as equivalent.) On the other hand, so few and far between are the occasions for forming notions whose specializations make up a continuous manifoldness, that the only simple notions whose specializations form a multiply-extended-magnitude are the positions of perceived objects and colors. More frequent occasion for the creation and development of these notions occur first in the higher mathematics.”

Then we come to the development of this n-ply extended manifoldness.

“Out of the general parts of the science of extended magnitude in which nothing is assumed but what is contained in the notion of it, it will suffice for the present purpose to bring into prominence two points; the first of which relates to the construction of the notion of a multiply-extended manifoldness, the second relates to the reduction of determinations of place in a given manifoldness to determinations of quantity, and will make clear the true character of an n-fold extent.”

The first question, that of the development of the notion of an n-ply manifoldness, is thus answered:

“If in the case of a notion whose specializations form a continuous manifoldness, one passes from a certain specialization in a definite way to another, the specializations passed over form a simply extended manifoldness whose true character is that in it a continuous progress from a point is possible only on two sides, forwards or backwards. If one now supposes that this manifoldness in its turn passes over into another entirely different, and again in a definite way, viz., so that each point passes over into a definite point of the other, then all the specializations so obtained form a doubly extended manifoldness. In a similar manner one obtains a triply extended manifoldness, if one imagines a doubly extended one passing over in a definite way to another entirely different; and it is easy to see how this construction may be continued. If one regards the variable object instead of the determinable notion of it, this construction may be describe as a composition of a variability of n + 1 dimensions out of a variability of one dimension.”

We shall omit the discussion of the second question which is “the study of the measure relations of which such a manifoldness is capable, and of the conditions which suffice to determine them.” We shall see what our author has to say about the second and third important conception in his system. We shall quote briefly.

“In the extension of space construction to the infinitely great, we must distinguish between unboundedness and infinite extent; the former belongs to the extent relations; the latter to the measure relations. That space is an unbounded three-fold manifoldness is an assumption which is developed by every conception of the outer world; according to which every instant the region of real perception is completed, and the possible positions of a sought object are constructed, and which by these applications is forever confirming itself. The unboundedness of space possesses in this way a greater empirical certainty than any external experience, but its infinite extent by no means follows from this; on the other hand, if we assume independence of bodies from position, and therefore ascribe to space constant curvature, it must necessarily be finite, provided the curvature has ever so small a positive value.”

We have quoted Riemann at some length, perhaps beyond what the importance of the subject demands. But we have deemed this expedient to show that our criticism is not exaggerated. No other representation of his thought could so well show the point we wish to bring out as the words of the author himself. In addition, we give the words of this author as indicative of the school of thought of which he is the founder, and whose followers have been such perfect imitators in this regard.

Anyone who can put down what he thinks, or thinks he thinks, in such cumbrous and learned language, deserves immediately to be received into the bosom of the mystic class of pundits, far apart from the ordinary humdrum vulgar. He simply belongs. It is true we are using only a translation, but it is a translation made by a prominent mathematician and close follower of Riemann, W. K. Clifford, as it appeared in Nature, Vol. VIII, in 1873. It is such pedantic and flatulent language (pure macrology) that has drawn a veil of mystery around a certain class of modern scientists, who seem to use language only to cloak their thoughts, or lack of thoughts, except to the initiated, and give to those outside the mystic realm the impression that the secrets thus mumbo-jumboed must contain profound and weighty mysteries, before which they can only stand agape or bow down in humble wonder. The Orphic, Eleusinian or Mithraic mysteries, or even the Poro of the African tribes, have nothing on this.

It is certainly tickling to the nerve of the absurd (when it does not become annoying, as shams always annoy at times) to read such solemn utterances and scan the meagre content, or weigh, if you can feel the weight, the thought contained. But it is still more humorous to see those whom the world considers the educated, supposed to have an open sesame to all knowledge, willingly stand outside the charmed circle and speak with bated breath of the wondrous wisdom concealed (evidently from them) in the weighty words of the association of pundits, who are after all not oblivious of the psychological effect of mystery on those without the esoteric circle:

“While words of learned length and thundering sound
  Amazed the gazing rustics ranged around.”

It is the same old scene, just one step higher in the cultural scale; that is all.

We do not wish to be taken as criticizing the literary style of the person or persons concerned. Every man’s style is his own; it is the work of both nature and art, and it would be too much to expect either to have been lavish in this respect to a mathematician, whose gifts are otherwise, and who can scarcely be expected with the fleeting time allotted him, to become perfectly acquainted with an art which itself is as long as life. But there is this about it. Every man that thinks clearly can at least express what he thinks to others so they know what he is thinking, and how he arrives at his thought. That is the purpose of language; language was never supposed to be a rubbish heap that we are expected to pick over bit by bit in the hope of finding a pearl. The coryphaeus of Relativity has recently been criticized by a prominent churchman among other things for his lack of clarity, because, as it was urged most justly, truth is always clear. With this phase of the criticism we can heartily concur, for while the discovery of truth may be left to the few, its possession is the right of the many.

After this interlude let us return whence we left off. We had just quoted Riemann. Now to show we do not consider our general criticism unfair, let us try to clarify the atmosphere befogged by the multitude of our author’s words, and see what is left. This may take a certain amount of space, but it is part of our undertaking to pronounce judgment on the basis of non-Euclidean Geometry. Let us begin at the beginning.

His first sentence is correct; geometry does assume, as he says, the notion of space and the first principles of construction in it. The same thing, however, cannot be said of the second. While it is true that the definitions of geometry are merely nominal, the true determinations do not appear in the axioms. The axioms are not geometric determinations at all, but are much broader. Riemann is thinking of postulates which are quite different. Now this is something which should not escape anyone who undertakes to write on geometry. An acquaintance with the first page of Euclid would be sufficient to avoid it. The Elements make a plain distinction between definitions, postulates, and axioms. Even before that, Aristotle had completely explained the difference. Now one is at liberty to dispute Aristotle’s and Euclid’s method, if one is able to bring forward a reason for doing so. To ignore it and simply pass over it is quite a different thing.

We are next told that the relation of these assumptions remains consequently in darkness. Notice the “consequently”; where is the consequence? Can anyone construct the argument of which this consequence is the conclusion? But listen again; because these determinations are in the form of axioms, we cannot perceive, whether their connection is necessary, or how far it is necessary, or, a priori, whether it is possible. The argument is on a par with the conclusion itself. What connections or relations are spoken of? Their connections with one another, with reality, or with the science of geometry? Just a statement in the air that can mean anything, and means nothing.

But in any case the axioms, which now become assumptions, are in darkness, and all the mathematicians from Euclid to Legendre could not dispel this darkness. Now the only darkness from Euclid to Legendre, if we omit the darkness poured out in floods by the neo-Geometers, was that covering the fifth postulate, if we except a few other objections, as that of Savile against the theory of proportion and Gauss on the theory of the plane.

But now we have a statement concerning a kind of universal darkness covering the relations or connections of the axioms, whatever that is, and this darkness is going to be dispelled by Riemann in working the general notion of multiply-extended-magnitudes. Let us then see him at work. We shall see further examples of brilliant thinking and still more brilliant reasoning sufficient to illuminate all the darkness.

First, he sets himself the task of constructing a notion of a multiply-extended-magnitude out of a general notion of magnitude. Well and good. But how? It will follow from this (notice thc word follow) that a multiply-extended-magnitude is capable of different measure relations, and consequently (notice again the consequently) space is only a particular case of a triply-extended-magnitude. In order to see the brilliance, note the reasoning, or what passes for it, the words “follow” and “consequently” that make the propositions at least look like conclusions. Here it is as it goes. Because he has set himself the task of constructing a notion of multiply-extended-magnitudes out of the general notion of magnitude, the multiply-extended-magnitudes are capable of different measure relations, and space itself becomes only a particular case of a triply-extended-magnitude. I wonder what would have happened if he had not set himself the task. Space would be then as we know it.

But there is still another consequence. That task was certainly fruitful in consequences. It follows as a necessary consequence that the propositions of geometry cannot be derived from general notions of magnitude, but the properties which distinguish space from other triply-extended-magnitudes are to be deduced from experience. That would certainly be a wonderful kind of deduction. Now anyone that can see the logical nexus of all these conclusions with the original task should certainly become an interpreter for Riemann. Where is the middle term connecting all these conclusions? As a matter of fact they are merely a series of statements unconnected logically, and the grammatical signs of logical connection are so much dust thrown into the eyes of readers. We shall analyze the thought objectively when we come to where he actually constructs this notion, and the brilliancy of all these conclusions will receive even greater lustre.

From all these conclusions, and we have had nothing else, arises a problem, how, we are not told; we are merely told it arises; that ought to be enough for us. But here is the problem. We must find the simplest matters-of-fact by which we may determine the measure relations of space. This is certainly a brilliant conception. Riemann has a whole series of systems of matters-of-fact. He certainly has been richly endowed. Most of us have to be satisfied with one. But the brilliancy increases. These matters-of-fact are of empirical certainty; they are hypotheses. How a matter-of-fact can be a matter of empirical certainty and an hypothesis at the same time he does not explain; but that is nothing; he is shedding light in Euclid’s darkness at any rate. Words mean nothing, or rather one way to shed light in obscurity is always to be sure never to use a word in its accepted sense, and never use an ordinary word for anything if you can discover a more mysterious one. We shall find plenty of examples. Now the upshot of the whole reasoning so far is that it leaves open to investigation whether the geometry of Euclid applies to the infinitely great and infinitely small. So far for the wonderful deductions drawn from this (evidently) more wonderful notion of a multiply-extended-magnitude. Let us watch the operation of its creation. But first he wishes to prepare us for the full wonder and mysteriousness of it. He shows us first how to get these notions.

The first wonder is that magnitude notions are only possible when there is an antecedent general notion which admits of different specializations. This is certainly either brilliant or profound, but if it were profound, it might be dark. There is really so much brilliancy in it that we can never hope to bring it all out. The first glaring thing is that we cannot have magnitude notions until we have some more general notion. What is it? We are not told. Perhaps it is not in the dictionary. All we are given to know about it is that it must admit different specializations (bestimmungsweisen). But at least we have examples; the point for the continuum, and the unit, or what Riemann calls the element, for the multitude. Now it is true that the unit is the bestimmungsweise, as I take it, the particular thing on which my idea of multitude is based, because I can arrive at multitude only through knowing unity. But is this true in the other case? Do we know the point before we know the continuum, or is the point the result of analysis of the continuum? It is evidently the latter, and therefore knowledge of magnitude is rather the bestimmungsweise of the point. Since this is precisely the essential point in Riemann’s system it is very important. As we shall see, his whole system is built on this conception. It thus becomes necessary to consider it more fully.

With all the beating around the bush, inaccurate statements, inconsequent reasoning, and improper use of words, of which there is a plethora, we arrive here at what is the germ of the Riemannian system, and the thought that doubtless he was struggling to express and justify by all the vicious reasoning that precedes. It does not seem to me that Clifford has given us the real meaning of the bestimmungsweise of Riemann. It seems that the latter wishes to point out that particular thing in something else that is, as it were, an antecedent and more general notion, which in the present case admits of some particular determination which will lead us to the notion of magnitude. This something is then that which determines our notion of magnitude, and our idea of magnitude is developed from our notion of this thing.

Now first with regard to the antecedent notion. There can be no antecedent notion unless it is either a generic or a transcendental notion. As there is no question here of transcendental notions (transcendental is here taken in the metaphysical sense), the antecedent notion must be generic. Now a generic notion becomes specific by the addition of a difference, or what is called the specific difference. Thus we obtain the notion of man when we add to the generic concept animal the difference reasonable. Now has magnitude such a genus? It has; and it is quantity. The differences of quantity are discrete and continuous. Continuous quantity is magnitude, and discrete quantity is multitude. But is this what is meant by Riemann? This is evidently not his bestimmungsweise for arriving at magnitude. Neither the point nor the unit are determinations of quantity in this sense.

Riemann is completely wrong, then, when he makes notions of quantity come from any antecedent general notion. The notion of quantity is one of the ten categories of Aristotle and is itself a first or ultimate genus. There is nothing beyond it but the transcendental notion of being. Accordingly there is no such antecedent generic notion; and there is no word for it in the dictionary either of a cultivated or any other language. Riemann therefore wisely, or shall I say cannily, omits trying to define it. He simply leaves it to the imagination to persuade us there must be such a thing. It is pure suggestion.

His argument then comes down to the bald statement that as the unit is the bestimmungsweise of the multitude, so the point is of magnitude. Let us use plain language and omit all fantastic expressions of n-ply magnitudes, manifoldnesses, etc. Now, as we have said, the unit is the basis of our knowledge of multitude in a certain sense, i.e., that of counting. Mathematically the unit is the base of multitude. But it is by no means so as far as our knowledge is concerned. We could not know multitude, or discrete quantity, through unity, or what Riemann calls the element. Multitude is quantity; quantity which can be numbered, because it is divisible, and actually divided or discrete. The knowledge of multitude is therefore prior to the knowledge of the unit of the multitude. We do not know the unit as unit unless we know it as a member of a class. A potato, an apple, an egg, a football, are not units unless they first fall under a class, such as things or certain kinds of objects. Riemann had a perception of this truth when he said of discrete quantity that, “any things being given, it is always possible to find a notion in which they are included.” In other words, the unit as unit is already included in a more general notion. But he immediately forgot what this meant. The given things are units only when we first have the including notion of which they are integral or distributive parts. The knowledge of discrete quantity does not come from what Riemann calls the elements of the manifoldness, but contrariwise.

But what of the point as a bestimmungsweise of continuous quantity, or of a continuous manifoldness? Riemann’s analogy is entirely false. A point does not bear the same relation to continuous quantity that the unit (Riemann’s element) does to discrete quantity. Discrete quantity is divisible into units, but continuous quantity is not divisible into points. There is then no parity.

The point is not quantity, but is reduced to quantity only generically, i.e., to quantity as a limit of quantity. It is the limit of a line having quantity. The line as quantity is divisible, but not into points. It is divisible into parts of lines which are continuous with one another. But the line or any of its parts has an infinite number of points, i.e., points are not an element in the quantity of the line at all. We can only make the point determine quantity in one way, that is by motion. But motion is unintelligible without an antecedent perception of quantity, or extension, and thus a point in motion necessarily supposes an antecedent concept of continuous quantity. It cannot therefore be a determinant of quantity in our cognition, for the determination is the other way around.

The whole concept of Riemann is therefore absolutely and completely false, and there is no basis in fact or in logic for his treatment of the question. He is handling a matter which he is not equipped to treat. His knowledge of abstract quantity is on a par with his reasoning. Most likely his jumbled treatment of the question comes from this incomplete and hazy knowledge. He seemed to be aware himself of his incompetence, for before developing his notion of an n-ply extended magnitude, he says: “In proceeding to attempt the solution of the first of these problems, the development of the notion of an n-ply extended magnitude, I think I may the more claim indulgent criticism in that I am not practised in such undertakings of a philosophical nature where the difficulty lies more in the notions themselves than in the construction.”

It is not merely a question of not being practised, but of complete incompetence. He is not entitled to indulgent criticism, for he had no business meddling with what he knew nothing about. A prudent, sensible man would not get beyond his depth and would stick to what he was competent to handle. His lack of knowledge and ability to cope with the matter he undertook to illustrate in such a way that he was going to do away with what he terms the darkness of two thousand years, has been the cause of introducing the grossest errors into science. Mathematicians, who are not philosophers as well, should learn to stick to their lasts.

But let us come to what he calls the construction of this n-ply manifoldness, where the difficulty is not in the notions themselves, and see if he fares any better. If anyone wishes to read a passage with a lot of buzz and sound and very little meaning, let him look over the passage where he does this. It is the passage beginning, “If in the case of a notion whose specialization, etc.” It would almost deserve repetition here as a sample of the style we described above. It is pure piffle aping depth by covering itself with obscurity. As it is, we shall simply translate it into everyman’s language. The whole thing simply amounts to what is plainly conveyed in any schoolboy textbook of geometry. A point in motion generates a line, or a magnitude of one dimension; a line in motion at right angles to itself generates a plane, or a magnitude of two dimensions; similarly, a plane in motion at right angles to itself generates a volume, or a magnitude of three dimensions. All true enough and put in plain language would convey no knowledge to a schoolboy, or to a dock-walloper for that matter. But the real masterpiece is to follow—the real quintessence of Riemannian fatuousness, the flaming symbol of all that Riemannians honor, the characteristic method they all follow. We shall give it in his own words: “And it is easy to see how this construction may be continued.” If that phrase can be beaten by anyone but a metageometer or a Relativist, the victor should deserve a prize of immortal laurel. Even the metageometers can not beat it, they can only imitate it closely.

Here we have explained for us in ponderous words what for no schoolboy would need explaining; and then just as Riemann comes to the unexplainable part, or at least, if his theory were right and there are really more than three dimensions, the point that would require to be explained, we are simply put off by his telling us it would be easy to continue. So much easier apparently even than developing three dimensions, for he develops the three for us as if we did not know how, and then tells us that four or more are easy. He stops there. He has thus proved a multiply-extended-manifold. This was the construction that did not have any philosophical difficulty in which he was not exercised. Most of us could pass up his explanation of the three dimensions; our difficulty is where the fourth and fifth and n-number come in. Either we are uncommonly stupid, or Riemann uncommonly fatuous.

This is the rubbish out of which Relativists have constructed their system. This writing of Riemann is the Bible of modern neo-Geometers; it is one of the canonical works of Relativity. Its author is everywhere acclaimed as a young miracle worker in the realm of mathematics, and the founder of a new school of science and philosophy. Einstein speaks of his “depth of thought,” and calls him “a mathematician far in advance of his time.”⁸ Great and original thinkers-do not usually think the manner of thought we have been analyzing.

Our criticism has already taken us beyond what we intended. There is scarcely a sentence that could not be picked to pieces. But just another observation about this n-ply manifoldness and then we shall hasten on to the other two foundation stones of Riemann’s system. We have seen Riemann claim that it is possible to construct different multiply-extended-magnitudes out of a general notion of magnitude. This of course is meaningless, since there is only one multiply-extended-magnitude and that is volume, or triple extension; and, furthermore, there is no generic idea of magnitude but that of quantity. Space is not then a particular case of a triply extended magnitude but is the only triply extended magnitude we know, if we understand by space real space. Geometry is not then to be differentiated from the science of other triply extended magnitudes (that do not exist) by one out of any number of systems of matters of fact. Riemann recognizes this in a way. He tells us in one place that “so few and far between are the occasions for forming notions whose specializations make up a continuous manifold-
ness, that the only simple notions whose specializations form a multiply-extended-magnitude are the positions of perceived objects and colors.” The same magnitude combination of space and color is also put forward by Helmholtz. In other words spatial relations and colors are the alleged examples of triple extension or three-ply extended manifoldnesses. Now color is not a magnitude at all; at most it is only accidentally connected with magnitude, inasmuch as it is only an extended surface that can receive color. But it is the surface that has extension, not the color; the color has only the dimensions of the surface, none of its own. It is then the surface that has magnitude, not the color. Multiply or n-ply extended dimensions do not belong to color as such. Who ever heard of or imagined, for instance, a one-dimensional color, or a three-dimensional color, to say nothing of a color of a greater number of dimensions or an n-ply extended color.

This is perhaps not quite the sense of Riemann. He has doubtless the same idea as Helmholtz, who also classifies colors along with space as manifesting a triply extended dimension. But his notion is that all the other colors are made up of the three primary colors, and in this sense are three-dimensional. But the argument returns to the same thing. Unless each of the primary colors is unidimensional, no combination of them can be three-dimensional. And if the primary colors are dimensional in extent, it can be understood in no other sense than space extension. This is of course not the concept of either Riemann or Helmholtz, because neither ever stopped to analyze his thought. They simply found a combination of three, and immediately jumped to the conclusion that they had not only an analogy for three-dimensional space, but even a particular case of a more generic triply extended manifold, of which space itself is the other species. But here the very idea of dimension is destroyed. It no longer means anything more than element. A color is a combination of three elementary colors; space is a combination of three elements; but there is no common genus that can be defined, to which we may attach the specific differences that distinguish compound colors from space. If this method of arguing is correct, then water is two-dimensional, because it is composed of the elements, hydrogen and oxygen.

Confusing color and space magnitude by including them under a common category ponderously called multiply-extended-magnitude is only another example of the confusion of thought that marks all the work of the neo-Geometers. The only triply extended manifoldness, or to use the commoner combination of terms, three-dimensional magnitudes, is that extension of bodies represented by our idea of space. Yet Riemann asserts (again it is pure assertion for which there is neither proof nor illustration) that many other three-dimensional magnitudes are conceivable. He attributes their creation and development to the higher mathematics. We have already seen how he constructed them. They are not matters of fact or experience but a pure arbitrary creation without even a definable concept back of them.

Now we come to the other two important discoveries of Riemann as incorporated in his Hypothesis on the Foundations of Geometry. We have already seen how he expressed it in the paragraph beginning, “In the extension of space construction, etc.” We shall take merely what we consider the root idea of the paragraph and put it into straightforward language.

Riemann is trying to distinguish between unboundedness and infinity with regard to space. The former he connects with what he calls the extent relations, the latter with the measure relations. To begin with, Riemann introduces two terms, extent relations and measure relations. Again we have confusion in both thought and language. What is meant by extent relations? by measure relations? what is the distinction between them? Extent is quantity and not relation; it therefore belongs to an entirely different logical category. It is true that measure is a relation; the unit of measure is related to the thing measured, or to the extent; but to what is the extent related? And what is an extent relation? But a proper knowledge of the categories would mean some knowledge of logic, and this would be altogether too much to expect from a non-Euclidean.

The whole paragraph is awry with twisted conceptions and warped logic. There is first the distinction between unbounded and infinite. Both mean the same thing, if we consider the form of the word—infinite, no limit; unbounded, no limit. It is true that we might distinguish between infinite and unbounded in metaphysical use; for instance, we should say that God is infinite, but not that He is unbounded. But with regard to space they have the same signification, extent without limit; that is, if we take the words in their actual signification.

That is something of course that Riemann seldom does. It does not help profundity to make things too plain. Still we could grant him the privilege of making an arbitrary distinction between them, provided he defined his distinction and then adhered to it. But he gives no definition, and the only distinction he points out is that the one concerns the extent relations, the other the measure relations. What the distinction is you are left to find out for yourself. But after all extent and measure have one and the same kind of determination, since it is the extent that is measured, and if the extent is infinite, so will be what he calls the measure relations; likewise, if the extent is limited, or finite, so will be the measure relation. There is then no possible way from this distinction of making distinctive definitions for each. So they not only mean the same thing originally, but even after the distinction he makes to show that they are distinct. Again, nothing but an argument of words—mere husks, no kernel.

But let us continue to probe what Einstein calls “depth of thought.” He makes the assumption (the assumption, mind you) that space is unbounded, and then goes on to state that this assumption is developed by our every conception of the outer world, according to which the region of real perception is completed. This is certainly a real psychological puzzle. If it means anything, it means that we start with an assumption that space is unbounded (to which we have no meaning attributed), then this assumption is developed by a conception (whatever that is), and then the region of real perception is completed according to this conception. In other words, our perception or experience of things about us is completed by a conception, which itself develops an assumption that space is unbounded. Let who can, explain the epistemology. Ordinary persons know that space is three-dimensional because their experience tells them so, and it is from this experience they form the idea or concept of space, and as far as unboundedness is concerned, space is for them practically unbounded, because they have never been able to measure to the end of it. We do not need to assume it; we have never been able to reach its limit. But let us leave aside epistemology, lest we be carried too far. Whatever Riemann’s unboundedness assumption is, it is an extent relation that is connected with our experience of space, since it is developed by our conception of the outer world which completes our everyday perception. Unboundedness thence turns out to be a measure relation, since our experience tells us nothing more about infinite extent than that our measuring powers cannot attain to the end of space.

Then he tells us that the infinite extent of space does not follow from this. Nothing could be truer; that is, if we take unbounded in the sense we have given, meaning unbounded relatively to us; for then unboundedness becomes entirely subjective, and thus it is something that does not belong to space at all, but to ourselves. It is evident then that this subjective unboundedness says nothing one way or the other of actual infinity; just as our inability to devour a whole ox at a sitting, or drink a tun of wine, makes them infinite, although they are unbounded as far as our appetite is concerned, because we cannot reach the limit.

But here is precisely where Riemann’s ideas are twisted. He wants a finite universe, and yet assumes that it is unbounded, which, if it belongs, as he says, to the extent relations, must mean unbounded or infinite in extent. His experience shows him it is unbounded only in the measure relation. So he assumes in reality an infinite universe, when he wants a finite universe. After this jumble, we have immediately an “on the other hand.” We are therefore leaving this hand as quite complete without more ado. But let us come to “the other hand.” Let us jump the fence and see what is there. Here we assume again. Quite an easy job. It saves thinking and arguing and proving. If we assume independence of bodies from position then we ascribe to space (another assumption or postulate) a constant curvature, and space must be finite. There is a grammatical connection between the if-clause and what follows, but where is the logical connection? In what way does ascribing curvature to space depend on assuming independence of bodies from position? But let us simply take the fact, never mind the connection. Riemann ascribes curvature to space, then space must be finite. Wonderful! So if he ascribed feet to space, space could walk; or a mouth, it would talk. Here we are suddenly dropped into the profoundest profundity of the whole system, dropped there without preparation or approach. We have curvature ascribed to space, not to paths in space, nor to lines in space, nor to surfaces in space, but to three-dimensional space itself. To this space we are told to ascribe a constant curvature, a thing no one can either perceive or conceive, so that there is not even a possible confirmation this time from either perception or conception. But if we ascribe it, the thing is done. What I admire in Riemann is his boldness. No intellectual difficulties ever stand in his way. He scorns proof. He scorns reason. Just ascribe whatever you will.

This developing of curvature is on a par with his development of an n-ply extended magnitude. The only development this got was to tell us it was easy. Here there is neither explanation of what curved space is, nor why it should be; it is simply ascribed. It does not take much effort to create such deep thought. As far as concerns these great discoveries, we are only told that for the one it is easy, for the other we simply ascribe it.

Great, certainly, is the power of assumption. But it is certainly a rather strong assumption for men who are so finicky of intellectual proof that they simply can’t stand Euclid’s assumption concerning the meeting of converging straight lines, and revolt because, forsooth, he cannot furnish an argument showing its metaphysical necessity. Evidently, they wish to cure themselves of their queasiness by extra strong doses of assumption. And the others, for whom Euclid’s simple postulate was as a whiplash on their sensitive intellectual skins, they are going to whip with scorpions and make them like it.

Such is the second foundation stone of Riemann’s mathematical temple upon which the rest is built. We have in this one of the two great discoveries of Riemann, curvature and finiteness of space. In postulating curvature of space Riemann takes leave of Euclidean geometry which is of straight line measurement. The school that ranks itself under his leadership tells us that Riemann challenged the infinity of the straight line, that he found this as fit a subject of dispute as the parallel postulate of Euclid. This is not correct. What he challenged was the straightness of the straight line. He first postulated curvature, and then only deduced from this curvature the finiteness of space, and hence the finiteness of the line that is the measure of the finiteness of space.

But before we leave this point, we wish to call attention to the methods of Riemannian Geometers in comparing their Geometry with that of Euclid. We are continually told that certain propositions will not hold in Riemann’s hypothesis, because these propositions sup- pose the infinity of the straight line. This has deceived even careful geometers. For instance, Heath, in a note on Euclid, I, 16, says: “As is well known, this proposition is not universally true under the Riemann hypothesis of a space endless in extent but not infinite in size. On this hypothesis a straight line is a ‘closed series’ and returns on itself; and two straight lines which have one point of intersection, have another point of intersection also, which bisects the whole length of the straight line measured from the first point on it to the same point again.”⁹ By dint of dinning this conception into the ears of the mathematical public, this public has ended by swallowing it whole.

But the question arises, is the statement true. Decidedly not. What is of consequence is not that the line is not infinite but that the line is not straight. This is clear from Riemann’s postulate of curvature. Moreover, the proof of Euclid’s proposition depends not in the slightest measure on the infinity or non-infinity of the line; it would hold just the same whether it were the one or the other. But what does matter is the straightness of the line. If the lines of space are all curved, none of Euclid’s propositions that require the angles of a triangle to be equal to two right angles for their proof to hold, will be valid; but if the lines of space are straight, the proofs hold, whether space is infinite or not. The only case where they would not hold is when finiteness of space is necessarily connected with curvature of space, which is quite different. If Riemann wanted this argument to follow he should have postulated finite space and then concluded to its curvature. But this would have been too absurd even for Riemann. It took his disciples to go that far. If we grant the fact that space is curved it will make it finite, but the fact that it is finite will not make it curved.

The statement is, therefore, pure hocus pocus, a mere throwing of dust into the eyes of readers. It is this kind of treatment and not the mere absurdity of the theory that is obnoxious and nettles. The whole question is settled by every writer that touches upon it by the repetition of the cabalistic words, “It does not hold in the hypothesis of Riemann because he rejects the infinite line.” There is no attempt made to prove or even explain it; the sense is simply imposed. And in fact the charm has worked; all were hypnotized into accepting it as gospel truth. The spirit of criticism that met Euclid’s slightest slip, fell into complete coma at the sound of this non-Euclidean incantation.

The procedure is not intellectually honest. If we were told plump and plain that non-Euclideans would not accept Euclid’s proof because they did not accept Euclid’s straight line, we might demur, but we could not complain of trickery and fraud. But that would be too clear-cut an issue. The ordinary person’s idea of straightness is too definite and too deep set to be thus easily disposed of; but it is different with infinity. He does not know much about it to begin with; there is always something mysterious in it; and when the supposedly more learned tell him something is connected with it, he will not question it so readily. The objection of non-Euclideans to Euclid’s proof of I, 16 is then nothing but artful dodging and unabashed chicanery. It resembles more the sharp practice of an unscrupulous attorney than the truth seeking of a scientist.

The next point is the question of fact. Riemann supposes that everyone else before his time considered space actually infinite, and he discovered its finiteness. At least the idea struck him (we shall show how it probably occurred) and he assumed it to be true. The complete answer to this is that his supposition is simply not true. No metaphysician of any account in the traditional philosophy of the ages, and none of the older mathematicians, ever held the actual infinity of space, or what amounts to the same, the infinity of the universe; for it is to the universe and not to space as such that the measure relations belong. Certainly Euclid would have been greatly surprised to be told that his proofs were objected to because he held the infinity of the straight line, which is exactly what he did not hold, for he did not admit the idea of the infinite in his geometry. Nor did Aristotle before him. Both held lines and space as infinite or unbounded in the sense that they are potentially infinite, the real sense of unbounded in the argument of Riemann, that is, we cannot put a limit to them. For in spatial relations, wherever we put a limit, we can still imagine space extending beyond it.

This is a matter of common knowledge that anyone at all acquainted with the subject should have known. Even if Riemann had the slightest acquaintance with the history of the question he was undertaking to solve, he would have found Proclus adducing Aristotle’s very argument by which the latter proves the finiteness of the universe, as his own proof of the parallel theory. So this is not a discovery of Riemann at all. It was common knowledge universally held for ages. It was Newton with his peculiar metaphysico-theological notions of infinity and space that first introduced the idea of infinite space into science. Just as he confounded time with the eternity of God, so he confounded space with God’s immensity, and therefore turned the unbounded conceptual space of the mind into a real infinite space which represented God’s ubiquity. And from this peculiar theology he introduced the notion into mechanics.

The next point is the peculiar notion that Riemann substitutes for the infinite space of Newton. He wants to have infinity and not have it at the same time. Therefore, he invents that grotesque mental distortion, a finite yet unbounded space. It is only when we get here that we can finally give a sense to unbounded as he uses it. But first let us look at the jugglery by which he arrives at it. First, we have a sense of unbounded put before our eyes which is the equivalent of Euclid’s and Aristotle’s potential infinite, that is, so great an extent that we are unable to put a limit to it in any direction. Then he hops to another sense of unbounded, the unboundedness of a path around a circle or a sphere. The two senses of unbounded are in no way connected, either logically, physically, or metaphysically. The first is that space is unbounded because it is not measurable by us on account of its vastness. The second means that a line or path returns on itself and thus has no end or limit. This is a beautiful example of equivocation, if one should wish to illustrate this fault in speech for a class in logic. The second is certainly a wonderful notion of unbounded. A cat chasing its tail is following such a path finite but unbounded. This sense of unbounded gives us an extraordinary way to cozen ourselves. Anything may be finite but unbounded in this sense so long as we can keep measuring round and round it, and never come to a limit that will stop us. If the Zeppelin that made the tour of the world had acted on such geometry, it would have to go on forever, since it would never arrive at the end. Jockeys and charioteers from time immemorial lost a great opportunity when they did not discover that they were following a finite but unbounded path. It would, however, have taken an agile steeplechaser to jump from that unboundedness to the other sense of the empiric unboundedness of the universe.

Attributing such a sense to a word and expecting others to swallow it is merely trying to mislead them with a rather shallow hoax. It certainly is not much of a tribute to their intelligence, or proof of the author’s own. It becomes really amusing to see Riemann’s follower, Einstein, later explaining this. He illustrates it in this wise. If we take a number of discs and spread them out on the surface of a sphere, the number that will fit on the sphere will be limited, and the sphere is thus shown to be finite. But if we take only one disc we can push it round and round and never come to a stop. In this sense the sphere is unbounded. Can it be that any one would ever fall for such rubbish? One might thus so fool an animal. A tiger might walk round and round in a cage apparently ever seeking an exit, but could any human being ever run round and round the same path and persuade himself it was unbounded?—not outside of a maison de santé.

This whole playing with the question of the infinite and unbounded is pure scientific and mathematical burlesque, which, because of the stolid seriousness with which it is put forth, lacks the drollery of the consciously facetious.

We now come to the other great discovery of Riemann, and that is that space is curved. Of course there is only one sense in which this is at all understandable, and that is that the lines in space are curved lines. But this is not what the school of Riemann holds. Riemann abolishes the straight line, and introduces the curved line as the only existing one; hence space itself is supposed to be curved. All that was left to do was to work out this curve analytically, and we have Riemann’s differential geometry. But let us leave this question for the time being, since we shall have to revert to it when we are discussing the curvature of space as postulated by Relativity.

After we have seen the chief points of his system, let us see what originality there is in the vaunted discoveries of Riemann. Whether and how much the idea of n-ply-extended-manifoldnesses and the analytic working out of their elements are original with Riemann, we cannot say. At least he claims it. But it would seem that he got his root idea elsewhere.

It is of course impossible to state how much Riemann took here or took there. All we have is the internal evidence and the external facts. Evidently he saw some chance to build on the second hypothesis of Saccheri. To hand also was the new or higher geometry known as Projective Geometry, as represented in the works well known at the time, Geometrie der Lage of V. Staudt, and the Ausdehnungslehre of Grassmann (Von Staudt, Geometrie der Lage, 1847; Grassmann Ausdehnungslehre, 1844; Riemann, Habilitationsschrift, 1854). In these we have the germ of his whole system, and more than the germ.

In the first place his notion of the infinite line. The treatment of the line in Euclidean and projective geometry is quite different. A straight line in projective geometry is always regarded in its entirety, stretching away in both directions to infinity. Euclid on the other hand never admits anything but finite lines as well as other finite quantities. The second point of contact is in the idea of parallel lines. In projective geometry parallel lines are defined as lines that meet at infinity. A corollary from this is that all points in a line at an infinite distance may be considered as one single point. This ideal point is called the point at infinity in the line. We attain this point at infinity by moving a point in either direction from a fixed point, so that a line appears closed by this point, and we can thus speak as if we could move a point from one position to another in two ways, either around through the point at infinity, or through finite points only.

Here we have the whole system of Riemann, even to the idea of the straight line turning to a curve at infinity. All he had to do was to consider the ideal point of projective geometry as real, and his system was made. He had his curved lines. Then if the curved lines were real, the space of these curved lines was finite. His notion of finite and unbounded is but an interpretation in reality of this conception of projective geometry. The straight line was abolished by the same step, for it now became a closed line. With such closed lines the second hypothesis of Saccheri became possible and even real. For the sum of the interior angles of a triangle would be greater than two right angles. The whole originality of Riemann’s scheme consisted in believing real the ideal conception of projective geometry. It was not a great intellectual discovery, but a young man’s taking for real a mere notion. What was a façon de parler for Kepler and Gauss, and also for Desargues and the projective geometers, became stark reality for Riemann, and the trick was turned. His conception is complete. The analytical method of treating geometry came from this system, so that when Klein reintroduced projective into non-Euclidean Geometry he was only bringing into relationship what was already naturally related.

We have now finished in a cursory way our appreciation of the work of the authors of the different non-Euclidean Geometries. Many more blunders and misconstructions could be pointed out, but we think we have sufficiently emphasized the weakness of the argument for the new Geometry, to show that it rests on no reasonable basis. This has been shown in a two-fold way: first, by an absolute proof of the Euclidean postulate from whose lack of proof non-Euclidean Geometry arose, and whose demonstration destroys at once the possibility of the contradictory propositions on which metageometry is built; secondly, by showing that neither the principles, nor the methods, nor the reasoning employed in developing non-Euclidean Geometry, have any claim to be termed scientific, but carry in themselves sufficient evidence that the whole theory lacks basic principle and consistency.

To begin with, since the truth of Euclid’s postulate has been established, even to its metaphysical necessity, there is no ground on which non-Euclidean Geometry may stand. This position will have to be accepted for Hyperbolic or Lobatschewskian Geometry, since the angle of parallelism is proved to be a right angle. There may be more of a quibble on the part of Riemannian Geometers, for they do not accept the possibility of the straight line, and if there is no straight line, but only curves, then the sum of the angles of a triangle will exceed two right angles. With them then the choice of a geometry simply means this. The Riemannian Geometry denies the existence of the straight line and admits only a curve. Euclidean geometry holds the possibility both of straight and curved lines. Of course it is not possible for the science of geometry to disprove this contention of Riemann otherwise than by showing its lack of reason, since it is a fundamental principle of the science, and no science proves its own principles. It is a question for a treatise on the foundations of geometry, which has no place here.

But even in that case the non-Euclidean position is false, for if non-Euclidean Geometry is true, there is only a choice between the two kinds of lines. Geometry should be either Euclidean or Riemannian. It could not be both so far as formal geometry is concerned. Curved lines, if they exclude straight lines, must ipso facto exclude all geometry of straight lines. The claim that it is the infinity of the straight line that makes the difference is an evident quibble. What makes the angles greater or less than, or equal to two right angles, is precisely the fact that the curvature of the line is either positive, negative, or zero; and as either of these is true, the other two are false, and we have respectively Riemannian, Lobatschewskian, or Euclidean geometry. But here again the neo-Geometers are illogical; they wish to keep both. They wish to retain Euclidean geometry for all practical purposes and use the Riemannian only for near-infinite spaces. Riemann only wants to question the justice of Euclidean geometry for the infinitely great and the infinitely small, and all his followers acquiesce in this position. They never had either the courage or the logic of their convictions to undertake to substitute their Geometry for that of Euclid all along the line. Of course they evade this to some extent by doing away with formal geometry, claiming all formal geometry only as near geometry. But we shall have to consider this question separately.

Since there can be no geometric argument with Riemannian Geometry, because it denies the very basis of geometry itself as founded on the first two postulates, the argument is entirely indirect, and consists in showing that there is nothing to the so-called science built on the opposite principles. In the first place it has not even the merit of originality. Its founders were men who have scarcely shown themselves of the intellectual calibre which we should expect from founders of an entirely new system of science. Those who expect by a brush of the pen to do away with the accumulated science of the human race, and substitute something entirely new in its place, should at least show that they are great and original thinkers. The new Geometry can show nothing of this kind. It consists of odds and ends picked up everywhere, and strangely sewn into a motley and outlandish crazy-quilt. The non-Euclidean Geometers are ornamented with spoils collected on every side, and they would be left pretty bare, if only what is originally theirs was left them. The little cornicula of Horace will find herself completely at home in the company of the neo-Geometers:

“Si forte suas repetitum venerit olim
Grex avium plumas, moveat cornicula risum.”¹⁰

Since this is the state of the case with regard to the originality of metageometry, it is no wonder that it is the indigest hodge-podge that we found it. Its intellectual weight is on a par with its originality. In seeking their odds and ends, by the very nature of the case, they could choose only the refuse thrown aside as useless for the living body of science; and hence they could only adopt the scientific absurdities, or what somehow could be turned into an absurdity, and thus the result was a fundamentally and thoroughly absurd system.

The indirect argument against non-Euclidean Geometry is then the heaped up absurdities that compose it, and its complete lack of anything like lucidity or logical connection. This we have shown chiefly in our criticism of Riemann. But the same can be said of the rest of the new Geometers. There is also another point to be considered, and that is, the entire lack of all proof for the principles of non-Euclidean Geometry. It is not merely that the proofs brought forward do not hold. None are brought forward. We have seen examples in Riemann. For his famous n-ply extended manifold, the sum total of the proof is that we are told it is easy to construct such a manifold. For the famous curvature of space we are merely told to assume it. This is of course quite easy, since it dispenses with all proof. But this is not Riemann’s attitude exclusively; it is that of all his followers, including the Relativists. They prove nothing, assume everything; and in large part they have gotten away with it, which one may take, if one wishes, as a forceful commentary on the intelligentsia of our epoch.

The nearest thing to a proof that has ever been urged for non-Euclidean Geometry consists of two statements: first, that the body of non-Euclidean Geometry can be developed without contradicting anything in Euclidean geometry that does not depend on the parallel theory; second that non-Euclidean Geometry is itself non-contradictory. Both of these are pure negative arguments. Even if granted, they would never prove the validity of non-Euclidean Geometry. At most, they would do away with objections. But as non-Euclideans seem very proud and confident of these assertions, let us see how true they are. We shall find they are just as true as all the other “truths” of metageometry.

First, let us tackle the claim of the non-Euclidean Geometers that their Geometry has been proved consistent and non-contradictory. From this they conclude immediately to the impossibility of proving the parallel postulate, for if the parallel postulate were true, non-Euclidean Geometry should show a contradiction with Euclidean. We now know what is to be thought of the impossibility of proving the parallel postulate, and hence also of the reasons by which the metageometers have arrived at it. Yet this is supposed to have been proved by Cayley, Klein, Beltrami, Hoüel, and by Poincaré.

Here we must revert to the history of non-Euclidean Geometry. It began with the idea that a logical and consistent geometry could be built up on the same foundation as that of Euclid, but substituting instead of the parallel postulate one incompatible with it. Up to the time of Lobatschewsky and Bolyai it was supposed that such a construction would lead to a contradiction. This, for instance, was the attitude of Saccheri. The inventors of non-Euclidean Geometry merely adopted the opposite belief, and on this their followers still base their arguments for its validity. They hold now that not only is no contradiction apparent in their system, but that none can ever be found.

The non-Euclideans are, therefore, supposed to have built up a body of geometry as consistent as that of Euclid, and therefore as unassailable. Is this true? We may answer that it i9, and it isn’t. There is first a distinction to be made between plane geometry and solid geometry. In plane geometry the non-Euclideans become consistent by being inconsistent with themselves. They begin with straight lines, and postulate certain properties for them that are inconsistent with Euclid’s defined and demonstrated properties of the straight line, and then immediately forget all about it, and make their straight lines identical with Euclid’s plane curves. They begin by denying the parallel postulate, and as a consequence they deny that the sum of the angles of the triangle is equal to two right angles. All this is with regard to Euclid’s straight line. When they begin to analyze their Geometry their lines turn out curved, and are not straight at all. Actually then non-Euclidean plane Geometry is identical with plane Euclidean geometry of curves, convex or concave. This portion of non-Euclidean Geometry is only a part of Euclidean geometry returning under another name:

“Naturam expellas furca, tamen usque recurret,
Et mala perumpet furtim fastidia victrix.”¹¹

The denial of the postulate of Euclid and the hypothesis changing the values of the interior angles of a triangle, instead of giving us a higher and more universal geometry, a metageometry, merely comes to this, that it is restricted to a simple portion of Euclidean plane geometry, that of curves. When worked out that is all there is to it:

“Parturiunt montes, nascetur ridiculus mus.”¹²

The relation of the two non-Euclidean Geometries to the peculiar curvature demanded by each was recognized even before the formation of these Geometries. As early as Lambert and Taurinus this fact was known. Lambert ascertained that the third hypothesis of Saccheri was realized on an imaginary sphere, now called a pseudosphere; and that the other hypothesis had its realization on an ordinary sphere. The latter corresponds to the Geometry of Riemann, the former to that of Lobatschewsky and Bolyai. Taurinus also, though convinced of the sole validity of Euclidean geometry, was quite clear that the geometry which could be built on the second hypothesis of Saccheri, and which he called the logarithmic-spherical, would have its realization on the sphere. These conclusions were also worked out later, in a frankly non-Euclidean sense by Beltrami. Even Riemann, as we have already seen, rejected the straight line outright, and in its place simply calculated his curved space. In plane geometry, therefore, his Geometry is that of the pure curve or the geometry of the spherical surface in Euclidean geometry.

In the domain of plane geometry, then, there is no contradiction, because it is pure Euclidean geometry of curves; only it was trussed up in a false phraseology. Thus, the triangle of Lobatschewsky and Bolyai is one really formed by three concave arcs, and hence the sum of its angles is less than two right angles. The triangle of Riemann is composed of three convex arcs of circles, and thus the sum of the angles of his triangle is greater than two right angles. But none of them contradict the Euclidean proposition that the sum of the angles are equal to two right angles, for the triangle to which this proof is attached is composed of straight lines. Thus the non-Euclidean Geometers remain consistent by being inconsistent with themselves. They started out with the notion of contradicting a necessary property of Euclid’s parallel straight lines, and they do nothing of the kind; they simply play with Euclid’s curved lines, and leave his straight lines alone whether parallel or converging. The non-Euclidean statement that they have another Geometry parallel to Euclid’s, but outside of it, because founded on a postulate contradictory to Euclid’s parallel postulate, is pure humbug as far as plane geometry is concerned.

There is, however, a real absurdity even in this non-Euclidean Geometry, although there may be no fundamental contradiction so far as plane geometry is concerned. This particularly concerns the Hyperbolic Geometry of Lobatschewsky and Bolyai. It consists in calling their curves straight lines while making them act as curves, and at the same time in introducing real straight lines alongside of them which they make act as straight lines. Hence arises necessary contradiction and absurdity.

We saw a very clear example of this in the propositions we criticized from the non-Euclidean Geometry of Carslaw concerning the quadrilateral of Saccheri. We saw the perpendicular lines there behaving in a most unmannerly way. That was because lines that were geometrically and analytically curved lines, were forced to act as straight lines. The middle line was also made to function as a straight line, but it was a straight line. Hence the contradiction. The two perpendiculars of the quadrilateral forming its vertical sides were curved lines that were forced into the definition of straight lines. No wonder they rebelled.

If, instead of this hybrid monster that non-Euclidean Geometry has tried to create, we did what the construction called for, and made the vertical sides concave curves and the middle line straight, we should have had a figure that was correct,

and one from which the desired conclusion could be drawn, and concerning which the demonstration given would be valid. Then, instead of a quadrilateral, whose vertical angles were actually obtuse angles and called acute, we should have had right angles on the base line, if the tangents to the curves where they intersected the base line, were at right angles to the base line; and also if the curve from that out were negative or concave, the vertical angles would be necessarily acute, and it would constitute a quadrilateral having right angles at the base, and two acute angles at the vertices. It could also be divided by a perpendicular straight line into two quadrilaterals each having three right angles and one acute angle. There would have been no contradiction or absurdity in it. But we should have been in plain Euclidean geometry. All that non-Euclidean Geometry did here was to jumble the notions of two different lines into one. In the quadrilateral of Saccheri we have both, each with different properties, yet entirely indistinguishable.

Now this of course is impossible in any kind of logic. If there are to be new Geometries, they can only be built on new conceptions of the properties of space. If the Euclidean conception of space is true, then we shall have Euclidean geometry as the sole valid one. If Riemann’s conception of space is true, then we shall have Riemann’s Geometry. If the conception of space of Bolyai and Lobatschewsky be true, then all geometry would be hyperbolic. But we can not have two kinds of space at the same time; it must be one or the other. If straight lines exist at all, then there is no room for either Riemann or Lobatschewsky. Yet such is the lack of consistency, and, one might say, even ordinary, everyday common sense, that these neo-Geometers employ both kinds of lines, and mix both geometries, as if contradictories could coexist at the same time.

So far as plane geometry is concerned, therefore, there is no fundamental contradiction to Euclidean geometry, but only the absurdities that people with a lack of logic and a peculiar faculty for mixing up the meanings of terms, have put into its arrangement. On the other hand, when non-Euclideans pretend to have anything new in plane geometry, they are simply fooling themselves, and others. Similarly, when they base any claim to establish non-Euclidean principles on this plane geometry, they base their argument on a delusion.

But the question is quite different when we come to solid geometry. Then we no longer have curved lines and curved surfaces, 6ut curved space itself, which is such an utter absurdity that no human being, no matter how much his brain has been affected by non-Euclidean Geometry, can explain, construct, bring within experience, or even conceive, in any possible way. We can conceive and construct curved lines and curved surfaces, but what is a curve in three-dimensional space, apart from curved lines and curved surfaces?

Here is where the contradiction should be, because such a conception is completely and irreductively non-Euclidean. But how show a contradiction when we cannot conceive the term, cannot represent it by a diagram, cannot form any geometric propositions concerning it? The neo-Geometers are therefore safe in challenging anyone to show its inconsistency. The conception is too intangible and inconceivable. It is a mental as well as a real nonentity. It cannot have any contradictory except the general contradictory that all non-being has—being. It is just as possible a conception as that of square-circle. What is the contradictory of a square-circle? There is none; its contradictoriness is in the very conception. The same with curved space. We cannot show its inconsistency; we can only deny it. We can conceive all kinds of curved lines in space; but if every path, or every geodesic, to use the language of the new Geometers, were a curve, space would be still three-dimensional and Euclidean, i.e., straight in its measure relations. Even a Riemannian sphere would have to have a diameter.

The entire claim to consistency is then an empty boast, void of meaning. For what is consistent is pure Euclidean; what is not Euclidean is inconceivable. This is so far admitted by the non-Euclideans themselves, that they attempt no formal geometry of a curved space. Its entire treatment is analytical. But analysis is one thing, geometry another. As far as analysis is concerned, it says nothing one way or the other about the geometric question.

The analytic representation of geometric figures is entirely conventional, and nothing can come out of the analysis but what we put into it. If we express a real geometric figure analytically, we can then perform any mathematical operation with the terms, and if we remain true to the conventional meaning, the result will be true geometrically; but to assume that because we can represent one, two, or three dimensions by the symbols x, x², x³, in the same way xⁿ may be taken to represent any number of dimensions, is mere unintelligent trifling; and to contend that because we can represent analytically a curved line or a curved surface, we can represent analytically a curved space, is an argument only for a dotard. If we cannot put in curved space formally, we cannot take it out by analysis. Analysis cannot decide the question of geometry.