Chapter III: , that any metrical Geometry, which should endeavour to (2)

II. _In three dimensions_, the results go still more against Helmholtz. Assuming free mobility only _within a certain region_, we have to distinguish two cases: _Either_ free mobility holds, within that region, absolutely without exception, _i.e._ when one point is held fast, _every_ other point within the region can move freely over a surface: in this case the axiom of Monodromy is unnecessary, and the first three axioms suffice to define our group as that of Euclidean and non-Euclidean motions. _Or_ free mobility, within the specified region, holds only of every point _of general position_, while the points of a certain line, when one point is fixed, are only able to move on that line, not on a surface: when this is the case, other groups are possible, and can only be excluded by Helmholtz's fourth axiom.

Having now stated the purely mathematical results of Lie's investigations, we may return to philosophical considerations, by which Helmholtz's work was mainly motived. It becomes obvious, not only that exceptions within a certain region, but also that limitation to a certain region, of the axiom of Free Mobility, are philosophically quite impossible and inconceivable. How can a certain line, or a certain surface, form an impassable barrier in space, or have any mobility different in kind from that of all other lines or surfaces? The notion cannot, in philosophy, be permitted for a moment, since it destroys that most fundamental of all the axioms, the homogeneity of space. We not only may, therefore, but must take Helmholtz's axiom of Free Mobility in its very strictest sense; the axiom of Monodromy thus becomes mathematically, as well as philosophically, superfluous. This is, from a philosophical standpoint, the most important of Lie's results.

=46.= I have now come to the end of my history of Metageometry. It has not been my aim to give an exhaustive account of even the important works on the subject--in the third period, especially, the names of Poincaré, Pasch, Cremona, Veronese, and others who might be mentioned, would have cried shame upon me, had I had any such object. But I have tried to set forth, as clearly as I could, the principles at work in the various periods, the motives and results of successive theories. We have seen how the philosophical motive, at first predominant, has been gradually extruded by the purely mathematical and technical spirit of most recent Geometers. At first, to discredit the Transcendental Aesthetic seemed, to Metageometers, as important as to advance their science; but from the works of Cayley, Klein or Lie, no reader could gather that Kant had ever lived. We have also seen, however, that as the interest _in_ philosophy waned, the interest _for_ philosophy increased: as the mathematical results shook themselves free from philosophical controversies, they assumed gradually a stable form, from which further development, we may reasonably hope, will take the form of growth, rather than transformation. The same gradual development out of philosophy might, I believe, be traced in the infancy of most branches of mathematics; when philosophical motives cease to operate, this is, in general, a sign that the stage of uncertainty as to premisses is past, so that the future belongs entirely to mathematical technique. When this stable stage has been attained, it is time for Philosophy to borrow of Science, accepting its final premisses as those imposed by a real necessity of fact or logic.

=47.= Now in discussing the systems of Metageometry, we have found two kinds, radically distinct and subject to different axioms. The historically prior kind, which deals with metrical ideas, discusses, to begin with, the conditions of Free Mobility, which is essential to all measurement of space. It finds the analytical expression of these conditions in the existence of a space-constant, or constant measure of curvature, which is equivalent to the homogeneity of space. This is its first axiom.

Its second axiom states that space has a finite integral number of dimensions, _i.e._ in metrical terms, that the position of a point, relative to any other figure in space, is uniquely determined by a finite number of spatial magnitudes, called coordinates.

The third axiom of metrical Geometry may be called, to distinguish it from the corresponding projective axiom, the axiom of distance. There exists one relation, it says, between any two points, which can be preserved unaltered in a combined motion of both points, and which, in any motion of a system as one rigid body, is always unaltered. This relation we call distance.

The above statement of the three essential axioms of metrical Geometry is taken from Helmholtz as amended by Lie. Lie's own statement of the axioms, as quoted above, has been too much influenced by projective methods to give a historically correct rendering of the spirit of the second period; Helmholtz's statement, on the other hand, requires, as Lie has shewn, very considerable modifications. The above compromise may, therefore, I hope be taken as accepting Lie's corrections while retaining Helmholtz's spirit.

=48.= But metrical Geometry, though it is historically prior, is logically subsequent to projective Geometry. For projective Geometry deals directly with that qualitative likeness, which the judgment of quantitative comparison requires as its basis. Now the above three axioms of metrical Geometry, as we shall see in Chapter III. Section B, do not presuppose measurement, but are, on the contrary, the conditions presupposed by measurement. Without these axioms, which are common to all three spaces, measurement would be impossible; with them, so I shall contend, measurement is able, though only empirically, to decide approximately which of the three spaces is valid of our actual world. But if these three axioms themselves express, not results, but conditions, of measurement, must they not be equivalent to the statement of that qualitative likeness on which quantitative comparison depends? And if so, must we not expect to find the same axioms, though perhaps under a different form, in projective Geometry?

=49.= This expectation will not be disappointed. The above three axioms, as we shall see hereafter, are one and all philosophically equivalent to the homogeneity of space, and this in turn is equivalent to the axioms of projective Geometry. The axioms of projective Geometry, in fact, may be roughly stated thus:

I. Space is continuous and infinitely divisible; the zero of extension, resulting from infinite division, is called a Point. All points are qualitatively similar, and distinguished by the mere fact that they lie outside one another.

II. Any two points determine a unique figure, the straight line; two straight lines, like two points, are qualitatively similar, and distinguished by the mere fact that they are mutually external.

III. Three points not in one straight line determine a unique figure, the plane, and four points not in one plane determine a figure of three dimensions. This process may, so far as can be seen _à priori_, be continued, without in any way interfering with the possibility of projective Geometry, to five or to _n_ points. But projective Geometry requires, as an axiom, that the process should stop with some positive integral number of points, after which, any fresh point is contained in the figure determined by those already given. If the process stops with (_n_ + 1) points, our space is said to have _n_ dimensions.

These three axioms, it will be seen, are the equivalents of the three axioms of metrical Geometry[65], expressed without reference to quantity. We shall find them to be deducible, as before, from the homogeneity of space, or, more generally still, from the possibility of experiencing externality. They will therefore appear as _à priori_, as essential to the existence of any Geometry and to experience of an external world as such.

=50.= That some logical necessity is involved in these axioms might, I think, be inferred as probable, from their historical development alone. For the systems of Metageometry have not, in general, been set up as more likely to fit facts than the system of Euclid; with the exception of Zöllner, for example, I know of no one who has regarded the fourth dimension as required to explain phenomena. As regards the space-constant again, though a _small_ space-constant is regarded as empirically possible, it is not usually regarded as probable; and the finite space-constants, with which Metageometry is equally conversant, are not usually thought even possible, as explanations of empirical fact[66]. Thus the motive has been throughout not one of fact, but one of logic. Does not this give a strong presumption, that those axioms which are retained, are retained because they are logically indispensable? If this be so, the axioms common to Euclid and Metageometry will be _à priori_, while those peculiar to Euclid will be empirical. After a criticism of some differing theories of Geometry, I shall proceed, in Chapters III. and IV., to the proof and consequences of this thesis, which will form the remainder of the present work.

FOOTNOTES:

[5] V. Mémoires de l'Académie royale des Sciences de l'lnstitut de France, T. XII. 1833, for a full statement of his results, with references to former writings.

[6] This bolder method, it appears, had been suggested, nearly a century earlier, by an Italian, Saccheri. His work, which seems to have remained completely unknown until Beltrami rediscovered it in 1889, is called "Euclides ab omni naevo vindicatus, etc." Mediolani, 1733. (See Veronese, Grundzüge der Geometrie, German translation, Leipzig, 1894, p. 636.) His results included spherical as well as hyperbolic space; but they alarmed him to such an extent that he devoted the last half of his book to disproving them.

[7] Klein's first account of elliptic Geometry, as a result of Cayley's projective theory of distance, appeared in two articles entitled "Ueber die sogenannte Nicht-Euklidische Geometrie, I, II," Math. Annalen 4, 6 (1871-2). It was afterwards independently discovered by Newcomb, in an article entitled "Elementary Theorems relating to the geometry of a space of three dimensions, and of uniform positive curvature in the fourth dimension," Crelle's Journal für die reine und angewandte Mathematik, Vol. 83 (1877). For an account of the mathematical controversies concerning elliptic Geometry, see Klein's "Vorlesungen über Nicht-Euklidische Geometrie," Göttingen 1893, I. p. 284 ff. A bibliography of the relevant literature up to the year 1878 was given by Halsted in the American Journal of Mathematics, Vols. 1, 2.

[8] Veronese (op. cit. p. 638) denies the priority of Gauss in the invention of a non-Euclidean system, though he admits him to have been the first to regard the axiom of parallels as indemonstrable. His grounds for the former assertion seem scarcely adequate: on the evidence against it, see Klein, Nicht-Euklid, I. pp. 171-174.

[9] V. Briefwechsel mit Schumacher, Bd. II. p. 268.

[10] f. Helmholtz, Wiss. Abh. II. p. 611.

[11] Crelle's Journal, 1837.

[12] Theorie der Parallellinien, Berlin, 1840. Republished, Berlin, 1887. Translated by Halsted, Austin, Texas, U.S.A. 4th edition, 1892.

[13] Frischauf, Absolute Geometrie, nach Johann Bolyai, Leipzig, 1872. Halsted, The Science Absolute of Space, translated from the Latin, 4th edition, Austin, Texas, U.S.A. 1896.

[14] Both Lobatchewsky and Bolyai, as Veronese remarks, start rather from the point-pair than from distance. See Frischauf, Absolute Geometrie, Anhang.

[15] Compare Stallo, Concepts of Modern Physics, p. 248.

[16] Gesammelte Werke, pp. 255-268.

[17] On the history of this word, see Stallo, Concepts of Modern Physics, p. 258. It was used by Kant, and adapted by Herbart to almost the same meaning as it bears in Riemann. Herbart, however, also uses the word _Reihenform_ to express a similar idea. See Psychologie als Wissenschaft, I. § 100 and II. § 139, where Riemann's analogy with colours is also suggested.

[18] Compare Erdmann's "Grössenbegriff vom Raum."

[19] Compare Veronese, op. cit. p. 642: "Riemann ist in seiner Definition des Begriffs Grösse dunkel." See also Veronese's whole following criticism.

[20] Vorträge und Reden, Vol. II. p. 18.

[21] Cf. Klein, Nicht-Euklid, I. p. 160.

[22] Since we are considering the curvature at a point, we are only concerned with the first infinitesimal elements of the geodesics that start from such a point.

[23] Disquisitiones generales circa superficies curvas, Werke, Bd. IV. SS. 219-258, 1827.

[24] Nevertheless, the Geometries of different surfaces of equal curvature are liable to important differences. For example, the cylinder is a surface of zero curvature, but since its lines of curvature in one direction are finite, its Geometry coincides with that of the plane only for lengths smaller than the circumference of its generating circle (see Veronese, op. cit. p. 644). Two geodesics on a cylinder may meet in many points. For surfaces of zero curvature on which this is not possible, the identity with the plane may be allowed to stand. Otherwise, the identity extends only to the properties of figures not exceeding a certain size.

[25] For we may consider two different parts of the same surface as corresponding parts of different surfaces; the above proposition then shows that a figure can be reproduced in one part when it has been drawn in another, if the measures of curvature correspond in the two parts.

[26] Crelle, Vols, XIX., XX., 1839-40.

[27] In this formula, _u_, _v_ may be the lengths of lines, or the angles between lines, drawn on the surface, and having thus no necessary reference to a third dimension.

[28] In what follows, I have given rather Klein's exposition of Riemann, than Riemann's own account. The former is much clearer and fuller, and not substantially different in any way. V. Klein, Nicht-Euklid, I. pp. 206 ff.

[29] See §§ 69-73.

[30] Grundlagen der Geometrie, I. and II., Leipziger Berichte, 1890; v. end of present chapter, § 45.

[31] Nicht-Euklid, I. pp. 258-9.

[32] Giornale di Matematiche, Vol. VI., 1868. Translated into French by J. Hoüel in the "Annales Scientifiques de l'École Normale Supérieure," Vol. VI. 1869.

[33] Crelle's Journal, Vols. XIX. XX., 1839-40.

[34] Nicht-Euklid, I. p. 190.

[35] This article is more trigonometrical and analytical than the German book, and therefore makes the above interpretation peculiarly evident.

[36] Such surfaces are by no means particularly remote. One of them, for example, is formed by the revolution of the common Tractrix

x = asin φ, y = a(log tan φ/2 + cos φ).

[37] "Teoria fondamentale degli spazii di curvatura costanta," Annali di Matematica, II. Vol. 2, 1868-9. Also translated by J. Hoüel, _loc. cit._

[38] See Klein, Nicht-Euklid, I. p. 47 ff., and the references there given.

[39] See quotation below, from his British Association Address.

[40] Compare the opening sentence, due to Cayley, of Salmon's Higher Plane Curves.

[41] V. Nicht-Euklid, I. Chaps. I. and II.

[42] See p. 9 of Cayley's address to the Brit. Ass. 1883. Also a quotation from Klein in Erdmann's Axiome der Geometrie, p. 124 note.

[43] Nature, Vol. XLV. p. 407.

[44] Nicht-Euklid, I. p. 200.

[45] I.e. the equation _AB_ + _BC_ = _AC_, for three points in one straight line.

[46] The formula substituted by Klein for Cayley's inverse sine or cosine. The two are equivalent, but Klein's is mathematically much the more convenient.

[47] Elements of Projective Geometry, Second Edition, Oxford, 1893, Chap. IX.

[48] Chap. III. Section B.

[49] See Nicht-Euklid, I. p. 338 ff.

[50] See his Geometrie der Lage, § 8, Harmonische Gebilde.

[51] The anharmonic ratio of four numbers, _p_, _q_, _r_, _s_, is defined as

(p - q).(r - s) / (p - r).(q - s).

[52] _I.e._ as transformable into each other by a collineation. See Chap. III. Sec. A, § 110.

[53] See Chap. III. Sec. A.

[54] It follows from this, that the reduction of metrical to projective properties, even when, as in hyperbolic Geometry, the Absolute is real, is only apparent, and has a merely technical validity.

[55] Sir R. Ball does not regard his non-Euclidean content as a possible space (_v. op. cit._ p. 151). In this important point I disagree with his interpretation, holding such a content to be a space as possible, _à priori_, as Euclid's, and perhaps actually true within the margin due to errors of observation.

[56] See Nicht-Euklid, I. p. 97 ff. and p. 292 ff.

[57] Newcomb says (_loc. cit._ p. 293): "The system here set forth is founded on the following three postulates.

"1. I assume that space is triply extended, unbounded, without properties dependent either on position or direction, and possessing such planeness in its smallest parts that both the postulates of the Euclidean Geometry, and our common conceptions of the relations of the parts of space are true for every indefinitely small region in space.

"2. I assume that this space is affected with such curvature that a right line shall always return into itself at the end of a finite and real distance 2_D_ without losing, in any part of its course, that symmetry with respect to space on all sides of it which constitutes the fundamental property of our conception of it.

"3. I assume that if two right lines emanate from the same point, making the indefinitely small angle _a_ with each other, their distance apart at the distance _r_ from the point of intersection will be given by the equation

s = 2aD/π sin rπ/2D.

The right line thus has this property in common with the Euclidean right line that two such lines intersect only in a single point. It may be that the number of points in which two such lines can intersect admit of being determined from the laws of curvature, but not being able so to determine it, I assume as a postulate the fundamental property of the Euclidean right line."

It is plain that in the absence of the determination spoken of, the possibility of elliptic space is not established. It may be possible, for example, to prove that, in a space where there is a maximum to distance, there must be an infinite number of straight lines joining two points of maximum distance. In this event, elliptic space would become impossible.

[58] For an elucidation of this term, see Klein, Nicht-Euklid, I. p. 99 ff.

[59] Cf. p. 9 of Report: "My own view is that Euclid's twelfth axiom, in Playfair's form of it, does not need demonstration, but is part of our notion of space, of the physical space of our experience, but which is the representation lying at the bottom of all external experience."

[60] The exception to this axiom, in spherical space, presupposes metrical Geometry, and does not destroy the validity of the axiom for projective Geometry. See Chap. III. Sec. B, § 171.

[61] Mathematicians of Lie's school have a habit, at first somewhat confusing, of speaking of motions of space instead of motions of bodies, as though space as a whole could move. All that is meant is, of course, the equivalent motion of the coordinate axes, _i.e._ a change of axes in the usual elementary sense.

[62] "Ueber die Grundlagen der Geometrie," Leipziger Berichte, 1890. The problem of these two papers is really metrical, since it is concerned, not with collineations in general, but with motions. The problem, however, is dealt with by the projective method, motions being regarded as collineations which leave the Absolute unchanged. It seemed impossible, therefore, to discuss Lie's work, until some account had been given of the projective method.

[63] Lie's premisses, to be accurate, are the following:

Let

x{1} = f(x, y, z, a{1}, a{2}...)
x{2} = φ(x, y, z, a{1}, a{2}...)
x{3} = ψ(x, y, z, a{1}, a{2}...)

give an infinite family of real transformations of space, as to which we make the following hypotheses:

A. The functions f, φ, ψ, are _analytical_ functions of

x, y, z, a{1}, a{2}....

B. Two points x{1}y{1}z{1}, x{2}y{2}z{2} possess an invariant, _i.e._

Ω(x{1}, y{1}, z{1}, x{2}, y{2}, z{2}) =
Ω(x{1′}, y{1′}, z{1′}, x{2′}, y{2′}, z{2′})

where x{1′}..., x{2′}..., are the transformed coordinates of the two points.

C. Free Mobility: _i.e._, any point can be moved into any other position; when one point is fixed, any other point of general position can take up ∞^{2} positions; when two points are fixed, any other of general position can take up ∞^{1} positions; when three, no motion is possible--these limitations being results of the equations given by the invariant Ω.

[64] On this point, cf. Klein, Höhere Geometrie, Göttingen, 1893, II. pp. 225-244, especially pp. 230-1.

[65] Axiom II. of the metrical triad corresponds to Axiom III. of the projective, and _vice versâ_.

[66] Cf. Helmholtz, Wiss. Abh. Vol. II. p. 640, note: "Die Bearbeiter der Nicht-Euklidischen Geometrie (haben) deren objective Wahrheit nie behauptet."