CHAPTER XX

THE FUNDAMENTAL CONCEPTS, TIME AND SPACE

=2001.= Kant’s Doctrine of Time.

I. Time is not an empirical concept deduced from any experience, for neither co-existence nor succession would enter into our perception, if the representation of time were not given _a priori_. Only when this representation _a priori_ is given, can we imagine that certain things happen at the same time (simultaneously) or at different times (successively).

II. Time is a necessary representation on which all intuitions depend. We cannot take away time from phenomena in general, though we can well take away phenomena out of time. In time alone is reality of phenomena possible. All phenomena may vanish, but time itself (as the general condition of their possibility) cannot be done away with.

III. On this _a priori_ necessity depends also the possibility of apodictic principles of the relations of time, or of axioms of time in general. Time has one dimension only; different times are not simultaneous, but successive, while different spaces are never successive, but simultaneous. Such principles cannot be derived from experience, because experience could not impart to them absolute universality nor apodictic certainty....

IV. Time is not a discursive, or what is called a general concept, but a pure form of sensuous intuition. Different times are parts only of one and the same time....

V. To say that time is infinite means no more than that every definite quantity of time is possible only by limitations of one time which forms the foundation of all times. The original representation of time must therefore be given as unlimited. But when the parts themselves and every quantity of an object can be represented as determined by limitation only, the whole representation cannot be given by concepts (for in that case the partial representation comes first), but must be founded on immediate intuition.--KANT, I.

_Critique of Pure Reason [Max Müller] (New York, 1900), pp. 24-25._

=2002.= Kant’s Doctrine of Space.

I. Space is not an empirical concept which has been derived from external experience. For in order that certain sensations should be referred to something outside myself, i.e. to something in a different part of space from that where I am; again, in order that I may be able to represent them as side by side, that is, not only as different, but as in different places, the representation of space must already be there....

II. Space is a necessary representation _a priori_, forming the very foundation of all external intuitions. It is impossible to imagine that there should be no space, though one might very well imagine that there should be space without objects to fill it. Space is therefore regarded as a condition of the possibility of phenomena, not as a determination produced by them; it is a representation _a priori_ which necessarily precedes all external phenomena.

III. On this necessity of an _a priori_ representation of space rests the apodictic certainty of all geometrical principles, and the possibility of their construction _a priori_. For if the intuition of space were a concept gained _a posteriori_, borrowed from general external experience, the first principles of mathematical definition would be nothing but perceptions. They would be exposed to all the accidents of perception, and there being but one straight line between two points would not be a necessity, but only something taught in each case by experience. Whatever is derived from experience possesses a relative generality only, based on induction. We should therefore not be able to say more than that, so far as hitherto observed, no space has yet been found having more than three dimensions.

IV. Space is not a discursive or so-called general concept of the relations of things in general, but a pure intuition. For, first of all, we can imagine one space only, and if we speak of many spaces, we mean parts only of one and the same space. Nor can these parts be considered as antecedent to the one and all-embracing space and, as it were, its component parts out of which an aggregate is formed, but they can be thought of as existing within it only. Space is essentially one; its multiplicity, and therefore the general concept of spaces in general, arises entirely from limitations. Hence it follows that, with respect to space, an intuition _a priori_, which is not empirical, must form the foundation of all conceptions of space....

V. Space is represented as an infinite given quantity. Now it is quite true that every concept is to be thought as a representation, which is contained in an infinite number of different possible representations (as their common characteristic), and therefore comprehends them: but no concept, as such, can be thought as if it contained in itself an infinite number of representations. Nevertheless, space is so thought (for all parts of infinite space exist simultaneously). Consequently, the original representation of space is an _intuition a priori_, and not a concept.--KANT, I.

_Critique of Pure Reason [Max Müller] (New York, 1900), pp. 18-20 and Supplement 8._

=2003.=

_Schopenhauer’s Predicabilia a priori._[11]

OF TIME OF SPACE

1. There is but _one time_, all 1. There is but _one space_, different times are parts of all different spaces are it. parts of it.

2. Different times are not 2. Different spaces are not simultaneous but successive. successive but simultaneous.

3. Everything in time may be 3. Everything in space may be thought of as non-existent, thought of as non-existent, but not time. but not space.

4. Time has three divisions: 4. Space has three dimensions: past, present and future, height, breadth, and which form two directions length. with a point of indifference.

5. Time is infinitely 5. Space is infinitely divisible. divisible.

6. Time is homogeneous and a 6. Space is homogeneous and a continuum: i.e. no part is continuum: i.e. no part different from another, nor is different from another, separated by something nor separated by something which is not time. which is not space.

7. Time has no beginning nor 7. Space has no limits end, but all beginning and [Gränzen], but all limits end is in time. are in space.

8. Time makes counting 8. Space makes measurement possible. possible.

9. Rhythm exists only in time. 9. Symmetry exists only in space.

10. The laws of time are _a 10. The laws of space are _a priori_ conceptions. priori_ conceptions.

11. Time is perceptible _a 11. Space is immediately priori_, but only by a perceptible _a priori_. means of a line-image.

12. Time has no permanence but 12. Space never passes but is passes the moment it is permanent throughout present. all time.

13. Time never rests. 13. Space never moves.

14. Everything in time has 14. Everything in space has duration. position.

15. Time has no duration, but 15. Space has no motion, but all duration is in time; all motion is in space; time is the persistence of space is the change in what is permanent in position of that which contrast with its restless moves in contrast to its course. imperturbable rest.

16. Motion is only possible in 16. Motion is only possible in time. space.

17. Velocity, the space being 17. Velocity, the time being the same, is in the inverse the same, is in the direct ratio of the time. ratio of the space.

18. Time is not directly 18. Space is measurable directly measurable by means of through itself and itself but only by means of indirectly through motion motion which takes place in which takes place in both both space and time.... time and space....

19. Time is omnipresent: each 19. Space is eternal: each part of it is everywhere. part of it exists always.

20. In time alone all things 20. In space alone all things are successive. are simultaneous.

21. Time makes possible the 21. Space makes possible the change of accidents. endurance of substance.

22. Each part of time contains 22. No part of space contains all substance. the same substance as another.

23. Time is the _principium 23. Space is the _principium individuationis_. individuationis_.

24. The now is without 24. The point is without duration. extension.

25. Time of itself is empty and 25. Space is of itself empty indeterminate. and indeterminate.

26. Each moment is conditioned 26. The relation of each by the one which precedes boundary in space to every it, and only so far as this other is determined by its one has ceased to exist. relation to any one. (Principle of sufficient (Principle of sufficient reason of being in time.) reason of being in space.)

27. Time makes Arithmetic 27. Space makes Geometry possible. possible.

28. The simple element of 28. The element of Geometry Arithmetic is unity. is the point. --SCHOPENHAUER, A.

_Die Welt als Vorstellung und Wille; Werke (Frauenstädt) (Leipzig, 1877), Bd. 2, p. 55._

[11] Schopenhauer’s table contains a third column headed “of matter” which has here been omitted.

=2004.= The clear possession of the Idea of Space is the first requisite for all geometrical reasoning; and this clearness of idea may be tested by examining whether the axioms offer themselves to the mind as evident.--WHEWELL, WILLIAM.

_The Philosophy of the Inductive Sciences, Part 1, Bk. 2, chap. 4, sect. 4 (London, 1858)._

=2005.= Geometrical axioms are neither synthetic _a priori_ conclusions nor experimental facts. They are conventions: our choice, amongst all possible conventions, is guided by experimental facts; but it remains free, and is only limited by the necessity of avoiding all contradiction.... In other words, axioms of geometry are only definitions in disguise.

That being so what ought one to think of this question: Is the Euclidean Geometry true?

The question is nonsense. One might as well ask whether the metric system is true and the old measures false; whether Cartesian co-ordinates are true and polar co-ordinates false.

--POINCARÉ, H.

_Non-Euclidean Geometry; Nature, Vol 45 (1891-1892), p. 407._

=2006.= I do in no wise share this view [that the axioms are arbitrary propositions which we assume wholly at will, and that in like manner the fundamental conceptions are in the end only arbitrary symbols with which we operate] but consider it the death of all science: in my judgment the axioms of geometry are not arbitrary, but reasonable propositions which generally have the origin in space intuition and whose separate content and sequence is controlled by reasons of expediency.--KLEIN, F.

_Elementarmathematik vom höheren Standpunkte aus (Leipzig, 1909), Bd. 2, p. 384._

=2007.= Euclid’s Postulate 5 [The Parallel Axiom].

That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.--EUCLID.

_The Thirteen Books of Euclid’s Elements [T. L. Heath] Vol. 1 (Cambridge, 1908), p. 202._

=2008.= It must be admitted that Euclid’s [Parallel] Axiom is unsatisfactory as the basis of a theory of parallel straight lines. It cannot be regarded as either simple or self-evident, and it therefore falls short of the essential characteristics of an axiom....--HALL, H. S. and STEVENS, F. H.

_Euclid’s Elements (London, 1892), p. 55._

=2009.= We may still well declare the parallel axiom the simplest assumption which permits us to represent spatial relations, and so it will be true generally, that concepts and axioms are not immediate facts of intuition, but rather the idealizations of these facts chosen for reasons of expediency.--KLEIN, F.

_Elementarmathematik vom, höheren Standpunkte aus (Leipzig, 1909), Bd. 2, p. 382._

=2010.= The characteristic features of our space are not necessities of thought, and the truth of Euclid’s axioms, in so far as they specially differentiate our space from other conceivable spaces, must be established by experience and by experience only.--BALL, R. S.

_Encyclopedia Britannica, 9th Edition; Article “Measurement.”_

=2011.= Mathematical and physiological researches have shown that the space of experience is simply an _actual_ case of many conceivable cases, about whose peculiar properties experience alone can instruct us.--MACH, ERNST.

_Popular Scientific Lectures (Chicago, 1910), p. 205._

=2012.= The familiar definition: An axiom is a self-evident truth, means if it means anything, that the proposition which we call an axiom has been approved by us in the light of our experience and intuition. In this sense mathematics has no axioms, for mathematics is a formal subject over which formal and not material implication reigns.--WILSON, E. B.

_Bulletin American Mathematical Society, Vol. 2 (1904-1905), p. 81._

=2013.= The proof of self-evident propositions may seem, to the uninitiated, a somewhat frivolous occupation. To this we might reply that it is often by no means self-evident that one obvious proposition follows from another obvious proposition; so that we are really discovering new truths when we prove what is evident by a method which is not evident. But a more interesting retort is, that since people have tried to prove obvious propositions, they have found that many of them are false. Self-evidence is often a mere will-o’-the-wisp, which is sure to lead us astray if we take it as our guide.--RUSSELL, BERTRAND.

_Recent Work on the Principles of Mathematics; International Monthly, Vol. 4 (1901), p. 86._

=2014.= The problem [of Euclid’s Parallel Axiom] is now at a par with the squaring of the circle and the trisection of an angle by means of ruler and compass. So far as the mathematical public is concerned, the famous problem of the parallel is settled for all time.--YOUNG, JOHN WESLEY.

_Fundamental Concepts of Algebra and Geometry (New York, 1911), p. 32._

=2015.= If the Euclidean assumptions are true, the constitution of those parts of space which are at an infinite distance from us, “geometry upon the plane at infinity,” is just as well known as the geometry of any portion of this room. In this infinite and thoroughly well-known space the Universe is situated during at least some portion of an infinite and thoroughly well-known time. So that here we have real knowledge of something at least that concerns the Cosmos; something that is true throughout the Immensities and the Eternities. That something Lobatchewsky and his successors have taken away. The geometer of to-day knows nothing about the nature of the actually existing space at an infinite distance; he knows nothing about the properties of this present space in a past or future eternity. He knows, indeed, that the laws assumed by Euclid are true with an accuracy that no direct experiment can approach, not only in this place where we are, but in places at a distance from us that no astronomer has conceived; but he knows this as of Here and Now; beyond this range is a There and Then of which he knows nothing at present, but may ultimately come to know more.--CLIFFORD, W. K.

_Lectures and Essays (New York, 1901), Vol. 1, pp. 358-359._

=2016.= The truth is that other systems of geometry are possible, yet after all, these other systems are not spaces but other methods of space measurements. There is one space only, though we may conceive of many different manifolds, which are contrivances or ideal constructions invented for the purpose of determining space.--CARUS, PAUL.

_Science, Vol. 18 (1903), p. 106._

=2017.= As I have formerly stated that from the philosophic side Non-Euclidean Geometry has as yet not frequently met with full understanding, so I must now emphasize that it is universally recognized in the science of mathematics; indeed, for many purposes, as for instance in the modern theory of functions, it is used as an extremely convenient means for the visual representation of highly complicated arithmetical relations.

--KLEIN, F.

_Elementarmathematik vom höheren Standpunkte aus (Leipzig, 1909), Bd. 2, p. 377._

=2018.= Everything in physical science, from the law of gravitation to the building of bridges, from the spectroscope to the art of navigation, would be profoundly modified by any considerable inaccuracy in the hypothesis that our actual space is Euclidean. The observed truth of physical science, therefore, constitutes overwhelming empirical evidence that this hypothesis is very approximately correct, even if not rigidly true.

--RUSSELL, BERTRAND.

_Foundations of Geometry (Cambridge, 1897), p. 6._

=2019.= The most suggestive and notable achievement of the last century is the discovery of Non-Euclidean geometry.--HILBERT, D.

_Quoted by G. D. Fitch in Manning’s “The Fourth Dimension Simply Explained,” (New York, 1910), p. 58._

=2020.= Non-Euclidean geometry--primate among the emancipators of the human intellect....--KEYSER, C. J.

_The Foundations of Mathematics; Science History of the Universe, Vol. 8 (New York, 1909), p. 192._

=2021.= Every high school teacher [Gymnasial-lehrer] must of necessity know something about non-euclidean geometry, because it is one of the few branches of mathematics which, by means of certain catch-phrases, has become known in wider circles, and concerning which any teacher is consequently liable to be asked at any time. In physics there are many such matters--almost every new discovery is of this kind--which, through certain catch-words have become topics of common conversation, and about which therefore every teacher must of course be informed. Think of a teacher of physics who knows nothing of Roentgen rays or of radium; no better impression would be made by a mathematician who is unable to give information concerning non-euclidean geometry.

--KLEIN, F.

_Elementarmathematik vom höheren Standpunkte_ aus _(Leipzig, 1909), Bd. 2, p. 378._

=2022.= What Vesalius was to Galen, what Copernicus was to Ptolemy, that was Lobatchewsky to Euclid. There is, indeed, a somewhat instructive parallel between the last two cases. Copernicus and Lobatchewsky were both of Slavic origin. Each of them has brought about a revolution in scientific ideas so great that it can only be compared with that wrought by the other. And the reason of the transcendent importance of these two changes is that they are changes in the conception of the Cosmos.... And in virtue of these two revolutions the idea of the Universe, the Macrocosm, the All, as subject of human knowledge, and therefore of human interest, has fallen to pieces.--CLIFFORD, W. K.

_Lectures and Essays (New York, 1901), Vol. 1, pp. 356, 358._

=2023.= I am exceedingly sorry that I have failed to avail myself of our former greater proximity to learn more of your work on the foundations of geometry; it surely would have saved me much useless effort and given me more peace, than one of my disposition can enjoy so long as so much is left to consider in a matter of this kind. I have myself made much progress in this matter (though my other heterogeneous occupations have left me but little time for this purpose); though the course which I have pursued does not lead as much to the desired end, which you assure me you have reached, as to the questioning of the truth of geometry. It is true that I have found much which many would accept as proof, but which in my estimation proves _nothing_, for instance, if it could be shown that a rectilinear triangle is possible, whose area is greater than that of any given surface, then I could rigorously establish the whole of geometry. Now most people, no doubt, would grant this as an axiom, but not I; it is conceivable that, however distant apart the vertices of the triangle might be chosen, its area might yet always be below a certain limit. I have found several other such theorems, but none of them satisfies me.--GAUSS.

_Letter to Bolyai (1799); Werke, Bd. 8 (Göttingen, 1900), p. 159._

=2024.= On the supposition that Euclidean geometry is not valid, it is easy to show that similar figures do not exist; in that case the angles of an equilateral triangle vary with the side in which I see no absurdity at all. The angle is a function of the side and the sides are functions of the angle, a function which, of course, at the same time involves a constant length. It seems somewhat of a paradox to say that a constant length could be given a priori as it were, but in this again I see nothing inconsistent. Indeed, it would be desirable that Euclidean geometry were not valid, for then we should possess a general a priori standard of measure.--GAUSS.

_Letter to Gerling (1816); Werke, Bd. 8 (Göttingen, 1900), p. 169._

=2025.= I am convinced more and more that the necessary truth of our geometry cannot be demonstrated, at least not _by_ the _human_ intellect _to_ the human understanding. Perhaps in another world we may gain other insights into the nature of space which at present are unattainable to us. Until then we must consider geometry as of equal rank not with arithmetic, which is purely a priori, but with mechanics.--GAUSS.

_Letter to Olbers (1817); Werke, Bd. 8 (Göttingen, 1900), p. 177._

=2026.= There is no doubt that it can be rigorously established that the sum of the angles of a rectilinear triangle cannot exceed 180°. But it is otherwise with the statement that the sum of the angles cannot be less than 180°; this is the real Gordian knot, the rocks which cause the wreck of all.... I have been occupied with the problem over thirty years and I doubt if anyone has given it more serious attention, though I have never published anything concerning it. The assumption that the angle sum is less than 180° leads to a peculiar geometry, entirely different from the Euclidean, but throughout consistent with itself. I have developed this geometry to my own satisfaction so that I can solve every problem that arises in it with the exception of the determination of a certain constant which cannot be determined a priori. The larger one assumes this constant the more nearly one approaches the Euclidean geometry, an infinitely large value makes the two coincide. The theorems of this geometry seem in part paradoxical, and to the unpracticed absurd; but on a closer and calm reflection it is found that in themselves they contain nothing impossible.... All my efforts to discover some contradiction, some inconsistency in this Non-Euclidean geometry have been fruitless, the one thing in it that seems contrary to reason is that space would have to contain a _definitely determinate_ (though to us unknown) linear magnitude. However, it seems to me that notwithstanding the meaningless word-wisdom of the metaphysicians we know really too little, or nothing, concerning the true nature of space to confound what appears unnatural with the _absolutely impossible._ Should Non-Euclidean geometry be true, and this constant bear some relation to magnitudes which come within the domain of terrestrial or celestial measurement, it could be determined a posteriori.

--GAUSS.

_Letter to Taurinus (1824); Werke, Bd. 8 (Göttingen, 1900), p. 187._

=2027.= There is also another subject, which with me is nearly forty years old, to which I have again given some thought during leisure hours, I mean the foundations of geometry.... Here, too, I have consolidated many things, and my conviction has, if possible become more firm that geometry cannot be completely established on a priori grounds. In the mean time I shall probably not for a long time yet put my _very extended_ investigations concerning this matter in shape for publication, possibly not while I live, for I fear the cry of the Bœotians which would arise should I express my whole view on this matter.--It is curious too, that besides the known gap in Euclid’s geometry, to fill which all efforts till now have been in vain, and which will never be filled, there exists another defect, which to my knowledge no one thus far has criticised and which (though possible) it is by no means easy to remove. This is the definition of a plane as a surface which wholly contains the line joining any two points. This definition contains more than is necessary to the determination of the surface, and tacitly involves a theorem which demands proof.--GAUSS.

_Letter to Bessel (1829); Werke, Bd. 8 (Göttingen, 1900), p. 200._

=2028.= I will add that I have recently received from Hungary a little paper on Non-Euclidean geometry, in which I rediscover all _my own ideas_ and _results_ worked out with great elegance, .... The writer is a very young Austrian officer, the son of one of my early friends, with whom I often discussed the subject in 1798, although my ideas were at that time far removed from the development and maturity which they have received through the original reflections of this young man. I consider the young geometer v. Bolyai a genius of the first rank.--GAUSS.

_Letter to Gerling (1832); Werke, Bd. 8 (Göttingen, 1900), p. 221._

=2029.= Think of the image of the world in a convex mirror.... A well-made convex mirror of moderate aperture represents the objects in front of it as apparently solid and in fixed positions behind its surface. But the images of the distant horizon and of the sun in the sky lie behind the mirror at a limited distance, equal to its focal length. Between these and the surface of the mirror are found the images of all the other objects before it, but the images are diminished and flattened in proportion to the distance of their objects from the mirror.... Yet every straight line or plane in the outer world is represented by a straight [?] line or plane in the image. The image of a man measuring with a rule a straight line from the mirror, would contract more and more the farther he went, but with his shrunken rule the man in the image would count out exactly the same number of centimeters as the real man. And, in general, all geometrical measurements of lines and angles made with regularly varying images of real instruments would yield exactly the same results as in the outer world, all lines of sight in the mirror would be represented by straight lines of sight in the mirror. In short, I do not see how men in the mirror are to discover that their bodies are not rigid solids and their experiences good examples of the correctness of Euclidean axioms. But if they could look out upon our world as we look into theirs without overstepping the boundary, they must declare it to be a picture in a spherical mirror, and would speak of us just as we speak of them; and if two inhabitants of the different worlds could communicate with one another, neither, as far as I can see, would be able to convince the other that he had the true, the other the distorted, relation. Indeed I cannot see that such a question would have any meaning at all, so long as mechanical considerations are not mixed up with it.--HELMHOLTZ, H.

_On the Origin and Significance of Geometrical Axioms; Popular Scientific Lectures, second series (New York, 1881), pp. 57-59._

=2030.= That space conceived of as a locus of points has but three dimensions needs no argument from the mathematical point of view; but just as little can we from this point of view prevent the assertion that space has really four or an infinite number of dimensions though we perceive only three. The theory of multiply-extended manifolds, which enters more and more into the foreground of mathematical research, is from its very nature perfectly independent of such an assertion. But the form of expression, which this theory employs, has indeed grown out of this conception. Instead of referring to the individuals of a manifold, we speak of the points of a higher space, etc. In itself this form of expression has many advantages, in that it facilitates comprehension by calling up geometrical intuition. But it has this disadvantage, that in extended circles, investigations concerning manifolds of any number of dimensions are considered singular alongside the above-mentioned conception of space. This view is without the least foundation. The investigations in question would indeed find immediate geometric applications if the conception were valid but its value and purpose, being independent of this conception, rests upon its essential mathematical content.--KLEIN, F.

_Mathematische Annalen, Bd. 43 (1893), p. 95._

=2031.= We are led naturally to extend the language of geometry to the case of any number of variables, still using the word _point_ to designate any system of values of n variables (the coördinates of the point), the word _space_ (of n dimensions) to designate the totality of all these points or systems of values, _curves_ or _surface_ to designate the spread composed of points whose coördinates are given functions (with the proper restrictions) of one or two parameters (the _straight line_ or _plane_, when they are linear fractional functions with the same denominator), etc. Such an extension has come to be a necessity in a large number of investigations, in order as well to give them the greatest generality as to preserve in them the intuitive character of geometry. But it has been noted that in such use of geometric language we are no longer constructing truly a geometry, for the forms that we have been considering are essentially analytic, and that, for example, the general projective geometry constructed in this way is in substance nothing more than the algebra of linear transformations.

--SEGRE, CORRADI.

_Rivista di Matematica, Vol. I (1891), p. 59. [J. W. Young.]_

=2032.= Those who can, in common algebra, find a square root of −1, will be at no loss to find a fourth dimension in space in which ABC may become ABCD: or, if they cannot find it, they have but to imagine it, and call it an _impossible_ dimension, subject to all the laws of the three we find possible. And just as √−1 in common algebra, gives all its _significant_ combinations _true_, so would it be with any number of dimensions of space which the speculator might choose to call into _impossible_ existence.

--DE MORGAN, A.

_Trigonometry and Double Algebra (London, 1849), Part 2, chap. 3._

=2033.= The doctrine of non-Euclidean spaces and of hyperspaces in general possesses the highest intellectual interest, and it requires a far-sighted man to foretell that it can never have any practical importance.--SMITH, W. B.

_Introductory Modern Geometry (New York, 1893), p. 274._

=2034.= According to his frequently expressed view, Gauss considered the three dimensions of space as specific peculiarities of the human soul; people, which are unable to comprehend this, he designated in his humorous mood by the name Bœotians. We could imagine ourselves, he said, as beings which are conscious of but two dimensions; higher beings might look at us in a like manner, and continuing jokingly, he said that he had laid aside certain problems which, when in a higher state of being, he hoped to investigate geometrically.--SARTORIUS, W. V. WALTERSHAUSEN.

_Gauss zum Gedächtniss (Leipzig, 1856), p. 81._

=2035.= _There is many a rational logos_, and the mathematician has high delight in the contemplation of _in_consistent _systems_ of _consistent relationships_. There are, for example, a Euclidean geometry and more than one species of non-Euclidean. As theories of a given space, these are not compatible. If our universe be, as Plato thought, and nature-science takes for granted, a space-conditioned, geometrised affair, one of these geometries may be, none of them may be, not all of them can be, valid in it. But in the vaster world of thought, all of them are valid, there they co-exist, and interlace among themselves and others, as differing component strains of a higher, strictly supernatural, hypercosmic, harmony.--KEYSER, C. J.

_The Universe and Beyond; Hibbert Journal, Vol. 3 (1904-1905), p. 313._

=2036.= The introduction into geometrical work of conceptions such as the infinite, the imaginary, and the relations of hyperspace, none of which can be directly imagined, has a psychological significance well worthy of examination. It gives a deep insight into the resources and working of the human mind. We arrive at the borderland of mathematics and psychology.

--MERZ, J. T.

_History of European Thought in the Nineteenth Century (Edinburgh and London, 1903), p. 716._

=2037.= Among the splendid generalizations effected by modern mathematics, there is none more brilliant or more inspiring or more fruitful, and none more commensurate with the limitless immensity of being itself, than that which produced the great concept designated ... hyperspace or multidimensional space.

--KEYSER, C. J.

_Mathematical Emancipations; Monist, Vol. 16 (1906), p. 65._

=2038.= The great generalization [of hyperspace] has made it possible to enrich, quicken and beautify analysis with the terse, sensuous, artistic, stimulating language of geometry. On the other hand, the hyperspaces are in themselves immeasurably interesting and inexhaustibly rich fields of research. Not only does the geometrician find light in them for the illumination of otherwise dark and undiscovered properties of ordinary spaces of intuition, but he also discovers there wondrous structures quite unknown to ordinary space.... It is by creation of hyperspaces that the rational spirit secures release from limitation. In them it lives ever joyously, sustained by an unfailing sense of infinite freedom.--KEYSER, C. J.

_Mathematical Emancipations; Monist, Vol. 16 (1906), p. 83._

=2039.= Mathematicians who busy themselves a great deal with the formal theory of four-dimensional space, seem to acquire a capacity for imagining this form as easily as the three-dimensional form with which we are all familiar.--OSTWALD, W.

_Natural Philosophy [Seltzer], (New York, 1910), p. 77._

=2040.=

Fuchs. Was soll ich nun aber denn studieren?

Meph. Ihr könnt es mit _analytischer Geometrie_ probieren. Da wird der Raum euch wohl dressiert, In Coordinaten eingeschnürt, Dass ihr nicht etwa auf gut Glück Von der Figur gewinnt ein Stück. Dann lehret man euch manchen Tag, Dass, was ihr sonst auf einen Schlag Construiertet im Raume frei, Eine Gleichung dazu nötig sei. Zwar war dem Menschen zu seiner Erbauung Die dreidimensionale Raumanschauung, Dass er sieht, was um ihn passiert, Und die Figuren sich construiert-- Der Analytiker tritt herein Und beweist, das könnte auch anders sein. Gleichungen, die auf dem Papiere stehn, Die müsst’ man auch können im Raume sehn; Und könnte man’s nicht construieren, Da müsste man’s anders definieren. Denn was man formt nach Zahlengesetzen Müsst’ uns auch geometrisch erletzen. Drum in den unendlich fernen beiden Imaginären Punkten müssen sich schneiden Alle Kreise fein säuberlich, Auch Parallelen, die treffen sich, Und im Raume kann man daneben Allerlei Krümmungsmasse erleben. Die Formeln sind alle wahr und schön, Warum sollen sie nicht zu deuten gehn? Da preisen’s die Schüler aller Orten, Dass das Gerade ist krumm geworden. _Nicht-Euklidisch_ nennt’s die Geometrie, Spotted ihrer selbst, und weiss nicht wie.

Fuchs. Kann euch nicht eben ganz verstehn.

Meph. Das soll den Philosophen auch so gehn. Doch wenn ihr lernt alles reducieren Und gehörig transformieren, Bis die Formeln den Sinn verlieren, Dann versteht ihr mathematish zu spekulieren. --LASSWITZ, KURD.

_Der Faust-Tragödie (-n)ter Teil; Zeitschrift für den math-naturw. Unterricht, Bd. 14 (1888), p. 316._

[Fuchs. To what study then should I myself apply?

Meph. Begin with _analytical geometry_. There all space is properly trained, By coördinates well restrained, That no one by some lucky assay Carry some part of the figure away. Next thou’ll be taught to realize, Constructions won’t help thee to geometrize, And the result of a free construction Requires an equation for proper deduction. Three-dimensional space relation Exists for human edification, That he may see what about him transpires, And construct such figures as he requires. Enters the analyst. Forthwith you see That all this might otherwise be. Equations, written with pencil or pen, Must be visible in space, and when Difficulties in construction arise, We need only define it otherwise. For, what is formed after laws arithmetic Must also yield some delight geometric. Therefore we must not object That all circles intersect In the circular points at infinity. And all parallels, they declare, If produced must meet somewhere. So in space, it can’t be denied, Any old curvature may abide. The formulas are all fine and true, Then why should they not have a meaning too? Pupils everywhere praise their fate That that now is crooked which once was straight. Non-Euclidean, in fine derision, Is what it’s called by the geometrician.

Fuchs. I do not fully follow thee.

Meph. No better does philosophy. To master mathematical speculation, Carefully learn to reduce your equation By an adequate transformation Till the formulas are devoid of interpretation.]