which when expanded gives It is therefore necessary that /i"o- a 2</o = - But, as this is homogeneous in the two component rotations involved, it does not follow that the separate terms of this equation must necessarily be zero. We satisfy this condition by writing kfi--ffo and ka. 2 =i . Let the body next receive any displacement, 8?;, 877 , and 877", then we have in general , v rfX . dX. , dX , SA= -, STJ+ T~? J ? + TT/OTJ . drj di di)" with similar equations for 8Y and 8Z. By substitution these equations become SX = - .j/tZ8ij - SY= - SZ= + If we multiply the first of these equations by X, the second by Y, and the third by kZ, and add, we find const. Here we have attained the fundamental property which the coordinates must satisfy. If k be equal to unity then we have the well-known condition of ordinary space and ordinary rectangulat coordinates, but it will be seen that there is nothing in the preced ing investigation to make it necessary that k should be unity. There are therefore a singly infinite variety of spaces in which it is possible for a rigid body to be displaced. The different values of k thus correspond to the different "curvatures" which a space might have while it still retained the fundamental property which is necessary for measurement by congruence. It will now be proper to study the special charac teristics of the space with which we are familiar. It has been called flat space or faomaloidal space to distinguish it from other spaces in which the curvature is not zero. It is manifest that the characteristic features of our space are not necessarily implied in the general notion of an extended quantity of three dimensions and of the mobility of rigid figures therein. 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. The special feature of our space, by which it is distin guished from spherical space on the one hand and pseudo- spherical space on the other hand, depends upon what Riemann calls the measure of curvature. If the sum of the three angles of a triangle is to be two right angles, and if the geometrical similarity of large figures and small figures is to be possible, then the measure of curvature must be zero. Now all measurements that can b.e made seem to confirm the axiom of parallels and seem to show the measure of curvature of our space to be indistinguish able from zero. It can be proved that the amount by which the three angles of a triangle would differ from two right angles in curved space depends upon the area of the triangle. The greater the area of the triangle the greater is the difference. To test the famous proposition, Euclid i. 32, it will there fore not be sufficient to measure small triangles. It might be contended that in small triangles the difference between the sum of the three angles and two right angles was so small as to be inextricably mixed up with the unavoidable errors of measurement. Seeing therefore that small triangles obey the law, it is necessary to measure large triangles, and the largest triangles to which we have access are, of course, the triangles which the astronomers have found means of measuring. The largest available triangles are those which have the diameter of the earth s orbit as a base and a fixed star at the vertex. It is a very curious circumstance that the investigations of annual parallax of the stars are precisely the investigations which would be necessary to test whether one of these mighty triangles had the sum of its three angles equal to two right angles. It must be admitted that the parallax-seeking astronomers have never yet found any reason to think that there is any measurable difference. If there were such a difference it would probably be inextricably mixed up with the annual parallax itself. Were space really pseudospherical, then stars would exhibit a real parallax even if they were infinitely distant. Astronomers have sometimes been puzzled by obtaining a negative parallax as the result of their labours. No doubt this has generally or indeed al ways arisen from the errors which are inevitable in inquiries of this nature, but if space were really curved then a nega tive parallax might result from observations which possessed mathematical perfection. It must, however, be remembered that even the triangles of the parallax investigations are utterly insignificant when compared with the dimensions of space itself. Even the whole visible universe must be only an uncon- ceivably small atom when viewed in its true relation to infinite space. It may well be that even with the parallax triangles our opportunities of testing the proposition are utterly inadequate to pronounce on the presence or absence of curvature in space. It must remain an open question whether if we had large enough triangles the sum of the three angles would still be two right angles. Helmholtz illustrates the subject by considering the representation of space which is obtained in a spherical mirror. A mirror of this kind represents the objects in front of it in apparently fixed positions behind the mirror. The images of the sun and of other distant objects will lie behind the mirror at a distance equal to its focal length, or, to quote the description of Helmholtz " 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 centimetres as the real man. And in general all geometrical measurements of lines or angles made with regularly varying images of real instruments would yield exactly the same results as in the outer world. All congruent bodies would coincide on being applied to one another in the mirror as in the outer world. All lines of sight in the outer world would be repre sented 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 Euclid s axioms. But if they could look out upon our world as we can 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 so far as I can see would be able to convince the other that he had the true, the other the distorted relations. Indeed I cannot see that such a question would have any meaning at all so long as mechani cal considerations are not mixed up with it." A very important contribution to this subject has been made by Professor Simon Newcomb, entitled " Elementary Theorems Relating to the Geometry of a Space of Three Dimensions and of Uniform Positive Curvature in the Fourth Dimension," see Jour. f. d. reine und angeivandte Math ., vol. Ixxxiii., Berlin, 1877. He commences by assuming the three following postulates : 1 " That space is triply extended, unbounded, without properties dependent either upon 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. " 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 2D without losing in any part of its course that symmetry with respect to space on all sides of it which constitutes the funda mental property of our conception of it." 3. That if two right lines emanate from the same point making
the indefinitely small angle a with each other, their distance apart