Robert Stawell Ball

“ When a plane rigid system is displaced upon itself there is one point of the system which remains unaltered, and all the circles which have as their centre remain unaltered also. It is quite natural to speak of this motion as a "rotation," and thus we may assert the truth in generalized measurement of the well-known theorem in ordinary geometry that Every displacement of a plane upon itself could have been prod need by a rotation of the plane around a certain point in the plane. ”

“ We suppose such two-dimensional beings to be absolutely limited to a certain surface. They could have no conception of space except as of two dimensions. The movement of a point would for them form a line, the movement of a line would form a surface. They could conduct their measurements and form their geometrical theories. They would be able to draw the shortest lines between two points, these lines being what we would call geodesies. ”

“ A doubly infinite number of geodesic surfaces can be drawn through every point. If a rigid body be divided into two parts by a geodesic plane, then no matter how the body be displaced the plane of section will still be geodesic. The plane of section may be made to pass through any point, and the body may then be given such an aspect as shall cause the section to coincide with any geodesic surface through the point, but this necessarily involves that the section shall fit each geodesic surface, in other words, that all the geodesic surfaces shall have a constant curvature. ”

“ The student of modern geometry is already accustomed to think of parallel lines as lines which intersect at infinity, or as lines whose inclination is zero. In speaking of the generalized geometry in a plane, we may define that two straight lines which intersect upon the fundamental conic are parallel. It thus follows that through any point two distinct parallels can be drawn to a given straight line. The only exception will arise in the case where the given line touches the fundamental conic. ”

“ 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 must thus be possible to pass continuously in an infinite number of ways from the point a to the point a. We thus assume that space is continuous when we have assumed that its points are represented by coordinates. It must be observed that we predicate nothing as to space which is not involved in the fact that to each point corresponds one group of three coordinates. To some extent the considerations now before us will apply to any other continuous manifoldness which requires three coordinates for the complete specification of its elements. ”

“ 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. ”

“ We are now in a position to inquire how the magnitude of an angle is to be expressed in the present system of measurement. Our definition of the magnitude of an angle must be made consistent with the supposition that when the angle is carried round by rotation about the vertex the magnitude shall remain unaltered. As anharmonic ratios are unaltered by the rotation, it follows that the anharmonic ratio of the pencil formed by the two legs of the angle and the two tangents to the fundamental conic must remain unaltered. ”

“ We here deal with the space with which Euclid has made us familiar, only observing that it is the measurements in that space which are to be conducted on a more general principle. We commence by assuming the existence of a certain surface called the "fundamental quadric," often called " the absolute." ”

“ The reader will at once perceive the analogy to a well-known phenomenon in ordinary geometry. The equation in rectangular coordinates a; 2 + 7/ 2 =0 denotes either a circle of which the radius is zero or the pair of lines in the latter case we are obliged to admit that the distance of any point on either of these lines from their intersection is equal to zero. We have now to consider the displacement of a rigid figure, and we shall for the present speak only of a plane movement. ”

“ 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. ”

“ It can therefore be easily shown that a sphere is a quadric which touches the fundamental quadric along its intersection with the polar plane of the centre of the sphere.
In discussing the general case of the displacement of a rigid system it will simplify matters to suppose that the fundamental quadric has real rectilinear generators. It must, however, be understood that the results are not on that account less general. A displacement must not alter the quadric, and must not deform a straight line. ”

“ Hence it follows that the only effect of a displacement upon a generator of the fundamental quadric will be to convey it to a position previously occupied by a different generator. We shall further suppose that the displacement is such that the two generators to which we have referred belong to the same system. Let A, B, C, D be four generators of the first system which by displacement are brought to coincide with four other generators A′, B′, C′, D′. Let X be one generator of the second system which the displacement brings to X′. ”

“ Take, for instance, the manifoldness which embraces all the circles that can lie in a given plane. The points of such a manifoldness are the circles. It is three- dimensioned, for two coordinates will be required for the centre of each circle and one for its radius. It is obviously a continuous manifoldness, for one circle may by infinitely graduated modifications pass into any other. Yet from this manifoldness also the conception of distance is absent. There is no intelligible relation of one circle to another which is analogous to the distance which we require to determine. ”

“ We do not now enter into the proof, but it is sufficiently obvious that a sphere of which the radius is the geometric mean between the greatest and least radii of curvature at each point will to a large extent osculate the surface, so that a portion of the surface in the neighbourhood of the point will, generally speaking, have the same curvature as the sphere. ”

“ 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 ”

“ We thus have a singly infinite number of displacements which are compatible with the condition that the two fundamental elements shall remain unaltered, and it is displacements of this kind which express the movements of a rigid system. ”

“ 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 at the distance Y from the point of intersection will be given by the equation 2aD . ”

“ The essential feature of a singly extended manifoldness is that a continuous progress of an element can take place only in two directions, either forwards or backwards. But a singly extended manifoldness may be regarded as itself an element in a manifoldness of a higher order. Thus the points on a circle form a singly extended manifold- ness, while the circle itself is one element of the manifold- ness which consists of a series of concentric circles. ”

“ As no finite movement can either bring a point to infinity or from infinity, it is obvious that the displacement must be such as to leave the fundamental conic unaltered. It is easily seen that the specification of the transforma tion in its general form requires eight constants ”

“ A point is to be defined for our present purpose as a group of three numerical magnitudes taken one from each of the three scales. This conception may be stated more generally. We can conceive n different numerical scales. Then a grottp of n numbers, one from each scale, will define an element of a continuous re-fold manifoldness. It will be obvious that unless the theory of distance possess a special character it will not be possible for a rigid body to exist. ”

“ 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. ”

“ 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. ”

“ The unit of force is termed the dyne, and it is defined to be the force which, acting upon a gramme of matter for a second, generates a velocity of a centimetre per second. The unit of work is the work done by this force working through a centimetre, and this unit is termed the erg. The unit of power is the power of doing work at the rate of one erg per second, and the power of an engine can be specified in ergs per second. By the prefixes deca, hecto, kilo, mega, we can express a magnitude equal to the unit multiplied by 10, 100, 1000, or 1,000,000 respectively. ”

“ 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. ”

“ It is important to observe that the elements of the manifoldness may be themselves objects of no little complexity. Thus, for instance, the conies forming a confocal group constitute the elements of a singly ex tended manifoldness. ”

“ It therefore follows that the anharmonic ratio in which four generators cut a fixed generator X′ is equal to the anharmonic ratio in which the four generators after displacement cut the same generator X′. ”

“ We therefore write 7^ = + sin (roots of the cubic for h must, therefore, be pure imaginaries, and thus we have the condition ffl + &! + c 2 = . Finally we have for the determination of x, y, z the following threo equations : I x + m y + n z = const. ”