line possessing this property is called by Leibnitz a straight line. It can be easily shown that a geodesic drawn on a figure will also be a geodesic when the figure is transferred to any other position. Suppose that the figure be divided into two parts A and B by the geodesic; then the part B can be moved round so as to lie upon A, and the boundary lines of the two portions will be coincident. Now let the two parts while superposed be translated to any other position, then the piece B may be slid off and back to its original position with regard to A. It must still fit, because the whole figure might have been translated before the subdivision took place. It follows that the division between A and B having been a geodesic in its original position will continue to be a geodesic
however the figure may be translated.In a similar way we obtain the conception of a plane. According to Leibnitz's definition a plane is a surface such that if a portion of the space contiguous thereto be slid along the surface it will continuously fit, and if the portion of space be transferred to the other side of the surface it will fit also. This definition has no meaning except we assume that the bodies may be translated in space without derangement of their dimensions. From any point we can imagine a doubly infinite number of geodesies radiating in all directions; if a plane be drawn through the point, then all the geodesies touching the plane at that point form what may be called a “geodesic surface.” It is shown that geodesic surfaces of this description can alone fulfil the conditions by which planes are to be defined. 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 point which we have now gained is one of very great importance. In our ordinary conceptions of space the geodesic surfaces are of course our ordinary planes, and the common curvature they possess is zero, but the condition that rigid bodies shall be capable of translation with unaltered features does not require that the curvatures shall be zero, it merely requires that the curvatures shall be constant. If we add, however, the postulate of similarity, then the curvatures must be zero. The postulate of similarity requires that it shall be possible to construct a figure on any scale and anywhere similar to a given figure. This practically includes the ordinary doctrine of parallels. Lobatchewsky developed the system of geometry on the supposition that the space had a constant curvature different from zero. In this geometry the parallels can be drawn through a given point to a given line, and, to quote Clifford—
“The sum of the three angles of a triangle is less than two right angles by a quantity proportional to the area of the triangle. The whole of this geometry is worked out in the style of Euclid, and the most interesting conclusions are arrived at, particularly in the theory of solid space, in which a surface turns up which is not plane relatively to that space but which for the purpose of drawing figures upon it is identical with the Euclidean plane.”
The most comprehensive mode of viewing the whole theory is that adopted by Riemann in his celebrated memoir “Ueber die Hypothesen welche der Geometrie zu Grunde liegen,” 1854 (Abhandl. der königl. Gesellsch. zu Göttingen, vol. xiii.).[1] The analytical treatment of this subject possesses one obvious advantage. The use of symbols only admits of deductions on purely logical principles. There is not therefore the risk of tacitly introducing other axioms in addition to those with which we started.
Magnitudes which have only one dimension present the theory of measurement in its simplest form. The length of a straight line may be taken as an illustration of a one-dimensioned magnitude. The velocity of a moving particle, the temperature of a heated body, the electric resistance of a metal, all these and many others are instances of one-dimensioned magnitude, the measure of which is to be expressed by a single quantity. But there may be magnitudes which require more than a single measurement for their complete specification. Take, for instance, a four-sided field which has been duly surveyed. Of what is the measurement of this field to consist? If the number of acres in the field be all that is required then the area is expressed by a simple reference to a number of standard acres. If, however, the entire circumstances of the field are to be brought into view, then a simple statement of the area is not sufficient. It can be easily shown that the surveyor must ascertain five independent quantities before the details of the shape of the field can be adequately defined. Four of these quantities- may naturally be the lengths of the four sides of the field, the fifth may be one of the angles, or the area, or the length of one of the diagonals. Speaking generally, we may say that five distinct measurements will be necessary to define the field adequately. The actual choice of the particular measurements to be made is to a great extent arbitrary. The only condition absolutely necessary is that they shall be all independent and free from ambiguity. Once these five quantities are ascertained then all the other features of the figure are absolutely determined. For instance, the four sides and the diagonal being ascertained by measurement, then the other diagonal, the four angles, and the area can all be computed by calculation. The five quantities would determine everything about the field except its actual position on the surface of the earth. If we further desired to have the field exactly localized certain other quantities must be added. The latitude and the longitude of one specified corner of the field would completely indicate that corner, while the azimuth of one side from that corner would complete the definition of its position. We are thus led to see that for the complete delineation of every circumstance relating to the shape of the field and its locality eight different measurements have been required. Two sets of eight measurements differing in any particular can never indicate the same field. It is very important to notice that the number of quantities required is quite independent of the particular nature of the measurements adopted. We might for instance have simply measured the latitude and the longitude of each of the four corners of the field. Once these quantities are known, then the shape of the field, its area, its angles, and its diagonals have all been implicitly determined. Here again we see that as two quantities are required to localize each of the four corners, so eight quantities will be required to fully determine the whole field.
accustomed to specify the position of a point by the relation which it bears to certain fixed axes. By means of certain quantities, either altogether linear or partly linear and partly angular, we are enabled to specify the position
of the point with absolute definiteness. These quantities
- ↑ A translation of this paper was published by Clifford in Nature (vol. viii. Nos. 183, 184, pp. 14–17, 36, 37), and has been reprinted in the collected edition of Clifford's Works, 1882, pp. 55–69. For a bibliography of higher-space and non-Euclidean geometry, see articles by George Bruce Halsted in the American Journal of Mathematics Pure and Applied, i. 261-276, 384, 385 ; ii. 65-70.
