we obtain a line which has all the properties of the straight lines of the Euclidean geometry. In particular, it easily follows that by laying off times the interval upon a straight line, an interval of length is obtained. A length, therefore, means the result of a measurement carried out along a straight line by means of a unit measuring rod. It has a significance which is as independent of the system of co-ordinates as that of a straight line, as will appear in the sequel. We come now to a train of thought which plays an analogous role in the theories of special and general relativity. We ask the question: besides the Cartesian co-ordinates which we have used are there other equivalent co-ordinates? An interval has [Pg 5] a physical meaning which is independent of the choice of co-ordinates; and so has the spherical surface which we obtain as the locus of the end points of all equal intervals that we lay off from an arbitrary point of our space of reference. If as well as ( from 1 to 3) are Cartesian co-ordinates of our space of reference, then the spherical surface will be expressed in our two systems of co-ordinates by the equations How must the be expressed in terms of the in order that equations (2) and (2a) may be equivalent to each other? Regarding the expressed as functions of the , we can write, by Taylor's theorem, for small values of the , If we substitute (2a) in this equation and compare with (1), we see that the must be linear functions of the . If we therefore put [Pg 6] then the equivalence of equations (2) and (2a) is expressed in the form It therefore follows that must be a constant. If we put = 1, (2b) and (3a) furnish the conditions in which = 1, or = 0, according = or ≠ . The conditions (4) are called the conditions of orthogonality, and the transformations (3), (4), linear orthogonal transformations. If we stipulate that shall be equal to the square of the length in every system of co-ordinates, and if we always measure with the same unit scale, then must be equal to 1. Therefore the linear orthogonal transformations are the only ones by means of which we can pass from one Cartesian system of co-ordinates in our space of reference to another. We see that in applying such transformations the equations of a straight line become equations of a straight line. Reversing equations (3a) by multiplying both sides by and summing for all the 's, we obtain The same coefficients, , also determine the inverse substitution of . Geometrically, is the cosine of the angle between the axis and the axis. [Pg 7]