Now, suppose there are two observers, A and B. They agree in their non-metrical projective geometry, _e.g._, what A calls a straight line so does B. They also both apply a Euclidean metrical system of measurement to this space. Their two metrical systems also agree in having the same plane at infinity, that is, lines which are parallel for A are also parallel for B. Furthermore, they have both successfully applied Newton's laws to the movement of matter, and agree in having the same sets of dynamical axes. But the framework (among these sets) which A chooses to regard as at rest is different from the frame (among the same sets) which B so regards. Without alteration of their respective judgment of rest, they choose their co-ordinate axes so that the origins (O for A, and O′ for B) are in relative motion along OO′, which is the axis of _x_ for both. Further, since OO′ is the line of symmetry of their diverse Euclidean systems, we assume that the two measure-systems agree for planes perpendicular to OO′, _i.e._, we assume a symmetry round OO′. Then if, for A at O, the distance OO′ be ξ, the relations at any instant between A's coordinates (_x_, _y_, _z_) and B's coordinates (_x′_, _y′_, _z′_) for the same point P are given by _x′_ = β(_x_ - ξ), _y′_ = _y_, _z′_ = _z_. Also, according to A's clock, O′ is moving forward with a uniform velocity _v_, and we measure A's time from the instant of the coincidence of O and O′. Thus ξ = _vt_, and _x′_ = β(_x_ - _vt_), _y′_ = _y_, _z′_ = _z_. We now consider B's clock, and ask for the most general supposition which is consistent with the fact that their judgments as to the fact of uniform motion are in agreement. We do not assume that events in various parts of space which A considers to be simultaneous are so considered by B. But we assume that at any point P, with coordinates (_x_, _y_, _z_) for A, there is a determinate relation between B's time T and _x_, _y_, _z_, _t_. Put T = ƒ(_x_, _y_, _z_, _t_). Write P = δT/δ_x_, Q = δT/δ_y_, R = δT/δ_z_, S = δT/δ_t_. Now suppose that the point P is moving, and that (_u_{1}, _u_{2}, _u_{3}) is its set of component velocities along the axes according to A's "space and clock" system, and (U_{1}, U_{2}, U_{3}) is its set of component velocities according to B's "space and clock" system. Then by mere differentiation it follows after a short mathematical deduction that U_{1} = {(_d_β/_dt_)(_x_ - _vt_) + β(_u_{1} - _v_)}/{P_u_{1} + Q_u_{2} + R_u_{3} + S}, U_{2} = _u_{2}/{P_u_{1} + Q_u_{2} + R_u_{3} + S}, U_{3} = _u_{3}/{P_u_{1} + Q_u_{2} + R_u_{3} + S}. But we have assumed that, whatever the direction of the resultant velocity (_u_{1}, _u_{2}, _u_{3}), the velocities (U_{1}, U_{2}, U_{3}) and (_u_{1}, _u_{2}, _u_{3}) are both uniform when either is uniform. Hence it is easily proved that β, P, Q, R, S are independent of the coordinates (_x_, _y_, _z_) and of the time _t_. In other words, they are constant. Hence we obtain