574 
Proceedings of the Royal Society 
small bodies along their straight courses are to be mutually propor- 
tional, and they are also to be proportional each of them to the 
simultaneous travel of the index on the dial. Now, if any other 
frame of three co-ordinate planes be arranged to exist with the 
point of their intersection keeping at any one of the moving points, 
and with the three planes maintaining changeless angles with the 
original three reference planes, any one of the moving points will 
either be at rest relatively to the new set of reference planes, or will 
generate, in relation to them, a straight line, and the simultaneous 
lengths traversed by the various points relatively to these new 
reference planes will be mutually proportional. It is convenient 
to notice, in preparation for subsequent reference of motions to true 
time or to a truly chronometric clock, that the simultaneous lengths 
traversed by the various points relatively to the new reference 
planes, and of the winch handle index relatively to its dial, will be 
mutually proportional. We may thus see that for the established 
set of motions of the points, there can exist as many sets of 
reference planes or frames as we please, differently moving and 
differently inclined, in reference to each of which every one of the 
points will generate a straight line with a quasi-velocity (or rate per 
dial-traveller progress) proportional to the quasi-velocity of every 
other along its own line. We are now perfectly entitled to speak 
of the motions of all these points as referred to any one of the 
frames and the original dial traveller, as being uniform rectilinear 
motions. The word uniform, it should be noticed, has, neither in 
its origin nor in its customary employment, any essential connection 
with progress of time. The notion besides of the dial-traveller as a 
standard to which the simultaneous travels of the various points 
may conveniently be referred, or rated, is not at all essential. We 
would be quite entitled, without knowledge of chrononietry, and 
without having recourse to the quasi-time indicated by the dial- 
traveller, to speak of the motions of all the points relatively to all 
the reference frames, as being uniform rectilinear motions. The 
uniformity in rate of progress would be in respect to rate of travel 
of any one of the points per simultaneous travel of any other one of 
them. In all that has been said in this inatter no assumption has 
been made as to any particular condition of rest or motion having 
belonged to the original reference frame. It may have been firmly 
