5 
to the three rectangular directions, West, South, and Up; 
and let this be called, provisionally, Forward, by contrast to 
the opposite direction, Backward, which must be assumed to 
be (in the same general sense) a fourth proportional to the 
directions of West, South, and Down. We shall then have, in- 
versely, Forward to Up as South to West ; and therefore, as 
West to North: if we admit, as it seems natural and almost ne- 
cessary to do, that (for directions, as for lengths) the inverses of 
equal ratios are equal; and that ratios equal to the same ratio 
are equal to each other. But again, Up is to South as South 
to Down, and also as North to Up: and we can scarcely avoid 
admitting, or defining, that (in the present comparison of di- 
rections) ratios similarly compounded of equal ratios are to be 
considered as being themselves equal ratios. Compounding, 
therefore, on the one hand, the ratios of Forward to Up, and 
of Up to South; and on the other hand the respectively 
equal (or similar) ratios of West to North, and of North to 
Up, we are conducted to admit that Forward is to South as 
West to Up. By a reasoning exactly similar, we find that 
Forward is to West as Up to South; and generally that if 
X, Y, Z denote any three rectangular directions such that 
A:X::Y:Z, A here denoting what we have expressed by 
the word Forward, then also A: Y:: Z: X (and of course, for 
the same reason, A: Z:: X: Y); so that the three directions 
XYZ may be all changed together by advancing them in a 
ternary cycle, according to the formula just written, without 
disturbing the proportionality assumed. But also, by the 
principle respecting proportions of directions in one plane, we 
may cause any two of the three rectangular directions XYZ 
to revolve together round the third, as round an axis, without 
altering their ratio to each other. And by combining these 
two principles, it is not difficult to see that because Forward 
has been supposed to he to Up as South to West, therefore 
the same (as yet unknown) direction ‘* Forward” must be 
supposed to be to any direction X whatever, as any direction 
Y, perpendicular to X, is to that third direction Z which is 
