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 difiicult to see that because Forward 

 has been supposed to be 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 



