6 
perpendicular to both X and Y, and which is obtained from Y 
by a right-handed (and not by a left-handed) rotation, through 
a right angle, round X; in the same manner as (and because) 
the direction West was so chosen as to be to the right of South, 
with reference to Up as an axis of rotation. Conversely we 
must suppose that if any three rectangular directions, XYZ, 
be arranged, as to order of rotation, in the manner just now 
stated, then Z: Y:: X: A; orin other words, we must admit, 
if we reason in this way at all, that the direction called 
already Forward, will be the fourth proportional to ZY X. 
And if we vary the order, so as to have Z to the left, and not 
to the right of Y, with reference to X, then will the fourth 
proportional to ZY X become the direction which we have 
lately called Backward, as being the opposite to that named 
Forward. 
Again, since Forward is to Up as South to West, that is 
in a ratio compounded of the ratios of South to East and of 
East to West, or in one compounded of the ratios of West to 
South, and of any direction to its own opposite ; or, finally, in 
a ratio compounded of the ratios of Up to Forward and of 
Forward to Backward, that is, in the ratio of Up to Back- 
ward, we see that the third proportional to the directions For- 
ward and Up is the direction Backward: and by an exactly 
similar reasoning, with the help of the conclusions recently 
obtained, we see that if X be any direction in tridimensional 
space, then A: X:: X:B; B here denoting, for shortness, 
the direction which has been above called Backward. 
The geometrical study of the relations between directions 
in space, combined with a few very simple and guiding prin- 
ciples respecting the composition of relations generally, might 
therefore have led to the conception or assumption of a certain 
pair of contrasted directions, namely, those which we have 
called Forward and Backward, and denoted by the letters A 
and B. And these are such that if we conceive a quantitative 
element to be combined with each, and give the name of 
POSITIVE UNITY to the unit of magnitude measured in the di- 
