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 ; or in 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 ZYX. 

 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 ZYX 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 

 clement to be combined with each, and give the name of 

 rosiTivE UNITY to the unit of magnitude measured in the di- 



