first, or the relation is reciprocal between the two.

We learn, in the next place, that motion in two transverse directions essentially constitutes motion in a third direction, conceived as intermediate between the former two. The cognizance of opposition to action, implies a consciousness of ability to perform the action resisted if the external obstacle were removed. Thus the resistance of which we are cognizant while moving our hand over the surface of a body, at the same time that it makes known to us the actual absence of motion in the direction of the normal, will suggest the possibility of motion in that direction, without further change in the conditions of the experiment than the removal of the bodily obstacle; that is to say, the same experience which made known to us the relation of transverseness, will lead us to conceive the possibility, in empty space, of advancing in the direction of the normal, without ceasing to move in the lateral direction in which the surface formerly extended, or in other words, of moving simultaneously in a given direction and in one transverse to it.

But when an object moving simultaneously in two directions is contemplated from without the sphere of the influences to which the separate movements are owing, it will appear to move in a third direction, related more or less nearly to the

direction of either of the component movements, according as the distance traversed in that direction is greater or less than the distance simultaneously traversed in the other.

An agent carried along in a certain direction by the motion of the rigid system in which he is placed, as in the cabin of a ship for instance, has the same freedom of action within the limits of the system as if the latter were at rest. He is capable of moving with the same facility in a direction transverse to that of the ship's motion as in the same direction with it. But if the real direction of his motion could be observed, with respect to external objects, while he is moving across the cabin, it would be found to lie in some intermediate direction, standing in a closer relation to that of the ship's motion or the transverse direction, according as the rate of the ship's motion is more or less rapid than that of his own walk across the cabin.

Thus, if an object be supposed to move independently in two directions, CA, CB, transverse to each other, and CP be the real direction of the motion in space, the relation of CP to CA and CB may be expressed by reference to the proportion in which motion in the direction CP admits of resolution in the directions CA and C B respectively. Now, let either of these directions as CA (Fig. 1) be taken as a standard, and let motion in

the direction CA be compounded with an indefinitely small proportion of motion in the direction CB. The result will be motion in a direction CP, differing extremely little from CA. Then by continually increasing the proportion of motion in the direction C B, and diminishing that in the direction CA, we shall obtain a series of intermediate directions varying in their relation to CA in every degree from coincidence to transverseness. In like manner, the combination of motion in the direction C B with C D, the opposite of the original standard, will furnish a similar series of directions intermediate between CB and CD, which will appear as the continuation of the former series. Beyond CD, on the other side, the series may be carried on by combination with motion in the direction of CE, the opposite of CB, and it will finally be brought back to the point from whence we started by combination of motion in the direction CE with motion in the original direction C A.

Thus, by combination of motion in a single direction, positively and negatively considered, and in a transverse direction taken in like manner in a positive and negative sense, we are enabled to distinguish a continuous circle of directions, each of which is placed between two neighbours differing from it by an indefinitely small amount of variation in opposite directions.

Now, the relation of transverseness being (as we have seen) primarily known as a relation of one direction to a multiplicity of others, let CA, CP (Fig. 1) be any two directions to both of which a third, CF, is transverse. Then motion in the direction CP may be resolved in a direction CA, and a direction transverse to it, C B. But as the directions C P and CA are both transverse to CF, motion in either of those directions will be wholly devoid of motion in the direction CF; that is to say, that motion in the direction C P, as well as one of the elements into which it may be resolved, will be wholly without effect in the direction C F, and, therefore, the remaining element, or the motion in the direction C B, must be equally ineffective in the same direction, or C B also will be transverse to CF. But motion in the directions CA and C B being both ineffective in the direction C F, the same must be true of every motion compounded of those elements; or in other words, every direction intermediate between CA and CB and their opposites will be included in the series transverse to C F. Thus it appears that if, in the series of directions CA, CP, etc., transverse to a given normal or standard direction C F, any individual, as CA, be taken as a second standard, the series will include a direction CB transverse to CA (and therefore constituting a third standard transverse to each of the other two),

together with every direction intermediate between CA and CB, positively and negatively considered.

Now let each individual of the series CA, CP, etc., arising from the combination of CA and CB, again be combined with the direction CF transverse to them all; the result will be a succession of series of the same kind, the aggregate of which will embrace every direction diverging from the point C, throughout an entire hemisphere of space. In like manner, the combination of the same primary series with the opposite to CF, will give the directions of the opposite hemisphere. Thus the entire scheme of directions diverging from a point in space, will be constructed by the combination of a single standard direction with the succession of series arising from the combination of two other directions, transverse to the original standard and also to each other; and any particular direction may be identified by the proportion in which distance in that direction is composed of distance in each of the three transverse standards or axes. If the vertical or up and down direction, for example, be taken as the primary axis of direction, the series transverse to it will be the directions of the horizontal plane; and if we take any one of these, as the right and left direction, for our second standard, and the transverse or the fore and aft direction for our third, the position of any direction