respectively, and the angle PAQ. Then AH p cose, and the volume = A P.AQ = rp cose, for r represents the number of units of area in D E F G. Hence, since a volume is a purely numerical quantity having only magnitude and no direction, we find that with this new convention the product of two vectors is a purely scalar quantity, or our new convention leads to a totally different result from the old.

Further, sincer and p are merely numbers, rp=pr, and thus AP A Q = r ρ cose = pr cos✪ = A Q. A P, if AQ be treated as the directed step which represents an area containing p units of area. Thus in this case the vector product obeys the commutative law, which again differs from our previous result. We can then treat the product of two vectors either as a vector and as a quantity not obeying the commutative law, or as a scalar and as a quantity obeying the commutative law. We are at liberty to adopt either convention, provided we maintain it consistently in our resulting investigations.

The method of varying our interpretation, which has been exemplified in the case of the product of two vectors, is peculiarly fruitful in the field of the exact sciences. Each new interpretation may lead us to vary our fundamental laws, and upon those varied fundamental laws we can build up a new calculus (algebraic or geometric as the case may be). The results of our new calculus will then be necessarily true for those quantities only for which we formulated our fundamental laws. Thus those laws which were formulated for pure number, and which, like the postulates of Euclid with regard to space, have been frequently supposed to be the only conceivable basis for a theory of quantity, are found to be true only within the limits

of scalar magnitude. When we extend our conception of quantity and endow it with direction and position, we find those laws are no longer valid. We are compelled to suppose that one or more of those laws cease to hold or are replaced by others of a different form. In each case we vary the old form or adopt a new one to suit the wider interpretation we are giving to quantity or its symbols.

§ 17. Position in Three-Dimensioned Space.

Hitherto we have been considering only position in a plane; very little alteration will enable us to consider the position of a point P relative to a point a as determined by a step A P taken in space.

We may first remark, however, that while two points A and B are sufficient to determine in a plane the position of any third point P, we shall require, in order to fix the position of a point p in space, to be given three points A, B, C not lying in one straight line. If we knew only the distances of P from two points A and B, the point P might be anywhere on a certain circle which has its centre on the line A B and its plane perpendicular to that line; to determine the position of P on this circle, we require to know its distance from a third point c. Thus position in space requires us to have at least three non-collinear points (or such geometrical figures as are their equivalent) as basis for our determination of position. Space in which we live is termed space of three dimensions; it differs from space of two dimensions in requiring us to have three and not two points as a basis for determining position.

Three points will fix a plane, and hence if we are given three points A, B, C in space, the plane through

them will be a definite plane separating all space into two halves. In one of these any point P whose position we require must lie. We may term one of these halves below the plane and the other above the plane. Let P N be the perpendicular from P upon the plane; then if we' know how to find the point N in the plane A B C, the position of P will be fully determined so soon as we have settled whether the distance P N is to be measured above or below the plane. We may settle by convention that all distances above the plane shall be considered positive, and all below negative. Further, the position of the point N, upon which that of P depends, may be

determined by any of the methods we have employed to fix position in a plane. Thus if N M be drawn perpendicular to A B, we have the following instruction to find the position of P: Take a step AM along AB, containing, say, a units; then take a step м N to the right and perpendicular to A B, but still in its plane, containing, say, y units; finally step upwards from N the distance NP perpendicular to the plane A B C, say, through z units. We shall then have reached the same point p as if we had taken the directed step A P. If a had been negative we should have had to step backwards from A; if y had been negative, perpendicular to A B only to the left; if z had been negative, perpendicular to the plane but

downwards.

The reader will easily convince himself that by observing these rules as to the sign of x, y, z he could get from A to any point in space.

and

Let i denote unit step along A B, j unit step to the right perpendicular to a в, but in the plane A B C, k unit step perpendicular to the plane A B C upwards, from foot to head. Then we may write

AP = x.i+y.j+z.k,

where x, y, z are scalar quantities possessing only magnitude and sign; but i, j, k are vector steps in three mutually rectangular directions.

N

M

FIG. 86.

The step A P may be regarded as the diagonal of a solid rectangular figure (a right six-face, as we termed it on p. 138), and thus we shall get to the same point P by traversing any three of its non-parallel sides in succession starting from A. But this is equivalent to saying that the order in which we take the directed steps x. i, y.j, and z. k is indifferent.

The reader will readily recognise that the sum of a number of successive steps in space is the equivalent to the step which joins the start of the first to the

finish of the last; and thus a number of propositions concerning steps in space similar to those we have proved for steps in a plane may be deduced. By dividing all space into little cubes by three systems of planes mutually at right angles, we may plot out surfaces just as we plotted out curves. Thus we shall choose any values we please for a and y, and suppose the magnitude of the third step related in some constant fashion to the previous steps. For example, if we take the rectangle under and some constant length a, always equal to the differences of the squares on æ and y, or symbolically if we take a z = x2-y2, we shall reach p by taking the step

The

The series of points which we should obtain in this way would be found to lie upon a surface resembling the saddle-back we have described on p. 89. above relation between z, x, and y will then be termed the equation to a saddle-back surface.

We cannot, however, enter fully on the theory of steps in space without far exceeding the limits of our present enterprise.

§ 18. On Localised Vectors or Rotors.

Hitherto we have considered the position of a point P relative to a point A, and compared it with the position of another point q relative to the same point

A.

Thus we have considered the ratio and product of two steps AP and a Q.

We have thereby assumed either that the two steps we were considering had a common extremity A, or at least were capable of being moved parallel to themselves