IV

Now, undoubtedly, this is the general aspect of the relation of the world of mathematical physics to our emotions, sensations, and thoughts; and a great deal of controversy has been occasioned by it and much ink spilled. We need only make one remark. The whole situation has arisen, as we have seen, from the endeavour to describe an external world "explanatory" of our various individual sensations and emotions, but a world

also, not essentially dependent upon any particular sensations or upon any particular individual. Is such a world merely but one huge fairy tale? But fairy tales are fantastic and arbitrary: if in truth there be such a world, it ought to submit itself to an exact description, which determines accurately its various parts and their mutual relations. Now, to a large degree, this scientific world does submit itself to this test and allow its events to be explored and predicted by the apparatus of abstract mathematical ideas. It certainly seems that here we have an inductive verification of our initial assumption. It must be admitted that no inductive proof is conclusive; but if the whole idea of a world which has existence independently of our particular perceptions of it be erroneous, it requires careful explanation why the attempt to characterise it, in terms of that mathematical remnant of our ideas which would apply to it, should issue in such a remarkable success.

It would take us too far afield to enter into

a detailed explanation of the other laws of motion. The remainder of this chapter must be devoted to the explanation of remarkable ideas which are fundamental, both to mathematical physics and to pure mathematics: these are the ideas of vector quantities and the parallelogram law for vector addition. We

have seen that the essence of motion is that a body was at $A$ and is now at $C$. This transference from $A$ to $C$ requires two distinct elements to be settled before it is completely determined, namely its magnitude (i.e. the length $AC$) and its direction. Now anything, like this transference, which is completely given by the determination of a magnitude 6 and a direction is called a vector. For example, a velocity requires for its definition the assignment of a magnitude and of a direction. It must be of so many miles per hour in such and such a direction. The existence and the independence of these two elements in the determination of a velocity are well illustrated by the action of the captain of a ship, who communicates with different subordinates respecting them: he tells the chief engineer the number of knots at which he is to steam, and the helmsman the compass

bearing of the course which he is to keep. Again the rate of change of velocity, that is velocity added per unit time, is also a vector quantity: it is called the acceleration. Similarly a force in the dynamical sense is another vector quantity. Indeed, the vector nature of forces follows at once according to dynamical principles from that of velocities and accelerations; but this is a point which we need not go into. It is sufficient here to say that a force acts on a body with a certain magnitude in a certain direction.

Now all vectors can be graphically represented by straight lines. All that has to be done is to arrange: (i) a scale according to which units of length correspond to units of magnitude of the vector–-for example, one inch to a velocity of $10$ miles per hour in the case of velocities, and one inch to a force of $10$ tons weight in the case of forces–-and (ii) a direction of the line on the diagram corresponding to the direction of the vector. Then a line drawn with the proper number of inches of length in the proper direction represents the required vector on the arbitrarily assigned scale of magnitude. This diagrammatic representation of vectors is of the first importance. By its aid we can enunciate the famous "parallelogram law" for the addition of vectors of the same kind but in different directions.

Consider the vector $AC$ in [figure]6 as representative