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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 characterize 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.

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