The answer lies again in the general principle of co-ordination, which in this very case receives a particularly cogent illustration. Three algebraic signs, x, y, and z, are co-ordinated with the three variable dimensions of space. First, the same independent and constant variability is ascribed to these signs, and, further, the same mutual relations are assumed to subsist between them as actually exist between the three-spacial dimensions. In other words, precisely the same kind of manifoldness is imparted to these algebraic signs as the spacial dimensions possess to which they are co-ordinated, and we may therefore expect that all the conclusions arising from these assumptions will find their corresponding parts in the spacial manifoldness. Accordingly, a co-ordinated spacial relation corresponds to every change of those algebraic formulas resulting from calculation, and if such changes lead to an algebraically simple form, then the spacial form corresponding to it must show an analogous simplicity. Here, therefore, we have a case such as was described under simpler conditions on p. 86 of operations undertaken with one group and repeated correspondingly in the co-ordinated group. And it is only the great difference in the things of which in this case the two groups are composed--spacial relations on the one side and algebraic signs on the other--that creates the impression of astonishment which was felt very strongly at the invention of this method, and which is still felt by students with talent for mathematics when they first become acquainted with analytical geometry. =41. Recapitulation.= Before we proceed to consider the fundamentals of other sciences, it is well to make a general résumé of the field so far traversed. Since the later sciences, as we have already observed, make use of the entire apparatus of the earlier sciences, the mastery of them must be assured in order to render their special application possible.