duced steps lead forward from the postulated premises to the announced end. Geometry therefore corresponds—so far as a correspondence can be traced between a mathematical and an observational science—chiefly to that part of geological or geographical investigation which is concerned with invention and deduction; for these processes, like the similar processes in geometry, can be performed by mental reflection in the dark, and have no close dependence on observation.
In observational sciences it is necessary to examine critically the different degrees of agreement that may exist between the deduced consequences of an hypothesis and the facts gathered by observation, in order to pass a safe judgment on the value of the hypothesis from which the consequences were deduced. This return to the facts is one of the most important as well as one of the most characteristic elements of scientific work.
If observation has discovered but few classes of simple facts, and if invention has brought forth only one hypothesis, which leads only to a few simple consequences, the value of the hypothesis must remain in doubt, even if its consequences agree rather closely with the facts; because agreement in such a case may be a matter of chance. Here no decided opinion as to the value of the hypothesis should be expressed; judgment must be suspended, and the mind held open for further light, either from observation, invention or deduction. Again, if, as above pointed out, the groups of consequences deduced from two or more rival hypotheses are about equally successful in matching the facts, no judgment must be pronounced in favor of either, however strong the investigator's desire to reach a conclusion may be. But if the peculiar and numerous consequences of a certain hypothesis agree to a remarkable extent with the highly specialized groups of abundant and varied facts, such an hypothesis is strongly commended thereby, for the possibility of accidental agreement is greatly diminished as the facts and consequences to be matched become more complicated, and as the number of agreements increases. Furthermore, if the facts, as at first collected, seem of arbitrary occurrence and unrelated distribution, and yet are afterwards found, by the suggestive aid of an hypothesis and its deduced consequences, really to possess a previously unsuspected order and many previously unseen relationships, the hypothesis which leads to this larger and clearer view is thereby greatly recommended; for it is highly commendable to a theory, if it leads to the discovery of reasonable system where confusion seemed to prevail.
But we must go further; for it often happens that, after an hypothesis has been invented, and after its consequences have been successfully confronted with the previously observed facts, new classes of facts may be discovered for which deduction had provided no appropriate