The philosophic issue under this head has been usefully cleared for English readers by Mr. A. J. Balfour in his _Defence of Philosophic Doubt_; and, in another sense, very usefully for rationalists by the same writer in his work _The Foundations of Belief_. The gist of the former treatise is an expansion of the proposition of Hume that all moral judgments, on analysis, are found to root in a sentiment or bias. In particular, Mr. Balfour argues that all scientific beliefs so-called, however immediately proved, rest upon general beliefs which are 'incapable of proof.' It is noteworthy that never through the whole treatise does Mr. Balfour analyse the concept of 'proof,' though his main aim is ostensibly to discriminate between proved and unproved propositions. It may be worth while, then, at this stage, to note the risks of intellectual confusion in connection with the term proof. The common conception, implicit in Mr. Balfour's argument, is that concerning a 'proved' thing either we have, or men of science say we have, a right of certainty, as it were, which we cannot have concerning anything not proved or not capable of proof. The simple fact is that the very idea of proof involves that of uncertainty you seek to prove that which is not unquestionable. To prove is to _probe_,[8] to test. The idea of 'demonstration,' which seems commonly to connote special certainty, carries us no further. It means a 'showing,' a 'letting you see with your own eyes.' In geometry, it stands for a chain of reasoning in which every step rests upon previous steps which ultimately rest upon axioms and definitions agreed upon. There the process is one of analysis--a showing that a proposition formerly unknown as such is really contained in or implied by propositions known. Certainty follows. Yet there is abundant record of 'proofs' or 'tests' which were fallacious, and of ostensible demonstrations which were flawed--modes of squaring the circle, for instance. The ultimate in the matter is the belief arrived at or evoked; and the significant fact for us is, that beliefs ostensibly so arrived at may be false, because the cited proof or evidence is erroneous or the demonstration inconsequent.