§ 2. The important doctrine of Dugald Stewart, which I have endeavoured to enforce, has been contested by Dr. Whewell, both in the dissertation appended to his excellent _Mechanical Euclid_, and in his more recent elaborate work on the _Philosophy of the Inductive Sciences_; in which last he also replies to an article in the _Edinburgh Review_, (ascribed to a writer of great scientific eminence,) in which Stewart’s opinion was defended against his former strictures. The supposed refutation of Stewart consists in proving against him (as has also been done in this work) that the premisses of geometry are not definitions, but assumptions of the real existence of things corresponding to those definitions. This, however, is doing little for Dr. Whewell’s purpose; for it is these very assumptions which are asserted to be hypotheses, and which he, if he denies that geometry is founded on hypotheses, must show to be absolute truths. All he does, however, is to observe, that they at any rate are not _arbitrary_ hypotheses; that we should not be at liberty to substitute other hypotheses for them; that not only “a definition, to be admissible, must necessarily refer to and agree with some conception which we can distinctly frame in our thoughts,” but that the straight lines, for instance, which we define, must be “those by which angles are contained, those by which triangles are bounded, those of which parallelism may be predicated, and the like.”(37) And this is true; but this has never been contradicted. Those who say that the premisses of geometry are hypotheses, are not bound to maintain them to be hypotheses which have no relation whatever to fact. Since an hypothesis framed for the purpose of scientific inquiry must relate to something which has real existence, (for there can be no science respecting non-entities,) it follows that any hypothesis we make respecting an object, to facilitate our study of it, must not involve anything which is distinctly false, and repugnant to its real nature: we must not ascribe to the thing any property which it has not; our liberty extends only to suppressing some of those which it has, under the indispensable obligation of restoring them whenever, and in as far as, their presence or absence would make any material difference in the truth of our conclusions. Of this nature, accordingly, are the first principles involved in the definitions of geometry. In their positive part they are observed facts; it is only in their negative part that they are hypothetical. That the hypotheses should be of this particular character, is however no further necessary, than inasmuch as no others could enable us to deduce conclusions which, with due corrections, would be true of real objects: and in fact, when our aim is only to illustrate truths, and not to investigate them, we are not under any such restriction. We might suppose an imaginary animal, and work out by deduction, from the known