4. I conceive, then, that the reviewer has failed altogether to disprove the doctrine that the axioms of geometry are necessary as a part of the foundations of the science. I had asserted further that these axioms supply what the definitions leave deficient; and that they, along with definitions, serve to present the idea of space under such aspects that we can reason logically concerning it. To this the reviewer opposes (p. 96) the common opinion that a perfect definition is a complete explanation of a name, and that the test of its perfection is, that we may substitute the definition for the name wherever it occurs. I reply, that my doctrine, that a definition expresses a part, but not the whole, of the essential characters of an idea, is certainly at variance with an opinion sometimes maintained, that a definition merely explains a word, and should explain it so fully that it may always replace it. The error of this common opinion may, I think, be shown from considerations such as these;--that if {112} we undertake to explain one word by several, we may be called upon, on the same ground, to explain each of these several by others, and that in this way we can reach no limit nor resting-place;--that in point of fact, it is not found to lead to clearness, but to obscurity, when in the discussion of general principles, we thus substitute definitions for single terms;--that even if this be done, we cannot reason without conceiving what the terms mean;--and that, in doing this, the relations of our conceptions, and not the arbitrary equivalence of two forms of expression, are the foundations of our reasoning. 5. The reviewer conceives that some of the so-called axioms are really definitions. The axiom, that 'magnitudes which coincide with each other, that is, which fill the same space, are equal,' is a definition of geometrical _equality_: the axiom, that 'the whole is greater than its part,' is a definition of _whole_ and _part_. But surely there are very serious objections to this view. It would seem more natural to say, if the former axiom is a definition of the word _equal_, that the latter is a definition of the word _greater_. And how can one short phrase define two terms? If I say, 'the heat of summer is greater than the heat of winter,' does this assertion define anything, though the proposition is perfectly intelligible and distinct? I think, then, that this attempt to reduce these axioms to definitions is quite untenable.