6. I have stated that a definition can be of no use, except we can conceive the possibility and truth of the property connected with it; and that if we do conceive this, we may rightly begin our reasonings by stating the property as an axiom; which Euclid does, in the case of straight lines and of parallels. The reviewer inquires (p. 92), whether I am prepared to extend this doctrine to the case of circles, for which the reasoning is usually rested upon the definition;--whether I would replace this definition by an axiom, asserting the possibility of such a circle. To this I might reply, that it is not at all incumbent upon me to assent to such a change; for I have all along stated that it is indifferent {113} whether the fundamental properties from which we reason be exhibited as definitions or as axioms, provided the necessity be clearly seen. But I am ready to declare that I think the form of our geometry would be not at all the worse, if, instead of the usual definition of a circle,--'that it is a figure contained by one line, which is called the circumference, and which is such, that all straight lines drawn from a certain point within the circumference are equal to one another,'--we were to substitute an axiom and a definition, as follows:-- _Axiom_. If a line be drawn so as to be at every point equally distant from a certain point, this line will return into itself or will be _one_ line including a space. _Definition_. The space is called a _circle_, the line the _circumference_, and the point the _center_. And this being done, it would be true, as the reviewer remarks, that geometry cannot stir _one_ step without resting on an axiom. And I do not at all hesitate to say, that the above axiom, expressed or understood, is no less necessary than the definition, and is tacitly assumed in every proposition into which circles enter. 7. I have, I think, now disposed of the principal objections which bear upon the proper axioms of geometry. The principles which are stated as the first seven axioms of Euclid's _Elements_, need not, as I have said, be here discussed. They are principles which refer, not to Space in particular, but to Quantity in general: such, for instance, as these; 'If equals be added to equals the wholes are equal;'--'If equals be taken from equals the remainders are equal.' But I will make an observation or two upon them before I proceed.