We there remarked, that the directly inductive truths of mathematics are few in number; consisting of the axioms, together with certain propositions concerning existence, tacitly involved in most of the so-called definitions.
Like other so-called definitions, these are composed of two things, the explanation of a name, and the assertion of a fact: of which the latter alone can form a first principle or premise of a science.
All the inductions involved in all geometry are comprised in those simple ones, the formulæ of which are the Axioms, and a few of the so-called Definitions.
The exact correspondence, then, between the facts and those first principles of geometry which are involved in the so-called definitions, is a fiction, and is merely supposed.
The rest of the premisses of Geometry consist of the so-called definitions, which assert, together with one or more properties, the real existence of objects corresponding to the names to be defined.
All the inductions involved in all geometry are comprised in those simple ones, the formulae of which are the Axioms, and a few of the so-called Definitions.
The above list will hopefully provide you with a few useful examples demonstrating the appropriate usage of "called definitions" in a variety of sentences. We hope that you will now be able to make sentences using this group of words.
