In Geometry, the proof of a theorem depends not only upon the definition of the figure or figures directly concerned, but also upon one or more axioms, and upon propria or constructions already established.  Thus, in Euclid’s fifth Proposition, the proof that the angles at the base of an isosceles triangle are equal, depends not only on the equality of the opposite sides, but upon this together with the construction that shows how from the greater of two lines a part may be cut off equal to the less, the proof that triangles that can be conceived to coincide are equal, and the axiom that if equals be taken from equals the remainders are equal.  Similarly, in Biology, if colouring favourable to concealment is a proprium of carnivorous animals, it is not deducible merely from their predatory character or any other attribute entering into the definition of any species of them, but from their predatory character together with the causes summarised in the phrase ‘Natural Selection’; that is, competition for a livelihood, and the destruction of those that labour under any disadvantages, of which conspicuous colouring would be one.  The particular coloration of any given species, again, can only be deduced by further considering its habitat (desert, jungle or snowfield):  a circumstance lying wholly outside the definition of the species.

The validity of an argument based partly or wholly on a definition depends, in the first place, on the existence of things corresponding with the definition—­that is, having the properties connoted by the name defined.  If there are no such things as isosceles triangles, Euclid’s fifth Proposition is only formally true, like a theorem concerning the fourth dimension of space:  merely consistent with his other assumptions.  But if there be any triangles only approximately isosceles, the proof applies to them, making allowance for their concrete imperfection:  the nearer their sides approach straightness and equality the more nearly equal will the opposite angles be.

Again, as to the things corresponding with terms defined, according to Dr. Venn, their ‘existence’ may be understood in several senses:  (1) merely for the reason, like the pure genera and species of Porphyry’s tree; the sole condition of whose being is logical consistency:  or (2) for the imagination, like the giants and magicians of romance, the heroes of tragedy and the fairies of popular superstition; whose properties may be discussed, and verified by appeal to the right documents and authorities (poems and ballads):  or (3) for perception, like plants, animals, stones and stars.  Only the third class exist in the proper sense of the word.  But under a convention or hypothesis of existence, we may argue from the definition of a fairy, or a demigod, or a dragon, and deduce various consequences without absurdity, if we are content with poetic consistency and the authority of myths and romances as the test of truth.