Here is an assumption that a figure, such as the definition expresses, may be described; which is no other than the postulate, or covert assumption, involved in the so-called definition. But whether that figure be called a circle or not is quite immaterial. The purpose would be as well answered, in all respects except brevity, were we to say, "Through the point B, draw a line returning into itself, of which every point shall be at an equal distance from the point A." By this the definition of a circle would be got rid of, and rendered needless; but not the postulate implied in it; without that the demonstration could not stand. The circle being now described, let us proceed to the consequence. "Since B C D is a circle, the radius B A is equal to the radius C A." B A is equal to C A, not because B C D is a circle, but because B C D is a figure with the radii equal. Our warrant for assuming that such a figure about the centre A, with the radius B A, may be made to exist, is the postulate. Whether the admissibility of these postulates rests on intuition, or on proof, may be a matter of dispute; but in either case they are the premises on which the theorems depend; and while these are retained it would make no difference in the certainty of geometrical truths, though every definition in Euclid, and every technical term therein defined, were laid aside.
It is, perhaps, superfluous to dwell at so much length on what is so nearly self-evident; but when a distinction, obvious as it may appear, has been confounded, and by powerful intellects, it is better to say too much than too little for the purpose of rendering such mistakes impossible in future. I will, therefore detain the reader while I point out one of the absurd consequences flowing from the supposition that definitions, as such, are the premises in any of our reasonings, except such as relate to words only. If this supposition were true, we might argue correctly from true premises, and arrive at a false conclusion. We should only have to assume as a premise the definition of a nonentity; or rather of a name which has no entity corresponding to it. Let this, for instance, be our definition:
A dragon is a serpent breathing flame.
This proposition, considered only as a definition, is indisputably
correct. A dragon is a serpent breathing flame: the word means that.
The tacit assumption, indeed (if there were any such understood
assertion), of the existence of an object with properties corresponding to
the definition, would, in the present instance, be false. Out of this
definition we may carve the premises of the following syllogism:
A dragon is a thing which breathes flame:
A dragon is a serpent:
From which the conclusion is,
Therefore some serpent or serpents breathe flame:
an unexceptionable syllogism in the first mode of the third figure, in
which both premises are true and yet the conclusion false; which every
logician knows to be an absurdity. The conclusion being false and the
syllogism correct, the premises can not be true. But the premises,
considered as parts of a definition, are true. Therefore, the premises
considered as parts of a definition can not be the real ones. The real
premises must be--
A dragon is a really existing thing which breathes flame:
A dragon is a really existing serpent:
which implied premises being false, the falsity of the conclusion presents
no absurdity.
If we would determine what conclusion follows from the same ostensible premises when the tacit assumption of real existence is left out, let us,
