EXAMPLE OF THE ANALYSIS OF A SYSTEM OF EQUATIONS BY THE METHOD OF REDUCTION TO A SINGLE EQUIVALENT EQUATION V = 0, WHEREIN V SATISFIES THE CONDITION V (1 − V) = 0.

1. Let us take the remarkable system of premises employed in the previous Chapter, to prove that “Matter is not a necessary being;” and suppressing the 6th premiss, viz., Motion exists,—examine some of the consequences which flow from the remaining premises. This is in reality to accept as true Dr. Clarke's hypothetical principles; but to suppose ourselves ignonorant of the fact of the existence of motion. Instances may occur in which such a selection of a portion of the premises of an argument may lead to interesting consequences, though it is with other views that the present example has been resumed. The premises actually employed will be—

1. If matter is a necessary being, either the property of gravitation is necessarily present, or it is necessarily absent.

2. If gravitation is necessarily absent, and the world is not subject to any presiding intelligence, motion does not exist.

3. If gravitation is necessarily present, a vacuum is necessary.

4. If a vacuum is necessary, matter is not a necessary being.

5. If matter is a necessary being, the world is not subject to a presiding intelligence.

If, as before, we represent the elementary propositions by the following notation, viz.:

1. x = Matter is a necessary being.

2. y = Gravitation is necessarily present.

3. w = Motion exists.

4. t = Gravitation is necessarily absent.

5. z = The world is merely material, and not subject to a presiding intelligence.

6. v = A vacuum is necessary.