iii. If equals be taken from equals the remainders will be equal.
iv. If equals be added to unequals the sums will be unequal.
v. If equals be taken from unequals the remainders will be unequal.
vi. The doubles of equal magnitudes are equal.
vii. The halves of equal magnitudes are equal.
viii. Magnitudes that can be made to coincide are equal.
The placing of one geometrical magnitude on another, such as a line on a line, a triangle on a triangle, or a circle on a circle, &c., is called superposition. The superposition employed in Geometry is only mental, that is, we conceive one magnitude placed on the other; and then, if we can prove that they coincide, we infer, by the present axiom, that they are equal. Superposition involves the following principle, of which, without explicitly stating it, Euclid makes frequent use:—“Any figure may be transferred from one position to another without change of form or size.”
ix. The whole is greater than its part.
This axiom is included in the following, which is a fuller statement:—
ix′. The whole is equal to the sum of all its parts.