great difficulty in showing him to be in error and in vindicating the right of the particular axiom in question to be ranked among explicative propositions.

I propose to make good this latter assertion by a demonstration depending upon some axioms and definitions, of the former of which different enunciations are given for the sake of greater clearness.

(1) A=A, B=B, C=C, D=D, E=E, F=F.

(2) The greater of two quantities is not less than the other nor equal.

(3) The less of two quantities is not greater than the other nor equal.

(4) The whole is greater than its part, and conversely the part is less than the whole.

(5) The whole is equal to the sum of its parts, and conversely the sum of the parts equals the whole.

Axiom. Everything is either A or Non-A.

(6) A and B are either equal in quantity or unequal.

(7) If A and B are unequal A is greater and B less, or B greater and A less.

Definition. Equal Quantities.

(8) Equal quantities are those of which whatever may be affirmed of the total quantity of one must be affirmed of the other, else they cease to be equal.

Definition. Unequal Quantities.

(9) Unequal quantities are those of which whatever may be affirmed of the total quantity of one cannot be affirmed of the other.'

Definition. Greater Quantities.

(10) One quantity is greater than another when a part of it is equal to that other.

Definition. Less Quantities.

(11) One quantity is less than another when it is equal to a part of that other.

To the definition of equality and inequality we shall presently recur; we will hence not comment upon it here.

Definition. Addition and Subtraction.

(12) Addition is the augmentation of one quantity by another; subtraction is the diminution of one quantity by taking away a part of itself.

Demonstration.

Let A and B be equal and c and D be equal also. added to C and B to D, A+C=B+D. Represent A+C by E, D by F; then E=F.

If E does not equal F, then it is greater or less, E and F being unequal (6) (7). Suppose it to be greater; one of the following suppositions must be made (Ax. 2) (4).

(a) A may be greater than B, C remaining equal to D. this is contrary to the supposition.

But

(b) A may be greater than B, and c greater than D. Both of these are contrary to the original supposition.

(c) ▲ may be greater than D Cleing equal to B. Now if c =B, C=A also, since A=B (8), and A>D, also C>D (8) which is contrary to the original supposition.

(d) A may be greater than D, C being greater than B. If then C> B, C > A, since A=B (8); and since A>D, also C> D.

C

For if

C>A, A<C (7); and if A<C, A=a part of c (11): let part and

= that

Xx

C

the remaining part of c; then a =

and c =

У

C C
+
x У

(5). And since A > D, D<A (7); and if D<A, D=a part of a

Α

(11): let A

A

D =

x

-=

x

and A=

C C

C = + ...C=D+

x У

A

the remaining part of A; then

+(8). But (D+

У У

> D (4); C>D.

This latter, however, is contrary to the original supposition. (e) A may be greater than D+ (B-C,) if c is less than B. For if CB, since CD by hypothesis, A must be greater than D not merely in order that E shall be greater than F, but it must also be greater than so much of B as shall be left after c is taken

out. But DC; hence A > D + B-D (8) or A > B which is contrary to the supposition.

(f) c may be greater than D, A being equal to B. This is contrary to the supposition.

(g) c may be greater than B, A being equal to D. If A=D, D=A (1) and also D=B since A=B (8). Hence C > D which is contrary to hypothesis.

(h) c may be greater than B, A being greater than D. This leads to contradiction in precisely the same manner as (d).

(i) c may be greater than B+ (D-A), if A is less than D. This is seen to be contradictory in the same manner as (e). (k) A may be greater than B+ D, which is contrary to hypothesis.

(1) c may be greater than B+D, which is contrary to hypothesis.

If then E>F and these eleven suppositions are all possible suppositions under which E can be greater than F, it is seen that to assume E to be greater than F involves contradiction. Therefore E cannot be greater than F. If we assume it to be less than F, then F is greater than E (7) and one of the same eleven hypotheses must be made with reference to this case (mutatis mutandis) which have just been given, and in any one with the same results. Hence every attempt to prove the sums of equals unequal ends in contradiction, and therefore it is true that the sums of equals are equal, and the axiom is exhibited as one having the same kind of necessity as The whole is greater than its part. It is therefore analytical and implied in the conceptions of equality. and inequality, addition and subtraction, greater and less.

The demonstration of the same truth by superposition, suggested by Mr. Mill, may be made with geometrical magnitudes. If E and F are equal, on being applied each to each they will coincide throughout. If not equal they will fail to coincide, and their parts will fail to coincide. A then will not be equal to B, nor C to D, which is contrary to supposition.

It may be said that we have not exhausted all possible conditions under which E may be greater than F. If not I leave to the critic the task of producing another hypothesis than the eleven given, and I will then examine the same with him to see if it does not lead to contradiction without making use of the axiom which we are essaying to prove. At present writing no other suppositions occur to me unless we take fractions of the quantities

before us, and in that event there would be no different results so far as I can see. We should only have different expressions of the same conclusions.

§ 20. In the somewhat tedious demonstration just gone over, we made use of two axioms in various forms equivalent to and included under the enunciations Whatever is, is (Law of Consistency) and Everything is either A or Non-A (Excluded Middle). We furthermore employed definitions of equality and inequality, greater and less, addition and subtraction. But we have really implied other axioms than we have stated; in fact we have been making use of the axiom Things equal to the same thing are equal to each other, and the axiom If A is greater than B and B is greater than C, A is greater than C. Let us look first to the axiom of mediate coincidence. If Mr. Mill's definition of equal magnitudes is accepted there is no escape from the implication that within its purview things equal to the same thing are equal to each other; and the proof by superposition is easy. I am not able to conceive any other notion of equality than that involved in Mr. Mill's definition. To Mr. Jevons's doctrine of the Substitution of Similars allusion has been made in several places, and with that doctrine I am in full accord. Equality means identity as to quantity, complete coalescence, interchangeability, power of substitution of one equal quantity for its fellow. As to geometrical plane figures, therefore, Mr. Mill's definition seems to me sufficient: but it does not go quite far enough for a general definition. Not all equal quantities are susceptible of proof by superposition. To be sure it may be said that we ascertain their equality in last resort only by showing coincidence of surfaces and lines. But I think a clearer and more striking statement of the meaning of equality in general may be found and which will cover those cases wherein superposition is not immediately practicable. If we consider that equality means identity of quantity, that equal quantities are those in which the quantity of quantity is, so to speak, the same, we may construct the definition before given (§ 19):

Equal quantities are those of which whatever may be affirmed of the total quantity of one must be affirmed of the other, else they cease to be equal. I say of the total quantity, for assertions might be made of one dimension of quantity in one which could not be made in another; one might be longer, shorter, or deeper than the other, though the total quantity would be the same.

From this definition it follows very evidently that Things which are equal to the same thing are equal to each other. For let A=B and CB; A=C also, since it may be affirmed of A and B alike that C equals them, A and B being equal and it being affirmed of one of these equals that C is equal to it.

§ 21. By the aid of the definitions and axioms before given the explicative character of the argumentum à fortiori is made apparent. That argument is analysed in (d) of § 19. It amounts simply to the truth: whatever is greater than the whole is greater than a part thereof, and, correspondingly, whatever is less than a part is less than the whole. Both these truths are explicative of our ideas of greater and less, whole and part, equality and inequality. It is not necessary again to go over the demonstration. Proof may also be made by superposition.

§ 22. We can now very shortly dispose of the five axioms of Euclid following the second, and which are admitted on all hands to be analytical. If equals are taken from equals the remainders are equal follows from the definitions of equals and unequals before given. So also, If equals be added to unequals the wholes will be unequal; If equals be taken from unequals the remainders are unequal; Doubles of the same are equals; Halves of the same are equal; are deducible from the first two axioms and follow from the definitions. The eighth, ninth, and tenth axioms need no further remark. (Things that coincide are equal; The whole is greater than its part; All right angles are equal.) The eleventh axiom has long been held a deduction from the definition of parallel lines.

§ 23. The twelfth axiom of Euclid is regarded both by Mr. Mill and Dr. Whewell as synthetical. Two straight lines cannot enclose a space is a proposition in the opinion of these gentlemen which is not implied in the meaning of straight line. Professor Bain's answer to this is ample. He remarks that the definition of two straight lines expresses the fact that when two lines are such that they cannot coincide in two points without coinciding altogether they are called straight lines; and coinciding altogether means that there shall be no intervening space: therefore to say that two straight lines can enclose a space is a contradiction in terms.' I cannot avoid expressing my wonder that Professor Bain is able to see anything more augmentative in the axioms of the sums of equals and of mediate coincidence than he sees in this Logic, Induction (London, 1870), p. 210.

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