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to apprehend it quite thoroughly; whereas reasoning in the logical whole with the Aristotelic syllogism as the unit form requires more mental discipline and maturity. Hamilton impetuously declares "mathematics not a logical exercise.' It would perhaps be wiser to hold with Coleridge that "Mathematics is no substitute for Logic," and to consider mathematical studies as the proper discipline preparatory to logical studies.

It will be well to observe that the distinction taken between logical and mathematical reasoning is not identical with the familiar distinction between moral reasoning and demonstration. Moral reasoning, better called dialectics, often occurs in the quantitative whole, and is then mathematical, yet it always involves more or less uncertainty. Demonstration is in many cases not quantitative or mathematical, but always carries with it certainty. The difference between these is that any dialectics involves to some extent empirical matter, and hence falls short of certainty; whereas demonstration is exclusively from intuitive principles, and carries their necessity along with it. This distinction, then, is not grounded on anything peculiar in the nature of the reasoning employed, which in all cases carries with it just the same approximation to certainty that belongs to the premises, but it is found in the nature of the premises themselves. According to its definition by Aristotle, demonstrative reasoning, producing scientific knowledge in the strictest sense, requires a conviction of the certainty of the propositions laid down. His scholastic followers devised the following syllogism as a specimen of the "Demonstratio potissima:"

Omne animal rationale est risibile; Omnis homo est animal rationale; ergo, Omnis homo est risibilis,

Here is complete identity in the terms, and the reasoning may be readily construed in the mathematical whole; but its major premise is empirical, not intuitive, not a priori, and hence it falls short of demonstration. In moral reasoning we proceed from what is granted

1

1 See in Discussions, Education, Article 1st, "On the Study of Mathematics as an Exercise of Mind." See also an article in the Athenæum for Aug. 24th, 1850.

2

The distinction drawn between mathematical and logical reasoning implies that the mathematical is not logical. The latter term, unfortunately, is used thus in a specific sense. In its general sense all reasoning is, of course, logical.

Anal. Post. i, 2, 1. Aristotle treats of demonstration in the Posterior Ana lytics, especially in chs. i-xiii, drawing his illustrations from pure mathematics.

by the disputant; the principia must first be allowed. In demonstrative reasoning there is no concession; or rather there can be no disputant. Pure mathematics, which is strictly demonstrative, furnishes the clearest illustrations of quantitative reasonings.

§ 2. Let us, then, turn our attention to pure mathematics, and therein to synthetical geometry, to observe the application of quantitative reasoning, and to ascertain how truly and best to exhibit its logical process. We find that geometry makes some use of qualitative reasoning, as when it has proved of triangles in general, or of the genus, that the three angles are together equal to two right angles, it afterwards applies this truth to the several species of triangle—the equilateral, the isosceles, the scalene. We find, also, that it sometimes uses comparative syllogisms, but that by far the greater part of its mediate inferences are in equivalent syllogisms.

Geometry, which is the science of spatial magnitudes, supplies itself at the outset with a series of technical terms by means of definitions analyzing our complex notions of various magnitudes. It then lays down certain postulates concerning these. Thirdly, it states indiscriminately certain axioms. These are, however, of two kinds: 1st, Certain synthetical judgments, stating the self-evident properties of spatial magnitudes, such as "Two straight lines cannot enclose a space" (Ax. x); and, 2d, Certain analytical judgments, such as "Things equal to the same are equal to each other" (Ax. i). According to Kant, the first are geometrical axioms proper, and must be assumed as intuitively evident before any of the more complex relations can be known by demonstration. They constitute the ultimate premises from which the science proceeds, and are, therefore, its peculiar basis. Those of the second class express general conceptions of equality and inequality relative to magnitudes. They are derived from the Primary Laws of Thought as applied to quantity, and, corresponding to the Canon and general rules of the qualitative syllogism, govern, in a mode entirely similar, our inferences in the quantitative whole.*

It has, however, been usual for logicians to regard these analytical

Axiom 1st of Euclid (given above) is the Canon of mediate inference. Nos. 6 and 7 are merely modified statements of the same. The other analytic axioms, Nos. 2 and 3, 4, 5, which are deducible from it, are Canons of immediate inference, corresponding to "complex conceptions" (Part 3d, ii, § 5). E. g., As from “A horse is an animal," and "Whatever is young is strong," we immediately infer "A young horse is a strong animal," so under the axiom "The sums of equals are equal," we can immediately infer from ab, and c=d, that a+c=b+d.

axioms, together with the synthetical, as ultimate premises in geometry, and, in exhibiting the logical analysis of a demonstration, to place one or the other as the major premise of nearly every syllogism. E. g.:

Magnitudes which are equal to the same are equal to each other;

Magnitudes equal to the adjacent exterior and interior angles of a triangle are equal to the same;

.. They are equal to each other.

Magnitudes equal to the adjacent exterior and interior angles of a triangle are equal to each other;

The three interior angles and two right angles are equal to the adjacent exterior and interior angles;

.. They are equal to each other.

All this is very true and formal, but very prolix and operose. Much in this way Mill exhibits the analysis of Euclid's Proposition v, bk. i;* and a similar analysis of the same proposition from certain old scholastic logicians may be found in Mansel's Aldrich."

Now it is very possible to exhibit an analysis of arguments in the logical whole in the same manner, making one of the dicta of Aristotle the major premise of the syllogism; but both process and result would be cumbersome and artificial. It is far simpler, clearer, and more natural to treat geometrical reasonings as we treat qualitative reasonings. Let us take the analytic axioms as canons governing the form and authorizing the process, and develop the demonstration by a direct chain of quantitative syllogisms. If you ask me to justify my Canon, I do it, as I justify Aristotle's dicta, by deducing it from the Primary Laws. The above syllogisms would then reduce to the one following:

The interior angles of a triangle are equal to an adjacent exterior and interior angle;

But these are equal to two right angles;

.. The interior angles are equal to two right angles.

The expression is rendered more facile by the use of a lettered figure, as is customary, whereby two or three letters take the place of a verbal description of a part. This method of exhibiting the logical analysis of a geometrical proof is not only far simpler, shorter, and more direct than the usual way, but it seems to me to correctly represent the actual mental process, which the other does not.

§ 3. "This simple reasoning," says Dr. Reid, "cannot be brought into any syllogism in mood and figure:

A is equal to B;

B is equal to C;

.. A is equal to C."7

8

And hence this eminent philosopher rejected Logic. It is remarkable that Bain uses the following language: "Logicians are aware that the form 'A equals B, B equals C, therefore A equals C,' is not reducible to the syllogism. So with the relation of 'greater than' in the argument a fortiori. Yet to the ordinary mind these inferences are as natural, as forcible, and as prompt as the syllogistic inference." He ought, then, to follow Dr. Reid, and give up Logic. Reid means to say that, taking the copula, according to approved logical rule, to be "is," and all that follows it to be the predicate, we have in this reasoning four terms: 1st, "A;" 2d, "equal to B;" 3d, "B;" 4th, "equal to C;" and this is unavoidable, so that this simple and unquestionably good inference is, according to the rules of your boasted Logic, the fallacy of Quaternio terminorum! Away with it!

The demand is to construe this quantitative reasoning as a qualitative syllogism subject to Aristotle's Dictum de omni. A and B are presumed to be two different things. But how much of A is here thought? Only one mark, its quantity. And so of B. Hence the first premise becomes "The quantity of A is equal to the quantity of B;" "The cost of the museum is equal to the university debt;" i. e., these two quantities are equal. But the mere quantity of a thing is a pure abstraction, and the two quantities, taken apart from all other attributes, are, if absolutely equal, indistinguishable in thought, and therefore are to us the same. Hence the true interpretation of the thought, and its full and accurate expression is, "The quantity of A is the quantity of B;" "The amount of the cost of the museum is (the same as) the amount of the university debt;" $75,000 is $75,000, indistinguishably. A mere form of words cannot bind Logic, which postulates to interpret and express the thought. Now, with our proposition in this form, no difficulty remains; for we may transfer to the logical whole, taking the terms as coextensive, and yet think the subject as contained under the predicate. Our syllogism, then, is Barbara. But all this should not be required. The phrase "is equal to" is

See Hamilton's Reid, p. 702.

8

Logic, p. 183. The treatment of this, and the cases discussed in the next section, by Mansel in his edition of Aldrich, Appendix, Note D, is quite unsatisfactory.

properly to be viewed as the copula interpreted. The same demand might be made to bring "A is contained under B," or "A is a kind or species of B," or "A has for one of its marks B," under the rule that the is is the copula, and what follows is the predicate. Then, upon the result, the demand might be repeated, and so ad infinitum. So far of quantitative reasoning in the equivalent degree, misnamed the "positive" degree.

§ 4. Propositions in the comparative degree have for their copula "is greater than," or its correlative, "is less than," for which the mathematical sign of inequality may be substituted. The typical form of the syllogism is:

A > B;

B > C;

.. A > C.

Simply converting these propositions, we invert the meaning of the copula and read:

B is less than A;

C is less than B;

.. C is less than A.

The planet Jupiter is greater than the earth;

The earth is greater than the moon;

.. The planet Jupiter is greater than the moon.

The axiom governing this class of syllogisms may be stated thus: What is greater than a greater is greater still than the thing.'

10

What was said in § 1 respecting the elimination of Conversion, Figure, and Mood is to be applied also to syllogisms of comparison. We cannot, however, say as much for the simplicity of this reasoning. For, be it observed, the premises authorize more than the strictly logical conclusion states. This excess is usually expressed thus:

.. By so much the more is A greater than C.

This sort of argument is called a fortiori, which may be understood to mean "by a stronger reason," and the conclusion expressed thus: Therefore a fortiori A is greater than C.

Such a conclusion can be reached only in the affirmative mood; so we may define the argument a fortiori to be a mathematical affirma

10 In pure mathematics this syllogism is used but rarely as compared with the syllogism of equivalence. We find, however, that Euclid demonstrates by aid of it Propositions vii, xvi, xvii, and others of his first book.