cised by hourly practice, it becomes gradually fit for higher efforts, and capable of the severer training required when it shall come to deal with wider subjects, or with abstract ideas. The field for comparison and judgment then becomes so extensive, and presents so many difficulties, the analogies are often so remote from casual perception, and the relations so compli cated, that, although the actual process of reasoning remains the same, long and careful training alone enables us to carry on that process securely. We have acquired sound habits of reasoning when this process has become familiar; when thought habitually works in that mode which brings most rapidly to light the true relations and consequences of whatever it contemplates.

Whenever we reason, we consider the relations of certain things to each other, in order to discover the consequence which must necessarily flow from those relations, according to certain fundamental principles, or to previously demonstrated or acknowledged truths. We either assume certain facts or propositions to be true, and from them draw a conclusion which the mind admits to be an inevitable consequence of them; or we state those facts or propositions as hypothetically true, and show the conclusion which must in like manner inevitably follow, if the hypothesis be correct. For example, we may say, "This woman is a good mother, therefore she will watch over her children's education "; or, "If this woman be a good mother, then she will watch over the education of her children." In the former statement, a fact is assumed as certain, and a consequence drawn conclusively from it; in the latter, the fact is stated so as to admit of doubt as to its truth, and the conse quence is shown to follow only if the truth of the fact be first proved.

All reasoning consists of this same process of drawing an inference or a conclusion from certain premises. Stated at full length, it takes the form of the logical syllogism; for in

stance:

All good mothers watch over their children's education;
This woman is a good mother;

Therefore, she will watch over her children's education.

In the briefer mode of stating it in common language, one of the premises is suppressed; but it evidently is in the mind, and is implied by the word therefore, which marks the conclusive nature of the inference drawn. The suppressed premise would, in fact, be the answer, should the use of that word be questioned. 66 Why do you say, she will therefore watch over her children's education? Answer. "Because all good mothers watch," &c. The general fact or proposition thus expressed in the major premise of the syllogism, and assumed though not stated in the common form of argument, is the groundwork, -the acknowledged, or proved, or self-evident truth, on which further conclusions are to be based. It may require itself to be proved by a process of reasoning, as in the instance above given, or it may contain a self-evident maxim, requiring no proof, as in the following:

Things that are equal to the same, are equal to one another;
A and B are each equal to C;

Therefore, A is equal to B.

But in either case the process by which the reasoning is carried on is the same.

The syllogism, then, is not a form for scholastic disputation, or a peculiar mode of reasoning adapted to particular subjects; but the detailed form of every process of reasoning, of that operation which the mind performs every time it draws an inference or a conclusion on any subject whatever. When we speak, therefore, of logical, or mathematical, or moral reasoning, we mean no more (as far as the expression is correctly used) than to distinguish between different classes of subjects, with reference to the different degrees of proof they are susceptible of; that is, to the nature of the premises we argue from, while the mode of reasoning remains the same in all. The identity of logical principles with geometrical axioms is a

further proof of this fact. When the logician says, "If two

terms agree with one and the same third, they agree with each other," it is evidently only another mode of expressing the axiom in geometry, referred to above, "Things which are equal

These are not pecu

to the same, are equal to one another." liar principles of particular sciences, but formulæ of the essential laws of thought itself.*

A mathematical proof, like every other argument, may then be expressed by a series of syllogisms, the difference between one kind of proof and another being in the nature of the prem ises. In a mathematical proof, however long and subtile it may be, however much beyond the grasp of ordinary minds to fol low it out, yet each distinct syllogism which forms a link in the demonstration consists of premises that are strictly deduced from self-evident principles; whereas in reasoning upon subjects which do not admit of rigid demonstration, the premises of each syllogism may be as much open to doubt and dispute as the point sought to be established upon them. The logical form cannot secure us against these sources of error; it can only insure that the process of reasoning itself shall be fairly conducted; that from the data given,- such as they are,the relation of ideas shall be clearly traced, and the conclusion fairly drawn. It strips a proposition of the disguise which either rhetoric or clumsy statement may have thrown over it, by showing in what part of the argument confusion or ambiguity may have crept in; it leads to the detection of their source, and thus exposes the fallacy or sophistry that may lurk under apparently coherent and undeniable propositions. It is for its use in this respect that some acquaintance with the prin ciples of logic is so essential to all who would train themselves to reason closely and correctly, and to guard against being misled by the inconclusive or sophistical reasoning of others. Error, in reasoning as in action, seldom stands alone; every false argument that we accept is too likely to become the ground of further conclusions, which in their turn must be equally false; it is a vicious link in the chain which connects all our knowledge on one subject, and vitiates, therefore, in a greater or less degree, the value of the whole.t

*See Whately's Logic, Introduction, and Book I. §§ 1, 2, 3, 4. See Whately's Logic, Book III., on Fallacies.

The perfect accuracy of mathematical terms, each being strictly defined, the closeness and indisputable grounds of the reasoning employed, the clearness and certainty of the results obtained, make mathematics the best school for strengthening the reason and giving precision and certainty to its operations. Even if other studies may be allowed to be of equal advantage in training the mind to acuteness, to searching investigation and power of abstraction, no other gives in the same degree the habit of accuracy, because none require it so imperatively; or rather, (for it is false to say that any serious subject requires less accuracy than another,) none detect so surely the want of it. Accuracy is the first and essentially necessary quality to aim at. Acuteness of reasoning, however valuable, however desirable, may yet be dispensed with by the generality even of educated persons; but accuracy is indispensably requisite for every moral as well as intellectual purpose of reasoning, the essential condition of sound judgment on whatever subjects the latter is exercised.

Nor is any severe course of mathematics necessary to aid us in this respect; if so it would be a vain recommendation, to the majority of women especially; the mental training this study affords may fortunately be attained at a far less cost of labor. The mere rudiments of mathematics are sufficient to answer the purpose in some degree, that is, to imbue the mind with the principles of just reasoning, to teach the nature of a proof, the value of exact definitions, the method by which strict investigation is carried on step by step, and rigorous conclusions drawn out. These are advantages which we may, perhaps, labor long in other branches of knowledge without acquiring, but which the study of one book of Euclid, with a view to attaining them, can hardly leave us without.

It is on this account that Locke says he would have every body learn something of mathematics, "not so much to make them mathematicians, as to make them reasonable creatures."* So, likewise, an eminent philosopher of our own day con

*

Essay on the Conduct of the Human Understanding, § 6.

siders some knowledge of abstract science "highly desirable in general education, if not indispensably necessary, to impress on us the distinction between vague and strict reasoning, to show us what demonstration really is.”* The same author also strongly exemplifies, in another point of view, the advantage of mathematics as a training for the reason, namely, as saving the mind from all groping and hesitation, from all confusion or possible misapprehension of terms. Speaking of the abstract sciences generally, he says, "Their objects are so definite, and our notions of them so distinct, that we can reason about them with an assurance that the words and signs used in our reasonings are full and true representatives of the things signified; and, consequently, that when we use language and signs in argument, we neither by their use introduce extraneous notions, nor exclude any part of the case before us from consideration. For example: the words space, square, circle, a hundred, &c., convey to the mind notions so complete in them. selves, and so distinct from every thing else, that we are sure when we use them we know and have in view the whole of our meaning. It is widely different with words expressing natural objects and mixed relations. Take, for instance, iron. Different persons attach very different ideas to this word. One who has never heard of magnetism has a widely different notion of iron from one in the contrary predicament. The vulgar who regard this metal as incombustible, and the chemist who sees it burn with the utmost fury, and who has other reasons for regarding it as one of the most combustible bodies in nature; the poet who uses it as an emblem of rigidity, and the smith and engineer, in whose hands it is plastic and moulded like wax into every form; the jailer who prizes it as an obstruction, and the electrician who sees in it only a channel of open communication, by which that most impassible of obstacles, the air, may be traversed by his imprisoned fluid, have all different, and all imperfect, notions of the same word. of such a term is like a rainbow,

The meaning

every body sees a different

* Sir J. Herschel's Discourse on the Study of Natural Philosophy, p. 22.