The most important question connected with tho mathematical sciences is the manner in which they should be taught as disciplines of the mind. This concerns all who consider any branch' of knowledge in that light ; and, as education spreads, this view of the subject will become of more and more consequence. Vitally essential as these sciences are to the advancement of the arts of life, we feel, in regard to this branch of their utility, in the situation of those who know that they must and will be attended to, because their cultivation is neces sary to the supply of wants which all can feel, and the promotion of interests which all can understand. It is not so with the first-mentioned object of their study, but rather the reverse ; for the wants of life being as easily supplied by the results of an illogical as of a logical system (provided only that vicious reasoning be not allowed to produce absolute falsehood), the facilities which laxity of reasoning affords in the mere attainment of results will always recommend it to those whose main object It is to apply the fruits of calculation to the uses of life. Such has been, and, wo are afraid, will continue to be, the tendency of the great advance which the last century made in application.
All we should positively contend for is tho necessity of making the entrance to the study as strict and rigorous as reason can make it, to all who are to receive liberal education. In the higher branches of mathematics many opinions prevail, and it would be impossible to make a formal standard of rigour. Add to this, that, with a certain degree of experience in the estimation of reasoning, it may be compa ratively immaterial which of several different methods is adopted ; either may be rigorous if properly understood, and if the habit of reducing looseness of phraseology, or dangerous abbreviation of logic, to strict definition and formal deduction, shall first have been well formed. To assist in gaining this end, we should propose— First, that no student should be allowed to enter upon the use of language in mathematical reasoning until he has acquired more acquaintance with the nature of assertion, denial, and deduction, than can be obtained from previous education as now given : this to be done by the study of the first elements of logic.
Secondly, that no consideration of facility or practical convenience should prevent the first study of arithmetic and geometry from being strictly demonstrative, and formally rigorous : rigour being defined to consist in explicit statement of every assumption, and logical treatment of every inference.
On the first of the preceding recommendations we shall only observe, that in order to distinguish between accurate and inaccurate inference, an acquaintance with the exact extent of meaning of the several modes of communication is absolutely necessary. This cannot be learnt from the ordinary use of language, which abounds in implications to be sug gested by the circumstances of the speaker, the context of the words, or the tone in which they are delivered. Before the phrases of demon
stration can be made to convey a meaning limited in both directions, the strict use of language must be made a study ; if this be neglected, the words of any book may pass between the teacher and the learner, but no precaution has been taken to secure their conveying the proper meaning, neither too much nor too little.
t On the second recommendation, we must first explain that we hold many points of controversy very cheap, so far as they concern the discipline given by the most elementary branches of mathematics. It matters nothing, in our view of the case, whether an axiom be really incapable of proof, or whether the substitution of another would or would not place the science on a more simple basis. The habit to be formed is that of tracing necessary consequences from given premises by elementary logical steps : the premises to be true or false, the con sequences to be true if the premises be true, and dubious (not neces sarily false) if the premises bo false. The only error which, at the stage in question, it is intended to avoid, is the deduction, as a neces sary consequence, of that which is not so. The mind of the learner, however, is allowed to dwell too much at the outset on the absolute truth or falsehood of the conclusions, to the neglect of their connection with the premises : hence it arises that when a process occurs in which it is essential to examine that connection for its own sake, it is the universal complaint that beginners find difficulty and obscurity. From what other cause arises the dislike of the indirect demonstration Unfortunately for the mental progress of the student, be is often allowed to use premises of an easy form, in cases whero complete preparation for the subject would require more extended first prin ciples and greater prolixity of deduction. To this, as before observed, no objection can be taken in itself, provided that no consequences be admitted except the legitimate ones. But something more is admitted: the pupil is presented, in consideration of his attention to one set of premises, with the consequences of another, and is allowed to make believe that he has come fairly by the latter. Thus, by a theory which applies only to the ratio of number to number, he is permitted to draw general conclusions upon all ratios. When, in opposition, we advise that the first studies should be demonstrative and rigorous, we do not imply, for instance, that the more difficult system should in all cases be preferred to the less complete but more simple : we confine ourselves to insisting that whatever the premises may be, the conclusions should really follow ; and that if the latter be necessarily of a limited character tho limitation should be stated.