EDUCATION.

I-ON THE STUDY OF MATHEMATICS,

AS AN EXERCISE OF MIND.*

(JANUARY, 1836.)

Thoughts on the Study of Mathematics as a part of a Liberal Education. By the Rev. WILLIAM WHEWELL, M.A., Fellow and Tutor of Trinity College. 8vo. Cambridge: 1835.

WE saw the announcement of this pamphlet with no ordinary interest, from the subject,-from the place of publication,-and from the author.

The subject is one of great importance in the science of education. Whether, and to what extent, the study of mathematics

*

[In French by M. Peisse; in Italian by S. Lo Gatto; in German, as a separate pamphlet, under the title,-"Ueber ben Werth und Unwerth der Mathematik, als Mittel der hoehern geistigen Ausbildung, Cassel, 1836." To this last there is an able preface; and the translator publishes the paper from an intimate and resistless conviction that the plan of study in some of our new gymnasia comprehends too great a variety of objects, and, especially, lavishes too much time and application on mathematical instruction ;—an instruction without interest to the majority of students, which, at the same time, pays no regard to the differences of natural disposition and future destination, overloads the memory, and compromises the development of the higher mental and moral capacities, while, more especially, it stunts the evolution of that free and independent activity of thought on which a utility for life and a susceptibility for its noblest avocations depend."-M. Peisse has likewise, in his preface, convincingly maintained the same position. This was also eloquently supported, with reference

conduces to the development of the higher faculties, is a question which, though never adequately discussed, has been very confidently and very variously decided. The stream of opinions, and the general practice of the European schools and universities, allow to that study, at best, only a subordinate utility as a mean of liberal education-that is, an education in which the individual is cultivated, not as an instrument towards some ulterior end, but as an end unto himself alone; in other words, an education, in which his absolute perfection as a man, and not merely his relative dexterity as a professional man, is the scope immediately in view. But, at the same time, it cannot be denied, that signs of a revolutionary tendency in popular opinion, touching the objects and the end of education, are, in this nation at least, becoming daily more and more obtrusive; and as the extended study of mathematics is that mainly proposed, in lieu of the ancient branches of discipline which our innovators would retrench, a professed inquiry, like the present, into the influence of this study on the intellectual habits, comes invested, independently of its general importance, with a certain local and temporary interest.

But the centre from which it proceeds, enhances also the interest of the publication. In opposition to the general opinion of the learned world,-in opposition to the practice of all other universities, past or present,-in opposition even to its oaths and statutes, and to the intention of its founders and legislators, the University of Cambridge stands alone in now making mathematical science the principal object of the whole liberal education it affords; and mathematical skill the sole condition of the one tripos of its honours, and the necessary passport to the other::

to this article, by S. Vallauri before, among other auditors, his Majesty of Sardinia; "Thomae Vallaurii De vitiis in puerili institutione vitandis oratio. Habita in Regio Taurinensi Athenaeo, III. nonas Novembres an. M.DCCC.LII." This article was attacked in a pamphlet published by Professor Chevallier of Durham, in the course of the year when it was published; but his opposition being either mere assertion or mere mistake, I do not find it necessary to say anything in reply. In fact, his defence of "The Study of Mathematics as conducive to the development of the Intellectual Powers," may suffice to show how little, even by an able advocate, can be alleged in vindication of their uti lity in this respect at all.

Certain statements in the criticism have also been controverted by Professor Boole in his very able "Mathematical Analysis of Logic," in 1847. I shall con

On Dr Whewell's rejoinder, see the end of the article.]

thus restricting to the narrowest proficiency all places of distinction and emolument in university and college, to which such honours constitute a claim;-thus also leaving the immense majority of its alumni without incitement, and the most arduous and important studies void of encouragement and reward. It is true, indeed, that the effect of this contracted tendency of the public university is, in some degree, tempered by certain favourable accidents in the constitution of more than one of its private colleges; but with every allowance for petty and precarious counteraction, and latterly for some very inadequate legislation, the University of Cambridge, unless it can demonstrate that mathematical study is the one best, if not the one exclusive, mean of a general evolution of our faculties, must be held to have established and maintained a scheme of discipline, more partial and inadequate than any other which the history of education records. That no Cambridge mathematician has yet been found to essay this demonstration, so necessary for his university, so honourable to his science, has always appeared to us a virtual admission, that the thesis was incapable of defence. A treatise, therefore, apparently on the very point, and by a distinguished member of the university, could not fail of engaging our attention; and this, whether it proposed to defend the actual practice of the seminary, or to urge the expediency of a reform.

From the character of its author, the pamphlet before us likewise comes recommended by no mean claim to consideration. Mr Whewell has already, by his writings, approved to the world, not only his extensive acquirements in mathematical and physical science, but his talent as a vigorous and independent thinker. To a narrower circle, he is known as the principal public tutor of the principal college of his university; and in this relation, his zeal, and knowledge, and ability have concurred in raising him to an enviable eminence. Though more peculiarly distinguished by his publications in that department of science so exclusively patronised by the university, he has yet shown at once his intelligence and liberality, by amplifying the former circle of studies pursued in the college under his direction; and, in particular, we are informed, that he has exerted his influence in awakening a new spirit for the cultivation of mental philosophy; in which department he has already introduced, or is in the course of introducing, a series of more appropriate authors than those previously in use.

In these circumstances it was with more than usual expectation that we received Mr Whewell's pamphlet. Its perusalmust we say it?-has disappointed us. The confession is unavoidable. Even the respect which we entertain for the character and talents of the author, compels us to be plain rather than pleasant with his work. As a writer, Mr Whewell has long out-grown the need of any critical dandling; the question he agitates is far too serious to tolerate the bandying of compliments; his authority, in opposition to our conviction, is too imposing to allow of quarter to his reasoning; whilst we are confident, that he is himself too sincere a champion of truth, to accept of any favour but what the interest of truth demands.

We say that we are disappointed with the pamphlet, and this on sundry accounts. We are disappointed, certainly, that its author did not here advocate for the university the liberal views which he had already extended to his college. But taking it for a vindication of mathematical study, as the principal mean in the cultivation of the reasoning faculty,-supposing also that the reasoning faculty is that whose cultivation is chiefly to be encouraged in the liberal education of a university,-considering it, in a word, from its own point of view alone, we say that we are disappointed with it, as failing signally in the accomplishment of what it proposes. In fact, had our opinion not previously been decided on the question, the perusal of this argument in defence of mathematical study, as a useful gymnastic of the mind, would have only tended to persuade us, that in this relation, it was comparatively useless.

Before entering on details, it is proper here, once for all, to premise:-In the first place, that the question does not regard, the value of mathematical SCIENCE, considered in itself, or in its objective results, but the utility of mathematical STUDY, that is, in its subjective effect, as an exercise of mind; and in the second, that the expediency is not disputed, of leaving mathematics, as a co-ordinate, to find their level among the other branches of academical instruction. It is only contended, that they ought not to be made the principal, far less the exclusive, object of academical encouragement. We speak not now of professional, but of liberal, education; not of that, which considers the mind as an instrument for the improvement of science, but of this, which considers science as an instrument for the improvement of mind.

Of all our intellectual pursuits, the study of the mathematical sciences is the one, whose utility as an intellectual exercise, when carried beyond a moderate extent, has been most peremptorily denied by the greatest number of the most competent judges; and the arguments, on which this opinion is established, have hitherto been evaded rather than opposed. Some intelligent mathematicians, indeed, admit all that has been urged against their science, as a principal discipline of the mind; and only contend that it ought not to be extruded from all place in a scheme of liberal education. With these, therefore, we have no controversy. More strenuous advocates of this study, again, maintain, that mathematics are of primary importance as a logical exercise of reason; but unable to controvert the evidence of its contracted and partial cultivation of the faculties, they endeavour to vindicate the study in general, by attributing its evil influence to some peculiar modification of the science; and thus hope to avoid the loss of the whole, by the vicarious sacrifice of a part. But here unfortunately they are not at one. Some are willing to surrender the modern analysis as a gymnastic of the mind. They confess, that its very perfection as an instrument of discovery unfits it for an instrument of mental cultivation, its formulæ mechanically transporting the student with closed eyes to the conclusion; whereas the ancient geometrical construction, they contend, leads him to the end, more circuitously indeed, but by his own exertion, and with a clear consciousness of every step in the procedure. Others, on the contrary, disgusted with the tedious and complex operations of geometry, recommend the algebraic process as that most favourable to the powers of generalization and reasoning; for, concentrating into the narrowest compass the greatest complement of meaning, it obviates, they maintain, all irrelevant distraction, and enables the intellect to operate for a longer continuance, more energetically, securely, and effectually.-The arguments in favour of the study, thus neutralize each other; and the reasoning of those who deny it more than a subordinate and partial utility, stands not only uncontroverted, but untouched,-not only untouched, but admitted.

Mr Whewell belongs to the class of thorough-going advocates; he would maintain the paramount importance of mathematical study in general; but willingly allows the worst that has been urged against it to be true of certain opinions and practices, to