siderable means of instruction, in having our education consist of ancient methods, which though sound and good, may be simplified, extended, and clothed in a deeper significance by newer methods which the teacher himself suggests. The new method comes as a commentary upon the old; and gives to the education so conducted the combined advantages of the stability of a fixed system, and the vivacity of a present reform.

114 It is, therefore, with no want of admiration for the subtilty and comprehensiveness of intellect which has been shown in many recent views of Algebra and Geometry, that I recommend our adhering to the ancient methods of treating such subjects, so far as the general purposes of a liberal education are concerned. Such views are truly admirable, as corrections, and extensions, or it may be as the antithesis of the established and traditional modes of treating the subjects; but if they were to become themselves established and traditional, they would be (as I have already endeavoured to show) far less effective in the discipline of the reason, than the older methods; besides depriving us of the continuity of that intellectual tradition which I have already spoken of, as one of the great ends of mathematical teaching in the course of a liberal education. I recommend the rejection, in our ordinary educational system, of the many novelties in notation and expression which have recently appeared in our Cambridge Mathematical works; but I admire the mathematical talents of those who have produced these works; and I think that such speculations are both very remarkable manifestations of mathematical skill and thought, and very fit subjects of attention for our mathematical students, where they reach the higher stages of their progress.

115 There is one leading question, in such an education as we are contemplating, on which I have already spoken, but on which it may not be useless to

add a few words :-I mean, the question whether both mathematical and classical instruction should be considered necessary in the case of every student. It is sometimes said that we shall educate men better, by encouraging in each that study for which he has talent and inclination;-not tormenting the man of classical taste with fruitless lessons of algebra, or the man of mathematical intellect with obscure passages of Greek. It is said, sometimes, that by such a genial education alone, do we really educate the man, or bring out his genius;-that the seeming of mathematical prowess, or of classical learning, which we wring by force from ungenial and unwilling minds, is of no value, and is no real culture. But to this we reply, that if men come really to understand Greek or Geometry, there is then, in each study, a real intellectual culture, however unwillingly it may have been entered upon. There can be no culture without some labour and effort; to some persons, all labour and effort are unwelcome; and such persons cannot be educated at all, without putting some constraint upon their inclinations. No education can be considered as liberal, which does not cultivate both the Faculty of Reason and the Faculty of Language; one of which is cultivated by the study of mathematics, and the other by the study of classics. To allow the student to omit one of these, is to leave him half educated. If a person cannot receive such culture, he remains, in the one case, irrational, in the other illiterate, and cannot be held up as a liberally educated person. To allow a person to follow one of these lines of study, to the entire neglect of the other, is not to educate him. It may draw out his special personal propensities; but it does not draw out his general human Faculties of Reason and Language. The object of a liberal education is, not to make men eminently learned or profound in some one department, but to educe all the faculties by which man shares in

the highest thoughts and feelings of his species. It is to make men truly men, rather than to make them men of genius, which no education can make them.

116 But even with regard to men of genius, it is not true that they have generally been men of one kind of cultivation only, or capable only of one kind of intellectual excellence. The case has been quite the reverse. During the middle ages, and down to the last century, the greatest mathematicians were almost invariably good classical scholars; and good scholars were almost invariably well acquainted with mathematical literature, and often very fond of it. And this connexion, in the main, has continued to our own day, so far as the mathematics and classics belonging to a liberal education are concerned. Not to speak of living persons, whose career at Cambridge might be adduced to prove this, the greatest Greek scholar of the last generation, Porson, was fond of Algebra, and was a proficient in it; and if we run over the highest wranglers of the last sixty years, we find at every period, men known to be well versed in classical literature, as Otter, Brinkley, Outram, Raincock, Wrangham, Palmer, T. Jackson, R. Grant, and many others.

117 Indeed, there can be no doubt but that the clearness of mind and vigour of character which make a man eminent in one line of study will also enable him to master the elementary difficulties of another subject, if it is fairly brought before him as something which must be done; although, if it be presented to him as a matter of choice whether he will make the attempt, caprice, fastidiousness, and the pleasure of doing what he can already do easily and well, may make him turn with repugnance from a subject in which he has not learned to feel any interest.

118 To which we may add, that to be able to command the attention and direct the mental powers, so as to master a subject which is not particularly

attractive to us, is a very valuable result of mental discipline. Whatever acuteness or sagacity a man may have on a special subject, if he be so helpless or so fastidious that he cannot employ his thoughts to any purpose or any other subject, we cannot consider him as a well cultured person: nor ought we to frame our education so as to give to men such an intellectual character.

We come back therefore to the doctrine stated sometime ago, that mathematical and classical studies, both Permanent and Progressive, are the leading and essential parts of a liberal Education for Englishmen. We have already, in some measure, pointed out the kind of Mathematics and of Classics which are to form the matter of such teaching as we contemplate; but we must now speak more at length of the methods according to which this teaching is to be conducted.

CHAPTER II.

ON THE METHOD OF TEACHING IN CLASSICS AND MATHEMATICS.

SECT. 1. Of College Lectures and Professorial

Lectures.

119 HAVING Considered in some measure what is to be taught to students, as the mathematical and the classical part of a liberal Education, I now proceed to make some remarks on the manner of teaching. Different methods may be adopted for this purpose; and these differences of method lead to such differences of character and result in the education so given, as are deserving of our very serious consideration.

The Classical and Mathematical Education, of which we have to speak, may be considered as consisting mainly in that given at School, and that given at the University. And first, I shall speak of Classical Education.

120 The teaching of Latin and Greek at school is necessarily in a great measure oral. After preparing himself by the aid of his dictionary and grammar, the boy says his lessons, in which he construes a portion of Latin or Greek aloud, and then, if required, parses some parts of it, being often corrected and informed by his master in points where he is wrong or ignorant. This being done by boys collected in classes, each boy is instructed, not only by what is said to himself, but also to his class-fellows. He may profit by the notice taken of their errours and defects, as well as of his own, and by their knowledge of what he does not know. Besides this, he is interested by the community