DISADVANTAGES OF MATHEMATICS.

In the first place, I think that the time which is devoted to these subjects viewed as a discipline is too long. While engaged in these pursuits a student is really occupied with a symbolical language, which is exquisitely adapted for the class of conceptions which it has to represent, but which is so very far removed from the language of common life that unless care be taken to guard against the evil, the mathematician is in danger of finding his command over the vernacular diminished in proportion as he becomes familiar with the dialect of abstract science. It must surely be in some degree disadvantageous to train clergymen and barristers for several years to familiarity with a refined and elaborate system of expression, for which they will have little direct use in after life, and to leave them without any cultivation of that ordinary language which is to be the main instrument of service in their future occupations. I estimate at a high value the influences of mathematical study, but I am sorry to see these benefits obtained by the sacrifice or at the peril of any of the qualifications which are necessary for success and for influence in practical life. There is especially one precaution that ought to be taken. The symbols of mathematics are so expressive that the meaning of an investigation can be discovered by a lenient examiner however slovenly or inaccurate the ordinary language may be which serves to connect the symbols. But the evil is so great which may arise from habitual carelessness in English composition, that examiners must be considered injudicious who do not rigidly maintain a good standard of excellence in this matter.

The increasing severity of our competitive examinations seems to aggravate the danger to which I refer. Formerly all students at the university were compelled by their colleges with more or less rigor to conform to a general course of study; ambitious mathematicians had to acquire at least a tincture of classical and general learning; while classical students, often sorely against their will, were compelled at Cambridge to undergo a mathematical training. Much of this system has been recently relaxed or dissolved. Many eminent scholars now regret the exemption of the classical students from nearly all their mathematical training; and it is I believe a still more serious evil if students intent mainly on mathematics are allowed to remain without a due counterpoise of other studies. I do not suppose that the candidates who attain to the highest places in the Mathematical Tripos are deficient in knowledge and interest in other subjects; but I fear that omitting these more distinguished men, the remainder frequently betray a rude ignorance in much that is essential to a liberal education. In our university mathematical training, there is, I think, what may be called a wearisome and unprofitable monotony. I speak of course with respect to the disciplinary character of the process. For example: a person gains a certain knowledge of a new subject, like the Differential Calculus; the fresh conceptions which this involves are important and instructive. But after gaining these conceptions, months may be spent in incessant practice in deductions and problems, so as to leave no weak spot which an examiner's lance could penetrate. Of course if the object is to produce a mathematical professor this elaborate drill may be accepted, though perhaps even then not absolutely necessary. But considered as a part of general education, although the minute accuracy which it tends to produce may be admired, yet it may well be doubted if the profit is commensurate with the expense of time and labor. The process seems a

modern innovation. In the study of mathematics, formerly, as a discipline, a general knowledge of the principles was all that was required; now we insist on a minute investigation of every incidental part of the subject. Exceptions and isolated difficulties seem to receive undue attention, on account of their utility for the examiner's purpose.

The great progress which mathematical science has made in late years, while increasing largely its power, considered as an instrument for the original investigator, has not necessarily promoted its educational value for ordinary students. I remember to have heard from the late Professor Boole, an opinion which I had also formed myself, that the increase in the resources of notation tended to diminish the importance of mathematics as a discipline. If we take, for example, the modern methods of abridged notation in algebraical geometry, while we must admire the genius which has created and developed such a remarkable system of investigation, yet we must see that when it is cultivated for examination purposes there is the great danger that the symbols may be used as substitutes for thought rather than as aids to thought.

EXPERIMENTAL PHILOSOPHY.

Experimental philosophy may be considered one of the most fashionable elements of education at the present time; though perhaps quite recently it has rather declined than advanced in public estimation. The assault which has been made in our time on the monopoly enjoyed by the older studies seems to have been a combined movement in favor of chemistry, natural philosophy, and natural history; and I apprehend that natural history will in the end secure the largest share in the conquest, if indeed it has not already done so. In schools it was doubtless more interesting to many boys to assist at a lecture in chemistry or natural philosophy than to work steadily at classics or mathematics; but chemistry and natural philosophy will in their turn be found dull when compared with natural history, which is associated with the love of outdoor exercise and the desire to catch the lower animals, which are so characteristic of English youth. To have these cherished pursuits elevated into serious duties, and dignified with the name of studies, must have been a joyful surprise to the generation of schoolboys who first encountered the welcome novelties.

I assert then that much of what is called experimental science has no claim whatever to the title; I hope I shall not be supposed to be merely trifling with words, for I believe myself that there is an important truth involved in the remark. The function of experiment, properly so called, in the investigation of the laws and processes of nature can hardly be unduly exalted; but it may be said of the experimenter, as of the poet, that he is born and not manufactured. The lecture rooms of professors of experimental philosophy must be devoted chiefly to the mechanical repetition of familiar processes; the spectators are told what they may expect to scc, and accordingly they see it with more or less clearness of conviction. The result of the whole performance may be that certain facts are impressed on the belief or on the memory, but it is difficult to secure any cultivation of the power of experimenting, or any mode of testing the existence of such a power. I am speaking with reference to teaching such subjects in large classes. It may as before be readily admitted that the force of the remarks will be much weakened in special cases. If, for instance, two or three pupils have the privilege of constant intercourse with some teacher eminent

for his original experimental power, it is very natural that a training of the highest value may thus be secured; but, setting aside such exceptional cases, it may be held generally that little of what is characteristically valuable in experimental philosophy is susceptible of transmission.

It would be absurd to recommend that any subject should be proposed in a purposely repulsive form to students, especially to youth: but, on the other hand, it seems to me a most encrvating practice to shrink from demanding even irksome attention whenever it is neccssary. The lesson that success in any pursuit demands serious toil must be learned eventually, and like most lessons is learned with least pain in early years. I have seen a sort of model lecture on a portion of natural science which was offered to a large public school, to which I should urge no objection if the time that it occupied were taken from play time, but which seemed to me a very unsatisfactory employment of an hour supposed to be devoted to study. Here I may venture to draw attention to the opinion held by the late Dr. Whewell, that natural history, chemistry, and physics should not be made part of the business of schools, but occasionally brought under the notice of the boys by lectures. These occasional lectures might be delivered by the eminent authorities of the period, and thus one serious difficulty would be obviated, namely that those who are absorbed in school work cannot maintain themselves at the current level of these fluctuating subjects, and thus are in the danger of teaching obsolete theories and demolished statements as part of a course the essential virtue of which resides in accuracy of information.

FUNCTIONS OF A UNIVERSITY.

There appear to be three distinct functions which are recognized as pertaining to the university: one is that of examination, one that of teaching, one that of fostering original research. The first of these three has practically been as yet most regarded; and many of us hope that it will in future decline either absolutely or relatively by the increased development of the other two. As to the matter of original research, without asserting that this is adequately regarded, yet we may say that there has been much improvement in recent times. The professorships have in various cases been augmented from some convenient funds, and thus elevated above the miserable pittances of which they formerly consisted; while the permission to retain a fellowship with them, notwithstanding marriage, has indirectly been a great boon to them. Moreover, many of the colleges have now the power to confer a fellowship on any person eminent in science and learning; and thus there is at least the opportunity, in cases where the will also exists, to encourage and assist those who devote themselves to unremuncrative intellectual pursuits.

But it seems to me that the most decided want in the place is an organized system of scientific instruction; and this remains although it would appear that various efforts have been made to supply the defect. For more than a quarter of a century the entire range of mixed mathematics has been represented in university public instruction by courses annually delivered on astronomical instruments, lunar theory, hydrostatics, and optics. Statutes have been drawn up with a view to secure the due distribution of the "various branches of mathematical science on which it is desirable that lectures should be given": and the duty of carrying the statutes into effect has been assigned to the Mathematical Board. The want of a suitable building and collection of instruments has been held to con

stitute the great obstacle to university instruction. The building, however, has been erected, and some advance made towards the formation of a collection of instruments. However the phantom of a well arranged and extensive cycle of public instruction seems still to elude the grasp; instead of it we have ever augmenting examinations. If there is no existing staff to which this public instruction can be assigned one should be forthwith called into existence, with due provision for effective work and reasonable remuneration.

Even if scientific lectures were not properly appreciated there still remains another mode of instruction which ought to be adopted, and which would perhaps be still more valuable than oral teaching; I mean the publication of works on the higher branches of mathematics which should combine, correct, and illustrate what has been scattered through the transactions of societies and the articles of scientific journals. I do not allude to mere academical compendia, of which in general there is a sufficient supply, but to works treating elaborately and fully the highest subjects. The history of science offers us splendid examples of such works; the Mécanique Céleste itself is one of them. For modern instances I may refer to the writings of Lamé, Chasles, Serret, Helmholtz, and Clebsch these men are eminent not only as original investigators and oral teachers, but as the authors of noble treatises. It will be highly advantageous if those who hold appointments which sccure leisure for research will accept it as a duty to compose one systematic work at least of the kind now indicated; it may be safely said that the result will do more for the advancement of science than the production of matter which is merely entombed in the memoirs of learned bodies. Amateurs may in some cases attempt to execute such tasks, but it is obvious that owing to the little leisure they can secure from their necessary avocations they must fall far below the standard which the professional cultivators of science can attain.

PROTEST AGAINST EXCESSIVE COMPETITIVE EXAMINATIONS.

I wish to join my protest, feeble as it may be, with that of many other persons both within and without the university, against the exorbitant development of the system of competitive examinations. We assume in all our arrangements that men will read only what will pay in examinations, and assume it, I believe, contrary to the evidence furnished by other universities, and by our own: and by showing how firmly we grasp this sordid creed ourselves we do our best to recommend it to others. We give our highest honors and rewards for success in special examinations; and thus we practically encourage not the harmonious development of all the faculties of the mind, but the morbid growth of some and the decay of others. We tempt our students to regard degrees and fellowships as the end of life, and not as incentives to manly exertion and aids to pure, unselfish service; we cannot wonder then that not a few who start in their course so well seem to fail; to use Bacon's simile, they resemble the fabled Atalanta who lost the race because she stooped to pick up the golden applès.

Are our students so buoyant after they have obtained their degrees that we can reproach ourselves with having left their craving for work unsatisfied, their energies unemployed? The opinion of many, I believe, is quite the reverse; they hold that we destroy the elasticity of our students by the incessant toil of examinations, that we squander with lavish prodigality the fresh energy of youth and early manhood, and suffer too often retribution in the languor and unprofitableness of maturer life.

First Steps in Teaching a Foreign Language-A Lecture.* Prof. Quick opens his lecture before the College of Preceptors thus: Those of us who have visited the Brighton or the Sydenham Aquarium well know the sight of the sea-anemones. The first impression one gets of them is, that they are merely enjoying themselves, or exhibiting their beauty as a peacock spreads his tail. But if we watch them till a tiny fish happens to stroll their way, we discover then that the anemones are not standing at ease or courting our admiration. No sooner is the fish within reach, than the hitherto placid anemone becomes all activity; the beautiful fibers disappear, and the little fish disappears with them. If we have the patience to await the result, we see the anemone at length open out again, and there reappears, not the fish, but just so much of it as the anemone finds indigestible. The rest has become anemone. Now here we see in a figure the proper attitude and action of the mind of a learner. It keenly desires knowledge; it is on the look-out for it; it seizes on whatever information comes within its reach, and it works upon this information, analyzes it, appropriates all the pith of it, and rejects the useless shell.

After stating the importance of a good method in teaching a foreign language, he reviews the methods of Ascham, Ratich, Hamilton, Jacotot, and Comenius, which we have already described, and concludes with an account of Robertson's method and Prendergast's Mastery System.

Robertsonian Method-Introduction.

The Robertsonian method is known chiefly in France, as a similar method, that of Langenscheidt, is in Germany. Robertson has framed his model book in such a way as to include all the main root words in the French language. When an author sets to work to employ a certain set of words rather than to convey any particular meaning, the composition can hardly turn out a great literary success. Robertson admits that, like Mrs. Malaprop, he forces into the service many poor words that would get their habeas corpus from any court in Christendom. I observe that a disciple of his, Dr. Boltz, who published, two or three years ago, a Robertsonian Introduction to German, has simply taken a tale written in that language, so that he is Robertsonian only in his treatment of the 'Stoff' selected. This treatment reminds one of Ascham's plan, but in some respects it is a great advance upon it. The text is split up into lessons-the early ones consisting of only two or three short sentences. Of each lesson we have three translations-the first a literal interlinear translation, the next one in fair English, and the third a translation, phrase by phrase, in parallel columns. This last is for practice in retranslation, and the pupil is required to study it till he can readily give the foreign equivalent for each phrase. Then the words of the lesson are used for what Mr. Prendergast would call variations—a very valuable feature in this system. Afterwards comes a lexicographical and grammatical commentary on the words of the lesson, about which a vast sea of information is given, altogether beyond the beginner's capacities and requirements. This part, says Robertson, may be omitted-must be, I should say; but some facts about the really important words of the lesson would no doubt be useful.

* The First Steps in Teaching a Foreign Language, with some accounts of celebrated methods. A Lecture delivered at the College of Preceptors (London), Feb. 11, 1875. By Rev. R. H. Quick.