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higher intellectual qualities, or to the peril of moral excellence itself. This may be an extreme case; but let us take a more moderate example. Soppose a youth to exhibit a fondness for imaginative exercises and literature, which may be the dawn of poetical genius. Moreover let us suppose that, in spite of the authority of an ancient sage, we find a function and a value for poets in our commonwealth; then we may conclude that we ought to stimulate the imagination: though perhaps it might be for the true happiness of the individual if the memory and the reason were trained rather than a faculty which is already unduly developed.
UXDUE INFLUENCE OF SPECIAL PRIZES AND EXAMINATIOX.
Although we have no such despotic power as to compel an individual to cultivate just that faculty which seems strongest, yet by our system of competitive examinations and prizes we tend to the same result. We take a boy at school who seems to exhibit an aptitude, say for mathematics, and foster that taste in every way we can. The boy comes to the University; he is already saturated with mathematics, and so must have almost exhausted the special benefit which that study is held to confer; at the same time, in other departments of knowledge, such as languages, history, natural science, he may be very deficient. Nevertheless he is kept for three or more years still at the old pursuits, exercising only those energies which have been abundantly developed, and leaving others dormant which have been too long neglected. If our object is to train mathematical teachers and professors this may be defended, though perhaps with only partial success; but if, as we commonly maintain, our object is to cultivate the mind so as to render it well fitted for future exertion in any direction which has to be followed, our arrangements are open to serious doubts.
The excessive cultivation for examination purposes of one department of knowledge to the exclusion of others seems to me one of the great cvils of our modern system of bribing students by great prizes and rewards to go through our competitive struggles. We are in danger of giving up all pretence of a general course of training for youth, and of allowing and even encouraging boys to select some special subject which they fancy they prefer, or rather perhaps which they least dislike. I should desire quite a contrary system; a scheme of study and examination should be drawn up after due deliberation, and all candidates be required to pass through this before the avenues to special distinction were opened. In theory, perhaps this is still attempted; but in practice we seem to deviate from such a course more and more every year at Cambridge. For instance, students of classics are no longer compelled, as they formerly were, to pass a mathematical examination for their degree; and for the most part undergraduates in the colleges are excused from attendance' at lectures on the subjects which they do not profess to cultivate. Even where an attempt is made to prescribe some general course the standard in each department is fixed so low as not to ensure more than the simplest rudiments of knowledge.
EXAMINATION VALUE OF SUBJECTS VALUED.
As we must employ some mode of testing the diligence of teachers and the attention of pupils, it seems inevitable that there must be processes of the nature of examinations ; hence it is important to pay some attention to the adaptability of subjects to the exigencies of examinations. It seems to me that the older subjects, classics and mathematics, are strongly to be recommended on the ground of the accuracy with which we can compare the relative performance of the students. In fact the definiteness of thesê subjects is obvious, and is commonly admitted. There is, however, another advantage, which I think belongs in general to these subjects, namely, that the examinations can be brought to bear upon what is really most valuable in the subjects. It is of course easy to say that the art of examination by long practice on these subjects has arrived at a degree of excellence far beyond what ought reasonably to be expected in the case of studies of quite recent popularity; but this does not seem to me to explain the matter completely. Take, for instance, mathematics, and observe how real and fresh the examination papers can be made; they in fact abound in new results which are quite commensurate in importance and interest with the theorems previously established and studied. Now, for a contrast, take the subject of history: this may be readily admitted to be important and instructive especially for the original inquirer who has to seek for evidence, to estimate its value, and to combine it in a consistent whole. But it may be seriously doubted whether the valuable parts of the subject can be developed in our usual systems of examination. From the cases, not, I admit, very numerous, which have fallen under my own notice, I have formed an unfavorable judgment on this matter; it appears to me that we find in examination papers chiefly dates and striking, obvious events, which form rather the skeleton of history than history itself; that the mere receptivity of the students is all that can be tested, to the exclusion of the faculties of comparison and of judgment; though these may be well developed by original researches in the subject. Thus, briefly, it seems to me that much of what constitutes the real value of mathematics can be tested by examinations, but in history there is little of this merit.
[Experimental and Natural Sciences arc considered by this author as not satisfactory for examination purposes.]
SPECIAL ADVANTAGES OF MATHEMATICS. Leaving aside such points as are weH known and obvious, I should like to draw attention to the inexhaustible variety of the problems and exercises which it furnishes; these may be graduated to precisely the amount of attainment which may be possessed, while yet retaining an interest and value. It seems to me that no other branch of study at all compares with mathematics in this. When we propose a deduction to a beginner wo give him an exercise in many cases that would have been admired in the vigorous days of the Greek geometry, Although grammatical exercises are well suited to ensure the great benefits connected with the study of languages, yet these exercises seem to me stiff and artificial in comparison with the problems of mathematics. It is not absurd to maintain that Euclid and Apollonius would have regarded with interest many of the elegant deductions which are invented for the use of our students in geometry; but it seems scarcely conceivable that the great masters in any other line of study could condescend to give a moment's attention to the elementary books of the beginner. The possibility of the early employment of the constructive and imaginative faculties is an important gain for many students who become weary of the prolonged and unvaried exercise of mere receptive attention. In the pursuit of a new language we may secure advantages of a similar kind but probably of inferior value; but in the early stages of most studies
there seems nothing to correspond : it is scarcely conceivable that examination papers in history or the natural sciences can offer any. tolerable equivalent in merit and importance to the problems of mathematics.
Another great and special excellence of mathematics is that it demands earnest voluntary exertion. It is simply impossible for a person to become a good mathematician by the happy accident of having been sent to a good school ; this may give him a preparation and a start, but by his own continued efforts alone can he reach an eminent position. The rough processes by which prizes are awarded to the possessors of knowledge regard only the results offered for inspection, and overlook the finer gradations of merit which depend on the mode of acquisition. Suppose, for example, that rewards are bestowed for the cultivation of modern languages; a person who obtains the reward may have earned his distinction by his own persevering application, mainly or exclusively, but on the other hand he may owe it to the fortunate incident of residence in a foreign country, or of habitual intercourse with those who spoke the language as their vernacular. The resulting amount of knowledge is no just index of the labor and perseverance which have been expended in gaining it; the credit to be properly assigned for the accomplishment may indeed belong to the successful candidate, but it may, and perhaps more justly, be attributed entirely to his friends and relatives.
A similar consideration applies, though with diminished force, to the study of the classical languages; the foundation of knowledge in these subjects can be laid in years so early that the pupil exerts but slightly his own will; his success is a combination depending indeed partly on his own ability and application, but still more on the judgment or kind fortune which deposited him in a good school.
We repeatedly see youths enter the universities whose position in the final classical examination is already practically assured; but distinguished success in the mathematical competition cannot be confidently expected, whatever be the ability of the candidate, unless he is willing to subject himself to steady and continued discipline. In whatever line of study distinction is sought the advantage of good teaching is great; but probably among all the pursuits of the University mathematics preeminently demand self-denial, patience, and perseverence from youth, precisely at that period when they have liberty to act for themselves, and when, on account of obvious temptations, habits of restraint and application are peculiarly valuable.
Nor do I know any study which can compete with mathematics in general in furnishing matter for severe and continued thought. Metaphysical problems may be even more difficult; but then they are far less definite, and, as they rarely lead to any precise conclusion, we miss the power of checking our own operations, and of discovering whether we are thinking and reasoning or merely fancying and dreaming. I speak now, as on former occasions, of studies as they present themselves to minds of average power and of ordinary conditions. For persons of exceptional ability any intellectual pursuit may prove stimulating and strengthening. In other words, discoverers and original geniuses form a class apart; we may admire them, but we should not inadvertently assume that their pursuits when adopted by inferior disciples will be as vivifying as to the great masters themselves.
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, 80 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 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 sce, 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