THE LIBRARY 
 
 OF 
 
 THE UNIVERSITY 
 OF CALIFORNIA 
 
 LOS ANGELES 
 
 Ex Libris 
 
 SIR MICHAEL SADLER 
 
 ACQUIRED 1948 
 
 WITH THE HELP OF ALUMNI OF THE 
 SCHOOL OF EDUCATION
 
 ON THE PRINCIPLES 
 
 or 
 
 ENGLISH 
 UNIVERSITY EDUCATION; 
 
 BY 
 
 THE REV. WILLIAM WHEWELL, M.A., 
 
 FELLOW AND TUTOR OF TRINITY COLLEGE, CAMBRIDGE ; AUTHOR OF 
 A HISTORY OF THE INDUCTIVE SCIENCES. 
 
 THE SECOND EDITION, 
 
 INCLUDING ADDITIONAL THOUGHTS ON THE 
 STUDY OF MATHEMATICS. 
 
 SiaS(a<rovtriv aAAijXois. 
 
 LONDON: 
 
 JOHN W. PARKER, WEST STRAND. 
 CAMBRIDGE : J. AND J. J. DEIGHTON. 
 
 M.DCCC.XXXV1II.
 
 IN this Edition, I have added a few 
 Reflexions tending to illustrate further the 
 nature of the intellectual training which the 
 study of Mathematics supplies ; and the mode 
 in which it must be conducted, after the first 
 stage, in order to answer its purpose. 
 
 TRINITY COLLEGE, \V 
 
 May 2f>, 1838. 
 
 662736
 
 CONTENTS. 
 
 FACE 
 
 PREFATORY REMARKS 1 
 
 CHAPTER I. OF THE SUBJECTS OF UNIVERSITY TEACHING. 
 
 Sect. I. Of the Distinction of Practical and Speculative 
 
 Teaching . . . . . . .5 
 
 Sect. 2. Of the Effect of Practical Teaching on the 
 
 Intellectual Habits 12 
 
 Sect. 3. Of the Effect of Practical and Speculative 
 
 Teaching on the Progress of Civilisation . 17 
 
 Sect. 4. Of the Learned Languages as Subjects of Uni- 
 versity Teaching 31 
 
 Sect. 5. Of the Necessity of combining Classical and 
 Mathematical Studies as Subjects of Univer- 
 sity Teaching ...... 37 
 
 Sect. 6. Of the Sciences as Subjects of University 
 
 Teaching 41 
 
 Sect. 7. Of the Moral Effect of Practical and Speculative 
 
 Teaching . . . . . . .45 
 
 CHAPTER II. OF PIRECT AND INDIRECT TEACHING. 
 
 Sect. 1. Of Examinations, and of College Teaching . 52 
 Sect. 2. Of Professorial Lectures . . . . .66 
 Sect. 3. Of Private Tutors 70 
 
 Sect. 4. Of the Combination of the University with the 
 
 College System 75
 
 Vlll CONTENTS. 
 
 CHAPTER III. Or DISCIPLINE. 
 
 f'AUK 
 
 Sect. 1. Of the Necessity of Discipline 78 
 
 Sect. 2. Of College Manners . . . . .80 
 
 Sect. 3. Of College Punishments 8!i 
 
 Sect. 4. Of Attendance at College Lectures . . !> 
 
 Sect. 5. Of Attendance at College Chapel . . . 104 
 
 Sect. 6. Of Fellows of Colleges lift 
 
 Sect. 7. Of the Free System 122 
 
 Sect. 8. Of Changes in the College System . . .127 
 
 APPENDIX. 
 
 THOUGHTS ON THE STUDY OF MATHEMATICS AS A 
 
 PART OF A LIBERAL EDUCATION . . . 135 
 
 Additional Thoughts on the Study of Mathematics . 177 
 
 A Letter to the Editor of the Edinburgh Review, 
 occasioned by the Review of the First Edition of 
 the Thoughts . . . . f . .186
 
 ON 
 
 THE PRINCIPLES 
 
 ENGLISH UNIVERSITY EDUCATION. 
 
 PREFATORY REMARKS. 
 
 THE considerations which I here offer to the 
 public on the subject of Education, have been 
 suggested by a long and somewhat laborious course 
 of researches on the principles and history of sci- 
 ence, and by many years' experience as a tutor in 
 a principal College of the University of Cambridge. 
 I trust, therefore, that I shall stand absolved from 
 all suspicion of approaching so important a subject 
 without due thought and preparation. I have for 
 some time intended, on the first occasion of com- 
 parative leisure, to state my views on the points 
 here treated of; and I should have done so, in the 
 same manner, and probably nearly at the same time 
 asj[ have done, whether or not other pamphlets on 
 questions connected with the English Universities 
 had appeared. I request the reader, therefore, not 
 to mix me up in his thoughts with any controversies 
 1
 
 Z ON THE PRINCIPLES OF 
 
 on such subjects which may happen to be going on 
 at this time. I mean not to express any disrespect to 
 persons engaged in such controversies ; but I must 
 take the liberty of saying, that I have neither sought 
 nor shunned the discussion of any questions on which 
 they may happen to have touched. 
 
 There is another controversy, to which some part 
 of the following pages may appear to have refer- 
 ence ; the question of the comparative value of 
 Mathematics, and of certain other studies which 
 have been termed Philosophy, as instruments of 
 Education. An Edinburgh reviewer, in a criticism 
 upon a former publication of mine, maintained that 
 the study of mathematics is, for such a purpose, 
 useless or prejudicial; and recommended the culti- 
 vation of "philosophy" in its place. In a letter to 
 the Editor of the Review, (which I published,) I 
 expressed my willingness to discuss the subject at 
 a future time ; and, referring to the mathematical 
 course of this University, as my example of mathe- 
 matical education, I requested to be informed, by 
 description, or by reference to books, what that 
 " philosophy" was, which the reviewer was prepared 
 to contend for, as a better kind of education. I 
 considered this as a proceeding, in the courtesy of 
 literary combat, equivalent to sending my opponent 
 the measure of my weapon, and begging to be fur- 
 nished with the dimensions of his. When, there- 
 fore, the reviewer, in reply, flatly refused* "to 
 perplex the question by a compliance with Mr 
 WhewelFs misplaced request," I certainly considered 
 myself as freed from any obligation to continue the 
 
 Edin. Rev. No. CXXV1I.
 
 ENGLISH UNIVERSITY EDUCATION. 
 
 controversy. No adherent of the reviewer could 
 expect me to refute a proposition which the author 
 himself did not venture to enunciate in an intelli- 
 gible form. And, therefore, in the present book, 
 I do not at all profess to discuss the question of 
 the value of mathematics, and other kinds of phi- 
 losophy, with reference to the reviewer's assertions, 
 but simply so far as it is brought before me by the 
 general course of my reflections. 
 
 I must also observe, that my remarks, at pre- 
 sent, will be bounded within the limits of my title. 
 I do not undertake to examine the subject of educa- 
 tion in general, but the Education of Universities ; 
 nor again, of Universities in general, but of English 
 Universities. Moreover, I am far from intending or 
 hoping to treat the subject, even thus limited, fully 
 and completely ; my purpose is merely to offer cer- 
 tain Considerations, having reference to the general 
 Principles on which the work of English Universi- 
 ties is and must be conducted, rather than to their 
 actual condition and their proceedings in detail. I 
 trust, however, that the Principles which I shall 
 endeavour to establish, are so far substantial and 
 practical, that their application to the real business 
 of Universities will be obvious and immediate. 
 
 A formal division of my subject might appear 
 as if I intended to exhaust it, and might mislead 
 the reader, since, as I have said, such is not my 
 purpose. But it will, I think, add to the clearness 
 of what I have to say, to divide my Considerations 
 into three chapters ; of which, the first will refer 
 to the matter taught, and mainly to the question 
 of what may be termed Practical and Speculative 
 1 2
 
 4 ON THE PRINCIPLES OF 
 
 Teaching : the second chapter will have reference 
 to the manner of teaching, and to the relative value 
 of the Direct and Indirect Methods of instruction: 
 the third chapter will treat of that superintendence 
 ana control, besides the mere teaching of the un- 
 derstanding, which, under the name of Discipline, 
 has hitherto been considered a part of the office of 
 English Universities. On all these subjects I trust 
 I shall be able to point out certain large and 
 weighty alternatives of principle, between which, in 
 all our Universities, old or new, we must necessa- 
 rily choose ; and I hope I shall be able to give 
 satisfactory reasons in favour of that choice which 
 I venture to recommend. I now proceed to my 
 task.
 
 ENGLISH UNIVERSITY EDUCATION. 
 
 CHAPTER I. 
 
 OF THE SUBJECTS OF UNIVERSITY TEACHING. 
 
 SECT. 1. OF THE DISTINCTION OF PRACTICAL AND 
 SPECULATIVE TEACHING. 
 
 THERE are two modes of teaching, which, in a 
 general view, may be broadly distinguished from each 
 other. In the one mode the lecturer merely expounds 
 to his audience the doctrines or results belonging to 
 some branch of knowledge ; he states the discoveries 
 and speculations of antecedent philosophers, or his 
 own, while the office of the audience is only to 
 attend to him ; they have to listen, to receive, 
 think on, and treasure up what the speaker delivers, 
 without being called upon themselves to take any 
 active part ; without being required to produce, to 
 test, or to apply the knowledge thus acquired. In 
 another mode of teaching, the learner has not 
 merely to listen, but to do something himself; not 
 merely to receive, but to produce his knowledge : 
 as when the mathematical student proves the pro- 
 position which is enunciated by his teacher, or solves 
 a problem proposed by him ; or when the classical 
 scholar renders Horace or Thucydides into English. 
 The former I call speculative, the latter, practical 
 teaching. And I must beg the reader to recollect 
 the manner in which I use these terms; namely, 
 with reference to the mode of teaching, not the 
 possible application of the subject taught. It is
 
 D ON THE PRINCIPLES OF 
 
 because geometry is taught thus practically, and not 
 because it is what is commonly called " practical 
 knowledge," that I designate the cultivation of ge- 
 ometry, in the manner which prevails in English 
 Universities, as Practical Teaching. In their marked 
 forms, these two kinds of teaching are very clearly 
 distinguished. Lectures uncombined with any ques- 
 tions or practical demands on the learner, are familiar 
 to us in our own Universities, in those of foreign 
 lands, in the metropolis, and in the provinces ; as 
 modes of treating of physics and metaphysics, geology 
 and political economy, taste and politics. All such 
 lectures I speak of as speculative teaching, since they 
 are employed in delivering to the hearer the doctrine 
 adopted by the teacher, in a speculative form. Prac- 
 tical teaching, where the scholar, with voice, pen, 
 or pencil, follows the track pointed out to him, and 
 is constantly brought back into it when he deviates, 
 are still more familiar ; for by this method we learn 
 every thing that, in the most peculiar sense, we learn 
 at all. It is by such a process that we become able 
 to read, to write, to cast accounts, to translate Latin 
 and Greek, to speak French and German, to solve 
 equations, to obtain our own results in the highest 
 branches of mathematics. The teaching of mothers 
 and fathers, of schools, and a great part of the teach- 
 ing of our English Universities, has hitherto been 01 
 this practical kind. 
 
 Now we may observe, that when we come to such 
 branches of literature and science as are likely to be 
 selected for the matter of University teaching, some 
 of these branches naturally and almost inevitably 
 require to be taught practically, while others as
 
 ENGLISH UNIVERSITY EDUCATION. 7 
 
 clearly are more fitted for the speculative mode of 
 teaching. Languages and mathematics are of the 
 former kind ; but many of the sciences, and those 
 especially which are wide and varied in their topics, 
 those which involve doubtful or newly-established prin- 
 ciples, those of which the foundations are constantly 
 undergoing changes, can hardly be taught otherwise 
 than speculatively. Such subjects are, for example, 
 geology, political economy, and, as appears to me, 
 metaphysics. In such subjects as these, the student 
 may listen, and may acquire such knowledge as the 
 teacher possesses; but he is not, and cannot be 
 called upon, as a part of the teaching, to do some- 
 thing which depends on the knowledge thus acquired. 
 He may follow with the clearest apprehension, and it 
 may be with full and well-founded conviction, the 
 views which are presented to him by the teacher ; 
 but still he is passive only ; he is a spectator, not 
 an actor, in the intellectual scene. He does not 
 interpret and employ a peculiar acquired language, 
 as he does in his classical reading, or his algebraical 
 calculations. 
 
 What I have called practical teaching prevails in 
 the Colleges of our English Universities. A large 
 portion of the teaching, in those institutions, has 
 always consisted, as it still does, of exercises, in 
 which the pupil translates his Greek or Latin author, 
 proves his proposition, or solves his equation, in the 
 hearing or under the eye of his tutor; or answers 
 interrogatories, in which he has to produce the 
 knowledge which he has acquired. I believe this to 
 have been the mode of teaching employed among us 
 from the earliest times. In that College, at least,
 
 ON THE PRINCIPLES OF 
 
 of which I know most, such a method is enjoined 
 in the statutes. Disputations are to be constantly 
 held in the chapel ; verses written and affixed in the 
 hall ; and the lecturers are to employ half an hour 
 in expounding their author, but a whole hour in 
 examining their class *. But besides these practical 
 lectures, we have always had lectures of the specu- 
 lative kind, delivered by the University professors. 
 Such lectures on history, morals, political economy, 
 law, medicine, anatomy, geology, botany, mineralogy, 
 chemistry, the mechanical sciences, and other sub- 
 jects, have constantly been going on in our Uni- 
 versities ; and have, especially of late years, often 
 excited very great attention. We may, therefore, 
 distinguish our practical and speculative teaching, 
 as college lectures and professorial lectures ; and 
 such a distinction corresponds to the phraseology 
 commonly in use among us. 
 
 It may be said, that with professorial lectures 
 examinations may be combined, and that such lec- 
 tures may thus be converted into practical teaching. 
 Nor do I intend to deny that, under certain con- 
 ditions, which I shall afterwards endeavour to de- 
 termine, this effect may be produced. But without 
 now entering into this subject, I trust that the 
 main features of the distinction, which I am trying 
 to point out, of the two kinds of teaching, are 
 already sufficiently clear. 
 
 Now it must be observed that, though all 
 branches of science and speculation, old and new, 
 
 * Lectorum singuli horam in dies singulos quibus legere tenentur 
 in classe sua examinanda consumant ; dimidiatam vero in authore in- 
 terpretando. Stat. Trin. Coll. Cant. cap. 9, De Lectorum Officio.
 
 ENGLISH UNIVERSITY EDUCATION. 
 
 fixed and moveable, may be made the subjects of 
 exposition in lectures, practical teaching is applicable 
 only to a limited range of subjects ; those, namely, 
 in which principles having clear evidence and stable 
 certainty, form the basis of our knowledge ; and in 
 which, consequently, a distinct possession of the 
 fundamental ideas enables a student to proceed to 
 their applications, and to acquire the habit of ap- 
 plying them in every case with ease and rapidity. 
 The ideas of space, of number, of the general rela- 
 tions of grammar and the force of language^ are 
 necessary and immutable parts of the furniture of 
 the human mind. And mathematics and languages, 
 which are the developement and working of these 
 ideas, can be practically taught, for we can appeal 
 to these ideas, and familiarize the mind with a series 
 of vast and varied, yet certain consequences, to which 
 they lead. But when we come to the wider physical 
 sciences, we can only present the facts as a matter 
 of observation, and the speculations as dependent on 
 the facts. Here there is no room for acquiring 
 habits of interpretation which can be tested by the 
 teacher. And in sciences which are not physical, as 
 morals or metaphysics, the philosophy of history, or 
 of taste, the instruction is still more inevitably of 
 the speculative kind. The teacher must be content 
 to tell, and the learner to receive, what has been 
 thought, or ought to be thought, on these subjects. 
 He does not, by learning them, acquire a new 
 faculty, which he must practically exercise. Such 
 subjects as I have just described, may, perhaps, 
 without impropriety, be distinguished by the col- 
 lective title of " philosophy ;" and if this be allowed, 
 1 5
 
 10 ON THE PRINCIPLES OF 
 
 it will, I think, appear, that philosophy is only fitted 
 for speculative, as mathematical and classical studies 
 are for practical, teaching. In saying this, I do not 
 at all profess to know, whether I am employing the 
 term "philosophy," in the sense attached to it by 
 other persons, who may have written on the subject ; 
 but it may, I think, designate appropriately a large 
 class of studies, all of which admit of the same mode 
 of communication to the student. 
 
 In such studies, moreover, even if examinations 
 be added to lectures, they can hardly constitute a 
 practical teaching ; for in such instances, the know- 
 ledge which lectures convey, is either merely retained 
 in the memory, or is employed as material for further 
 speculation by the student, and is not assimilated 
 and converted into a practical habit of intellectual 
 action. Examinations, therefore, in these cases, may 
 test the goodness of the memory, and the clearness 
 of the apprehension or general faculties ; and we 
 may also conceive examinations of a higher kind, 
 that call out the powers of original thought, and 
 detect the activity of talent and genius. But to 
 require proof of mere memory and clearheadedness 
 is not practical teaching, in the same sense as it 
 is so to ascertain that a power of interpretation or 
 calculation has been acquired ; and the higher kind 
 of examination which we have mentioned, in which 
 the student is called upon to give evidence of his 
 own speculative talent, presupposes that practical 
 teaching of which we here speak, and is not to be 
 confounded with it. And thus, even with the ad- 
 dition of examinations on subjects of general phi- 
 losophy, there will still remain, between those studies
 
 ENGLISH UNIVERSITY EDUCATION. 
 
 and the mathematical and classical pursuits of the 
 English Universities, that difference which I describe 
 by calling the former speculative teaching. 
 
 Thus the distinction of speculative and practical 
 instruction, which at first sight appears to be a 
 difference of the manner of teaching, is found, on 
 examination, to imply a difference of the subjects 
 taught. When we have determined that we will 
 teach practically, we have decided that we must 
 lecture, not on philosophy, not on metaphysics or 
 speculative morals, or political economy, but on 
 subjects of a different kind ; on the works of Greek 
 and Latin authors ; the properties of space and 
 number; the laws of motion and force. 
 
 Of course, I mean only, that so far as we teach 
 practically, we must select such subjects. Nothing 
 prevents us, and as I have said, we have not been 
 prevented, from giving, in addition to our college 
 courses, professorial lectures on all the other subjects 
 which I have mentioned. But it is not on that 
 account the less important to my purpose, to keep 
 the consideration of the two kinds of study distinct. 
 It is obvious also, that, in many cases, the same 
 subject admits of being dealt with in both ways. We 
 may not only ascertain that our pupils can translate 
 Sophocles, but we may present to them the widest 
 speculative views at which critics have arrived, 
 respecting the history and structure of the Greek 
 language, or the Greek drama. We may enter into 
 discussions respecting the metaphysical grounds of 
 the axioms of geometry, the processes of algebra, the 
 laws of motion. Such speculations and discussions 
 are of the highest interest and value ; but it is easy
 
 12 ON THE PRINCIPLES OP 
 
 to see that they are something in addition to the 
 teaching of Greek and mathematics. They add im- 
 mensely to the value of the practical acquisition of 
 language and mathematical habits, but they pre- 
 suppose the acquisition ; and when these philosophi- 
 cal views are substituted for the practical instruction, 
 they are altogether empty and valueless as means of 
 education. 
 
 But I do not here insist upon this point. In 
 the present section, my object has been to distinguish 
 the two systems, before I compared them. Trusting 
 that the distinction is now sufficiently clear, I pro- 
 ceed to the comparison. And this I shall consider 
 with reference to such points as these : the effect 
 on the intellectual and on the moral character of 
 those who are educated, and on the general progress 
 of national culture and civilisation. 
 
 4 
 
 SECT. 2. OF THE EFFECT OF PRACTICAL TEACHING 
 ON THE INTELLECTUAL HABITS. 
 
 THE advantages which belong to the study of 
 mathematics, as an intellectual discipline, have been 
 often stated by various persons. I may repeat 
 language which I have already used : " In mathe- 
 matics, the student is rendered familiar with the 
 most perfect examples of strict inference ; he is 
 compelled habitually to fix his attention on those 
 conditions on which the cogency of the demonstra- 
 tion depends ; and, in the mistaken or imperfect 
 attempts at demonstration made by himself or 
 others, he is presented with examples of the most 
 natural fallacies, which he sees exposed and cor-
 
 ENGLISH UNIVEKSITY EDUCATION. 13 
 
 rected." My Edinburgh reviewer * expressed a wish, 
 that these latter " novel assertions had been explain- 
 ed and exemplified ;" and obviously, was really at 
 a loss to understand them, although they refer to 
 the daily occurrences of the lecture-room. This is 
 a curious proof how entirely practical teaching is 
 logt sight of, amid the speculations of his school. 
 I may observe, too, as I have done elsewhere -f% that 
 reasoning, as a practical habit, is taught with pe- 
 culiar advantage by mathematics, because we are, 
 in that study, concerned with long chains of reason- 
 ing, in which each link hangs from all the pre- 
 ceding. " The language contains a constant suc- 
 cession of short and rapid references to what has 
 been proved already ; and it is justly assumed, that 
 each of these brief movements helps the reasoner 
 forwards in a course of infallible certainty and 
 security. Each of these hasty glances must possess 
 the clearness of intuitive evidence, and the certainty 
 of mature reflection : and yet must leave the rea- 
 soned mind entirely free to turn instantly to the 
 next point of his progress. The faculty of perform- 
 ing such processes well and readily, is of great 
 value ;" and this faculty can hardly be acquired and 
 cultivated in any other way, than by the study of 
 mathematics. 
 
 I shall not pursue the consideration of the bene- 
 ficial intellectual influence of mathematical studies. 
 It would be easy to point out circumstances, which 
 show that this influence has really operated; for 
 
 * Review of Thoughts on the Study of Mathematics, p. 12J. 
 t Mechanical Euclid, with Remarks on Mathematical Reasoning, 
 p. 144.
 
 14 ON THE PRINCIPLES OP 
 
 instance, the extraordinary number of persons, who, 
 after giving more than the common attention to 
 mathematical studies at the University, have after- 
 wards become eminent as English lawyers. It would 
 be easy, also, to gather together a " cloud of wit- 
 nesses," who have spoken with admiration and en- 
 thusiasm of mathematics as a discipline of the mind. 
 But this would be a very idle mode of treating the 
 subject ; for it might be possible also, to adduce a 
 large bulk of similar testimony on the other side. 
 And what could be inferred from this array of cloud 
 against cloud? Except we can get some clear in- 
 sight into the subject ourselves, we can never know 
 whether the authors we adduce, are riot speaking 
 from views as vague and confused as our own. 
 When any one one will point out any other study, 
 as a mode of practically teaching reasoning, which 
 he maintains to be preferable to mathematics, we 
 may be tempted to make the comparison ; but this 
 has not been done, so far as T know. 
 
 It may be said, that mathematical reasoning is 
 but one kind of reasoning, and that the study and 
 practice of this alone, ought not to be spoken of 
 as the cultivation of the reasoning power in general. 
 To this, I reply, that the faculty of reasoning, so 
 far as it can be disciplined by practical teaching, 
 receives such a discipline from mathematical study. 
 If, for instance, any one says, " Why do you not 
 cultivate the habit of inductive as well as of deductive 
 reasoning?" I answer, that the only cultivation of 
 which inductive reasoning admits, is that which is 
 supplied by deductive reasoning. For when we col- 
 lect a new truth by induction from facts, what is
 
 ENGLISH UNIVERSITY EDUCATION. 15 
 
 the process of our minds ? We acquire a new and 
 distinct view, or hit upon a right supposition ; and 
 we perceive that, in the consequences of our new 
 notions, the observed facts are included. The former 
 part of this process, the new and true idea suited 
 to the emergency, the happy guess, no teaching can 
 give the student. All that we can do is, to fix the 
 idea when he has it, and to teach him to test his 
 hypothesis by tracing its consequences. And this, 
 the cultivation of deductive habits does. We cannot 
 teach men to invent new truths; we cannot even 
 give them the power of guessing a riddle. But those 
 who have been inventors, have always had, not only 
 that native fertility of mind which no education can 
 bestow, but also a talent of clearly and rapidly ap- 
 plying their newly-sprung thoughts, in which talent 
 half their power consisted, and which is precisely 
 that faculty which mathematical habits may improve. 
 And the distinctness of the fundamental ideas (a 
 state of thought essential alike to sound reasoning 
 from old truths, and to the discovery of new,) is not 
 unprovided for by the study of mathematics ; for 
 though deductive habits do not give distinct funda- 
 mental ideas they demand them ; and, by the con- 
 stant appeal to such ideas, they fix and develope 
 them. A perception of the truth of mathematical 
 axioms cannot be conveyed into the mind by reason- 
 ing ; but still, the mathematical reasoner usually sees 
 more clearly than other men, the necessary truth of 
 his axioms *. Other persons may have the idea of 
 space, as well as the geometer; the idea of force 
 
 * On this subject, see the Remarks at the end of the Mechanical 
 Euclid.
 
 16 ON THE PRINCIPLES OF 
 
 and matter, as well as the mechanician ; but these 
 ideas shine with a clearer and steadier light in the 
 minds- of those who constantly work by such lamps, 
 and therefore, carefully tend and trim them. 
 
 Since the study of mathematics is thus useful, 
 not only in teaching habits of deduction, which are 
 exemplified in its proofs, but also in leading men 
 to the distinct ideas which are expressed in its defi- 
 nitions and axioms, we learn a lesson respecting the 
 kind of mathematics which we may most advan- 
 tageously introduce in our education. For since 
 those clear ideas upon which the several mathe- 
 matical sciences depend are a valuable mental pos- 
 session, both on their own account, and as examples 
 of such a class of elements of truth, we ought not to 
 be content with one or two such ideas and their 
 consequences, but should introduce the student to 
 a wider range of mathematical proof. We shall 
 thus succeed best in repressing the evil consequences 
 which might arise from confining ourselves to one 
 kind of reasoning. We ought, therefore, to include 
 in our course, not only pure mathematical sciences, 
 geometry, arithmetic, and algebra, the consequences 
 of the fundamental ideas of space, number, and 
 quantity ; but we ought also admit the consequences 
 of other ideas, which lead to rigorous mathematical 
 sciences, such as the ideas of pressure and matter, of 
 rigidity and fluidity, of velocity and force ; of wln'ch 
 ideas the developements are found in the sciences of 
 Mechanics and Hydrostatics. This maxim I have al- 
 ready urged, in a former publication on this subject*. 
 
 * Thoughts on the Study of Mathematics ; reprinted at the end 
 of this volume.
 
 ENGLISH UNIVERSITY EDUCATION. 17 
 
 And I rejoice to say, that a recent alteration in 
 the examinations of the University of Cambridge, 
 by which certain portions of Mechanics and Hydro- 
 statics are introduced into the lower Examinations 
 for Degrees, has made our system, what appears to 
 me, on the grounds just stated, a better intellectual 
 education than it was before. 
 
 I shall not here dwell upon the intellectual effect 
 of the practical teaching of Greek and Latin, but 
 proceed to consider the effect of the two systems 
 of instruction in another point of view. 
 
 SECT. 3. OF THE EFFECT OF PRACTICAL AND SPECU- 
 LATIVE TEACHING ON THE PROGRESS OF CIVILISATION. 
 
 IF I were to begin by asserting that the pro- 
 gress of civilisation is essentially connected with the 
 prevalent education, the assertion would probably be 
 assented to ; but at the same time, it would pro- 
 bably also be understood in so general and indistinct 
 a manner, that no real use could be made of it in 
 our argument. The connexion is, indeed, generally 
 acknowledged; for instance, Dr. Diesterweg's pam- 
 phlet, in which he so deeply deplores the present 
 diseased condition of the German Universities, (a 
 subject which has recently excited much remark in 
 that country,) is entitled*, " The Vital Question of 
 Civilisation." But some definite statement of the 
 nature of this connexion is requisite, in order to 
 enable us to draw any inferences from it. The 
 subject is far too large to be treated generally here; 
 
 * Die Lebensfrage der Civilisation ; oder : Ueber das Verderben 
 auf den Deutschen Universitaten.
 
 18 ON THE PRINCIPLES OP 
 
 but there is one view of it to which I hope I shall 
 obtain the reader's assent. 
 
 Among the elements and indications of civilisa- 
 tion, I think it will be allowed that a generally 
 diffused faculty of speculative thought forms a lead- 
 ing point. Such a faculty, and its habitual exercise, 
 forms a main distinction between the most refined 
 and the rudest nations ; as, in a broader sense, it 
 does between man and lower animals. For even 
 the brutes have practical powers of thought ; they 
 have a practical notion of space and force ; a prac- 
 tical sense of things good and bad, of things which 
 they may and which they may not do ; but man 
 alone has a geometry and a mechanics, an idea of 
 happiness and of a moral law. And the clearness 
 of the ideas which speculation requires and uses, is 
 one of the most essential features of the progress of 
 intellectual refinement. This appears in matters, 
 which at first seem trivial, as, for instance, in the 
 importance attached to correct speaking and writing. 
 Why is it that a false spelling or wrong accent in 
 our own language, is considered a mark of a vulgar, 
 or, at least, unpolished, mind? Why is it that a 
 false quantity, or a false concord, is looked upon 
 with horror by the thorough scholar, as something 
 offensive and ridiculous? Why is it that men are 
 more angry at being accused of bad reasoning than 
 of erroneous opinions? Clearly because all these 
 faults imply an incomplete and ill- conducted culti- 
 vation of the speculative faculty, in reference to 
 language or to reasoning. The errors may be tri- 
 fling, but they seem to disclose faults of intellectual 
 training : they are, in the world of literature and
 
 ENGLISH UNIVERSITY EDUCATION. 19 
 
 thought, what violations of good breeding are in 
 manners. A single fault of grammar may betray a 
 want of perception of the analogy of language ; a 
 single fault of logic may shew that the speaker has 
 no distinct apprehension of the force of demonstra- 
 tion ; and when this judgment is formed of him, he 
 immediately appears to sink below the standard point 
 of cultivation and connexion of thought. He is less 
 cultured than those who detect his deficiencies ; 
 less refined in his intellectual character, because he 
 is less distinct and connected in his intellectual habits. 
 And thus we have, in the common judgments of man- 
 kind, an evidence that they consider distinctness and 
 clearness of the speculative faculty as one of the 
 elements of civilisation. 
 
 But we may take a larger view. Probably all 
 persons will acknowledge, that those nations by whom 
 great advances in knowledge are made, and among 
 whom such advances are widely diffused and well 
 understood, have the pre-eminence in civilisation. 
 Great scientific discoveries, along with a general 
 national interest and intelligence respecting such 
 matters, are circumstances peculiar to the most high- 
 ly cultured times. Now this consideration will lead 
 us by a different road to the same element of civilisa- 
 tion which we have already pointed out. For by 
 a history of each of the sciences in. succession*, I 
 have proved, I hope satisfactorily, that their pro- 
 gress depends upon the distinctness of certain fun- 
 damental ideas; and that these ideas, being first 
 clearly brought into view by the genius of great 
 
 * History of the Inductive Sciences, from the earliest to the 
 present Times.
 
 20 ON THE PRINCIPLES OP 
 
 discoverers, become afterwards the inheritance of all 
 who thoroughly acquire the knowledge which is thus 
 made accessible. In highly cultered nations, a large 
 portion of society will thus attend to the progress 
 of knowledge ; so as to obtain a just view, at least 
 of the general nature of the treasures which are 
 thus placed within their reach, and of the triumphs 
 which their intellectual leaders have achieved. And 
 thus we are again brought to the principle, that 
 distinct speculative ideas generally diffused, are an 
 essential part of our conception of civilisation. 
 
 Having reached this point, we have to inquire 
 whether practical or speculative teaching, distin- 
 guished as in the preceding section, be the best 
 instrument for that kind of culture on which civilisa- 
 tion thus depends. I do not think it would be 
 difficult to shew, from general considerations, that 
 it is only by the practical teaching of mathematics, 
 that the fundamental ideas of science can become 
 distinct among men in general. But a more inter- 
 esting mode of deciding this point will be, to look 
 back at the history of the world ; for the whole 
 history of the world has been one grand experi- 
 ment on this subject. Let us take a general view 
 of the result. Our question is, whether practical 
 or speculative teaching most promotes civilisation ; 
 which question, as we have seen, may be decided, 
 by inquiring whether an education in mathematics, 
 or in general philosophy, is most favourable to the 
 progress of science, and to the general diffusion of 
 the knowledge which this progress brings. A rapid 
 survey of the history of education, with this view, 
 will repay us.
 
 ENGLISH UNIVERSITY EDUCATION. 21 
 
 > 
 
 Of the Greek education, up to the time of Plato, 
 we know enough to be able to assert, that it was in 
 the main practical teaching. The " music 11 (MOUO-I/CJJ) 
 which constituted the principal part of this, was 
 taught unquestionably in 'a practical manner; and 
 if the occasion admitted, it might be shown, both 
 from the elements which it included, and from 
 the way in which it was conducted, that it had 
 nearly the same effect that the practical teaching 
 of mathematics has, in giving distinctness to the 
 ideas, independently of its other and collateral 
 influences. But in the time of Aristophanes*, a 
 change took place in the instruction of the Greek 
 youth. The sophists and philosophers were extraor- 
 dinarily admired and followed ; and to acquire an 
 acquaintance with their doctrines and systems came 
 to be considered as the most essential part of a liberal 
 education. This was still more the case among the 
 Romans, when they attempted to take a place among 
 cultivated nations. Their youth listened to what 
 " Chrysippus and Grantor taught," and were thus 
 supposed to be filled with all learningf . The study 
 of philosophy, in the general sense, that is, of the 
 moral, metaphysical, and physical doctrines of the 
 framers of universal systems, was, as we know, the 
 highest conception of the Greeks and Romans in 
 their aims at intellectual culture, till civilisation it- 
 self sickened and declined. It was so, too, among 
 the Neoplatonists, the schoolmen, the theologians 
 of the middle ages; till in the monasteries there 
 again grew up a method of practical teaching from 
 
 " See The Clouds. f See the beginning of Cicero's Offices.
 
 22 ON THE PRINCIPLES OF 
 
 which the system of the English universities had 
 its origin. 
 
 Such is the course of education ; now what is the 
 corresponding course of knowledge? The answer is 
 well worth notice. The progress of science corresponds 
 to the time of practical teaching ; the stationary, or 
 retrograde, period of science, is the period when phi- 
 losophy was the instrument of education. At the time 
 of Plato, the Greek education had been for a long 
 period virtually mathematical ; a fact, of which the 
 very term mathematics is the record. At that time 
 the greatest scientific discovery of the ancient world, 
 the resolution of celestial phenomena into circular 
 motions, was caught sight of by Plato, and soon 
 after fully brought out by Hipparchus. At a similar 
 stage of Greek culture, although at a later time and 
 in a different country, the science of mechanics was 
 established by Archimedes, on foundations fitted to 
 endure to eternity. What might have been the 
 history of civilisation if the Greek education had 
 continued to be practically mathematical, we cannot 
 tell. Speaking according to human views of proba- 
 bility, perhaps the Greeks might, in that case, have 
 anticipated the discoveries of modern times by a 
 thousand years; and the places of Galileo, and 
 Kepler, and Newton, might have been preoccupied 
 by citizens of Athens and Alexandria. But the 
 speculative study of philosophy prevailed. From 
 that time no material advance was made in science. 
 What great men had already taught mankind, was 
 perverted or forgotten by their degenerate followers. 
 The schools of the philosophers resounded with sys- 
 tems old and new, with wranglings and boastings ;
 
 ENGLISH UNIVERSITY EDUCATION. 23 
 
 but this availed not to urge on the intellectual 
 progress of man, or even to prevent his sliding 
 backwards. The simple geometrical conceptions of 
 the school of Plato were debased and weighed down 
 by a cumbrous apparatus of crystalline spheres. 
 The mechanical truths brought to light by Archi- 
 medes, were, like his tomb, overgrown with the rank 
 and unprofitable vegetation of later days, till they 
 were lost sight of; and were not resumed and 
 pursued till a thousand years, and half a second 
 thousand, had elapsed. It is a manifest mistake to 
 ascribe the decay of science to the incursions of the 
 northern nations. Science was dead, and literature 
 mortally smitten, before the external pressure was 
 felt. But the study of speculative philosophy, as 
 the business of cultured men, survived. Still the 
 intellectual world grew darker and darker. " Light 
 after light goes out, and all is night." In vain do 
 the schoolmen of the middle ages build system upon 
 system, as the schoolmen of Athens and Alexandria 
 had done before. The centuries roll on, and bring 
 no day. But in the mean time the religious orders 
 have established among themselves a system of 
 practical teaching. They introduce mathematics into 
 their course with especial attention. The principle 
 of progress is soon felt to be again at work. A 
 Franciscan friar lifts up his voice against the sway 
 of Aristotle, and points to the far-off temple of 
 science, declaring that mathematics is its gate and 
 its key*. His announcement is found to be true. 
 
 * Harum scientiarum porta et clavis est mathematica, quatn 
 sancti a principle mundi invenerunt, etc. Roger Bacon, Specula 
 Mathematica, cap. i.
 
 24 ON THE PRINCIPLES OP 
 
 From the like mathematical schools proceed the 
 luminaries of a new dawn, Copernicus, Galileo, 
 Kepler, Newton, are the founders of a fresh era of 
 knowledge, because they are well-trained mathema- 
 ticians. The universities of Europe assume a form 
 in which such a training goes on ; thus the cultured 
 classes become capable of receiving and appreciating 
 the great discoveries by which man's intellectual posi- 
 tion is advanced ; and we reach the present condition 
 of the civilised world. 
 
 But we have not yet done with the survey of this 
 great experiment. In one country of Europe the 
 universities give up their habits of practical teaching, 
 and return to the speculative method. They make 
 philosophy their main subject. Their professors de- 
 liver from their chairs system after system to ad- 
 miring audiences. The listener may assent or criticise ; 
 but he is not disturbed by any demands on his mind, 
 such as the teaching of mathematics gives rise to. 
 And what is the class of men thus produced, in their 
 bearing upon the progress of sure and indestructible 
 knowledge? ' They are such men as to be utterly 
 incapable even of comprehending and appreciating 
 the most conspicuous examples of the advance of 
 science. Those who are universally allowed to be 
 the greatest philosophers of our own day in the 
 German universities, Hegel and Schelling, cannot 
 understand that Newton went further than Kepler 
 had gone in physical astronomy, and despise New- 
 ton's optical doctrines in comparison with the vague 
 Aristotelian dogmas of Gothe respecting colours*. 
 
 * See Hegel's Encyclopaedia, and Schelling's Lectures.
 
 ENGLISH UNIVERSITY EDUCATION. 25 
 
 Thus, the experiment on education, which has 
 been going on from the beginning of Greek civilisa- 
 tion to the present day, appears to be quite distinct 
 and consistent in its result. And the lesson we learn 
 from it is this ; that so far as civilisation is connected 
 with the advance and diffusion of human knowledge, 
 civilisation flourishes when the prevalent education is 
 mathematical, and fades when philosophy is the sub- 
 ject most preferred. We find abundant confirmation 
 of the belief, that education has a strong influence 
 upon the progress of civilisation ; and we find that the 
 influence follows a settled rule: when the education 
 is practical teaching, it is a genuine culture, tending 
 to increased fertility and vigour ; when it is specula- 
 tive teaching, it appears that, however the effect is 
 produced, men's minds do, in some way or other, 
 lose that force and clearness on which intellectual 
 progression depends. 
 
 I cannot go on to the next point of my argument 
 without an observation founded on the view which 
 has been presented. It is impossible, after the survey 
 we have just made, not to reflect of what immense 
 importance the question of the two kinds of educa- 
 tion is. The reform of the European universities, a 
 subject which is now exciting so much interest in 
 England, France, and Germany, is, in truth, what it 
 has been termed, the Vital Question of Civilisation. 
 Upon the decision of that question may depend, 
 whether Europe, and America, which must follow the 
 intellectual fortunes of Europe, .shall, for the next 
 thousand years, be in the condition of the later Greeks 
 and Romans, having for their mental aristocracy, a 
 class of philosophical system-builders, commentators, 
 2
 
 26 ON THE PRINCIPLES OF 
 
 and mere metaphysicians ; or shall go on to exhibit 
 that healthy vigour and constant effort at real pro- 
 gress and improvement, which has characterized this 
 quarter of the globe for the last three hundred years. 
 This is no slight matter. And let no one attempt to 
 make it less momentous, by persuading himself that 
 civilisation must advance ; that we cannot run back 
 into an inferior condition of culture and thought. 
 The history of the world shows that we have no such 
 security. Civilisation, in its best sense, may too 
 surely decline. Greece and Rome had wasted by 
 their own folly almost all that was most valuable 
 in their intellectual inheritance, before the foreign 
 spoiler came. The civilisation of the eastern and 
 southern shores of the Mediterranean, once the fairest 
 spots in the world of literature and art, where is it, 
 and how is it vanished ? It is not enough to say that 
 the barbarising storm of Mahommedan conquest has 
 swept over and destroyed it. The Mahommedans 
 did not barbarise Spain or Persia. And to whatever 
 violent external causes we may ascribe this deplorable 
 change, it shows, at least, that in some countries 
 civilisation takes deeper root than in others ; and 
 warns us to use our best endeavour, that, so far as 
 we are concerned, our country and the world may 
 lose nothing of that real civilisation which, combined 
 with morality and religion, constitutes the brightest 
 glory and most precious treasure of the human 
 race. 
 
 It is difficult to mark out, even in conjecture, the 
 path of the future progress of mental civilisation. 
 Yet some light we may gather from the history of 
 the past. One idea after another, of those which
 
 ENGLISH UNIVERSITY EDUCATION. 27 
 
 constitute the basis of science, becomes distinct, first 
 in the minds of discoverers, then in the minds of all 
 cultured men, till a general clearness of thought illu- 
 minates the land ; and thus the torch of knowledge 
 is handed forwards, thousands upon thousands light- 
 ing their lamps at it as it passes on ; while still from 
 time to time some new Prometheus catches a fresh 
 light from heaven, to spread abroad among men in 
 like manner. Thus the opening of Greek civilisation 
 was marked by the production of Geometry ; the 
 idea of space was brought to a scientific precision. 
 Of that step we still inherit the benefits; for example, 
 all educated Europeans conceive the relation of the 
 various parts and lines on the terrestrial globe with 
 a distinctness in which the rude savage or uncultured 
 boor has no share. The opening of the civilisation 
 of modern Europe was distinguished, in the same 
 way, by the production of a new science, Mechanics, 
 which soon led to the Mechanics of the Heavens. And 
 this step, like the former, depended on men arriving 
 at a properly-distinct fundamental idea. The revival 
 of the scientific idea of force (an idea which had been 
 brought to light by Archimedes, and extinguished 
 again amid the mists of Greek philosophy,) was, as 
 I have elsewhere shown, the essential condition to 
 which this step was due*. This idea, too, has now 
 been communicated to persons of education in general, 
 as the general reception of the Newtonian theory of 
 the universe proves ; while, at the same time, the 
 very indistinct views which men of considerable culti- 
 vation often entertain of the mechanism of the uni- 
 verse, proves that the fundamental ideas on which 
 
 * History of the Inductive Sciences, Book VI. chap. i. sect. 2. 
 2 2
 
 28 ON THE PRINCIPLES OF 
 
 a clear apprehension of the doctrine of universal at- 
 traction depends, have hitherto been very imperfectly 
 diffused through the atmosphere of the literary world. 
 And the cause of this remaining imperfection probably 
 is, that elementary mechanics has not hitherto been 
 made an essential portion of a liberal education, as 
 for centuries elementary geometry has. Nothing for- 
 bids us to look forward to the time, when not only 
 this deficiency shall be supplied*, but when men's 
 minds shall have been carried much further in the 
 same track. We may imagine a future period of 
 mental culture, when the elements of chemistry and 
 natural history shall be fundamental parts of a good 
 education, as the elements of mathematics now are ; 
 and when consequently the ideas on which our know- 
 ledge of the composition of bodies, or our estimate 
 of the natural classes of organized beings, depends, 
 shall be as clear in cultivated minds, as the concep- 
 tion of universal attraction is now, in the mind of 
 a thoroughly educated man ; or as the conception of 
 the circles of latitude and longitude, in the thoughts 
 of a well-taught child. And if we add to this, the 
 possibility that the ideas which are the bases of sound 
 criticism, morals, and politics, may become equally 
 distinct and equally diffused, by means of an appro- 
 priate education, we catch a glimpse of the grand 
 and boundless vista of possible and probable intel- 
 lectual refinement and civilisation which the future 
 offers. 
 
 * Partly for the reasons here suggested, I have published a work 
 on Elementary Mechanics (The Mechanical Euclid), such as I hope 
 may fit that branch of knowledge for taking its due place in education 
 by the side of geometry.
 
 ENGLISH UNIVERSITY EDUCATION. 29 
 
 Whether or not the reader may assent to the 
 view thus presented of the nature and prospects of 
 civilisation, he will, I trust, sympathize with another 
 reflection which offers itself to us at this point of our 
 survey. If the destinies of the highest civilisation of 
 man, be thus closely connected with the progress of 
 truly liberal education; and if it depend upon the 
 constitution and conduct of educational institutions, 
 whether such civilisation shall continue to advance, 
 or shall become retrograde : it is impossible not to 
 reflect, how grave and weighty is the office of those, 
 on whom it falls to found and to put in action new 
 institutions of liberal education, intended to meet 
 the requirements of present and future ages. To do 
 this, is a great, and we may say, a solemn task. 
 Those who are engaged in it, must act as men build- 
 ing for eternity. We see no reason to believe other- 
 wise, than that this great nation, hitherto so highly 
 favoured with outward and inward gifts, (and with it, 
 its vigorous progeny, which, while peopling and civilis- 
 ing the other side of the globe, is involved in the 
 intellectual fortunes of its parent,) is destined by 
 Providence for an advance yet to be long continued, 
 in civilisation and refinement of the best and highest 
 kind. To what consummation the world is reserved 
 by its Governor we know not, nor whether he has 
 decreed, that, before the final close of all things, the 
 brightness of civilised England must wane and be* 
 come dim, as that of Greece and Rome has done 
 before. But this we know, that it would be the most 
 fantastical presumption of system-making, for any 
 one to predict and reckon in centuries the calculated 
 time and rate of the declension of Britain. We know,
 
 SO ON THE PRINCIPLES OF 
 
 too, that if such a declension menaced us, the wisest, 
 as well as the noblest course, would be to seek and 
 apply our remedies in the spirit of considerate and 
 hopeful regard for the future. And surely, even if 
 our final declension were certain, and if we could yet, 
 by our exertions, so retard its progress, that, during 
 the ensuing three hundred years, our condition should 
 be no worse than for the last three hundred years it 
 has been, this were a blessing, and a distinction 
 among the nations of the earth, well worth the best 
 resolves and exertions the nation can bring to the 
 task. When, therefore, we attempt to construct in- 
 stitutions of education for the countless youth of 
 centuries still to come, we enter upon a task full of 
 solicitude and responsibility, but full also of hope and 
 promise. And in this spirit should the office be dis- 
 charged ; all narrow interests, and little jealousies, 
 and limited regards, being laid aside, and the great 
 object itself, the transmission of the best portion 
 of our own culture to the Britons of ages now far 
 removed, being steadily kept in view. With this 
 object we should guard especially against bringing 
 down the standard of our system to the level which 
 transient and partial circumstances, or popular pre- 
 judices, may suggest. That education which will 
 secure to the future the civilisation of the past and 
 present, is that which the country really requires ; 
 and modes of education which may attract for a mo- 
 ment, but can produce no effect of this kind, are of 
 no value for the real purposes of education, and can 
 satisfy none of the real wants of the age. 
 
 But a part of this subject requires a separate 
 section.
 
 ENGLISH UNIVERSITY EDUCATION. 31 
 
 SECT. 4. OF THE LEARNED LANGUAGES AS SUBJECTS 
 OF UNIVERSITY TEACHING. 
 
 IT has appeared in a preceding Section, that the 
 decision of the question, whether our teaching shall 
 be practical or speculative, in the sense already 
 explained, in a great measure decides the subjects 
 taught ; since certain subjects only can be made the 
 basis of practical education, and certain other sub- 
 jects are peculiarly fitted for speculative discussion. 
 But there are some other questions concerning the 
 matter taught, which may be considered here ; for 
 instance, the proposal to include in it modern lan- 
 guages and their literature, instead of, or along with, 
 the ancient authors of Greece and Rome; and to 
 introduce the modern sciences, as general physics, 
 chemistry, natural history, and geology, along with 
 pure and mixed mathematics. I will say a word on 
 each of these questions. 
 
 It is one of the characters of the present time, 
 alarming to many persons, but, if we use the occasion 
 well, a blessing rather than an evil, that doctrines 
 which have hitherto passed unquestioned, and on 
 which the frame of the institutions of European states 
 is founded, are unscrupulously and rudely assailed. 
 The propriety of the use of what are called the 
 learned languages (Greek and Latin), among the 
 main instruments of education, is a doctrine of this 
 kind. And the question whether, in modern educa- 
 tion, these languages are to retain their ancient 
 supremacy, or whether, on the contrary, the lan- 
 guages and literature of modern Europe are to be
 
 32 ON THE PRINCIPLES OF 
 
 placed by their side, or before them, has been re- 
 cently discussed with reference to educational insti- 
 tutions, both in this and other countries. In France, 
 for example, this has been the subject of animated 
 debates in the Chamber of Deputies ; and that dis- 
 tinguished man of science, M. Arago, is reported, on 
 such an occasion, to have expressed himself to the 
 following effect : 
 
 " I ask for classical studies : I require them : I 
 consider them as indispensable ; but I do not think 
 that they must necessarily be Greek and Latin. I 
 wish that, in certain schools, these studies should be 
 replaced, at the pleasure of the municipal councils, by 
 a thorough study of our own tongue. I wish that in 
 each college, it should be permitted to put, in the 
 place of Greek and of Latin, the study of a living 
 language. I require even, that this language may be 
 different according to the situation of the place ; that 
 at Perpignan and at Bayonne, for example, it may be 
 the Spanish ; at le Havre, the English ; at Besancon, 
 the German." 
 
 He then proceeds to answer certain objections, 
 of which I shall only notice the one which more 
 peculiarly concerns our subject. 
 
 " It is urged," he says, " that Greek and Latin 
 must be the principal classical studies, for they are 
 the true culture of the mind? 
 
 To this he makes the following reply : 
 
 " What does this mean ? Are Pascal, Fenelon, 
 Bossuet, Montesquieu, Rousseau, Voltaire, Corneille, 
 Racine, Moliere the incomparable Moliere, are 
 these writers deprived of the privilege which is so 
 liberally granted to the ancient authors, of enlight-
 
 ENGLISH UNIVERSITY EDUCATION. S3 
 
 ening, of unfolding the mind, of touching the heart, 
 of putting in vibration the springs of the soul ! 
 God preserve me from insulting you, by refuting in 
 detail a heresy such as this !" 
 
 In opposition to the opinion thus expressed, I 
 maintain that Greek and Latin are peculiar and 
 indispensable elements of a liberal education; and 
 it is my business to shew, that the study of the 
 modern authors just enumerated, and of others, 
 however admirable their works may be, does not 
 produce that kind of culture of the mind, which 
 is the true object of a liberal education. 
 
 This culture of the mind consists in sharing in 
 the best influences of the progressive intellectual 
 refinement of man. The present age is not inde- 
 pendent of those which have preceded it. On the 
 contrary, it is the heir of all the past. Its wealth, 
 intellectual and material, may have been improved 
 in the hands of the present holders, but the value 
 of what we have added is small, compared with the 
 amount of what we found already accumulated. In 
 thought and language, as well as in arts and the 
 products of art, we inherit an inestimable fortune 
 from a long line of ancestors. In literature, we 
 are the children of the early Greeks ; 
 
 Ka'o/iou TOV fa\ai vea Tpo<f>i'i. 
 
 But thoughts can be inherited, and words, in all 
 their force, transmitted, only by those who are con- 
 nected with their ancestors in the line of thought 
 and understanding, as well as in the mere succession 
 of time. And how is this connexion of generations, 
 thus requisite to the transmission and augmentation, 
 of mental wealth, to be kept up ? 
 2 5
 
 34 ON THE PRINCIPLES OF 
 
 The cultivated world, up to the present day, has 
 been bound together, and each generation bound to 
 the preceding, by living upon a common intellectual 
 estate. They have shared in a common develope- 
 ment of thought, because they have understood 
 each other. Their standard examples of poetry, 
 eloquence, history, criticism, grammar, etymology, 
 have been a universal bond of sympathy, however 
 diverse might be the opinions which prevailed re- 
 specting any of these examples. All the civilised 
 world has been one intellectual nation ; and it is 
 this which has made it so great and prosperous a 
 nation. All the countries of lettered Europe have 
 been one body, because the same nutriment, the 
 literature of the ancient world, was conveyed to 
 all, by the organization of their institutions of edu- 
 cation. The authors of Greece and Rome, familiar 
 to the child, admired and dwelt on by the aged, 
 were the common language, by the possession of 
 which each man felt himself a denizen of the commu- 
 nity of general civilisation ; free of all the privileges 
 with which it had been gifted from the dawn of 
 Greek literature up to the present time. 
 
 What can the best authors of modern days do in 
 the way of filling such an office I Even if their lan- 
 guage were universally familiar in cultured Europe, 
 how do they connect us with the past ? How do 
 they enable us to read the impress which was 
 stamped upon thought and language in the days 
 of Plato and Aristotle, in virtue of which it is still 
 current ? How do they enable us to understand 
 the process by which the language of Rome con- 
 veyed the culture, the philosophy, the legislation of
 
 ENGLISH UNIVERSITY EDUCATION. 35 
 
 the ancient civilised world into the modern ? How 
 do they enable us to understand the thoughts and 
 feelings to which they themselves appeal ? If the 
 Greek and Latin languages were to lose their fa- 
 miliar place among us, Montesquieu and Bossuet, 
 Corneille and Racine, would lose their force and their 
 charm. Those who read and admire these authors 
 constantly make a reference in their minds to the 
 works of the ancients, which they know immediately 
 or through a few steps of derivation. If this know- 
 ledge were taken away, many of the strings would 
 be broken in the instrument on which those artists 
 played. And though, so long as a liberal education 
 continues what it has been, the well-educated diffuse 
 to others a general admiration of the "classical 
 authors 1 '' of their own language ; if Greek and Latin 
 were to cease to be parts of general culture, the 
 admiration of the classical authors of England and 
 France would become faint and unintelligent, and, 
 in a few generations, would vanish. 
 
 The same may be said of language. The languages 
 of ancient Greece and Rome have, through the whole 
 history of civilisation, been the means of giving dis- 
 tinctness to men 1 s ideas of the analogy of language, 
 which distinctness, as we have seen, is one main ele- 
 ment of intellectual cultivation. The forms and 
 processes of general grammar have been conveyed to 
 all men's minds by the use of common models and 
 common examples. To all the nations of modern 
 Europe, whether speaking a Romance language of 
 not, the Latin grammar is a standard of compari- 
 son, by reference to which speculative views or 
 grammar become plain and familiar.
 
 36 ON THE PRINCIPLES OF 
 
 And then, as to the derivation of the modern 
 European languages : Those who are familiar with 
 Greek and Latin cannot but feel, in every sentence 
 they read and write, that the whole history of the 
 civilised world is stamped upon the expressions they 
 use. The progress of thought and of institutions, 
 the most successful labours of the poet, the philo- 
 sopher, the legislator, have, in thousands of cases, 
 operated to give a meaning to one little word. 
 Those who feel this, have a view of the language 
 which they speak, far more intelligent, far more 
 refined, than those who gather the force of words 
 from blind usage, without seeing any connexion or 
 any reason. What does intellectual culture mean, 
 if it does not mean something more than this ? 
 What does it mean, but that insight, that distinct- 
 ness of thought with regard to the terms we employ, 
 which saves us from solecisms, not by habit but by 
 principle, which shows us analogy where others see 
 only accident, and which makes language itself a chain 
 connecting us with the intellectual progress of all ages. 
 
 In what a condition, should we be, if our connexion 
 with the past were snapped; if Greek and Latin 
 were forgotten ? What should we then think of our 
 own languages? They would appear a mere mass 
 of incoherent caprice and wanton lawlessness. The 
 several nations of Europe would be, in this respect at 
 least, like those tribes of savages who occupy a vast 
 continent, speaking a set of jargons, in which scarcely 
 any resemblance can be traced between any two, or 
 any consistency in any one. The various European 
 languages appear to us obviously connected, mainly 
 because we hold the Latin thread which runs through
 
 ENGLISH UNIVERSITY EDUCATION. 37 
 
 them ; if that were broken, the pearls would soon 
 roll asunder. And the mental connexion of the 
 present nations with each other, as well as with the 
 past, would thus be destroyed. What would this be 
 but a retrograde movement in civilisation? 
 
 In nations as in men, in intellect as in social 
 condition, true nobility consists in inheriting what is 
 best in the possessions and character of a line of 
 ancestry. Those who can trace the descent of their 
 own ideas, and their own language, through the race 
 of cultivated nations ; who can show that those whom 
 they represent, or reverence as their parents, have 
 everywhere been foremost in the fields of thought and 
 intellectual progress, those are the true nobility of 
 the world of mind ; the persons who have received 
 true culture ; and such it should be the business 
 of a liberal education to make men. 
 
 With these views, I cannot conceive it possible 
 that any well-constituted system of University teach- 
 ing, in any European nation, can do otherwise than 
 make the study of the best classical authors of 
 Greece and Rome, one of its indispensable and car- 
 dinal elements. But before I proceed, I cannot 
 refrain from pointing out the evil of making such an 
 element, or any one element alone, too exclusive or 
 too large a part of our system of instruction. 
 
 SECT. 5. OF THE NECESSITY OF COMBINING CLASSICAL 
 AND MATHEMATICAL STUDIES AS SUBJECTS OF UNI- 
 VERSITY TEACHING. 
 
 THE arguments which we have urged in support 
 of the necessity of the ancient languages as pro- 
 minent parts of the teaching of our Universities.
 
 38 ON THE PRINCIPLES OF 
 
 proceed upon the ground of their usefulness as in- 
 struments of mental culture. And this effect has 
 been contemplated, as resulting, not only from the 
 familiarity which the student of classical literature 
 may acquire with the works and style of the bright- 
 est periods of ancient civilisation ; but also, from the 
 clear views to which he may be led, by such studies, 
 of the principles and history of grammar, language, 
 and literary thought. Now it must not be forgotten, 
 that in classical as in any other literature, the reader 
 who merely flutters through a series of authors such 
 as catch his fancy, who studies them only as a 
 literary amusement, without severe thought, or steady 
 perception of the principles of language and compo- 
 sition, cannot receive from them such a culture as 
 we have supposed ; any more than from any other 
 line of reading, suggested and directed by mere 
 caprice and personal taste. Indeed, since principles 
 are disclosed and illustrated by the reading of poets 
 and orators, far more obscurely and vaguely than 
 by most other studies, classical literature so pursued 
 is entirely inefficient for any purpose of genuine 
 mental cultivation. It will only produce a taste, 
 fastidious, indeed, but superficial and arbitrary, with- 
 out any distinct and developed apprehension of ana- 
 logies and reasons. And even if the classical authors 
 be studied profoundly and thoroughly, as examples 
 of language, composition, and thought, still they only 
 supply one occasion among many, for the cultivation 
 of the more exact operations of the mind ; and in 
 this, or in any other way, the adoption of one in- 
 strument alone, for such a purpose, will make the 
 resulting culture extremely partial and deficient.
 
 ENGLISH UNIVERSITY EDUCATION. 39 
 
 The mode in which this defect may most effectually 
 be remedied, is by combining, with the study of 
 classics, the study of the elementary portions of 
 mathematics. For these severer studies will bring 
 into play that class of intellectual faculties, which the 
 pursuit of elegant literature alone leaves unexercised. 
 We may add, too, that the mental powers so de- 
 veloped will react upon the study of classical authors ; 
 and the perception of general relations, of grounds and 
 reasons, even in matters of grammar and taste, will be 
 far more likely to arise in the mind of a student thus 
 disciplined, than in that of the mere elegant scholar. 
 Every person of mathematical cultivation is necessarily 
 an analyst of conditions and connexions : the ana- 
 lytical power thus awakened will commonly exercise 
 itself upon language, as well as upon mathematical 
 quantity ; and thus a familiarity with the best models 
 of composition will become such a discipline of 
 distinct ideas, with regard to the principles of lan- 
 guage and thought, as, for our purposes, we require 
 it to be. 
 
 The study of elementary mathematics, therefore, 
 along with the study of classical authors, ought to 
 be imperatively required by all Universities. To 
 separate these two branches of study, and to allow 
 students to neglect one of them, because some per- 
 sons have a taste for one, and some persons for 
 the other, is to abdicate the functions of educa- 
 tion altogether. Universities and Colleges do not 
 exist merely for the purpose of enabling men to do 
 what they best like to do ; or for the purpose of 
 offering and awarding prizes for trials of strength, in 
 modes selected by the combatants. Their business
 
 40 ON THE PRINCIPLES OF 
 
 is the general cultivation of all the best faculties 
 of those who are committed to their charge, and 
 the preservation and promotion of the general cul- 
 ture of mankind. And it is certain, that of all the 
 persons who derive advantage from a University 
 education, none are more benefited than those who, 
 with a great general aptitude for learning, are pre- 
 vented, by the requisitions of such institutions, from 
 confining their exertions to one favourite channel. 
 The man of mathematical genius who, by the de- 
 mands of his College or his University, is led to 
 become familiar with the best Greek and Latin 
 classics, becomes thus a man of liberal education, 
 instead of being merely a powerful calculator. The 
 elegant classical scholar, who is compelled, in the 
 same way, to master the propositions of geometry 
 and mechanics, acquires among them habits of rigour 
 of thought and connexion of reasoning. He thus 
 becomes fitted to deal with any subject with which 
 reason can be concerned, and to estimate the pro- 
 spects which science offers ; instead of being kept 
 down to the level of the mere scholar, learned in the 
 literature of the past, but illogical and incoherent in 
 his thoughts, and incapable of grappling with the 
 questions which the present and the future suggest. 
 To neglect to demand a combination of these two 
 elements, would be to let slip the only machinery 
 by which Universities, as the general cultivators of 
 the mind, can execute their office.
 
 ENGLISH UNIVERSITY EDUCATION. 41 
 
 SECT. 6. ON THE SCIENCES AS SUBJECTS OF 
 UNIVERSITY TEACHING. 
 
 FROM what has already been said of the use of 
 mathematics in University education, it will easily 
 be inferred, that we cannot find, in any of the more 
 modern physical sciences, any thing that can fitly be 
 substituted for that study. The effect of the clear 
 insight of geometry or mechanics cannot be efficient- 
 ly replaced by sciences which exhibit a mass of ob- 
 served facts, and consequent doubtful speculations, 
 as geology; or even by other sciences, as chemistry 
 and natural history, which, though they involve 
 philosophical principles, can only be learnt by pre- 
 senting numerous facts to the senses. But though 
 such sciences cannot do the work of mental culti- 
 vation, they are highly valuable acquisitions to the 
 student, and may very beneficially engage his at- 
 tention during the later years of his University 
 career. For although they do not constitute the 
 culture, they belong to the information of the well- 
 educated man; though his habits of thought must 
 be formed among other subjects, they may be bene- 
 ficially employed on these. And it is advantageous 
 for the general sympathy and mutual understanding 
 of the cultivated part of mankind, that such persons 
 should have many subjects of common interest. 
 
 But we may say much more than this. A 
 considerable general knowledge of the modern 
 progressive sciences, is as requisite to connect the 
 educated man with the future, as a thorough ac- 
 quaintance with ancient literature is to connect him
 
 42 ON THE PRINCIPLES OP 
 
 with the past. Except he know what has been done 
 and is doing, in the way of extending our knowledge 
 of earth, its elements, and its inhabitants, how can 
 he judge what are the probable prospects of our 
 knowledge? And if he be indifferent to this, how 
 can he feel that interest in the future fortunes of his 
 race, which becomes a person of his lofty extraction ? 
 Some insight into the progressive sciences is an es- 
 sential part of a liberal education, in any large sense 
 of the term. 
 
 This consideration is in some measure connected 
 with the choice of our course of mathematical read- 
 ing. For mathematics has been one of the great 
 instruments of the progress of the physical sciences ; 
 and the constant use of this instrument, and the 
 efforts to make it more effective, lead to frequent 
 modifications of its form and expansions of its 
 powers ; and thus occasion, from time to time, 
 changes in the current system of mathematics. This 
 change, so far as the use of mathematics in education 
 is concerned, is an evil rather than a good : .for in 
 our design we do not wish our pupils to possess 
 mathematics principally as information, nor even as 
 an instrument, although in that way it is of great 
 advantage to many persons ; but as an intellectual 
 discipline ; and this end is best answered by teaching 
 the stable system of a demonstrative science. But 
 at the same time we must recollect, that we cannot 
 give to our mathematical studies their true dignity, 
 without showing the place they hold in the progress 
 of science ; and, for the reason which we have men- 
 tioned, namely, in order to nourish in our more 
 advanced pupils an interest in that progress, we do
 
 ENGLISH UNIVERSITY EDUCATION. 43 
 
 not hesitate to introduce to them all new mathe- 
 matical methods, which have a prominent and per- 
 manent importance. But this is salutary to the 
 more advanced pupils only. No deviation from 
 the plain instructive circle of ancient elementary 
 mathematics is fitted for the first stages of the 
 students University progress. The mode in which 
 his elementary mathematical studies best produce 
 their effect upon him is, when they are presented in 
 the luminous simplicity in which the Greek intellect 
 contemplated them, not when they are disguised and 
 obscured by being translated into the modern lan- 
 guage of symbols. To learn this language, (a valu- 
 able lesson for the mind, if it be rightly taught,) is 
 best made an ulterior step in the student's advance. 
 
 But the physical sciences are useful, not only as 
 belonging to the information of the educated man, 
 but also as supplying him with examples of Inductive 
 Reasoning. The general rules and conditions of 
 such reasoning have hitherto been very imperfectly 
 pointed out ; but a knowledge of the sciences gives 
 any one the means of speculating for himself on the 
 subject ; a subject of the strongest interest with 
 reference to the future progress of knowledge. I 
 have already said, that a practical instruction in 
 Inductive Reasoning is not possible, except so far 
 as it depends on the cultivation of the Deductive 
 Faculty. We may lead men to feel the force of 
 demonstration, but we cannot teach them to discover 
 new truths. And I may repeat my observation, that 
 the value of this practical teaching of the reason by 
 means of mathematics may be much enhanced by 
 a proper selection of our mathematical studies;
 
 44 ON THE PRINCIPLES OP 
 
 namely, by not confining ourselves to pure mathe- 
 matics, and, least of all, to the pure mathematics of 
 symbols. On the subject, considered in this point of 
 view, I, a little while ago, published a few remarks, 
 under the title of "Thoughts on the Study of Mathe- 
 matics, as a part of a Liberal Education ;" and as 
 these remarks may serve to illustrate further what 
 is here said, I reprint them at the end of the present 
 book. 
 
 It will be observed, that I intentionally omit, at 
 present, all reference to professional education ; and, 
 consequently, to lectures, examinations, and degrees, 
 in Divinity, Medicine, and Law. All such studies 
 should be subsequent to the intellectual culture of 
 which I now speak ; and the professions to which 
 these studies belong will derive the greatest part of 
 their real dignity and refinement, from their being 
 built on such a foundation. I omit, also, all con- 
 sideration of what is called " practical knowledge," 
 such as civil engineering, practical arts and trades, 
 and the like ; which are sometimes recommended as 
 useful elements of education. If these are wanted 
 as professional knowledge, they must be learnt, as, 
 in fact, they are learnt, among professional men and 
 practical applications. If they are wished for as 
 information, they stand on the same ground as the 
 higher physical sciences, of which we have spoken 
 above, so far as the arguments there employed are 
 applicable to them. But this " practical knowledge" 
 can never stand in the place of a really liberal edu- 
 cation, nor in the smallest degree supersede the 
 necessity of the studies we have pointed out, if such 
 an education be our object.
 
 ENGLISH UNIVERSITY EDUCATION. 45 
 
 SECT. 7. ON THE MORAL EFFECT OF PRACTICAL AND 
 SPECULATIVE TEACHING. 
 
 BESIDES the communication to the student of the 
 matter taught, there are certain collateral effects of 
 the two kinds of teaching, which are well worthy of 
 notice. I will speak briefly of a few of these. 
 
 Mathematical doctrines are fixed and permanent ; 
 no new system of geometry can supersede the old. 
 The old truths will always be true, and always 
 essential. Not only so, but even the old books re- 
 main in use. Euclid has never been superseded, and 
 never will be so without great detriment to educa- 
 tion. And if Archimedes had written a treatise on 
 Mechanics, in extent and form similar to that of 
 Euclid on Geometry, such a work would probably 
 have been one of our best instruments of education 
 at the present day. 
 
 In philosophical doctrines, on the contrary, a 
 constant change is going on. The commentator 
 supersedes the original author, or at least becomes 
 equally important : the systematiser is preferred to 
 him who first threw out the same thoughts in a less 
 regular form. Or else a revolution takes place : the 
 old system is refuted ; a new one is erected, to last 
 its little hour, and wait its certain doom, like its 
 predecessor. There is nothing old, nothing stable, 
 nothing certain, in this kind of study. Change is 
 constantly taking place ; change is constantly looked 
 for. Novelty is essential, in order to command at- 
 tention or approbation. The ship sails on ; old 
 objects glide back ; the point of view changes. The
 
 46 ON THE PRINCIPLES OP 
 
 student knows, or at least cannot but suspect, that 
 his teacher and his teacher's creed are but for a 
 day; and that what is demonstrated to be true, 
 will be found hereafter to be a truth so imperfect, 
 that it is best put out of sight. 
 
 Now I conceive it cannot be doubted that the 
 mind of a young man employed mainly in attending 
 to teachers of this latter kind, must fail to acquire 
 any steady and unhesitating conviction of the im- 
 mutable and fixed nature of truth, such as the study 
 of mathematics gives. This constant change in the 
 system of received doctrines must unsettle and en- 
 feeble his apprehension of all truths. He has no 
 time, he has no encouragement, to take up the 
 doctrines that are placed before him, and to study 
 them till he is firmly possessed of them, secure that 
 their certainty and value can never alter. He lives 
 among changes, and has not the heart to labour 
 patiently for treasures that may be ravished from 
 him by the next revolution. The state of Germany, 
 for instance, has of late years been as unfavourable 
 to the intellectual welfare of its students, as the 
 condition of the most unstable government of the 
 East is to the material prosperity of its subjects. 
 A great philosophical conquest is made by Kant, and 
 a universal empire is supposed to be on the point of 
 being established. But Fichte, who began with being 
 a follower of Kant, ends by deposing him. Schelling 
 carries away the allegiance of Germany from Fichte ; 
 and then Hegel becomes more powerful than any of 
 his predecessors; and a younger Fichte raises the 
 standard against all these rulers. And thus, with dire 
 shedding of ink, revolution after revolution succeeds.
 
 ENGLISH UNIVERSITY EDUCATION. 47 
 
 Now amid all this change and fear of change, 
 how can any man eat tranquilly of the fruit of his 
 own field, under his own vine and fig-tree ? How 
 can he cultivate his own thoughts, and possess in 
 a tranquil and even spirit the knowledge and the 
 habits of mind which he has acquired ? He cannot 
 feel or relish old and familiar truths, such as mathe- 
 matical sciences deal with. He cannot be content 
 with such conclusions as can be obtained by the way 
 of demonstration. He becomes almost inevitably 
 himself a wide and restless speculator ; criticising 
 what has already been done in philosophy ; attempt- 
 ing to guess what will be the next step ; and desti- 
 tute, not only of those clear ideas, and those habits 
 of exact thought, through which alone any real 
 advances in knowledge can be appropriated by the 
 student, but devoid also of that steady belief in the 
 permanent nature and value of speculative truth, 
 which is an essential virtue of the understanding. 
 
 Again ; another mode in which this speculative 
 teaching operates unfavourably, as I conceive, upon 
 students, is this ; it places them in the position of 
 critics instead of pupils. In mathematical and other 
 practical teaching, the teacher is usually, and almost 
 necessarily, much the superior of his scholar in the 
 knowledge which they cultivate together ; and the 
 scholar cannot but feel this, and must consequently 
 be led to entertain a docile and confiding disposition 
 towards his instructor. On the other hand, when a 
 system is proposed, as only offering its claims to him, 
 and asking his assent, which he may give or refuse, 
 he feels himself placed in the situation of an equal 
 and a judge, with respect to his professor. And if,
 
 48 ON THE PRINCIPLES OP 
 
 as is very likely to be the case with active-minded 
 young speculators, he goes through several phases of 
 philosophical opinion, and gives his allegiance to a 
 succession of teachers, he can hardly fail to look 
 upon them with a self-complacent levity, which in- 
 volves little of respect. He will probably think of 
 his masters much as the poet speaks of the objects 
 of his transient admiration whom he chronicles : 
 
 The gentle Henrietta then, 
 
 And a third Mary next did reign, 
 
 And Joan, and Jane, and Audria; 
 And then a pretty Thomasine, 
 And then another Katharine, 
 
 And then a long et cetera. 
 
 Now this want of docility, confidence, and respect, 
 when it prevails in the student towards his teacher, 
 cannot, I think, be looked upon otherwise than as a 
 highly prejudicial feeling, and one which must destroy 
 much of the value and usefulness of the education 
 thus communicated. 
 
 The difference of the subjects which are recom- 
 mended by different persons as suitable for University 
 teaching, does in fact depend upon an entire differ- 
 ence in the views and temper of the authors of the 
 recommendations. In the teaching of universities, 
 a spirit of respect, or a spirit of criticism, may be 
 appealed to. According to the first system, we must 
 select subjects which consist of undoubted truths, 
 and works of unquestioned excellence, and must re- 
 quire the student to familiarise himself with these. 
 Such subjects are mathematical studies, and the best 
 classical authors. According to the other system, we
 
 ENGLISH UNIVERSITY EDUCATION. 49 
 
 take subjects in which we endeavour to draw the 
 student's attention by our mode of treating them, 
 and to carry his conviction with us by our argu- 
 ments. In this system, we invite him to inquire for 
 himself; to accept or reject according to his best 
 judgment ; to examine all doctrines boldly and tho- 
 roughly. This critical system it is which rejoices ta 
 have philosophy for its subject, and has shown alike 
 its vigour and its tendency by the rapid succession 
 of prevalent systems. 
 
 I do not at all hesitate to say, that the respectful 
 system appears to me the proper line of education. 
 I conceive that the student ought to have, placed 
 before him, something which is of a stable and per- 
 manent kind ; in which it is a good mental exercise 
 to struggle with the apparent objections, because it 
 is certain that by effort and practice they may be 
 overcome; and in which it has been ascertained 
 that admiration is not the result of novelty, or of 
 some transient bearing upon the feelings of the age. 
 The critical system seems to me to be properly 
 addressed, not to students who are undergoing educa- 
 tion, but to philosophers who have already been com- 
 pletely educated. If this course educate a man for 
 anything, it educates him to be a judge of philosophi- 
 cal systems ; an office which so few Englishmen will 
 ever have to fill, that it does not appear wise or rea- 
 sonable to make it the main object of our education. 
 Nor can I believe, that to put young men in the 
 position in which that system of teaching places 
 them, at a period of their lives when they ought to 
 be quietly forming their minds for future action, can 
 have any other result than to fill them with a shallow 
 3
 
 50 ON THE PRINCIPLES OF 
 
 conceit of their own importance ; to accustom them 
 to deliver superficial and hasty judgments ; and to 
 lead them to take up new systems, with no due ap- 
 preciation of the knowledge, thought, and gravity of 
 mind, which are requisite for such a purpose. 
 
 I believe that this opinion of the effect of the 
 two modes of university education has been confirm- 
 ed by the actual result. The practical education of 
 the English Universities has produced men fitted im- 
 practical life. I need not dwell upon this. I have 
 already noticed how well the training of the college 
 appears to prepare men to become good lawyers. I 
 will add, that I conceive our physicians to be the 
 first in the world, and that I ascribe their excellence 
 mainly to the practical course of general culture 
 which they receive in the Universities ; which does 
 what no merely professional education can do ; and 
 of which the effects are seen, when the professional 
 employments bring into play the intellectual habits. 
 Our clergy derive inestimable advantages from the 
 cast of their university education; and if clerical 
 education among us be capable of improvement, this 
 certainly will not be brought about by the substi- 
 tution of the philosophy of Schelling and Hegel for 
 the mathematics of Euclid and Newton. That our 
 Universities educate men to be legislators, statesmen, 
 and magistrates of some practical power and skill, 
 no one can doubt, except he who thinks that this 
 little island has, for the last three hundred years, run 
 an unprosperous course, and held an undistinguished 
 place in Europe. For the fortunes of nations are 
 determined, under Providence, by their practical lead- 
 ers, and men are formed by their education.
 
 ENGLISH UNIVERSITY EDUCATION. 51 
 
 In Germany and France, we are told that there 
 prevails among the young men of the Universities a 
 vehement and general hostility to the existing insti- 
 tutions of their country. I know not how truly this 
 is said ; but I conceive that such a consequence may 
 naturally flow from an education which invokes the 
 critical spirit, and invites it to employ itself on the 
 comparison between the realities of society and the 
 dreams of system-makers. 
 
 I shall not here prosecute this subject further, 
 since my object is to hasten on to some principles 
 which apply more intimately to that process of in- 
 struction which has hitherto existed in the English 
 Universities. But I hope I have made it appear 
 that, distinguishing the two systems of education as 
 I have done, we may, with nearly equal propriety, 
 treat of them as practical and speculative teaching ; 
 or on the one hand mathematics combined with classics. 
 and on the other philosophy ; or college lectures, and 
 professorial lectures; and may look upon them as 
 exemplifying a respectful and a critical spirit. And 
 I hope I have satisfied the reader that (allowing fully 
 the value and use of philosophy and of professorial 
 lectures in their due place, of which I may afterwards 
 speak,) we could not abandon the practical teaching, 
 the mathematical and classical studies, and the College 
 lectures of our Universities, without great loss to the 
 intellectual training of our youth, without destroying 
 highly beneficial feelings which exist between them 
 and their teachers, and without putting in serious 
 and extensive jeopardy the interests of the civilisation 
 of England and of the world. 
 
 3 2
 
 52 ON THE PRINCIPLES OF 
 
 CHAPTER II. 
 
 OF DIRECT AND INDIRECT TEACHING. 
 
 SECT. 1. OF EXAMINATIONS, AND OF COLLEGE 
 TEACHING. 
 
 BY indirect teaching, I mean a course of educa- 
 tion in which the student's exertions are directed 
 mainly towards examinations, disputations, or some 
 other public trial of his acquirements ; and in which 
 he is led to acquire knowledge, principally by the pro- 
 spect of the distinctions, honours, or advantages, which 
 attend upon success in such trials. I distinguish such 
 teaching from that direct teaching, in which in- 
 structions are given as claiming the student's atten- 
 tion on the ground of their own value ; and in which 
 they are recommended to him by his own love of 
 knowledge, by the advice and authority of his in- 
 structor, and the general sympathy of the body in 
 which he lives. 
 
 In the English Universities, there has always, I 
 believe, been a combination of these two kinds of 
 teaching ; and such a combination is, I conceive, the 
 best scheme of education. In the selection and ma- 
 nagement of each of these elements, however, there 
 are some considerations which appear to me of great 
 importance, and which I will briefly state. 
 
 The College lectures, and other College instruc- 
 tions, appear to have been, till recently, of the nature
 
 ENGLISH UNIVERSITY EDUCATION. 53 
 
 of direct teaching. The studies thus presented to the 
 pupil were considered as sufficiently recommended by 
 the injunctions of the College and the parental au- 
 thority* of the tutors, without reference to ulterior 
 objects. The public disputations and theses, which 
 must be performed in order to obtain a degree, form- 
 ed a scheme of indirect teaching; and the College 
 teaching was consistent with this, but was far from 
 being considered as merely ministerial to it. Several 
 subjects were introduced in the courses of College 
 instruction, and, indeed, are still, which have no re- 
 ference to these public University trials, and which 
 were selected by the authorities of the College, be- 
 cause they were considered as valuable for their own 
 sake, and proper parts of a liberal education. But 
 though this is the case, a strong disposition has mani- 
 fested itself of late years, in the University of Cam- 
 bridge at least, to give a great preponderance to the 
 indirect system; to conduct our education almost 
 entirely by means of examinations, and to consider 
 the lectures given in the Colleges as useful only in 
 proportion as they prepare the student for success in 
 the examinations. On this point I will offer a few 
 remarks. 
 
 As I have already said, a combination of direct 
 and indirect instruction appears to be desirable. The 
 love of knowledge, and the love of distinction with 
 the fear of disgrace, are the two mainsprings of all 
 
 * Pupilli tutoribus pareant, honoremque paternum et reverentiam 
 deferant, quorum studium, labor et diligentia in illis ad pietatem et 
 scientiam informandis ponitur. Tutores sedulo quae docenda sum. 
 doceant, qnaeque etiam agenda, instituant moneantque Stat. Trin. 
 Coll. Cant. cap. x.
 
 54 ON THE PRINCIPLES OP 
 
 education, and it does not appear wise or safe to try 
 to dispense with either of them. Moreover the Uni- 
 versity must, in the discharge of its proper functions,- 
 have tests of proficiency, to be applied before her 
 degrees and honours are granted. There must, there- 
 fore, be University examinations. On the other hand, 
 it must always be recollected that examinations are 
 a means, not an end; that a good education, a 
 sound and liberal cultivation of the faculties, is the 
 object at which we ought to aim ; and that examina- 
 tions cease to be a benefit, where they interfere with 
 this object. 
 
 That such a danger is possible, a very little re- 
 flection will show. The knowledge which is acquired 
 for the purpose of an examination merely, is often of 
 little value or effect as mental culture, compared with 
 that knowledge which is pursued for its own sake. 
 When a man gives his mind to any subject of study 
 on account of a genuine wish to understand it, he 
 follows its reasonings with care and thought; ponders 
 over its difficulties, and is not satisfied till all is clear 
 to his mental vision. On the other hand, when he 
 studies for an examination only, he does not wish to 
 understand, but to appear to understand ; he cares 
 not for unsolved difficulties in his mind, if the ex- 
 aminer detect them not ; he wishes to see clearly, 
 only in order that he may express himself clearly. 
 He may thus lose much of what is best in the in- 
 fluence of those studies which, when more faithfully 
 pursued, tend to educe distinct ideas and sound 
 reasoning habits. Again : what is acquired for an 
 examination is likely to be soon forgotten : the mind 
 is bent upon it with an effort, which, though strong
 
 ENGLISH UNIVERSITY EDUCATION. OO 
 
 at the time, is felt to be temporary, and is followed 
 by a relapse into comparative apathy and oblivious- 
 ness. The student soon lets slip what he has thus 
 collected for a special purpose ; just as the busy 
 advocate forgets the circumstances of his client's case 
 almost as soon as he has pleaded it. Again: 
 the habit of preparing for examinations makes the 
 studies which are not recommended by an obvious 
 reference to such an object, appear flat and insipid. 
 The mind craves for the excitement to which it has 
 been accustomed : it becomes restless and volatile ; 
 loses the appetite for quiet thought and patient 
 study, and the trust in advantages which must be 
 waited for. Again: if examinations become too 
 frequent, all good courses of study are interfered with. 
 For it is impossible to arrange public examination so 
 as to point out a succession of subjects which forms 
 a good system for all. That which must be required 
 of every one is far too little to employ and exercise 
 the more powerful and active minds. They, therefore, 
 when they have to conform their studies to require- 
 ments constructed for smaller intellects, are thwarted 
 and inteiTupted in their more genuine pursuits. 
 
 I urge these objections, not to show that we ought 
 not to have examinations, but in order to point out 
 that the use of examinations is exposed to dangers 
 which must be guarded against, if we would not for- 
 feit some of the best effects of University education. 
 Mankind are always ready to transfer their solicitude 
 from the end to the means. When examinations 
 have become a prominent part of our system, when it 
 it is seen how much the effect of the system depends 
 upon the mode in which they are conducted, it may
 
 56 ON THE PRINCIPLES OF 
 
 easily happen that men may turn all their attention 
 to the arrangements and circumstances of examina- 
 tions, as if this were the supreme object of the 
 legislation of a University. This would be to disci- 
 pline soldiers, not for the battle, but for the review. 
 We cannot make the examinations every thing to 
 our students, without making the love of knowledge 
 nothing. 
 
 The University of Cambridge is proud, and with 
 much justice, of the acknowledged purity of her 
 examinations. They are free from all taint of sinis- 
 ter motives and practices ; above partiality, and 
 above the suspicion of partiality. This is an in- 
 estimable advantage ; an elevation of character 
 which she has reached by long years of vigilance 
 and scrupulousness, and which must be vigilantly 
 and scrupulously guarded. But this virtue must 
 not be regarded as if it were a sufficient pledge for 
 all others. The examinations may be perfectly fair, 
 and yet useless for all the best purposes of education. 
 I have no doubt, that when disputations were as 
 important parts of the University proceedings as 
 examinations now are, disputations were usually 
 moderated with perfect fairness by the presiding 
 officer. But that circumstance alone would not save 
 them from being useless trifling, and perverse sub- 
 tilty. Examinations, though perfectly impartial, may 
 come to deserve as little respect as most persons 
 now give to the logical exercises of the days of dis- 
 putation. 
 
 The recommendations of the system according to 
 which the University teaches by examinations, are 
 obvious enough, and deserve to be attended to.
 
 ENGLISH UNIVERSITY EDUCATION. 57 
 
 Besides that, as I have said, the application of some 
 test of proficiency is a necessary part of the functions 
 of a University, examinations such as now exist 
 among us are more agreeable both to most teachers 
 and to most pupils, than the system of direct teach- 
 ing. Teachers often prefer the former system, be- 
 cause it relieves them from the constantly repeated 
 effort and anxiety which accompanies direct in- 
 struction, at least, when bestowed on unwilling or 
 unintelligent pupils. If all solicitude about the 
 students daily attendance, his daily progress, his 
 transient difficulties, his fluctuating diligence, can 
 be rendered superfluous, by examining, at last, what 
 has been the general result of his study, they are 
 naturally glad to escape so easily a burden 1 so op- 
 pressive. With such recommendations, the labour 
 of examination, although not light, is readily under- 
 taken. On the other side, the student often prefers 
 the examination to other modes of instruction, be- 
 cause he is there impelled by the desire of distinction, 
 by the stir of contest, by the play of hopes and fears, 
 sympathy and novelty. In the examination-hall he 
 is not passive, but active. He is there one of the 
 principal actors in the piece, not a subordinate cha- 
 racter, as he is in the lecture-room. It is not 
 wonderful, therefore, that examinations, at least 
 voluntary examinations, are crowded by hosts of 
 our willing pupils. 
 
 What has been said hitherto refers to voluntary 
 examinations, which students are induced to enter 
 by the love of distinction. The effect of compulsory 
 examinations, also, requires notice. These, or some- 
 thing equivalent, must exist in a University ; but 
 35
 
 58 ON THE PRINCIPLES OF 
 
 when they are considered as the only means of 
 University education, it is easily seen that the edu- 
 cation must be bad. For their requisitions must be 
 lowered to the level of the average power of mind 
 and of application which young men possess, in order 
 that University degrees may be the general mark 
 of a liberal education : and, hence, the substance of 
 such examinations cannot be sufficient to exercise 
 and improve the quicker and more capacious in- 
 tellects. Moreover, for reasons already stated, the 
 knowledge which is acquired for examinations ope- 
 rates less as culture, than that which is obtained 
 under other circumstances. And when the exami- 
 nation is a compulsory one, there is a servile and 
 ignoble influence breathing about it, since it acts 
 not on the hopes, but on the fears ; and holds dis- 
 grace and degradation before the eyes of the can- 
 didate. Such examinations may be necessary, but 
 they never can be more than a necessary evil ; and 
 that system would, indeed, be unworthy a great and 
 highly-civilised nation, in which the machinery of 
 education was all of this structure. 
 
 Besides the University examinations, there exist, 
 in all the Colleges of Cambridge, College examina- 
 tions, on which I will make a few remarks. These 
 examinations approach much more nearly to the 
 character of instruments of direct instruction, than 
 the University examinations. They are different in 
 each year of the student's progress ; and, taken in 
 connexion, they conduct the pupil through a course 
 of instruction selected with care and judgment. The 
 appointment of such a course of study for the em- 
 ployment of the successive periods of the pupil's
 
 ENGLISH UNIVERSITY EDUCATION. 59 
 
 residence, is indeed part of the original structure of 
 most College codes. The subjects which are thus 
 enjoined in the ancient statutes are, in many cases, as 
 will readily be imagined, now obsolete ; in Trinity 
 College, of which I can best speak, the system has 
 been repeatedly accommodated to the changes oc- 
 casioned by the progress of science, but always with 
 a careful reference to the general spirit and purport 
 of the statutes. The mathematical course, even 
 now, deviates very little from that described in the 
 original code*. The College lectures take up these 
 subjects in order ; at the end of the academic year, 
 an examination in these subjects takes place in each 
 College, and is conducted by some of the fellows of 
 the College, selected according to certain general 
 rules. 
 
 The College course exhibits that element of in- 
 struction which I have called direct teaching ; since 
 not only a certain proficiency is required, but the 
 circumstances, means, and order of the student's 
 progress, are prescribed. This is, I conceive, the 
 proper place and office of the Colleges ; an office 
 of the highest importance to the cause of education 
 in our Universities ; since without it, the bad con- 
 sequences of mere examinations, which I have point- 
 ed out, would become most grievous evils, and the 
 University system would lose the whole of its real 
 value. The College examinations should therefore 
 be conducted with reference to this office of the 
 
 * Lector autem Mathematicus doceat primum Arithmeticam , 
 deinde Geometricam, turn cognitionem Sphaerae et Cosmographiam, 
 deinde Astronomiam, postremo Musicam. Stat. cap. ix. Theoretical 
 Harmonics, which is meant by Musica, is not yet abandoned as a 
 subject of lectures.
 
 60 ON THE PRINCIPLES OF 
 
 Colleges ; and are by no means to be carried on 
 as if they were merely preparatory to the exami- 
 nations required by the University. 
 
 A question here occurs, which, though it may 
 appear a point of mere detail, when considered with 
 reference to the English Colleges, involves a prin- 
 ciple of considerable extent and importance, and 
 may concern all institutions which have education 
 for their object. The question is this: Whether 
 the same persons ought to teach and to examine 
 in the same subjects? 
 
 This can occur as a practical question, only in 
 those institutions in which there are other persons 
 besides the teachers, among whom examiners may be 
 found. But this is the case at most of our Colleges ; 
 in consequence of the existence, in each, of a body 
 of Fellows, provided for, to a certain extent, by 
 the foundation, and not necessarily engaged in the 
 tuition. And there are obvious advantages in com- 
 mitting examinations into such hands. For such 
 persons are free from the prepossessions which tutors 
 may be supposed to have with respect to their own 
 pupils, and which may bias their judgment in es- 
 timating the performances of the candidate at an 
 examination. Such examiners form a kind of tri- 
 bunal which may check caprice and carelessness in 
 the tutors. They also strengthen the hands of the 
 teaching body, since they appear as an independent 
 class, sanctioning the studies and instructions of the 
 teachers, and keeping up an acquaintance with such 
 literary and scientific pursuits as they recommend, 
 without the same professional obligation to do so. 
 Again, it is advantageous to the cause of general
 
 ENGLISH UNIVERSITY EDUCATION^ 61 
 
 cultivation of mind, that there should be a motive 
 supplied, such as is afforded by the duties of the 
 examiner's office, to induce a number of persons, 
 whose previous training has made such studies easy, 
 to keep their hold on science and literature, and 
 to acquaint themselves with the highest point that 
 knowledge has reached. 
 
 But these advantages, however desirable, are not 
 always attainable. It is not easy, except in Colleges 
 which have a considerable number of resident fellows, 
 to find a sufficient body of independent examiners. 
 Even where there are persons on the spot, fitted for 
 the duty, it may often happen that they may not 
 have energy and public spirit enough to prevent their 
 feeling such an office as intolerably irksome and 
 laborious. They may, too, want skill in examining, 
 especially if each holds the office a short time only. 
 But one of the greatest inconveniences which attends 
 the employment of such examiners is, the almost 
 inevitable want of any close correspondence between 
 the instructions given by the tutors, and the questions 
 asked ; and the consequent diminution of the efficacy 
 of the teaching process. In almost all subjects of 
 study, it is possible that a long string of questions, 
 not in themselves objectionable, and a very good* 
 course of instruction, may lie wide of each other ; so 
 that the student who has followed the path of study 
 pointed out to him by his tutor, may be unable to 
 appear with advantage in the examination. And 
 though this entire discrepancy of the lectures and the 
 examination is an extreme case, which may not often 
 occur, the want of a mutual bearing in the two, 
 which will always exist, will destroy in the student
 
 62 ON THE PRINCIPLES OF 
 
 that vivid interest and attention, which he would 
 bestow upon his tutor's communications, if he felt 
 assured that the love of knowledge would lead to the 
 same point as the love of distinction. Again : not 
 only may the independent examiner's questions be 
 devoid of relation to the teacher's view of the sub- 
 ject, but they may be bad in themselves. A man 
 cannot conduct an examination well, except he be 
 quite master of his subject. If, instead of being an 
 habitual study, in which he is familiar with all that 
 has been done in its various forms, the subject be 
 a portion of literature or science which he takes 
 up merely as an examiner, his acquaintance with it 
 will probably be meagre, unconnected, and partial. 
 His knowledge will have no fulness, richness, depth, 
 or variety; and he will not give to those who are 
 really accomplished in the study in question, the 
 opportunity of doing themselves justice, by showing 
 the extent of their acquirements. Again : an ex- 
 amination conducted by a person who does not 
 habitually pursue the study, will not be brought 
 up to the recent progress of knowledge. In classics, 
 such a person will be disposed to confine himself to 
 a mere interpretation of the language, without re- 
 quiring any knowledge of modern critical researches ; 
 in mathematics, he will omit the recent forms and 
 results of scientific investigations. He will deal only 
 with that part of knowledge which, once acquired, 
 has abided by him in spite of neglect. The person 
 who is most likely to acquaint himself with the last 
 advances in any study, is he who habitually teaches 
 the subject ; and who is thus impelled himself to 
 learn, by his interest in the results, by his famili-
 
 ENGLISH UVIVERSITY EDUCATION. 63 
 
 arity with the methods, and by the desire he must 
 have to preserve the respect of his pupils, and his 
 superiority over them. It is true, that the pro- 
 fessors of letters and sciences may be bigoted to an 
 established system, and may thus resist improvement 
 and progress ; but the operation of this feeling rarely 
 continues longer than is requisite duly to balance the 
 love of novelty and change in another class ; and, in 
 fact, almost all general improvements in literature 
 and science have been introduced and diffused by 
 professors. The progress of knowledge runs far less 
 risk of being retarded by the bigotry of teachers, 
 than by the ignorance, indolence, and carelessness of 
 unprofessional examiners. And with regard to the 
 objection derived from the possible partiality and 
 caprice of the teachers, it must be observed that per- 
 sons in so important and responsible a situation as 
 theirs must necessarily be treated with a confidence 
 proportioned to the fidelity which is expected of 
 them. It is not by detached and external checks 
 that they can be made to do their duty, if they 
 have not a due feeling of it themselves : and if their 
 partiality and caprice be excluded from examinations, 
 it may still find abundant room to produce its effect, 
 except the teacher is so far controlled as to have 
 this usefulness materially damaged. The position of 
 teachers who are themselves examiners, would gene- 
 rate, under any tolerable system of selection of teach- 
 ers, a mutual observance and a self-watchfulness, 
 which would be far better safeguards than any ex- 
 ternal and occasional interference. 
 
 The establishment of a board of examiners inde- 
 pendent of the teachers, converts the system from one
 
 64 ON THE PRINCIPLES OP 
 
 of direct to one of indirect teaching ; and must be 
 avoided or modified, except we are prepared to give 
 up direct instruction altogether. If we do this, and 
 trust entirely to the force of examinations, using 
 only the honour and disgrace which they bring, as 
 the means of stimulating indolence and calling forth 
 exertion, we come to an intelligible system, but one 
 very different from any which has ever prevailed in 
 the English Universities. We need then make no 
 demand for attendance at lectures, nor even for resi- 
 dence. One final examination, or several examina- 
 tions at certain intervals, must be all the evidence we 
 require of the student's proficiency, and of his fitness 
 to receive the stamp of University approbation. In 
 this system, all the influences of our direct College 
 teaching, both those which have been mentioned, and 
 others which are still to be spoken of, are entirely 
 abandoned. There may be persons who would think 
 this an advantage; who would prefer the uncolle- 
 giate system of foreign universities to ours ; or who 
 would think that we might sufficiently supply any 
 deficiencies which may exist in them, by university 
 examinations properly devised. Such a system is 
 quite intelligible; but it behoves us to understand 
 what it is, before we decide in its favour. It is 
 right that we should see clearly that it never has 
 been our system ; and that when we talk of its 
 establishment among us, we propose, not the im- 
 provement, but the destruction of our College prac- 
 tices; not a modification, but a revolution in our 
 English University education. 
 
 If, on the other hand, we wish to retain the ad- 
 vantage of direct instruction, we must preserve or
 
 ENGLISH UNIVERSITY EDUCATION. 65 
 
 reform the habits of our colleges in the manner which 
 is requisite to make them efficacious in their proper 
 sphere. We must consider College lectures as im- 
 portant parts of our process, and College examinations 
 as part of the same scheme of direct teaching; we 
 must rather make our examinations auxiliary to the 
 effect of our lectures, than the lectures merely sub- 
 servient to the examinations. And if it appear, as 
 there seems much reason to believe, that this can be 
 done only by giving the lecturers a considerable 
 practical influence upon the examinations, it would 
 be very weak and inconsistent to hesitate to adopt 
 that arrangement, in consequence of any such fears 
 and suspicions as have been spoken of. 
 
 If we were thus to connect our lectures and our 
 examinations, we should, I conceive, do much, not 
 only to produce a body of cheerful and active teach- 
 ers, and of willing and attentive hearers, but also 
 to favour the rapid adoption and diffusion of all that 
 the progress of knowledge brings to light. For a 
 lecturer would prepare for his audience, with plea- 
 sure and with spirit, all that was likely to interest 
 the most zealous and intelligent among them, when 
 he was secure of having his instructions carefully 
 attended to and treasured up. And the student 
 would, in such a system, go on with all his impulses 
 tending the same way ; urged forwards alike by his 
 love of knowledge and his emulation, his sympathy 
 with his teacher and his desire of honour.
 
 66 ON THE PRINCIPLES OF 
 
 SECT. 2. OF PROFESSORIAL LECTURES. 
 
 I HAVE spoken hitherto of our College teaching; 
 but it must be recollected that this, though the 
 peculiar and characteristic, is not the only part of 
 our University system. Besides this, there are in 
 the English Universities not only university exami- 
 nations, but, as in other universities, professors of 
 various departments of science, literature, and spe- 
 culation. These professors have been, and are, 
 some of the most eminent men of their time in 
 their respective provinces ; and their lectures have 
 been rich in instruction and interest ; but a few 
 remarks upon them will suffice for our present 
 purpose. 
 
 Reproaches have often been cast upon the English 
 Universities, in consequence of several of their pro- 
 fessors discontinuing their lectures ; but the cause 
 has not been duly attended to. In almost all cases 
 in which this has been done, the discontinuance of 
 the lectures has arisen from the defection of the audi- 
 ence. It may be said, that this shows the lectures to 
 have been bad; that they would have been attended 
 if they had been worth attending. I think this as- 
 sertion may be shown to be unfounded. 
 
 In the first place, I would observe, that professo- 
 rial lectures appear to have very small attraction 
 for the greater part of Englishmen. The German 
 student pursues the career of his professor with 
 avidity through the most thorny and abstruse paths 
 of speculation, without any motive except his taste 
 for such an employment ; and even writes down his
 
 
 ENGLISH UNIVERSITY EDUCATION. 67 
 
 lectures, with a diligence which must be considered 
 as somewhat superstitious, since it goes to the ex- 
 tent of fixing his attention principally upon the 
 mechanical process of writing at the time, and his 
 voluminous "heft" is often turned to small account 
 afterwards. I do not think that, under any cir- 
 cumstances, English students would be brought to do 
 this, or anything like this. Even when the matter 
 is interesting, and the manner striking, how rarely 
 does the lecturer collect and keep together a volun- 
 tary audience in England ! And if his topic be a 
 subject of exact science or critical research, we are 
 certain that his hearers will soon be reduced to a 
 very few students, and perhaps a few personal friends. 
 In the metropolis, most persons have known of many 
 admirable lectures, delivered in various institutions, 
 on subjects even of great popular interest, as geology 
 or political economy, where general neglect was be- 
 stowed, so undeservedly, as to be a matter of grief 
 and indignation to those who attended. We may 
 explain this as we can, but the fact is certain. It 
 might appear as if our countrymen were too prac- 
 tical to love knowledge and speculation for its own 
 sake, and to bestow time and systematic thought 
 upon it, except it leads to something of profit or dis- 
 tinction. We have seen evidence of the same temper 
 in various circumstances connected with the institu- 
 tions for academic education recently established. It 
 may be possible for a lecturer to draw together an 
 audience by treating some popular subject in a strik- 
 ing manner; but he must have very crude or very 
 visionary notions, who thinks that a solid, unam- 
 bitious course of lectures, on a subject in which
 
 68 OX THE PRINCIPLES OF 
 
 little of novel views or striking exposition could be 
 introduced, would be diligently and regularly followed 
 by voluntary students. 
 
 But I observe further, that even if there were in 
 this country at large a disposition to attend profes- 
 sorial lectures which are really instructive and valu- 
 able, such an attendance could not be expected to be 
 generally given by the students of our Universities. 
 For their time is engaged, and their efforts are drawn 
 away from such occupations, by the demand which 
 the University makes upon them, and especially by 
 the examinations. As the examinations are now con- 
 stituted, they require all the study and intellectual 
 exertion which many of the students are able to give. 
 The influence of such requirements on voluntary study 
 is not a matter of opinion or conjecture. A few years 
 ago a new University examination was established 
 at Cambridge, to take place in the second year of the 
 student's residence. Not only has its effect in inter- 
 fering with the attention to the College courses been 
 very decided, but the University professors at once 
 felt its operation. The lectures of Professor Smyth 
 upon modern history, eloquent and thoughtful disqui- 
 sitions, which had long enjoyed great popularity, and 
 drawn together, year after year, a crowded lecture- 
 room, immediately lost half their audience. Some- 
 thing of the same kind happened to others of the 
 University professors. With men of moderate talents 
 and application, the demands of the examinations are 
 a familiar, and I believe a just, excuse for not pur- 
 suing with earnestness any voluntary study. And 
 even those of most active minds, and of the most 
 lively interest in matters of taste and knowledge, can
 
 ENGLISH UNIVERSITY EDUCATION. 69 
 
 only with difficulty snatch from the course of study 
 into which the University prizes impel them, frag- 
 ments of time to give to other pursuits. It is true, 
 that in many cases the energy of youthful zeal and 
 love of knowledge breaks through all difficulties ; and 
 that our lectures on geology, botany, chemistry, and 
 other sciences, are pursued, as they well deserve to 
 be, by a body of intelligent and persevering voluntary 
 students, and even by persons who aim at high 
 honours in our examinations. But these are young 
 men happily constituted in intellect and disposition, 
 whom systems cannot spoil, as systems cannot make 
 them. It is very certain that the great body of young 
 Englishmen during their residence at the Universities, 
 will never, till their characters and dispositions have 
 undergone a material change, derive any great portion 
 of their education from their voluntary attendance on 
 the public lectures of the University. They bring to 
 the Universities no tastes, or ambition, or prepara- 
 tion, which lead them to follow the speculations 
 which are thus placed before them ; and though they 
 do, I conceive, under a proper administration of our 
 institutions, derive from them very great advantages, 
 these advantages would be reduced to nothing, if they 
 were made to depend upon the voluntary pursuit of 
 literature and science, carried on with the assistance 
 of the University professors. 
 
 Some of the University lectures at Cambridge, 
 particularly those of the Regius Professor of Greek 
 and the Plumian Professor of Experimental Phi- 
 losophy, have been so directed as to be of great use 
 to those who are candidates for the honours which the 
 University bestows ; and on that account, as well as
 
 70 ON THE PRINCIPLES OF 
 
 for their other merits, have been attended with avidity. 
 But this attraction, and the consequent benefits, apply 
 only to students of active minds and generous ambi- 
 tion ; a class about whose education there is nowhere 
 any difficulty. Professorial lectures of this kind can 
 exercise no influence upon a class who are numerous 
 among the students of all universities, the compara- 
 tively dull and inert intellects. And yet these latter 
 are a body which must not be neglected as of no con- 
 sequence ; since not only it contains a great number 
 of estimable, right-minded, and useful men, but also 
 such men are capable of a very large share of mental 
 cultivation, which it is most important to them and 
 to society they should receive. Many of these per- 
 sons will pass none but compulsory examinations, and 
 for such examinations professorial lectures can never 
 be an effective preparation. For these, therefore, if 
 College teaching be supposed absent, the system will 
 become entirely one of indirect instruction. They 
 will derive their knowledge from private study, or 
 from private tutors, under the expectation of the 
 examination which the University insists upon their 
 passing. But before I proceed, I must say a few 
 words on one of the subjects just mentioned. 
 
 SECT. 3. OF PRIVATE TUTORS. 
 
 I BELIEVE that private tutors, under various forms, 
 exist in all universities. Objections have often been 
 made to them, some of which I will consider. 
 
 It has been objected, that they encumber the 
 student with a large and unreasonable expense ; but 
 this allegation I shall not dwell upon. For if private
 
 ENGLISH UNIVERSITY EDUCATION. 71 
 
 tuition be really an advantage, both parents and 
 pupils will desire to have it ; and it would be absurd 
 to legislate against their sacrificing a pecuniary to 
 an intellectual benefit. And if private tuition be 
 prejudicial to the pupil, or to the university system, 
 it ought to be opposed on better grounds than the 
 mere consideration of expense. 
 
 The objections to private tuition, which principally 
 require notice, are, that it generates intellectual 
 dependence and superficial knowledge ; and that it 
 interferes with public teaching. 
 
 It cannot be denied, that the reader who over- 
 comes for himself the difficulties which occur in his 
 studies, and who thinks over his subject by himself 
 till he sees it clearly and fully, is likely to acquire 
 both more distinct ideas, and more conscious power 
 of thought, than he who carries his difficulties to 
 a private tutor as soon as they arise, and passively 
 accepts his explanation. At the same time, it must 
 be observed, that many students would never over- 
 come their difficulties at all, or free themselves from 
 the little embarrassments of learners, 'without some 
 special assistance. The best way of securing the ad- 
 vantage without the disadvantage, in this case, appears 
 to be, that the student should have some persons 
 capable of explaining what he finds obscure, whom he 
 may consult occasionally when his perplexities occur, 
 but upon whom he shall not be in habitual and close 
 dependence for his progress. Such persons the pub- 
 lic tutors in the universities ought always to be. 
 
 But this view supposes, that the student's object 
 is merely to master the subject of his study. In 
 general, however, the student's temptation to engage
 
 72 ON THE PRINCIPLES OF 
 
 a private tutor arises from the wish to prepare him- 
 self for an examination; either a voluntary examina- 
 tion which he enters from a wish for distinction, or 
 a compulsory one which he is required to pass. In 
 these cases, it may be true, that the dependence on 
 the private tutors much diminishes the value of the 
 knowledge acquired ; but still, so long as it is found 
 or thought that such instructions give a person a su- 
 perior chance of success in an examination, they will 
 be sought for : legislation against such tuition will be 
 felt to be a rigid measure, and will, probably, be 
 evaded. Such a law did exist some years ago, in the 
 University of Cambridge, but has since been repealed, 
 for such reasons as have been mentioned. It does not 
 appear advisable, therefore, to prohibit private tutors 
 on such grounds, in a system of indirect instruction. 
 
 But let us consider the other objection; that 
 private tuition interferes with the direct instruction 
 of College tutors. It is evident that this will be the 
 case, if there be, besides the College lectures, a Col- 
 lege examination, not founded upon the lectures. For 
 the same temptation will exist in this case as in the 
 case of University examinations, to obtain and depend 
 upon the assistance of a private tutor, in preparing 
 for the examination. And here, as before, although 
 the College lectures, by assuming a professorial cast, 
 may be useful and valued among the best intellects 
 of the class, those who are more dull, or indolent, or 
 ill-prepared, will not be able to profit much by such 
 lectures, and will prefer to study, if they study at all, 
 under the direction of a private tutor. Thus, the 
 indirect teaching, in such a case, will stifle the direct 
 teaching.
 
 ENGLISH UNIVERSITY EDUCATION. 73 
 
 But it may be said, the instructions of private 
 tutors, are direct teaching; therefore, it might be 
 added, let those who please be left to such teaching, 
 and not placed under a College tutor, by whose 
 lectures in general they cannot profit. 
 
 To this I answer, that private tuition is not direct 
 University teaching. If the pupil be allowed to 
 prepare himself for the examination by the aid of a 
 person who is in no ostensible and responsible situa- 
 tion, he is not in a state of pupilage to College or 
 University at all. The examination is, then, the one 
 point of contact between the educating body and the 
 person educated ; and if this be the case, there re- 
 mains no longer any reason why the pupil's presence 
 at the university should be required, in the intervals 
 of the examinations. The private tutor may, with 
 equal reason, be a clergyman in a distant country 
 village, or a young man in London, if he is not 
 responsible for his charge, any further than he is 
 interested in the result of the examination. And the 
 student, if he is left entirely to himself, his private 
 tutor not professing to direct his efforts towards any 
 thing except the pupil's fitness for examination, will, 
 in most cases, be much better situated in any place 
 removed from the University, than among a body of 
 young men of the most various fortunes and charac- 
 ter ; where it must often happen that, if no super- 
 intendence be exercised, the extravagant and idle 
 will set the fashion, and the worse will corrupt the 
 better, in habits, manners, and morals. 
 
 But, it may be said, the private tutors must not 
 be indifferent about the general character and con- 
 duct of their pupils. They must exercise a general 
 4
 
 74 ON THE PRINCIPLES OF 
 
 superintendence and control over their expenses and 
 habits. To this I reply, that to make such a con- 
 dition effective, the tutors must be placed in some 
 direct official position, and their authority and re- 
 sponsibility clearly declared and enforced. This 
 being granted, as it can hardly be denied, they be- 
 come precisely the " Tutores" of our College statutes, 
 and differ from the actual College Tutor only in the 
 number of their pupils. 
 
 But our present business is with private tutors, 
 who have not this official place and responsibility ; 
 and I think it has been shown, that they cannot be 
 considered as teaching directly on the part of the 
 College, or of the University. By their natural ten- 
 dencies, they belong to a system of indirect teaching ; 
 and a system of direct teaching in Colleges can be 
 maintained, only by prohibiting private tutors, or by 
 making them subordinate to the College system. 
 
 The latter alternative appears to be much the 
 preferable one, when we consider, that among those 
 who come to Universities, there must be many, who 
 from slowness of intellect, or previous defects of 
 education, cannot, without such assistance, derive 
 advantage from any course of lectures. And pri- 
 vate tutors being thus allowed in some cases, the 
 proper mode of securing their subordination to the 
 College system of teaching appears to be, to place 
 them under the control of the College teachers. 
 We may, for instance, direct, that the public tutors 
 shall determine or advise whether a private tutor 
 be needed, shall select the person, and shall com- 
 municate with him from time to time on the progress 
 of his pupil.
 
 ENGLISH UNIVERSITY EDUCATION. 75 
 
 If this practice be adhered to, (as, in fact, some- 
 thing like it has hitherto generally prevailed,) and 
 if the College examinations be made to support the 
 College lectures, the system of direct College teach- 
 ing may be preserved; but if not, the system of 
 indirect teaching will absorb everything, and the 
 public tutors will become an empty name. The most 
 able of the students will desert the public lectures 
 for the instructions of those who specially prepare 
 them for the highest examinations ; the slower minds 
 will seek those private tutors who may enable them 
 to pass the inevitable ordeal ; the Class will be 
 broken into pieces, and almost into individuals, by 
 the diversity of men's powers and courses ; with their 
 influence on the studies of their pupils, the Tutors 
 will lose their influence on their character and con- 
 duct in general ; and the system will cease to be that 
 of the English Universities. 
 
 If, on the other hand, we wish to preserve the 
 English College system of direct teaching, since we 
 have to combine it with the system of indirect 
 teaching, which is almost essential to a University, 
 we must consider how this combination can be 
 effected. 
 
 SECT. 4. OF THE COMBINATION OF THE UNIVERSITY 
 WITH THE COLLEGE SYSTEM. 
 
 I SHALL dwell very briefly on this subject, for 
 I do not here intend to propose changes in detail 
 in any particular University. I will only offer a few 
 general considerations. 
 
 The continued superintendence and control which 
 direct College teaching implies, are most easy and 
 4 2
 
 76 ON THE PRINCIPLES OF 
 
 natural in the earlier part of the pupil's residence 
 in the University. I should judge it desirable that 
 the two first years of his career were employed in 
 this kind of study ; and I think it would be a great 
 advantage, if no University examinations, compulsory 
 or emulous, disturbed the even tenour of this course 
 of instruction. In the University of Cambridge, this 
 maxim has been of late departed from. Besides 
 various prizes, which are open to the junior as well 
 as the senior students, an examination, termed " the 
 previous examination," was established in 1830, at' 
 which all the students are required to acquit them- 
 selves in a satisfactory manner, in the middle of 
 their second year. The ground of this step was a 
 persuasion that in some of the Colleges the progress 
 of the students was not sufficiently insisted upon ; and 
 the belief that it was the duty of the University 
 to secure an attention to study, in the first years 
 of residence, by a compulsory examination, as she 
 is supposed to effect the same object, at a later 
 period, by the final examination which precedes the 
 admission to an academic degree. It is impossible 
 not to respect the motives which actuated the authors 
 of this measure, and I believe it is conceived to 
 have answered its purpose to a very great extent. 
 It has, however, interfered very seriously with the 
 College instruction, in Colleges where no such new 
 compulsion was needed; and I should gladly see 
 it removed out of the second year of the College 
 course. 
 
 After two years spent under the influence of 
 College rules, I conceive the student might, with 
 advantage, be given over to the motives and em-
 
 ENGLISH UNIVERSITY EDUCATION. 77 
 
 ployments which the University offers ; though still, 
 as before, under the general superintendence and 
 direction of his College tutor. By this time, the 
 list of mathematical subjects would have been gone 
 through, so far as it is important to follow them in 
 a definite order, and with a discussion of principles ; 
 and classical subjects might easily be framed into a 
 progressive system, fitted to the same period. Aftei 
 this, the forward student would be prepared for the 
 highest exertions and the widest speculations, for 
 competitions and professorial lectures ; while the slow 
 learner would have exhausted all that could easily 
 be communicated to him directly ; and might be well 
 employed in preparing for the final examination ; 
 and, if possible, in also voluntarily attending the 
 lectures of some professor on a subject of general 
 interest ; and it would be quite consistent and 
 reasonable, that this, the University period of the 
 student's career, should be opened by a previous exa- 
 mination, fitted to ascertain that he was in the 
 state of proficiency which the system supposed. 
 
 I give this sketch merely with a view of pointing 
 out the mode in which the University and College 
 systems may be combined, and the great inconve- 
 niences avoided, which may arise, and have arisen, 
 from their conflicting tendencies ; by no means in- 
 tending to offer it as a scheme of which the details 
 are matured. But this, or something of this kind, 
 is, I think, a plan well worthy the notice of all 
 English Universities.
 
 78 ON THE PRINCIPLES OF 
 
 CHAPTER III. 
 
 OF DISCIPLINE. 
 
 SECT. 1 OF THE NECESSITY OF DISCIPLINE. 
 
 THAT the teaching of the intellect alone is in- 
 sufficient to prepare man for his place in society, 
 and for all the higher purposes of his destination, 
 is allowed by all who have thought seriously on 
 education. This declaration has, for instance, been 
 repeated again and again by some of the wisest 
 among those patriotic men who, in France, are 
 trying to remodel in a beneficial manner the national 
 education. We must, say they, educate, not instruct 
 merely ; we must infuse a sense of moral and re- 
 ligious responsibility, as well as mere knowledge ; we 
 must form the principles of conduct as well as the 
 intellect. 
 
 But how is this to be done ? or can it be done 'I 
 What selection of the matter or of the mode of com- 
 munication, can affect the moral nature ? What kind 
 of knowledge can give habits of self-government and 
 a sense of duty? We may make the boy learn by 
 memory, maxims and rules, prayers and creeds ; but 
 the memory does not sway the heart. And when 
 we have placed the youth in the independent po- 
 sition of the student at a university, how shall we
 
 ENGLISH UNIVERSITY EDUCATION. 79 
 
 teach his light mind and impetuous spirit to recollect 
 that his condition is one of grave responsibility ; that 
 he must act with considerate reference to external 
 regards and internal convictions of duty; and that 
 the religion taught to his boyhood is intended to 
 form an unbroken part of the business of his life ? 
 
 The answer which the principles of English edu- 
 cation direct us to give to such inquiries, is this ; 
 that the meaning and the value of the moral and 
 religious maxims which are taught to the boy, are 
 to be impressed upon his heart by the personal 
 exhortation of parents and other instructors ; and 
 that the student at the university is not to be un- 
 controlled, but is to be in such a condition that he 
 is never allowed to forget, that the demands of 
 society and the rules of duty must direct his habits 
 of action and shape his manners. 
 
 This lesson, which cannot be taught through the 
 memory alone, is conveyed by the position in which 
 the student is placed at the English Universities. 
 For he is subjected to many rules, and put under 
 governors and monitors, who, by their institution, 
 are invested with a combination of parental and 
 official authority. And hence he acts in a little 
 world, which is constituted of definite relations and 
 duties, and requires a certain self-restraint and self- 
 regulation at every step ; and thus is a fit school to 
 prepare him for the world of real action. Whereas. 
 without such a constitution of the University, the 
 student's academical career is a period of unbounded 
 freedom from restraint and responsibility ; which 
 may be full of enjoyment, and, to many, of the 
 occasions of great intellectual developement ; but
 
 80 ON THE PRINCIPLES OF 
 
 which, to the greater number, must be a portion 
 of life perfectly heterogeneous with all the rest ; 
 a scene governed by its own principles, these being 
 such as can by no means be admitted into the 
 general business of society. 
 
 We shall now consider some of the rules and 
 arrangements which thus constitute the Discipline 
 of the English Universities. Each University, in 
 its collective capacity, has its institutions and rules, 
 constructed with reference to the preservation of 
 good manners, morals, and religion ; but I shall 
 speak principally of the discipline of the English 
 Colleges, since I shall thus sufficiently bring into 
 view the principles of the subject. 
 
 SECT. 2. OF COLLEGE MANNERS. 
 
 WHAT the manners of Colleges were intended 
 and expected to be by the founders of Colleges, 
 I cannot show better than by a few extracts from 
 the Statutes of Trinity College, Cambridge ; and 
 especially from the chapter, " De modestia et morum 
 honestate colenda." 
 
 " Whereas there is nothing which more adorns 
 men of letters than modesty and purity of manners ; 
 we therefore decree and ordain, that all inferiors 
 behave themselves towards their superiors in a sub- 
 miss and reverent manner; the Scholars towards 
 the Bachelors, the Bachelors towards the Masters of 
 Arts, these towards Bachelors and Doctors of Di- 
 vinity, and all towards the Master as the supreme 
 governor, and also towards the eight Seniors as 
 fathers and leaders. And if any shall be shown to
 
 ENGLISH UNIVERSITY EDUCATION. 81 
 
 have offended in this point, let them be punished 
 according to the will of the Master, or, in his ab- 
 sence, of the Vice-Master and Senior Dean. Let 
 none of the Bachelors or Scholars go into the town 
 without taking some one with him to be, as it were, 
 the witness of his proper conduct : let no one in the 
 hall, in the court, or elsewhere within the College, 
 neglect to take off his cap in the presence of a 
 Master of Arts, or one of higher degree. And if 
 any one be found to have gone into the town alone^ 
 for the first offence let him be deprived of one week's 
 commons ; for the second, of two weeks ; for the 
 third, of a month ; for the fourth, by the consent of 
 the Master and eight Seniors, or the greater part 
 of them, let him be removed from College. But 
 Bachelors who have finished the second year after 
 their degree, we allow to go into the town without 
 a companion. Let no one of the Fellows, or of the 
 Scholars, or of the other residents, frequent houses 
 suspected, or of ill name ; and if after two admo- 
 nitions from the Master, he abstains not, let him 
 lose the College in the manner above described. 
 Let the authors of domestic sedition, detraction, 
 dissension, or wrangle, for the first offence lose a 
 month's commons ; for the second, three months ; 
 for the third, let them, as we have said, be expelled 
 from the College. . . . We also decree, ordain, 
 and exhort, that the Master, Fellows, Scholars, and 
 other residents in the College, do use their utmost 
 endeavour to nourish, cherish, and preserve concord, 
 unity, peace, and mutual charity ; and avoid, in word 
 and deed, scurrility, ribaldry, scoffs, whisperings, 
 reproaches, and scandals. (Scurrilitatem, obscoena 
 4 5
 
 82 ON THE PRINCIPLES OP 
 
 verba, scommata, susurros, probra, scandala, verbo 
 et facto vitent.) . . . Let no one keep dogs, 
 ferrets, hawks, or singing birds, in the College ; nor 
 be immoderately given to hunting or hawking ; and 
 if any one transgress, let him be punished as above." 
 In the same manner, it is directed that no one shall 
 spend the night elsewhere than in his chamber ; or 
 dine or sup in his chambers, except with the per- 
 mission of the Master ; or go without his cap and 
 gown ; or depart out of College without leave grant- 
 ed ; with many similar rules. And again, (De Cu- 
 biculorum Distributione) : " We will and decree 
 that each person conduct himself with propriety in 
 his own chamber; and do not, by immoderate cla- 
 mour, or loud laughter, or singing, or noise, or 
 dancing, or musical instruments, keep his neighbour 
 from sleep, quiet, or study ; and also that he abstain 
 from late revels, and from potations." And with 
 like views, it is provided that no Bachelor, Scholar, 
 Pensioner, or Sizer, be without a Tutor; and not 
 only the relation of Tutor and Pupil is described in 
 the way we have already noticed (p. 53), but it is 
 directed that all the bills due to the College shall be 
 paid by the Tutor. 
 
 It may be asked whether these rules have been 
 observed, and what the effect has been. And to 
 this we answer, that both in this, and in other Col- 
 leges of the English Universities, these and similar 
 rules have been in force, in a great measure literally; 
 while in other cases, where the change of circum- 
 stances and manners required a modification of the 
 forms, the spirit and tendency has been preserved, 
 and in all well-conducted Colleges has regulated the
 
 ENGLISH UNIVERSITY EDUCATION. 83 
 
 temper and principles of their usual proceedings. 
 We may add, also, that the effects of this discipline 
 have been in the highest degree beneficial ; and have 
 shown that such a system, if earnestly and faithfully 
 administered, may, in a great measure, lead to a 
 general prevalence of that respectful temper, that 
 moral character, those good manners and orderly 
 habits, at which it aims. 
 
 The impression made on the disposition by such 
 a system of discipline, is stronger than might be 
 expected by most persons. Even tempers of great 
 levity and stubbornness, if they are met at every 
 turn of their extravagant and self-willed motions 
 with the calm, but severe countenance of a system 
 of rules like these, imposing punishment for trans- 
 gression, so long as it can be ascribed to thought- 
 lessness, but pointing constantly to the door, if 
 transgression is persisted in ; are awed and tamed ; 
 and in a little while moulded to their position : 
 while the great body of young Englishmen, of the 
 condition of those who come to the Universities, 
 conform, with a generous obedience of spirit, to rules 
 which are the very essence of the institution in which 
 they are placed, and of which all the better natures 
 among them see and feel the value. I am quite per- 
 suaded that no one could become acquainted with 
 the temper of the students of our Universities to- 
 wards their College discipline, and towards those 
 who administer it, without forming a strong affection 
 and admiration for them, and a steady hope and 
 trust in the beneficial influence of the College sys- 
 tem upon their manners, temper, and practical 
 character.
 
 84 ON THE PRINCIPLES OF 
 
 How, indeed, should it be otherwise, considering 
 what is the extraction and the previous condition of 
 our students? The youth of England, those whose 
 childhood has been nurtured in the homes of England ; 
 who come to us from the arms of English mothers, 
 and from the side of English sisters ; whose memory 
 is stored with the history, and with the poetry, and 
 with the prayers of their native land ; these are not 
 men to bring to us a churlish and captious spirit, 
 looking for and finding nothing but evil in the body 
 in which they are placed. These are not men to 
 set themselves in array against the mild and equal 
 restraints of beneficial rules ; these are not men to 
 think resistance and insurrection glorious, against 
 laws and authorities however necessary. Who can 
 suppose that Englishmen are not ready to conform to 
 a wise system of discipline, to enter into it with all 
 their hearts, and by this means to bring forth the 
 very utmost strength of their strenuous national cha- 
 racter, who can suppose this, that looks at what 
 they have done and do, in this very manner, in our 
 army, and in that unparalleled example of the com- 
 bined force of strong character and rigorous rule, our 
 navy? I have not the smallest doubt that the full 
 and cheerful obedience of the pupils of the English 
 Colleges to any portion of the requirements I have 
 above quoted, which, on a considerate revision, it was 
 thought proper to preserve, might be obtained, with 
 moderate vigilance and care. 
 
 It is long since an opinion was expressed, that 
 Discipline was declining in the English Universities. 
 The reader of Cowper will recollect that this subject 
 is pursued at some length in The Task.
 
 ENGLISH UNIVERSITY EDUCATION. 85 
 
 In Colleges and Halls in ancient days, 
 When learning, virtue, piety, and truth, 
 Were precious and inculcated with care, 
 There dwelt a sage called Discipline. * * 
 
 * * 
 
 If e'er it chanced, as sometimes chance it must, 
 That one among so many overleap'd 
 The limits of control, his gentle eye 
 Grew stern, and darted a severe rebuke : 
 His frown was full of terror, and his voice 
 Shook the delinquent with such fits of awe* 
 As left him not, till penitence had won 
 Lost favour back again, and closed the breach. 
 But Discipline, a faithful servant long, 
 Declined, at length, into the vale of years, 
 A palsy struck his arm. 
 
 I do not know how faithful, as a general representa- 
 tion, was this poet^s censorious picture of the state of 
 Colleges and Halls in his time ; but I have little 
 doubt that, in regard to morals, and, as to essentials, 
 in regard to manners, the condition is better now 
 than it was then. At least, I am very certain that 
 the consummation which he describes has not yet 
 taken place, and, I trust, never will, when " Disci- 
 pline, at length, fell sick and died." In some cases, 
 the alteration of the national manners, in others, the 
 changed circumstances of the students, are such as to 
 make a literal enforcement of the original rules ab- 
 surd. For example, in the passage above quoted, the 
 same reason which made it fit to allow Bachelors to 
 go into the streets of the town at their own discre- 
 tion, applies now to under-graduates, who are as 
 advanced in age at the present day, as Bachelors 
 were when the statutes were given. It may be, too, 
 that with an improvement in many respects, there is
 
 86 ON THE PRINCIPLES OP 
 
 an increased laxity in the observance of rules on 
 other points ; for example, the constant use of the 
 academic dress. But in most such cases, the fault is 
 in those of superior position, who ought to enforce 
 the observance of rules, and who are not sufficiently 
 vigilant and earnest in this part of their duty. " Nos, 
 nos consules desumus." If, for instance, all persons 
 in the Universities, who have pupils under their care, 
 were persuaded that the academic dress is a valuable 
 remembrancer of the duties and obligations of the 
 student's position, and were to enjoin its use on all 
 occasions, and to rebuke its absence, there can be 
 little doubt that omission in this respect might soon 
 be rendered as rare as it ever was. But it is not my 
 intention, at present, to propose special reforms. 
 
 The tendency of one of the directions above men- 
 tioned may not be obvious as a matter of discipline : 
 I mean, that which directs that the College bills of 
 the pupils shall be discharged by the tutor. This is 
 now extended, in most or all of the Colleges in Cam- 
 bridge, into a practice of passing, not only the College, 
 but also the tradesmen's bills, and the payment of 
 them, through the hands of the tutor. I have no 
 doubt that this practice tends much, by the super- 
 intendence and opportunity of remonstrance and 
 control which it introduces, to moderate, both the 
 extravagance of thoughtless young men, and the 
 demands of unscrupulous tradesmen ; and that, even 
 when it is not extended to the whole of the expenses, 
 it keeps up among the students a general feeling of 
 the necessity of probity and punctuality in pecuniary 
 matters, which might soon be seriously impaired, if 
 they were left to frame their own code of honour on
 
 ENGLISH UNIVERSITY EDUCATION. 87 
 
 such subjects, with no check to their inexperience, 
 levity, and caprice. The vulgar prejudice, that in 
 such an arrangement the tutors improperly gain 
 much by paying the tradesmen's bills long after they 
 receive the money, though they may lose by paying 
 the College bills before they can receive anything, as 
 they are compelled to do, is not worthy notice in a 
 survey of principles like this. Nor, perhaps, ought 
 the great labour and solicitude thus occasioned to 
 the tutors to stand in the way of the continuance of 
 a rule so beneficial. 
 
 In our view of the influence of English University 
 education upon the manners of the students, it would 
 not be right to omit to consider the effect the stu- 
 dents have in forming each other. This, indeed, is, 
 and must be, one of the most important points in all 
 systems of social education. And the operation of 
 the English Universities in this way, is one of their 
 most important and beneficial functions. Young men 
 of all classes, from the highest to very lowly ones, 
 are brought together, and made to feel that a common 
 participation in a liberal education puts them, to a 
 certain extent, on a footing of equality, and establishes 
 an obligation of mutual respect. The peasant's son, 
 who becomes the country clergyman, is thus elevated 
 in feelings to the level of his office ; for he is made 
 to feel that, in the obligation to good manners and 
 straightforward conduct, and in the right, too, to 
 judge of such matters, he is no way behind the 
 duke, who was his contemporary at College, and is 
 his neighbour. At the University, also, the young 
 Englishman receives the most decisive part of that feel- 
 ing which he is acquiring during his whole education,
 
 88 ON THE PRINCIPLES OF 
 
 and which it is one of the most essential offices of his 
 education to unfold; namely, the feeling that he is 
 an Englishman ; a knowledge of the principles by 
 which the actions of his fellow-citizens are regulated, 
 and by reference to which his own will be judged of, 
 joined with a sympathy with their objects, and a 
 habit of balancing himself among their impulses. 
 Finally, at that crisis of life, when the vigour of 
 manly thought blends with the warmth of youthful 
 susceptibility, engaged in the enjoyments of lite- 
 rature, speculation, companionship, or competition, 
 with his fellow-countrymen of his own age and 
 position, he acquires a number of subjects of com- 
 mon interest, of agreeable retrospect, of endearing 
 recollection; and these points of union bind together 
 the Universitymen of the same standing, by a tie 
 which rarely loses its hold, or its charm, during their 
 lives. 
 
 We must, however, look at the dangers of this 
 mutual influence, as well as its advantages. Many 
 of these dangers are obvious enough ; and against 
 these, the College system, with its discipline and its 
 tutors, appears to me, if well administered, to provide 
 as far as human institution can provide. For we 
 must recollect, that evil cannot be entirely removed; 
 that men can be taught to act, only by being in a 
 great measure free agents ; and that when all act 
 freely, some will act ill. Some may become vicious 
 or extravagant at the universities, but the contagion 
 of such men is far less baneful under the system 
 there pursued, than it would be in a similar body 
 of young Englishmen, collected under almost any 
 other circumstances.
 
 ENGLISH UNIVERSITY EDUCATION. 89 
 
 Besides the operation of example and private 
 intercourse, which students exercise on each other ; 
 there occur sometimes occasions in which their mutual 
 influence appears in a more public manner. But 
 such manifestations cannot proceed far, without 
 coming in conflict with obvious maxims. If the 
 actions of the students of English universities ever 
 assume a character of studied independence, or con- 
 cert, or tumult, implying a forgetfulness that they 
 are under a system of superintendence, control, and 
 discipline ; in such a case I conceive that the 
 English university system is infringed ; and that the 
 governing body, or the pupils, or both, have lost sight 
 of the rules which alone can make their relation 
 permanently sound and beneficial. 
 
 SECT. 3. OP COLLEGE PUNISHMENTS. 
 
 THE subject of punishments in education is one 
 of extreme difficulty. Even in the earlier stages of 
 youth, how hard it is to find any infliction which 
 represses faults, without exciting a spirit of evasion 
 and hostility. And when we have to do with youths 
 who have acquired, in a great measure, the character 
 and the privileges of manhood, how shall we invent a 
 punishment, which they will feel sufficiently to make 
 it effective, and to which they will submit ? By what 
 severities shall the teacher repress and constrain those 
 -who almost feel in themselves a right to independence, 
 without introducing extreme punishments for those 
 small offences which must often occur? Corporal 
 chastisement, though habitually referred to in the 
 statutes of English universities, constructed when the
 
 90 ON THE PRINCIPLES OF 
 
 age of students was different from what it is now, 
 is of course out of the question. Personal restraint, 
 though not so utterly disused, is not favourable to 
 health of body or mind, and is not easily enforced 
 with rigour, except by a force organized for the pur- 
 pose. Tasks for the memory vary much in their 
 efficacy, according to the habits of him who performs, 
 and of him who imposes them; besides being a serious 
 infliction on the latter, as well as the former : more- 
 over, except supported by some possible ulterior 
 punishment, they are evaded or neglected. The 
 same may be said of tasks for the pen, which, be- 
 sides, offer additional modes of evasion. Fines are 
 felt little by the son of a rich parent, or by the 
 thoughtless son of any parent. Taking away days 
 or weeks, as it is technically called, that is, not 
 allowing them to be reckoned to the student's re- 
 quisite residence, besides sharing the inconveniences 
 of other punishments, leads to procrastination, and 
 to a habit of speculating on the chances of the 
 future ; since the evidence of residence is only ex- 
 amined at certain epochs of the student's career. 
 Temporary banishment may be useful in some cases, 
 but it is not applicable to small offences. And final 
 extrusion is a sharp sword, which cuts, but does not 
 untie, the knot ; which must be considered as sacri- 
 ficing one to many, and therefore must not be lightly 
 brought into play. 
 
 Moreover, how shall we induce the teacher to 
 inflict these punishments ? to incur ill-will and 
 anger from those who are almost his equals ; with 
 whom, perhaps, he is living on terms of familiar 
 intercourse ; and with whom, at any rate, such a
 
 ENGLISH UNIVERSITY EDUCATION. 91 
 
 hostile position must impair confidence and regard \ 
 And this too, perhaps, for some neglect of what 
 appears a mere form ; for if forms be not insisted on, 
 rules crumble away. It is as difficult to find a hand 
 to hold the rod, as it is to construct the rod itself. 
 
 How have our colleges attempted to deal with 
 this difficulty, and how have they succeeded? It 
 appears to me, that they have devised a plan, which, 
 if faithfully and consistently acted on, would pro- 
 duce the true results of punishment, all so desirable, 
 yet so difficult to reconcile ; which would repress 
 small offences without extreme severity, and would 
 be severe, when necessity was, without forfeiting 
 respect and regard. 
 
 This plan has already appeared in the quotations 
 which have been made from college laws. Its general 
 character may be briefly stated : it is this : Every 
 college punishment is an expression of the disappro- 
 bation of the college ; this disapprobation is increased 
 by every successive offence ; and, carried to a certain 
 point, makes removal from the college necessary. 
 
 Thus, we have seen in the above extracts; for 
 the first offence, let him forfeit one month's com- 
 mons ; for the second, three months' 1 ; for the third, 
 let him be expelled the College; and the same 
 kind of formula is used in almost every penal ap- 
 pointment. It will easily be seen, that in this 
 manner, punishments, which are slight as inflictions, 
 are serious as warnings. A small fine, or the for- 
 feiture of a college allowance, or some restraint on 
 the pupil's motions, or an exercise of the memory, 
 or of the pen, which in themselves might be thought 
 lightly of, receive efficiency from the consideration
 
 92 ON THE PRINCIPLES OF 
 
 of their possible consequences. They do not, when 
 gone through, leave the delinquent where he was 
 before : there is a shade on his prospect : he has 
 to re-establish a character, without which he is in 
 a situation of increased danger. He may dispel 
 the shade ; he may resume his original position of 
 safety ; but that must be by showing the will to 
 avoid further offence. Without this, the frown of 
 the body, under whose regard he is, becomes con- 
 tinually darker and darker, and at last the sen- 
 tence of rejection is uttered. 
 
 Such appears to be the intended spirit of College 
 punishments. Their efficacy in fact, will depend 
 upon their being administered in this spirit. If any 
 penal process is ever put forward as an intended 
 infliction of so much inconvenience for so much trans- 
 gression, the general meaning of the proceeding is 
 changed, the language of College law is misconstrued, 
 and its words lose their force and significance. The 
 purpose of the college was, not retribution, but warn- 
 ing; the demand was, not pain, but amendment; the 
 process was intended to operate, not on the passive, 
 but on the active powers of the offender ; not to task 
 his patience, but to teach his will. 
 
 And hence, in estimating transgression, the will is 
 mainly to be attended to. For instance, in reference 
 to the neglect of any formal rule, the question is not 
 so much, Did the pupil exactly conform to the rule ? 
 as, Did he endeavour to conform to it? Did he make 
 regularity an object, among the things which are 
 really his objects ? Did he allow himself no inten- 
 tional omissions? were there no deviations arising 
 from obstacles which he would have overcome, or
 
 ENGLISH UNIVERSITY EDUCATION. 93 
 
 thoughtlessness which would not have occurred, on 
 a matter in which he was in earnest ? If he was 
 thus blameless in purpose, the violation of the rule 
 in form need not be insisted on. In such a case, 
 impunity will lead to no bad consequences. 
 
 It becomes necessary, on this view of the case, 
 that the infliction of punishment should be usually 
 accompanied with personal intercourse between the 
 officer and the culprit, for otherwise the character of 
 the offence cannot be rightly judged of. And this 
 intercourse, to a considerable extent, is absolutely 
 requisite to the efficacy of College punishments. It 
 is important, not only in order to enable the officer 
 to ascertain that temper of the offender on which the 
 amount of the transgression depends, but still more, 
 on another account ; namely, to give him an oppor- 
 tunity of explaining its meaning and tendency 
 according to the principle above laid down, and of 
 enforcing its influence by such remonstrance, rebuke, 
 and warning, (or encouragement, if necessary,) as the 
 situation of the pupil and of the officer may give 
 opportunity for. The occasion which the institutions 
 of our Colleges afford for intercourse of this kind, is 
 among the most valuable parts of the structure, and 
 must be a main source of their beneficial influence, 
 till their principles are utterly changed, and English 
 University education entirely subverted. 
 
 Undoubtedly such intercourse as this is often very 
 distasteful and irksome, both to officers and to stu- 
 dents ; and especially to those who have never been 
 led to consider, that the whole usefulness and import- 
 ance of the Colleges, as distinguished from the Uni- 
 versity, resides in the relations between the College
 
 .94 ON THK PRINCIPLES OF 
 
 officers and the students ; from which relations the 
 duty of remonstrance and warning necessarily flows. 
 Instead of this plan, which unavoidably produces soli- 
 citude, and often very unsatisfactory conversations and 
 unpleasant feelings, many persons would doubtless 
 much prefer a system in which certain fixed punish- 
 ments should be applied according to certain fixed 
 rules; and in which no perplexity or irritation on 
 either side need exist, since all that happened must 
 have been covenanted and foreseen. But the proper 
 reply to the proposal of such a scheme would be, that 
 there are no punishments, which, so administered, 
 can answer the purpose of punishment. For suppose 
 the lighter inflictions to be disregarded : nothing 
 then remains, but to enforce them by heavier se- 
 verities ; and so on, till we come to some punish- 
 ment which is a real misfortune to the student. And 
 to suppose the teacher careless of this, would be to 
 suppose in him such an indifference to the welfare of 
 his pupils, as is a very unfit temper for an English 
 University. The teacher, then, must warn his pupil 
 of the ultimate consequences of transgression, and 
 endeavour to induce him to avoid them ; and thus 
 we come back to the plan of personal remonstrance. 
 College rules, and the existence of rule at all, ne- 
 cessarily lead us to this conclusion, whenever the 
 teachers and the pupils are living in mutual good 
 will and friendly intercourse. 
 
 But the business of remonstrance and personal 
 intercourse, with regard to violations of rule, is not 
 only irksome and repugnant to those who do not 
 sympathize with the spirit of College institutions ; 
 it is also, in a large body, oppressive and difficult,
 
 ENGLISH UNIVERSITY EDUCATION. 95 
 
 from the time, and energy, and vigilance which it 
 requires of the officers, and from the solicitude and 
 pain which it occasions to them. And this may be 
 so serious an evil, that it may be well worth consi- 
 deration whether it cannot be in some measure 
 alleviated. And especially if the true principles of 
 College government should have been partially lost 
 sight of, or misunderstood, by officers or by pupils, 
 the task of restoring them to their proper authority 
 and efficacy by personal intercourse, would require 
 exertions which it might not be easy to obtain. In 
 such a case, perhaps something might be done by 
 putting upon paper admonitions which are suited to 
 the most common offences, and which might, without 
 trouble, be used when any occasion occurred. For 
 example, a paper to the following effect might be 
 conveyed to the pupil, preceded by a statement of 
 the rules laid down by the College for his conduct. 
 
 " These are the rules which you are required to 
 observe by the College of which you have been 
 admitted a member. You are to understand that 
 the intention of obeying such rules as exist in the 
 College, was taken for granted by the College in 
 consenting to your admission. You are to under- 
 stand, also, that the constant intention of obeying 
 such rules is taken for granted so long as you con- 
 tinue a member of the College ; and that as soon as 
 it becomes clear to us that you have not such an 
 intention, we cannot permit you to remain among 
 us. 
 
 " In case of your deviating from the rules by 
 thoughtlessness or accident, you are liable to such 
 punishment as the officers of the College may impose.
 
 96 ON THE PRINCIPLES OF 
 
 And you are to understand such punishment, not as 
 a satisfaction to the College, but as an expression of 
 the disapprobation of the College towards you. If 
 such punishments are repeatedly incurred, you are to 
 understand that each of them expresses an increasing 
 conviction on the part of the College, that the pur- 
 pose of obedience and conformity, which is the con- 
 dition of your being allowed to remain among us, 
 does not exist. If this conviction is not removed or 
 diminished by your own conduct, any additional con- 
 firmation of it by future offences will be considered 
 as making your removal necessary. 
 
 " You are to recollect that the intention of obey- 
 ing the rules constitutes your duty, and that the 
 College officers are to judge of this intention. It is 
 therefore no palliation of any omissions to say that 
 equal or greater transgressions are committed by 
 others ; or that you are not very irregular ; or that 
 you observe some rules though you transgress others; 
 or that you did not expect to have absolute regu- 
 larity required. The intention of absolute regularity 
 is required. 
 
 " Such are the conditions of your position here : 
 without attention to these on your part, we do not 
 hope to be of any use to you ; nor can we allow your 
 connexion with us to continue, when that hope is 
 gone. And every occasion which we may have to 
 remind you of these conditions, you must consider as 
 an admonition and a warning, which, if not duly 
 attended to, will be followed by more serious conse- 
 quences." 
 
 Such a paper as this might be conveyed to the 
 student on each transgression ; and perhaps it might
 
 ENGLISH UNIVERSITY EDUCATION. 97 
 
 be useful to mark, in the list of rules, that rule, the 
 neglect of which had occasioned the admonition's 
 being sent. Such a communication would remove all 
 reasonable ground of complaint and misunderstand- 
 ing ; and in all the better disposed, would probably 
 produce the effect of bringing them to a due sense of 
 their position and duty. 
 
 There is one mode of influencing the students, 
 which I have not mentioned yet, but which ought 
 not to be overlooked ; I mean the step of the tutor's 
 communicating with the student's parents or friends 
 in serious cases of offence, representing to them his 
 pupil's errors and danger, inviting their aid, or sug- 
 gesting a voluntary temporary removal. As connected 
 with punishment, this step may be considered as a 
 more forcible remonstrance and warning, and may be 
 very beneficial. 
 
 I believe that, in some institutions for education, 
 the difficulty of providing punishments and enforcing 
 rules has been got rid of, by calling in the parent's 
 interposition at the very first. For example, the list 
 of the student's attendance at his teacher's lectures 
 may be sent at once to the parent, and thus the 
 lecturer may escape the troublesome and disagreeable 
 task of compelling attendance from the unwilling 
 student. 
 
 This plan appears to me to be one which, from 
 misunderstanding or despairing of the powers of a 
 College, misapplies and throws away the parent's 
 influence. For the remonstrance, displeasure, and 
 pain of a parent might be strong motives with a 
 young man, if they were reserved for very serious 
 cases ; if the mere fact of applying to the parent
 
 98 ON THE PRINCIPLES OF 
 
 were evidence how gravely the tutor or the College 
 looked upon the matter ; and if such a step were 
 never taken in consequence of the neglect of rules, 
 till the failure of the lighter forms of remonstrance, 
 and the neglect of the common punishments of the 
 College, had aggravated the case. Such an appeal 
 might affect even a stubborn spirit, if it were made 
 the catastrophe of a tragedy of which the previous 
 acts took place in College ; but if it appear in the 
 first scene, it is lightly thought of. The student may 
 easily satisfy an indulgent parent by excuses which 
 would not satisfy a College officer ; or may disregard 
 his dissatisfaction : and in either case, not only the 
 rule, which we must suppose a salutary one, is evaded, 
 but also a general habit of disregard, both of regu- 
 larity and of authority, is nourished. 
 
 We must now say a few words of the application 
 of this kind of discipline to particular cases. 
 
 SECT. 4. OF ATTENDANCE AT COLLEGE LECTURES. 
 
 IT will easily be supposed that constant attendance 
 at the College lectures is one of the rules of such in- 
 stitutions, and is, therefore, enforced by such a disci- 
 pline as we have described. Express directions on 
 this point are found in the statutes which I have 
 already quoted: "If any one is absent from the 
 lectures, let him be fined ; but if he is absent often, 
 let him be brought before the Master of the College.'"'' 
 
 Since the lectures constitute the teaching in the 
 College, the necessity of their being regularly carried 
 on, both by teachers and pupils, is obvious. But an 
 objection to this is sometimes made, to this effect :
 
 ENGLISH UNIVERSITY EDUCATION. 99 
 
 -i What is the use of compelling attendance in the 
 lecture-room, when you cannot compel the pupil to 
 give his thoughts to the lecture I If he is interested 
 in the subject, he will come without compulsion ; if 
 he is not, you can do him no good." 
 
 To this objection we might obviously reply, on 
 the principles of the last section, that a willingness, 
 not only to repair to the lecture-room regularly, but 
 to give attention to the lecture, is supposed in the 
 student ; and that where there is no disposition on 
 his part to avail himself of the means of teaching 
 which the College employs, there is no longer any 
 reason for his connexion with the College ; that is, 
 no reason which the College can recognise. 
 
 But we may say further, that the assertion, that 
 though you can compel the student's attendance, you 
 cannot compel his attention, is altogether frivolous. 
 You do not compel, indeed, but by the habits which 
 you establish, and the occasions and inducements 
 which you offer, you are almost certain of engaging 
 his attention in no small degree. Placed for an hour 
 under the eye of his tutor, who presents to his notice 
 a subject urged upon him by its known place in the 
 College course, and perhaps by its occurrence in an 
 examination which he must undergo; surrounded 
 by his fellow-students, who are actively employed on 
 the same subject; perhaps personally addressed by 
 his tutor; his path directed, his mistakes pointed 
 out ; the student ought not to refuse, and in most 
 cases will not refuse, to give his mind to that which is 
 thus recommended by duty, sympathy, and circum- 
 stance. The use, or at least one use, of the habit of 
 College lectures is, like the use of all habits, that by 
 5 2
 
 100 ON THE PRINCIPLES OF 
 
 bringing an employment before us in a regular and 
 familiar manner, it takes away the disposition to 
 deliberate whether or not we shall engage in it. We 
 enter into it, according to the common expression, 
 as a matter of course. And thus, under a system of 
 compulsory lectures well administered, the pupils will 
 devote many an hour to real study, which otherwise 
 might have been spent in other employments. 
 
 It may be said that lectures can be of little use, 
 since the student has access to books ; and since it is 
 probable that the College tutor will not know so 
 much as the best modern authors have known. But 
 it is easy to see, that though he may not know so 
 much, he may communicate more : he may collect 
 from many books all that best suits the immediate 
 object ; and books have none of the power of obtain- 
 ing attention from the thoughtless which, as we have 
 seen, the lecture-room supplies. The mere existence 
 of books is of little use to those who have no wish to 
 read. Of writers assuredly we may say, if not of 
 College lecturers, that they may offer to teach, but 
 cannot compel others to learn. 
 
 Again, the meeting in the lecture-room is of use, 
 not only in enabling the tutor to explain the subject, 
 and direct the learner's studies, but also in making 
 him acquainted with the student's powers and habits 
 of thought. He sees what his pupil does, in the way 
 of study, and how. The frequent and oral inter- 
 course which a College lecture-room allows, enables 
 the teacher to obtain an acquaintance with the pro- 
 gress of his hearers, quite different from that which 
 the professor in his chair can acquire. And this in- 
 creased acquaintance with the students 1 ability and
 
 ENGLISH UNIVERSITY EDUCATION. 101 
 
 character may enable the teacher not only to make 
 his instructions more efficacious, but to give them 
 that turn which best suits their object as a part of 
 a liberal education. 
 
 It is not easy to point out in detail how this is to 
 be done ; but some points are tolerably obvious. 
 Mathematics, for example, which is to operate as a 
 discipline of the reasoning faculties, must be treated 
 in a manner suitable to this purpose. No false rea- 
 soning must be introduced or tolerated, however true 
 or important the result ; and it must be constantly 
 ascertained, by question or trial, that the student 
 apprehends the force of the proof; that he sees 
 the demonstration as demonstration. 
 
 There is another point, not so obvious, but not 
 unimportant. The tutor may usefully dwell on the 
 grounds of the reasoning ; and here his employment 
 will require some thoughtfulness on his part : for the 
 common mathematical works, very properly, omit any 
 detailed discussion of the grounds of the elementary 
 truths on which their reasonings are built. Such dis- 
 cussions belong rather to metaphysicians, and have 
 been with them a favourite employment. It is among 
 them that we find attempts to resolve the questions 
 respecting the foundation of the axioms of geometry, 
 the proofs of the laws of motion, the generality and 
 interpretation of algebraical language. The mathema- 
 tician goes forwards from these elementary principles, 
 and travels in the light ; the metaphysician tries to 
 trace backwards the origin of these principles in a 
 region of comparative obscurity. But still it becomes 
 him who possesses mathematical knowledge as an ele- 
 ment of mental cultivation, not to take his principles
 
 102 ON THE PRINCIPLES OF 
 
 upon trust from others. And though he need not go 
 deeply into metaphysical disquisitions on the nature 
 and origin of our knowledge, he ought, at least, to 
 turn his thoughts to this subject, so far as to appre- 
 hend the difference of necessary truths and mere 
 observed facts. To do this, he must be led to per- 
 ceive the full force of the axioms of those sciences 
 which he cultivates ; and this the tutor may enable 
 him to do, by turning his attention to the origin of 
 these elementary convictions, at the same time that 
 he traces their consequences. And such speculations 
 may serve to educe, in due distinctness and clearness, 
 the fundamental ideas which the sciences involve, and 
 the possession of which, by the student, constitutes, 
 as we have said, the cultivation of mind which scien- 
 tific studies give. 
 
 It may be thought difficult to give such lectures as 
 shall be suitable to the whole of a large class, since 
 there must be such wide differences of ability, quick- 
 ness, and previous instruction among them. And if 
 the lecture be considered merely as a preparation for 
 an independent examination, this difficulty is formid- 
 able : nor can we carry on a successful struggle against 
 it by any subdivision of our classes. But if the 
 teaching of lectures be considered as valuable for its 
 own sake ; if the examination which succeeds the 
 lectures be one in which the lecturers are the ex- 
 aminers, or shape the course of the examination ; and 
 if the subjects be properly chosen ; I conceive the 
 difficulty may be very nearly conquered. For this 
 purpose, the subjects should be such as require little 
 previous knowledge, and demand only patient thought 
 and clear notions. Such a course of mathematics
 
 ENGLISH UNIVERSITY EDUCATION. 103 
 
 might, for instance, be framed, by taking Geometry as 
 the business of the first year ; elementary Mechanics, 
 treated with the plainness and rigour of geometry, for 
 the subject of the second year; and Hydrostatics, 
 Optics, and Astronomy, treated in the same manner, 
 for the employment of the succeeding time. If these 
 subjects are considered to be the basis of the College 
 lectures, even slow learners may be carried on in the 
 College course : and if the tutor provide himself with 
 a good selection of problems and methods, not con- 
 tained in the common treatises, which he may pro- 
 pose to the more advanced students, as matter for 
 the employment of their own ingenuity and skill, he 
 may, I think, furnish them with abundance of pro- 
 fitable occupation ; while at the same time even the 
 quickest and most accomplished students will gain a 
 benefit by being brought back occasionally to the 
 proof of their elementary propositions. And in this 
 system, the community of occupation of the whole 
 class will give the students opportunity to form judg- 
 ments of one another, which produce a beneficial 
 influence. The slowest and most thoughtless will 
 learn duly to respect, and in some measure to un- 
 derstand, the character of the better intellects. 
 
 These latter remarks must be understood rather 
 as suggestions of what the spirit of College lectures 
 ought to be, than as any distinct proposition. For 
 such a system might easily be carried so far as to 
 sacrifice the higher students to the lower; a pro- 
 ceeding by all means to be avoided; since it would 
 impair the dignity of our education, and cloud the 
 prospects of the progressive sciences in England.
 
 104 ON THE PRINCIPLES OP 
 
 SECT. 5. OF ATTENDANCE AT COLLEGE CHAPEL. 
 
 THE subject of the enforcement by rule of the 
 student's attendance at the religious services of his 
 College, is one in which the discussion of opposite 
 opinions can hardly be carried on without offence 
 and pain. The sacred character and deep interest 
 of all which belongs to our religious concerns, lead 
 men almost inevitably to look at such questions with 
 excited minds : while the very wide and profound na- 
 ture of all the principles on which religious questions 
 must be discussed, renders it difficult to argue closely 
 on this subject. What I it is said, on the part of 
 those who object to the enforcement of rules in such 
 cases, do you undertake to make men pious by com- 
 pulsion ? Do you not know that the consciousness of 
 a perfect freedom is requisite in order that a man 
 may turn his thoughts in any fitting manner to his 
 Heavenly Master? Would you degrade the ordi- 
 nances, which ought to be so sacred, into mere lifeless 
 formalities? Would you constrain men, with averse 
 minds, and unregulated thoughts, to go through the 
 external ceremonies of prayer? Would you make 
 that which pretends to be the worship of God, be 
 in reality a muster for the sake of order? Would 
 you force men to come together in the name of God, 
 when you have no higher object than to interrupt 
 their idle or foolish employments ? Constraint, which 
 in matters of study is merely absurd, is here profane. 
 
 It will readily be supposed that the ancient College 
 system, in this, as in other points, has taken the side 
 of discipline, in opposition to that of perfect freedom.
 
 ENGLISH UNIVERSITY EDUCATION. 105 
 
 A daily public worship in the College Chapel is 
 established, or supposed, in the English Colleges; and 
 the attendance of the students at this service is in 
 most, I believe in all, enforced by means similar to 
 those already mentioned. At the period to which 
 these institutions owe their origin, it does not appear 
 to have been conceived possible, that thoughtful and 
 pious persons should wish to be liberated from this 
 obligation of daily public prayer. The thoughtless- 
 ness of youth might require to be admonished of its 
 duty; the infirmity of age might require indulgence; 
 but the general rule, that the Christian members of 
 a great institution, collected for the most serious 
 purposes, and living together after the manner of a 
 large family, should have a practice of daily prayer 
 in common, appears to have presented itself to the 
 minds of the legislators of that time as a self-evident 
 maxim. Nor can we wonder at this ; when we con- 
 sider that, after the Reformation, pious men of our 
 own Church, most averse in their minds to mere 
 hollow show and lifeless ceremony, were most earnest 
 in recommending a rigorous and formal regularity 
 of time, place, and circumstance, in the conduct of 
 men^s devotions. And when we consider further, 
 that in most large families, of which the heads have 
 strong religious feelings, daily regular family worship 
 is in our own time kept up, and is considered as a 
 practice highly important, both as it testifies their 
 sense of their relation to God, as it tends to sober 
 and direct each person's own thoughts, and still more, 
 as it proceeds upon the hope of obtaining for the 
 whole family, the blessings promised to common 
 supplications ; when we consider this, we cannot be 
 5 5
 
 106 ON THE PRINCIPLES OP 
 
 surprised that College worship, which may be looked 
 upon as only another kind of family worship, has 
 always been insisted on. 
 
 But, it may be said, by all means preserve the 
 daily celebration of divine service in the College, but 
 let it be attended only by those who repair to it 
 voluntarily. You will thus have a genuine worship ; 
 and those who go will really derive from it religious 
 benefits, 
 
 To this I should reply, by asking whether those 
 who practise daily common prayer in their own 
 families, leave it to their children and their servants 
 to attend the service or not, as if it were a matter of 
 indifference ? whether, if any member of the family 
 were to absent himself constantly, I do not say out 
 of religious scruples, but out of mere want of serious- 
 ness and impatience of constraint, whether, in such 
 a case, remonstrance and urgency would not be the 
 proceeding and the duty of a pious head of a family ? 
 
 But it may be said, remonstrance and advice are 
 very different from College punishment for absence 
 from the College chapel. And to this I reply, by 
 referring to what I have already said, respecting 
 College punishments. They ought always to be so 
 administered and understood, as to have the character 
 of remonstrance and warning. College punishments, 
 however slight, however formal, are an expression 
 of the disapproval of the College. Every person 
 in authority, and every governing body, have their 
 respective modes of expressing their approbation and 
 disapprobation. The parent, or the master of a fa- 
 mily, does it in one way ; the College does it in 
 another ; but the meaning and object are the same
 
 ENGLISH UNIVERSITY EDUCATION. 107 
 
 in all such cases. If, then, we allow the propriety 
 of interference in the case of the family, on what 
 grounds do we deny it in the case of the College? 
 
 Or, take the matter another way. Suppose that 
 we leave attendance at the College chapel voluntary ; 
 and suppose, as would probably be the result, that a 
 great number of the students appear there very rarely, 
 or absent themselves altogether. Suppose, too, we 
 know (as may easily be known,) that this practice is 
 not accompanied by any voluntary religious exercises 
 of any other kind, but arises from mere want of 
 serious thought, and aversion to regular habits. Are 
 we to allow this state of things to continue, without 
 interposing in any way? It appears to me that we 
 cannot do so. The whole system of our Colleges is 
 framed upon the principle of not allowing those habits 
 to continue unchecked and unopposed, of which we 
 disapprove. We could not see our pupils going on, 
 week after week, and year after year, with no recog- 
 nition of their Christian duties and hopes, without 
 some expression of our sorrow and dissatisfaction. 
 Let it be supposed, then, that we must remonstrate. 
 But what will verbal remonstrance effect, or who can 
 employ it adequately, in a body, it may be, of several 
 hundred young men ? Remonstrance, then, must be 
 represented, or at least supported, by College punish- 
 ment. 
 
 But it may be said, that if we thus obtain the 
 form, we lose the substance ; that the persons thus 
 brought together by compulsion, have no devotion in 
 their thoughts, and are not bettered by the practice. 
 I acknowledge, with regret, that a College chapel is 
 not, in the sincerity and earnestness of its devotions.
 
 108 ON THE PRINCIPLES OF 
 
 all that the friend of religion would wish it to be; 
 but is the Parish Church ? In both places there are 
 the cold and careless ; in both, the serious and pious. 
 I trust that many a heartfelt prayer arises to heaven 
 in the daily services of our Colleges ; and that many, 
 even of the thoughtless and callous, have their 
 thoughts calmed and solemnized by its stillness and 
 order. 
 
 Institutions can be bound to do only what is pos- 
 sible. They keep up the laudable practice; they give 
 the daily occasion ; they prevent manifest neglect and 
 transgression. They can hardly do more ; they can- 
 not control men's wandering minds, or drive the spirit 
 of prayer to their hearts, or breathe warmth and life 
 into their supplications. It is most inconsistent to 
 blame institutions for confining themselves to the en- 
 forcement of forms. What else can be enforced but 
 forms? And is this a reason for enforcing nothing \ 
 
 But when we say that institutions can enforce 
 nothing but forms, do we, therefore, allow that they 
 can do no real good? Far from it. Though they 
 can compel forms only, this very existence of forms, 
 according to the common laws of human habit, pre- 
 serves and supports the substance. As we have 
 already said, in speaking of attendance at College 
 lectures, the occasion, the knowledge of sympathy and 
 mutual approval, the recognised duty, the satisfac- 
 tions which arise from acting under its influence ; 
 these and similar inducements make the institution 
 of habitual public worship far from inoperative, in 
 exciting and maintaining real piety of heart. And 
 there certainly is not, in the demands of College 
 services, anything which need destroy this operation.
 
 ENGLISH UNIVERSITY EDUCATION. 109 
 
 It has been asked, and so far as I am aware, no 
 answer has ever been given to the question, how the 
 consciousness of perfect freedom, which is supposed 
 to be inconsistent with the religious customs of our 
 Colleges, is reconciled with the various restraints and 
 requisitions and formalities which occur in all our 
 national religious observances ; with a rigorously 
 defined formula of prayer, scarcely allowing of any 
 deviations ; with the obligation of precise religious 
 ceremonies on all the great occasions of life ; as 
 birth; marriage; death; admission into the Christian 
 Church ; into its ministry ; into many of the situa- 
 tions of our social scheme, especially in our Colleges 
 and Universities. When any one has published an 
 attempt to draw the line which shall include these 
 appointments as laudable, and exclude the rules of 
 College chapel as inadmissible, we shall better know 
 how to estimate the opinions which are delivered on 
 this subject. 
 
 When it is said that the rules of attendance at 
 chapel are enforced for the sake of objects extraneous 
 to religion, as regularity and sobriety of conduct ; I 
 conceive there is no force in the argument, except it 
 could be shown that the rules are enforced on this 
 account alone, which is not the ground on which I 
 have been maintaining their value. For I presume 
 the most sincerely and devoutly pious man, who 
 thinks and feels daily worship to be of inestimable 
 value as a really religious exercise, will still think it 
 an additional advantage in such a course, that regular 
 habits of daily worship are inconsistent with extreme 
 listlessness, or frequent revelry, or wild extravagance 
 of demeanour. In these respects it is surely no
 
 110 ON THE PRINCIPLES OP 
 
 reproach to godliness that she has the promise of 
 the world which now is, as well as of that which 
 is to come. 
 
 Indeed, I know not why we should hesitate to 
 urge as one reason for insisting upon the College 
 observances of which I have spoken, the circum- 
 stance so constantly remarkable, that the students 
 of the best regulated minds, and the most admira- 
 ble for temper, character, and morals, are those 
 who give to College rules, in all respects, the most 
 cheerful and scrupulous obedience. A person who 
 is irregular and refractory in respect of the rules 
 respecting the College lectures, and prayers, and the 
 like, is the most likely not to be blameless in matters 
 about which there can be no difference of opinion. 
 In almost every instance, such a person carries away 
 from College a part only of the advantages which its 
 institutions are fitted to communicate to the willing 
 pupil. Nor is it unimportant to remark, that the 
 constant meetings with the pupil, to which this regu- 
 larity gives rise, enable the officers of the College to 
 form an opinion of his character and manners, which, 
 though resting on apparently trifling indications, can 
 hardly be erroneous when these are daily repeated. 
 
 It may, perhaps, be said, that after all, the system 
 of College daily worship has not succeeded in fact ; 
 and that many of those who attend it have little 
 reverence or devotion in their thoughts. I think that 
 this aspect of the facts may, at present, very easily be 
 overstated ; but I should be sorry to enter upon a 
 ground of defence, which might appear as if we were 
 satisfied with ourselves in such points ; in which 
 Christian men ought always, forgetting the things
 
 ENGLISH UNIVERSITY EDUCATION. Ill 
 
 which are behind, and striving after the things which 
 are before, to press forward toward the goal of their 
 high calling. I will only say, that, of persons having 
 authority in our Colleges, if it would be unbecoming 
 in those who have laboured to make the ordinances 
 efficacious, to speak of what they have done, it would 
 be still more unreasonable in those who, though 
 involved in like obligations, have not taken any part 
 in such endeavours, to object upon the ground of what 
 remains undone. 
 
 There appear to be one or two maxims with re- 
 gard to College ordinances of this kind, which may 
 serve still further to illustrate their operation. All 
 exterrtal decency and decorum should be as much as 
 possible insisted on. In all cases, and especially in an 
 assemblage of young persons, among whom will occur 
 ready sympathies, light thoughts and unstable cha- 
 racters, any absence of gravity of manner in those 
 who are present, is contagious, extending from one to 
 another, and from the countenance to the mind. The 
 habit by which we adopt the attitude of inward sup- 
 plication, or follow the well-known service along the 
 printed page with our eyes, is no unmeaning formality : 
 it excludes the wandering gaze and mutual glance 
 which unsettle the serious thought. In a College of 
 which I have already spoken, the appearance, and it 
 may be hoped the efficacy, of the worship was much 
 improved, by prohibiting a common practice of enter- 
 ing the chapel when the service was some way ad- 
 vanced. Again; the regulations should as much as 
 possible be such as admit of a devotional meaning. 
 To require attendance a certain number of days in 
 the week, is, on this account, not a desirable rule :
 
 112 ON THE PRINCIPLES OP 
 
 but to enjoin a participation in morning prayer every 
 day, is a fit ordinance among Christians ; and to 
 admit of attendance at the evening worship, as a 
 mark of willing obedience, when that of the morning 
 has been neglected, is conformable to the usual course 
 of College administration. 
 
 It may be said, that, after all, this is a cold and 
 lifeless aspect of religion ; that we have hardly gone 
 further than to show, that our forms cannot con- 
 sistently be abolished ; that we have shown little in 
 our ordinances likely to excite vital religion ; that 
 we may defend our system, but that we cannot induce 
 really pious men to look upon it with admiration. 
 And this we are ready to grant, not entirely, but 
 partly, in the sense in which really pious men would 
 urge it. We know that forms of themselves can do 
 nothing ; we know, that,, when we defend forms, we 
 are urging points which zealous men may assent to, 
 but do not deeply care for. We know that we ought 
 to go beyond these beggarly elements. But what is 
 that to the purpose ? To go beyond, we must go 
 through them. If our forms cannot be abolished 
 without intolerable evil, and do all that forms can 
 do, what more is needed for their justification ? 
 
 But we have spoken of going beyond these forms ; 
 how, it may be asked, is that to be done ? This is a 
 question to which no definite general answer can be 
 given. Who shall say, what other ways, besides 
 formal rules, best promote and maintain a true sense 
 of religion in a community ? Here, at least, a free- 
 dom of thought and action are requisite. What 
 observances, besides those of College worship, what 
 solitary or social readings, lectures, or exercises, on
 
 ENGLISH UNIVERSITY EDUCATION. 113 
 
 religious subjects, shall be selected, as fit instruments 
 for such a work, must be left to teachers and stu- 
 dents, according to their stations, to decide. Those 
 who endeavour to frame a scheme of self-regulation 
 or mutual edification, with an earnest and serious 
 mind, will hardly hit upon a plan that will not be 
 profitable. 
 
 Yet one maxim may here be offered. A regular 
 recurrence of observances is, in such things also, a 
 wholesome and beneficial practice. In those persons, 
 who have had the inestimable blessing of an early 
 religious education, the feelings of piety are often 
 best sustained by the continuance of early religious 
 habits. The prayers that they have first been taught, 
 the religious writers whom they have admired from 
 childhood, are the oil which may best feed their lamp. 
 The established devotional habits which can be re- 
 tained, should not be lost. But here the student's 
 case brings some difficulties, arising from his change 
 of circumstances in coming to the University. For 
 example, his Sunday, with its stated habits and 
 religious employments at his home, may have been 
 one of the main events of his Christian life. While 
 he is at College, these habits can no longer be 
 nourished by the arrangements and intercourse of his 
 family. Nor can his College, from the very nature 
 of the case, supply this want, except by forms, of 
 which we acknowledge the insufficiency. Here, there- 
 fore, the student must provide for himself; and, as 
 he values his Christian welfare, must give such a 
 character of sobriety, and thoughtfulness, and re- 
 ligion, to the employment of his Sunday, as becomes 
 a Christian man. And though the habits of home
 
 114 ON THE PRINCIPLES OP 
 
 may be changed, the gentle and kindly pressure of 
 habit may be retained. Supposing, for example, 
 that the student attend the morning and evening 
 worship of his College chapel, and one of the sermons 
 which are every Sunday preached before the English 
 Universities, there still remains a large portion of the 
 day, which may be moulded into a shape by suitable 
 habits. If some of the vacant time which the 
 cessation of his ordinary studies leaves, be usually 
 occupied in intercourse with friends, these may best 
 be one or two only, and, as much as possible, always 
 the same. And to attend habitually some one of the 
 parish Churches within his reach, may keep up the 
 recollection of the Church of his home, and preserve 
 the social sympathy of English worship. The day, 
 too, may have its own books and studies. But I do 
 not here dwell upon such points, wishing only to 
 point out in a general manner how the student may 
 best feed and foster, during his University career, 
 that religious faith, and hope, and love, which we may 
 trust he will bring from his mother's knee and his 
 father's hearth ; and which, if he hold it fast, will be 
 his best guide and support in those wider and busier 
 scenes, of which the University is the portal. It is 
 such a character of mind, too, which will give him a 
 cultivation of a higher kind than that of which I have 
 spoken ; which will not merely transmit a refined and 
 elevated tone of thought to another generation, but 
 will prepare the soul, by a progress in holiness and 
 love, for its own future place in a higher region of 
 purity and blessedness.
 
 ENGLISH UNIVERSITY EDUCATION. 115 
 
 SECT. 6. OF FELLOWS OF COLLEGES. 
 
 IN the Colleges of the English Universities, the 
 administration of the education, as well as of the 
 property, and other interests and duties of those 
 institutions, is, by their statutes and constitution, 
 committed to the Fellows of the College, along with 
 the Master. These persons, thus invested with im- 
 portant possessions, privileges, and offices, are those 
 upon whom the working of the College system de- 
 pends ; and it is important, in order to judge of the 
 tendency of the system, to consider, both what their 
 functions are, according to the plan of these insti- 
 tutions, and what their actual character is. I think 
 I shall assist the reader in forming a judgment upon 
 these subjects, by quoting the statements of an in- 
 telligent and accomplished American traveller, de- 
 servedly much admired and esteemed in his own 
 country, who visited England a short time ago, and 
 whose account is in the main correct and impartial. 
 
 " As there are no fellowships in our American 
 colleges, you may perhaps be gratified by some ac- 
 count of them as they exist here. In this University 
 (Cambridge) there are about one hundred and fifty 
 fellowships attached to the different colleges, most of 
 which are given on examination, and to merit only. 
 They vary in value, from 150/. to 300/. a-year, besides 
 free commons and apartments. They are held dur- 
 ing 4ife, or until marriage. Many of the occupants 
 become tutors of colleges, private tutors, &c. ; but 
 nothing is absolutely required of them. Much has 
 been said of the indolence of Fellows ; of their dis-
 
 116 ON THE PRINCIPLES OP 
 
 position to quarrel and petty intrigue ; and of their 
 fondness for guzzling ale, tippling port, and playing 
 whist. Such things were. Nay, since such are the 
 natural consequences of a want of ambition to be 
 useful or distinguished, a want of occupation, and a 
 want of that most practical stimulant, dire necessity, 
 such things are. The cases, however, are unfrequent. 
 The Fellows to whom I had the honour to be intro- 
 duced were men of a different stamp. They were 
 gentlemen, in the highest sense of that high term, 
 and bore about them no traces of their somewhat 
 monastic system. Their conversation smelt a little of 
 the shop ; was sometimes a little too mathematical, 
 at least for me ; but was throughout the most 
 thoroughly intellectual that I ever enjoyed. Their 
 reunions, after a plain but well-cooked dinner on the 
 dais of their College-hall, either in the common sit- 
 ting-room, or in the apartments of some individual 
 member, left on my mind a delightful impression. It 
 was such as literary society should be, composed only 
 of men of real learning ; of friends confiding in the 
 mutual esteem entertained by all, undisturbed by 
 ambitious quacks or impudent pretenders*." 
 
 From the class to which belong the persons thus 
 described, are selected, not only the tutors, but all 
 the College officers engaged, both in education and 
 administration. The Master and Fellows are, in fact, 
 the College. And these institutions expect from their 
 Fellows, not only care and fidelity in the discharge 
 of office, but active zeal and lively affection. Accord- 
 ing to the statutes which I have already quoted, a 
 
 * Remains of the Rev. Edward Griffin (of New York), Vol. n. 
 p. 259.
 
 ENGLISH UNIVERSITY EDUCATION. 117 
 
 Fellow of the College, on his admission, swears " that 
 he will be faithful and friendly to the College ; that 
 to it, and to all the fellows and scholars, and to the 
 master, he will bear good-will and give help, not only 
 while he lives therein, but afterwards, to the best of 
 his power, when occasion may arise ; that he will 
 never cause loss or trouble to the College, and that 
 the designs, combinations, conspiracies, plots, deeds, 
 and words of others, which may bring damage and 
 evil fame to the College, he will repel as far as he 
 can, and will denounce them to the officers of the 
 College who are to take cognizance and give judg- 
 ment concerning such things ; that even if he is 
 formally expelled, he will not bring any action against 
 the College or its members; that he will obey the 
 master, vice-master, seniors, and officers, in all lawful 
 and honest matters, and pay them due reverence and 
 honour ; and finally, that he will take upon himself 
 all offices imposed by the master and seniors, and will 
 administer them with the utmost fidelity and dili- 
 gence." 
 
 Nor have these provisions been without their 
 effect. The Fellows of Colleges not only undertake 
 the offices which the College imposes upon them, (of 
 course, with no unnecessary disregard of their private 
 convenience,) but they do also, almost universally, 
 bear a domestic affection to the body, or, as it is 
 often familiarly called, "the house." This is not 
 difficult to understand; for there is among the Fellows 
 of the same College something approaching to a family 
 bond. They have common possessions, a common 
 home, a common table, a common interest to de- 
 liberate upon ; they have among them numerous close
 
 118 ON THE PRINCIPLES OF 
 
 ties of intimacy and regard, formed in the period of 
 life and under the circumstances which render friend- 
 ships most durable and dear. And thus it is very 
 natural that there should prevail among them a 
 general sympathy and unity of feeling, with regard 
 to the fundamental principles of the institution of 
 which they are all children. 
 
 It will be understood, therefore, that the ad- 
 ministration of the College system of discipline, such 
 as we have described it, may generally be committed 
 to such hands with every prospect of its being faith- 
 fully carried into effect. The Fellows cannot but 
 feel that the enforcement of such a system, by their 
 joint and separate influence, is one of the main ob- 
 jects of their institution. In this view, there is an 
 immense advantage in the existence of a body of 
 men, who, though only some of them are teach- 
 ers or officers, all sympathise with the teachers and 
 officers, discountenance irregularity and indocility, 
 and assist in carrying rules into effect. The use of 
 such a body in supplying examiners in addition to the 
 teachers, is also great ; for though, as I have already 
 urged, there appear to be strong reasons against 
 making the College examinations independent of the 
 tutors, there are also strong reasons against com- 
 mitting them to the tutors entirely ; and against de- 
 stroying that acquaintance and sympathy with the 
 College course of instruction, which the Fellows in 
 general acquire by taking a share in the examina- 
 tions. And for the discharge of those most important 
 offices by which, in addition to the superintendence of 
 the tutors, College discipline is preserved, (the offices 
 of Deans, &c.,) Colleges are dependent entirely on
 
 ENGLISH UNIVERSITY EDUCATION. 119 
 
 the willing and persevering exertions of their resident 
 Fellows. If this support should fail them, the system 
 must fall into ruin. 
 
 Besides these advantages, the institution of Fellow- 
 ships has other uses which must not be overlooked. 
 They are prizes of considerable value ; and these and 
 other minor prizes of the same kind (scholarships, 
 &c.,) being given to merit according to the test of 
 examinations, supply a constant encouragement to 
 those academic studies, which although, as we have 
 seen, they are essential to the preservation and pro- 
 gress of intellectual civilisation, would not be ade- 
 quately supported by the demands of practical life 
 and popular opinion. The institution of Fellowships, 
 as they now exist, extends this influence in a peculiar 
 manner to the learned professions ; for since most of 
 the Fellowships may be held for a certain period 
 without residence, they are objects of great competi- 
 tion to those who intend to be lawyers, or physicians, 
 or clergymen ; and they always give something of a 
 literary and speculative tone to the mind ; both by 
 the preparation which they require, and by the op- 
 portunity which they afford of lingering a little while 
 in the region of letters and science, before the busi- 
 ness of professional life absorbs the powers of thought 
 and action. 
 
 Such appears to be the proper intention, and such 
 has hitherto been in a great measure the actual 
 operation, of the Fellowships of our English Uni- 
 versities. But since I have, throughout this essay, 
 examined the actual principles of our Universities, 
 by weighing them against their alternatives, let 
 us consider what would be the effect of the decay
 
 120 ON THE PRINCIPLES OP 
 
 or absence of the College spirit which I have endea- 
 voured to describe. If the Fellows should lose their 
 interest and sympathy in the general business of the 
 instruction and discipline of the College; if they 
 should discontinue the traditional practices of being 
 often present in the Chapel and the Examination- 
 hall; if they should endeavour to evade offices which 
 bring any labour and trouble, or should discharge 
 them with a cold and scanty service, barely fulfilling 
 the forms, and taking no pains to make them effec- 
 tive ; if this should ever happen, we cannot doubt 
 but that the effect of the College system must be 
 grievously crippled and weakened. 
 
 But we may go further in supposition. It is not 
 very easily reconcilable with our view of the inten- 
 tion of Colleges, that those persons who compose 
 them should openly declare themselves against the 
 College system; for it would seem that he who is 
 member of a legitimately-constituted body, if he dis- 
 approve of its practices, should endeavour to reform 
 them by efforts within the body, and not by calling 
 in the stranger. But suppose that any of the govern- 
 ing members of a College (and all the Fellows are 
 governing members, since all in turn may take offices 
 of administration,) should disapprove of the College 
 system of instruction or discipline, as we have de- 
 scribed it above, and should express his opinions 
 openly. It cannot be concealed that this would be a 
 great evil ; and if such opinions came to be held, or 
 were supposed to be held, by several, beyond doubt 
 such a difference of sentiment among the governors 
 must needs throw a chill and gloom over the general 
 administration of the College. For those of the Fel-
 
 ENGLISH UNIVERSITY EDUCATION. 121 
 
 lows who, thinking their own College system the 
 most beneficial and excellent of all systems of educa- 
 tion, would labour with unstinted efforts and with 
 cheerful hope to make it efficacious, if they conceived 
 themselves to be supported by the general good-will, 
 sympathy, and confidence of their brethren and daily 
 companions, would, if they were compelled to see 
 that this sympathy did not exist on very important 
 points, be driven to take refuge in a joyless sense of 
 duty ; and though they might not labour the less, 
 would lose the hopeful and buoyant spirit which 
 makes labour easy and efficient. If we might use a 
 comparison somewhat too large for the subject, we 
 might say that they would have to nourish the sted- 
 fast but melancholy resolve of a patriot, struggling 
 for what he knows to be a sinking state. But our 
 Colleges, we trust, will never come to this condition. 
 No additional supposition which we can make 
 would bring us to any increased difficulty. For if 
 we conceive some one of the members of our Col- 
 leges, in recommending changes, to show none of 
 that affection and reverence to the institution, which 
 is the bond of union among its members, or if we 
 suppose him to urge accusations against it in that 
 scoffing and scurril tone of which the statutes above 
 quoted express their detestation, the consequence 
 may easily be foreseen. He would excite among all 
 around him, even those who themselves wished for 
 change, no feeling but dislike and repugnance; and 
 the finger of public scorn, and the shudder of silent 
 disgust, would make him feel at the bottom of his soul, 
 coarse though it might be, how much he had mistaken 
 the character of the society which he blemished. 
 
 6
 
 122 ON THE PRINCIPLES OP 
 
 To return, therefore, to the case before supposed, 
 of a difference of opinion among the governing body 
 of a College, moderately and considerately expressed, 
 but carried so far as to damage that co-operation and 
 community of purpose in which the strength of the 
 body consists : it may be asked whether there be 
 any means of preventing or of remedying such a 
 grievous distemper. But upon this inquiry I shall 
 not here enter. It is easy to see that the remedy, if 
 it exist, must be difficult to discern and to apply, 
 since the disorder is one affecting the spirit, and not 
 the forms of the institution. Perhaps something 
 might be suggested : but as my object in the present 
 work is rather to consider the operation of the prin- 
 ciples which have hitherto prevailed in Colleges than 
 to propose changes, I shall not at present pursue the 
 subject. 
 
 SECT. 7 OF THE FREE SYSTEM. 
 
 BY the Free System of Universities, I mean that 
 system in which there is no discipline such as I have 
 described, the students being left to act without con- 
 trol, both as to attendance upon the lectures and 
 other appointments of the College, and as to man- 
 ners and conduct in general. It is to be observed 
 that this free system is perfectly consistent with 
 examinations and public trials at certain periods ; for 
 these processes are so far from implying discipline, 
 that they do not even imply the student's residence 
 in the intervals between the public occasions. 
 
 The free system is understood to be the one 
 which prevails in a large portion of the universities
 
 ENGLISH UNIVERSITY EDUCATION. 123 
 
 out of England. The Germans consider " Academic 
 Freedom" as one of the main principles of their 
 universities; but this has never been the system of 
 English Universities. It is, however, the system 
 towards which we tend by every relaxation in the 
 enforcement of our discipline, and by every resistance 
 to discipline on the part of our students. It may 
 therefore be useful to point out some of its features. 
 
 The free University system is founded on the 
 doctrine that there is no University control over the 
 private and social conduct of the student. He is left, 
 like any other citizen, to be guided by his own sense 
 of propriety, and controlled by the law of the land. 
 But it will readily be supposed that a large body of 
 young men, just emerging from boyhood, most of 
 them belonging to families of some property, and 
 given over for the first time in their lives to the 
 unrestrained exercise of their own propensities and 
 judgments, will conduct themselves in a manner very 
 different from the same number of citizens of any 
 other class. Their newly-felt freedom will need to be 
 exhibited in a conspicuous manner ; and an opinion 
 of their own superiority, which will naturally arise 
 from sympathy and companionship, will make them 
 despise all who are not of their body. And thus, if 
 by ancient usage the students wear a peculiar dress, 
 their position will generate the turbulence and the 
 pride of the Gown. If they are not so distinguished 
 from their fellow-townsmen, they will soon find means 
 themselves of marking the difference between the 
 Bursch and the Philister. 
 
 That a body of young men, who conceive that no 
 right to control them exists, will receive with the 
 
 6 2
 
 124 ON THE PRINCIPLES OF 
 
 utmost indignation all attempts to subject them to 
 any kind of rule, may readily be imagined. They 
 will consider the cause of resistance as the sacred 
 cause of liberty, and will, with the greatest self-com- 
 placency, speak of, and behave to those who would 
 impose any restraints upon them, as narrow-minded 
 oppressors. We have examples of such a spirit, in 
 the conduct of the medical students of our own 
 metropolis. Among these persons, we are told*, any 
 attempt made to ascertain the regularity of the stu- 
 dent's attendance at the lectures, is held up to scorn 
 as the " lecture-room spy system."" But this spirit 
 does not confine itself to resistance to rule ; it soon 
 assumes a right to interfere in the appointment of 
 professors and similar matters ; or at least to resent, 
 by active proceedings, any such measures, when they 
 do not suit the taste of the students. Thus, there 
 has recently been a tumult on a subject of this kind, 
 in one of the schools of Paris ; and it is well known 
 that in Germany, the students have often manifested 
 the offence they have taken on such occasions, by 
 organized migrations, or by putting some particular 
 University under the interdict of the clubs of the 
 young men. 
 
 These may appear to be extreme cases, but they 
 serve to show, what that free University system is, 
 which is opposed to the discipline system ; and it is 
 useful to recollect, that we cannot recede from the 
 one, without approaching the other. It is easy to 
 conceive many other proceedings, which, without 
 assuming the principles of the free system in their 
 full vigour, are still inconsistent with the condition 
 
 * British Magazine, April, 1837.
 
 ENGLISH UNIVERSITY EDUCATION. 125 
 
 of persons under discipline. Such are, for example, 
 all acts of a political kind, as meetings or organiza- 
 tions for political purposes, public petitions, or public 
 discussions on the agitated questions of the day, and 
 the like. Under the discipline system, the student 
 cannot act at the University, except in the capacity 
 of a pupil. In like manner, public and tumultuous 
 exhibitions of opinion by the cries of a crowd, (al- 
 though, in the excitement to which crowds are sub- 
 ject, even members of the governing body may be 
 drawn on to join in the shout,) cannot, in any calm 
 view of propriety, be reconciled with the pupil's posi- 
 tion. Indeed, the exaggerated and insatiable manner 
 in which public applause is given by bodies of stu- 
 dents, is of itself an evidence that in such cases, they 
 feel that they are not merely expressing an opinion, 
 but are also gaining a victory over some expected 
 restraint by the mode of expression. It often hap- 
 pens, that meetings under such circumstances inflame 
 themselves by their own tumult, till all order, deco- 
 rum, and common sense are lost sight of; and the 
 cries that are uttered, and the manner in which they 
 are received, show how decided the tendency of such 
 proceedings is, to make the actors in them lose all 
 regard for good manners, as well as for subordination. 
 It can hardly be doubted, I think, that the ten- 
 dency of the free system, if introduced into the English 
 Universities, would be to corrupt the character, and 
 deprave the manners of the students. If they felt 
 themselves out of the reach of control and authorita- 
 tive monition on points of general conduct and habits, 
 they would feel that the regulation of the fashions, 
 manners, and principles of the academic body was
 
 126 ON THE PRINCIPLES OP 
 
 committed to their hands ; and they would not fail 
 to devise a system of their own. They would make 
 their own rules of morals, their own codes of honour 
 and honesty ; and though I do not doubt that these 
 would bear traces of the general morality, probity, 
 and good sense of the English character, I fear they 
 would be below the discipline standard. The teachers 
 too, among a crowd of imperious, self-willed and self- 
 satisfied young men, who would be, by the constitution 
 of the academic body, their equals in all but knowledge, 
 would be held, in the opinion of the young men them- 
 selves, to be an inferior class to the students. I do 
 not suppose, that the generous spirit of young English- 
 men of the better orders would ever take the tone 
 which is said to prevail among the medical students 
 of London*; of representing the teachers as inter- 
 ested in increasing the burdens of the students, and 
 entertaining no other views towards them, than to rob 
 them of as much money as they can ; but I do not 
 think that we could have that cordial respect and 
 sincere deference of pupils to tutors, which at present 
 generally prevails in our Colleges, except under a 
 system in which the privilege of remonstrance and 
 admonition which age, experience, and reflection give, 
 was sanctioned and dignified, by being made at the 
 same time a right and a duty. 
 
 I conceive, therefore, that any one who seriously 
 wishes the education of our Universities to continue 
 and extend its beneficial influences, cannot hesitate, 
 on every occasion which may occur, to lend his best 
 exertions to the preservation of that system of Col- 
 lege and University discipline, of which the only 
 
 * British Magazine, as above.
 
 ENGLISH UNIVERSITY EDUCATION. 127 
 
 ultimate alternative is a system of entire misrule, and 
 the unbounded sway of youthful caprice, extrava- 
 gance, and turbulence. 
 
 SECT. 8. OF CHANGES IN THE COLLEGE SYSTEM. 
 
 THE system of education in our Colleges, like all 
 other systems, may require some modification from 
 time to time, for the very purpose of causing it the 
 better to answer its genuine beneficial purposes. For 
 example, an alteration in the aspect of some branches 
 of science and learning, produced by the progress of 
 knowledge, may render changes in the course of read- 
 ing and examination desirable. A few brief general 
 remarks on the principles which apply to such changes, 
 must close our present task. 
 
 In the first place, I observe, that the common 
 topics which are sometimes used in favour of changes 
 in general, are to be listened to with great suspicion, 
 when applied to the provisions of the College sys- 
 tem: such as, that the world is constantly advancing, 
 and that we must accommodate ourselves to its pro- 
 gress; that the present generation is more wise, 
 more enlightened, more free from- prejudice than its 
 predecessors, and that, therefore, we must not bind 
 it in fetters which they constructed. Without here 
 dwelling on the doubts which might be urged against 
 these assumptions, we must consider that the office 
 which Colleges have to discharge, is inconsistent with 
 any hasty or frequent application of the maxims 
 founded upon them. Universities, so far as they are 
 schools of general cultivation, represent the permanent, 
 not the fluctuating elements of human knowledge.
 
 128 ON THE PRINCIPLES Of 
 
 They should be progressive, for otherwise, they cannot 
 be permanent ; but the progress in which they ought 
 to share, is not one which can be estimated from year 
 to year, but rather is reckoned by centuries. They 
 have to transmit the civilisation of past generations 
 to future ones, not to share and show forth all the 
 changing fashions of intellectual caprice and subtlety. 
 They ought not, therefore, rapidly or easily to intro- 
 duce changes into the subjects of their study. They 
 ought to wait till novelties have been well discussed, 
 and firmly established, before they adopt them into 
 their elementary course. I am here, of course, not 
 speaking of professional education; for an accomplish- 
 ed and thoroughly educated man, of any profession, 
 will possess the most advanced knowledge, and be 
 acquainted with the best literature, which belongs to 
 his department. But in that fundamental education, 
 of which I have principally treated, the old ways are 
 not lightly to be abandoned. As I have already said, 
 I should be sorry to see Euclid lose his ancient place, 
 or even his ancient form, in our system. 
 
 In the statutes of several of our Colleges, not 
 only the subjects, but the books, are prescribed, which 
 are to be the subject of study. This may appear an 
 inconvenient and importunate interference with the 
 progress of literature, by which one book constantly 
 supplants another. Probably such a rule would not 
 be introduced by persons legislating at the present 
 day, when we are more familiar with changes in 
 literary currency. Nor would I recommend such 
 laws. But we may observe, that provisions of this 
 kind appear to have been suggested by a regard for 
 that permanence of the course of culture which is
 
 ENGLISH UNIVERSITY EDUCATION. 129 
 
 necessary to the utility of our Colleges. And the 
 subject is not without its difficulty ; for if we pre- 
 scribe no fixed course, and leave the determination 
 of the subjects of study to the teachers of the moment, 
 we incur the danger of capricious changes and sudden 
 revolutions of doctrine, which would throw the sys- 
 tem into confusion and annihilate its effect as culture. 
 And if our laws point out subjects only and not 
 authors, they do not escape the evil of instability, 
 except where the subjects have long assumed a per- 
 manent form, in which case, the authors also have 
 generally become classical. If, for instance, we adopt 
 Political Economy as one of our subjects, who can tell 
 us what kind of science political economy will be fifty 
 years hence ? whether it will be most connected with 
 metaphysics, mathematics, or history ? 
 
 The course adopted by the legislators of our 
 Colleges has, in practice, worked well. The pre- 
 scribed list of authors has not been adhered to ; but 
 in the deviations from it, care has been taken, that 
 the new books introduced should, as much as possible, 
 represent those which they supplanted ; and that the 
 new plan should, in its general bearing and spirit r 
 conform to the original scheme : and in this way we 
 have courses of mathematical and classical reading, in 
 which the books, and many of the modes of treating 
 the subject, are modern, but the subjects themselves 
 are little changed from the foundation of the Colleges. 
 In other cases, new subjects have been introduced. 
 The power of making such alterations has been sup- 
 posed to reside in the governing body of each College, 
 and has been exercised under an habitual sense of 
 the importance of the duty : nor does it appear that 
 
 6 5
 
 130 ON THE PRINCIPLES OP 
 
 any better authority can be set up for this purpose. 
 The authority of the University may prescribe sub- 
 jects of examination for its degrees ; but to confine 
 the education of the Colleges within this limit, would 
 be to deprive it of its best advantages. Nor need any 
 one fear that, under such government as we describe, 
 the reluctance to admit change will be too great. I 
 believe most of those who have had much to do with 
 the administration of education at Cambridge, will 
 agree with me, that we have, of late years, admitted 
 changes in our system rather with too much than 
 with too little facility. 
 
 An objection will, perhaps, be started to changes 
 in College courses, introduced in the manner I have 
 described, on the ground that the governing body 
 have sworn to obey the original statutes, and, there- 
 fore, have no right to alter them. I confess I cannot 
 assent to the soundness of such a decision. It appears 
 to me, and I think will appear to many persons, that 
 the substitution of a better book of elements for a 
 worse on the same subject, might well be included in 
 a conscientious interpretation of conformity to the 
 statutes. But if the removal of such discrepancies 
 will satisfy the minds, either of those persons who 
 have taken such oaths, or of any others who are fit 
 judges, by all means let the attempt at a closer 
 reconciliation between precept and practice be made. 
 Only let this be done with due attention to the con- 
 siderations we have pointed out ; the necessity of 
 great permanence in our College courses, in order 
 that they may form a national education ; and the 
 difficulty of establishing such permanence by a mere 
 enumeration of the subjects to be studied.
 
 ENGLISH UNIVERSITY EDUCATION. 131 
 
 Above all, let no one who wishes to have our 
 Universities really useful and respectable, and who, 
 therefore, would remove such defects as they may 
 labour under, bring to his work of reformation a 
 hostile and excited temper. If any one shall apply 
 himself to this subject, believing that great chasms 
 have been formed in our system, and are to be re- 
 paired ; that our practice is far removed from our 
 statutes ; that our system is much behind the con- 
 dition of the age ; that we have gone on in our 
 beaten track, careless and inobservant of what has 
 been passing in the world about us; that we feel 
 no strong sense of duty and responsibility; if any one 
 shall come, believing this, and having, therefore, in 
 his breast, a magazine of suppressed indignation and 
 contempt, which a single spark may explode ; such 
 a person is, by his own state of mind, not only utterly 
 unfit to be our judge, but probably incompetent to 
 investigate the subject of University education at all. 
 He probably does not know that Universities and 
 Colleges have for their office, not to run a race with 
 the spirit of the age, but to connect ages, as they 
 roll on, by giving permanence to that which is often 
 lost sight of in the turmoil of more bustling scenes. 
 He can hardly be aware in what temper all reform 
 of Universities ought to be undertaken. In order to 
 introduce real improvements, we must bring to the 
 task a spirit, not of hatred, but of reverence for the 
 past ; not of contempt, but of gratitude towards our 
 predecessors. If we are able to go beyond them, it 
 must be by advancing in their track, not by starting 
 in a different direction. We must continue their line 
 of instruction, and study their academic constitutions.
 
 132 ON THE PRINCIPLES OP 
 
 What the vocation of the present age for such 
 legislation may be, I know not. The men of our 
 day will deserve no small admiration if they succeed 
 in framing laws which shall operate as beneficially, 
 and endure as long, as those of our English Colleges 
 have done. And if they are to do this, it must be, 
 not by rejecting and despising, but by adapting and 
 improving the older codes. If, instead of such a 
 Reform of the English Universities, we attempt to 
 remodel those institutions on some foreign or ima- 
 ginary plan, we shall justly incur the condemnation 
 of all wise men, and shall deserve a sorrowful and in- 
 dignant remembrance from our successors. 
 
 " Let not England forget her privilege of teaching 
 the nations how to live" This was proudly said ; per- 
 haps too proudly, for nations do not always profit by 
 adopting even the better institutions of another na- 
 tion. But it was said with a patriot pride ; and 
 those who administer our institutions in the spirit 
 which this sentence expresses, will hardly fail to 
 refine and elevate the character of their countrymen. 
 Nor is the exhortation without a solid ground ; for 
 has England not taught the nations how to live ? By 
 an admirable combination of active and original in- 
 tellect, with unequalled practical sagacity and force 
 of character, England has constantly impelled the 
 progress of thought and of institutions in Europe, 
 while, at the same time, she has held back from the 
 extravagances and atrocities to which the progressive 
 impulse has urged more unbalanced nations. The 
 bold and vigorous metaphysicians of England first put 
 in action the speculative movement of modern Ger- 
 many; but England refuses to follow the wild onward
 
 ENGLISH UNIVERSITY EDUCATION. 133 
 
 whirl of system after system, to which this movement 
 has led. English teachers of political freedom, and 
 the free institutions of England, called up the spirit 
 which has broken the bonds of more than one despot- 
 ism ; but England, (thank God !) was never hurried 
 into the democratic madness of her nearest neighbour. 
 England had a Reformation of religion without the 
 abolition of her Church ; she had a Revolution of 
 dynasty without the destruction of her Loyalty ; she 
 has had a Reform of her Parliament, we trust, with- 
 out any fatal wrench to her Constitution. We have, 
 therefore, no small reason to confide in the practical 
 sense and fixed sobriety of the English character, and 
 to believe that England may still continue to teach 
 the nations how to live ; how to preserve and how 
 to reform ancient and beneficent institutions. And 
 this she may do, not by new modelling her ancient 
 Universities, the sources of so much benefit, and the 
 objects of so much love ; not by an abrogation of the 
 functions, or a revolution of the constitution, of her 
 Colleges ; but, when need shall be, by a calm and 
 serious revisal of their laws ; above all, by a revival 
 of their genuine spirit, and a fit estimate of their 
 exceeding importance and value. 
 
 This I urge, not in a spirit of inflexible adherence 
 to the past, but of care for and hope in the future. 
 I trust it has appeared in the preceding reflections, 
 as I trust it has appeared also by other indications, 
 that I hold opinions respecting the improvement of 
 our modes of education, which are not dependent 
 upon any recent or external suggestions of the need 
 of reform. And it is because I consider that the 
 right administration of the Universities of the land
 
 134 ON THE PRINCIPLES OF, ETC. 
 
 involves the welfare of countless generations of Eng- 
 lishmen yet unborn, and of centuries of English 
 civilisation yet only in the germ, that I warn all 
 whom it may concern, against attempting this task 
 in a hasty or angry spirit ; and remind them that 
 they cannot exercise any wise deliberation concerning 
 the future, without a sober and reverent regard to 
 the past. 
 
 So may our mother flourish while the name 
 Of England holds its proud pre-eminence 
 Among the nations : in her ancient halls 
 And venerable cloisters be our youth 
 Invigorated by salubrious draughts 
 Of free and fervent thought, and let the mind 
 Of our great country, like a mighty sea, 
 Be fed and freshened with perpetual streams 
 Of pure and virtuous wisdom, from those springs 
 Gushing unceasingly.
 
 THOUGHTS 
 
 THE STUDY OF MATHEMATICS, 
 
 AS A PART OF 
 
 A LIBERAL EDUCATION. 
 
 TO WHICH IS ADDED, 
 
 A LETTER TO THE EDITOR OF THE EDINBURGH REVIEW, 
 
 OCCASIONED BY THE REVIEW OF THE FIRST 
 
 EDITION OF THE THOUGHTS.
 
 THOUGHTS 
 
 STUDY OF MATHEMATICS, 
 
 THE various opinions which we hear and read, on the 
 use and effects of the Study of Mathematics, show us 
 that the subject is far from generally understood. There 
 must be in the world a good deal of misapprehension 
 and confusion of thought respecting this matter, or we 
 should hardly meet with such different and opposite 
 judgments as perpetually come in our way. On the one 
 hand we are told that mathematics is a most admirable 
 mental discipline ; that it generates habits of strict rea- 
 soning, of continuous and severe attention, of constant 
 reference to fundamental principles : on the other side it 
 is asserted, that mathematical habits of thought unfit a 
 man for the business of life; make his mind captious, 
 disputatious, over subtle, over rigid ; that a person 
 inured to mathematical reasoning alone, reasons ill on 
 other subjects, seeks in them a kind and degree of proof 
 which does not belong to them, becomes insensible to 
 moral evidence, and loses those finer perceptions of fit- 
 ness and beauty, in which propriety of action and deli- 
 cacy of taste must have their origin. 
 
 Any view of this subject which would show us how 
 far and under what circumstances each of these opinions 
 is true, would probably help us to see how we must 
 regulate our studies so as to make them most beneficial ;
 
 138 THOUGHTS ON THE 
 
 and might therefore be of interest to some persons, even 
 though the argument should be of a nature somewhat 
 metaphysical; as, indeed, having to treat of the effect 
 which certain processes of thought produce upon the 
 mind, it can hardly avoid being. It is in this belief that 
 the few reflections which follow have been written. 
 
 The most obvious point of view under which the study 
 of mathematics, as a mental discipline, offers itself to us, 
 is as an example and exercise of exact reasoning. It will 
 probably be allowed by all, that the power, at least, of 
 tracing securely and readily the necessary consequences 
 of assumed principles is a desirable acquisition. Yet it 
 can hardly be denied, that in a great part of mankind 
 such a power requires to be confirmed and strengthened 
 by education, since by nature it exists in a low degree 
 and confused form only. Men's minds are full of con- 
 victions which they cannot justify by connected reason- 
 ing, however reasonable they may be. Nothing is more 
 common than to hear persons urge very foolish argu- 
 ments in support of very just opinions; and what has 
 been said of women is often no less true of the sex 
 which pretends to have the more logical kind of head: 
 namely, that if they give their judgment only, they are 
 not unlikely to be right, but if they add their reasons for 
 it, those will most probably be wong. There prevails 
 very widely an obscurity or perplexity of thought, which 
 prevents men from seeing clearly the necessary con- 
 nexion of their principles with their conclusions. 
 
 Now, though there is this chance of being practically 
 right and speculatively wrong, it will probably be al- 
 lowed that this is not a state of mind in which those can 
 acquiesce contentedly whose object is the mental culture 
 of man. It may be very possible to act well and to judge 
 justly, while we think confusedly and argue ill; but we 
 cannot on that account see no harm in confused thought
 
 STUDY OF MATHEMATICS. 139 
 
 and bad reasoning. The object of a liberal education is 
 to develop the whole mental system of man, and thus to 
 bring it into consistency with itself; to make his specu- 
 lative inferences coincide with his practical convictions; 
 to enable him to render a reason for the belief that is 
 in him, and not to leave him in the condition of Solo- 
 mon's sluggard, who is wiser in his own conceit than 
 seven men that can render a reason. 
 
 This complete mental culture must, no doubt, consist 
 of many elements ; but it is certain that an indispensable 
 portion of it is such a discipline of the reasoning power 
 as will enable persons to proceed with certainty and 
 facility from fundamental principles to their conse- 
 quences. And this part of the cultivation of the mind 
 is what I shall at present especially consider. Let us 
 suppose it established, therefore, that it is a proper object 
 of education to develop and cultivate the reasoning fa- 
 culty. The question then arises, by what means this 
 can be done ; what is the best instrument for educating 
 men in reasoning ? 
 
 There are two principal means which have been used 
 for this purpose in our Universities; the study of Mathe- 
 matics and the study of Logic. These may be considered 
 respectively as the teaching of reasoning by practice and 
 by rule. In the former study, the student is rendered 
 familiar with the most perfect examples of strict infer- 
 ence ; compelled habitually to fix his attention on those 
 conditions on which the cogency of the demonstration 
 depends ; and in the mistaken and imperfect attempts at 
 demonstration made by himself or others, he is presented 
 with examples of the most natural fallacies, which he 
 sees exposed and corrected. In studying logic, on the 
 other hand, a person finds the conditions formally stated 
 under which an inference is legitimate ; he is enjoined to 
 see that in any given case these conditions are satisfied ;
 
 14-0 THOUGHTS ON THE 
 
 and if a fallacy exists, he is provided with rules by which 
 it may be condemned and made more glaringly wrong. 
 
 Now I venture to say that the former kind of teaching 
 is, in my opinion, likely to be the more efficacious of the 
 two. For reasoning a practical process must, I think, 
 be taught by practice better than by precept, in the same 
 manner as fencing or riding, or any other practical art, 
 would be. Our object, or at least our first object, in this 
 case, is that the student should conduct his train of de- 
 duction securely, yet without effort, just as in the riding- 
 school the object is that the learner should proceed firm- 
 ly and easily upon his steed. It is desirable, not so 
 much to define good arguments, as to feel their force; 
 not so much to classify fallacies, as to shun them; just as 
 the horseman tries to obtain a good seat rather than to 
 describe one, and rather avoids falling than considers in 
 how many ways he may fall. To cultivate logic as an 
 art (for I do not now speak of the theory) appears to 
 resemble learning horsemanship by book ; and though 
 such learning is not without its use, it cannot supersede 
 the necessity of habitual exercise, which the pursuit of 
 mathematics supplies. In reasoning, as in other arts, we 
 are not masters of what we have to do, till we do it both 
 well and unconsciously. Now, this advantage a judi- 
 cious cultivation of mathematics supplies. It familiar- 
 izes the student with the usual forms of inference, till 
 they find a ready passage through his mind, while any- 
 thing which is fallacious and logically wrong at once 
 shocks his intellectual habits, and is rejected. He is 
 accustomed to a chain of deduction, where each link 
 hangs from the preceding; and thus he learns continuity 
 of attention and coherency of thought. His notice is 
 steadily fixed upon those circumstances only in the sub- 
 ject on which the demonstrativeness depends ; and thus 
 that mixture of various grounds of conviction, which is
 
 STUDY OF MATHEMATICS. 141 
 
 so common in other men's minds, is rigorously excluded 
 from his. He knows that all depends upon his first 
 principles, and flows inevitably from them ; that, how- 
 ever far he may have travelled, he can at will go over 
 any portion of his path, and satisfy himself that it is 
 legitimate ; and thus he acquires a just persuasion of the 
 importance of principles, on the one hand, and, on the 
 other, of the necessary and constant identity of the con- 
 clusions legitimately deduced from them. Logic, on the 
 contrary, forcing upon our notice the rules which we 
 follow when we reason well, hardly allows them to be- 
 come so habitual as to escape our consciousness; nor 
 does she familiarize us with long trains of strict reason- 
 ing, since she generally gives special deductions only as 
 examples of forms of argument. And thus the conti- 
 nuity and concentration of thought, and the quick sense 
 of demonstration, which it is our aim to educe, are not 
 taught so well by this study as by that of mathematics. 
 
 Supposing, then, that we wish to consider mathe- 
 matics 6s an element of education, and as a means of 
 forming logical habits better than logic itself; it becomes 
 an important question, how far the study thus recom- 
 mended is justly chargeable with evil consequences such 
 as have been already mentioned. Does it necessarily 
 make men too little sensible to other than mathematical 
 reasonings? Does it teach them to require a kind of 
 fundamental principles and a mode of deduction which 
 are not in reality attainable in questions of morals or 
 politics, or even of natural philosophy ? If it does this, 
 it may well unfit men for the most important employ- 
 ments of the human mind ; for the power of reasoning, 
 however cultivated, can be of no use on any particular 
 subject, if we cease to be able to appreciate the just 
 principles of the subject. To educate the logical power 
 in such a way, would be to strengthen the cable, but to
 
 142 THOUGHTS ON THE 
 
 lose the anchor: it would be to learn to read all lan- 
 guages on condition of understanding one only. 
 
 But is this, in fact, usually the case ? And if it happen 
 sometimes, and sometimes only, under what circum- 
 stances does it occur ? This latter question has, I think, 
 important practical bearings, and I shall try to give 
 some answer to it. 
 
 I would reply, then, that if mathematics be taught 
 in such a manner that its foundations appear to be laid 
 in arbitrary definitions, without any corresponding act 
 of the mind : or if its first principles be represented as 
 borrowed from experience, in such a manner that the 
 whole science is empirical only ; or if it be held forth 
 as the highest perfection of the science to reduce our 
 knowledge to extremely general propositions and pro- 
 cesses, in which all particular cases are included; so 
 studied, it may, I conceive, unfit the mind for dealing 
 with other kinds of truth. For if there be any portions 
 of human speculation which depend upon mental facul- 
 ties and operations of a peculiar kind ; if they require 
 special and appropriate conceptions, which definitions 
 may fix, if they are present, but cannot convey if they 
 are absent ; the student, whose mind has been entirely 
 formed by such mathematics as we have just described, 
 will have no aptitude for seizing the principles of such 
 subjects; he will be discontented with the supposition 
 that any science should require a mental process which, 
 in the most perfect of sciences, does not occur; and will 
 be constantly attempting to reduce his new study to a 
 dependence on definitions alone, and thus to give it the 
 simplicity and independence which formed the charm of 
 his mathematical speculations. Again : if he be left to 
 suppose that mathematical truths depend ultimately upon 
 the evidence of the senses, he will look in other subjects 
 for evidence equally palpable ; and will not bring away
 
 STUDY OF MATHEMATICS. 143 
 
 from mathematics that lesson which another mode of 
 pursuing the study might impart to him, that there exist 
 vast and solid edifices of truth, the foundations of which 
 are not laid in the information which our external senses 
 give us. And again : the habit of ever aiming at ex- 
 treme generality would further aggravate such tendencies 
 as I have mentioned, since it would draw his thoughts 
 away from all peculiarities of principle and reasoning in 
 particular parts of his science; and would thus remove 
 the last chance of the applicability of the analogy of 
 mathematical reasoning to other subjects. Moreover, 
 all these injurious effects of the study of such a school of 
 mathematics would evidently be much inflamed, if these 
 peculiar doctrines of the school should be false in their 
 mathematical reasoning, as well as inapplicable to other 
 matters ; for then the sensibility of the intellect to sound 
 principles must be blunted by the perverse habit of not 
 seeing them under the most favourable conditions. Add 
 to this, that all which concerns the nature of first prin- 
 ciples operates, far more than any other portion of the 
 mathematical discipline of the mind, when we apply 
 ourselves to other subjects : for in most other provinces 
 of speculation the question is mainly what principle 
 can be admitted and maintained ; and it is only in few 
 cases that we find long trains of continuous inference, 
 in which considerable powers of mathematical deduction 
 are brought into play. If, therefore, by the cultivation 
 of mathematics, we impart false or inapplicable notions 
 of the nature of first principles, at the same time that we 
 teach men to reason well when the principles are given, 
 our discipline is of imperfect character and of doubtful 
 advantage. And if it appear, as I think it will, that 
 such a kind of mathematics as I have mentioned be most 
 likely thus to mingle evil with the good, it will probably 
 be allowed that we have strong reasons for adhering
 
 144 THOUGHTS ON THE 
 
 rather to a different school ; and the more especially, as 
 I have already said, if a different set of doctrines from 
 those to which I allude be right in mathematics as well 
 as useful in philosophy. 
 
 In order to be understood more distinctly, I will 
 notice some of the points in which we may trace, in ma- 
 thematical teaching, such tendencies as I have noticed. 
 
 In doing this I shall not fear giving offence by my 
 criticisms of the mode in which mathematical principles 
 may be or have been presented by those who have 
 written on the subject. The authors in whom such 
 views occur as those which I have to point out, will, 
 I am persuaded, rather rejoice than grieve at the dis- 
 cussion of their principles. But, in fact, it is not to 
 particular authors that I have to refer. The characters 
 of a peculiar school of mathematical speculation are to 
 be found scattered through various branches of the 
 science in various forms, often implied rather than ex- 
 pressed, but still so far characteristic and connected as 
 to make them fit instances of that which it is my object 
 to describe. 
 
 It will be recollected that the first character which I 
 noticed, in the school of which I speak, was "the study- 
 ing mathematics in such a manner that its foundations 
 appear to be laid in arbitrary definitions, without any 
 corresponding act of the mind." I will give several ex- 
 amples where this has been done, and, as I conceive, 
 erroneously. 
 
 I. I will first speak of the grounds of Elementary 
 Geometry ; not because I have anything to blame in the 
 way in which these are commonly taught ; but because 
 I can thus, perhaps, best explain my meaning. 
 
 It has been a question much discussed among meta- 
 physicians and mathematicians, whether the truths of 
 geometry depend on axioms or on definitions. It has
 
 STUDY OP MATHEMATICS. 145 
 
 been asserted by some that they result from definitions 
 alone ; and that on this circumstance depends their ne- 
 cessary and demonstrable truth. On the other hand, 
 they who maintain that axioms also are requisite, chal- 
 lenge their opponents to produce a system of geometry 
 without axioms ; and though much ingenuity has been 
 expended in the attempt, this has never yet been satis- 
 factorily performed, or the challenge fairly answered. 
 
 But supposing this could be done supposing we 
 could get rid of geometrical axioms altogether, and de- 
 duce our reasoning from definitions alone it must be 
 allowed, I think, that still our geometrical propositions 
 would properly depend, not on the definitions, but on the 
 act of mind by which we fix upon such definitions ; in 
 short, on our conception of space. The axiom that two 
 straight lines cannot enclose space, is a self-evident truth, 
 founded upon our faculty of apprehending the properties 
 of space, and of conceiving a straight line. We cannot 
 find a form of words which will express at the same time 
 the nature of the line and the resulting truth. We have, 
 consequently, in our books of geometry, a definition ap- 
 parently useless, and a principle apparently unproved. 
 But who does not see that the axiom does really depend 
 on the definition? or rather, that we see the axiom to be 
 true, precisely because we conceive perfectly the nature 
 of a straight line, which the definition imperfectly ex- 
 presses ? 
 
 Although, therefore, I do not here deny that it may 
 be possible to approximate to a system of "Geometry 
 without Axioms*;" and although I am willing to allow 
 that the more simple and the more obviously coherent 
 our first principles (whether definitions or axioms) are 
 made, the more perfect our system becomes ; I still hold 
 
 * Colonel Peyronnet Thompson has published an ingenious work 
 with this title.
 
 146 THOUGHTS ON THE 
 
 that it is no philosophical blemish in our geometry that 
 it rests upon axioms as well as definitions, since the one 
 may serve, as well as the other, to express those proper- 
 ties of the fundamental conception of space, to which our 
 demonstrations must refer. And we should present a 
 false view of the nature of geometrical truth if we were 
 to represent it as resting upon definitions, and were to 
 overlook or deny the faculty of the mind and the intel- 
 lectual process which is implied in our fixing upon such 
 definitions. The foundation of all the properties of 
 straight lines is certainly not the definition, but the 
 conception of a straight line; and in the same manner 
 the foundation of all geometrical truth resides in our 
 general conception of space. 
 
 This doctrine may appear to some too plain and ob- 
 vious to require to be insisted on. Whether or no that be 
 the case, it will be found, I think, not to be barren of con- 
 sequences in its application to other parts of mathematics. 
 
 II. If it be thus clear that mathematical truths depend 
 for their evidence and certainty upon some fundamental 
 general conceptions, such as that of space; it will follow, 
 that, in order to comprehend aright any portion of mathe- 
 matical science, we must apprehend steadily and clearly 
 the fundamental conceptions belonging to that portion. 
 And any attempt to supersede this necessity by defi- 
 nitions borrowed from another subject, will inevitably 
 leave us with imperfect and insecure principles, and with 
 false notions of the nature of truth. 
 
 I would notice, as an example of this error, the modes 
 in which the doctrine of Proportion is usually presented. 
 It is clear that we are capable of readily forming a dis- 
 tinct conception of proportion, and that a very slight 
 suggestion is sufficient to call up this conception in our 
 minds. We may begin by defining proportional quan- 
 tities to be those in which the first magnitude is the same
 
 STUDY OF MATHEMATICS. 147 
 
 multiple, part, or parts of the second, which the third is 
 of ihe fourth ; but we find that when we have done this, 
 we cannot abstain from applying the same conception of 
 proportionality to cases in which one quantity is neither 
 multiple, part, nor parts of another; for we readily allow 
 that the diagonals of two squares are proportional to their 
 sides. Thus the conception of proportion is immediately 
 extended beyond the limits of the definition ; an unde- 
 niable proof that the conception is not the creature of the 
 definition. This conception being thus formed, certain 
 self-evident properties are seen to belong to it. The sim- 
 plest of these properties appear to be such as those which 
 have recently been noticed as the fundamental principles 
 of the subject*; " That when magnitudes are expressed 
 by the number of equal units they contain, their ratio is 
 not altered by altering the magnitude of the unit ;" and the 
 like. But the principle which was used by the ancient 
 geometers, and which still holds its place in our treatises 
 on geometry, is that stated in Euclid's fifth Definition of 
 the fifth Book; That equimultiples, any whatever, being 
 taken under certain conditions, a certain relation of greater 
 and less obtains. This definition, and the demonstrations 
 founded upon it, are usually considered by learners as 
 obscure and confused. I conceive that this impression 
 arises, in part at least, from the attempt which, in this 
 case, was made by Euclid himself, to reduce the subject 
 to definition alone. For if proportion had been separately 
 defined, so as to bring the conception of it distinctly be- 
 fore the mind, I conceive that the assertion of this fifth 
 definition would be assented to without difficulty as an 
 axiom. But then, what definition of proportion shall we 
 take ? It is clear that, in this case, we must have one 
 which is liable to the same objections as Euclid's defini- 
 tion of a straight line ; namely, first, that the defining 
 
 * Peacock's Algebra, Art. 355. 
 72
 
 148 THOUGHTS ON THE 
 
 term itself requires definition (for evenly is not more 
 intelligible than straight} ; and secondly, that the defini- 
 tion is useless (for the demonstrations rest on the axiom.) 
 Now the observations which have been made help us out 
 of this difficulty; for they teach us that the objections 
 just stated are not fatal to the utility of our definition; 
 since its use is, not to afford a verbal proposition which 
 may enter into our demonstrations, but to direct us to 
 that conception by which we may seize the fundamental 
 necessary truths. If the definition of proportion does 
 this, it answers its purpose. If it give us such a notion 
 of proportion as enables us to see the axiomatic truth of 
 the characteristic property contained in the " fifth defini- 
 tion" of Euclid, it makes our system complete: and this 
 it is not difficult to obtain. A few examples of propor- 
 tions will, at any rate, produce the impression ; and per- 
 haps the definitions which are usually rejected may be 
 considered as sufficiently clear ; namely, " ratio is the 
 mutual relation of two magnitudes in respect of quantity," 
 and "proportion is the similarity of ratios;" or, as it has 
 been expressed by others, " ratio is the relation of two 
 magnitudes in respect ofquotity," (how -many -times-ness,} 
 and " proportion is the equality of ratios." These defini- 
 tions, I say, though apparently indefinite and insulated, 
 would hardly fail to answer their purpose ; for they would 
 give such a conception of proportion as would entitle us 
 to make the " fifth definition" an axiom. 
 
 If we take any other course, we either run into the 
 apparent confusion and complexity which, as has been 
 stated, arises from mixing in the " fifth definition" the 
 character of definition and of axiom; or, on the other hand, 
 by taking as our definition the one first mentioned, which 
 describes one quantity as a multiple, part, or parts of the 
 other, and reasoning from it in cases to which it does not 
 apply, we transform our mathematics from a praxis of
 
 STUDY OF MATHEMATICS. 149 
 
 logic to an example of the most loose and inconsequent 
 reasoning possible. We give pretended demonstrations 
 where the only material difficulty in the proof is slurred 
 over in silence; and thus present geometry as a system 
 in which our sole object is to get, by some probable mode 
 of conjecture, right results ; and where we have not, nor 
 can have, any fixed and certain method by which we 
 prove them to be right. 
 
 That such a mode of treating the most perfect of spe- 
 culative sciences must both degrade the reasoning habits 
 of the student, and enfeeble and pervert his taste and 
 relish for speculative truth in general, cannot, I think, be 
 doubted. 
 
 III. I will notice, as the next example of the errone- 
 ous system to which I refer, the mode in which the first 
 principles of the Differential Calculus are sometimes pre- 
 sented. There are, no doubt, various methods, all satis- 
 factory, in which this may be done ; but there are other 
 methods which, according to the view I am now taking, 
 cannot be allowed to be sound and philosophical, because 
 they endeavour, by definitions and other artifices, to 
 evade all reference to the real fundamental principle of 
 the subject. That fundamental principle is the concep- 
 tion of a Limit. And this conception, like those already 
 mentioned, may be expressed either by means of defini- 
 tions, or axioms, or both ; but whatever course is taken, 
 the foundation on which our conclusions rest is the idea 
 itself. Newton appears to have been the person who 
 first clearly developed the properties of this idea, al- 
 though we can trace its influence upon the reasonings of 
 geometers at an earlier period, and even among the an- 
 cients. The first Section of the Principia contains the 
 results of the conception of a limit or an ultimate ratio, 
 stated in various forms, as definitions, axioms, and pro- 
 portions; and precisely because they all contribute to
 
 150 THOUGHTS ON THE 
 
 express a fundamental idea, these different forms are by 
 no means kept clearly distinct ; but, on the contrary, are 
 all grouped together under the title of Lemmas. In 
 attempting to exhibit in a clearer light the properties 
 thus asserted, we find that we may make various arrange- 
 ments of them, and refer them, in various ways, to the 
 fundamental idea, and to collateral definitions and axioms. 
 But we can in no way evade the necessity of recurring 
 to the peculiar conception of a limit, which, though as 
 difficult to define as a straight line, or proportion, is, like 
 them, soon distinctly formed by the student, and easily 
 made subservient to mathematical reasoning. Having 
 once arrived at this conception, we may state in various 
 ways the axiomatic results that flow from it; for instance, 
 we may employ one general and familiar axiom, that 
 " what is true up to the limit is true at the limit ;" an 
 axiom of most extensive and important application in 
 entering upon the higher mathematics. 
 
 Of this conception of a limit and its consequences, the 
 Differential Calculus is the symbolical exhibition: and I 
 do not hesitate to say, that it is impossible to present the 
 doctrines of that calculus in a logical manner, without 
 referring to this fundamental idea in some part of the 
 process. In speaking of the calculus I include, of course, 
 its applications to curves, mechanical forces, and other 
 problems, since it is, in fact, principally valuable as an 
 instrument for proving propositions on such subjects : 
 and I maintain, that though it may be possible to esta- 
 blish, by means of algebraical operations, without any 
 new principle, a branch of analytical calculation which 
 we may call the Differential Calculus, we cannot justifiably 
 use it as the calculus is used by mathematicians, till we 
 have introduced into it the notion of a limit. The neces- 
 sity of this step may be shifted from one part of our rea- 
 soning to another, but never be altogether evaded. I
 
 STUDY OF MATHEMATICS. 151 
 
 shall proceed to show that we are, in such applications of 
 the Differential Calculus, driven to the doctrine of ulti- 
 mate ratios, or to some equivalent method. 
 
 1. In order to apply the Calculus to the Rectification 
 of Curves, we may assume the axiom of Archimedes, that 
 a curve is less than any broken line which has the same 
 extremities and includes the curve. This assumption is 
 made by Lagrange in his Theorie des Fonctions, (partie 
 2% chap. vi. no. 29) ; in the title-page of which work he 
 professes to give " the Principles of the Differential Cal- 
 culus, disengaged from all consideration of infinitely small 
 quantities, of vanishing quantities, of limits, and of 
 fluxions." Now I say, that without the consideration of 
 limits, this assumed axiom is not only not evident, but is 
 not true ; for a sinuous curve may be greater than a 
 broken line, with the same extremities, which includes it. 
 And if it be replied that the axiom is asserted only of 
 curves which have their concavity turned throughout to 
 the same side, I demand how the property can be evidently 
 true with the condition, and not evident or not true with- 
 out it, if we do not see some evident bearing of the con- 
 dition upon the property ? and I ask what bearing this 
 is ? It may be answered, that a broken line composed of 
 straight lines, which has its concavity turned throughout 
 the same way, is less than another such broken line which 
 includes it ; and that both the truth and the evidence of 
 this property are secured by the condition respecting the 
 concavity ; and it may be added, that we cannot conceive 
 a curve to differ in this respect from such a broken line. 
 To this I assent ; but I require further, that we should 
 put in the simplest and most exact form, the expression 
 of the necessary conviction that the properties of a curve 
 must be the same as the universal properties of a broken 
 line composed of straight lines. Now this, I maintain, 
 can be done in no other way than by a reference, explicit
 
 152 THOUGHTS ON THE 
 
 or implicit, to the conception of a curve as the limit of a 
 broken line ; and to the general principle of limits, that 
 " what is true up to the limit is true at the limit." If it 
 is imagined that the necessity of the condition with 
 respect to the concavity can be shown in any other way, 
 let the requisite axioms be stated; but if this is not done, 
 the principle of Archimedes, as far as it is true, is by no 
 means of an axiomatic character. As a foundation of the 
 applications of the Differential Calculus to curves, it 
 would be much simpler to assume at once Newton's 
 seventh Lemma, that the arc, the chord, and the tan- 
 gent are ultimately equal. 
 
 2. Similar remarks, as to the want of axiomatic evi- 
 dence in the geometrical principles usually assumed, apply 
 still more clearly to the mode in which the Calculus is 
 applied to find the areas of surfaces. For in this case, 
 when we have two curved surfaces bounded by common 
 borders, it is far from easy, even to conjecture, which is 
 the greater of the two. For instance*, if a curve, and the 
 broken line made up of the two tangents at its two extre- 
 mities, revolve about any axis, which of the two generates 
 the greater surface, the curve or the broken line ? Pro- 
 bably the young geometer will not at once guess the con- 
 ditions which determine the answer. Here, therefore, 
 we shall not only reason loosely, but perhaps make false 
 assertions, if we endeavour to establish the application of 
 the Differential Calculus to surfaces, by means of assumed 
 axioms which do not refer to limits. How difficult it is 
 for a scrupulous mathematician to satisfy himself in the 
 attempt may be seen in Maclaurin's Fluxions. 
 
 But it may be said, the axioms assumed by writers on 
 these subjects have always led to right results. I grant 
 that they not only have done so, but probably always will 
 do the same. And why is this? Precisely because they 
 are applied only at the limit. It is there only that they
 
 STUDY OF MATHEMATICS. 153 
 
 are wanted, it is there only that they are used, and there 
 they are always true. Whatever may be the relations of 
 the broken lines and curve lines, of the broken surfaces 
 and curve surfaces, as to greater or less, while they are 
 finite, when we come to the limit, the relation of greater 
 and less ceases, and they are equal. Newton's seventh 
 Lemma for curves, and a similar proposition for curve sur- 
 faces, are then indisputably true; and this is sufficient for 
 all the purposes of analytical and geometrical reasoning. 
 
 I presume it will not be maintained that correctness 
 of logic is of no importance in our mathematical teach- 
 ing; and that, if the conclusion be right, it need not 
 disturb us that our fundamental principles are false, and 
 our course of deduction inconclusive. I conceive rather 
 that all are agreed that soundness of reasoning is the first 
 and indispensable merit of such instruction. If therefore 
 the conclusions, in treatises of the Differential Calculus 
 which do not employ the principles of limits, are true 
 only in virtue of the principle which they reject, and can 
 have no adequate support in the reasonings on which 
 they profess to rest, it is not unimportant to point out 
 the existence of this paralogism. 
 
 Whatever course of sound demonstration we follow, 
 the properties of the limit will be the foundation of it. 
 If we employ the assumption of Archimedes, the essen- 
 tial part of the assumption is, that it should be true at the 
 limit. In this case the assumption coincides nearly with 
 Newton's seventh Lemma, which obviously and profess- 
 edly depends on the ultimate condition of the figure. We 
 may treat the subject either in these or in other ways. 
 We may, for instance, define a curve to be the limit of a 
 polygon of many sides, and may transfer to curves the 
 properties of polygons, in virtue of the axiom that " what 
 is true up to the limit is true at the limit." And in the 
 same manner we may get rid of the difficulties respecting 
 75
 
 154 THOUGHTS ON THE 
 
 tangents, by defining a tangent, to be the limit of a line 
 cutting the curve in two points, which limit is obtained as 
 the points coalesce. In defining the radius of curvature, 
 the conception of limits is introduced by most writers. 
 But in all cases, if we fairly trace back to elementary 
 principles the properties of curves, we shall find that we 
 must include in our reasonings the conception of a limit, 
 and an axiom or axioms founded upon this concep- 
 tion. 
 
 3. The application of the Differential Calculus to 
 Mechanics gives room for the same observations which 
 have been made respecting its use with regard to curves. 
 It is impossible to justify this application, without rea- 
 soning upon the principle of limits. And we may intro- 
 duce this principle, by means of certain axioms respecting 
 the spaces and velocities, in various ways. We may, for 
 instance, assert, as axioms, that when a body moves with 
 an increasing velocity, the space described in any time is 
 greater than the space which would be described in the 
 same time with the initial, and less than that which would 
 be described with the final velocity. Such an axiom 
 would be analogous to that of Archimedes respecting 
 curves. It would express nothing but what is very ob- 
 viously seen to follow from our general notion of velocity, 
 and would not require any precise definition of velocity. 
 It would however appear, on a more scrupulous considera- 
 tion, as in the case of curves already discussed, that the 
 evidence of this property of velocity resides in our being 
 able to follow the axiom up to the limit; and that the 
 security from error in its application arises from our ap- 
 plying it only at the limit; for how, without recurring to 
 the conception of a limit, shall we find a criterion to dis- 
 tinguish cases in which the velocity constantly increases, 
 from those in which it alternately increases and diminishes ? 
 Similar remarks might be made concerning force.
 
 STUDY OF MATHEMATICS. 155 
 
 Another method, and perhaps a better, of treating this 
 subject is, to introduce the conception of a limit into the 
 definition of velocity and of force ; which being done, we 
 can reason from our definitions without the need of any 
 special axiom. We thus purify our fundamental principles 
 at the same time from the superfluous notions of greater 
 and less, from insulated and useless definitions of velocity 
 and force, and from apparently unsupported assumptions 
 in our axioms. This we effect by defining velocity as the 
 limit of the ratio of the increments of space, and time, and 
 force, as the limit of the ratio of the increments of velocity 
 and time. This method corresponds to that which I have 
 suggested in the other case, of defining a tangent as the 
 limit of a straight line cutting the curve in two points, or 
 a curve as the limit of a polygon. 
 
 The principal objection which offers itself to this 
 mode of presenting these subjects is, that the definitions 
 thus appear to become too remote from the common con- 
 ceptions of the things defined. Tangents, curves, ve- 
 locity, force, it may be said, are notions which we can 
 form, without considering anything so abstract as a limit. 
 To this I would reply, that I do not think we can form 
 these notions with sufficient distinctness for the purposes 
 of our reasonings, without introducing the conception of 
 limits ; and that the proof that we cannot do so is, that all 
 attempts to reason correctly on these subjects without the 
 aid of the doctrine of limits have failed. It will be found 
 to be the general or universal process in forming defini- 
 tions for the purposes of science, that we substitute for 
 the common and current notions, which are always in 
 some degree vague, other more exact notions which in- 
 clude them, and are so much more distinct as is requisite 
 for the purposes of strict reasoning. Our reasoning in 
 the cases now before us is conducted by the principle of 
 limits; and our definitions are accordingly shaped on the
 
 156 THOUGHTS ON THE 
 
 conception of limits, so as to allow the principle to come 
 into play. In this method the conception of a limit is 
 employed in constructing our notions as well as in regu- 
 lating our inferences. 
 
 But whether we. take this course or any other, in 
 treating of velocity and force by symbolical reasoning, we 
 shall find ourselves compelled (if we are at all scrupulous 
 in our reasoning) to introduce the conception of a limit, 
 either into our definitions or into our axioms; nor can 
 any refinement of geometry or algebra in any degree 
 diminish this necessity, however they may serve to dis- 
 guise it by complicating and confusing the subject. 
 
 The doctrine of limiting ratios may be considered as 
 the key to all the abstruser parts of mathematical science; 
 and if a boundary were to be drawn between the Elemen- 
 tary and the Higher Mathematics, the employment of this 
 doctrine might most properly be made the frontier line. 
 And if, when the student arrives at this critical point of 
 his career, we neglect to educe, or even endeavour to 
 discard, the fundamental conception on which all his 
 ulterior knowledge must in fact depend, we not only sow 
 the seeds of endless obscurity and perplexity during all 
 his future advance in this science, but we also weaken 
 his reasoning habits and disturb his perception of specu- 
 lative truths; and thus make our mathematical discipline 
 produce, not a wholesome and invigorating, but a dele- 
 terious and perverting effect upon the mind. 
 
 IV. The next example which 1 shall notice of the 
 dependence of the branches of mathematics on some fun- 
 damental conception, is an important one. The concep- 
 tion of force or pressure is the groundwork of the whole 
 doctrine of Mechanics. However we may define force, 
 it is necessary, in order to understand the elementary 
 reasonings of this portion of science, that we should con- 
 ceive it distinctly. Do we wish for a test of the distinct-
 
 STUDY OF MATHEMATICS. 
 
 157 
 
 ness of our conceptions ? The test is, our being able to 
 see clearly the necessary truth of the axioms on which 
 our reasonings rest. These axioms may be different ac- 
 cording to different ways of treating the subject. If we 
 begin from the properties of the lever, we have such 
 axioms as this, that when two parallel forces act on a 
 lever, the pressure on the fulcrum is equal to their sum. 
 If on the other hand we begin with the composition of 
 forces at a point, and hence prove the lever, it will be 
 found that we inevitably assume such principles as these, 
 that forces, any how resolved into their components, 
 may have their components substituted for them in all 
 cases ; again, that a force may be supposed to act at any 
 point of its direction, which introduces the conception of 
 a perfectly rigid body; and again, that a pressure ne- 
 cessarily supposes an equal and opposite pressure. These 
 principles are all perfectly evident as soon as we have 
 formed the general conception of pressure ; but without 
 that act of thought, they can have no evidence whatever 
 given them by any form of words, or by reference to 
 other truths; by definitions, or by illustrations from 
 other kinds of quantity. 
 
 In the present state of our mathematical studies, I 
 cannot so properly say that this necessity of the concep- 
 tion of force as a foundation of mechanical reasoning 
 is denied, as that it is too vaguely and slightly acted 
 upon. Force is usually and rightly described as the 
 cause which produces or tends to produce motion ; and 
 is thus referred to the more general conception of cause. 
 But this particular kind of cause, which is thus de- 
 scribed as producing a tendency to motion, (a defini- 
 tion which it will easily be seen is liable to the same 
 remarks as the definition of a straight line,) is so con- 
 ceived as to give rise to certain axioms, some of which 
 we have seen above ; and the conception, thus rendered
 
 158 THOUGHTS ON THE 
 
 available for the purposes of deduction, is the inevitable 
 foundation of mechanical reasoning. The student can- 
 not advance in this field steadily or clearly, except he 
 feel distinctly that he is on new ground; that he is 
 dealing with a set of notions different from those of 
 pure mathematics, and resting on peculiar principles. 
 Of this, perhaps, he is not always apprized. He is 
 led, or left to imagine, that his new study is merely 
 a developement of his notions acquired in the previous 
 parts of mathematics ; that the quantities which he has 
 to do with now are of the same kind as the quantities he 
 had to do with then ; that the addition and mutual de- 
 struction of forces can be understood, by understanding 
 addition and mutual destruction of algebraical quantities, 
 without considering specially what force is, and what 
 peculiarities belong to this new kind of quantities. 
 Should he approach the subject with such an impres- 
 sion, it will be no wonder if his notions always remain 
 mere algebraical abstractions, without mechanical value 
 or meaning ; and if he himself continue, to the end of 
 his career, incapable of applying them to any really 
 mechanical question. 
 
 Some persons, indeed, appear to go so far as not 
 only practically to neglect, but speculatively to deny, 
 any essential peculiarity of character in the notion of 
 force. Thus Professor Playfair, in his "Outlines of 
 Natural Philosophy," Art. 86, says that "force has, in 
 
 reality, no other signification than -j- ; and that thus 
 
 an entire treatise of dynamics might be written, in 
 which the word force would not once occur." If he 
 had attempted to write such a work, he should have 
 found the necessity of referring, at any rate, to the 
 conception of force. How, without such a reference, 
 would he have shown that action and reaction are
 
 STUDY OF MATHEMATICS. 159 
 
 equal ? When two equal bodies act on each other, why 
 
 d i) ft'V 
 
 must + -7- in the one be equal to =- in the other ? 
 ilt at 
 
 The student who recurs to this conception of pressure, 
 can readily deduce such a result; but without that, 
 he would not be able to advance a step in the train 
 of deduction. 
 
 It was nothing but the want of this distinct idea of 
 pressure, or the loss of it, after Archimedes, which 
 made all the attempts of the ancients in mechanical 
 reasoning entirely futile up to the time of Stevin. It 
 was, for instance, impossible that men could reason on 
 these subjects to any purpose, so long as their minds 
 were in such a state that they could accept the fable 
 of the Remora or Echineis, a fish half a foot long, 
 which, merely by sticking to the bottom of a ship, 
 without any other hold, could prevent its going on, 
 though impelled by the strength of four hundred rowers; 
 and though the fish, when caught and laid on the deck, 
 possessed no such power*. Any one, whose notion of 
 force had acquired any degree of distinctness, would at 
 once see the absurdity of this; since the ship and its 
 rowers must pull the fish, as much as the fish the ship ; 
 a result which it would however be difficult to deduce 
 
 d v 
 
 from the definition of force -r- , without some accom- 
 
 at 
 
 panying idea. 
 
 That this distinctness of mechanical conception is not 
 always sufficiently inculcated, we have often very clear 
 evidences in our Examinations. Students whose talents 
 and proficiency in pure mathematics are very consider- 
 able, are frequently found quite incapable of solving a 
 very simple mechanical problem so as to bring out any 
 significant result. It is not difficult to see whence this 
 
 * Plin. Nat. Hist. ix. 41. xxxn. 1.
 
 160 THOUGHTS ON THE 
 
 arises. Their minds have never been taught to grapple 
 with the real mechanical elements of pressure and re- 
 action ; they have been accustomed to trust the process of 
 deduction to certain representative symbols and the rules 
 of their combination. These, rightly used and well 
 understood, may, no doubt, be useful aids : but such 
 representations can never be employed to any purpose 
 by those who have no distinct conceptions of what they 
 represent. And even in the most favourable cases, it is 
 to be recollected, that by this mode of treating a problem 
 we make two operations necessary ; the obtaining our 
 symbolical result, and the interpreting it. The student 
 who can rightly combine his symbols, may read them 
 wrongly, or not at all. His familiar oracle may deliver 
 a response indeed, but it may be in a language which he 
 cannot understand ; his devotion must be very blind 
 if he is satisfied with such a reply; yet this may 
 happen if we, his instructors, acquiesce in these empty 
 rites. 
 
 If we neglect to inculcate and require distinctness in 
 the student's apprehension of the idea of force and the 
 resulting principles, the evil in reference to the discipline 
 of the mind is so much the more grievous, in that we 
 thus let slip one of the most favourable opportunities for 
 showing the wide range of mathematical reasoning ; for 
 teaching that the strictest logical necessity, and the most 
 remarkable continuity of deduction, are not confined to 
 the domain of space and number only, but belong to all 
 subjects on which our notions are sufficiently distinct and 
 precise; and that the mode in which this rigorous and 
 scientific character may be extended to new subjects is, 
 by making our fundamental ideas so steady and clear, 
 that they may generate in the mind principles, whether 
 definitions or axioms, which may serve as the starting- 
 point of our reasonings.
 
 STUDY OP MATHEMATICS. 161 
 
 V. My object is to illustrate, not to exhaust the 
 subject ; and I will therefore only mention one other 
 instance of the importance of possessing aright the fun- 
 damental conception of each branch of mathematics : I 
 now speak of the conception of fluidity as the basis of the 
 doctrines of Hydrostatics. It has often been proposed 
 as a definition of a fluid, that it is a body having parts 
 moveable among each other with perfect ease; and many 
 attempts have been made to deduce from this definition 
 the general property of fluids, that they transmit pres- 
 sure unaltered in all directions. But in all such attempts 
 there is and must be a latent reference to something not 
 expressed, which is really the foundation of the science. 
 It is manifest a priori that the definition just given can- 
 not be a sufficient basis for the doctrines of the pressure 
 of fluids ; for how can we evolve, out of the mere notion 
 of mobility, which includes no conception of force, the 
 independent conception of pressure ? All such attempts, 
 as far as they lead to true results, do so in virtue of a 
 tacit assumption of that part of our conception of fluids 
 which is, hydrostatically, the essential part, namely, that 
 they transmit pressure in all directions; if we press a 
 full bladder downwards at the top, it presses outwards 
 at the side. This is clearly supposed in our conception 
 of fluidity: and from this we may prove that the pres- 
 sure is equal in all directions, which is the most material 
 proposition of the science*. But without thus including 
 pressure as well as mobility in our conception of fluids, 
 we can have no sufficient grounds for our hydrostatical 
 demonstrations. 
 
 The way in which a neglect of clearness and com- 
 pleteness in framing the fundamental conceptions of our 
 sciences leads to bad reasoning, and consequently to bad 
 mental habits, may be seen in the subject now under 
 
 See Mechanical Euclid, B. n. Props. 2, 3.
 
 162 THOUGHTS ON THE 
 
 consideration. We shall probably find among our ma- 
 thematicians (for I do not pretend to criticise particular 
 books,) such reasoning as the following: "A fluid is a 
 body, the parts of which are perfectly moveable in all 
 directions : if therefore a force act in any direction upon 
 any particle of it, there must be, acting on the same par- 
 ticle, equal forces in all other directions." Now this is 
 palpably a fallacy. If a particle be kept at rest by 
 forces acting on it, the only consequence which follows 
 from the laws of mechanics is, that it must be acted on 
 by pairs of equal and opposite forces : we cannot hence 
 infer the smallest necessity that the lateral forces should 
 be equal to the vertical ones. But, it may be said, the 
 particle is a fluid particle, and thus a vertical pressure 
 will be resisted by a lateral pressure. This, no doubt, is 
 a reason for the inference above drawn, or may be de- 
 veloped so as to become one. But then, it is exactly 
 that reason which we assert to be the only proper one ; 
 namely, the conception of fluidity as implying the trans- 
 mission of pressure in all directions. 
 
 To apprehend the idea of a fluid in such a manner as 
 to deduce from it all the remarkable conclusions which 
 form the science of Hydrostatics, has always been justly 
 considered, from the time of Stevin and Galileo to our 
 own, as an important and critical step in our philoso- 
 phical knowledge of nature; and hence, the science 
 which teaches us to do this, may be a valuable portion 
 of our intellectual discipline. But this effect it can 
 never produce, if those properties of fluids, on which 
 all the reasoning depends, are, at the very outset, so 
 expeditiously and completely wrapped up in symbols 
 that the student never has a distinct view of them, or 
 perhaps is not aware of their existence as a separate 
 object for the employment of his thoughts. The hy- 
 drostatic paradox may cease to be considered as a
 
 STUDY OP MATHEMATICS. 163 
 
 paradox in two ways ; in one, by our seeing the dif- 
 ficulty and its solution ; in the other, by our seeing 
 neither. It is not difficult to judge which is the more 
 instructive view of the subject. 
 
 Thus I have, I think, shown, by a sufficient number 
 of instances, that there is, Avith some mathematicians, a 
 tendency to exclude or overlook the peculiar funda- 
 mental principles of several portions of mathematical 
 science, substituting for them mere verbal definitions, 
 or forcing a way, in defiance of all maxims of sound 
 reasoning, to the point where symbolical calculation 
 begins. I will not now dwell further on the injurious 
 effects which such practices must produce when these 
 portions of mathematics are used as instruments of edu- 
 cation, but will proceed to explain another of the ob- 
 jections which I have stated in general terms. 
 
 I mentioned it as likely to make the study of mathe- 
 matics less beneficial as a mental discipline than it might 
 otherwise be, if the first principles of our knowledge 
 be represented as borrowed from experience, in such a 
 manner that the whole science becomes empirical only. 
 
 I will not suppose that any person who has paid any 
 attention to mathematics does not see clearly the differ- 
 ence between necessary truths and empirical facts ; be- 
 tween the evidence of the properties of a triangle, and 
 that of the general laws of the structure of plants. The 
 peculiar character of mathematical truth is, that it is 
 necessarily and inevitably true; and one of the most 
 important lessons which we learn from our mathematical 
 studies is a knowledge that there are such truths, and a 
 familiarity with their form and character. 
 
 This lesson is not only lost, but read backwards, if 
 the student is taught that there is no such difference, 
 and that mathematical truths themselves are learnt by
 
 164 THOUGHTS ON THE 
 
 experience. I can hardly suppose that any mathema- 
 tician would hold such an opinion with regard to geo- 
 metrical truths, although it has been entertained by 
 metaphysicians of no inconsiderable acuteness, as Hume. 
 We might ask such persons how Experience can show, 
 not only that a thing is, but that it must be; by what 
 authority she, the mere recorder of the actual occur- 
 rences of the past, pronounces upon all possible cases, 
 though as yet to be tried hereafter only, or probably 
 never. Or, descending to particulars, when it is main- 
 tained that it is from experience alone that we know that 
 two straight lines cannot enclose space, we ask, whoever 
 made the trial, and how ? and we request to be informed 
 in what way he ascertained that the lines with which he 
 made his experiment were accurately straight. The fal- 
 lacy is, in this case, I conceive, too palpable to require 
 to be dwelt upon. 
 
 When we pass from the properties of space and num- 
 ber to those of force and matter, we shall perhaps find a 
 larger class of persons who will hold our fundamental 
 principles to be derived from experimeiit. That experi- 
 ment has had a share in the establishment of such prin- 
 ciples, the history of science very clearly shows ; but if 
 this share be loosely and obscurely conceived, such an 
 opinion may still lead to very unphilosophical views, 
 and very illogical habits. By what experiments is it 
 found that the pressure on the fulcrum is equal to the 
 sum of the weights? Who supposes that Archimedes 
 thought it necessary to verify this result by actual trial ? 
 or if he had done so, what practical mode could he have 
 employed, of showing forces to be equal, which did not 
 take for granted some principle, requiring proof as much 
 as the axiom does? I shall not here dwell upon the 
 share which experience really has in the formation of 
 our mechanical principles, having already attempted to
 
 STUDY OF MATHEMATICS. 165 
 
 explain elsewhere* the manner in which external phe- 
 nomena call up and interpret the intellectual conceptions 
 on which science depends. 
 
 An imperfect exhibition of the fundamental concep- 
 tion of Hydrostatics may render the elementary rea- 
 soning of that science illogical, as we have seen ; 
 some mathematicians have been driven, by the apparent 
 insufficiency of such reasoning, to throw themselves 
 entirely upon experiment for the foundations of hydro- 
 statical science. " Fluids/' say they, "are found by ex- 
 periment to press equally in all directions." There would 
 be no difficulty in pointing out experiments which are 
 sufficient to illustrate the proposition ; and to remove any 
 difficulty which the first announcement of it might occa- 
 sion : but I do not think that all the experiments of this 
 kind which ever were made, could be held sufficient to 
 establish a proposition so general and so rigorous, if we 
 saw no antecedent reason for it. And in fact, it cannot 
 be doubted that good mathematicians, in tracing the 
 consequences of this general property of fluids, do rea- 
 son from their perception of its connexion with the 
 nature of fluidity, and do not find themselves obliged 
 to refer to the contingent and limited conditions of an 
 experimental proof. 
 
 This disposition to ascribe such an empirical origin to 
 truths which have in fact a deeper root in our minds, is 
 injurious in its influence upon our speculations on other 
 subjects; for it tends to make us suppose that the foun- 
 dations of other sciences and bodies of demonstration are 
 to be sought in plausible maxims, collected from expe- 
 rience by the same kind of loose and casual observation, 
 and assented to with the same slight attention to the facts 
 on which they profess to rest. It tends moreover to lead 
 us to imagine, that when our common experience sug- 
 
 Trans. Camb. Phil. Soc. Vol. iv.
 
 166 THOUGHTS ON THE 
 
 gests to us no maxims possessing this kind of apparent 
 evidence and generality, we may despair of ever arriving 
 at a connected system of truth and certainty ; and it 
 teaches us to believe that any appeal to experience, how- 
 ever vaguely made, is to be held inevitable and final, and 
 is to put an end to all speculation on the subject; instead 
 of suggesting to us how difficult the interpretation of our 
 experience is, and how much caution and thought are 
 required in order to infer anything from it. It is easy to 
 see how fatal to our success and welfare in other pro- 
 vinces of speculation impressions of this kind must be. 
 
 I must now again briefly notice a third character of 
 the school of mathematics which I am venturing to de- 
 scribe; that it is there held forth as the highest perfec- 
 tion of our knowledge, to reduce it to extremely general 
 propositions and processes, in which all particular cases 
 are included. 
 
 This love of generality is intact a natural consequence 
 of the obliteration of all the peculiar fundamental concep- 
 tions which belong to different parts of mathematics. If 
 there be no essential difference in the ideas with which 
 we have to deal, there is no reason why we should not 
 embrace as many of them as possible in one form of ex- 
 pression. And thus we may neglect the peculiarities 
 which belong to our conception of space, and identify 
 our geometrical propositions with the corresponding pro- 
 perties of numbers, and of symbols which originally re- 
 present numbers. We may then consider the difference 
 between plane and solid figures (so peculiar a difference 
 in itself) as merely the difference between the combina- 
 tions of two and of three quantities. We may express 
 the properties of forces in like manner by algebraical 
 symbols ; for forces may be measured by lines and by 
 numbers ; and he who sees nothing peculiar in the prin-
 
 STUDY OF MATHEMATICS. 167 
 
 ciples of this measurement, runs forwards with no phi- 
 losophical scruples to the point where it is supposed to 
 be already established. The reasons for all peculiar con- 
 sideration of any special classes of figures or quantities 
 being thus pushed aside, the most general views are the 
 most simple : straight lines are merely a kind of curve ; 
 plane curves must not be considered as confined to their 
 plane, but as existing under the most general form of 
 reference; positions, paths, forces, motions, are all best 
 described by means of algebraical elements ; and these, 
 though called three rectangular co-ordinates, are consi- 
 dered as depending on the conception of space, only at a 
 few points of the calculation, all the intermediate steps 
 being determined by the rules of the combination of 
 symbols alone. 
 
 To the beauty of mathematical generalisations no one 
 can be insensible who is capable of deriving any pleasure 
 from the study of mathematics ; and this process, and the 
 habit of thus generalising, are of very instructive effect 
 on the mind, even for the purposes of philosophy of all 
 kinds. But this act of generalisation, to be of any value, 
 must be founded on particulars rightly and distinctly 
 conceived; and must be capable of being verified by 
 inversion, that is, by carrying back the general proposi- 
 tion at will into any number of particular cases. The 
 great geniuses of the mathematical world have always 
 delighted in the widest generalisations, because they have 
 by nature possessed this distinctness of particular know- 
 ledge, and have thus been able to perform the ascent to 
 generalities and the descent to particulars with a secure 
 rapidity ; but these are feats of strength and agility which 
 it is not given to all to imitate. The talent of generalisa- 
 tion is the last which is developed in the mathematical 
 student; and if he attempt the process while his posses- 
 sion of particulars is still obscure and wavering, he can-
 
 168 THOUGHTS ON THE 
 
 not fail to reduce his knowledge to a state of confusion 
 and worthlessness. The attempt is much like under- 
 taking to read the lyrical poetry of a language in which 
 we have not yet mastered the simplest phrases of prose. 
 When a mathematician has apprehended the properties 
 of curves as distinctly as Monge, or the conceptions of 
 mechanics as clearly as Lagrange, he may pursue, and 
 imitate if he can, the beautiful speculations of the Geo- 
 metric Descriptive, and the Mecanique Analytique. And 
 even in a far smaller degree, so far as the student has the 
 materials and the power, let him seize the most general 
 point of view within his reach, as one of his stations. 
 But let him be certain that he knows the path up to this 
 summit and down from it ; and let him shun the common 
 fault of taking indistinctness as evidence of generalisa- 
 tion, for this is to suppose ourselves on a mountain be- 
 cause we are in a mist. 
 
 As I have said, the love of generalisation, so far as it 
 is a fault, arises from a neglect of the peculiarities which 
 belong to the reasonings of particular subjects, and is 
 prejudicial to the student, because it prevents his rightly 
 apprehending these reasonings, and their foundations. 
 And thus the evil consequences of the disposition of 
 which I now speak, in its effects upon the intellectual 
 habits, are nearly the same as those which have before 
 been described. 
 
 I trust that I have now shown that there exist certain 
 modes of treating the study of mathematics, and certain 
 views concerning its foundations, which must diminish 
 its benefits as a mental discipline and a preparation for all 
 other branches of philosophical speculation. It is a mat- 
 ter of great difficulty to catch with precision, to fix in 
 words and to make the subject of reasoning, anything so 
 indefinite and evanescent as the general effect of abstract 
 pursuits and studies upon the mind ; but still it appears
 
 STUDY OP MATHEMATICS. 169 
 
 to me, on the grounds already stated, difficult to deny 
 that a school of mathematics, which should seek to re- 
 duce the foundations of the subject to verbal definitions, 
 or to the evidence of the senses, and the processes of the 
 science to undistinguishing generalities, must tend to 
 produce such impressions as have been pointed out. It 
 must tend to make men believe that all certain knowledge 
 is reducible to identical propositions, or to facts obvious 
 to all the world; and that logical clearness, acuteness, 
 and invention, are all that any branch of speculation 
 requires ; thus concealing the necessity of that research 
 and meditation which are requisite to acquire a posses- 
 sion of the first principles of all philosophical subjects, 
 and which are far more important than mere logical en- 
 dowments. And on subjects on which no hope of cer- 
 tainty offers itself in definitions or in familiar maxims, 
 minds so formed may naturally cease to hope or care for 
 truth, however weighty be the interests which our opi- 
 nions affect. If, besides this, the reasoning of this school 
 of mathematics be in itself, as I have endeavoured to show 
 that it is, indefensible and false, the logical habits as well 
 as the philosophical capacities of the student will be per- 
 verted ; and mathematical study will lose, as a discipline, 
 that solitary advantage which has been allowed to it even 
 by those who have held it to be injurious in other re- 
 spects. 
 
 However slight may be the degree hi which the ten- 
 dencies of which I have spoken exist in the mathema- 
 tical studies of this University, it would be desirable to 
 remedy them, and to secure to our course of education 
 those advantages which a sounder and better system 
 would produce. It is with the desire of inculcating this 
 object, that the present reflections are published. 
 
 Some persons may, perhaps, think that it is only an 
 idle imagination to expect from the mathematical teach- 
 8
 
 170 THOUGHTS ON THE 
 
 ing which students receive in the University, effects on 
 the mind, such as have been here spoken of. They may 
 suppose that, in by far the greater number of cases, the 
 knowledge of mathematics which is acquired is too 
 superficial, and too soon forgotten when the academic 
 course is finished, to leave any deep or permanent im- 
 pression. And it would not be difficult, by speaking 
 of the problems which are brought under the student's 
 notice here, as useless abstractions and fantastical subtle- 
 ties, or by putting in print the familiar or slighting 
 terms in which these matters are often spoken of in 
 conversation, to throw an air of ridicule upon the doc- 
 trine of the formation of the mind by such means. No 
 one, however, who is capable of serious thought on such 
 subjects, would be moved by trifling of this kind. How- 
 ever obscure the process, however trivial the details, 
 however uncertain the result of individual cases, we 
 know that, in fact, the employments of the youth have a 
 great share in forming the character of the man; and 
 that pursuits leave traces of their indirect effect on the 
 habits, long after they~ themselves have ceased to exist. 
 But in this case we are not left to reasoning and conjec- 
 ture. No one who knows the recent history of this Uni- 
 versity can doubt that the mathematical studies of its 
 members have produced a very powerful effect on the 
 general character of their mental habits. Any one who 
 is acquainted with the lawyers, or men of business, or 
 statesmen, whom the University has produced in our 
 own and the preceding generation, knows, from obser- 
 vation of them and from their own declarations, that 
 their mathematical pursuits here have in no small degree 
 regulated their mode of dealing with other subjects. 
 With respect to lawyers, indeed, we have the evidence 
 of their practical success to the reality of this connexion; 
 and, among them, the extraordinary coincidence of pro-
 
 STUDY OF MATHEMATICS. 171 
 
 fessional eminence in after-life with mathematical dis- 
 tinction in their University career, shows that our 
 studies may be an admirable discipline and preparation 
 for pursuits extremely different from our own. That 
 mathematical habits do, or have done, so much to make 
 men good lawyers, is not an unimportant consideration 
 with respect to that profession; but it is far more im- 
 portant as showing what such a training may effect in 
 reference to other and wider studies. 
 
 I will not undertake the invidious task of pointing 
 out cases in which the tendencies of that peculiar school 
 of mathematics on which I have animadverted, may be 
 supposed to have shown themselves in the mode of treat- 
 ing other subjects. But I may venture to observe, that 
 the recent mathematical literature of the University ap- 
 pears to me to illustrate the views expressed in these 
 pages. The number of new treatises on various portions 
 of the mathematics which have been published by au- 
 thors who had recently passed through their under- 
 graduate career, shows that the mathematical studies 
 in which men here are engaged do not leave their minds 
 inert and passive, but rouse them to speculate for them- 
 selves. The subjects of speculation in these treatises 
 have generally been such as bear directly upon some 
 of the fundamental conceptions above spoken of: for 
 instance, one of the most usual subjects of such treatises 
 is the first principles and processes of the Differential 
 Calculus. The frequency of such publications may, I 
 think, be explained, by the consideration above stated; 
 that there prevails among us a mode of teaching the 
 subject, in which the fundamental conception is slightly 
 referred to or entirely eluded. For, this being done, the 
 impression left on the student's mind will necessarily be 
 unsatisfactory; and when, in a season of leisure, and 
 with some skill in calculation, he begins to settle his 
 8 2
 
 172 THOUGHTS ON THE 
 
 own views, it is probable that some new aspect of the 
 subject will appear to him more clear than that of his 
 masters. The vessel not being ballasted by any definite 
 principle, rolls into a new position of transient equi- 
 librium with every new wave. I am far from wishing 
 to speak with disrespect of such treatises, and have 
 great obligations to several of them; but it must have 
 appeared, from what I have already said, that I con- 
 ceive they must fail in their object, where it is to 
 make algebraical definitions alone the foundation of 
 the Calculus; and that they can do no service to the 
 cause of education, if their merit be their generality. 
 
 I come now to the practical inferences which I 
 would draw from the above reflections. The first of 
 these obviously is, that we should, in the mathema- 
 tical knowledge which we teach or require, avoid the 
 evil and choose the good; that we should not encou- 
 rage by our lessons, or books, or questions, those ten- 
 dencies of which the disadvantageous character has, I 
 hope, been made apparent. It is highly for the inter- 
 est of the University that the overdisposition to ana- 
 lytical generalisations should not be fostered; that a 
 clear acquaintance with first principles in all subjects 
 should be demanded, and that to each subject its own 
 proper principles should be assigned ; that particular 
 as well as general propositions, special problems as 
 well as comprehensive formulae, should be made a part 
 of the trial; so as to ascertain that generalisations are 
 not introduced without a thorough acquaintance with 
 their lower steps, and a power of applying them to 
 the cases which they include. Works in which sim- 
 ple considerations are, by the aid of steady thought, 
 made to lead to important and apparently remote con- 
 clusions, (such as Professor Airy's " Gravitation,") are
 
 STUDY OF MATHEMATICS. 
 
 173 
 
 much to be preferred to works of mere calculation. 
 And, without pretending to decide between the rival 
 claims of geometrical and algebraical modes of reason- 
 ing in cases where either may be used, we should re- 
 quire that the mode which is selected be so presented 
 as to show that the meaning of the expressions em- 
 ployed is distinctly understood by the student. 
 
 It will, perhaps, be objected to the opinions here 
 maintained, that the kind of study which I recom- 
 mend is much more laborious than the practice of 
 symbolical calculations, while it leads only to the same 
 results. That such a kind of study will require and 
 occasion more thinking than the mere application of 
 technical rules, I by no means deny; it is precisely 
 because it does this, that it is a useful instrument of 
 education. I can hardly believe that any one would 
 expect to produce any beneficial effect upon the mind, 
 without urging it to activity and effort. It Avould be 
 a very strange and futile imagination, if Ave were to 
 suppose that there is any branch of science, of which 
 the results, when adopted as a matter of tradition, 
 without being appropriated by any act of thought of 
 our own, can be of any value, either as discipline or 
 as knowledge. A scheme of study which escapes or 
 tries to escape the labour of thinking, will answer 
 none of the purposes at which we ought to aim. 
 
 But whether such a system of mathematical study 
 shall or shall not prevail in this University, must depend 
 entirely upon our Examiners. It is neither surprising nor 
 blameable that our candidates for mathematical honours 
 should be content with general formulae as the solutions of 
 all problems, and general equations as the representatives 
 of all principles, if they find that their Examiners appear 
 to desire nothing more. It is only by demanding a know- 
 ledge of real principles, and by rewarding the power of
 
 174 THOUGHTS ON THE 
 
 applying them, that our mathematics can discharge its 
 proper office in our system of teaching. 
 
 If those who conduct the examinations for university 
 and for college honours were to regulate their conduct by 
 maxims like these ; and if the public opinion of the Uni- 
 versity, which is the guardian of our system of education, 
 were to adopt these maxims, and to make them clearly 
 heard on all occasions, I do not hesitate to say that the 
 effect of our mathematical teaching, as a logical and philo- 
 sophical discipline of the mind, would be greatly elevated, 
 and that the beneficial results would be felt in the im- 
 proved intellectual character of our most active-minded 
 students. 
 
 But it appears to me, further, that it would be neither 
 desirable nor necessary to confine the benefits of such an 
 improvement to those students who are candidates for 
 the mathematical honours of the University. We owe it 
 to the intellectual interests of the country which are com- 
 mitted to us, that we should bestow as complete a mental 
 culture as circumstances allow, on all on whom we confer 
 an academical degree. I believe that the mathematical 
 study to which men are led by our present requisitions 
 has an effect, and a very beneficial effect, on their minds : 
 but I conceive that the benefit of this effect would be 
 greatly increased, if the mathematics thus communicated 
 were such as to dissipate the impression, that mathe- 
 matical reasoning is applicable only to such abstractions 
 as space and number. For this purpose, let a knowledge 
 of some portion of Mechanics and Hydrostatics be intro- 
 duced among the requisites for a degree; and, if neces- 
 sary, let the knowledge of Algebra be required no longer, 
 for I can hardly believe that this part of our mathematical 
 teaching is of much value in any point of view. Very 
 plain and easy systems of Mechanics and Hydrostatics 
 would answer the purpose which I recommend, but they
 
 STUDY OF MATHEMATICS. 175 
 
 must be of an appropriate character. They must not, for 
 instance, be of what is called a popular nature, consisting 
 of assertions without reasoning, and propositions of the 
 most different kind and evidence jumbled together. Such 
 systems as would here be wanted, must contain none but 
 the most rigorous demonstrations, although none but the 
 most simple calculations. Unquestionably such systems 
 are possible. Perhaps at present no treatises exist which 
 would exactly answer the purpose, but I have no doubt 
 that if the recommendation just given were once adopted 
 by the University, the requisite books could easily be 
 produced. 
 
 My object has been to speak of the study of mathe- 
 matics as a logical and philosophical discipline of the 
 mind, in which office I consider that study to hold a highly 
 important place among us. It is on this ground that I 
 now urge the introduction of Mechanics and Hydrostatics 
 into all our examinations for the Bachelor of Arts' degree, 
 looking upon these as the most instructive and philo- 
 sophical, and also the most simple branches of mathe- 
 matics, after Plane Geometry. I do not wish now to 
 enter upon all the other reasons which might, as I con- 
 ceive, be urged in favour of such a measure. The great 
 practical utility of these sciences would weigh with many 
 persons, and their application to a vast variety of ques- 
 tions which come under the notice of every one, and often 
 demand our thoughts even in the course of common life. 
 And a consideration far more worthy to be attended to, 
 in my opinion, is, that these sciences are examples, and 
 excellent examples, of that great system of physical 
 knowledge which has been steadily advancing ever since 
 the revival of learning in Europe, and with the character 
 and nature of which no liberally educated man ought to 
 be unacquainted. It is no small part of the value of the 
 mathematical studies which we place before the candi-
 
 176 THOUGHTS ON THE, ETC. 
 
 dates for the highest honours, that by the cultivation of 
 those, they are brought into an acquaintance with the 
 most recent and profound researches, and thus feel them- 
 selves called upon to sympathize with the struggles and 
 successes, the hopes and anticipations of the great men 
 of their time, whose names and discoveries are to be an 
 inheritance to the latest generations. There is no reason 
 why this advantage should be confined to a few only. 
 The mechanical principles on which the doctrine of uni- 
 versal gravitation depends may be presented in a very 
 simple form ; and even some of their most abstruse and 
 remarkable conclusions may be obtained, as has recently 
 been shown*, with very .little of the apparatus of tech- 
 nical calculation. Whether or not it were thought ad- 
 visable to conduct the great portion of students so far as 
 this, they could not be led, even to enter on the path 
 which leads through Mechanics to the Theory of the 
 Universe, without feeling that they had acquired some 
 view of the nature of that remarkable portion of human 
 knowledge ; and thus, as their classical reading brings 
 them in contact with that which is best in the mental 
 productions of the ancient world, their mathematical 
 studies would place them within sight of the noblest 
 effort of modern intellect. 
 
 That the study of mathematics, conducted according 
 to the maxims which I have thus urged, is a most valuable 
 and essential part of a liberal education, I hope I have 
 convinced the reader. And I have no doubt that the 
 alteration in our examination which I have above recom- 
 mended, is, so far as it is desirable for the purposes here 
 considered, practicable without material inconvenience to 
 any parties. 
 
 TRINITY COLLEGE, 
 October 1, 1835. 
 
 * In Prof. Airy's "Gravitation."
 
 ADDITIONAL THOUGHTS 
 
 ON 
 
 THE STUDY OF MATHEMATICS 
 
 AS A PART OF A LIBERAL EDUCATION- 
 
 THE following remarks occur in the Preface of a 
 work which I have recently published : and as they 
 contain a further developement of the views delivered 
 in the preceding pages, I insert them here. The work 
 in which they were originally published is entitled 
 " The Doctrine of Limits, with its Applications ; namely, 
 Conic Sections, The First Three Sections of Newton's 
 Principia, The Differential Calculus." 
 
 The Treatise on the Doctrine of Limits is intended 
 to contain all those portions of a mathematical course 
 of reading which it appears desirable to recommend to 
 students, as their employment during the second year of 
 their residence in the University; with the exception of 
 the Elements of Mechanics, on which I have published 
 separate works. It will, I think, be found convenient to 
 have this group of subjects thus brought together ; be- 
 cause it may not only save the trouble of reference to 
 many books, but may prevent the omissions and repeti- 
 tions which occur when the system is made up of de- 
 tached fragments; and may secure that connexion and 
 coherence in the reasoning, which are often wanting 
 when a number of different persons undertake to con- 
 struct each his section of the logical chain. 
 85
 
 178 ADDITIONAL THOUGHTS ON THE 
 
 But though these considerations appear to render 
 such a work as the present desirable, they would proba- 
 bly not have been sufficient to induce me to encounter 
 the labour and difficulty of writing it, if I had not thought 
 that it would illustrate and promote the genuine objects 
 of the study of Mathematics as part of a University edu- 
 cation. I have already endeavoured to define those 
 objects, and to extend our means of attaining them, 
 and I will now make a few further remarks on the 
 subject, with a special reference to the Treatise just 
 mentioned. 
 
 All exact knowledge supposes the mind to be able to 
 apply, steadily and clearly, not only the processes of rea- 
 soning, but also certain fundamental ideas ; and it is one 
 main office of a liberal education to fix and develope these 
 ideas. The ideas of Space and of Number are the sub- 
 ject matter of Geometry, of Arithmetic, and of Algebra 
 in its character of Universal Arithmetic: and since all 
 our knowledge, relative to the external world, must be 
 subject to the conditions of space and number, the ele- 
 mentary portions of mathematics just mentioned are, 
 rightly and necessarily, made the basis of all intellectual 
 education. If we advance further in mathematical study, 
 with the view of its thus serving as an intellectual disci- 
 pline, what other ideas do we thus bring into activity 
 and use ? I reply, that the main general ideas which we 
 have next to introduce, and which consequently should 
 be the governing principles of the studies of the second 
 stage of a liberal education, are the following : the me- 
 chanical ideas of Force and Body, with their various 
 modifications; the idea of the Symmetry of symbolical 
 expressions ; the idea of the Universal Interpretation of 
 symbols, including as an important branch of this, the 
 Application of Algebra to Geometry ; and the idea of a 
 Limit, In the present volume, I trust there will be found
 
 STUDY OF MATHEMATICS. 179 
 
 exemplifications and applications of all these ideas, suffi- 
 cient to form the basis of a system of study, such as our 
 object requires; although the consequences of the last of 
 them, the Doctrine of Limits, form the leading .subject 
 of the book. 
 
 It will be easily seen that, according to the course of 
 study hitherto usually prevalent in this University, after 
 the first year's studies are completed, all the ideas just 
 noticed have been brought into use, although they may 
 not always have been contemplated distinctly and se- 
 parately. Mechanics has usually been a prominent por- 
 tion of the studies of the second year. Of the ideas which 
 form the groundwork of this science, and of the mode in 
 which they may be evolved into a scientific system, I 
 have elsewhere given my views*. The idea of Sym- 
 metry is introduced when we consider the Symmetrical 
 Functions of the Roots of Equations, and is thus the 
 basis of the whole Theory of Equations: it is moreover ot 
 extensive use and influence in the application of Algebra 
 to Geometry. The Rules of Interpretation of Symbols 
 are well exemplified by the investigations in which co- 
 ordinates are used ; and the principles of such rules, in 
 their most general form, have of late years been exhi- 
 bited in an able and striking mannert. The Doctrine or 
 Limits has usually been studied in its principles in the 
 First Section of Newton's Principia; and in its appli- 
 cations in the Second and Third Sections, and in the 
 sequel to the student's progress, in the remainder of that 
 immortal work. The idea of a Limit is also, as I have 
 elsewhere shewn:}:, necessarily involved in every applica- 
 tion of the Differential Calculus, whether to curves or to 
 mechanical problems. And thus our course of study in 
 
 See The Mechanical Euclid. 
 
 t See Professor Peacock's Algebra. 
 
 f See above, p. 155.
 
 180 ADDITIONAL THOUGHTS ON THE 
 
 the second year, in this University, has really been an 
 intellectual discipline of a high order; exercising the 
 student's mind to a definite apprehension and ready use 
 of the abstruse and comprehensive ideas of which we 
 have spoken ; and opening to him the large and rich 
 storehouses of knowledge, of which these ideas supply 
 the key. 
 
 In the Treatise on the subject I have somewhat ex- 
 tended the usual domain of the Doctrine of Limits, by 
 including in it all those portions of Conic Sections which 
 depend upon Tangents, and consequently all which 
 concerns Conjugate Diameters. It appears to me that 
 no other satisfactory basis can be found for any part 
 of our knowledge of curves (except so far as our pro- 
 positions only refer to straight lines intercepted by the 
 curves). The Axiom of Archimedes, as it is called, 
 respecting the length of arcs of curves, cannot be ren- 
 dered entirely satisfactory in any other manner, as I 
 have elsewhere urged*. And although the usual nega- 
 tive Definition of a Tangent may be made the foundation 
 of the properties of some figures by reasoning ex ab- 
 surdo ; the direct proof which offers itself when we 
 define the tangent as the Limit of a line cutting in two 
 points, resulting when the two become one, appears to 
 be a more genuine demonstration. 
 
 With this view I found it necessary to separate the 
 properties of Conic Sections into two parts ; the first 
 part (Book i.) being independent of the idea of a limit, 
 (which forms the subject of Book n.) and, besides its 
 being necessary to the proof of the following portions of 
 my subject, serving to illustrate the ideas of Symbolical 
 Symmetry, and Interpretation, and the use of Co-ordi- 
 nates. The other part of the properties of Conic Sec- 
 tions (Book in.) comes after the Doctrine of Limits, on 
 * See above p. 157.
 
 STUDY OF MATHEMATICS. 181 
 
 which it depends; and besides its value in itself, sup. 
 plies those propositions which are used in treating of 
 Central Forces. (Book iv.) 
 
 It thus happens, that my aim in writing a Treatise 
 on Conic Sections has been different from that of most 
 of those mathematicians who have preceded me in such 
 an attempt. Their object has in general been to render 
 their work as homogeneous and symmetrical as possible. 
 They have endeavoured to produce a system which 
 should be rigorously geometrical, with no taint of sym- 
 bolical reasoning; or else one which should be purely 
 analytical : and they have sought to make the demon- 
 strations for the ellipse and hyperbola correspond as 
 exactly as possible. My aim has been to employ such 
 demonstrations as should illustrate some principle or 
 some process of extensive use ; and hence I have sought 
 variety rather than uniformity in the kind of proof. I 
 have demonstrated algebraically the properties of Conic 
 Sections, which do not depend upon tangents; while I 
 have given geometrical proofs of several other proper- 
 ties. I have deduced the property of the rectangles of 
 oblique co-ordinates in the ellipse from the consideration 
 of the circular projection of the ellipse; a mode of con- 
 sidering the ellipse, which it is very useful for the 
 student to be acquainted with ; while in the hyperbola 
 I have deduced the same proposition from the general 
 proportionality of the segments of parallel lines ; a pro- 
 perty which is true of all the Conic Sections. Thus 
 the equality of all the tangential parallelograms, in the 
 ellipse and in the hyperbola, is connected with two quite 
 different sets of properties in the two curves ; although 
 this identity of property results from a clear analogy of 
 definition, which connection, by adopting another line 
 of reasoning, might be brought into view. By the 
 method which I have adopted, it appears to me that
 
 182 ADDITIONAL THOUGHTS ON THE 
 
 the subject of Conic Sections is rendered more com- 
 pendious than it can otherwise be made. 
 
 In Book iv, which contains the application of the 
 Doctrine of Limits to the direct Problem of Central 
 Forces, I have followed almost exactly the steps of 
 Newton in the second and third Sections of the Prin- 
 cipia. It appeared to me that little could be gained by 
 any deviation, in the way of evidence or convenience : 
 and the interest which belongs to these problems as 
 features in the history of the greatest scientific discovery 
 ever made, renders it desirable that the student should 
 see the solutions in the form in which they came from 
 the hand of the great master. I have only modified them 
 so far as to obtain an equation instead of a proportion 
 for the value of the force. I have, however, added to 
 Newton's propositions a few others, which it is inconve- 
 nient to the student not to know. 
 
 The Differential Calculus has, from the time of Da- 
 lembert, most commonly been treated as an application 
 of the Idea of a Limit, as I have treated it in Book v. 
 The attempt made by Lagrange to evade the use of this 
 idea, however ingenious, is quite incapable of being 
 realized, if this Calculus is to be employed in the solu- 
 tion of the problems which give it its meaning and value. 
 The temporary favour which the project found in the 
 eyes of some mathematicians, arose, as I conceive, from 
 the persuasion that mathematical truths are exhibited in 
 their most genuine shape when they are made to depend 
 upon definitions alone ; an opinion of which I hope I 
 made the falsity apparent *. I have endeavoured to avoid 
 the incoherencies and defects which are often introduced 
 in the attempts to develope the fundamental idea into a 
 systematic Calculus. 
 
 * See Remarks, &c. in the Mechanical Euclid.
 
 STUDY OF MATHEMATICS. 183 
 
 I have added a book (Book vi.) on the Integral Cal- 
 culus, which may conveniently be studied to a certain 
 extent in the student's second year of residence. I have 
 given certain processes of integration, but principally as 
 examples of the methods which may be used. I have 
 avoided following out this subject in a complete and 
 systematic manner; a task which would have led me 
 into too wide a field; and also, beyond the stage to 
 which I wish to conduct my readers. 
 
 I have not attempted to avoid coincidences with other 
 books in the processes and proofs which I have given. 
 We who at this time of day write on subjects so often 
 treated of before, are all too dependent on our prede- 
 cessors, whether we know it or not, to have much right 
 to complain of those who borrow from us. In further 
 explanation of such coincidences, I may observe, that a 
 few peculiar points which may have appeared in print 
 before, as the proof of the properties of the ellipse by 
 means of the circular projection, and the Rules for the 
 Reduction of Integrals (Bookvi. Art. 31.) were originally, 
 or at least independently, devised by me, and communi- 
 cated to friends many years ago. 
 
 If I have succeeded in producing a manual which 
 brings together the mathematical studies of the second 
 year in a convenient and satisfactory form, and in such 
 a manner as to facilitate their operation in the way of 
 intellectual discipline, I shall think my labour well be- 
 stowed. It is not likely that I shall hereafter endeavour 
 to carry on to an ulterior point the scheme of mathema- 
 tical education of which the Doctrine of Limits and the 
 Mechanical Euclid may be considered as portions. But 
 I may take the liberty of stating here, in what direction 
 I should prosecute my labours, if I should ever be led to 
 make the attempt. My aim would still be to furnish the 
 student's mind with some additional wealth, from among
 
 184 ADDITIONAL THOUGHTS ON THE 
 
 those comprehensive and fundamental Ideas on which 
 science depends. I should wish, for example, to enable 
 our young men to master two ideas of great importance 
 and interest: that of a Wave, and that of Polarity. The 
 former of these offers itself in the course of the sciences 
 of Hydronamics and Optics : the latter occurs in va- 
 rious forms as the basis of a large portion of Optics, 
 Magnetism, Electricity of all kinds, and Chemistry. 
 My object would be, not to use any complex mathe- 
 matical processes in tracing the consequences of these 
 ideas ; but to apply them in such cases, as without much 
 mathematical reasoning might tend to make them fixed 
 and clear in the reader's mind. It would be easy to 
 shew how far this has been from a common case. The 
 general notion of a wave, as a transfer of form distinct 
 from a transfer of matter, has been a stumbling block 
 to most unmathematical reasoners; and the confusion 
 and perplexity which have arisen from the attempts 
 to grapple with this notion, have appeared on various 
 occasions. Now this notion is the basis of the Undu- 
 latory Theory of Light, as well as of all reasonings 
 concerning the waves of water; and it is capable of 
 being illustrated in a very elementary manner, to those 
 who will give the subject a steady attention. Again ? 
 the notion of Polarity has been so far obscure among 
 men in general, that many, and even mathematicians 
 of note, have asserted that the phrase " Polarization of 
 Light," was an arbitrary and unmeaning term. Yet it 
 will be seen by a little consideration, that this is the 
 only just phrase for the case; since Polarity, or Po- 
 larization, is the universal mode of describing an op- 
 position of properties depending upon an opposition of 
 positions. The same idea, variously modified, might be 
 illustrated by the fundamental doctrines of Magnetism, 
 Galvanism, and the like ; and when the student has
 
 STUDY OF MATHEMATICS. 185 
 
 gone through such elementary studies, he would be 
 far better prepared, than without them he can be, to 
 derive intellectual profit from the beautiful analysis 
 which is applicable to such cases*. 
 
 By extending our mathematical studies in this man- 
 ner, always insisting upon the most rigorous synthetical 
 exposition of the elementary principles of each added 
 portion, our mathematical education might become an 
 intellectual discipline, such as the world has never yet 
 seen, and such as it has not even learnt to hope for. 
 Yet such an education alone can provide for the future 
 progress and diffusion of real knowledge, and for the 
 prosperity and advance of that civilization which de- 
 pends upon the diffusion of knowledge. It has been, 
 and is, my ambition to contribute my efforts to the 
 object of securing to this University a large share in 
 the merit and glory of preserving, improving, and 
 rendering effectual, an education thus truly liberal. 
 
 * See Mr Airy's Tracts, for the Undulatory Theory ; and 
 Mr Murphy on The Principles of Heat and Electricity, for the 
 other subjects.
 
 EDITOR OF THE EDINBURGH REVIEW. 
 
 UNITED UNIVERSITY CLUB, 
 January 23, 1836. 
 
 MY DEAR SIR, 
 
 I WAS gratified to find that a little 
 pamphlet which I recently published as " Thoughts on 
 the Study of Mathematics," had excited so much notice 
 as to give it a place at the head of an article in the 
 Edinburgh Review; and in regard to the manner in 
 which the Reviewer has spoken of me, I have certainly 
 no reason to be dissatisfied ; nor am I at all disposed to 
 complain of the way in which he has urged his own 
 opinions. But I think the article is likely to give 
 rise to a misapprehension which ought to be cor- 
 rected, and for that purpose I trouble you with this 
 letter. 
 
 I wrote my pamphlet in order to enforce certain 
 views respecting the conduct of our mathematical ex- 
 aminations at Cambridge. The question on which I 
 threw out a few " Thoughts" was, what kind of mathe- 
 matics is most beneficial as a part of a liberal education. 
 That this was the question to which I was trying to give 
 some answer I stated in a passage (quoted by the Re- 
 viewer) at page 8 of the pamphlet. The previous seven
 
 LETTER TO THE EDITOR. 187 
 
 pages, in which, among other matter, I had said a tew 
 words on the question, whether mathematics in general 
 or logic is the better mental discipline, were obviously 
 only an introduction to the discussion of certain proposi- 
 tions which, as the Reviewer observes, "occupy the 
 remainder of the pamphlet." 
 
 It was therefore with no slight surprise that I looked 
 at the magnificent manner in which the Reviewer has 
 spoken of the small portion of these seven small pages 
 which refers to the more general question. He calls it 
 " a treatise, (a Treatise /) apparently on the very point" 
 (p. 410), "a vindication of mathematical study" (p. 411); 
 and having thus made me work at a task of his own 
 devising, he repeatedly expresses great disappointment 
 that I have executed it so ill; that, "so little is said on 
 the general argument." I should have thought that this 
 circumstance might have helped him to perceive that it 
 was not my general argument. 
 
 I see nothing but the convenient and blameless 
 practice of reviews in making the title of my book the 
 occasion of publishing an Essay on a subject only 
 slightly connected with mine; but it appears to me 
 that to attempt to gain a victory by representing a page 
 or two of my " Thoughts" as containing all that can be 
 said by an able, earnest, official, advocate on the other 
 side, is not a reasonable treatment of the question. The 
 writer proclaims that he means to give "no quarter to 
 my reasonings," but this proceeding looks rather like 
 making an unexpected attack on a point when he thinks 
 himself well prepared, on the arbitrary pretext that the 
 truce has been broken by the adversary. 
 
 I should have no disinclination, on a convenient 
 occasion, to discuss the very important and interesting 
 question which is the subject of the Review. I cannot, 
 however, look forward with confidence to the prospect of
 
 188 LETTER TO THE EDITOR OF 
 
 my being able to take it up for a considerable period, 
 and shall probably leave the Reviewer in possession of 
 his self-chosen field of battle for several months, it may 
 be years. But if I should return to the subject, I should 
 wish to know as definitely as is possible what are the 
 questions at issue between us; and I would therefore 
 beg from the Reviewer information on the following 
 points. 
 
 The Works which form our examples of Mathema- 
 tical reasoning are well known: I wish to know also 
 what works of "Practical Logic" on other subjects 
 (p. 413) the Reviewer is willing to propose as rival 
 instruments of education. 
 
 I wish to have some distinct account of the nature of 
 that "Philosophy" which is by the Reviewer put in 
 contrast to Mathematical study ; (p. 422,) and if possible 
 to have some work or works pointed out, in which 
 this philosophy is supposed to be presented in such 
 a way as to make it fit to be a cardinal point of 
 education. 
 
 I may remark also, that all the Reviewer's arguments, 
 and, I believe, the judgments of all his " cloud of wit- 
 nesses," are founded upon the nature and processes of 
 pure mathematics only ; on a consideration of the study 
 of the mere properties of space and number. My sug- 
 gestion of the means of increasing the utility of mathe- 
 matical studies was directed mainly to this point; that 
 we should avoid confining ourselves to pure mathe- 
 matics; that we should resort to departments in which 
 we have to deal with other grounds of necessary truth, 
 as well as the intuitions of space and time : so far, there- 
 fore, the Reviewer and I have a common aim, and I 
 notice this with the more pleasure, since we have so far 
 a better prospect of understanding each other in any 
 future discussion.
 
 THE EDINBURGH REVIEW. 189 
 
 I will not now trespass further on your patience. In 
 order to remind my Cambridge readers of the state of 
 the question, I shall probably place before them some- 
 thing to the same effect as what I have now written. 
 
 Believe me. my dear Sir, 
 
 Yours very faithfully, 
 
 W. WHEWELL.
 
 A 000 1 62 998 9 
 
 LIBRARY USE 
 ONLY