The philosophy of education; or, The principles and practice of teaching

Front matter

Book__

Educational Foundations

A Monthly ^ejci-'BooK of Pedagogy and General Culture for Teachers, Normal Schools, Training Classes, Reading Circles and Home Study 000

Preface To The American Edition

The growing desire for treatises on education that is manifesting itself is a most encouraging feature. It indicates real progress. Tate's Philosophy of Education has long been held in high esteem by thoughtful teachers; and though published nearly thirty years ago no more lucid and compact a statement of principles has been made, and it well deserves to be reprinted. It was in this volume those educational precepts were found that in these modern days are so. famikr to the teacher : "From the Known to the Unknown;" " From the Simple to the Complex;" " From the Concrete to l^he'iV^'stract," etc.

The work displays a vast amount of pedagogical knowledge, and gives proof of the prolonged and careful study of the subject by the author. It cannot but aid the practical teacher, for a marked feature of the present stage of progress in education is the desire for the principles that govern the art of teaching.

That portion of the book devoted to psychology is the least attractive, though it fairly represents the ideas that held sway during the first half of this century— the period that Tate represents. Dissent has been made where it was deemed needful. The notes which have been added arc not intended as criticisms, but to in<Ucate here and there the agreement or dis-

IV PREFACE TO THE AMERICAN EDITION.

[Agreement of the statements of the volume with the prevailing views on psychology and pedagogy.

Parts III., IV., and V. contain suggestions that were valuable thirty years ago, and might very properly have been omitted; but in order to make the volume complete they are reproduced, but in small type.

The republication of Tate's Philosophy of Education by Messrs. E. L. Kellogg & Co. is one of many efforts they are making to elevate the rank and dignity and usefulness of the teacher's work, and deserves cordial appreciation.

Edward E. Sheib, President Louisiana State Normal School,

Baltimore, Aug. 12, 1885.

Preface

This work is the result of the labor and reflection of many years ; it, in fact, embodies the experience of my life as a practical educator. It contains an exposition of all the leading principles upon which my other works on education have been written ; and in order to understand fully the drift and purpose of the one, the teacher must study the expositions and principles of the other. Wherever I have adopted the ideas of others, I have always, to the best of my recollection, made a due acknowledgment of the obligation,

I am not acquainted with any work which really treats of the philosophy of education in connection with the practice of it. Our books on education are either too purely speculative or too exclusively empirical, and, so called, practical.

My most earnest desire is that this work may be the means of directing the attention of the practical educator to the philosophy of education, and to the development of those systems and methods which are best calculated to establish in our schools a thoroughly sound and enlightened education.

T. Tate.

CHAPTER II. Importance of Method

Method in Education— in Art— in Science— in Poetry— in Oratory— in Nature. A distinguished Teacher must have Method. 25-28.

A Glance at the History of Method.

Socrates— Euclid— Bacon— Newton— Archimedes. Primary Education- Locke — Rousseau — Pestalozzi — Lancaster — Bell — Government Scheme of Education. 28-32.

Present Condition and Future Progress of Education. Necessity of further Progress. Educators divided into two Classes. The Baconian Philosophy considered in relation to the Progress of Modern Education. 32-40.

Philosophy of Method.

Education based on an Induction of Facts. The Principles of Method coneidei-ed subjectively as well as objectively. Difficulties peculiar yiji CONTENTS.

to the Inquiry. Importance of Definitions. Collection of Facts. To distinguish between Facts and Opinions. Comparison and Classification of Facts. Relation of Cause and Effect. General Principles. Evils of implicit Confidence in Method. Experiments required to test Systems and Methods. To estimate the Results of Method. 40-58. '

Chapter III

to asce'^^tain the nature of the being to be educated. general

Facts relating to the Development of the

Intellectual Faculties 59-62.

Primttive Intelligence as shown in Perception and Intuition, considered AS the Basis of Development. 62-68.

ScnsiUi )n, Reflection, and Intuition. The infant Soul contains implicitly all the Faculties of the developed Intelligence.

Classification of the Faculties of the Mind.

Four distinct Stages of Development. Classification of the Faculties of the Mind as a whole. Classification of the Intellectual Faculties. Explanatory Remarks. First Stage— the Perceptive Faculties.— Second Stage— the Conceptive or Representative Faculties. — Third Stage— the Cognitive Faculties.— Fourth Stage— the Cogitative Faculties. 68-79.

Essential Points to be considered in relation to Method as APPLIED to Education.

1. Nature of the Faculties. (1) The peculiar Function of each Faculty.

(3) Mutual Relation of the Faculties— Relation of Succession— Relation of Assimilation— Relation of Aggregation. (.3) The Faculties considered with respect to their simultaneous Action and Cultivation.

2. The Subjects best adapted for the Cultivation of the different Facul-

ties.

3. Nature of Motives acting on each Class of Faculties.

4. The Habits of Action to be established in relation to each Class of

Faculties.

5. The Methods of Instruction adapted to each Class of Faculties.

6. Application of Results to the different Periods of Education ; Five

Educational Periods— Infancy— Early Childhood— Childhood— Early Youth— Youth. 79-91.

CONTENTS. ix

CHAPTER IV. General Principles of Teaching, or Elements of Method

We should follow out the Intentions of Nature. Piinciple of Utility and Development. Principle of Harmonious Development. Instruction should be progressive, i' Principle of Self development. We should appeal to the Senses. The Reasoning Faculties should be cultivated on an enlarged Basis. Teaching from the Simple to the Complex. Facts taught before Causes, etc. The Concrete before the Abslraet. Constructive Teaching. Principles before Rules. Oral and Collective Teaching— Principles of School Classification. Instruction should give Pleasure— to secure the Attention— the Principle of School Routines— First or Preliminary Lessons— The Infant-School System— Imposition of Tasks— School Discipline. Thorough Teaching—Reproduction of Lessons— Examples and Applications— Reiterations of Lessons. Cultivation of Habits. 91-141. » ,,

CHAPTER I. Preliminary Notions

Importance of Psychological Analysis in relation to Teaching. A Glancg at our Childhood and Early Youth. A Cursory View of our Intellectual and Moral Faculties, as regards their Mode of Development. 143-153.

CHAPTER II. Cultivation of the Intellectual Faculties— Cultivation of the

Perceptive Faculties and of the Faculties of Primitive

Judgment, Conception, Imitation, Abstraction,

AND Language.

The Senses. Knowledge derived from Experience. The Cultivation of the Senses necessarily includes the Cultivation of the Perceptive Faculties. Certain Properties detected by different Senses. Children should express in Language the Results of their Observations and Judgments. The Conceptive Faculties should be cultivated v;ith the Perceptive Faculties. Notes of a Lesson for cultivating thg

CONTENTS.

Conceptive Faculties. Definition of Terms— of Form, etc., how siven. Children should write their Ideas in their own Language. 152-161.

CHAPTER ni.

Cultivation of the Intellectual Faculties, continued— Cultivation OF the Faculty of Attention.

Importance of the Habit. Attention should be voluntary. Suggestive Teaching. Causes which tend to destroy the Habit. Fresh Motives, etc. Mode of treating Boys of different Tempers, Tastes, and Talents—the Feeble— the Sluggish— the Volatile— the Timid— the Quick. A Digression on Thought, Language, and Genius. 161-176.

Chapter IV

Cultivation op the Intellectual Faculties, continued— Cultivation OF Memory and Recollection.

Memory influenced by Attention, Habits, and Associations. Philosophical Associations. Rules for the Cultivation of Memory, applied to various Subjects of Instruction, in the Course of which the Method of Contrast and Comparison, and that of picturing out Scenes, are fully explained. 177-209.

Chapter V

Cultivation op the Intellectual Faculties, continued— Imagination and Taste.

Imagination dependent on Culture. The Picture Style of Teaching. The Imagination cultivated by Poetry, Fables, and Tales. The Sentiment of the Beautiful cultivated by Drawing and Music. 209-215.

Chapter VI

Cultivation op the Intellectual Faculties, continued— Reason and Judgment; Wit and. Invention.

General Principles for the cultivation of the Reasoning Powers. Relations of Things and Events, viewed in six Distinct Aspects. How Processes of Reasoning should be analyzed. Sources of False Reasoning pointed out. Rules for the Conduct of the Under.standiiig. How to foster the Development of the Inventive Powers. 215-237.

CONTENTS. Xi

CHAPTER Vn.

Cultivation op the Moral Faculties.

General Principles. Moral Training based on Religion. The Sentiments of Veneration and Faith. The Benevolent Affections. Habits of Action. Influence of Example. The three Cardinal School Virtues: Truthfulness— Honesty— Humility. Classification of Subjects in relation to the Cultivation of the Intellectual and Moral Faculties. 238-254.

Part Iii.— On The Comparative Advantages Of Different Methods And Systems

OF INSTRUCTION.

Systems of Instruction.

The Individual and Collective Systems. Home Education. The Pupil- Teacher System. The Mixed System. 255-260.

Methods op Instruction.

Synthetic and Analytic Methods. Examples of the Synthetic and A.nalytic Methods of Teaching. Interrogative or Catechetical Method. Principles and Rules common to the two Forms of Interrogation—Special Rules for Examination Questions— Special Principles and Rules relative to Suggestive Interrogation. Examples of good and bad Examination Questions.— Examples of Suggestive Interrogations. — The Simultaneous Method— Examples of Simultaneous Teaching after the Catechetical Method— The Elliptical Form of Teaching— Examples. The Constructive Method. The Illustrative Method. The Lecturing Method. Mixed Method. On the Repx'Oduction of Lessons in Writing. On certain Plans or Artifices for economizing Time, etc.— An Examination Lesson on Spelling— An Examination Lesson on Arithmetic. Respective Advantages of the three Great Methods of Examination. On the Preparation of Lessons—Notes of a Lesson. On the Periodical Examination of Classes and the Registration of Progress. On the Qualifications of the Schoolmaster in relation to his Professional Duties— The Teacher's Attainments— The Teacher's Capabilities and Character— Aptitude for Teaching. On School Registers for recording the Results of Different Methods of Instruction, and also for testing the Capabilities of Teachers in relation to these Methods. General Conclusions derived from the Writer's Registration of the Results of Methods, etc. 260-287.

Xii CONTENTS.

Part Iv.— On The Application Of Different Systems And Methods To The Various

BRANCHES OF ELEMENTARY EDUCATION.

The Scriptures; History; etc.

Reading and Spelling; Etymology; Grammar. Specimen of a Reading Lesson. The Look and Say Plan— The Phonic Plan. On teaching the Alphabet, etc. Grammai- more fully considered—Lessons on Grammar— Lessons on Composition and the Analysis of Sentences. 288-;^95.

Arithmetic.

Lessons on the Addition of Fractions. Lesson on Rule of Three. Mental Arithmetic. 295-297.

Geography, 297, 298.

Drawing.

General Principles and Rules. Model Drawilig -Dupuis' System. A simple Piece of Apparatus for Teaching the Principles of Perspective. Model Lessons on Drawing. 298-309.

Practical Geometry and Mensuration.

Drawing Instruments, etc. Lessons on Geometry — Observations relative to Familiar Modes of Exposition. 310-312.

Algebra. A Lesson on Equations. 312-314.

Mechanical and Physical Science. A Lesson on Chemistry. 314, 315.

Natural History. 315-317.

PART v.— ON SCHOOL ORGANIZATION AND DIS- CIPLINE

School Buildings and Fittings. School Apparatus.— List of Apparatus for General Use— Specimen Forms of School Registers— Specimen of a Time-Table and Routine of Lessons. 318-325.

Classification. The Pupil- Teachers. 326. 327.

School Discipline, Order, etc. 327-33-

THE

OR,

THE PRINCIPLES AND PRACTICE OF TEACHING.

Introduction

As man is not only a phj'sical, but also a thinking and an accountable being, so therefore education, in its comprehensive sense, may be viewed in three aspects — that is, in relation to our physical, intellectual, and moral nature. I here propose to consider the last two departments of education ; to determine, if possible, the best methods whereby our nature may be educated intellectually and morally. The end of all educatioa should be, to promote man's happiness, not only during his present transitory existence, but throughout the eternity which is to follow.

The principal means of education in this country are — school instruction, books, public lee lures and discourses, and exhibitions of works of science and art. But the efficiency of all the popular means of education are dependent upon, and in fact inseparably connected with, the primary instruction of the schoolroom. The treasures of our literature and science are inaccessible to him who has not been taught the first rudiments of language. Hence it is, that the brilliant productions of the poetic genius, or the gigantic creations of the science of any particular age, afford us no data for estimating the state of education among the mass of the people of that age.'^

The schoolmaster must begin the work of education. The subject of method, therefore, should be treated chiefly in relation to the work of the schoolmaster.

Education is a Science as Well as an Art. — Practical teachers, as well as the public generally, had, until recently, regarded education more as an art than as a science, consisting merely of a few arbitrary and empirical rules which may be modified or altered to suit the tastes and attainments of the teacher, or to answer the opinions and circumstances of the managers of schools. This unfortunate prejudice has, no doubt, had its origin, to a great extent, in the fact that the greater portion of teachers were unfit for their office. Few minds were capable of viewing education apart from its miserable and unworthy representatives, or dissociating it from the operation of the schools which came within the sphere of their own immediate observation.

Twenty years ago, anybody was considered good enough for a schoolmaster. If a tradesman failed in business, he was thought to be learned enough for a schoolmaster; a feeble, sickly youth, who was not considered strong enough to practise any regular trade, was thought to be sufficiently qualified to undertake the duties of school-keeping ; if a mechanic happened to get a limb fractured he would, as a matter of course, save himself from starvation by opening a school ; when a man who had seen better days applied to the parish officers for out-door relief, they gravely debated the question, whether it was more

* " Long after the brilliant show of talent, and the creation of literary supplies for the national use, in the early part of the last century, the deplorable mental condition of the people remained in no very great degree altered. To pass from beholding that bright and sumptuous display in order to see what there was corresponding to it in the subsequent state of the popular cultivation, is like going out from some magnificent apartment, with its lustres, music, refections, and assemblage of elegant personages, to be beset by beggars in the gloom and cold of a winter night."

EDUCATION AS A SCIENCE, ETC. 15

expedient to send him to the quarry to break stones, or to confer upon him the office of parish schoolmaster. Such was the low estimate formed of the qualifications requisite for a schoolmaster. This state of things, doubtless, tended to retard the progress of education both as a science and an art, for the odium attached to the office, as well as the insufficiency of the remuneration, prevented properly qualified persons from undertaking the duties. But within the last fifteen years a change in public opinion has been gradually taking place: the working and middle classes have been led to see the value of a sound -elementary education, and thereby to estimate more highly the difficulties and importance of the duties of the common school master. This salutary change is in a great measure due to the government schemes of education. I confidently hope that the day is not distant when the force of public opinion will elevate education into the rank of a recognized science.

Elementary education has two great ends: 1. To develop the intellectual and moral faculties ; or, in other words, to develop the faculties of the perfect man ; 2. To communicate to the pupil that sort of knowledge which is most likely to be useful to him in the sphere of life which Providence has assigned him.*

The science of education must be based upon the nature of the being to be educated ; that is to say, upon the laws which govern the development of the intellectual and moral faculties. These laws may be determined as well by observation as by psychological analysis. Every faculty of our nature has its proper period and peculiar mode of development.

Now the philosophical educator will always suit his methods of instruction to the age of his pupils, or rather to the state of the intellectual and moral development of the faculties of his pupils ; and;' he will also administer to them that intellectual aliment, both as to kind and degree, which is best calculated to promote the growth of the faculties at their different stages of

* The object of education must remain imperfectly defined so long as there is not a clearly expressed intention of making the future man or woman a moral power by conferring true worth upon the individual -E. S.

IQ PHILOSOPHY OF EDUCATION.

development.* Method, and the principles of method, therefore, necessarily become to him distinct and all-important matters of iuquir3^

A good teacher, before laying down any plans for the management of his school, makes himself acquainted with the tempers, habits, capabilities, and attainments of his pupils. He then asks himself the two great questions. What shall I teach? How shall I teach? He is well aware that these questions cannot be satisfactorily answered without a thorough knowledge of the nature of the beings whom he has to teach, as well as a comprehensive acquaintance with the various methods whereby instruction may be communicated.

All artificial and unnatural methods of insti-uction, violating the laws of mind, necessarily demand the use of unhealthful stimulants. There is always a want of organization in schools where the plans and methods of the master are framed without any regard to the constitution of the human mind, or the peculiar tempers, tastes, and capabilities of the pupils : such masters always blame their pupils for the failures of their system, but never seem to be aware that the excellence of a system depends upon its adaptation to the intellectual and moral condition of these pupils.

A teacher, who is ignorant of human nature, is like an engineer who sets to work to erect a bridge before he has made himself acquainted with the properties of the material employed in the structure ; when his work is completed, he finds, perhaps, that the material is ruptured by the pressure, or by the expansion due to heat ; it is true, he might console himself with the reflection that his plan would have been excellent if it had not been for the peculiar properties of the material. A wise engineer would first make himself acquainted with the nature

* It is a leading idea of the Herbart School of Philosophy that the development of the individual proceeds in a line parallel to that progress observable in the history of the development of the human race ; that theiefoie the material for instruction must in each instance be boi^ rowed from such stages in the development of the race as corre^pond with the existing mental conditions of the cTiild.— E. S.

EDUCATION AS A SCIENCE, ETC. 17

and properties of his material, and then, knowing the difficulties which he would have to encounter, he would provide against them accordingly.

The teacher who is thoroughly acquainted with the laws regulating the juvenile mind suits his methods of instruction to the soul which he has to rear, and, fully foreseeing the difficulties which he has to encounter, lays his plans accordingly : he is quite prepared to supply strength to what may be weak, and to introduce a self- corrective agency to meet any ebullitions of temper or waywardness of disposition.

Our ignorance of mental philosophy has hitherto led us into various erroneous methods and systems of education.

The teacher shows an ignorance of the tastes and capabilities of the infant mind when he overtasks his juvenile pupils with the dull, dry details of technical learning, in the place of communicating to them that kind of knowledge which is best calculated to foster the development of their perceptive and observing faculties.

Teachers at one time believed that the first object of primary instruction is to cultivate the verbal memory of their pupils, when, in fact, the verbal memory is one of the few faculties of our nature which need no cultivation. This erroneous opinion led to the adoption of the task system : little boys had to commit to memory frightful columns of spelling, long paragraphs of geography, abstiact grammatical definitions, declensions of nouns, and conjugations of verbs. The debasing system of rewards and punishments formed a necessary adjunct to this unnatural system of instruction. In this system the cultivation of the reasoning powers was entirely disregarded, and the aids of philosophical memory, or the faculty of association, were never called in requisition.

The same eiToneous opinion of human nature led to the adoption of the rule and rote system of instruction, whereby the pupil had to work out results by formulas and dogmas rather than by the independent and healthful exercise of his own reasoning powers. For example, in the teaching of arithmetic 2 and practical geometry, the pupil was required to work out his problems by a rule appealing to his memory and simple apprehension, rather than by the exercise of his own reasoning powers.

These unnatural methods of instruction gave rise to debasing systems of discipline. Under a proper system of teaching, children rarely require any other motive to attention than the pleasure which the acquisition of knowledge affords them ; but what natural motive can induce a child to study what is above his capacity, or to commit to memory what he cannot comprehend ? hence the teacher's only resource was to act upon the vanity or the fear of his pupils.

The art of education consists in the practice of its principles. It stands in the same relation to the science of education that any other art does to the scientific principles of that art.

A man may be thoroughly acquainted with the principles of any particular art, without being an adept in the practice of it ; in order to have this, he must practise the art until he has acquired the requisite amount of tact and skill. At the same time, it must be observed that the highest amount of skill can only be obtained by a thorough knowledge of the principles of the art, combined with the constant application of these principles. Thus, for example, a man may be thoroughly acquainted with the principles of architectural construction, and yet he may not be able to frame a door or to build a shed.

A man may be intimately acquainted with all the leading principles of education, and yet at the same time he may not be able to give efficient instruction to a Class of little boys. It is a lamentable error to suppose that if a man has knowledge he must necessarily possess the art of communicating that knowledge. In order that a man may become a good teacher, he must not only be thoroughly acquainted with the various branches of elementary education, and intimately acquainted with the great leading scientific principles of education, but he must also acquire that tact and skill in the management of numbers and classes, and that fluency of diction, power of illustration, and facility of availing himself of contingent cir-

DIVISION OF THE SUBJECT. 19

cumstances, which can only be attained by long practice and patient study.

The art of education, without a due regard to its science, degenerates into empiricism ; and the science, without the practice of the art, becomes little better than a code of barren abstractions without the vital principle of development. The philosophy of education should go hand in hand with the practice of it : every step of advance taken by the one should be followed by a corresponding progress of the other ; philosophy should suggest plans and theories, art should test them and try them ; philosophy should build up a structure of general principles and rules, art should supply the facts— the materials — by which, and upon which, this structure should be reared.

Division op the Subject.

The philosophy of education may be divided into five parts :

1. On method as applied to education.

2. On the cultivation of the intellectual and moral faculties.

3. On the comparative advantages of different systems and methods of education.

4. On the application of different systems and methods to the various branches of elementary education.

5. On school organization and discipline.

Chapter I. Methods And Systems Of Instruction — Definition Of Terms

Different Methods and Systems of Education. — By a METHOD of Education is meant the peculiar way in which a subject is taught ; and by a system is meant those peculiar arrangements, both as to organization and modes of teaching, whereby instruction generally may be given to the pupils of a school. A system is the development of a method applied to certain objects.

* It would seem that previous to all consideration of method the object and the aims of education should be clearly presented. For the method must depend, 1, upon the object we have in view ; 2, upon the possibility of attaining that object ; 3, upon the nature of the subject that is to be educated. Nations, religions, and ages have differed respecting the object of education. It is also important that the possibility of moulding the child's mind and character should be demonstrated. The limits of this pliability should also be pointed out, and the fact that with the advancing years the child's nature assumes more firmness and as it approaches maturity opposes further attempts to direct and mould its mind. Finally, the selection of a method must be preceded by a consideration of man's moral and psjchical nature, as it must conform to the laws governing this mental and moral development. Tate speaks of this time and again, but in connection with other thoughts ; so that the subject does not assume the important place which in truth belongs to it.— E. S.

SYNTHESIS AND ANALYSIS. 21

A difference of opinion at present exists relative to tbe use of the term "method" as applied to education. According to some v/riters, method simply means the way in which a subject of aistruction may be treated ; so that there are only two methods of education, namely, Synthesis and Analysis. Such a restrictive use of the term is not only based on a contracted view of the subject, but it does not give the entire conception usually associated with the term. We use the term in a more comprehensive sense : A method of teaching comprehend?, not merely the wfiy in which the subject-matter is treated, but also the means, artifices, forms of expression, etc., that are employed in conveying instruction to a class of children in a common school.

There are two great methods whereby a subject may be treated, viz., Syis thesis and Analysis. By the former method vv'e put the parts of a subject together ; by the latter we take the subject-matter to pieces. The method of synthesis is the method of inducticTn, whereby we ascend step by step from the simple to the complex — from the particular to the general formula ; the method of analysis is the method of deduction, whereby we descend from the abstract principle to the various particular forms which it comprehends.

As both methods are employed in the discovery of tiTith, so both methods may be used in the exjiosition of truth. The expcrijnentalist may show the composition of water synthetically by holding a tumbler over the flame of a candle (or a flame of hydrogen gas), at the same time calling attention to tlie moisture that is formed on the interior surface of the glass; or, more exactly, by detonating, by means of the electric spark, the proper mixture of hydrogen and oxygen: in these experiments water is formed by the combination of its elements. He may also show the composition of water analytically by means of the galvanic battery; in this case the poles of the battery analyze or decompose the water, that is, reduce it to its simple elements, the hydrogen being attracted by the one pole, and the oxygen by the other. We teach arithmetic deductively, or analyticalh% when we lay down a general rule and require our pupils to work out the particular example by that rule, for in this case we proceed from the general formula to the particular example — from the abstract principle to its special application. On the contrary, we teach arithmetic inductively, or sjmthetically, when we proceed at once to work out, step by step', the particular example from certain simple, known elementary principles, without taking any abstract rule for granted: in this case the pupils are led to prove the rule for themselves.

The method of synthesis is constructive; by this method the skilful teacher builds up thought upon thought — truth upon truth— until his pupils have, almost insensibly, acquired a vast accumulation of knowledge. I have called the method of synthesis a constructive method, because it is analogous to the way in which mechanical contrivances are completed: thus, the ingenious builder lays stone upon stone, beam upon beam, until he has reared a vast and beautiful structure, exciting, it may be, the wonder and admiration of the world: in this way, too, surprising results may be attained in education. Synthesis may be called a suggestive method of instruction; because it is progressive, proceeding, step by step, from the known to the unknown, — from the simple to the complex.

By far the larger number of the great physical laws of nature were discovered by induction, and even many of our leading mathematical theorems and principles were established by the same process. Now if it be true (and we have reason to believe that it is true) that the method of exposition should correspond to the method of discovery, it follows that the method of induction or synthesis is, for the most part, the more eligible for primary instruction. At the same time, it must be observed, that there are certain subjects of knowledge which may be efficiently taught by the method of analysis. But this subject will hereafter receive a more adequate consideration.