REMARKS ELEMENTARY EDUCATION IN SCIENCE. AN INTRODUCTORY LECTURE, DELIVERED AT Till: ol'l:\l\. Of Tilt CLASSES OF MATHEMATICS, PHYSICS, AND ClIKMISTIt V. l>f TIIK UNIVERSITY OF LONDON, NOVEMBER , I BY AUGUSTUS DE MORGAN, PROFLSSOR Of MAT I- I MAT1CI l\ I IN- tM VI. KM TV )| I ONDON. LONDON: PHIS I I I) i:V THOMAS DAV19OW, I (HI !OH\ TAYLOR, ."(, riTKK <;<>\VKK S'J'KKKT, BOOKSELLER AM> IT It 1. 1 -ill \< I it MI i. r M v i; n-> 1 1 N . 1830. p\t- 4i KK MARKS ON ELEMENTARY EDUCATION IN SCIENCE. Two years have now elapsed since the first lecture was delivered within the walls of this institution two years, which, independently of the rise of an University in the metropolis have given hirth to events \vhieh will render tin-in ivmurkable in the history of education. Already we' see similar establishments in process of completion, and may onallv prophesy that no long time will clap.se before iv considerable town in thi> country will have taken measures to bring home to its inhahitants those facilities for the BOqili] i '1'it lit of the higher hranrhes of knowledge, which ha\e hitherto falUn to the lot of capital cities only. But even this i> not the most singular feature of the times. The .spirit of improvement has ended where she should have begun, by laying her hand upon the systems of preparatory education. !i the notion that all useful knowledge was to be acquired from the study of the learned lan;;-u,-i has been some time extinct, it is only now that it begins to be admitted and acted upon, that habits of inquiry and of rect judgment are more useful than any knowledge of facts, however extensive, and that literature and science are to the young mind, not the end of its education, so much as tin- IIK,!!!> of promoting the above-mentioned objects. 4 avoid making continually. They form part of the occupa- tion of every child from the time when the development of its ideas commences. They are readily made ; their results are easily retained; and so simple are the last in their nature, that it would have been thought childish to put them into words, had it not become necessary to embody them as the principles and groundwork of the mathematical sciences. The fact is that the terms observation and ex- periment are closely connected. The first is the faculty of the mind by which the second is conducted, and all the results of observation are so many experiments, whether they are derived by the senses alone, or by the assistance of the telescope or the thermometer. For example, we learn to distinguish by the sight and touch that which we call a straight line. If any proof were necessary that our ideas on this subject are derived from observation alone, it would be found in the impossibility of defining this term. If any one were to ask me what is meant by the word, I could only answer by referring him to some visible object. By the evidence of our senses, and by no other means, we find that a straight line is such that two of this species can only coincide or meet in one point. The moment they are made to coincide or meet in two points, they become one and the same straight line. Again, sup- pose a straight line drawn and a point taken without it. Through this point let any other straight lines be drawn in all directions. We can demonstrate that there is one of these straight lines which will never meet the first line, though ever so far produced or lengthened in either direction. But it is to experiment or observation that we are indebted for the knowledge of the fact, that there is ONLY one straight line which is in this predicament, since no demon- stration of this assertion has ever been given. These propositions are the groundwork of geometry, and may fairly be said to be derived in the same manner as the results obtained from the most complicated apparatus ; the difference being, that they are observations which we all must have made so often and so early, that we forget the time when we first gathered information from them. I do not know whether I shall not be considered as pushing tliis assertion to the verge of paradox when I maintain that our first ideas on the subject of number spring from the same source. The notions of one, or of more than one. are not only obtained from visible and tangible objects, but all the first questions of arithmetic are performed by their assixtancc. So little do the operations of the mind, unassixU'd by daily experience and repetition of numerous objects, assist in giving extended ideas of numeration, that tho>e nations whose social wants do not demand large num- bers, never acquire the ideas of them. We are acquainted with >e\cral tribes who cannot reckon as far as one hundred, and one i- mentioned whirh has no names for numbers be- yond five. Hut \\liate\cr maybe the opinions upon this subject. thi> much is certain, that children do constantly resort to the u>e of some tangible objects, that is, to experi- nent, to acquire the first notions of number, and to work the fir f arithmetic. i regard to Natural Philosophy and Chemistry, I need hardly say, that from experiment, and experiment only, our first knowledge of these sciences is derived. Far i there being any tendency to mistake in this point, the costliness and unusual nature of the apparatus employed, the time and skill required for many of the processes, and, above all, the interesting and popular nature of the K ad many who are unacquainted with the real state he case to suppose that these branches of knowledge are wholly dependent upon experiment, and not at all upon reasoning and demonstration. Thus, while, in the notions usually entertained of the mathematics, their difficulties are magnified by the supposition that all is exertion of the in- tellect, and that the evidence of the senses goes for nothing; on the other hand, the physical sciences, and especially inistrv, are supposed to consist "m the exhibition of a iking and curious results, unconnected by any 6 train of reasoning: a notion which, if true, would place them, as a discipline of the mind, in no higher rank than the tricks of a juggler, however useful they might be in their application to the arts of life. I now proceed to apply these observations to the early study of these sciences, or of preliminary matters connected with them. It would be worse than useless to attempt to teach them to the youngest students in all their extent and rigour. But it does not therefore follow that no pains are to be taken to prepare the way for this knowledge, and to render its acquisition more easy, more interesting, and more accurate. In what way this is to be done lies the question which I propose to discuss. A young person seizes an idea more readily, and applies it with more facility, when a distinct appellation is given to it. It is not that he has actually acquired knowledge by naming that which was already in his mind, but that he has rendered what he possesses available for the acquisition of more. He has thereby fixed upon it the stamp which will enable it to pass from himself to others, and with interest from others to himself again. He can now form other combinations of ideas, communicate new difficulties, and receive new explanations. Every one must recollect how hard he found it at one time to express what he wished to say when it had any reference to form or shape. Many can remember when their whole geometrical nomenclature consisted in calling a figure which had corners square, and one without them round, the last term applying equally to the figure of the earth and to that of the wheel of a carriage. Such confusion is no trifling disadvantage. It is a hindrance both to the teacher and the learner. To remedy it, early and correct attention should be given to the meaning of scientific terms. There is no reason whatever why the names angle, parallelogram, circle, rectangle, plane, &c., should not be as familiar, and convey as distinct ideas, as the words long, short, or heavy. The same remark extends also to the terms used in physical science. A child might be made to understand, and to acquire new ideas by understanding, tlu' distinction between such words as fluid and liquid, force and resistance-. &c. He could receive and retain more cor- notion^ than are v ually given on such expressions as prcs-mv. attraction. fri. don, &c. In chemistry, he might taught the use of common terms with more reference to the usual classification of bodies. He would thus come to the studv of the with a large stock of preliminary ideas, and with no incorrect notions to alter, arising from the vague and confused use of names. I may appear to ome to have been trifling, in entering into such detail, but only those who have taught can know how much depends on the accurate knowledge of simple words, and how effectu- ally the pupil's progress is retarded by the evil for which I have been suggesting the remedy. Again, the observations which are made in early age, though sufficiently conclusive as to the facts they embrace, -nits in an accurate form, and indeed are littl smt in ideas in which proportion or measure- ment of any kind are involved. The power of comparing -timating relations of magnitude is one which shows itself BUch later than those which we have- already noticed, if indeed it would lie developed at all, in the greater part of man thout the aid of instruction. An assertion which expresses a positive fact, if it be simple, is easily confirmed <>r fl fated] one \\hich implies the comparison of two things sort requires more attention, more knowledge, and m< us habitude of thinking. Thus, for ex- ample, no one, bowcref young, will, when the proposition IB named to him even for the first time, conceive himself not to have known that two different straight lines can meet only in one jx>int. He will, even if ignorant to this extent before, find uithin himself such a readiness to assent to the is he can conceive to arise from nothing but previous knowledge. JUit the assertion that two sides of a triangle than the third, is one which will require a littl ieration to make it evident to the senses; 8 that the exterior angle of every triangle is greater than either of the interior and opposite angles, will probably appear doubtful ; while, that this exterior angle is exactly equal to the two interior angles, will appear to have no evidence at all. Yet all these propositions might, by a little management, be made so apparent to the senses as to require no further demonstration, as far as moral certainty is concerned. Had no proof of them existed, they might have been received as axioms or first principles ; and we may venture to say, that had such been the case, no more objection would have been raised to them at this day, than is experienced by the unproved proposition alluded to in the former part of this lecture, which is in effect the same as the last axiom of Euclid. Perhaps at one time these and many such propositions were viewed in this light. The proverb says, that Rome was not built in a day, and we know that there was a Rome of huts and cabins, which preceded the city of palaces. I could as easily conceive, that the first essay of savages in the art of building would be a magnificent castle, with vaulted roofs and winding staircases, as that the first geometers imagined that there even existed such a connexion between their several pro- positions as would make one an infallible consequence of another. It is more likely that actual measurement was at one time the only method of discovery or verification ; that by these means a large mass of undoubted truths was collected ; that accident or more attentive consideration first discovered the connexion between two or more propositions ; that men of superior minds, thus drawn to a new view of the subject, began to entertain the suspicion that these empirical truths were the links of one chain, the connexion of which might be traced by reasoning ; and that they ultimately succeeded in reducing the whole to the form in which it appears in the immortal work of Euclid. The traditions of the history of the sciences, imperfect as they are, yet bear out this sup- position in several points. This early history exhibits in a remarkable manner the same succession of errors, the same i' of imperfections and redundancies, which mark the progress of a young student in our days, and should be studied by all who desire to qualify themselves for the task of instructing others. A\ hen we come to consider the subject of arithmetic, we shall find that the results derived from simple observation are >oth in number and importance. Not only is the science than that of geometry, but we are not at an early age so conversant with the objects of the former : the latter. \c\ erthelos, by draw inh new and remarkable ideas ; and while they show that the J arithmetic is not what it is usually considered :he extreme of d ulness and dry ness, will OMTcise the mind in one of the most useful faculties which sesses, that oftgeMnJMfig by induction. As an ex- ainpl. successive odd numbers, IK Binning from unity, be added together, the result will always be a square number, that is, one which i> formed by multiplying a number by This proposition is usually reserved for the student uh<> le some progress in algebra; but there is no reason why it should not be established in the mind more by observation and induction. Many such facts, not only curious and pleasing in themselves, but connected closely with \arious phenomena of natural philosophy, as is the one just noticed, might be made known at an early age, and their acquisition would lighten the labour of the student in commencing algebra in two ways, since a pre- k i low ledge of such points would render interesting a study, which is usually considered the driest part of mathe- s, and the habits of calculation so formed, would n the drudgcrv which i> inseparable from its pre- liminary operation^. The t Innentarv proportions of natural philosophy, and those of dicniistrv, are now considered as calculated for the understandings of the youngest students. Many of 10 their simplest truths come into the mind as early as the first principles of geometry, which have fallen under our notice. But here the same remark occurs which I have already had occasion to make. Prominent and absolute facts are seized and recollected, while those which imply comparison or re- lation excite no curiosity. Thus while each one observes that a stone falls more quickly in the second instant of its descent than in the first, few inquire how much more quickly, or what relation the space described in the second instant bears to that in the first. The general fact connected with all machines, that a small weight is by the disposition of their parts made to balance a larger one, excites attention, while the explanation of the numerical law is viewed with indiffer- ence. It is this disposition which, passing unnoticed in the child, lays the foundation of superficial and smattering habits in the man, and brings shame upon what are nevertheless most justly called the improvements in education. It has been objected, and with truth, that the extensions of ele- mentary instruction have not had that tendency to promote real and accurate knowledge which their supporters claim for them. A triumphant answer will never be given to this assertion as long as it is considered sufficient to fill the young mind with pleasing facts stripped of their rigorous form, and therefore incapable of leading to any consequences. Yet who can wonder that, as far as the physical sciences are con- cerned, such habits should prevail, when so many are allowed in their youth to skim the surface of natural philosophy, to abstract only what is pleasing and prominent, to fix their attention on simple facts, without being led to observe the actual relations which prevail among them, the laws, as they are called, which regulate the phenomena, by the observation and verification of which alone real instruction is to be de- rived from experiment. Strip the results of physics of their accurate and numerical character, let them be any thing else but exercises of the powers of comparison and measurement, and a show of fireworks will be as useful a study, and as good a discipline for the mind. In what I have been saying, I 11 would not be understood as condemning the methods usually employed in the instructive of older students. Real and ititir instruction it is not difficult to find; but the habits formed in childhood, of which I have been endeavouring to tnuv tl in the minds of many persons an inability to estimate- the advantages of correctness, and a dis- inclination for the thought and study which are necessary to arrive at it. Still less would I have ventured before you with thl I not certain that the manner of liiiU the physical sciences in this university is in strict accoi ith the principles which I have endeavoured to maintain, and is not in the least adapted to feed the appetite hose who hunger for amusement at the expense of such as (I ;d instruction. Many may be here present, with knouK-i: Tin a judgment, who have inspected the upparat: ^ d for the purposes of natural philosophy mistrv in thi> institution; tlu-v cannot but have re- that the whole is, as much as it possibly can be, in manv instances by novel methods, adapted to plan before ti numerical laws which were at first con- H mathematical results. It is not con.sidered suf- the mathematical student should deduce these hisions bv reasoning from the first principles; in addi- tion to this, the experimental vcrilications are such, as by \es would amount to demonstration. This marks the the instruction here- given; and to enable the student to appreciate and find delight in such apparently di \ minuteness, I have recommended that he should begin early to follow the track into which he must turn, before he' can be said to be in the path of knowledge*. What is the reason, it may be asked, thai the sciences in general, and particularly the- mathematics, which are the foundation of accurate knowledge in all, are delayed till what is comparatively so late a period in life ? I say, so late a period, because, while it is considered that at the age of all children mav with advantage commence the tnd Latin, the formal puiMiit of mathematic- 12 is pushed off to the age of fourteen or fifteen. Is it that the latter study is more difficult than the former ? Does it re- quire the intellect of a more advanced age ? or may any method be devised by which a pursuit so generally acknow- ledged to be of the highest utility, may occupy its fair place in the system of education ? I shall endeavour to answer these questions at as much length as our time will admit. The difficulties of mathematics are much exaggerated; I mean the necessary difficulties, for I do not deny that to the student of fourteen years of age, whose mind is utterly un- prepared for the study, they offer obstacles of as serious a nature as any elementary branch of knowledge whatever. This necessary preparation is seldom made, for which the following explanation may account. The groundwork of the mathematics is the science of arithmetic. It is the necessary forerunner of algebra, of which indeed it is, or ought to be, a constituent part. It is therefore evident that the clearness of the student's per- ceptions of the first doctrines of algebra must be most ma- terially influenced by the manner in which he has learnt the principles of arithmetic. Now it is a well known fact, that these principles are not, in nine cases out of ten, a part of the preparatory education of those who commence algebra. It is not that abundance of time is not bestowed upon the subject in our schools, since in most cases, what goes by the name of arithmetic, forms a part of the studies of many years, The defect, which is so notorious, that it is hardly requisite to allude to it in distinct terms, is that the attention is entirely directed to such practical rules as are of most frequent occurrence in commercial operations, and to the rules only, not to the principles on which they are established. Reasoning and reflection are entirely excluded from a science, which of all is the most adapted for the development of these faculties in the young mind. I am not underrating the im- portance of readiness in calculation, a habit so essential to the every day pursuits of life. But surely it would be no bar to the attainment of this habit, to mix up with the 13 methods which are to form it, actual demonstrations of the rules by which the student is to be guided. There is no- thing \\hatever in the fundamental operations of arithmetic, of which tlie demonstration is not simple, practical, and suited to the capacity of the young. I would even go so far as to say, that the .science of arithmetic is more easy than the art, and that the labour usually required for the at- tainment of practical correctness might be very materially lessened In the introduction of theoretical principles. It has bcvn well observed by Condillac, in treating of this very ect, that a rule is like the parapet of a bridge; it may less passenger from tumbling over, but will not help him to walk forward. While I admit the necessity of such rules, it is as tin- result* of the student's own reflection and coi',\iction not of tin- oiders of a master, or the au- thority of a book. 1 should require to apologize for an insult to the ttodaPStafedingl of those who hear me, were I to enlarge any further upon the inutility of the present iiods. In ai those, if there be any, who uphold the present system on account of the importance of commer- cial operations to a "Teat part of the world, and who there- think that all should be saddled with the whole routine he COUnthlg-house, I will only observe, that a rule is established in other cases u Inch it behoves them to show should not extend to this. General education is intended to develope the faculties, ao as to enable the individual to apply the whole strength of mind with which nature has gifted him, to tin purposes of life, and, in particular, to the call hich he is intended. Common sense, therefore, dictates the postponement of professional instruction till a led. at least in all case*, wherein its prosecution in- terferes with the great objects to which we have alluded. 1 Ie would be held to be joking who should propose that all should read Celsus at ten years of age, because many :ined for the- profession of physic: nevertheless, as singular an anomaly prevails in education, as far as arith- (1. In the remaining branches, the same 14 defects are discernible though not to the same extent, and from this system springs the dislike with which the mathe- matics are regarded by the majority of those who commence the study. What taste for the acquisition of languages would have existed, had it been the custom to permit no one to speak until he had attained the age of fourteen ? And how can it be expected that a study which requires thought, reasoning, and logical comparison of ideas, should find favour or be regarded with any feeling but disgust, when, by a careful prohibition of the previous exercise of these faculties, the task is rendered one of very great difficulty ? And let no one suppose, because he finds no embarrassment in fol- lowing a train of reasoning, that it would have been an easy task had the habit been left to be v formed till nearly the completion of his education. I know it for fact, that there is no trouble which many students will not take to avoid following a demonstration, when their previous habits have not made it comparatively attainable. I have known the propositions of Euclid committed to memory by rote, even to the right arrangement of the numerous combinations of letters, by those who found such an exercise more easy than the usual method of mastering a demonstration. It is cus- tomary to explain so extreme a case by saying, that the in- dividual has no turn for mathematics, a phrase much in use, and which I conceive would be rightly interpreted, by saying that he has turned to them too late. To return to the subject If the system of teaching arithmetic as just described were amended, much time would be saved, part of which might be devoted to further prosecution of mathematical studies. The student would find algebra an easy generalization of arithmetic. He would not see any great difficulty in applying to num- bers in general, represented by letters for the sake of ab- breviation, those principles which had become familiar to him by constant application to particular cases. If, in addition to this, the suggestion made in a former part of this lecture had been acted upon if he were pre*. 15 pared with some knowledge of the propositions to which he must look forward, the road would be more short and more pleasant. To know what is coming, to be able to anticipate the result, adds sensibly to the pleasure of .uiring knowledge : we may say, indeed, that there is little satisfaction without it. ;iu-try. \\ Inch is a science more pleasing to the majority of learner* than algebra, and which is, for the purposes of the many, the more useful of the two, might be taught at an . 14 e than is the custom at present. I do not say that the imt rigorous form of the science is that which nld be first adopted, since there is much in it which 1 not he appreciated by the youngest students. Undoubtedly, geometry is made a part of the usual course of education, on account of the strictness of its rea- soning, and the absence of all ci re urn. stances which have any iincv to introduce doubt or balance of probabilities into its conclusions. Kvery plan, therefore, which is proposed fnr teaching geometry is valuable or not, according as it increases the perception of the force of the reasoning, or the iar\. It is not true that demonstration, however un- answerable, brings immediate conviction to the mind of one unusi-d to it. All must come by experience only to an apprei iation of its value ; they must, in fact, learn to reason. Anv <>iu -ingle argument drawn from an elementary pro- position, would certainly, when the terms were explained, obtain an immediate assent : it could hardly be otherwise when we consider that few theorems contain individual reasonings more difficult than the following: Two mag- nitudes are each of them equal to a third; therefore, they are equal to one another. But a connected chain , in which, perhaps, at every second step, a new though self-evident principle is introduced, increases in difficulty more than in proportion to its length. I feel that the opinion which I am about to hazard will appear new to many; I know not whether I may not say to all, since 1 am not aware of any authority for it. As far 16 as I am concerned, it is my own conviction, drawn from what I have observed in the course of my duties in this place. Most people would say, that the arguments of geometry which are incontrovertible, were the means of establishing conclusions which, before the admission of the reasonings, were doubtful. I am of opinion .that the reverse is the case : that the conclusions themselves, in most instances of the elements of this science, are so natural, so probable, so evident to the senses, as not to derive any additional sanc- tion from reasoning so new and strange as that of geometry must appear to the beginner. On the contrary, I hold that confidence in the complicated reasonings is taught by ob- serving that they do, at last, lead to evident and undoubted results. Those who are unused to rigorous proof have more faith in the simple and well-tried method of ocular demonstration than in the new and apparently cumbrous and superfluous machinery of mathematical argument ; just as the sling and stone was preferred to the armour of Saul, because it had not been proved by him who was to use it. I have been told repeatedly by pupils, on ending some of the most elementary propositions, that they knew that before; and I have found the difficulty to consist in con- vincing them that they were now to know it in a different way, which would, when understood, lead them to things which they did not know beforehand. Those, then, who will agree with me in opinion that, in the first elements, the results lend force to the reasoning, arid not the reasonings to the result, will come with me to the conclusion that the preliminary step to be taken, in teaching geometry, is to increase the certainty of the results by those means which nature first points out. I mean that the first course of geo- metry should be experimental ; that the student should be guided by ocular demonstration to a knowledge of the facts, to the connexion of which he is afterwards to apply his rea- soning powers. There is no proposition in the elements of Euclid, at least in the part which relates to plane geometry, but what might readily be proved in this manner. An 17 additional advantage would be, the complete establishment of the meaning of the terms, which would thus be prac- tically fixed in the student's mind, before he enters on the new track which it is the ultimate object of geometry to point out. A youth thus prepared in the elements of mathematics, might commence the study of physics in a manner some- what stricter than is usually thought advisable. The itest obstacle in the way of an earlier attainment of the first principles, both of physics and chemistry, is the expense i difficulty of procuring suitable apparatus, both of which considerably overrated. The costly and minutely exact rmnents with which the cabinet of a public lecturer is (\ are by no means indispensable for the purposes of pre- liminary instruction: they are made to be distinctly seen throughout tin- >}>ace of a room such as that in which we now are ; have all the additional finish which the best work- men can bestow; and are made to fulfil, as far as such instruments can do, the conditions of perfect exactness. The apparatus required for our purpose might be simple, cheap, and not containing many instruments: the principles which it i^ intended to illustrate are few in number; and it would be useless to attempt a degree of correctness, which thoM-, for whose benefit the whole is intended, would beun- te. It would tend greatly to the reduction lie number of experiments if the student were commonly versed in tin- results, not to say the reasonings, of geometry. Truths of this description, by what means so- (1, are a species of actual knowledge which tend to tin better comprehension of the phenomena of physics. The same may be said of the first principles of algebra; a which oreatly extends the power of deducing some phenomena from others, and of establishing the con- 11 between them. But though the student should grow up unacquainted with the mathematics, to years when time and inclination the pursuit, he should, nevertheless, if it be in hi.s- power, turn his attention to the phenomena of nature, as c 18 collected and displayed in the sciences of mechanics, in the widest sense of the word, astronomy and chemistry; and this he may do with perfect confidence, that although he will never compare with the mathematician in the extent and profundity of his views, and the facility of applying his prin- ciples, he may, nevertheless, be sure of reaping such a harvest as will amply repay him for his trouble, and render him an object of envy to all who have never paid any atten- tion to these subjects. It ought not to diminish the satis- faction which he feels in wandering over the ample fields that lie .open before him, that there are others which he cannot tread, except by another path, to the gate of which r want of time or opportunity has prevented his approach ; for he should recollect that those who have been more fortunate and have reached a point which to him appears distant, are nowise nearer to the possible limit which bounds the human power of discovery. Facts and principles which Newton or Galileo would have gladly known, are spread before him in abund- ance : the researches of the mathematician, and the labours of the observer, are now divested of technicalities, and re- duced to such a form that he can seize results which, not many years ago, were concealed in the covering of mathe- matics under which they were born and nourished. A century has not elapsed since an affectation universally pre- vailed, which was equivalent to a declaratory law, that he who did not know all should know nothing. Although the use of a learned language was, in a great measure, abolished, there still remained a method, as effectual, of concealing the stores of natural science ; and this, without any overt act against the progress of knowledge, which the liberal-minded of that day would have scorned as much as if they had lived in the present age. Their error consisted in taking no pains to simplify elementary knowledge ; in not endeavour- ing to clear the entrance of the building which they took so much pride in erecting. The case is altered now: men of the highest acquirements are not ashamed of the task of placing before the world, in a shape intelligible to all, as Ml many of the truths of natural philosophy as will admit of it. This is thf a^v of elementary works as well as of remarkable but among the many advantages which it pos- sesses, and which, in spite of all that has been said of them, have not been UK) much boasted, there is one rising defect, which all who value real knowledge should exert themselves interact It i- natural to man. when he gains what he feels to-be an advantage, by an unusual or unexpected method, to look down upon ami deerv the means of which others are making use for the attainment of the same ; even though the latter should be acknowledged to be, in the long run, more dfectual. Tin-, if I mistake not, is beginning to be the ease at present : while the simplification and consequent ex- of natural philosophy has made many well informed thinker- on this subject, and thousands of superficial readers, a notion begin- to he formed as if the mathematics were un_ IMffj , and as if all requisite knowledge could be gained without them. rnnecessarv we have shown them to be in in sense, inasmuch as valuable information can be gained without them ; but what comparison is therebetween 1 the mathematician and natural philosopher, and the latter char ;iarated from tlu- former? less than between the scholar who had read In- authors in their native language, and the man w ho has contented himself with a translation. u allowing the re-ults to be the same in both cases, which I am far from granting, supposing that both of the first class understand equally well the phenomena of the motion of a plain -t, and both of the second the events of the in or lMo}H)i.iH-ian war it is the manner of arriving at these facts which will plainly mark the distinction be- tween the two the moment they are brought together: the has improved his mind and extended his faculties, by following tin- track of discovery; should he even forget what he kiuw>, he has not lo-t all he cannot, if he would, rid him-elf of the habits he has acquired: the other has .1 tin ivcital of the voyage has added to his stock of 20 information, but not to the powers of his understanding; he receives what he is told, but cannot certainly know whether it is right or wrong. In estimating the advantages of such attention to real science, we must not forget that time will be thereby saved in two ways ; the path itself will be more easy to travel, and the period at which it is entered upon will be earlier. Thus the elements of many other sciences may be added to the usual course of education, which at present are excluded from it by considerations of the want of time for their cultivation. The study of the structure of our globe, so prominent an object of curiosity at the present day, so use- ful in the study of geography, as it should be, might, to a small extent at least, form a part of elementary education. The elements of mineralogy, as far at least as they are ne- cessary to the study of geology, of which the first is the gram- mar, would offer no difficulty to any one acquainted with the first principles of chemistry. Natural history, a study so pleasingto the young, and so conversant with familiar objects, that it has often been proposed as one of the first objects of attention for children, would enlarge the mind by calling the attention to the countless examples which there exist of the adaptation of means to their end. No one of these sciences is destitute of instances of strict reasoning and sound prac- tical theories. Neither the time nor my own knowledge of these subjects will permit of my entering into the manner in which they should be taught. One thing is certain : that as there is no one science which does not aid in the advance- ment of all the rest, so there is no useful habit of mind acquired from the study of any one, which is not beneficially felt in proceeding to the others. The cautious habit of ob- servation acquired from a correct study of physics and che- mistry, finds ample exercise in those of geology or natural history. The correct, close, and sustained reasoning of mathematics is useful throughout. Nor is this connexion observable only between those parts of knowledge which go by the name of sciences. The study of languages, so 21 necessary to them all, receives from them, in its turn, illus- tration and ornament. History, metaphysics, political eco- nomy, frequently require some collateral knowledge, which he derived from the phenomena of nature. But it is Hess to further pursue truths which will find an almost universal assent Thus much have I said upon the study of science, as a part of the education of all, and as tending to promote the great object, the development of the faculties of the mind. All that I have hitherto observed is grounded upon consi- derations quite independent of the practical utility of the nces, and would lose none of its force were they mere speculations, in nowise affecting the progress of the arts or the increase of 'the conveniences of life. I now come to other considerations. We are all aware that mathematics and physics, so useful to all, from the training they afford, are to many a part of their professional education, bearing closely ujxm matters of the highest concern to a commercial and manufacturing nation. I therefore request your atten- tion to some observations which are of importance at any time, hut which have struck me particularly as deserving, in a remarkable degree, the attention of those who live in an era -uch as our own. A wide line of di>tinction has been drawn between two branches of knowledge which have been called by the names iy and practice. No appellations have been more -used, as indeed we may suspect when we recur to the that thev have becom the watchwords of parties in science. According to some, a theorist is a person who indulges in general and speculative views, without giving hi- attention to useful objects, and such as are of immediate concern to public intere>ts; while a practical man is one whose knowledge directly relates to those parts of science which can be immediately turned to account in naviga- tion, machinery, and commerce, or some other direct ap- plication. According to others, a theorist is one who has Iy Miidied the matters in que>tion well and scientifically ; uhile a practical man is like the common soldier, whose' employment it is to carry into effect that which others have projected. Much discussion has ensued upon these names, into which it is not my purpose to enter. The fact is, that theory, in the proper sense of the word, is that without which practice cannot thrive, and the two must be united, in what- ever w r ay that effect may be produced, before any real good can result. The first order of talent is divided among men in various ways. In some, it turns towards those branches which can be pursued in the retirement of the closet; in others it engages the mind in the pursuits of busy life, and the applications of the phenomena discovered by the first class ; while to the reputation of both it would be of the worst consequence could their disputes upon words effect an actual separation between them. This country has long held the first rank in the useful arts, which depend upon the applications of science. This is not a boast, since it is a fact candidly admitted by foreigners^ and demonstrable to the senses of all whom travel and observation have enabled to verify it. Never- theless, it is as true that there are other countries where theoretical knowledge is more generally diffused. This has given occasion, I doubt not to many, to depreciate the latter species of knowledge as a professional disqualification, on the ground that our country, in which it is less cultivated, has nevertheless kept its station, and even gained upon others in useful applications and practical arts. But these, I think, do not consider that, independently of our being able to draw from the stores of others, our relative political situations have been different. To make this apparent, we must consider what has been the reception of science in this country and others. For political opinions, this is not the place ; but admitted facts, when they bear upon the subject now in question, are open to my use; and I shall avail myself of them, to show in what situation we may probably stand, if we neglect the diffusion of theoretical knowledge. When the sciences were first revived in Europe, the despotic governments of that time, well aware what enemy had made his appearance, and knowing that their system of }x)licy was based upon the universal ignorance of the times', u>cd all their efforts to crush its supporters and to prevent its diffusion. This very ignorance supported them in their undertaking, by attributing to sorcery all results of know- ledge which were hevond the comprehension of the vulgar. Gradually, however, this state of things ceased to exist, and Miencc, which hail survived all attempts at its destruction, became an object of respect ; but still its course, in Britain and abroad, was differently modified by the character of the people and their government s. On the continent, the higher powers began to conceive that \\hat could not be conquered as an enemy might be UM-ful as an allv, and they took into their pay, if I may so re>s it, tin- sciences and their followers. Discoveries warded, and those who made them were connected in >ome manner with the government they lived unucr. To it they were made 1 to owe their means of subsistence, and thus placed bevond the cares of life, they WCW enabled to turn their attention to that which they n I, to tin- advancement of the sciences and the perfert'on of the without having to consider whether thai -;iich they were occupied could be turned to immediate amount or ictical application was in high repute as far as it could be .rous to the ruling powers. The go- M-I : u the only customer which wa* well served: its armies, its public works, displayed all that skill united with labour could effeet. The people, on the other hand, de- prived of useful knowledge, and of all .stimulus to improve- ment and exertion. Mink into a li>tless dependence on the anw for the supply of all public wants. To this day such i- tin case over a great part of the continent, even among acute and enterprising nations. The practical arts (1 not he expected to flourish in such a state of things, and accordingly they have been stationary, or advanced with tardy Mrp- win n compared with the rapid march which they have made m Kngland In thi> country the more popular nature of the govern- ment has given another species of impulse to scientific knowledge. The commercial character of its people, and its division into little parishes, each attending to its own concerns, and paying its own expenses, has, with other circumstances, thrown upon every individual the necessity of attention to all matters which immediately concern him, as well of a more public as of a domestic nature. Thus each one, secure of no interruption from the jealousy of government, in endeavouring to convert any improvement in art to his own benefit, has been left to employ all his talents in any line to which his interest might induce him to turn. The consequence is, the high state of practical per- fection in which we see the useful arts ; the readiness with which every hint, that holds out a fair prospect, is caught at ai^d adopted; and the facility with which capital to any amount is collected for any public application of science of which the want is felt. But in the meanwhile little en- couragement is held out for the pursuit of abstract know- ledge. All will naturally direct their attention to that which offers the most immediate prospect of advantage. As a pro- fession, therefore, the pursuit of theoretical science has sunk in importance ; and though we are not, nor ever have been, without our share of those whose labours have been successful in this department, yet the number of them has borne no proportion to what might naturally have been expected in a nation whose very existence may be said to depend upon the arts, of which theoretical science is the parent. Nor will the introduction of the illustrious names which our history presents advance an opposite argument ; for in ap- preciating the state of theoretical knowledge of any parti- cular country, I do not look at a Newton, a Lagrange, or an Euler, of whom one country produces but few, and at distant epochs ; but at the knowledge of the many, who are to direct the exertions of their fellow-countrymen ; to the ranks from out of which such men must spring. These are in my opinion the causes of the advancement of theory abroad, and its stationary character at home. But 25 s TOmbination of circumstances in our favour appears I8e. Wo shall have rivals on the continent of character different from those who have hitherto . To all the advantages of a wide diffusion of :al knowledge, prepared for them in the days of de- pendence on their governments, they will add the energies of a people unfettered by the restrictions of what was sup- be state policy, and awakened to the desire of being themselves the conductors of their own affairs. There are many among them who are well aware of the causes of our superiority, and whose knowledge of the manner in which to overtake us is equalled by their desire to show the path to their fellow countrymen. What then should be done to .municate among ourselves an impulse similar to that which they are now receiving ? Nothing but to follow their put in operation the means of remedying the derived from superior information, in the only war from which both parties retire winners, that of rivalry in the diffusion of knowledge. THE END. 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