Chapter VII: The Sources of the Principle of Energy (6)

When a set of facts comes into apparent conflict with another set of facts, and a problem is presented, its solution consists ordinarily in a more refined distinction or in a more extended view of the facts, as may be aptly illustrated by Newton's solution of the problem of dispersion. When a new mathematical or scientific fact is _demonstrated_, or _explained_, such demonstration also rests simply upon showing the connexion of the new fact with the facts already known; for example, that the radius of a circle can be laid off as chord exactly six times in the circle is explained or proved by dividing the regular hexagon inscribed in the circle into equilateral triangles. That the quantity of heat developed in a second in a wire conveying an electric current is quadrupled on the doubling of the strength of the current, we explain from the doubling of the fall of the potential due to the doubling of the current's intensity, as also from the doubling of the quantity flowing through, in a word, from the quadrupling of the work done. In point of principle, explanation and direct proof do not differ much.

He who solves scientifically a geometrical, physical, or technical problem, easily remarks that his procedure is a _methodical_ mental quest, rendered possible by the economical order of the province--a simplified purposeful quest as contrasted with unmethodical, unscientific guess-work. The geometer, for example, who has to construct a circle touching two given straight lines, casts his eye over the relations of symmetry of the desired construction, and seeks the centre of his circle solely in the line of symmetry of the two straight lines. The person who wants a triangle of which two angles and the sum of the sides are given, grasps in his mind the determinateness of the form of this triangle and restricts his search for it to a certain group of triangles of the _same form_. Under very different circumstances, therefore, the simplicity, the intellectual perviousness, of the subject-matter of mathematics and natural science is felt, and promotes both the discipline and the self-confidence of the reason.

Unquestionably, much more will be attained by instruction in the mathematics and the natural sciences than now is, when more natural methods are adopted. One point of importance here is that young students should not be spoiled by premature abstraction, but should be made acquainted with their material from living pictures of it before they are made to work with it by purely ratiocinative methods. A good stock of geometrical experience could be obtained, for example, from geometrical drawing and from the practical construction of models. In the place of the unfruitful method of Euclid, which is only fit for special, restricted uses, a broader and more conscious method must be adopted, as Hankel has pointed out.[124] Then, if, on reviewing geometry, and after it presents no substantial difficulties, the more general points of view, the principles of scientific method are placed in relief and brought to consciousness, as Von Nagel,[125] J. K. Becker,[126] Mann,[127] and others have well done, fruitful results will be surely attained. In the same way, the subject-matter of the natural sciences should be made familiar by pictures and experiment before a profounder and reasoned grasp of these subjects is attempted. Here the emphasis of the more general points of view is to be postponed.

Before my present audience it would be superfluous for me to contend further that mathematics and natural science are justified constituents of a sound education,--a claim that even philologists, after some resistance, have conceded. Here I may count upon assent when I say that mathematics and the natural sciences pursued alone as means of instruction yield a richer education in matter and form, a more general education, an education better adapted to the needs and spirit of the time,--than the philological branches pursued alone would yield.

But how shall this idea be realised in the curricula of our intermediate educational institutions? It is unquestionable in my mind that the German _Realschulen_ and _Realgymnasien_, where the exclusive classical course is for the most part replaced by mathematics, science, and modern languages, give the _average_ man a more timely education than the gymnasium proper, although they are not yet regarded as fit preparatory schools for future theologians and professional philologists. The German gymnasiums are too one-sided. With these the first changes are to be made; of these alone we shall speak here. Possibly a _single_ preparatory school, suitably planned, might serve all purposes.

Shall we, then, in our gymnasiums fill out the hours of study which stand at our disposal, or are still to be wrested from the classicists, with as great and as varied a quantity of mathematical and scientific matter as possible? Expect no such proposition from me. No one will suggest such a course who has himself been actively engaged in scientific thought. Thoughts can be awakened and fructified as a field is fructified by sunshine and rain. But thoughts cannot be juggled out and worried out by heaping up materials and the hours of instruction, nor by any sort of precepts: they must grow naturally of their own free accord. Furthermore, thoughts cannot be accumulated beyond a certain limit in a single head, any more than the produce of a field can be increased beyond certain limits.

I believe that the amount of matter necessary for a useful education, such as should be offered to _all_ the pupils of a preparatory school, is very small. If I had the requisite influence, I should, in all composure, and fully convinced that I was doing what was best, first greatly curtail in the lower classes the amount of matter in both the classical and the scientific courses; I should cut down considerably the number of the school hours and the work done outside the school. I am not with many teachers of opinion that ten hours work a day for a child is not too much. I am convinced that the mature men who offer this advice so lightly are themselves unable to give their attention successfully for as long a time to any subject that is new to them, (for example, to elementary mathematics or physics,) and I would ask every one who thinks the contrary to make the experiment upon himself. Learning and teaching are not routine office-work that can be kept up mechanically for long periods. But even such work tires in the end. If our young men are not to enter the universities with blunted and impoverished minds, if they are not to leave in the preparatory schools their vital energy, which they should there gather, great changes must be made. Waiving the injurious effects of overwork upon the body, the consequences of it for the mind seem to me positively dreadful.

I know of nothing more terrible than the poor creatures who have learned too much. Instead of that sound powerful judgment which would probably have grown up if they had learned nothing, their thoughts creep timidly and hypnotically after words, principles, and formulæ, constantly by the same paths. What they have acquired is a spider's web of thoughts too weak to furnish sure supports, but complicated enough to produce confusion.

But how shall better methods of mathematical and scientific education be combined with the decrease of the subject-matter of instruction? I think, by abandoning systematic instruction altogether, at least in so far as that is required of _all_ young pupils. I see no necessity whatever that the graduates of our high schools and preparatory schools should be little philologists, and at the same time little mathematicians, physicists, and botanists; in fact, I do not see the possibility of such a result. I see in the endeavor to attain this result, in which every instructor seeks for his own branch a place apart from the others, the main mistake of our whole system. I should be satisfied if every young student could come into living contact with and pursue to their ultimate logical consequences merely a _few_ mathematical or scientific discoveries. Such instruction would be mainly and naturally associated with selections from the great scientific classics. A few powerful and lucid ideas could thus be made to take root in the mind and receive thorough elaboration. This accomplished, our youth would make a different showing from what they do to-day.[128]

What need is there, for example, of burdening the head of a young student with all the details of botany? The student who has botanised under the guidance of a teacher finds on all hands, not indifferent things, but known or unknown things, by which he is stimulated, and his gain made permanent. I express here, not my own, but the opinion of a friend, a practical teacher. Again, it is not at all necessary that all the matter that is offered in the schools should be learned. The best that we have learned, that which has remained with us for life, outlived the test of examination. How can the mind thrive when matter is heaped on matter, and new materials piled constantly on old, undigested materials? The question here is not so much that of the accumulation of positive knowledge as of intellectual discipline. It seems also unnecessary that _all_ branches should be treated at school, and that exactly the same studies should be pursued in all schools. A single philological, a single historical, a single mathematical, a single scientific branch, pursued as common subjects of instruction for all pupils, are sufficient to accomplish all that is necessary for the intellectual development. On the other hand, a wholesome mutual stimulus would be produced by this greater variety in the positive culture of men. Uniforms are excellent for soldiers, but they will not fit heads. Charles V. learned this, and it should never be forgotten. On the contrary, teachers and pupils both need considerable latitude, if they are to yield good results.

With John Karl Becker I am of the opinion that the utility and amount for individuals of every study should be precisely determined. All that exceeds this amount should be unconditionally banished from the lower classes. With respect to mathematics, Becker,[129] in my judgment, has admirably solved this question.

With respect to the upper classes the demand assumes a different form. Here also the amount of matter obligatory on all pupils ought not to exceed a certain limit. But in the great mass of knowledge that a young man must acquire to-day for his profession it is no longer just that ten years of his youth should be wasted with mere preludes. The upper classes should supply a truly useful preparation for the professions, and should not be modelled upon the wants merely of future lawyers, ministers, and philologists. Again, it would be both foolish and impossible to attempt to prepare the same person properly for all the different professions. In such case the function of the schools would be, as Lichtenberg feared, simply to select the persons best fitted for being drilled, whilst precisely the finest special talents, which do not submit to indiscriminate discipline, would be excluded from the contest. Hence, a certain amount of liberty in the choice of studies must be introduced in the upper classes, by means of which it will be free for every one who is clear about the choice of his profession to devote his chief attention either to the study of the philologico-historical or to that of the mathematico-scientific branches. Then the matter now treated could be retained, and in some branches, perhaps, judiciously extended,[130] without burdening the scholar with many branches or increasing the number of the hours of study. With more homogeneous work the student's capacity for work increases, one part of his labor supporting the other instead of obstructing it. If, however, a young man should subsequently choose a different profession, then it is _his_ business to make up what he has lost. No harm certainly will come to society from this change, nor could it be regarded as a misfortune if philologists and lawyers with mathematical educations or physical scientists with classical educations should now and then appear.

* * * * *

The view is now wide-spread that a Latin and Greek education no longer meets the general wants of the times, that a more opportune, a more "liberal" education exists. The phrase, "a liberal education," has been greatly misused. A truly liberal education is unquestionably very rare. The _schools_ can hardly offer such; at best they can only bring home to the student the necessity of it. It is, then, his business to acquire, as best he can, a more or less liberal education. It would be very difficult, too, at any one time to give a definition of a "liberal" education which would satisfy every one, still more difficult to give one which would hold good for a hundred years. The educational ideal, in fact, varies much. To one, a knowledge of classical antiquity appears not too dearly bought "with early death." We have no objection to this person, or to those who think like him, pursuing their ideal after their own fashion. But we may certainly protest strongly against the realisation of such ideals on our own children. Another,--Plato, for example,--puts men ignorant of geometry on a level with animals.[131] If such narrow views had the magical powers of the sorceress Circe, many a man who perhaps justly thought himself well educated would become conscious of a not very flattering transformation of himself. Let us seek, therefore, in our educational system to meet the wants of the present, and not establish prejudices for the future.

But how does it come, we must ask, that institutions so antiquated as the German gymnasiums could subsist so long in opposition to public opinion? The answer is simple. The schools were first organised by the Church; since the Reformation they have been in the hands of the State. On so large a scale, the plan presents many advantages. Means can be placed at the disposal of education such as no private source, at least in Europe, could furnish. Work can be conducted upon the same plan in many schools, and so experiments made of extensive scope which would be otherwise impossible. A single man with influence and ideas can under such circumstances do great things for the promotion of education.

But the matter has also its reverse aspect. The party in power works for its own interests, uses the schools for its special purposes. Educational competition is excluded, for all successful attempts at improvement are impossible unless undertaken or permitted by the State. By the uniformity of the people's education, a prejudice once in vogue is permanently established. The highest intelligences, the strongest wills cannot overthrow it suddenly. In fact, as everything is adapted to the view in question, a sudden change would be physically impossible. The two classes which virtually hold the reins of power in the State, the jurists and theologians, know only the one-sided, predominantly classical culture which they have acquired in the State schools, and would have this culture alone valued. Others accept this opinion from credulity; others, underestimating their true worth for society, bow before the power of the prevalent opinion; others, again, affect the opinion of the ruling classes even against their better judgment, so as to abide on the same plane of respect with the latter. I will make no charges, but I must confess that the deportment of medical men with respect to the question of the qualification of graduates of your _Realschulen_ has frequently made that impression upon me. Let us remember, finally, that an influential statesman, even within the boundaries which the law and public opinion set him, can do serious harm to the cause of education by considering his own one-sided views infallible, and in enforcing them recklessly and inconsiderately--which not only _can_ happen, but has, repeatedly, happened.[132] The monopoly of education by the State[133] thus assumes in our eyes a somewhat different aspect. And to revert to the question above asked, there is not the slightest doubt that the German gymnasiums in their present form would have ceased to exist long ago if the State had not supported them.

All this must be changed. But the change will not be made of itself, nor without our energetic interference, and it will be made slowly. But the path is marked out for us, the will of the people must acquire and exert upon our school legislation a greater and more powerful influence. Furthermore, the questions at issue must be publicly and candidly discussed that the views of the people may be clarified. All who feel the insufficiency of the existing _régime_ must combine into a powerful organisation that their views may acquire impressiveness and the opinions of the individual not die away unheard.

I recently read, gentlemen, in an excellent book of travels, that the Chinese speak with unwillingness of politics. Conversations of this sort are usually cut short with the remark that they may bother about such things whose business it is and who are paid for it. Now it seems to me that it is not only the business of the State, but a very serious concern of all of us, how our children shall be educated in the public schools at _our_ cost.

FOOTNOTES:

[Footnote 113: An address delivered before the Congress of Delegates
of the German Realschulmännerverein, at Dortmund, April 16, 1886.
The full title of the address reads: "On the Relative Educational
Value of the Classics and the Mathematico-Physical Sciences in
Colleges and High Schools."

Although substantially contained in an address which I was to have
made at the meeting of Natural Scientists at Salzburg in 1881
(deferred on account of the Paris Exposition), and in the
Introduction to a course of lectures on "Physical Instruction in
Preparatory Schools," which I delivered in 1883, the invitation of
the German Realschulmännerverein afforded me the first opportunity
of putting my views upon this subject before a large circle of
readers. Owing to the place and circumstances of delivery, my
remarks apply of course, primarily, only to German schools, but,
with slight modifications, made in this translation, are not without
force for the institutions of other countries. In giving here
expression to a strong personal conviction formed long ago, it is a
matter of deep satisfaction to me to find that they agree in many
points with the views recently advanced in independent form by
Paulsen (_Geschichte des gelehrten Unterrichts_, Leipsic, 1885) and
Frary (_La question du latin_, Paris, Cerf, 1885). It is not my
desire nor effort here to say much that is new, but merely to
contribute my mite towards bringing about the inevitable revolution
now preparing in the world of elementary instruction. In the opinion
of experienced educationists the first result of that revolution
will be to make Greek and mathematics alternately optional subjects
in the higher classes of the German Gymnasium and in the
corresponding institutions of other countries, as has been done in
the splendid system of instruction in Denmark. The gap between the
German classical Gymnasium and the German Realgymnasium, or between
classical and scientific schools generally, can thus be bridged
over, and the remaining inevitable transformations will then be
accomplished in relative peace and quiet. (Prague, May, 1886.)]

[Footnote 114: Maupertuis, _Oeuvres_, Dresden, 1752, p. 339.]

[Footnote 115: F. Paulsen, _Geschichte des gelehrten Unterrichts_,
Leipsic, 1885.]

[Footnote 116: There is a peculiar irony of fate in the fact that
while Leibnitz was casting about for a new vehicle of universal
linguistic intercourse, the Latin language which still subserved
this purpose the best of all, was dropping more and more out of use,
and that Leibnitz himself contributed not the least to this result.]

[Footnote 117: As a rule, the human brain is too much, and wrongly,
burdened with things which might be more conveniently and accurately
preserved in books where they could be found at a moment's notice.
In a recent letter to me from Düsseldorf, Judge Hartwich writes:

"A host of words exist which are out and out Latin or Greek, yet are
employed with perfect correctness by people of good education who
never had the good luck to be taught the ancient languages. For
example, words like 'dynasty.' ... The child learns such words as
parts of the common stock of speech, or even as parts of his
mother-tongue, just as he does the words 'father,' 'mother,'
'bread,' 'milk.' Does the ordinary mortal know the etymology of
these Saxon words? Did it not require the almost incredible industry
of the Grimms and other Teutonic philologists to throw the merest
glimmerings of light upon the origin and growth of our own
mother-tongue? Besides, do not thousands of people of so-called
classical education use every moment hosts of words of foreign
origin whose derivation they do not know? Very few of them think it
worth while to look up such words in the dictionaries, although they
love to maintain that people should study the ancient languages for
the sake of etymology alone."]

[Footnote 118: Standing remote from the legal profession I should
not have ventured to declare that the study of Greek was not
necessary for the jurists; yet this view was taken in the debate
that followed this lecture by professional jurists of high standing.
According to this opinion, the preparatory education obtained in the
German Realgymnasium would also be sufficient for the future jurists
and insufficient only for theologians and philologists. [In England
and America not only is Greek not necessary, but the law-Latin is so
peculiar that even persons of _good_ classical education cannot
understand it.--_Tr._]]

[Footnote 119: In emphasising here the weak sides of the writings of
Plato and Aristotle, forced on my attention while reading them in
German translations, I, of course, have no intention of underrating
the great merits and the high historical importance of these two
men. Their importance must not be measured by the fact that our
speculative philosophy still moves to a great extent in their paths
of thought. The more probable conclusion is that this branch has
made very little progress in the last two thousand years. Natural
science also was implicated for centuries in the meshes of the
Aristotelian thought, and owes its rise mainly to having thrown off
those fetters.]

[Footnote 120: I would not for a moment contend that we derive
exactly the same profit from reading a Greek author in a translation
as from reading him in the original; but the difference, the excess
of gain in the second case, appears to me, and probably will to most
men who are not professional philologists, to be too dearly bought
with the expenditure of eight years of valuable time.]

[Footnote 121: "The temptation," Judge Hartwich writes, "to regard
the 'taste' of the ancients as so lofty and unsurpassable appears to
me to have its chief origin in the fact that the ancients were
unexcelled in the representation of the nude. First, by their
unremitting care of the human body they produced splendid models;
and secondly, in their gymnasiums and in their athletic games they
had these models constantly before their eyes. No wonder, then, that
their statues still excite our admiration! For the form, the ideal
of the human body has not changed in the course of the centuries.
But with intellectual matters it is totally different; they change
from century to century, nay, from decennium to decennium. It is
very natural now, that people should unconsciously apply what is
thus so easily seen, namely, the works of sculpture, as a universal
criterion of the highly developed taste of the ancients--a fallacy
against which people cannot, in my judgment, be too strongly
warned."]

[Footnote 122: English: "In the beginning God created the heaven and
the earth. And the earth was without form and void; and darkness was
upon the face of the deep. And the spirit of God moved upon the face
of the waters."--Dutch: "In het begin schiep God den hemel en de
aarde. De aarde nu was woest en ledig, en duisternis was op den
afgrond; en de Geest Gods zwefde op de wateren."--Danish: "I
Begyndelsen skabte Gud Himmelen og Jorden. Og Jorden var ode og tom,
og der var morkt ovenover Afgrunden, og Guds Aand svoevede ovenover
Vandene."--Swedish: "I begynnelsen skapade Gud Himmel och Jord. Och
Jorden war öde och tom, och mörker war pä djupet, och Gods Ande
swäfde öfwer wattnet."--German: "Am Anfang schuf Gott Himmel und
Erde. Und die Erde war wüst und leer, und es war finster auf der
Tiefe; und der Geist Gottes schwebte auf dem Wasser."]

[Footnote 123: Compare Herzen's excellent remarks, _De
l'enseignement secondaire dans la Suisse romande_, Lausanne, 1886.]

[Footnote 124: _Geschichte der Mathematik_, Leipsic, 1874.]

[Footnote 125: _Geometrische Analyse_, Ulm, 1886.]

[Footnote 126: In his text-books of elementary mathematics]

[Footnote 127: _Abhandlungen aus dem Gebiete der Mathematik_,
Würzburg, 1883.]

[Footnote 128: My idea here is an appropriate selection of readings
from Galileo, Huygens, Newton, etc. The choice is so easily made
that there can be no question of difficulties. The contents would be
discussed with the students, and the original experiments performed
with them. Those scholars alone should receive this instruction in
the upper classes who did not look forward to systematical
instruction in the physical sciences. I do not make this proposition
of reform here for the first time. I have no doubt, moreover, that
such radical changes will only be slowly introduced.]

[Footnote 129: _Die Mathematik als Lehrgegenstand des Gymnasiums_,
Berlin, 1883.]

[Footnote 130: Wrong as it is to burden future physicians and
scientists with Greek for the sake of the theologians and
philologists, it would be just as wrong to compel theologians and
philologists, on account of the physicians, to study such subjects
as analytical geometry. Moreover, I cannot believe that ignorance of
analytical geometry would be a serious hindrance to a physician that
was otherwise well versed in quantitative thought. No special
advantage generally is observable in the graduates of the Austrian
gymnasiums, all of whom have studied analytical geometry. [Refers to
an assertion of Dubois-Reymond.]]

[Footnote 131: Compare M. Cantor, _Geschichte der Mathematik_,
Leipsic, 1880, Vol. I. p. 193.]

[Footnote 132: Compare Paulsen, _l. c._, pp. 607, 688.]

[Footnote 133: It is to be hoped that the Americans will jealously
guard their schools and universities against the influence of the
State.]