A real understanding of the world and its civilisation, however, is not the only result of the study of mathematics and the physical sciences. Much more essential for the preparatory school is the formal cultivation which comes from these studies, the strengthening of the reason and the judgment, the exercise of the imagination. Mathematics, physics, chemistry, and the so-called descriptive sciences are so much alike in this respect, that, apart from a few points, we need not separate them in our discussion.

Logical sequence and continuity of ideas, so necessary for fruitful thought, are par excellence the results of mathematics; the ability to follow facts with thoughts, that is, to observe or collect experiences, is chiefly developed by the natural sciences. Whether we notice that the sides and the angles of a triangle are connected in a definite way, that an equilateral triangle possesses certain definite properties of symmetry, or whether we notice the deflexion of a magnetic needle by an electric current, the dissolution of zinc in diluted sulphuric acid, whether we remark that the wings of a b.u.t.terfly are slightly colored on the under, and the fore-wings of the moth on the upper, surface: indiscriminately here we proceed from observations, from individual acts of immediate intuitive knowledge. The field of observation is more restricted and lies closer at hand in mathematics; it is more varied and broader but more difficult to compa.s.s in the natural sciences. The essential thing, however, is for the student to learn to make observations in all these fields. The philosophical question whether our acts of knowledge in mathematics are of a special kind is here of no importance for us. It is true, of course, that the observation can be practised by languages also. But no one, surely, will deny, that the concrete, living pictures presented in the fields just mentioned possess different and more powerful attractions for the mind of the youth than the abstract and hazy figures which language offers, and on which the attention is certainly not so spontaneously bestowed, nor with such good results.[123]

Observation having revealed the different properties of a given geometrical or physical object, it is discovered that in many cases these properties depend in some way upon one another. This interdependence of properties (say that of equal sides and equal angles at the base of a triangle, the relation of pressure to motion,) is nowhere so distinctly marked, nowhere is the necessity and permanency of the interdependence so plainly noticeable, as in the fields mentioned. Hence the continuity and logical consequence of the ideas which we acquire in those fields. The relative simplicity and perspicuity of geometrical and physical relations supply here the conditions of natural and easy progress. Relations of equal simplicity are not met with in the fields which the study of language opens up. Many of you, doubtless, have often wondered at the little respect for the notions of cause and effect and their connexion that is sometimes found among professed representatives of the cla.s.sical studies. The explanation is probably to be sought in the fact that the a.n.a.logous relation of motive and action familiar to them from their studies, presents nothing like the clear simplicity and determinateness that the relation of cause and effect does.

That perfect mental grasp of all possible cases, that economical order and organic union of the thoughts which comes from it, which has grown for every one who has ever tasted it a permanent need which he seeks to satisfy in every new province, can be developed only by employment with the relative simplicity of mathematical and scientific investigations.

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 ill.u.s.trated 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 quant.i.ty 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 quant.i.ty 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 circ.u.mstances, 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 const.i.tuents of a sound education,--a claim that even philologists, after some resistance, have conceded. Here I may count upon a.s.sent 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 inst.i.tutions? It is unquestionable in my mind that the German Realschulen and Realgymnasien, where the exclusive cla.s.sical 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 cla.s.sicists, with as great and as varied a quant.i.ty 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 acc.u.mulated 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 cla.s.ses the amount of matter in both the cla.s.sical 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 formulA, 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 a.s.sociated with selections from the great scientific cla.s.sics. 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 acc.u.mulation 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 lat.i.tude, 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 cla.s.ses. With respect to mathematics, Becker,[129] in my judgment, has admirably solved this question.

With respect to the upper cla.s.ses the demand a.s.sumes a different form. Here also the amount of matter obligatory on all pupils ought not to exceed a certain limit. But in the great ma.s.s 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 cla.s.ses 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 cla.s.ses, 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 h.o.m.ogeneous 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 cla.s.sical 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 cla.s.sical 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 inst.i.tutions 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 circ.u.mstances 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 compet.i.tion 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 cla.s.ses which virtually hold the reins of power in the State, the jurists and theologians, know only the one-sided, predominantly cla.s.sical 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 cla.s.ses 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 a.s.sumes 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 rAgime 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 RealschulmAnnerverein, at Dortmund, April 16, 1886. The full t.i.tle of the address reads: "On the Relative Educational Value of the Cla.s.sics 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 RealschulmAnnerverein afforded me the first opportunity of putting my views upon this subject before a large circle of readers. Owing to the place and circ.u.mstances 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 inst.i.tutions 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 cla.s.ses of the German Gymnasium and in the corresponding inst.i.tutions of other countries, as has been done in the splendid system of instruction in Denmark. The gap between the German cla.s.sical Gymnasium and the German Realgymnasium, or between cla.s.sical 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 Da.s.seldorf, 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 cla.s.sical 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 cla.s.sical 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 unsurpa.s.sable 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 G.o.d 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 G.o.d moved upon the face of the waters."--Dutch: "In het begin schiep G.o.d den hemel en de aarde. De aarde nu was woest en ledig, en duisternis was op den afgrond; en de Geest G.o.ds 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 Ade och tom, och mArker war pA djupet, och G.o.ds Ande swAfde Afwer wattnet."--German: "Am Anfang schuf Gott Himmel und Erde. Und die Erde war wAst und leer, und es war finster auf der Tiefe; und der Geist Gottes schwebte auf dem Wa.s.ser."]

[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 a.n.a.lyse, Ulm, 1886.]

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

[Footnote 127: Abhandlungen aus dem Gebiete der Mathematik, WArzburg, 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 cla.s.ses 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 a.n.a.lytical geometry. Moreover, I cannot believe that ignorance of a.n.a.lytical geometry would be a serious hindrance to a physician that was otherwise well versed in quant.i.tative thought. No special advantage generally is observable in the graduates of the Austrian gymnasiums, all of whom have studied a.n.a.lytical geometry. [Refers to an a.s.sertion 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.]

APPENDIX.

I.

A CONTRIBUTION TO THE HISTORY OF ACOUSTICS.[134]

While searching for papers by Amontons, several volumes of the Memoirs of the Paris Academy for the first years of the eighteenth century, fell into my hands. It is difficult to portray the delight which one experiences in running over the leaves of these volumes. One sees as an actual spectator almost the rise of the most important discoveries and witnesses the progress of many fields of knowledge from almost total ignorance to relatively perfect clearness.

I propose to discuss here the fundamental researches of Sauveur in Acoustics. It is astonishing how extraordinarily near Sauveur was to the view which Helmholtz was the first to adopt in its full extent a hundred and fifty years later.

The Histoire de l'AcadAmie for 1700, p. 131, tells us that Sauveur had succeeded in making music an object of scientific research, and that he had invested the new science with the name of "acoustics." On five successive pages a number of discoveries are recorded which are more fully discussed in the volume for the year following.

Sauveur regards the simplicity of the ratios obtaining between the rates of vibration of consonances as something universally known.[135] He is in hope, by further research, of determining the chief rules of musical composition and of fathoming the "metaphysics of the agreeable," the main law of which he a.s.serts to be the union of "simplicity with multiplicity." Precisely as Euler[136] did a number of years later, he regards a consonance as more perfect according as the ratio of its vibrational rates is expressed in smaller whole numbers, because the smaller these whole numbers are the oftener the vibrations of the two tones coincide, and hence the more readily they are apprehended. As the limit of consonance, he takes the ratio 5:6, although he does not conceal the fact that practice, sharpened attention, habit, taste, and even prejudice play collateral rAles in the matter, and that consequently the question is not a purely scientific one.

Sauveur's ideas took their development from his having inst.i.tuted at all points more exact quant.i.tative investigations than his predecessors. He is first desirous of determining as the foundation of musical tuning a fixed note of one hundred vibrations which can be reproduced at any time; the fixing of the notes of musical instruments by the common tuning pipes then in use with rates of vibration unknown, appearing to him inadequate. According to Mersenne (Harmonie Universelle, 1636), a given cord seventeen feet long and weighted with eight pounds executes eight visible vibrations in a second. By diminishing its length then in a given proportion we obtain a proportionately augmented rate of vibration. But this procedure appears too uncertain to Sauveur, and he employs for his purpose the beats (battemens), which were known to the organ-makers of his day, and which he correctly explains as due to the alternate coincidence and non-coincidence of the same vibrational phases of differently pitched notes.[137] At every coincidence there is a swelling of the sound, and hence the number of beats per second will be equal to the difference of the rates of vibration. If we tune two of three organ-pipes to the remaining one in the ratio of the minor and major third, the mutual ratio of the rates of vibration of the first two will be as 24: 25, that is to say, for every 24 vibrations to the lower note there will be 25 to the higher, and one beat. If the two pipes give together four beats in a second, then the higher has the fixed tone of 100 vibrations. The open pipe in question will consequently be five feet in length. We also determine by this procedure the absolute rates of vibration of all the other notes.

It follows at once that a pipe eight times as long or 40 feet in length will yield a vibrational rate of 12, which Sauveur ascribes to the lowest audible tone, and further also that a pipe 64 times as small will execute 6,400 vibrations, which Sauveur took for the highest audible limit. The author's delight at his successful enumeration of the "imperceptible vibrations" is unmistakably a.s.serted here, and it is justified when we reflect that to-day even Sauveur's principle, slightly modified, const.i.tutes the simplest and most delicate means we have for exactly determining rates of vibration. Far more important still, however, is a second observation which Sauveur made while studying beats, and to which we shall revert later.

Strings whose lengths can be altered by movable bridges are much easier to handle than pipes in such investigations, and it was natural that Sauveur should soon resort to their use.

One of his bridges accidentally not having been brought into full and hard contact with the string, and consequently only imperfectly impeding the vibrations, Sauveur discovered the harmonic overtones of the string, at first by the unaided ear, and concluded from this fact that the string was divided into aliquot parts. The string when plucked, and when the bridge stood at the third division for example, yielded the twelfth of its fundamental note. At the suggestion of some academician[138] probably, variously colored paper riders were placed at the nodes (noeuds) and ventral segments (ventres), and the division of the string due to the excitation of the overtones (sons harmoniques) belonging to its fundamental note (son fondamental) thus rendered visible. For the clumsy bridge the more convenient feather or brush was soon subst.i.tuted. . While engaged in these investigations Sauveur also observed the sympathetic vibration of a string induced by the excitation of a second one in unison with it. He also discovered that the overtone of a string can respond to another string tuned to its note. He even went further and discovered that on exciting one string the overtone which it has in common with another, differently pitched string can be produced on that other; for example, on strings having for their vibrational ratio 3:4, the fourth of the lower and the third of the higher may be made to respond. It follows indisputably from this that the excited string yields overtones simultaneously with its fundamental tone. Previously to this Sauveur's attention had been drawn by other observers to the fact that the overtones of musical instruments can be picked out by attentive listening, particularly in the night.[139] He himself mentions the simultaneous sounding of the overtones and the fundamental tone.[140] That he did not give the proper consideration to this circ.u.mstance was, as will afterwards be seen, fatal to his theory.

While studying beats Sauveur makes the remark that they are displeasing to the ear. He held the beats were distinctly audible only when less than six occurred in a second. Larger numbers were not distinctly perceptible and gave rise accordingly to no disturbance. He then attempts to reduce the difference between consonance and dissonance to a question of beats. Let us hear his own words.[141]

"Beats are unpleasing to the ear because of the unevenness of the sound, and it may be held with much plausibility that the reason why octaves are so pleasing is that we never hear their beats.[142]

"In following out this idea, we find that the chords whose beats we cannot hear are precisely those which the musicians call consonances and that those whose beats are heard are the dissonances, and that when a chord is a dissonance in one octave and a consonance in another, it beats in the one and does not beat in the other. Consequently it is called an imperfect consonance. It is very easy by the principles of M. Sauveur, here established, to ascertain what chords beat and in what octaves, above or below the fixed note. If this hypothesis be correct, it will disclose the true source of the rules of composition, hitherto unknown to science, and given over almost entirely to judgment by the ear. These sorts of natural judgment, marvellous though they may sometimes appear, are not so but have very real causes, the knowledge of which belongs to science, provided it can gain possession thereof."[143]

Sauveur thus correctly discerns in beats the cause of the disturbance of consonance, to which all disharmony is "probably" to be referred. It will be seen, however, that according to his view all distant intervals must necessarily be consonances and all near intervals dissonances. He also overlooks the absolute difference in point of principle between his old view, mentioned at the outset, and his new view, rather attempting to obliterate it.

R. Smith[144] takes note of the theory of Sauveur and calls attention to the first of the above-mentioned defects. Being himself essentially involved in the old view of Sauveur, which is usually attributed to Euler, he yet approaches in his criticism a brief step nearer to the modern theory, as appears from the following pa.s.sage.[145]

"The truth is, this gentleman confounds the distinction between perfect and imperfect consonances, by comparing imperfect consonances which beat because the succession of their short cycles[146] is periodically confused and interrupted, with perfect ones which cannot beat, because the succession of their short cycles is never confused nor interrupted.

"The fluttering roughness above mentioned is perceivable in all other perfect consonances, in a smaller degree in proportion as their cycles are shorter and simpler, and their pitch is higher; and is of a different kind from the smoother beats and undulations of tempered consonances; because we can alter the rate of the latter by altering the temperament, but not of the former, the consonance being perfect at a given pitch: And because a judicious ear can often hear, at the same time, both the flutterings and the beats of a tempered consonance; sufficiently distinct from each other.

"For nothing gives greater offence to the hearer, though ignorant of the cause of it, than those rapid, piercing beats of high and loud sounds, which make imperfect consonances with one another. And yet a few slow beats, like the slow undulations of a close shake now and then introduced, are far from being disagreeable."

Smith is accordingly clear that other "roughnesses" exist besides the beats which Sauveur considered, and if the investigations had been continued on the basis of Sauveur's idea, these additional roughnesses would have turned out to be the beats of the overtones, and the theory thus have attained the point of view of Helmholtz.

Reviewing the differences between Sauveur's and Helmholtz's theories, we find the following: 1. The theory according to which consonance depends on the frequent and regular coincidence of vibrations and their ease of enumeration, appears from the new point of view inadmissible. The simplicity of the ratios obtaining between the rates of vibration is indeed a mathematical characteristic of consonance as well as a physical condition thereof, for the reason that the coincidence of the overtones as also their further physical and physiological consequences is connected with this fact. But no physiological or psychological explanation of consonance is given by this fact, for the simple reason that in the acoustic nerve-process nothing corresponding to the periodicity of the sonant stimulus is discoverable.

2. In the recognition of beats as a disturbance of consonance, both theories agree. Sauveur's theory, however, does not take into account the fact that clangs, or musical sounds generally, are composite and that the disturbance in the consonances of distant intervals princ.i.p.ally arise from the beats of the overtones. Furthermore, Sauveur was wrong in a.s.serting that the number of beats must be less than six in a second in order to produce disturbances. Even Smith knows that very slow beats are not a cause of disturbance, and Helmholtz found a much higher number (33) for the maximum of disturbance. Finally, Sauveur did not consider that although the number of beats increases with the recession from unison, yet their strength is diminished. On the basis of the principle of specific energies and of the laws of sympathetic vibration the new theory finds that two atmospheric motions of like amplitude but different periods, a sin(rt) and a sin[(r + [rho])(t + [tau])], cannot be communicated with the same amplitude to the same nervous end-organ. On the contrary, an end-organ that reacts best to the period r responds more weakly to the period r + [rho], the two amplitudes bearing to each other the proportion a: [phi]a. Here [phi] decreases when [rho] increases, and when [rho] = 0 it becomes equal to 1, so that only the portion of the stimulus [phi]a is subject to beats, and the portion (1-[phi])a continues smoothly onward without disturbance.

If there is any moral to be drawn from the history of this theory, it is that considering how near Sauveur's errors were to the truth, it behooves us to exercise some caution also with regard to the new theory. And in reality there seems to be reason for doing so.

The fact that a musician will never confound a more perfectly consonant chord on a poorly tuned piano with a less perfectly consonant chord on a well tuned piano, although the roughness in the two cases may be the same, is sufficient indication that the degree of roughness is not the only characteristic of a harmony. As the musician knows, even the harmonic beauties of a Beethoven sonata are not easily effaced on a poorly tuned piano; they scarcely suffer more than a Raphael drawing executed in rough unfinished strokes. The positive physiologico-psychological characteristic which distinguishes one harmony from another is not given by the beats. Nor is this characteristic to be found in the fact that, for example, in sounding a major third the fifth partial tone of the lower note coincides with the fourth of the higher note. This characteristic comes into consideration only for the investigating and abstracting reason. If we should regard it also as characteristic of the sensation, we should lapse into a fundamental error which would be quite a.n.a.logous to that cited in (1).

The positive physiological characteristics of the intervals would doubtless be speedily revealed if it were possible to conduct aperiodic, for example galvanic, stimuli to the single sound-sensing organs, in which case the beats would be totally eliminated. Unfortunately such an experiment can hardly be regarded as practicable. The employment of acoustic stimuli of short duration and consequently also free from beats, involves the additional difficulty of a pitch not precisely determinable.

FOOTNOTES: [Footnote 134: This article, which appeared in the Proceedings of the German Mathematical Society of Prague for the year 1892, is printed as a supplement to the article on "The Causes of Harmony," at page 32.]

[Footnote 135: The present exposition is taken from the volumes for 1700 (published in 1703) and for 1701 (published in 1704), and partly also from the Histoire de l'AcadAmie and partly from the MAmoires. Sauveur's later works enter less into consideration here.]

[Footnote 136: Euler, Tentamen novae theoriae musicae, Petropoli, 1739.]

[Footnote 137: In attempting to perform his experiment of beats before the Academy, Sauveur was not quite successful. Histoire de l'AcadAmie, AnnAe 1700, p. 136.]

[Footnote 138: Histoire de l'AcadAmie, AnnAe 1701, p. 134.]

[Footnote 139: Ibid., p. 298.]

[Footnote 140: Histoire de l'AcadAmie, AnnAe 1702, p. 91.]

[Footnote 141: From the Histoire de l'AcadAmie, AnnAe 1700, p. 139.]

[Footnote 142: Because all octaves in use in music offer too great differences of rates of vibration.]

[Footnote 143: "Les battemens ne plaisent pas A l'Oreille, A cause de l'inAgalitA du son, et l'on peut croire avec beaucoup d'apparence que ce qui rend les Octaves si agrAables, c'est qu'on n'y entend jamais de battemens.

"En suivant cette idAe, on trouve que les accords dont on ne peut entendre les battemens, sont justement ceux que les Musiciens traitent de Consonances, et que ceux dont les battemens se font sentir, sont les Dissonances, et que quand un accord est Dissonance dans une certaine octave et Consonance dans une autre, c'est qu'il bat dans l'une, et qu'il ne bat pas dans l'autre. Aussi est il traitA de Consonance imparfaite. Il est fort aisA par les principes de Mr. Sauveur qu'on a Atablis ici, de voir quels accords battent, et dans quelles Octaves au-dessus on au-dessous du son fixe. Si cette hypothAse est vraye, elle dAcouvrira la vAritable source des RAgles de la composition, inconnue jusqu'A prAsent A la Philosophie, qui s'en remettait presque entiArement au jugement de l'Oreille. Ces sortes de jugemens naturels, quelque bisarres qu'ils paroissent quelquefois, ne le sont point, ils ont des causes trAs rAelles, dont la connaissance appartient A la Philosophie, pourvue qu'elle s'en puisse mettre en possession."]

[Footnote 144: Harmonics or the Philosophy of Musical Sounds, Cambridge, 1749. I saw this book only hastily in 1864 and drew attention to it in a work published in 1866. I did not come into its actual possession until three years ago and then only did I learn its exact contents.]

[Footnote 145: Harmonics, pp. 118 and 243.]

[Footnote 146: "Short cycle" is the period in which the same phases of the two co-operant tones are repeated.]

II.

REMARKS ON THE THEORY OF SPATIAL VISION.[147]

According to Herbart, spatial vision rests on reproduction-series. In such an event, of course, and if the supposition is correct, the magnitudes of the residua with which the percepts or representations are coalesced (the helps to coalescence) are of cardinal influence. Furthermore, since the coalescences must first be fully perfected before they make their appearance, and since upon their appearance the inhibitory ratios are brought into play, ultimately, then, if we leave out of account the accidental order of time in which the percepts are given, everything in spatial vision depends on the oppositions and affinities, or, in brief, on the qualities of the percepts, which enter into series.

Let us see how the theory stands with respect to the special facts involved.

1. If intersecting series only, running anteriorly and posteriorly, are requisite for the production of spatial sensation, why are not a.n.a.logues of them found in all the senses?

2. Why do we measure differently colored objects and variegated objects with one and the same spatial measure? How do we recognise differently colored objects as the same in size? Where do we get our measure of s.p.a.ce from and what is it?

3. Why is it that differently colored figures of the same form reproduce one another and are recognised as the same?

Here are difficulties enough. Herbart is unable to solve them by his theory. The unprejudiced student sees at once that his "inhibition by reason of form" and "preference by reason of form" are absolutely impossible. Think of Herbart's example of the red and black letters.

The "help to coalescence" is a pa.s.sport, so to speak, made out to the name and person of the percept. A percept which is coalesced with another cannot reproduce all others qualitatively different from it for the simple reason that the latter are in like manner coalesced with one another. Two qualitatively different series certainly do not reproduce themselves because they present the same order of degree of coalescence.

If it is certain that only things simultaneous and things which are alike are reproduced, a basic principle of Herbart's psychology which even the most absolute empiricists will not deny, nothing remains but to modify the theory of spatial perception or to invent in its place a new principle in the manner indicated, a step which hardly any one would seriously undertake. The new principle could not fail to throw all psychology into the most dreadful confusion.

As to the modification which is needed there can be hardly any doubt as to how in the face of the facts and conformably to Herbart's own principles it is to be carried out. If two differently colored figures of equal size reproduce each other and are recognised as equal, the result can be due to nothing but to the existence in both series of presentations of a presentation or percept which is qualitatively the same. The colors are different. Consequently, like or equal percepts must be connected with the colors which are yet independent of the colors. We have not to look long for them, for they are the like effects of the muscular feelings of the eye when confronted by the two figures. We might say we reach the vision of s.p.a.ce by the registering of light-sensations in a schedule of graduated muscle-sensations.[148]

A few considerations will show the likelihood of the rAle of the muscle-sensations. The muscular apparatus of one eye is unsymmetrical. The two eyes together form a system which is vertical in symmetry. This already explains much.