these evidences of defective scholarship were glossed over at the time of the examination by some mechanical contrivance. How far these young men were familiar with the personages, the thoughts, and the forms of the ancient world, whether they had any sense of our dependence on the ancients, and of our being their intellectual descendants -for that is the sum and substance of humanism-I of course have had no opportunity for determining. Nor did I get any systematic information of their historical knowledge. However, their indifference toward broad ideas and historic sequence makes it difficult for me to believe that they are permeated with the spirit of antiquity, or that they had received a sound historical training.

To all this I must add another deplorable fact. For the most part these young people wrote in ungrammatical and inelegant German. Owing to the unsettled state of our orthography, our word-formation, and our construction of sentences, instruction in the mother-tongue is more difficult among ourselves than among people who have a settled usage in language. But the young people, as a rule, did not even suspect that any one could care about purity of language and pronunciation, force of expression, brevity, or pointedness of style. One is ashamed, as a German, of such barbarism as this, knowing what care instruction in the mother-tongue receives from the French and English, in whose eyes an infraction of its rules appears to be, as it were, a sacrilege. The more closely this blemish in our educational practice is connected with a deep-lying national defect, the more is it to be wished that the gymnasia had been successful in removing it. This neglect of the mother-tongue is, in the youth of the present day, aecompanied by a lack of acquaintance with the German classics that is oftentimes astounding. Time was when, in Germany, no one any longer quoted from the first part of "Faust," because quotation had been overdone. Is the time now coming when it can no longer be quoted, because no one would understand the allusion?

With respect to instruction in mathematics, I know that but few masters succeed in advancing all their pupils equally. Clearly there are minds highly gifted in other respects, but to which mathematics is a book with seven seals. I would only remark upon the mathematical programme prescribed by tradition and convention for the highest class in our gymnasia. In a semi-official plan of studies this programme is given as follows: "Geometry of solids, with mensuration of surfaces and volumes; geometrical and stereometrical problems; problems in algebra, particularly in its application to geometry; indetermined equations; continued fractions; the binomial theorem." Though under "problems in algebra, particularly in its application to geometry," analytical geometry might be included, that branch is omitted from the gymnasial plan of studies by a ministerial decision of ancient date, but still in full force, and the mathematical programme of the highest realschulen surpasses in this respect that of the gymnasia.

The influence of mathe

Now, this I hold to be a serious error. matics as an educating force is not fully exerted till the student passes from these elementary studies to analytical geometry. No doubt, even simple geometry and algebra accustom the mind to strict quantitative reasoning, and to assuming as true nothing but axioms or demonstrated propositions. But the representation of functions by curves or surfaces opens a new world of ideas, and teaches us the use of one of the most fruitful methods whereby the human mind has increased its own powers. What the invention of this method by Viète and Descartes was to mankind, that will initiation into it still be to every mind that has any turn for such studies-namely, an illumination marking an epoch in life. This method has its roots in the profoundest depths of the human intellect, and hence is of far higher importance than the most ingenious analytical processes which are applicable only to a particular case. True, trigonometry is analytical geometry; as taught in the gymnasia, trigonometry, like stereometry, as both these terms indicate, has to do rather with mensuration, and its use is restricted to a certain class of problems. On the other hand, between any two quantities whatsoever, of which the one can be regarded as dependent on the other, there never exists a relation so complicated but that it may be represented by a curve; of this fact Quetelet has furnished an instructive demonstration-as, for example, where he represents by curves criminal tendency, literary talent, etc. This mode of representing the mutual dependence of things is of as much advantage to the government functionary and the political economist as to the physicist and the meteorologist.

But in medicine it is indispensable. In the preface to my "Untersuchungen über thierische Elektricität," which bears the date of March, 1848, I spoke in commendation of it as a means of bringing mathematics to bear on physiology, even in cases where the complexity is so great as to preclude the possibility of measuring, of weighing, or of calculating time. I then first laid an absciss-axis in a nerve, while Ludwig made the blood-circulation itself trace in curves its variations of pressure, and Helmholtz made the muscle in like manner trace its own contractions. Nowadays, thanks mainly to the labors of Marey, there is scarcely any department of experimental physiology or pathology that does not yield, through the graphical method, results of high importance. But, as our students of medicine may have quit the gymnasium without ever having so much as heard of a system of coördinates, I am compelled, at the opening of my lectures on physiology, to make my hearers acquainted with the elements of analytical geometry.

From the reasons assigned for the above-quoted decision of the ministry, whereby conic sections are excluded from the gymnasium course of study, it is plain that its author was unacquainted with the general scope of the branch of science he put under ban, and that he considered

it as too difficult for the prima.' This is a mistake. On the contrary, there are minds which, though highly gifted, and of a philosophical turn, yet lack the sort of subordinate attention that is necessary in order, for instance, to carry out a long trigonometrical calculation, and which find themselves much more at ease in analytical geometry. The fact that analytical geometry prepares the way through the differential and integral calculus to the last and highest aims of mathematics, and hence to their most difficult portion, should only form one reason more for beginning the study of it in the gymnasium. And, not to pass by unanswered an objection which might be raised, I would remark that, owing to the flourishing state of mathematical instruction in our universities for a long time past, the present masters of mathematics in the higher classes of our gymnasia are, almost without exception, qualified to teach analytical geometry, and would even be glad were they authorized to teach that branch. Many of the highest living authorities in this department share in the views which I have here expressed. Then, too, in several gymnasia of non-Prussian Germany, analytical geometry is already taught.

I will not now dwell on the fact that the freshmen in our medical classes, who, in the course of their studies, and later, in the practice of their art, have to depend largely on a right use of their senses, bring from the gymnasium only a very defective training in this respect. I omit the consideration of this, because we have not to do here with the medical student as such, but only in so far as he typifies the student in general; and I take him as a type because my observations on the work of the gymnasium are based principally on the results seen in him. Here the question arises, whether the gymnasium attains its end better in the case of students belonging to the other faculties. To a certain extent it does. With those who later devote themselves to the intellectual sciences, natural disposition and home-surroundings will oftentimes be more favorable to humanistic studies than with those who are impelled by hereditary realism toward medicine and the investigation of Nature. Besides, students of theology and jurisprudence are more favorably situated for retaining their humanistic culture than are students. of medicine, who from their first semester have to do with a world of things which have no connection, save through their terminology, with classical studies. Hence the average degree of humanistic culture among medical students is a very good test for determining how far the gymnasium is in a condition to oppose the encroachments of realism.

But even when we take into account all the youths who receive a gymnasium education, however diverse their tendencies as regards branches of study, we do not find in them so quick an interest in classical studies as would justify us in seriously expecting from it a reaction in the idealistic sense. Not reckoning philologists, who of course are not within the scope of our remarks just now, there are but few stu1 Highest class in the gymnasium.

dents, indeed, who in later years ever open an ancient author. So far from having any warm love for the classics, most persons regard them with indifference; not a few with aversion. They are remembered only as the instruments by means of which they were made familiar with the rules of grammar, just as the only conception they retain of universal history is that of learning by rote insignificant dates. Was it for this that these youths sat for thirty hours weekly on a school-bench till their eighteenth or twentieth year? Was it for this that they devoted most of their time to studying Greek, Latin, and history? Is this the result for the attainment of which the gymnasium remorselessly englooms the life of the German boy?

In view of this state of things, we ask whether everything is going on aright; whether it is not time, and whether it is not worth our while, to make an effort at reform? Here as elsewhere it is easier, especially for outsiders, to find fault than to determine how to repair the defect. Here, as is so often the case in complicated questions of administration and of human life in general, there are many causes in operation. We take into consideration one, while ten others of no less importance escape unnoticed. Still, though I expose myself to this danger, I will not refrain from expressing my views.

Without meaning any offense to the distinguished men who have taken an active part in organizing our gymnasia, or who are still so engaged, I cannot conceal my conviction that the spirit of the gymnasium does not sufficiently keep pace with the development of the human mind in modern times. As is evident from what has already been said, I am fully alive to the perils with which our intellectual culture is threatened by an excess of realism. At the same time, we cannot shut our eyes to the fact that natural science has given a new aspect to human existence. We should be imitating the ostrich in burying its head in the sand were we to deny the mighty revolution described above, and it were a vain and perilous thing to try to stop the rolling wheel of such a process of development. But hitherto the gymnasium has not taken this development sufficiently into account. Despite a few concessions, which are apparent rather than réal, it is still what the Reformation made it, when as yet there was no natural science—namely, a learned school essentially designed as a means of preparing for the study of the intellectual sciences.

In this backwardness of the gymnasium and its refusal to comply with the demands of the time lies the strength of the realschule. I do not propose to enter here on the intricate question of the competencies proper to each of these two kinds of institutions. For the rest, I agree with the views of those who desire only one species of higher schools, which should fit their pupils equally for the university, the industrial or architectural academy, the army, etc. Plainly, this would be simply the humanistic gymnasium transformed so as to meet these new requirements. Apart from measures of administration, all that is needed to

put an end to the rivalry of the realschule is, that the gymnasium should sacrifice to the needs of the time some of its time-honored but antiquated claims, and conform itself somewhat more to the tendencies of the modern world. So soon as the gymnasium becomes imbued bona fide with a new spirit, and insures fitting preparation even to those who devote themselves to other than intellectual sciences, this rivalry will cease of its own accord. The much-mooted question of the admission of realschulen pupils to faculty classes would thus be settled, for then the realschule would revert to its original intent, and be an industrial school— an institution of great importance in its proper sphere.

What, then, do I demand of the gymnasium so that it shall appear to meet the requirements of the time? Essentially, very little, indeed. First, I demand more mathematics. The mathematical course must include the discussion of equations of the second degree, and a few other plane curves, and must also give an introduction to differential calculus through the theory of tangents. To this end, a greater number of hours must of course be given to mathematics-six or eight, instead of four. In the examinations for advancement and graduation, mathematics must really stand on an equality with the ancient languages and history. The equality of the teacher of mathematics with the teachers of the other branches would in this way be made an actual fact.

It will now, perhaps, be expected that I will further demand a large increase in scientific instruction. But I do not at all purpose to convert the gymnasium into a school for science-teaching. All that I ask is that as much shall be done to meet the wants of the future physician, architect, or military officer, as those of the future judge, or preacher, or teacher of classical languages. Thus, I ask for only so much natural history in the lower classes of the school as will awaken the faculty of observing, and that facilities be given for familiarizing the lads with the method of classification, which is rooted in the depths of the understanding, and whose educational force is so eloquently described by Cuvier. Let Darwinism, of which I am myself an adherent, be excluded from the gymnasium. In the higher classes, for the reasons assigned in my report, I should like to have taught, not physics and chemistry with experiments, but mechanics, the elements of astronomy, also of mathematical and physical geography-to which studies one hour more than heretofore could be devoted without injury.

But how are we to find time for these innovations? In the prima two hours might be gained by doing away with the religious instruction. We cannot understand the use of such instruction in a class whose Protestant pupils have all been confirmed; and yet, in the semiofficial plan of studies already quoted, more than half a page of fine print is expended in setting forth the subject-matter of this instruction, while five lines suffice to dispatch the mathematical programme. On reading this half-page and the corresponding half-page for the upper second class, one imagines he has before him the programme of a theological