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collective impression produced by the locality examined, to an active effort to distinguish this impression into its component parts. The great compound picture of the district about him divides into innumerable little ones, of towns, men, animals, trees, flowers, and in like manner do the mountains-for instance, their minerals, and their structure. What has been said of the method of geognostic study, both of its rudiments and of its ultimate purpose, is applicable, as we shall see, to other branches of natural science.


"As the susceptible painter, the ingenuous poet, rejoice in the heavens and the earth, so does the youthful heart." And, I may add, the future geognosist. But, it will be asked, does this laborious and prosaic workman proceed from the same initial point of education as the passionate and delicate painter? I answer, decidedly, Yes; and, I add, other departments of art begin, in like manner, coincidently with other departments of science. If a boy loves flowers, he may become equally a botanist or a flower-painter. The celebrated painter of animals, Paul Potter, the author of "Reynard the Fox," as well as the great zoologist, Cuvier, all, as boys, took delight in animals, and had an eye susceptible to them. A liking for beautiful mathematical bodies may characterize a future mineralogist, or mathematician, or architect. Susceptibility to colors indicates a future painter or a future optician; and an ear for music, either a musician or an acousticist. Nor do the different roads of the artists and naturalists, who proceed from the same point, ever become entirely separate. Michael Angelo was a great anatomist; Durer wrote on perspective, and on the relations of the human body; Otto Philip Runge constructed a theory of colors. Goethe sang of flowers, and wrote his valuable "Metamorphoses of Plants;" he had an eye seldom equaled for the beauty of mountains, and he both observed and described them in a masterly manner, according to their geognostic character. A man who is endowed with susceptibility to beauty, and the artist's power of representation, and also with clear and energetic thought, will produce scientific works containing beauty, and artistic works of profound thought. It is not only true that we find united, in extraordinary men, great capacity both for science and art, and that the first rudiments of scientific and artistic training are frequently the same, but we see that many arts need the aid of science, and many sciences of the arts. The architect must understand mechanics; the painter, perspective, anatomy, and the chemistry of colors: botany and zoology require good pictures of plants and animals; and mineralogy, clear and accurate drawings of crystals.

Science seeks principally truth; but art, beauty. While the botanist endeavors to establish as correctly and completely as possible the idea of the species Rose, the painter tries to present his ideal of a Rosa centifolia; and the poet leads us, through the gardens of poetry, to roses of unimaginable beauty. While the Greek sculptor carved the Lions of St. Mark, Cuvier gave us an excellent description of the king of beasts. From the school of Werner came scientific works on mineralogy and mining, and likewise the miners' songs of Novalis.

I have lengthened this discussion, in order to bring out a pedagogical rule to which I have already referred in speaking of teaching geognosy. It is, to have constant reference, not only at the beginning but throughout all the course of instruction in natural science, to the beauty of God's works; to cultivate the pupils' susceptibility to this beauty; and to develop, along with the receptive faculty, however directed, the power of representing as perfectly as possible the thing seen so that, for example, the boys shall learn not only to examine and recognize plants and crystals but to draw them. It is more necessary to refer to this, because the beauty of which I speak is so wholly indifferent to so many teachers. They make no endeavor to learn whether their pupils take such pleasure in flowers, and examine them with the same penetrating attention that a flowerpainter uses. They rather make their tyros analyze them, pull them to pieces, physically and mentally count their anthers and pistils, &c. Before the boys have even gained a thorough and familiar idea of the flower, they are made to endeavor to get an idea of its species in this destructive manner.

Especial haste is used, in those departments of natural science which are based on mathematics, in proceeding from observation by the senses to abstract mathematical theory. It is no wonder that this is the case in our day, when atomistics and mechanics, in a mathematical form, are every where forcing themselves forward, and where so many are seeking after mere bare truth only, without any reference at all to beauty.


Mathematics are the root and blood of a knowledge of the laws of nature and of art.* It reveals the laws of crystallization and of chemical unions; the number of petals and of anthers; the figure, size, and motions of the stars. It is the soul of the firmness of mighty cathedrals, of harmony in music; it gives the painter proportion and

"The form was in the archetype before it was in the work; in the divine mind before it was in the creature."-Kepler, "Harmon. Mundi," I.

grouping, and lives in the hexameters of Homer and the choral measures of the tragedians.

But can we for such reasons, when instruction is required in music, drawing, &c., answer, Yes! we teach mathematics, and shall thus at least indirectly prepare the pupil for the studies which you wish! By no means; and as little would it serve where instruction in natural science is required. These considerations lead to the very important question of the relations between mathematical instruction and instruction in drawing, music, natural science, &c. On this point there are two opposite opinions; one of which would place mathematics at the beginning of the courses, and the other at the end.

In support of the former of these doctrines, it may be said, "If we grant that mathematics form the theory of laws of nature and art, what could be more appropriate than to begin with it? When the scholars have gained a thorough acquaintance with pure mathematics, they thus become capable of easily mastering any natural science, or of acquiring knowledge and skill in the arts. In the pure mathematics is the point for setting the lever which will move the world; it is the center from which light radiates to innumerable points on the circumference-to innumerable sciences and arts. Should the teacher rather choose to select from their multitude one point or a few, and thence seek to reach the center?"

This view is plausible, but untenable.

The history of the arts and sciences is opposed to the idea of beginning with instruction in pure mathematics. The course of development of the human race has not confirmed its propriety, either. The fact was not that minds of a purely speculative character, operating entirely within themselves, developed pure mathematical truth, which others afterward applied to nature and art. In this sense, there have been almost no applied mathematics. The truth is, that a gradual and deliberate apprehension of purely mathematical relations has developed in such departments as music, surveying, architecture, drawing, astronomy, geology, &c.,* from a beginning of purely material conceptions, yet guided by the principles of mathematics, hidden within them as a human instinct. From this heterogeneous world of phenomena its common elementary spirit, the spirit of pure mathematics, arose subsequently. This succession of the sciences can not be too carefully remembered, for every scholar has to go through one more or less similar.

It is also a great error to believe that a person thoroughly grounded

How completely new is the world of beautiful inter-related mathematical bodies which has arisen from the investigations into natural crystals, and how utterly were the great early mathematicians without an a priori knowledge of it!

in pure mathematics is thus fully prepared for all the arts and sciences which are based on mathematics-that he can juggle with them by means of his formulas. Is it supposed that one who has learned general bass-the mathematical basis of music-has by that means trained his feelings and his ear? Does knowledge of perspective make a painter; or of metrics, a poet? Is one who knows. how to calculate a crystal a mineralogist?

On the contrary, the reason, during those years when it is dormant, but the senses are active and hungry, is powerfully stimulated by pure mathematics, and developed at the expense of the senses. The boy, under an unnatural mental excitement, and thrown into this wholly subjective train of thought-this activity of the reason exclusively within itself-loses his quiet, peaceful, and natural bodily sensitiveness to the material creation. He will even, in time, lose the humility with which he sought after the laws of God's world, with self-sacrifice and sincere industry, and with which he felt a pious joy in discovering them; and he imperceptibly becomes a scientific egoist, having no feeling for faith in any thing except in his own mind and mental labor; and who, even if he discovers a natural law, can only rejoice in it as in the child of his own intellect-as if he were a lawgiver to the creation. I am not exaggerating. Only consider any one of many trained naturalists, who have been educated in this way, whether they are not such as I have said.

If, now, we would preserve a natural and proper susceptibility to nature in our pupils—if we would protect them against such a premature and bald forcing of the growth of the understanding-we must permit them to begin their studies with the natural and easy observation and practice of youth; and gradually bring them forward from this to a properly pure mathematical mode of investigating and training.

Mathematical instruction, too early put in the place of physical observation of nature, is so far from compensating for it that it is injurious to it. Bacon's observation is here eminently in point: "Mathematics should terminate the study of natural philosophy; it should not introduce or create it."


With Werner opened a new era not only in the science of mineralogy but also in the method of instructing it. Before him, scientific mineralogy was scarcely known; or the thorough knowledge, description, or classification of minerals. Naturalists were satisfied with un

* What has here been said will be illustrated by subsequent examples. Further details will be found in the chapter on Geometry.

derstanding and teaching such of their peculiarities as were most obvious. Gold, they said, is yellow, bright, and heavy. But these same terms might be used to describe copper pyrites, or iron pyrites—as in Messing. Werner perceived how defective were such descriptions; and how far they were from being sufficient to describe the peculiarities of a mineral or a species—and still more to distinguish with entire certainty one mineral, or one species, from another. He believed that not merely this or that prominent characteristic of a mineral, but all of its characteristics, the most obvious and the most recondite alike, should be understood and expressed. It was in this belief that he wrote his "Theory of External Characteristics," (Lehre von den Äussern Kennzeichen.) What he here aimed at was, in fact, an exhaustive statement of the sensible characteristics of minerals; though all that he stated himself to seek was the best, fittest, and most invariable expressions for their characters, their species, and their grades. The motto of his book was "Be not facile in choice of words; in order that you may agree in things." And he arranged these characteristics in a definite and well-adjusted order.

In describing all the peculiarities of a mineral, he paid all his attention to the order, clear comprehension, and expression of its external characteristics. He endeavored to set forth in words the whole of the peculiarities of the mineral, in the most correct manner, so that his description should fully state the elements of the whole impression made by the mineral upon the senses.

In a similar manner he described a species of minerals; but with this difference, that, whereas the single stone has one definite color, one definite mode of crystallization, &c., the species to which it belongs usually includes a variety of related colors and crystals, which must be described.

Not to enlarge upon the brief general theory of classification with which Werner began, he commenced his mineralogical lectures proper with instruction in the external marks. This was followed by a description of the species closely connected with it, and by a rapid exhibition of the groups described. His oral lecture, which was of great value in itself, was the prominent feature; and the actual display of the groups of minerals was quite subordinate.

"Words are good," says Goethe, "but not best." This was true in the present case. I have already mentioned how we strove in vain not to be confined to a mere description of the minerals, but to ob

It is this defectiveness in descriptions which leaves us so often at a loss to know what mineral the carly writers-Pliny, for instance-meant by any given name.

This work appeared in 1774, and was translated into various languages. Werner was twenty-four when he wrote it.

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