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later period. In examining this cube, for instance, it would appear at once that its sides were equilateral and equiangular; and that one of its horizontal sides was bounded by four vertical ones. But that its edge, diagonal of a side, and axis are to each other as 11:72: ✓ 3 could not be perceived with the bodily eye, but appears by the help of the Pythagorean problem.
The demonstrations, as is sufficiently evident, must have begun with such as were concrete, simple, and visible, and proceeded to such as were more comprehensive, abstract, and beyond the scope of the senses. For instance, the application of the Pythagorean problem to all right-angled triangles would scarcely have been undertaken at the beginning. But in the case of isosceles right-angled triangles, inspection would show, by a very simple demonstration, that the squares of the sides were together equal to the square of the hypothenuse. If this were proved, the question was then easily suggested, Is it true of all right-angled triangles? If a square were divided by a diagonal into two triangles, it was evident that each of them contained one right angle and two half right angles, the sumn of the three being two right angles; and then the question would naturally occur, Is this true of all triangles ?
, In the same manner it would be necessary to proceed from the simplest and most regular solids and figures to the more complicated and less regular; from those most easily seen by the eye to the more abstraet, requiring the use, not of the senses, but of the reason. When at last the most comprehensive demonstration and definition had been learned, there would be no further mention of the previous concrete cases, which had been an introduction to the study of the more abstract ones, but the cases to consider would now be those involved in the definition and demonstration last found.
It has repeatedly been observed that the teacher of a science must adhere to its proper course of development, and must in his instructions follow it more or less strictly. Every pupil ought once to follow this path, which its first discoverers and investigators worked out after
• A B C-Book of Crystallography," (A B C-Buch der Krystallkunde,) pp. IX., XI., XXIII., and 164 ; and Harnisch, "Mannil of the German Common School System," (Handbuch über das deutsche Folkschulwesens.) Ist ed., 1820, p. 232.
+ The demonstration may be somewhat as follows:
A B C, isosceles right-angled triangle. ABD E. the square of its hypothenuse, contains eiglut small triangles, and the squares on its sides together coutaiu also eight, and all of these small triangles are of the same size and shape.
'so many and long-enduring errors, but which the present pupils, with their teacher's aid, now find out in a shorter time, and with certainty.
According to these principles, to which I subscribe, I consider it natural to begin teaching geometry with treating of solids, with which it is highly probable that the actual development of the science began; and to proceed from that point, by abstraction, to the elements. It is here that Euclid's method should be adopted, and that we should proceed by demonstrations, from the elements up to solids. In the former course, it is instruction that leads, and reason silently follows; in the latter, the reason speaks, and the intuition must place faith in it.
Many mathematicians are now agreed that Euclid's demonstrative course of instruction should be preceded by an introduction of an intuitional character. In the theory of forms brought forward by Pestalozzi and his school, in particular, was discovered a preparatory course in geometry, in which intuition was the chief actor, as is the reason in geometry proper.*
Still, however, the beginning was not made with solids, but, in accordance with a method of elementarizing which was pushed even to caricature, with points—unmeasurable, dimensionless points. Lines come next, and were taught in innumerable and aimless combinations. Lastly, surfaces were discussed; for of solids Schmid's well-known Theory of Form, the predecessor of many more, scarcely spoke at all, and what little was said was not worth mentioning:
The necessity was afterward felt of beginning with a solid—the cube, for instance; but merely with the design of showing from it the process of abstraction by which to proceed from the solid to the point. As soon as this had been briefly done, they then commonly proceeded to the combination of points, lines, &c., and to other operations, as were just alluded to. How important soever this theory of form may seem to me, and however much I may honor the intelligence, industry, and effort with which this new course of discipline was worked out by able pedagogues, still I can not possibly recognize the method which they followed as the right one. I
What I would recommend is, that instruction in geometry should begin, not with such a brief analysis of one or another solid into its geometrical elements, but with a continued study, at some length, of many mathematical solids. And now, if solids are to be both the beginning and the end of the elementary study of geometry, the
* Part 2, p. 101.
t1 entirely agree with the acute and able judgment parsed by Curtmapp on the study of Forni in common schools, and on Froebel's " eccentric proposal to use geometrical combinations as a principal amusement for children." See Curtmann's "School and Life," (Die Schule und das Leben,) p. 62.
question naturally comes up, What bodies ? Shall they be those of which every stereometry treats--the prism, pyramid, sphere, cone, and cylinder ? Shall it be the five regular solids ?
The opinion of Montücla, already given on this point, might perhaps alarm us, even if inclined toward an affirmative. He compares the theory of the five regular bodies to ancient mines, which are neglected because they cost more than they produce. “Geometers," he continues, “ will use thein at most for a leisure amusement, or as suggestive of some singular problem.” But such old mining works are opened again, and afford great profits; and the merest leisure sometimes is the occasion of solemn earnestness. Of many of the solids which the ancient mathematicians constructed, with scientific geometrical skill,* the originals have been found in nature in our own times; and, besides these, an innumerable multitude of other beautiful forms, in wbich are revealed laws of which no mathematician ever dreamed.
It is mineralogy which has opened to us this new geometrical world--the world of crystallography. With this I first became
. acquainted, as I have already mentioned, in Werner's school, at Freiberg. When I afterward came to Yverdun, in 1809, and made myself acquainted with Schmid's Theory of Forms, this latter appeared to me the most uncouth of all possible opposites of crystallography.
This theory of forms consisted of endless and illimitable combinations. The object seemed to be to find at how many points a line could be intersected; but no reference was made to the question whether the figures resulting froin such combinations were beautiful or ugly. But, in the absence of a sense of mathematical beauty, great caution must be used in approving a course of mathematical instruction which consists principally of mathematical intuitions. Nothing of any value, as I have mentioned, was said of solids.. Every thing seemed designed to keep the boys in incessant, intense, and even overstrained productive activity, without any care whether the product was of any geometrical value. A formal result, it might be said, was the chief thing sought.
But how diametrically opposite was the study of crystallography at Freiberg to this unnatural and endless production of mathematical misconceptions! It began with a silent ocular investigation of the wonderfully beautiful crystals themselves; works of Him who is the "Master of all beauty." A presentiment of unfathomable, divine geometry came upon us; and how great was our pleasure as we gradually became acquainted with the laws of the various individual forms,
*Including several of the thirteen Archimedean solids.
and their relations. Nobody thought of any special formal usefulness in his study of crystals; it would have seemed almost a blasphemy to us had any one told us to use the crystals for our education. We quite forgot ourselves in the profundity and unfathomable wealth of our subject; and this beneficial carelessness seemed to us a much greater formal benefit than could have been obtained by any restless running and hunting after such a benefit.
The opposite impressions thus received at Freiberg and Yverdun are indelibly impressed upon my mind. And I readily admit that all my inclinations drew me toward a quiet investigation of God's works; an inward life from which my actual knowledge should gradually grow. In proportion as I have experienced the blessing of this peaceful mode of activity, I find an incessant, restless, overstrained activity more repulsive to me, and I am frightened at the pedagogical imperative mood, “Never stand still!” It is to me as if all beautiful Sundays and their sacred rest were entirely abrogated, and as if I were forced to hasten forward, restlessly and forever, without once delaying for peaceful contemplation, though the road should lead through the summer of paradise.
But to return to my subject.
When, twenty-four years ago, I wrote my “ Attempt at an A B CBook of Crystallography," (Versuch eines A B C-Buch der Krystallkunde,) I remembered, while employed on that common ground of mathematics and mineralogy, Schmid's Theory of Forms, and expressed the hope that a scientific crystallography, proceeding according to the laws of nature, might accomplish, in a regular manner and with a clear
purpose, what the theory of forms of Pestalozzi's disciples bad endeavored to do without regularity or definite purpose.
I was convinced that such a connection with the subject of crystals must give to the treatment of the theory of forms a character entirely new, and entirely opposite to that previously usual. Wherever begioners were required to practice this incessant combination and production, they would now be employed in becoming familiar with natural crystals and inodels of them. They should not be confined exclusively to models, lest they should fall into the error of supposing theinselves to have to do only with human productions; and of imagining that there are no other mathematics except those of man. Natural crystals lead the pupil to a much profounder source of mathematical knowledge; to the same source from which Plato, Euclid, and Kepler drank.*
* Mohl's valuable work on the forms of grains of pollen shows that among them are several mathematical ones; as octahedrons, tetrahedrons, cubes, and pentagonal dodecahedrons. Mohl's Coutributions, Plate I., 3; Pl. 11.. 30, 34, 35; Pl. VI., 17, 18: &c.) Schkuhr had already described dodecahedrons and icosahedrons. Thus mathematical forms are found also in the mathematical world.
I will here give some details to show that proper instruction in crystallography will serve the same purpose which was sought by the theory of forms. Every solid, I would first say,* fills a certain space, and the questions to ask respecting it are,
, 1. What is the form of the solid (or of the space which it fills ?)
2. What is its magnitude, (or the magnitude of the space which it fills )
Similar questions arise respecting limited superficies. If now we compare two solids, or two surfaces, they may be either,
a. Alike in form and magnitude, or congruent; as, for instance, two squares or cubes of equal size. The squares will cover each other, the cubes would fill the same mold.
b. Alike in form but unlike in magnitude, or similar; as two squares or cubes of different sizes. Of two similar but unequal solids, the smaller, A, may be compared with the larger, B, in a decreasing proportion. If any line of A equals, for instance, one-half of the corresponding line of B, all the other lines of A are to the corresponding ones of B in the same proportion.
c. Unlike in form but alike in magnitude, or equal; as a square and a rhomboid of equal base and higbt; a square prism and a crystal of garnet, where the side of an end of the prism equals the short diagonal of one of the rhombic surfaces of the crystal, and a side edge of the prism is twice as long as the same diagonal.
d. Unlike in form and magnitude.
The theory of form, as its name indicates, is chiefly concerned with the forms of bodies and surfaces; and so is crystallography. The latter deals only incidentally with the materials of bodies, and treats chiefly of the shape of single crystals, and the comparison of different ones, with the design of discovering whether they vary from each other or not.
I was occupied many years with elementary instruction in crystallography; and from these labors resulted the “ Attempt at an A B C-Book of Crystallography," which I have already mentioned.
In the course of this instruction I found by experience how much not only older persons but even boys of ten or twelve are attracted by these beautiful matheniatical bodies, and how firmly their forms were impressed on their minds ; so firmly that the more skillful of them could go accurately through the successive modifications of related forms, without using any models.
Any one who has studied elementary crystallography, as an introduction to geometry, will find this course a great assistance to the understanding of the ancient Greek geometers. He will not ask, as
“ A B C-Book of Crystallography," p. 162.