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 which 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 from 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 had 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 beginners were required to practice this incessant combination and production, they would now be employed in becoming familiar with natural crystals and models of them. They should not be confined exclusively to models, lest they should fall into the error of supposing themselves 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 Contributions, Plate I., 3; Pl. II.. 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 hight; 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 mathematical 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

*See my "A B C-Book of Crystallography," p. 162.

the modern mathematicians do, what is the use of investigating the regular solids? And he will find himself much better able to study in the method of the ancients; a method the neglect of which has been lamented by Fermat, Newton, and Montücla. A later writer has described this method as one which speaks to the eyes and the understanding, by figures and copious demonstrations. And he laments that the more recent mathematicians have allowed themselves to be carried to a harmful extreme by the extraordinary facility of the algebraic analysis. "In fact," he says, "the ancient method had certain advantages, which must be conceded to it by any person only even moderately acquainted with it. It was always lucid, and enlightened while it convinced; instead of which, the algebraic analysis constrains the understanding to assent, without informing it. In the ancient method, every step is seen; and not a single link of the connection between the principle and its furthest consequence escapes the mind. In the algebraic analysis, on the other hand, all the intermediate members of the process are in a manner left out; and we merely feel convinced in consequence of the adherence to rule which we know is observed in the mechanism of the operations in which great part of the solution consists."*

Speaking pedagogically, no one can doubt, after the descriptions thus given, whether the geometrical method of the ancients has the advantage, in regard to form, over the analytical one of the moderns. I have shown elsewhere how harmful it is to give the boys formulas, by whose aid they can easily reckon out what they ought to discover by actual intuition; as in the case where a pupil, who scarcely knows how many surfaces, edges, and angles a cube has, computes instantly by a formula, by a mere subtraction, what is the number of angles of a body having 182 sides and 540 edges, without having the least actual knowledge of such a body.

'An instance of the predominance of the analytic method is found in the "Mécanique Céleste" of Lagrange, which appeared in 1788. In this, the author says, "The reader will find no drawings in this work. In the method which I have here employed, neither constructions nor any other geometrical nor mechanical appliances are needed; nothing but purely algebraical operations."

IX. ARITHMETIC.

{Translated from Raumer's "History of Pedagogy," for the American Journal of Education.]

THE difference between ancient and modern methods of instruction is remarkably clear in the case of arithmetic.

By way of describing the ancient method, I will cite portions of one of the oldest and best reputed of German school-books-the "Elementa Arithmetices" of George Peurbach.* This author was, in his time, the greatest mathematician in Germany; and one of his pupils was the great Regiomontanus.t

Peurbach's arithmetic began with the consideration of numbers. "These," he แ says, are divided by mathematicians into three kinds: into digits, which are smaller than ten; articles, (articuli,) which can be divided by ten without a remainder; and composite numbers, consisting of a digit and an integer. Unity is however no number, but the rudiment of all numbers; it is to number what a point is to a line. In arithmetic it is usual, after the manner of the Arabs, who first invented it, to work from right to left. Every figure, when standing in the first place at the right hand, has its own primitive value; that in the second place has two times its primitive value, in the third place a hundred times, in the fourth one thousand times, and so on."

The second chapter is on addition. "To unite several numbers in one, write them so that all the figures of the first place (units) shall stand under each other, and in like manner of the second place, and so on. Having arranged them in this way, draw a line under them, and then begin the work at the right hand by adding together all the numbers of the right column. The sum resulting from this

"Elements of Arithmetic. An algorithm of whole numbers, fractions, common rules, and proportions. By George Peurbach. All recently edited with remarkable faithfulness and diligence. 1536. With preface by Philip Melancthon." (Elementa Arithmetices. Algorithmus de numeris integris, fractis, regulis communibus, et de proportionibus. Autore Georgio Peurbachio. Omnia recens in lucem edita fide et diligentia singulari. An. 1536. Cum præfacione Phil. Melanth) Peurbach was born in 1423, and died 1461.

"This philosophy of celestial things was almost born again in Vienna under the auspices of Peurbach. This whole department of learning, (astronomy,) after having lain in dishonor for centuries, has of late flourished anew in Germany, under the restoring hands of two men, Peurbach and Regiomontanus. Their very achievements testify that these two heroes were raised up, for the promotion of that branch of learning. by some wonderful power of divine appointment." This is Melancthon's opinion, as given in his preface to the "Sphæra” of Sacro Bosco. Comp. Montücla, “History of Mathematics,” part 3, book 2; also Schubert's "Peurbach," &c.