at the rapidity of the former. I kept a diary, in which I daily entered briefly the work of each pupil, and how he had done it. This is of the greatest use in tracing and guiding their development. If the number of pupils was large, I found the following arrangement very convenient. I had all the more difficult crystals numbered, according to Hauy's plates, and the number lay with each one. The pupils, who had made sufficient progress, made a written description of the crystals, and laid their paper next to the described crystal. Thus only a very brief comparison of their description with my own was necessary. If they agreed, well; if not, the pupil studied the crystal further, until the descriptions coincided-unless, indeed, there had been an error on my part. Of such an occurrence I am never ashamed. I do not desire to be to my pupils an undisputed authority, but a teacher who understands his duty to them; and his first duty is love of truth.

VIII. GEOMETRY.

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

THE school-days of the writer fell in the latter years of the last century. At that time the opinion prevailed that but few scholars had a talent for mathematics; an opinion, indeed, which seemed to be supported by the usually trifling results of mathematical instruction. Later defenders of this department of study, however, controverted this doctrine. It is not the pupils, they said, who are deficient in capacity for learning mathematics; it is the teachers, who have not the talent for teaching it. If the teachers would follow the proper method, they would learn that all boys have more or less capacity for mathematics.

When I remember how often even the more talented of my companions fell into despair from finding themselves, with the best inclination, unable to follow the instructions of their mathematical teacher, I find myself ready to agree with these defenders.

At the end of my university course, I went to Freiberg. At the mining school there, under the able instruction of Werner, I first became acquainted with crystallography, which had inexpressible attractions for me. The more I advanced in this study, and the greater my love of it, the more clearly I saw that crystallography was for me the right beginning, the introduction, to geometry. What if this is the case, I reflected, with others also; especially for students of a more receptive tendency, who are repelled by the rigors of logical demonstrations?

No one can quite escape from himself; and the reader will forgive me if, in the following views upon elementary instruction in geology, I exhibit too much of the course of my own studies in it. He can, however, abstract what is merely personal from what is applicable to others.

And now to my subject.

Formerly geometry and Euclid were synonymous terms. To study Euclid was to study geometry; he was the personification of geometry. His "Elements," a school-book for two thousand years, is much the oldest scientific school-book in the world. Composed three hundred years before Christ, for the Museum at Alexandria, it was exclusively

used in ancient times, and in modern times also, down to the eighteenth century.

To this imposing permanent eminence of Euclid's "Elements," for two thousand years, corresponds its great diffusion among civilized and even half-civilized nations. This is shown most strikingly by the great number of translations of it. It has been translated into Latin, German, French, English, Dutch, Danish, Swedish, Spanish, Hebrew, Arabic, Turkish, Persian, and Tartar.*

With few exceptions, there is the utmost harmony in praise of Euclid. Let us hear the evidence of a few authors. Montücla, the historian, says, "Euclid, in his work, the best of all of its kind, collected together the elementary truths of geometry which had been discovered before him; and in such a wonderfully close connection that there is not a single proposition which does not stand in a necessary relation to those preceding and following it. In vain have various geometers, who disliked Euclid's arrangement, endeavored to break it up, without injuring the strength of his demonstrations. Their weak attempts have shown how difficult it is to substitute, for the succession of the ancient geometer, another as compact and skillful. This was the opinion of the celebrated Leibnitz, whose authority, in mathematical points, must have great weight; and Wolf, who has related this of him, confesses that he had in vain exerted himself to bring the truths of geometry into a completely methodical order, without admitting any undemonstrated proposition, or impairing the strength of the chain of proof. The English mathematicians, who seem to have displayed most skill in geometry, have always been of a similar opinion. In England, works seldom appear intended to facilitate the study of the sciences, but in fact impede them. There, Euclid is almost the only elementary work; and England is certainly not wanting in geometry."

The opinion of Lorenz agrees entirely with that of Montücla. In Euclid's works, he says, "Both teacher and pupil will alike find instruction and enjoyment. While the former may admire the skillful association and connection of his propositions, and the judgment with which his demonstrations are joined to each other and arranged in succession, the latter will enjoy the remarkable clearness and (in a certain sense) comprehensibility which he finds in him. But this ease of comprehension is not of that kind which is rhetorical rather than demonstrative, and this absolves from reflection and mental effort; such an ease, purchased at the expense of thoroughness, would be beneath the dignity of such a science as geometry. And more

Montücla, I., 24. The list of editious and translations of Euclid's “Elements" occupies, in the fourth part of Fabricius' "Bibliotheca Graca," sixteen quarto pages.

*

over, Euclid himself was so penetrated with a sense of the derivation of the value of geometry, from the strict course pursued in its demonstrations, that he would not venture to promise even his king any other way to learn it than that laid down in the 'Elements.'* And in truth, the strictly scientific procedure, which omits nothing, but refers every thing to a few undeniable truths by a wise arrangement and concatenation of propositions, is the only one which can be of the greatest possible formal and material use; and authors or teachers, who lead their readers or pupils by any other route, do not act fairly either to them or to the science. Nor have the endeavors, which have at various times been made, to change Euclid's system, and sometimes to adopt another arrangement of his propositions, sometimes to substitute other proofs, ever gained any permanent success, but have soon fallen into oblivion. Geometry will not come into the so-called 'school method,' according to which every thing derived from one subject—a triangle, for instance-is to be taken up together.. Its only rule of proceeding is to take up first what is to serve for the right understanding of what comes afterward."

Thus Lorenz considered Euclid's work unimprovable, both as a specimen of pure mathematics and as a class-book. Kartner thought the same. The more the manuals of geometry differ from Euclid, he said, the worse they are. And Montücla, after the paragraph which I have quoted, proceeds to detail the defects of the correctors of Euclid. Some, disregarding strictness of demonstration, have resorted to the method of inspection. Others have adopted the principle that they will not treat of any species of magnitude-of triangles, for instance-until they have fully discussed lines and angles. This last, Montücla calls a sort of childish affectation; and says that, to adhere to the proper geometrical strictness in this method, the number of demonstrations is increased as much as it would be by beginning with any thing of a compound nature, and yet so simple as not to require any succession of steps to arrive at it. And he adds: "I will even go further, and am not afraid to say that this affected arrangement restricts the mind, and accustoms it to a method which is quite inconsistent with any labors as a discoverer. It discovers a few truths with great effort, when it would be no harder to seize with one grasp the stem of which these truths are only the branches."†

"There is no royal road to geometry."

The

This reads as if Montücla had read many of the modern mathematical works. abridgment and alteration of the "Elements" began as early as in the sixteenth century, and in the second half of the seventeenth the number of altered editions increased. Such were "Eight books of Euclid's 'Elements.' arranged for the easier understanding, by Dechales," (Euclidi's elementorum libri octo,ad faciliorem captum accommodati auctore Dechales,) 1660; and "Euclid's Elements,' demonstrated in a new and compendious manner," (Euclidi's elementa nova methodo et compendiarie demonstrata,) Sens, 1690, &c. Montücla may also have had

The opinions of the admirers of Euclid seem to agree in this: that the "Elements" constitute a whole, formed of many propositions, connected with each other in the firmest and most indissoluble connection, and that the order of the propositions can not be disturbed, because each is rendered possible by, and based upon, the preceding, and again serves to render possible and to found the next. As a purely mathematical work, and as a manual of instruction, Euclid's "Elements" are so excellent that all attempts to improve it have

failed.

On reading these extracts it might be imagined that all the world was quite unanimous on the subject of instruction in geometry, and that all acknowledged as their one undoubted master this author, who has wielded for two thousand years the scepter of the realm of geometry. But far from it. We find strange inconsistencies prevailing on the subject, which are in the most diametrical opposition to these supposed opinions respecting Euclid. For how can we reconcile the discrepancy of finding the same men who see in Euclid such a closely knit, independent, and invariable succession of propositions, omitting, in instruction, whole books of the "Elements?" If they make use of the whole of the first book, this only proves that they consider that book as a complete and independent whole. Others go as far as through the sixth book, omitting, however, the second and fifth; and still others take the first, sixth, then the seventh, and then the eleventh and twelfth, entirely omitting the thirteenth. Can a book of the supposed character of this be treated in such a way, losing sometimes five, sometimes nine, and sometimes twelve of its thirteen books?

But how, I ask again, can we reconcile such treatment with such descriptions of Euclid's "Elements?" If we closely examine these descriptions, however, we shall see that, notwithstanding the lofty tone of their laudations, they still lack something. All praise the thorough and close connection of the book, but nothing more. It is as if, in representing a handsome man, he should be made only muscular and strong-boned; or, as if the only thing said in praise of Strasburg Minster should be that its stones were hewed most accurately, and most closely laid together. But is there nothing in the work of Euclid to admire except the masterly, artistic skill with which he built together so solidly his masonry, his mathematical proposi

reference to the "New Elements of Geometry," (Nouveaur clémens de géometrie, Paris, 1667. This was by Arnauld, of the celebrated school of Port-Royal. Lacroix says of it, "It is, as I believe, the first work in which the geometrical propositions were classed according to abstractions; the properties of lines being treated first, then those of surfaces, and then those of bodies" "Essays on instruction generally and in mathematics in particular," (Essais sur l'enseignement en général et sur celui des mathématiques en particulier.) By Lacroix, Paris, 1816, p. 289. Unfortunately, I have been unable to examine Arnauld's work. By Lacroix's description, it would seem to have been a forerunner of the Pestalozzian school.