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 moreMontücia, I., 24. The list of editions and translations of Euclid's “Elements" occupies, in the fourth part of Fabricius' “ Bibliotheca Græca,” sixteen quarto pages.

but

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, 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. Aş 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," (Nouveaux elémens 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.

tions? Is there not very much beauty in the scientific thought, so proprofound, so comprehensive, and so thoroughly diffused through every part of the work? The great Kepler was even inspired by this beauty, and was exceedingly enraged at Ramus' attack on Euclid, especially against the tenth book of the "Elements." Ramus said that he had never read any thing so confused and involved as that book; whereupon Kepler answers him thus: "If you had not thought the book more easily intelligible than it is, you would never have found fault with it for being obscure. It requires great labor, concentration, care, and special mental effort, before Euclid can be understood. You, who in this show yourself the patron of ignorance and vulgarity, may find fault with what you do not understand; but to me, who am an investigator into the causes of things, the road thereto only opened itself in this tenth book." And in another place he says, "By an ignorant decision this tenth book has been condemned not to be read; which, read and understood, may reveal the secrets of philosophy."

*

Kepler also further attacks Ramus, for not subscribing to the assertion of Proclus-although it is evidently true-that the ultimate design of Euclid's work, toward which all the propositions of all the books tend, was the discussion of the five regular bodies.* And Ramus has put forth the singularly rash assertion that those five bodies are not forthcoming at the end of Euclid's "Elements." And by thus destroying the purpose of the work, as one might destroy the form of an edifice, there is nothing left except a formless heap of propositions. "They seem to think," says Kepler, further, "that Euclid's work was called 'Elements' (σroxɛîa) because it affords a most various mass of materials for the treatment of all manner of magnitudes, and of such arts as are concerned with magnitudes. But it was rather called 'Elements' from its form; because .each subsequent proposition depends upon the preceding one, even to the last proposition of the last book, which can not dispense with any preceding one. Our modern constructors treat him as if he were a contractor for wood; as if Euclid had written his book to furnish materials to every body else, while he alone should go without any house."

Kepler's estimate differs materially from those first given, in that he does not only praise Euclid's skill in building firm and solid masonry, but the magnificence of his whole structure, from foundationstone to ridge-pole. But later mathematicians have found fault with Proclus and Kepler for bringing into such prominence the five regular

Except those which treat of perfect numbers, Proclus says, in his commentary on the first book of Euclid, "Euclid belonged to the Platonic sect, and was familiar with that philosophy, and accordingly the whole of his elementary course looked forward to a consid. eration of the five beautiful bodies' of Plato."

bodies, and finding in them the ultimate object of Euclid's work. Even Montücla and Lorenz do this, although, as we have seen, they agree wholly with Kepler and others in finding that the chain of propositions in Euclid's "Elements" is a most perfect one, and that no proposition is stated which is not based upon a previous one. But it would have been impossible for Euclid to construct such a chain, had he not at the beginning of it seen clearly through its whole arrangement; had he not, during the first demonstration of the first book, had in his eye the last problem of the thirteenth. For no architect can lay the first foundation-stone of his building until he has clearly worked out his drawings for the whole.

The most superficial observation will show that Euclid begins with the simplest elements, and ends with the mathematical demonstration of solid bodies. He commences with defining the point, line, and surface; treats of plane geometry in the first six books, and comes to solids only in the eleventh. The first definition in this book, that of bodies, follows on after the former three. Lorenz gives us the reason why Euclid inserted between plane and solid geometry, that is, between the sixth and eleventh books, four other books. "The consideration of the regular figures and bodies," he says, "presupposes the doctrines laid down in the tenth book on the commensurability and incommensurability of magnitudes; and this again the arithmetical matter in the seventh, eighth, and ninth books."

The five regular solids, in point of beauty, stand altogether by themselves among all bodies; Plato calls them the "most beautiful bodies." We need not therefore wonder at Euclid for taking, as the crown of his work, the demonstration of their mathematical nature and of their relations to the most perfect of all forms, the sphere. In the eighteenth proposition of the thirteenth book, the last of the whole work, he demonstrates the problem. To find the sides of the five regular bodies, inscribed in a sphere. If this proposition was not the intended object, it is at least certainly the keystone of the

structure.

Many things show that the demonstration of the five regular bodies, and of their relations to the cube, was really the final object of the "Elements." The Greeks, from their purely mathematical sense of beauty, and remarkable scientific tendencies, admired and studied this select pentade of bodies, which played a great part first in the Pythagorean and afterward in the Platonic school. But that Euclid, who seems to have been instructed by pupils of Plato, followed Pythagoras and Plato in this respect, if we are not convinced of it by the "Elements," is clearly enough shown by the quotation given from Proclus, and by the following ancient epigram: