dust; and the living are called upon to revere their memory and to emulate their virtues. For Paris and its people the prayers of Christians all over the globe should ascend to heaven; and all whose duty or privilege it may be to visit. the great city should give as much of their time and influence as they can to the small but devoted band of laborers who are earnestly 'contending for the faith once delivered to the saints,' and valiantly fighting against the combined hosts of sensuality, superstition, and atheism.

ART. IV.-Elements of Geometry. By A. M. Legendre. With Additions and Modifications, by M. A. Blanchet, Éléve of the Polytechnic School; Director of Studies of SainteBarbe. Translated from the eleventh French edition, by Francis H. Smith, A. M., Superintendent and Professor of Mathematics in the Virginia Military Institute, Lexington, Virginia. Baltimore: Kelly & Piet. 1867.

Professor Smith is, unquestionably, a man of no ordinary genius. No one can possibly doubt this, who has the pleasure of a personal, or a literary, acquaintance with that distinguished gentleman. His genius is, indeed, preëminent, rising high, and shining brightly, above a generation of pigmies and blunderers.

We do not mean by this, however, that his genius is universal. Indeed, we have very little respect for your universal geniuses, for your admirable Crichtons, who know all things,

1 It is very natural that Professor Smith should magnify the author whom he translates, and whose work he wishes to sell. But this is no reason why he should mistranslate the title-page of the work. According to the translation, M. Blanchet is 'director of studies of Sainte-Barbe,' which leaves the reader free to imagine that he presides over some great university or college, known in France under the name of 'Sainte-Barbe.' But the original says, that M. Blanchet is 'director of the studies of the preparatory school of Sainte-Barbe.' We have no doubt that there are, at this time, much better preparatory schools in the State of Virginia, whose directors' are better qualified than is M. Blanchet to make improvements in the geometry of Legendre.

and who do all things equally well. They are apt to be a little like General Andrew Jackson, as described by Richard Henry Wilde, who replied to the inquiry of a certain Grand Duke, that he might write to the General, then President of the United States, in any language he pleased - Latin, Greek, German, Spanish, Italian, French, or English-for he understood them all equally well! This is not the case with Professor Smith. Omniscience is not one of his foibles. He may, perhaps, think so himself, but he will find no one to agree with him. His capacity, his genius, is limited to certain directions; but, then, in its own particular direction, it is transcendent, it is wonderful, it is without a parallel a peer.

We are not sure, however, that he understands his own forte. It is not every man of genius who knows what he is good for; and, sadder still, it is not every man of genius who is aware of his own deficiencies. It is right here, indeed, that the great weakness of Professor Smith lies. He may know what he is good for; he certainly does not know what he is not good for. All men agree that he has a marvellous genius for management that there is not, in the whole South, another person who could have displayed as great skill as he has shown in the building up of the Virginia Military Institute. All honor to his genius, we say, for his great achievement. It is marvellous in our eyes. We have not the most remote idea how this wonder has been wrought by him. But somehow or other, we know not how, he always contrives to get around the Legislature of Virginia, and obtain all he wants. for his Institute. We have never read nor heard of anything like it, except in the story of the single man, who 'dispersed in the woods,' surrounded 'a parcel of Indians,' and 'took them all prisoners.' Some persons, because they could not understand how Professor Smith gets around the Legislature of Virginia and takes them captive, have given him the credit of a very great genius for circumnavigation. But we have no opinion to express on the subject. We simply admire and applaud the wonderful achievement, without pretending to the least insight into the modus operandi of the magician. We could, indeed, as easily imagine how Shakspeare created

one of his great dramas, as how Professor Smith has built up and preserved the Virginia Military Institute.

The mathematical career of Professor Smith has filled us with as great admiration and wonder as his administrative ability. The work before us, for example, displays a most marvellous genius; but then, since the truth must be spoken, it is a genius for blundering. How he could have written such a book, is more than we can possibly conceive. He must have originally possessed some talent for the science of mathematics, at least for the acquisition of other people's ideas, for he graduated at est Point, and even stood high in his class. But the time has past when graduation at the National Military Academy can, without other evidences of proficiency, preserve a man's reputation as a teacher of mathematics. A skeep skin, even from that academy, can no more make a great mathematician than it can make a great general. If, indeed, Professor Smith ever had a genius for mathematics, or any solid attainments therein, he must have lost them, somehow or other, in his manœuvres to 'disperse himself,' with a view to surround and capture the Legislature of the Old Dominion. We propose, in this paper, to illustrate, by an appeal to the book before us, his marvellous genius for blundering in the simplest elements of geometry. We have no wish to dispel the very happy hallucination, under which he has so long labored, that it is his mission to elevate and bear aloft the standard of mathematical education in the South; but we do intend, if possible, to protect the interests of Southern science, and Southern education, against the inroads and ravages of his very extraordinary genius.

His work is, indeed, disfigured, from beginning to end, with blunders in definitions, blunders in first principles, blunders in theorems, and blunders in demonstrations. It is the production of a marvellous genius. Let us now see if this statement, however strong, is not literally and strictly true.

To begin with the Preface, the author there says: 'Geometry rests upon self-evident truths; and from these, by the rigid processes of deduction, the student of mathematics is afforded a valuable mental discipline, which supplies an important

correction for some of the evils resulting from an exclusive devotion to analysis. Hence its importance as an essential branch of academic instruction." Here the author falls into the vulgar error, that self-evident truths,' or axioms, are the first principles of the science of geometry, from which, by rigid processes of deduction, its discoveries are made. The truth is, however, that the properties of triangles, or quadrilaterals, or circles, or cones, or cylinders, or spheres, are deduced, not from self-evident truths or axioms, but from the definitions of those several figures. In other words, definitions, and not axioms, are the first principle of the science, from which all its beautiful truths are deduced. If Professor Smith had only read Plato, or Locke, or Stewart, or any other writer, on the first principles of geometry, except Davies, he would not have fallen into so gross a blunder. But this is only the beginning of sorrows.' It is, moreover, a compartively pardonable blunder, because it relates, not so much to the science as to the philosophy of geometry. But, surely, it be hooves every teacher of the science, especially every one who aspires to the high office of the teacher of teachers, to understand what are its first principles or premises. Self evident truths or axioms are not the first principles or premises of geometry, but only the connecting links by which its conclusions are fastened, as by a chain of adamant, to its first principles or premises. These first principles, or premises, or 'postulates,' are its definitions.

Geometryes its origin, as a science, to the inventive genius of the Greeks." Nor is this true. Geometry was a science long before the Greeks had an existence. If Montucla may be trusted, this noble science was cultivated in China, India, Egypt, and Chaldea, long before its earliest dawn in Greece. The Greeks, in fact, learned the science from their predecessors, to which, however, they made many valuable additions of their own. No one ever supposed certainly no historian of the science ever supposed that Euclid was the inventor of the Elements. He only collected them from all quarters whether of Chinese, Hindoo, Egyptian, or Chaldean

1 Preface, p. 1.

2 Preface, p. 1.

origin and, for the benefit of all ages, embodied them in his immorta work.

The demonstrations left by the Greek geometers,' says Professor Smith, ' are models of accuracy, clearness, and elegance, and are admirably adapted to training the mind to habits of close reasoning and luminous arrangement." This is true. But it is evidently a truth which our author has only learned and repeated by rote; for, instead of seeing this truth for himself, he blunders most egregiously, as we shall presently demonstrate, precisely in those portions of the science in which the Greek geometers have been most remarkable for their accuracy, clearness, and elegance.' Admirable as these things are for 'training the mind to habits of close reasoning' and accurate thinking, little benefit has our author seemed to derive from them.

'Soon after the revival of letters, the principal works of the Greek geometers were translated in Italy by Commandine, embracing fragments of Apollonius' Conics and the Collections of Pappus. Toward the close of the sixteenth century, Vieta restored the lost Tract of Tangencies, and Snellius reproduced the Plane Loci. Soon afterward Viviani, the disciple of Galileo, supplied the fifth book of Apollonius.'

92

In reading this very learned passage, the question forces itself upon our minds, For what earthly purpose was it written? It certainly could rot have been intended for those who had read the most meagre sketch of the history of mathematics that was ever written, since it could not possibly give one particle of information even to such readers. For whom was it intended, then?-for those who knew nothing? If so, it left them just where it found them, with the exception of a few learned and utterly unintelligible names. Apollonius' Conics,' the Collections of Pappus,' 'the lost Tract of Tangencies,' 'the Plane Loci,' and 'the fifth book of Apollonius.' Alas! and what are these? we seem to hear the unlearned and astonished reader exclaim. We seem to see him, moreover, open his mouth and gape with amazement at the vast erudition of the learned writer. He certainly does not, and cannot, derive 1 Preface, p. 1. 2 Ibid., pp. 1 and 2.