he assumes to be a teacher of teachers,' and 'the grand Mogul of Southern Science.' He is profoundly ignorant of the structure, or the rationale, of the first differential co-efficient of the Calculus; and yet has he waked all the mountains of Virginia, and all its hollows, too, with his crowings about his superior knowledge of that branch of the mathematics. He is not likely, however, to suffer much from a sense of his deficiency, for what he lacks in science he makes up in self-compla

cency.

The second important theorem, which we have promised to notice, is in these words of Professor Smith's translation of M. Blanchet: 'Two rectangular parallelopipedons A G, A L, having the same base, A B C D, are to each other as their altitudes A E, A I.' (Book VI, Prop. IX.) If the altitudes have a common measure, the demonstration is easy. But suppose the altitudes are incommensurable, what then? Why, then, M. Legendre has recourse to the reductio ad absurdum of the Greek geometers. Not so M. Blanchet and his Virginia disciple. If,' say they,' the altitudes A E and A I were incommensurable, we might prove, by the method explained in Book II, Prop. XVIII, that their ratio would always be the same as that of the parallelopipedons.' As the proof is the same, so is our reply the same. It is built on a false foundation.

6

We approve the object at which M. Blanchet aims. The object, namely, to simplify the demonstrations of Legendre, by substituting the first principles of the infinitesimal method for the reductio ad absurdum of the ancients. But the man who undertakes to introduce these first principles into the elements of geometry should first take the pains to understand them. Otherwise his attempt will be, like M. Blanchet's, a budget of blunders. Many modern geometers have, indeed, aimed at the same object as M. Blanchet, but among all the disastrous failures in this 'high endeavor,' we know of none more signal or more disgraceful than that of the directeur de l'école préparatoire de Sainte-Barbe.' There were, in his own country, many writers and thinkers who might have been of great service to him, such as D'Alembert, Carnot, and Duhamel. But if he has ever read the works of these men

at all, it must have been with that easy, careless, and negligent inattention, which did not leave the least trace of their great and luminous ideas on his memory. He may have a genius for geometry; he certainly has no genius for its philosophy. He may, for aught we know, be admirably qualified to manipulate the formula of the Calculus, or to work it as a practical engine; he certainly does not comprehend the very first principles of its internal mechanism or rationale. Hence he was not the man to introduce modifications,' or ' ameliorations,' into the admirable work of Legendre. He has, indeed, excluded the reductio ad absurdum from that work, and thereby relieved the memory and the intelligence' of his pupils from the demands of that tedious and operose method; but he has only substituted bosh in its place. We are sorry, sincerely and profoundly sorry, that the students of the Virginia Military Institute are doomed to use such stuff, instead of science, in the cultivation, training, and development of their minds. We have long delayed the duty of reviewing the performance of Professor Smith. When, more than five years ago, it was handed to us by the publishers for notice, we informed them that we could not possibly notice it favorably. We afterward gave the same information to Professor Smith himself, when he called to see us, and introduced the subject of our opinion of his book. We assured him, however, at the same time, that we should be in no hurry to publish a criticism of his Blanchet. We also added that, in a little work then passing through the press,' we had criticised the principle of his book in advance; that if, after examining the little work referred to, he should consider us in the right, he might have ample time to correct quietly, and without notice, what we conceived to be the fallacies of his geometry; but if, on the contrary, he should consider our views incorrect, or if, on any ground, he should conclude to retain his text-book as it is, it would then be time enough to notice it. We have now waited five long years, and more, for the revision, but waited in vain. In the meantime, his work had been almost entirely banished from our minds, by the pressure of other duties; but, recently, 1 The Philosophy of Mathematics.

the presentation of another work on geometry, by a Northern author, and nearly as faulty as itself, has forcibly recalled it to our recollection, and reminded us of our duty as reviewers. Science is not sectional, and, as duty begins at home, so we determined to bestow our first service on the Virginia Military Institute.

We intended to devote only a short notice to the errors of Professor Smith. But when, upon examination, we discovered that they had been sanctioned by M. Blanchet, we deemed them worthy of a more extended review. Hence the present article. This has been called for, as it seems to us, by the nature and the consequences of those errors. In the first place, the fundamental principle on which M. Blanchet builds the doctrine of the circle, the cylinder, the cone, and the sphere, strikes a fatal blow at the foundation of the whole Differential Calculus. It is, in the second place, clearly and demonstrably false.

I. It aims a fatal blow at the whole foundation of the Differential Calculus. For, if two variable quantities are ultimately equal, because, according to the law of change to which they are subjected, they may be made to approach each other as nearly as we please, or to differ from each other by less than any assigned quantity of the same species, then is the whole foundation of the Differential Calculus utterly swept away. This may be rendered perfectly obvious. For, according to the definition of all geometers, all indefinitely small quantities, or 'infinitesimals,' have zero for their limit. Hence, as they may be made to approach as nearly as we please to zero, so may they be made to approach as nearly as we please to each other. Are they, therefore, always ultimately equal? If so, then are their 'ultimate ratios' always, or in all cases, equal to the same constant quantity I. How, then, in the name of common sense, or reason, can their ultimate ratios' be made to yield that infinite variety of values, which constitute the very basis of the Differential Calculus? They can yield only I. Hence, if the principle of M. Blanchet be true, the Differential Calculus is utterly without a foundation. It is merely the baseless fabric of a vision.'

geom

II. But the principle of M. Blanchet is clearly and demonstrably false.' Its fallacy is demonstrated, by means of etry, in the little work already referred to.1 It is there demonstrated geometrically that the limit of the ratio of two indefinitely small quantities may be either infinity or zero' (p. 225), or any quantity between those two extremes. The same thing may be just as easily demonstrated by means of one of the very simplest processes of algebra:

Let i, for example, stand for an indefinitely small variable, or 'infinitesimal,' whose limit is zero. But if i is indefinitely

small, is much smaller. Hence, we have in

ratio of two indefinitely small quantities.

22

i

i

or in

the

22

whose limit is evidently o. On the other hand,

whose limit is oc (infinity). Thus, the ratio of two indefinitely small quantities, which may be made to approach as near as we please to zero, or to differ from each other by less than any assigned quantity, may vary from o to ∞ (from zero to infinity). E. E. D.

Yet, in spite of this easy and obvious demonstration, or else in ignorance of it, M. Blanchet builds on the principle, that if two quantities may be made to approach each other as nearly as we please, or to differ in value by less than any assigned quantity, they are ultimately equal;' so that their 'ultimate ratio,' or the ratio of their ultimate values, must be equal to the constant quantity I.

We repeat, in conclusion, that we approve of the design or object of M. Blanchet, but not of the bungling manner in which that design has been executed. There have been many other geometers-Davies, Hockley, Ray, Whewell, Todhunter, and a host of others-who have aimed at the same object with M. Blanchet. But, as we have elsewhere shown, without success. We do not despair, however, of seeing this object

1 See Philosophy of Mathematics, Chap. VIII, pp. 223, 224, 225.

2 Philosophy of Mathematics, Chap. I.

as representations of the modes in which weak minds and credulous temperaments are harassed and unsettled by vague prognostications in regard to matters impenetrably shrouded from human ken. Mr. Bickersteth's poem, Yesterday, To-day, and Forever, is a chiliastic argument, displaying much of the obscurity of the Sibylline oracles, and no greater credibility. It is a curious cross between Pollock and Milton, possessing more characteristics of the former than of the latter. It contains some gorgeous passages of rhetorical verse, and, in the First Book, some tender sentiments and pathetic descriptions gracefully expressed. If it avoids the censure of the Paradise Lost by the black-letter lawyer, that it proves nothing, it is more censurable by undertaking a poetical demonstration and failing in its aim, while destroying all poetry by entertaining such an aim. An argument for the approach of the Millennium in learned and elegant verse is a strange resuscitation of the Sibyls for the waning of the nineteenth century.

The Rev. Mr. Baxter falls into no such poetical heresy. His language is the plainest and baldest of prose. He proceeds patiently and unmethodically in the track of interpretation customary with those who would wrest the prophecies to the establishment of their own hallucinations in regard to the final day. Even the designation of the late Emperor of the French as preeminently the man of sin and Antichrist, who was to usurp supreme dominion and blasphemous ascendancy, was in consonance with the usual routine before Haguenau, and Sedan, and Metz had dissipated this particular delusion. We have copied the greater part of Mr. Baxter's long and breathless rubric; we have patiently and painfully reproduced the larger half of it-verbatim, literatim et punctuatim—notwithstanding its vermicular articulations, its awkward convolutions, and its ungrammatical expressions, because the titlepage contains the marrow and essence of the work, and presents in one confused view the entire doctrine which we intend to question, and strenuously to disclaim. The frontispiece is not merely emblematic of the book, but may be appropriately considered the book. The curtain is the picture. The portico is the temple. Behind the façade are only chaotic