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 mayor 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 rationde. Hence he was not the man to introduce modifications,' or 'ameliorations,' into the admirable work of Legendre. He has, indeed, excluded the reductio a 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 criticed 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, I The Philosophy of Mathematics.

t

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 Virgmia 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 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 infiniteimals,' 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.'

6

II. But the principle of M. Blanchet is clearly and demonstrably false.' Its fallacy is demonstrated, by means of geometry, in the fittle work already referred to.' 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 smalle. Hence, we have in

ratio of two indefinitely small quantities.

i

or in

the

22

On the other hand,

whose limit is a (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 oc (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.

most clearly, fully, and perfectly accomplished. For, after reviewing the attempts of the above-named geometers, we have not been afraid to say: 'But if we reject the notion, that the inscribed regular polygon ever becomes equal to the circle, or coincides with it, what shall we do? If we deny hat they ever coincide, how shall we bridge over the chasm between them, so as to pass from a knowledge of right-lined figures and volumes to that of curves and curved surfaces? Shall we, in order bridge over this chasm, fall back on the reductio ad absurdum of the ancients or can we find a more short and easy passage without the sacrifice of aerfect logical rigor in the transit? This is the question. This is the very first problem which is, and always has been, presented to the cultivators of the infinitesimal method. Is there, then, after the lapse and the labor of so many ages, no satisfactory solution of this primary problem? It is certain that none has yet been found which has become general among mathematicians. I believe that such a solution has been given, and that it only requires to be made known [and clearly demonstrated] in order to be universally received, and become a possession forever aytiμasadei more precious than even the gift of Thucydides.'

ART. V.-1. Armageddon, or a Warning Voice from the last Battle field of Nations. By Beale. London. 1837-8. 3 vols., 8vo.

2. Louis Napoleon, the Destined Monarch of the World and personal Antichrist; foreshown in prophecy to confirm a seven years' covenant with the Jews about or soon after 1864-5, and (after the resurrection and translation of the Wise Virgins has taken place, two years and from four to six weeks after the covenant,) subsequently to become completely supreme over England and most of America, and fiercely to persecute Christians during the latter half of the seven years, until he finally perishes at the descent of Christ at the battle of Armageddon, about or soon after 1872-3, etc., etc., etc. 13th thousand. By the Rev. M. Baxter, late Missionary of the Episcopal Church. Philadelphia. 1866.

3. Yesterday, To-day, and Forever. By the Rev. Edward Bickersteth. New York: Robert Carter & Brother.

1870.

Our world has not yet perished, although both public and private sis might indicate that the harvest was ready for the reaper. Quosque tandem? How long, O Lord, how long? Our earthly habitation still wheels along its appointed orbit, observing the successions of the seasons, and serenely repeating the alternations of night and day, while its multitudino populations follow each other like the waves of the sea, or start into light and disappear with perplexing rapidity and brief duration, like the spark of a scroll in the last moments of conflagration. Earthquakes convulse, floods desolate, and droughts famish different regions, but the round globe still rolls on, and men of profound science speculate more on the exhaustion of the central luminary that enlightens, warms, and vivifies the system to which we belong, and the consequent freezing out of the terrestrial tubes, than on the final inflammation. But all men are not men of profound science, and all cannot indulge with equal placidity in these cool calculations. The opening year is the last of the two assigned by the Rev. Mr. Baxter for the consummation. During the past summer, some uneasiness, but more curiosity, was excited by the hoax ascribed to Professor Plantamour, which announced that on the 12th August a portentous comet would rush through the heavens, smite our planet, and reduce it to dust and ashes. The 12th August came and went away without producing a jar. The comet did not appear, though shooting stars were numerous during the month, and a few nights before the day of expected doom there was a remarkable aurora in the West

a broad, bright, white streak in the heavens, like an enormous comet's tail. Such apprehensions, such announcements, come and go with each generation, and almost with each year and if the delusion is mortal, it is as vivacious as the heads of the hydra.

The writings named at the head of our article, and a host of others of similar character and tendency, demonstrate this fact. They are significant in themselves, and more significant