lege of the City of New York, and that, in 1873, Charles Davies published a book where he also repudiated the absurdum reasoning. "For nearly twenty years mathematicians and myself have been at logger. heads on the issue made by me about the circle. I now propose to set at rest all doubts against the demonstration published by me in 1860 and 1862." In more recent years Mr. Benson's efforts to revolutionize mathematics have been unabated.

Dr. A. Martin tells us of a quadrator who deposited with him a manuscript, in 1885, proving that the long sought for ratio is exactly 3. Mr. Faber, the writer of it, distinguished himself also in other branches of mathematical inquiry. In a phamphlet of thirty-four pages, in 1872, "Theodore Faber, a citizen of the United States, New York," makes the "extraordinary and most significant discovery of a lacking link in the demonstration of the world-renowned Pythagorean problem, utterly disproving its absolute truth, although demonstrated as such for twentythree centuries." In justice to Mr. Benson, it should be remarked that he, too, is waging war against Euclid, I, 47.*

*Since writing the above, we have received from Dr. Artemas Martin a copy of the Notes and Queries, Vol. V, Nos. 6 and 7, June and July, 1888, giving a Bibliography of Cyclometry and Quadratures. From this article we see that Theodore Faber has appeared in print also on the subject of the quadature of the circle. The article gives over twenty publications, besides the ones mentioned above, by American writers who believe that they have found the true and exact ratio.

APPENDIX.

BIBLIOGRAPHY OF FLUXIONS AND THE CALCULUS.

TEXT-BOOKS PRINTED IN THE UNITED STATES.

HUTTON, CHARLES. Course of Mathematics, in two volumes.

American editions, revised by Robert Adrain, appeared in 1812, 1816 (?), and 1822. Evert Duyckinck brought out an edition in New York in 1828. Another edition appeared in 1831. The second volume contains a short account of the doctrine of fluxions, using the Newtonian notation.

VINCE, REV. S. The Principles of Fluxions, first American edition, corrected and enlarged. Philadelphia: Kimber & Conrad. 1812. Pp. 256. Employs the Newtonian notation.

BEZOUT. First Principles of the Differential and Integral Calc lus, or the Doctrine of Fluxions, from the Mathematics of Bezout, and translated from the French for use of students of the University at Cambridge, New England. Boston, 1824. Pp. 195. This book forms a part of Farrar's Cambridge Mathematics. It is the first work published in this country employing the notation of Leibnitz and the infinitesimal method. "In the Introduction, taken from Carnot's Réflexions sur la Métaphysique du Calcul Infinitesimal, a few examples are given to show the truth of the infinitesimal method, independently of its technical form." This is done by explaining that there is a "" compensation of errors."

RYAN, JAMES. The Differential and Integral Calculus. New York, 1828.* Pp. 328. "The works which I have principally used in preparing this treatise are Lacroix, Lardner, Boucharlat, Garnier, and Du Bourguet's Differential and Integral Calculus; Lagrange's Calcul des Functions, Simpson's Fluxion's, Peacock's Examples on the Differential and Integral Calculus, and Hirsch's Integral Tables" (advertisement). The first section of the book is given to "preliminary principles," in which the three methods of Newton, D'Alembert, and Lagrange are explained. The method adopted by the author is that of limits, but no formal definition of the term "limit" is given. The symbol (0), indicating the absence of quantity, is everywhere treated with the same courtesy and implicit confidence as though it were actually a quantity. The inquiry as to whether the laws of analysis are really applicable to this foreign intruder into the society of actual magnitudes, or whether it has to be governed by laws of its own, is nowhere deemed necessary. These remarks apply with equal force to other works on calculus and to works on algebra.

YOUNG, J. R. Elements of the Differential Calculus, comprehending the general theory of surfaces, and of curves of double curvature. Revised and corrected by Michael O'Shannessy. Philadelphia, 1833. Pp. 255.

In the preparation of this American edition, the editor was assisted by Professor Dod, of Princeton College.

In the explanation of the process of differentiation, he makes h absolutely zero, in an expression like this:

y' =Y=3x2+3xh+h2.

* Ryan's Calculus is now a rare book. The copy we have before us was kindly lent to us by Prof. W. Rutherford, of the University of Georgia.

395

"In both these cases (of which that given here is one), as indeed in every otherthe respective differential co-efficients are only so many particular values of the gen, always reduces, when h=0." In the above example

eral symbol to which

y'-y
h

dy 3x2. "The expressions and dx

dz dy

have, we see, the advantage over the symbol

of particularizing the function and the independent variable under consideration

dy dz
dx dy

and this, it must be remembered, is all that distinguishes or

from o, for dy,

dz, dx, are each absolutely 0." "These differentials, although each 0, have, nevertheless, as we have already seen a determinate relation to each other (!); thus, in the last example, this relation is such that dy = 2b (a + bx) dx, and, although this is the same as saying that 0=2b (a + bx) 0; yet, as we can always immediately obtain from this form the true value of dy or we do not hesitate occasionally to make

dx'

use of it." It will thus be seen that the author has no hesitation whatever in breaking up the differential co-efficient.

YOUNG, J. R. The Elements of the Integral Calculus, with its applications to geometry and to the summation of infinite series. Revised and, corrected by Michael O'Shannessy. Philadelphia, 1833. Pp. 292.

u'-u

DAVIES, CHARLES. Analytical Geometry and Differential and Integral Calculus. 18-. Elements of the Differential and Integral Calculus. 1836. Several editions of Davies' calculus appeared. In the improved edition of 1843 (pp. 17 and 18) the author says that 2ax is the limit toward which the ratio h 2ax+ah approaches in proportion as h diminishes, and hence "expresses that particular ratio which is independent of the value of h." Bledsoe objects to this, saying, "Shall they (teachers) continue to seek and find what no rational beings have ever u' И found, namely, that particular value of which does not depend on the value of h h? That is to say, that particular value of a fraction which does not depend on its denominator!" Davies represents by dx "the last value of h, which can not be diminished, according to the law of change to which h or x is subjected, without becoming 0." "It may be difficult," says the author, "to understand why the value which h assumes in passing to the limiting ratio is represented by dx in the first member and made equal to 0 in the second." To this Bledsoe says: "Truly this is a most difficult point to understand, and needs explanation. For if h be made absolutely zero, or nothing on one side of the equation, why should it not also be made zero on the other side?" "Why should 'a trace of the letter x' be preserved in the first member of the equation and not in the second? The reason is, just because dx is needed in the first member and not in the second to enable the operator to proceed with his work."

As regards the conception of the term "limit," Davies believed that a variable actually reached its limit. "The limit of a variable quantity is a quantity toward which it may be made to approach nearer than any given quantity, and which it reaches under a particular supposition.'

Davies believed that by the definition of M. Duhamel, according to which a variable never reaches its limit, there seemed to be placed an impassable barrier" between a variable quantity and its limit. "If these two,quantities are thus to be forever separated," says he, "how can they be brought under the dominion of a common law, and enter together in the same equation?"t

* Nature and Utility of Mathematics, by Charles Davies, New York, 1873, p. 291. † Ibid., p. 326.

PEIRCE, BENJAMIN. An Elementary Treatise on Curves, Functions, and Forces. Volume I, containing analytic geometry and the differential calculus. Boston and Cambridge, 1841. Pp. 301. Volume II, containing calculus of imaginary quantities, residual calculus, and integral calculus. Boston, 1846. Pp. 290.

The method followed in these volumes is the infinitesimal, of which the author was a great admirer. The differential co-efficients are here denoted by D, D', etc. The second volume treats of many rather advanced subjects, such as imaginary infinitesimals, imaginary logarithms, imaginary angles, the imaginary angle whose sine exceeds unity, potential functions, residuals, definite integrals, elliptic integrals, method of variations, linear differential equations, Riccati's equation, and particular solutions of differential equations.

CHURCH, ALBERT E. Elements of the Differential and Integral Calculus. New York, 1842.

This is in many respects a good work, but the explanation of fundamental principles therein contained is too brief, and fails to convey a philosophic knowledge of them. The difficulties which a student is likely to encounter in a treatise like this have been well stated by a writer in the Nation of October 18, 1888: "What vexes and perplexes him (the student) is that he seems to himself to comprehend very clearly what he is doing, and to be doing what all his previous training had taught him he must not do. It all seems very easy, very simple, and very absurd. He is told to 'take the limit' of one side of his equation by striking out a quantity because it is approaching zero,' while on the other side the same quantity must be carefully preserved, because it is one of the terms of the ratio which is the very essence of the whole process."

MCCARTNEY, WASHINGTON. Principles of the Differential and Integral Calculus, and their application to Geometry. Philadelphia, 1844. Pp. 340.

The author makes use of the doctrine of limits, but retains the language of infini-.

dy

tesimals. dx is used as a mere symbol to denote the ultimate ratio,

dy dx

0

66

being in reality

But inasmuch as the rules for differentiating and the geometrical application of ultimate ratios are more readily understood by regarding the increments of the ordinate and abscissa as indefinitely small, we will call these increments in their ultimate state, indefinitely small quantities." "For the sake of convenience," the student is asked to call dy and dx what he has just been told that they really are not. Such an exposition of a fundamental principle is quite apt to fail to give satisfaction to beginners. McCartney's Calculus is a book possessing several good features.

LOOMIS, ELIAS. Analytical Geometry and Calculus. 1851.

Later the Calculus was published in a separate volume and much enlarged. The unfolding of fundamental principles, as given in the improved edition of 1874, is less objectionable than that in the preceding works which adopt the method of limits. The term "limit of a variable" is here subjected to definition, but the student is not dy informed whether or not the variable ever reaches its limit. The symbol is made dx

incr. y

to represent the limiting value of incr. x Confusion is apt to arise in the mind of the

student from the fact that dx is "put for the incr. x in the limiting value" (which value is zero), and is afterward said to be "indeterminate" in value, "either finite or indefinitely small."

SMYTH, WILLIAM. Elements of the Differential and Integral Calculus. 1854.

The author uses the infinitesimal method, but says (p. 229) that "as a logical basis of the calculus, the method of Newton and especially that of Lagrange have some advantage. In other respects the superiority is immeasurably on the side of the method of Leibnitz."

COURTENAY, EDWARD H. Treatise on the Differential and Integral Calculus and on the Calculus of Variations. New York, 1855. Pp. 501.

The exposition of the method of limits, as given in this in many respects admirable work, is likewise open to objection. de is pronounced to be "indefinitely small" and equal to h, but when h=0 at the limit, de continues to remain indefinitely small. ROBINSON, HORATIO N. Differential and Integral Calculus, 1861.

Some of Robinson's elementary works on mathematics became popular, but not so his advanced works. His calculus and astronomy met with able but severe criticism in the Mathematical Monthly. Robinson's work did not appear in a second edition, but the work of Quinby was added to "Robinson's Series" in place of it.

DOCHARTY, GERARDUS BEEKMAN. Elements of Analytical Geometry and of the Differential and Integral Calculus. New York, 1865. Pp. 306.

The part on the calculus covers 204 pages.

The method of limits is employed and treated in the manner customary with us at the time the book was written.

SPARE, JOHN. The Differential Calculus: with Unusual and Particular Analysis of its Elementary Principles, and Copious Illustrations of its Practical Application. Boston, 1865. Pp. 244.

This work I have never seen. Dr. Artemas Martin, who kindly sends me its title, calls it a unique work, as may be seen from the following, which he quotes from its preface: "The calculus being algebra, a strictly numerical science, the present treatise claims to have labored successfully in putting on the true character as such. No insinuation is allowed to prevail that it is any part whatever of analytical geometry or that it is other than the natural sequel and supplement of common algebra; useful, indeed, as an appliance, to borrow, in investigation of the few kinds of geometrical quantity."

QUINBY, I. F. A New Treatise on the Elements of the Differential and Integral Calculus. New York, 1868. Pp. 472.

Here, as in other works based on the method of limits, the student encounters at

0

the outset the perplexing statement that, where O denotes "absolute zero,” is equal to some particular quantity.

STRONG, THEODORE. A Treatise on the Differential and Integral Calculus. New York. 1869. Pp. 617.

This work was printed, but, we understand, never published. The author died while the work was in press. Theodore Strong was professor at Rutgers College from 1827 to 1863, and enjoyed the reputation of being one of the very deepest and most erudite mathematicians in America. He was a very frequent contributor to our mathematical periodicals. To students who possessed taste for mathematical investigation he was a good teacher, but to those who had no taste he was unintelligible. He had an unconscious tendency to diverge into regions where the ordinary student could not follow him. This same tendency is exhibited in his Calculus, and also in his Elementary and Higher Algebra, published in 1859. Both works possess many original features, but the novelties contained in them are not always improvements. These books are defective in arrangmeent, and not at all suited for use in the class-room. In his general view of the calculus Strong follows Lagrange, but his mode of presentation is quite new. He believed that his treatment divested the calculus of all its old metaphysical encumbrances. He attempted to show how the foundations of this science could be established without the intervention of any of the antiquated hypotheses. "It is hence clear," says he, "that the differential and integral calculus are deducible from what has been done, without using infinitesi mals or limiting ratios" (p. 271).