On the Walter Reed Memorial. At a meeting of the association held in Washington, a committee was appointed, of which I was made chairman, to take such measure as might be found wise for securing a permanent memorial of Major Walter Reed, U. S. A., in recognition of his important services to humanity. Acting under this authority, it was at length found expedient, after several preliminary meetings, to form an incorporation in the city of Washington to hold such funds as might be contributed. This incorporation is now endeavoring to raise the sum of $25,000, of which the income may be paid to Mrs. Reed and the principal may be devoted to a permanent memorial of Dr. Reed. More than $13,000 has been subscribed already, a large part of this amount coming from the medical profession. This is all in addition to the action of Congress, which has given, on the representations of your committee, an unusual pension to Mrs. Reed. The effort is now making to secure the additional sum of $12,000, and the cooperation of all members of the American Association for the Advancement of Science is urgently desired. Yours respectfully, DANIEL E. GILMAN, On the Relations of the Association to the Journal 'Science.' We beg to report that the arrangement by which SCIENCE publishes the official notices and proceedings of the association and is sent free of charge to the members in regular standing on payment of two dollars for each appears to give satisfaction. We recommend that the contract with The Macmillan Company be renewed for the year 1905. SIMON NEWCOMB, Chairman, CARROLL D. WRIGHT, L. O. HOWARD, R. S. WOODWARD, J. MCK. CATTELL, G. K. GILBERT. The following members of the association were elected fellows: Section A: Hayes, Ellen, Wellesley, Mass. Milham, Willis I., Williamstown, Mass. Quinn, John Jones, Warren, Pa. Section B: Davis, Bergen, New York City. Lewis, E. Percival, University of California. Pegram, George Braxton, Columbia Univ., New York City. Section C: Dorr, Allen Wade, Washington, D. C. Section D: Bissell, Geo. W., Ames, Iowa. Blanchard, A. H., Providence, R. I. Greene, Arthur Maurice, Jr., Columbia, Mo. Section E: Aguilera, Jose G., Mexico, Mex. Bawell, Joseph, 105 Bishop St., New Haven, Conn. Bayley, W. S., Waterville, Me. Berkey, C. P., New York City. Bien, Julius, 140 Sixth Ave., New York. Collier, Arthur James, Washington, D. C. Grimsley, Geo. Perry. Hayes, C. Willard, Washington, D. C. Heilprin, A., Academy Natural Sciences, Phila. Lyman, Benj. S., Philadelphia. Merriam, John C., Berkeley, Calif. Penfield, S. L., Yale University, New Haven, Conn. Tower, Ralph Winfred, American Museum of Natural History, New York City. Section F: Allis, E. P., Menton, France. Bailey, Vernon, Washington, D. C. Bawden, H. Heath, Vassar College, Poughkeepsie, N. Y. Beebe, C. W., New York City. Birge, E. A., Madison, Wis. Blake, Joseph A., 601 Madison Ave., New York City. Ames, Oakes, North Easton, Mass. Banker, Howard J., Greencastle, Ind. Blodgett, Frederick H., College Park, Md. Cannon, W. A., Tucson, Ariz. Duval, Joseph W., Washington, D. C. Fitzpatrick, Iowa City, Iowa. Holferty, George M., St. Louis, Mo. Pond, Raymond H., 87 Lake St., Chicago, Ill. Bair, Joseph H., Boulder, Colo. Churchill, William, New Haven, Conn. Burton, Theodore E., Cleveland, O. Stokes, Anson Phelps, New York City. Section K: Abbott, Alexander C., University of Pennsylvania. Burton-Opitz, Russell, New York City. Dexter, E. G., Urbana, Ill. Flexner, Simon, Rockefeller Institute, New York City. Lindley, Ernest H., University of Indiana, Loeb, Leo, University of Pennsylvania. Smith, Allen J., University of Pennsylvania. SCIENTIFIC BOOKS. Elements of the Differential and Integral Calculus. By W. A. GRANVILLE. Boston, Ginn and Company. Pp. xiv + 463. A characteristic feature of mathematics in the last half century is the increasing attention paid to the foundations and rigorous development of this science. In analysis this movement began with Gauss, Cauchy and Abel in the early years of the nineteenth century and found its greatest exponent in Weierstrass. The movement thus begun has been continued by such men as Riemann, Dedekind, Hankel, Cantor, Jordan, Dini, Stolz, Harnack, Peano and a host of younger men. As a result of these investigations it was found that much of the reasoning hitherto employed and in current use among mathematicians was either worthless or required to be modified, restricted or completed. It thus became necessary to rewrite textbooks on analysis or to prepare new ones more in harmony with the new teachings. In this way arose the new edition of Jordan's 'Cours d'Analyse' and Harnack's edition of Serret's Calcul,' as well as the new works of Stolz, ' Allgemeine Arithmetik,' and 'Grundzüge'; Tannery, 'Théorie des fonctions d'une variable'; Dini, 'Fondamenti per la teorica delle funzioni di variabili reali.' In England and America more progressive teachers have felt for some time the need of a modern text-book on the calculus, which is at once rigorous and elementary. The task of writing such a work is not easy. On the one hand, it is necessary to avoid the worthless and even vicious forms of reasoning which mar so many elementary treatises and which are simply intolerable to one educated according to modern standards of rigor. On the other hand, the author must not introduce subtilties of reasoning and logical refinements beyond the needs and comprehension of those who are to use the book. The volume under review is an attempt to solve this difficult problem. To our mind the efforts of its author have been abundantly crowned with success. In perusing Dr. Granville's book one feels throughout that the author has in mind the requirements of modern rigor. The demonstrations, it is true, often rest on intuition; but this is necessary in a first course, as all will admit. They are, however, usually correct as far as they go, and free from the defects we have mentioned above. We believe the present volume is eminently a safe book to put in the hands of the beginner. He will get no false notions which afterwards will have to be eradicated, with much difficulty; he will, on the other hand, acquire a considerable acquaintance with the principles of the calculus and a good working knowledge of its methods. We make now a number of criticisms and suggestions. The definition of limit given in § 29 is not the one given by Cauchy and Weierstrass and now universally accepted. Looked at carefully, we see it supposes that all variables are functions of an auxiliary variable, the time. This leads to unnecessary complications in the definition of the limit of a function in § 32. We believe the strict Weierstrassian definition should be given and used. As an aid to comprehension, the author's notions in these articles might prove useful. In § 34 the notion of a graph is explained; but not with sufficient care, to our mind. How is the reader to know from their graphs that x and log x are continuous functions? The three properties of the exponential function given in this article result from their arithmetical properties and not from their graph, as the author seems to imply. The definition of the derivative given in § 41 is not satisfactory; what the author really defines is the differential coefficient at a point. It is their aggregate that forms the derivative. In § 55 the author has avoided an error which is very prevalent. His passage to the limit is, however, not completely justified. He has yet to show that He should see that there can be no need of making the further assumption, dx/dy 0: for if it were, the equation (1) could not exist. In § 133 the author introduces a double limit without any explanation. As such limits are used in connection with double integrals, § 231, seq., they should be explained with care. The footnote on page 194 is unintelligible to us and certainly will give rise to misapprehension. The theory of total differentiation does not meet our approval at all. The author has treated the subject from the standpoint that the variables x, x, x are all functions n of some one variable. Instead of true total differentials, he gets total derivatives. The du in § 137 are not total differentials, but differtials of functions of one variable. In the differentiation of implicit functions the author assumes merely the existence of the partial derivatives. He should assume also their continuity. The form of demonstration is bad, as it requires him to assume (tacitly) the existence of the very thing he is seeking, viz., dy/dx. In the treatment of envelopes, § 141, the author does not as usual give sufficient conditions for the validity of his reasoning, but contents himself with the vague statement in a footnote that the process is all right in all applications made in this book.' This blemish, which a few lines will remedy, should be removed in another edition. The definition of an infinite series given in § 147 is not felicitous. In avoiding the lax definition usually given the author has gone to the opposite extreme. The simplest way seems to be to consider as a symbol to which a meaning is attached as to other symbols, as > <=, etc. The solution of Ex. 3, § 152, is not quite rigorous, as it postulates the covergence of G. In § 160 undefined arithmetical operations are performed on series. We can not agree with the author that the remainder in Taylor's series for several variables is too complicated to be given. The treatment of maxima and minima can be made much more complete without complications or difficulty. The reasoning given at the bottom of page 248 can be made not only plausible,' but entirely conclusive, using no more space that that required by the author. In the reduction of indefinite integrals the author proves the trivial formulæ tion is constantly employed, even in establishing important theorems. Two chapters, XXIX. and XXX., are devoted to definite integrals. In the first we arrive at the notion of a definite integral by means of the notion of area; in the second, by means of the limit of a sum. In our opinion the first treatment is not only superfluous, but should be entirely omitted on several counts. The relatively few blemishes in this work, the reviewer is glad to state, will be removed in the next edition. JAMES PIERPONT. YALE UNIVERSITY. The Study of the Atom, or the Foundations of Chemistry. By F. P. VENABLE. Easton, Pa., The Chemical Publishing Co. Pp. 290. The history of an important scientific theory is an interesting study, where it is possible, as it often is, to trace the orderly development of that theory from stage to stage. The evolution of the atomic theory is a subject which has claimed the attention of many writers, and the story has been told so often and so well in works on the history of chemistry, that one wonders whether it is not familiar to most chemists. A careful perusal of this book does not disclose any new point of view, or anything new in the method of treatment, though the matter is generally presented in a satisfactory manner, especially Chapter V., which deals with the periodic system. In the last chapter of the book the author considers the most recent hypotheses regarding the constitution of matter by J. J. Thomson, Rutherford and others. The book is generally clear, conservative in tone and, on the whole, well-proportioned, though 75 pages, or one fourth of the contents, seems rather too much to devote to the conception of the atom before the time of Dalton, especially as this material must be taken entirely from secondary sources. The book may be commended as a good summary for students. E. T. ALLEN. SOCIETIES AND ACADEMIES. NEW YORK ACADEMY OF SCIENCES. SECTION OF GEOLOGY AND MINERALOGY. THE section was called to order at 8:15 P.M., November 21, 1904, with Vice-president Kemp in the chair and forty persons present. The first paper of the evening was by Professor J. J. Stevenson, upon The Island of Spitzbergen and its Coal,' and was illustrated by lantern slides. In introducing his subject, the speaker described briefly the coast of northern Norway and its geology, and referred in some detail to Bergen, Hammerfest and other cities. Spitzbergen was then taken up, and its coals and their geological relations were passed in review. The coal beds are of Jurassic age, and the coal is peculiar in that it partakes of the characters of the lignites as well as of the true coals. The second paper on the program was by Professor James F. Kemp, on The Titaniferous Magnetite in Wyoming.' On account of the lateness of the hour, the speaker presented his topic only in abstract. The magnetite occurs in two places, fifteen and twenty miles north of Laramie, Wyoming, the former and smaller occurrence being near the Shanton ranch, the latter and larger being on Chugwater Creek. Both are in wall-rock of anorthosite which is practically indistinguishable from anorthosite occurring in the Adirondacks. The ores range from 20 per cent. to 40 per cent. TiO,. Thin sections show that they contain green spinels, and one slide presents much olivine. They can be most reasonably explained as intrusive dikes. In this view the speaker agreed with Waldemar Lindgren, who has published a brief note regarding them. JAMES F. KEMP, Secretary pro tem. THE section held a special meeting December 2, 1904, with Vice-president Kemp in the chair and two hundred members and visitors in attendance. The meeting was called to order at 8:25 P.M. and the program of the evening was at once taken up. This consisted of a lecture by Professor Albrecht Penck, of the Imperial University at Vienna, who is an honorary member of the academy. The speaker discussed' The Glacial Surface Features of the Alps,' and gave a brief sum- · mary of some of the results of the twenty years of masterly work which has been done by him and under his direction in the Tyrol. Professor Penck described in popular language the nature of the valleys of the Alps and showed by means of lantern slides and a diagram how the glaciers have widened and deepened portions of their rocky basins and produced lakes. After a vote of thanks to the distinguished guest of the evening, the section adjourned. EDMUND OTIS HOVEY, Secretary. THE PHILOSOPHICAL SOCIETY OF WASHINGTON. THE 592d meeting was held December 10, 1904. The first paper was read by invitation by Mr. H. H. Kimball, of the Weather Bureau, on Variations in Insolation and in the Polarization of Blue Sky-light, during 1903 and 1904.' Observations with an Angström pyrheliometer have been maintained by the Weather Bureau at Washington since April, 1903. Comparison with previous observations at Providence, R. I., and Asheville and Black Mountain, N. C., indicate that the quantity of solar. radiation reaching the surface of the earth on clear days during 1903 was considerably less than during 1902 and 1904, the deficiency from April to September being 16 per cent. as compared with 1902, and 9 per cent as compared with 1904. Observations with a Pickering polarimeter indicate that there have been corresponding fluctuations in the polarization of blue skylight, the percentage of polarization at a point on a vertical great circle passing through the sun and 90° from it, having averaged 49.6 per cent. from May to October of 1904, as compared with 40.6 per cent. during the same months of 1903. The work of the astrophysical observatory of the Smithsonian Institution and numerous European observations were quoted, showing similar deficiencies in insolation, in the transmissibility of the atmosphere, and in the polarization of blue sky-light, during 1903. The subject was considered to be one well worthy of investigation by meteorologists. Mr. J. F. Hayford, of the Coast and Geodetic Survey, presented some recent results on 'The Computation of Deflections of the Vertical from the Surrounding Topography.' |