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obtain their derivatives. Real roots of equations of higher than the second degree are approximated by synthetic division without shifting the y-axis. The chapters on analytic geometry (41 pages) are devoted chiefly to the straight line and conic sections. These are treated very compactly, but with the completeness of many texts. Space is saved in the chapter on conics by leaving much of the theory to be worked out as exercises. For example, the equation of an ellipse is derived in the form

(1 − e2)x2

2еpx + y2 = 0,

and the following exercises given.

"2. Move the origin to the point (ep/(1 - e2), 0) to remove the term in the first power of x."

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"3. Call ep/(1 — e2) e2) a and (1 — e2)a2 = b2 in the equation obtained in example (2) and show that the equation reduces to

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Nearly a third of the book (88 pages) is given to the calculus. The e, d form of proof is used for the preliminary theorems on limits, the derivative introduced in a purely formal way, and the theorems necessary for differentiating algebraic functions, implicit as well as explicit, are obtained before it comes to light that the derivative has any concrete significance, either as the slope of a tangent line or as a rate. Series receive an extended treatment. Maclaurin's series is obtained by differentiating an assumed expansion and determining the coefficients. The function e is defined as a series, and the expansion of eAu is the basis for obtaining the derivative of e". Integration is considered as the inverse of differentiation, and also as the limit of a sum. The applications of the calculus most emphasized are tangent lines, maxima and minima, velocity and acceleration, plane areas, volume of a solid of revolution, work, and centroids.

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Much of the material is presented in a very compact form, so that, as stated in the preface, "The teacher will find an opportunity for originality in developing the text and at times a necessity for more details." The authors' style is sometimes abrupt, as in the sentence "Let student solve and verify," and a few sentences are open to more serious criticisms. For example: "Let x be a variable and a some constant. Then if a x assumes its sequence of values in order, there comes a stage, such that, for all subsequent values of x, the numerical value of a becomes and remains less than any assigned small value €, then a is called the limit of x" (page 64). "If, at a given instant of time, the speed of a body becomes uniform, then the distance As, passed over in the time At, is the instantaneous speed at the given instant" (page 214). The treatment of instantaneous speed which follows the latter quotation is unsatisfactory in that it confuses two different meanings of As.

BRIER HILL, N. Y.,

August, 1919.

ARTHUR SULLIVAN GALE.

NOTES.

a

"In 1880 the Cambridge University Press began the republication in collected form of Stokes's Mathematical and Physical Papers. In this publication he introduced for the first time the solidus notation for division, originally introduced by De Morgan in his article on the Calculus of Functions in the Encyclopaedia Metropolitana. If a fraction like or a differential coefficient such as dy, is mentioned in the text, the printing of such expressions requires a good deal of dx' "justification" on the part of the compositor. To avoid this expense and the loss of space Stokes introduced the linear notation a/b and dy/dx.” [A. Macfarlane's lecture on "Sir George Gabriel Stokes" in Lectures on Ten British Physicists of the Nineteenth Century (New York, 1919), pp. 100-101.]

The Mathematical Association of Japan for Secondary Education issued the first number of Volume 1 of its Journal under date of April, 1919. It is published by M. Kaba, Tokyo, as chief editor, assisted by M. Kuroda, M. Watanabe, S. Nagao and H. Furnkawa. While all papers in this issue were printed in Japanese, the announcement is made that contributions in English are accepted. The contents of the initial issue are as follows:

Treatise

Presidential Address by T. Hayashi.
Miscellanies

On the graph of log x by M. Kaba; list of market prices and some technical terms in business circles by M. Kaba; on some syllabi of mathematics in the European and American secondary schools by M. Kuroda.

Problems for Consideration

On entrance examination by M. Kaba; on the equipment for teaching of mathematics by H. Furnkawa; on the teaching of arithmetic in the middle school course by M. Kuniyeda; on the teaching of graphs in the middle school course by M. Kuroda.

Mathematical Problems by M. Watanabe; Book Reviews; Questions; Miscellaneous Reports; Social Column; Special Reports.

ARTICLES IN CURRENT PERIODICALS.

...

BULLETIN DES SCIENCES MATHEMATIQUES, volume 54, March, 1919: Review by R. Garnier of E. Turrière's Sur le calcul des objectifs astronomiques de Fraunhofer (1917), 49-50; "Sur un théorème relatif a l'extension du théorème de Rolle aux fonctions de plusieurs variables" by E. Gau, 50-51; "Sur le calcul des perturbations" by H. Vergne, 51-72; Revue des publications, 17-24.-April: Review by A. Boulanger of P. Duhem's Etudes sur Léonard de Vinci. Troisième série (Paris, 1913), 73-77; "Sur le calcul des perturbations" (suite et fin) by H. Vergne, 78–79; "La série + Į + √ + & +17 + 15 + 2 + 3 + 1 + 15 + 3 + 4 + où les dénominateurs sont 'nobres premiers jumeaux' est convergente ou finie" by V. Brun, 100-104 (à suivre). BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, volume 26, no. 1, October, 1919: "In memory of Gabriel Marcus Green" by E. J. Wilczynski, 1-13 [Quotation: "In the six short years of his mathematical career, from 1913-1919, he enriched geometry with so many new ideas and important results as would suffice to excite our admiration if they had been spread over all of a normal life time. In his death we have suffered a heavy loss, but his life and work will continue to be, for many of us, an everlasting source of strength and inspiration"]; "Reduction of the elliptic element to the Weierstrass form" by F. H. Safford, 13-16; "A note on 'continuous mathematical induction"" by Y. R. Chao, 17-18; "On the number of representations of 2n as a sum of 2r squares" by E. T. Bell, 19-25; "Some functional equations in the theory of relativity" by A. C. Lunn, 26-34; "Formulas for constructing abridged mortality tables for decennial ages" by C. H. Forsyth, 34-38; Review by F. W. Owens of Dowling's Projective Geometry (New York, 1917), 39-40; Review by C. F. Craig of McClenon's Introduction to the Elementary Functions (Boston, 1918), 40-41; "Notes" and "New Publications," 41-48.

BULLETIN OF THE FIRST DISTRICT NORMAL SCHOOL, Kirksville, Mo., volume 15, no. 9, September, 1915; Mathematics Series no. 1: "The use of the graph in geography teaching" by B. Cosby, 7-8; "A plan for the study of stocks and bonds" by W. H. Zeigel, 9-14; "The history of mathematics as an incentive to mathematical study" by G. H. Jamison, 15-20; "The value of a life setting to algebra problems" by B. Cosby, 21-25; "Analytic geometry in our high schools" by W. H. Zeigel, 26-32; "Sets of orthogonal functions and their oscillation properties" by C. A. Epperson, 33-38; "A teacher's library" by G. H. Jamison, 38-39; "Henry Ward Beecher and mathematics," 40-Volume 16, no. 10, October, 1916; Mathematics Series no. 2: "Building and loan associations explained" by W. H. Zeigel, 5-10; "An experiment in teaching algebra" by C. A. Epperson, 11-13; "Some suggestions and observations on the teaching of high-school mathematics" by G. H. Jamison, 14-20; "The purpose and content of high-school arithmetic" by B. Cosby, 21-30; "The cube root of a binomial surd" by W. H. Zeigel, 31-38-Volume 18, no. 12, December, 1918 (published August, 1919). Mathematics Series No. 31: "A geometrical problem -proof and application" by W. H. Zeigel, 5-10; "Keeping abreast of the times" by G. H. Jamison, 11-13; "Investments" by Byron Cosby, 14-20; “A suggested problem for classes in analytic geometry and surveying" by C. A. Epperson, 21-23; "The rôle of assumptions and definitions in high school mathematics" by G. H. Jamison, 24-30.

JOURNAL OF THE UNITED STATES ARTILLERY, Volume 50., June 1919: "A method of computing differential corrections for a trajectory" by G. A. Bliss, 455-460 [reprinted from corrected proof in volume 51 (October), 445-449]-Volume 51, August: "Equations of differential variations in exterior ballistics" by W. E. Milne, 154-159-September: "The new ballistics" by R. S. Hoar, 285-295; "The use of adjoint systems in the problem of differential corrections for trajectories" by G. A. Bliss, 296-311; "Effect of the earth's rotation upon the point of fall" by P. Field, 328-329-October: "Rotating bands" by O. Veblen and P. L. Alger, 355–390.

man.

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Nature, volume 104, September 25, 1919: "Mathematics at the University of Stressburg" by H. B. Heywood, 74; The "algebraic cube," 79 [model in eight pieces illustrating the formula (a + b)3 a3 +3a2b+3ab2+b3. "The blocks are supplied in a neat cubical box, 10 cm. to the edge, by Messrs. Barnes and Morris, Ltd., scientific instrument makers, Audrey House, Ely Place, London, E. C. 1]; The University of Edinburgh Mathematical Institute, 87-October 9: Review by S. Brodetsky of T. R. Running's Empirical Formulas (New York, 1907), H. B. Phillips's Differential and Integral Calculus (New York, 1916-17), W. P. Milne and G. S. B. Westcott's First Course in the Calculus, Part 1 (London, 1918), R. C. Fawdry's Dynamics, Part 2 (London, 1919), and R. S. Heath's Solid Geometry (London, 1919), 109-110; P. E. B. Jourdain, 117 [“With the mathematician Philip Edward Bertrand Jourdain there died on October 1 a truly remarkable Jourdain lived only thirty-nine years, but the amount and value of the work that he accomplished, considering the disability under which he labored, are almost incredible. . . . He went up to Cambridge in 1898, then already a cripple. During his course at Cambridge he spent some time in Germany and became a fluent and scholarly linguist, speaking and reading several European languages. In 1904, though now physically quite incapacitated, he was awarded the Allen mathematical scholarship for research, and throughout the remainder of his short career his main activities were directed to the prosecution of mathematical investigations. His most important work was the discovery of certain series of infinite numbers. Working with Russell and Whitehead, he showed that certain arithmetical processes could be applied to them, and thus he obtained new and interesting results. He continued on this line of research, and even a few days before his death, of the imminence of which he was fully aware, he succeeded in demonstrating the existence of a previously unsuspected series of infinities. . . Jourdain contributed extensive mathematical articles to the last edition of the Encyclopaedia Brittanica. He founded and edited the International Journal of Ethics. He was for some years the English editor, and since the death of Carus in 1918, the chief editor of the Monist. He also made a number of translations for the Open Court Publishing Co. Jourdain took the liveliest interest in the movement for encouraging the history of science. He was a contributor to Isis, and at the time of his death he had in preparation an article for the Studies in the History and Method of Science which it is hoped he may have left in a state ready for publication"]; "Hindu Spherical Astronomy," 119 ["Mr. G. R. Kaye has published a paper on 'Ancient Hindu spherical astronomy' in the Journal and Proceeding of the Asiatic Society of Bengal (Vol. 15). In this he summarises, with the aid of modern mathematical formulae, the fundemantal portions of the principal classical astronomical texts, which date from between A.D. 498 (the Aryabhatiya) and about A.D. 1000, when the redaction of the Surya Sidd1 For this number the Bulletin was called Bulletin of the State Teachers College.

hanta now extant was written. Indian trigonometry is, like Indian astronomy, of Greek origin, but the Indians developed the methods received from the Greeks in various ways. There seems to be no doubt that the Indians were the first to introduce the use of sines instead of chords, and to compute the table of sines. But they never went further, and did not make use of the tangent function. They never gave a proof of any rule they enunciated. .."]

OPEN COURT, volume 33, September, 1919: [a Paul Carus number with portrait] "The ideals of the life and work of Paul Carus" by P. E. B. Jourdain, 521-523 [Quotation: "At the Gymnasium of Stettin he came under the influence of the great mathematician Hermann Grassmann, of whom he always spoke with affectionate respect. Later he studied at the Universities of Strassburg and Tübingen. Owing to the need he felt so strongly for keeping his independence of thought, he resigned a teaching post in Germany and came first to England and then to America"].

REVUE DE MÉTAPHYSIQUE ET DE MORALE, 26. année, July-August, 1919: "A propos de la démonstration géométrique: Réponse à M. Goblot" by L. Rougier, 517-521.

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REVUE GÉNÉRALE DES SCIENCES PURES ET APPLIQUÉES, 30. année, nos. 15-16, August, 1919: "La vie et l'oeuvre de Léonard de Vinci. A propos du quatrième centenaire de sa mort by F. Bottazzi, 465-477; Review by R. D'Adhémar of P. Boutroux's Les Principes de l'analyse mathématique. Exposé historique et critique, volume 2 (Paris, 1919), 492.

494.

REVUE SCIENTIFIQUE, August 16-23, 1919: "Le temps et sa mesure" by M. Moulin, 486

SCHOOL SCIENCE AND MATHEMATICS, volume 19, no. 7, October, 1919: "What graphical and statistical material should be included in the ninth-grade mathematics course?" by L. E. Mensenkamp, 595-598; "Developing ability to solve the verbal problem; the basic aim of the ninth grade course" by Elsie G. Parker, 599–604; Problems and solutions, 655–658; "The theorem of Nichomachus" by U. P. Davis, 663 ["Cubical numbers are sums of consecutive odd integers"]; "Central Association of Science and Mathematics Teachers" by C. S. Winslow, 664-665.

SCIENCE, new series, volume 50, August 30, 1919: "New activities in the history of science" by L. C. Karpinski, 213-214-September 12: "Not ten but twelve!" by W. B. Smith, 239-242 [advocating duodenary scale of notation].

SCIENTIA, volume 26, no. 3, September, 1919: "La teoria di relatività nel suo sviluppo storico. Porte Ia: La relatività della prima maniera" by A. Palatini, 195–207, supplément, 59-72; "Le danger de l'application du calcul des probabilités aux sciences de la nature et en particulier à l'astronomie" by E. Belot, 242-246; Review by G. Scorza of Montessus de Ballore's Leçons sur les fonctions elliptiques en vue de leurs applications (Paris, 1917), and Exercices et leçons de mécanique analytique (Paris, 1915), 247-248; Review by U. Ricci of G. H. Knibbs's The Mathematical theory of population (Melbourne, 1917), 260 [The review: "L'éminent directeur de la statistique officielle de l'Australie a voulu donner une preuve de sa bravoure en réunissant en un gros volume une collection très étendue de formules mathématiques permettant de mesurer les phénomènes démographiques. Les instruments analytiques propres à définir et à décrire la consistance et les variations d'une population dans ses divers aspects (distribution d'une population par sexe et par âge, masculinité, c'est-à-dire proportion entre hommes et femmes, natalité, nuptialité, fécondité, mortalité, migrations) sont forgés, adaptés et appliqués avec grande richesse, nous dirons presque avec luxe. De très nombreuses courbes et surfaces sont dessinées dans le volume, et côte-à-côte avec la théorie s'avancent les exemples tirés du recensement australien de 1911 et d'autres statistiques australiennes, sans parler de comparaisons établies avec les données empruntées à d'autres pays. Tous ceux qui s'occupent de statistique générale et de démographie consulteront avec profit ce volume qui est une mine de précieuses et minutieuses études"].

SCIENTIFIC AMERICAN SUPPLEMENT, volume 88, September 20, 1919: "Derivation of new magic squares" by "Weg," 191.

SCIENTIFIC MONTHLY, volume 9, no. 4, October, 1919: "Linkages" by F. V. Morley,

366-378.

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, volume 20, no. 3, July, 1919: "Certain types of involutorial space transformations" by F. R. Sharpe and V. Snyder, 185-202; "On a new treatment of theorems of finiteness" by O. E. Glenn, 203-212; "On the theory of developments of an abstract class in relation to the calcul fonctionnel" by E. W. Chittenden and A. D. Pitcher, 213-233; "On the influence of keyways on the stress distribution in cylindrical shafts" by T. H. Gronwall, 234-244; "Some convergent developments associated with

irregular boundary conditions" by J. W. Hopkins and D. Jackson, 245-259; "Groups possessing a small number of sets of conjugate operators" by G. A. Miller, 260-270.

ZEITSCHRIFT FÜR MATHEMATISCHEN UND NATURWISSENSCHAFTLICHEN UNTERRICHT, volume 50, no. 4-5, April, 1919: "Ein neues elementares Verfahren zur Lösung von Extremaufgaben" by H. Dörrie, 153–177; "Ueber die Konstruction der Ellipse" by E. Wiedemann, 177–181; "Aufgaben-Repertorium," 189-192.

AMERICAN DOCTORAL DISSERTATION

J. H. Weaver, "Some extensions of the work of Pappus and Steiner on tangent circles." AMERICAN MATHEMATICAL MONTHLY, January, 1920, volume 27, pp. 2-11; also reprinted. (Pennsylvania, 1916.)

UNDERGRADUATE MATHEMATICS CLUBS.

EDITED BY U. G. MITCHELL, University of Kansas, Lawrence.
CLUB ACTIVITIES.

THE MATHEMATICS CLUB OF BROWN UNIVERSITY, Providence, R. I. [1918, 33-34, 459; 1919, 167].

April 18, 1919: "The mathematical theory of investments" by Professor Clinton H. Currier.

May 15: "Mathematics in chemistry" by Esther A. Brintzenhoff '19; "Isosceles trigonometry" by Chauncey D. Wentworth '20; "History of calculating machines" by Mr. W. L. Morden, New England manager of the Monroe Calculating Machine Company. Election of officers. Taking of club photograph.

June 4: Picnic.

The average attendance at the meetings for the year 1918-19 was 47. The officers elected for the year 1919-20 were:

Chairman of Club, Professor Roland G. D. Richardson;

Committee on Program, Professor Raymond C. Archibald, Alice F. Hildreth Gr., Pauline A. Barrows '21, Chauncey D. Wentworth '20, Daniel E. Whitford '20;

Committee on Arrangements, Professor Ray E. Gilman, Frances M. Merriam '20, Constance W. Haley '21, Raymond L. Wilder '20, Marshall H. Cannell '22, Bruce H. McCurdy '22.

THE MATHEMATICS CLUB OF CONNECTICUT COLLEGE, New London, Conn. [1918, 270, 460].

Active membership in this club is limited to students pursuing courses in mathematics beyond the regular freshman requirement. There were ten members of the club in 1918-19 and due to influenza and diphtheria quarantines as well as to war conditions only four formal meetings were held.

The officers for 1918-19 were: President, Margaret Maher '19; secretary, Justine McGowan '20; treasurer, Louise Avery '21. These officers constitute

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