Geschichte der Elementar-Mathematikby Johannes Tropfke

  • Published on
    08-Jan-2017

  • View
    215

  • Download
    1

Transcript

  • Geschichte der Elementar-Mathematik by Johannes Tropfke Review by: Solomon Gandz Isis, Vol. 29, No. 1 (Jul., 1938), pp. 167-169 Published by: The University of Chicago Press on behalf of The History of Science Society Stable URL: http://www.jstor.org/stable/225963 . Accessed: 09/05/2014 18:44 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. . The University of Chicago Press and The History of Science Society are collaborating with JSTOR to digitize, preserve and extend access to Isis. http://www.jstor.org This content downloaded from 169.229.32.138 on Fri, 9 May 2014 18:44:13 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/action/showPublisher?publisherCode=ucpress http://www.jstor.org/action/showPublisher?publisherCode=hss http://www.jstor.org/stable/225963?origin=JSTOR-pdf http://www.jstor.org/page/info/about/policies/terms.jsp http://www.jstor.org/page/info/about/policies/terms.jsp
  • REVIEWS I67 incorporation in this technical treatise. It is a concept demanding the type of scientific analysis CARNAP always favors. Certain other limitations to the professed tolerance and generality aimed at in this work persist; for example, CARNAP aims in the general syntax, rather than in a special syntax, to organize and improve the theory of logical types. Several excellent contributions are made to this theory; but there is at the same time no provision made for possible languages that would avoid or eliminate every scheme of logical types. Again, the two languages developed by CARNAP are distinctive and illustrative of the possibilities for variety in languages, in their illustration of the " coordinate-language " form as opposed to the usual " name-language " form. To establish the very possibility of such coordinate languages and to investigate their logic and feasibility is a great liberalizing advance. But again, the advance actually made suggests further liberalizations that CARNAP has not recognized. Thus his general syntax is largely worked out with the coordinate pattern in mind. Further the variety possible within the general coordinate scheme is not suggested. What seems to the reviewer an arbitrary prominence is given to spatio- temporal concepts. This is of course one of the distinguishing marks of his physicalistic theory, possibly tenable and worth investigation, but out of place when it narrows the perspective of the pure logician. The review of this truly excellent book must not end, however, on a note of disparagement; the volume develops too many highly successful and valuable concepts. It breathes an atmosphere of freedom, of progress in cooperative inquiry. Duke University. HENRY S. LEONARD. Johannes Tropfke. - Geschichte der Flementar-Mathematik. Dritter Band: Proportionen, Gleichungen. Dritte, verbesserte und ver- mehrte Auflage. IV+239 pages. WALTER DE GRUYTER & Co., Berlin und Leipzig, 1937. (Price RM io). More than thirty-six years ago, in 1902, the first edition of TROPFKE'S Geschichte der Elementar-Mathematik appeared in print. Ever since that time the tireless author continued collecting and correcting, adding and sifting his material, so that we have before us now, in the third edition, the mature fruits of the work of a lifetime. In contrast to the great historical work of MORITZ CANTOR, TROPFKE'S history of mathematics is arranged systematically, according to the subjects and topics. TROPFKE is less concerned with the mathemati- cians, their life-history and the historical background, than with the history and progress of the mathematical ideas and theories, special This content downloaded from 169.229.32.138 on Fri, 9 May 2014 18:44:13 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp
  • I68 ISIS, XXIX, I attention being given also to the development of the mathematical language and script, the terminology and the symbolism of mathematics. The third volume deals with proportions and equations, but while the theory of proportions is disposed of in twenty pages, the whole bulk of the book, more than two hundred pages, is devoted to the discussion of the history of equations. The third edition was greatly improved, and also enlarged by the inclusion of a considerable amount of new material, particularly in the fields of Babylonian, Egyptian and Arabic algebra. Thus TROPFKE was able to make use of the algebra of ABO KAMIL SHUJA' (c. 900), which is now available in a German translation made by DR. JOSEF WEINBERG from the Hebrew version of MORDECAI FINZI (c. I470). ABO KAMIL$s work is of great importance for the history of algebra and deserves further investigation. ABC KAMIL SHUJuM had deeply influenced LEONARDO FIBONACCI (c. I2oo) and through him mediaeval mathematics in general. He is the first known to us to relate Oriental algebra to Greek geometry, by referring to EUCLID, II 5-6, as furnishing the demonstration for the solution of the quadratic equations. The geometric proofs in AL- KHUWARIZMI'S algebra are basically different from the Euclidean demon- strations and have nothing to do with Greek geometry. ABUj KAMIL'S algebra is based upon AL-KHuwARIzMI's algebra and was intended to be an elaboration and further development of the same. ABU KAMIL not only introduces the new Greek demonstrations of the Euclidean geometric algebra, but he also makes an effort to preserve the ancient Babylonian type-forms of equations and methods of solution, which were discarded by AL-KHUWARIZMI. In the last decade there grew up a rich crop of newly discovered material in the field of Babylonian algebra. As usual in such cases, the decipherment, translation and edition of the texts must pave the way for their proper evaluation and interpretation from the point of view of the historian of mathematics. Already in his extensive monograph "Zur Geschichte der quadratischen Gleichungen iiber drei einhalb Jahrtausende" published in the 7ahresbericht der deutschen Mathematiker- Vereinigung, 1933-34, TROPFKE grapples with the difficulties of Baby- lonian algebra and with the problems confronting us in the history of the quadratic equations. The reviewer has devoted a rather com- prehensive study to the same subject in Osiris III, pp. 405-557, under the title "The Origin and Development of the Quadratic Equations in Babylonian, Greek, and Early Arabic Algebra". (A short summary of it appeared in Scientia, I937, pp. 249-257). In this essay the reviewer arrived at results which differ, considerably, from those reached by TROPFKE in his present volume and in his monograph just mentioned. This content downloaded from 169.229.32.138 on Fri, 9 May 2014 18:44:13 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp
  • REVIEWS I69 The reviewer, in his essay, advanced a new theory concerning the original type-forms of equations and methods of solution developed by the ancient Babylonians, and also concerning their relations to Greek and Arabic algebra. The student who is interested in the deeper in- vestigation of the subject is herewith referred to these studies. Professor TROPFKE is to be congratulated on his great achievement, and we heartily wish him to succeed in soon bringing out the third edition of his remaining volumes. His book has become a standard work in History of Mathematics which no serious student of the subject can afford to miss in his library. The reviewer believes that the book well deserves to be made available in an English translation, so that the English scholar, too, may easily enjoy the great wealth of information assembled by TROPFKE from the widely scattered literature which grew up during four thousand years in the history of the discipline so closely connected with the history of our civilization. New York City, SOLOMON GANDZ. P. W. Wilson*-The romance of the calendar. 351 P., 13 figs. London, GEORGE ALLEN and UNWIN Ltd., 1937. (Cloth, io s. 6 d.) According to the Foreword, " there does not seem to be any book of convenient size-at any rate, in the English language-that surveys the development and the significance of the calendar as a whole... No book, certainly no book so full of factual and speculative material as this, has ever been published without a percentage of what experts -sometimes differing from one another-will consider to be error. On any point of the kind I shall be happy to stand corrected... Friendly criticism in the reading of the manuscript and proofs has come from such eminent authorities as..." (the list includes the names of a lord, a bishop, two astronomers, and four other scientists or scholars). The reader of the Foreword might think that here, at last, is an authoritative English book on comparative calendariography-not as complete and technical as GINZEL'S three-volume Handbuch der Chronologie, of course, but a readable condensation of it, at least. The reader of WILSON's book will be disappointed; to judge by the thirteen-page Index, the author never heard of GINZEL. The aiuthor uses source material of questionable quality and uncertain vintage, and shows a regrettably uncritical attitude toward his sources. The result is a " popular " book of little value to serious readers. I did not take the trouble of reading the book as a whole-neither, for that matter, could the eight eminent authorities mentioned in the Foreword have read the whole MS and all the proofs; I have read several chapters, and I have picked up a considerable amount This content downloaded from 169.229.32.138 on Fri, 9 May 2014 18:44:13 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Article Contents p.167 p.168 p.169 Issue Table of Contents Isis, Vol. 29, No. 1 (Jul., 1938), pp. 1-306 Volume Information [p.1] Front Matter [pp.3-8] Preface to Volume Twenty-Nine:... Primum herbam deinde spicam deinde plenum frumentum in spica [pp.9-14] The Tomb of Omar Khayyâm [pp.15-19] Souvenirs concernant le Géomètre Yougoslave Marinus Ghetaldi, conservés à Doubrovnik, en Dalmatie [pp.20-23] Descartes et le théorème de Poincaré [pp.24-28] Sir Samuel Garth--A Court Physician of the 18th Century [pp.29-42] The First Table of the Normal Probability Integral: Its Use by Kramp, Who Constructed It [pp.43-48] Nils Gabriel Sefström--The Sesquicentennial of His Birth [pp.49-57] The Writings of J. K. Fotheringham [pp.58-68] Astronomy according to the Jews [pp.69-71] Some Notes on Chinese Musical Art [pp.72-94] Le Primitif et le Savant [pp.95-97] Notes and Correspondence [pp.98-103] Reviews untitled [pp.104-108] untitled [pp.108-110] untitled [p.110] untitled [pp.110-113] untitled [pp.113-115] untitled [p.116] untitled [pp.116-118] untitled [pp.118-119] untitled [pp.119-121] untitled [pp.121-123] untitled [pp.123-126] untitled [pp.126-127] untitled [pp.127-129] untitled [pp.129-132] untitled [pp.132-133] untitled [pp.133-134] untitled [pp.134-135] untitled [pp.135-138] untitled [pp.138-140] untitled [pp.140-141] untitled [pp.141-142] untitled [pp.142-143] untitled [pp.143-145] untitled [pp.145-148] untitled [pp.148-150] untitled [pp.150-153] untitled [pp.153-155] untitled [pp.155-157] untitled [pp.157-158] untitled [p.159] untitled [pp.160-163] untitled [pp.163-167] untitled [pp.167-169] untitled [pp.169-172] untitled [p.172] untitled [pp.172-176] untitled [pp.176-179] untitled [pp.179-181] untitled [pp.182-183] untitled [p.183] untitled [pp.183-184] untitled [pp.184-185] untitled [pp.185-186] untitled [pp.186-188] untitled [pp.188-189] untitled [pp.189-191] untitled [pp.191-192] untitled [pp.192-194] untitled [pp.194-196] untitled [pp.196-197] untitled [pp.197-200] untitled [pp.200-203] Fifty-Third Critical Bibliography of the History and Philosophy of Science and of the History of Civilization (to End of January 1938--With Special Reference to Sections 31 to 33) [pp.204-300] Indianapolis Meeting of HSS, 1937 [pp.301-302] The History of Science Society [pp.303-305] Back Matter [pp.306-306]