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Historia Mathematica 26 (1999), 181–198 Article ID hmat.1999.2240, available online at http://www.idealibrary.com on

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The purpose of this department is to give sufficient information about the subject matter of each publication to enable users to decide whether to read it. It is our intention to cover all books, articles, and other materials in the field. Books for abstracting and eventual review should be sent to this department. Materials should be sent to Prof. David E. Zitarelli, Department of Mathematics, Temple University, Philadelphia, PA 19122, U.S.A. (E-mail: [email protected]) Readers are invited to send reprints, autoabstracts, corrections, additions, and notices of publications that have been overlooked. Be sure to include complete bibliographic information, as well as translit- eration and translation for non-European languages. We need volunteers willing to cover one or more journals for this department. In order to facilitate reference and indexing, entries are given abstract numbers which appear at the end following the symbol #. A triple numbering system is used: the first number indicates the volume, the second the issue number, and the third the sequential number within that issue. For example, the abstracts for Volume 20, Number 1, are numbered: 20.1.1, 20.1.2, 20.1.3, etc. For reviews and abstracts published in Volumes1 through 13 there are an author index in Volume13, Number 4, and a subject index in Volume 14, Number 1. The initials in parentheses at the end of an entry indicate the abstractor. In this issue there are abstracts by Joe Albree (Montgomery, AL), Thomas L. Bartlow (Villanova, PA), Donald W. Bushaw (Pullman, WA), John G. Fauvel (Milton Keynes), Paulus Gerdes (Maputo), Leon Harkleroad (Poughkeepsie, NY), Graham Jagger (Milton Keynes), Victor J. Katz (Washington), Marvin Knopp (Philadelphia), James V. Rauff (Decatur, IL), Daniele C. Struppa (Fairfax, VA), and David E. Zitarelli.

Abeles, Francine F. Some Mathematical Letters of Charles L. Dodgson, 1866–1867, in #26.2.149, pp. 60–67. An account of correspondence among Dodgson, William Spottiswoode, and others during the time when Dodgson was working on determinants. (TLB) #26.2.1

Ackerberg-Hastings, Amy. How Straight Must a Straight Line Be for a Straight Line to Be Straight? The Teaching of Euclidean Geometry in the Early Republican United States, in #26.2.149, pp. 68–77. An exploration of British and French inﬂuences on the selection and preparation of geometry textbooks in the early decades of the 19th century in the United States. (TLB) #26.2.2

Adams, Rebecca. Mathematics and Phyllotaxis, in #26.2.149, pp. 27–33. Describes efforts to model “the posi- tioning of leaves and other organs of plants around an axis of growth” using the Fibonacci sequence in former times, and ideas of fractals in modern times. (TLB) #26.2.3

Alcantara, Jean-Pascal. La théorieleibnizienne du changement en 1676: Une interpretation du dialogue Pacidius Philalethi àla lumièrede la Caracteristique geom´ etrique´ , Theoria (San Sebastian) 12 (1997), 225–255. An interpretation of Leibniz’s dialogue Pacidius Philalethi (1646) which asserts that Leibniz’s “theory of the continuum allows one to deduce the topological conditions of universal changes.” See the review by Eberhard Knobloch in Mathematical Reviews 98k:01007. (JVR) #26.2.4

Anderson, J. Milne; Drasin, David; and Sons, Linda R. Wolfgang Heinrich Johannes Fuchs (1915–1997), Notices of the American Mathematical Society 45 (1998), 1472–1478. A biographical sketch of W. H. J. Fuchs, the specialist in complex function theory who ﬂed Germany for England in 1933 and spent most of his professional career at Cornell University. (DEZ) #26.2.5

181 0315-0860/99 $30.00 Copyright C 1999 by Academic Press All rights of reproduction in any form reserved. 182 ABSTRACTS HMAT 26

Anglin, W. S. The History of the Bachet Equation: From Diophantus to Baker, in #26.2.149, pp. 78–101. Describes methods of attacking the Bachet equation beginning with Diophantus’s problem VI.17 and culminating in the work of Alan Baker and others from the 1960s to the 1990s. (TLB) #26.2.6

Artmann, Benno. See #26.2.136.

Barabashev, A. G. Frege’s Triangle and the Existence of Mathematical Objects [in Russian], Istoriko-Matemati- cheskie Issledovaniya 37 (1997), 292–313. A discussion of formalism, logicism, intuitionism, and mathematical Platonism, for which the author provides a priority system with the help of a graphic scheme based on Frege’s triangle. (DEZ) #26.2.7

Barrantes, Hugo, and Ruiz, Angel.´ La historia del comite´ interamericano de educacion´ matematica´ [The His- tory of the Inter-American Committee on Mathematics Education], Santaféde Bogotá,Colombia: Academia Colombiana de Ciencias Exactas, F´ısicasy Naturales, 1998, viii + 94 pp., softbound. An examination of “the diverse and concurrent processes” (p. vii) in the 35-year history of the Inter-American Committee on Math- ematics Education (IACME), with an English translation following the Spanish original. After setting the so- ciohistorical and ideological background, the book describes the nine conferences sponsored by IACME, two in detail, in Bogotáin 1961 and in Santiago in 1965. Also described are the topics discussed at the conferences and the protagonists who organized the conferences, with a special section devoted to Marshall Stone. (DEZ) #26.2.8

Barrow-Green, June. History of Mathematics: Resources on the World Wide Web, Mathematics in School 27 (4) (1998), 16–22. A 12-part categorization of Web sites, with guidance on searching and critical evaluation of what may be found. (JGF) #26.2.9

Barrow-Green, June, and Siu, Man-Keung. History of Mathematics at the International Congress of Mathemati- cians, BSHM Newsletter 38 (1998), 12–16. Separate reports on various aspects of the history of mathematics at the International Congress of Mathematicians held in August 1998 in Berlin, including the three invited addresses, workshops, short communications, special sessions, poster sessions, short communications, and an exhibition. (DEZ) #26.2.10

Barrow-Green, June. See also #26.2.101.

Bashmakova, I. G. Sofya Alexandrovna Yanovskaya (1896–1966): Reminiscences [in Russian], Istoriko- Matematicheskie Issledovaniya 37 (1997), 105–108. Reminiscences by the author of the eminent Russian historian of mathematics, philosopher of mathematics, and specialist in mathematical logic, S. A. Yanovskaya. See also #26.2.154. (DEZ) #26.2.11

Bass, Hyman; Cartan, Henri; Freyd, Peter; Heller, Alex; and Mac Lane, Saunders. Samuel Eilenberg (1913– 1998), Notices of the American Mathematical Society 45 (1998), 1344–1352. Personal reminiscences of “Sammy” Eilenberg, the Polish-born topologist, Bourbakist, and art collector who ﬂed Warsaw for the U.S. in 1938 and spent most of his professional career at Columbia University. (DEZ) #26.2.12

Blay, Michel. Reasoning with the Infinite: From the Closed World to the Mathematical Universe, trans. M. B. DeBevoise, Chicago: Univ. of Chicago Press, 1998, 216 pp., hardbound, $30. The author traces the origins of the mathematization of astronomy that enabled natural philosophers to explain and predict cosmic motions. He begins with the metaphysical debate over infinity, which undermined Galileo’s claim that Euclidean geometry was the key to universal knowledge and spurred Leibniz’s new differential and integral calculus, then considers the philosophical consequences of submitting the infinite to rational analysis. (DEZ) #26.2.13

Boissonnade, Auguste, and Vagliente, Victor N., eds. Analytical Mechanics: Translated from the Mecanique´ analytique, nouvelle edition´ of 1811, Dordrecht/Boston/London: Kluwer, 1997, xlvi + 594 pp., hardbound; $210, £143, Dﬂ 325. The editors have translated the famous Mecanique´ analytique by J. L. Lagrange into English. There is also a preface by Craig G. Fraser. In this work, Lagrange used the “Principle of virtual work” as the foundation for all of mechanics and thereby brought together statics, hydrostatics, dynamics, and hydrodynamics. (DEZ) #26.2.14 HMAT 26 ABSTRACTS 183

B¨olling, Reinhard. Georg Cantor—Ausgew¨ahlte Aspekte seiner Biographie, Jahresbericht der Deutschen Mathematiker-Vereinigung 99 (1997), 49–82. Selected aspects of the biography of Georg Cantor. See the review by Roger L. Cooke in Mathematical Reviews 98b:01027. (DEZ) #26.2.15

Bollobás,Béla. Paul Erdös—Lifeand Work, in #26.2.58, pp. 1–41. A brief account of the life of Paul Erdös with a comprehensive survey of his work by one of his friends and collaborators. See the review by Jerrold W. Grossman in Mathematical Reviews 98b:01028. (DEZ) #26.2.16

Brigaglia, Aldo, and Roero, Clara Silvia. La storia della matematica (II): Dal Rinascimento all’Illuminismo [History of Mathematics (II): From the Renaissance to the Enlightenment], Lettera Matematica 27–28 (1998), 26–34. This second part of a series provides an annotated bibliography of mathematics from the Renaissance to the birth of the Enlightenment. The authors present a series of commentaries between the various bibliographical references. (DCS) #26.2.17

Bryden, D. J. Capital in the London Publishing Trade: James Moxon’s Stock Disposal of 1698, a “Mathematical Lottery,” The Library 19 (1997), 293–350. In the later 17th century many booksellers disposed of stock and raised capital through the popular craze for lotteries. James Moxon’s lottery proposal provides details that allow the minimum quantity of his stock to be computed. This paper also provides a printing history of each item included in the lottery. (GJ) #26.2.18

Buhmann, Martin D., and Fletcher, Roger, eds. M. J. D. Powell’s Contributions to Numerical Mathematics, in M. D. Buhmann and A. Iserles, Approximation Theory and Optimization, Cambridge, UK: Cambridge Univ. Press, 1997, pp. 1–30. A review of the work of Michael J. D. Powell in approximation theory and optimization. (DEZ) #26.2.19

Cartan, Henri. See #26.2.12.

Caveing, Maurice. La constitution du type mathematique´ de l’idealit´ e´ dans la pensee´ grecque. Vol. I. Essai sur le savoir mathematique´ dans la Mesopotamie´ et l’Egypte´ anciennes, Lille: Presses universitaires de Lille, 1994, 419 pp., softbound, 240 F. See the extended review by Jens Hyrup in Mathematical Reviews 98b:01006. (DEZ) #26.2.20

Cermák,Libor;ˇ Nedoma, Josef; and Zen´ˇ ıˇsek, Alexander. In Memoriam Professor MiloˇsZlámal, Czechoslovak Mathematics Journal 48 (1998), 185–191. A summary of the work of M. Zlámalwith a list of his publications. (DEZ) #26.2.21

Chemla, Karine. Reﬂections on a World-Wide History of the Rule of False Position, or: How a Loop Was Closed, Centaurus 39 (2) (1997), 97–120. After a discussion of the two differing cases of the rule of double false position that are explicated in the Chinese text, Nine Chapters of the Mathematical Art, the author concludes that there was transmission of these ideas to the West, but only of one of the methods. See the review by Victor Katz in Mathematical Reviews 98b:01008. (DEZ) #26.2.22

Cheng, Shiu-Yuen;Li, Peter; and Tian, Gang, eds. A Mathematician and His Mathematical Work:Selected Papers of S. S. Chern, River Edge, NJ: World Scientiﬁc, 1996, xvi + 707 pp., hardbound, $86. This volume begins with a collection of articles containing biographical and scientiﬁc information about the geometer, Shiing S. Chern. For a complete list of the contents, see the review by Harold Donnelly in Mathematical Reviews 98b:01035. See also #26.2.163. (DEZ) #26.2.23

Cifarelli, Donato Michele, and Regazzini, Eugenio. De Finetti’s Contribution to Probability and Statistics, Sta- tistical Science 11 (1996), 253–282. An analysis of De Finetti’s role in the history of Bayesian probability and statistics, with an extensive bibliography of his work, plus a discussion of work that inﬂuenced him and work that depended on him. See the review by Henry E. Kyburg, Jr., in Mathematical Reviews 98b:01033. (DEZ) #26.2.24

Cload, Bruce. Why We Must Teach the History of Mathematics, Bulletin CSHPM/SCHPM 23 (1998), 7–11. The author describes a one-year course he taught in the history of mathematics that produced a “new-found appreciation 184 ABSTRACTS HMAT 26 of the importance of the history of mathematics and ... the signiﬁcant advantages of a historical perspective in mathematics instruction.” (DEZ) #26.2.25

Cohen, Edward L. A Short History of Gregorian Calendar Reform, in #26.2.149, pp. 102–116. Efforts to improve the Gregorian calendar began in the 18th century, but this paper focuses on the calendar reform movement in the 20th century. (TLB) #26.2.26

Cohen, I. Bernard. Howard Aiken on the Number of Computers Needed for the Nation, Annals of the History of Computing 20 (1998), 27–32. Howard Aiken is reputed to have said that only a small number of computers would be needed for the needs of the whole world. But he seems to have been referring to proposed computers for the Bureau of the Census or the Bureau of Standards. (JGF) #26.2.27

Cohen, Morris, ed. His Glassy Essence: An Autobiography of Charles Sanders Peirce, Lincoln, NE: Univ. of Nebraska Press, 1998, xlvi + 318 pp., softbound, $15. Originally published in 1923, this version contains an introduction by Kenneth Laine Ketner that discusses the life, thought, and intellectual milieu of C. S. Peirce (1839–1914) by weaving components of his biography scattered throughout Peirce’s published and unpublished writings into a novelistic account. Also included are autobiographical musings, biographical notations from the letters of family and friends, and selections from Peirce’s philosophical treatises. (DEZ) #26.2.28

Cooke, Roger L. See #26.2.15.

Corry, Leo. Les étapesou les paradigmes des mathématiques?in Elena Ausejo and Mariano Hormigón,ed., Paradigms and Mathematics, Madrid: Siglo XXI de EspañaEditores, 1996, pp. 169–191. An analysis of peri- odization in the history of mathematics from historians, Montucla, Cantor, Struik, and Yushkevich, and modern mathematicians, Dieudonné,Bell, and Kline. The issue of whether mathematics proceeds in stages or paradigms is then discussed. (DEZ) #26.2.29

D’Ambrosio, Ubiratan. Etnomatematica: Lo stato dell’arte [Ethnomathematics: The State of the Art], Lettera Matematica 27–28 (1998), 4–12. The author, one of the founders and theoreticians of ethnomathematics, reﬂects on the discipline as the intersection of mathematics and cultural anthropology. (DCS) #26.2.30

Davies, Barry. Historical Threads for “Seeing” in a Curved Space, Bulletin CSHPM/SCHPM 23 (1998), 5–6, 20. The author describes his use of notions from Euclid, Riemann, Einstein, and Levi-Civit`ain his Web site on curved space, and appeals for help from historians on this project. See http://www.uvw.com/ppr. (DEZ) #26.2.31

Day, Jane. Robert C. Thompson, 1931–1995, Linear and Multilinear Algebra 43 (1997), 13–18. A tribute to R. C. Thompson. See also #26.2.111. (DEZ) #26.2.32

Deakin, Michael A. B., and Lausch, Hans. The Bible and Pi, Mathematical Gazette 82 (1998), 162–166. It has been suggested that biblical references that seem to imply a -value of 3 in fact encode, in Hebrew, a more accurate value of 333/106. More plausibly, this is a coincidence: analogous procedures can show that Queen Victoria wrote In Memoriam, and in any case 355/113 is far more accurate. (JGF) #26.2.33

Demidov, Sergei S. On the Paper “On the History of Qualitative Methods in the Theory of Differential Equa- tions” by I. B. Pogrebyssky [in Russian], Istoriko-Matematicheskie Issledovaniya 37 (1997), 281–292. The author provides the background for a 1968 paper written by the eminent historian of science, I. B. Pogrebyssky (1906– 1971), on the history of differential equations. The paper, reproduced on pp. 283–292, presents a good example of the interaction of two types of inﬂuence, the internal logic of the development of mathematics, and the external needs of theoretical mechanics and technology. (DEZ) #26.2.34

Demidov, Sergei S. See also #26.2.159.

Dhombres, Jean, and Sakarovitch, J., ed. Desargues en son temps, Paris: Blanchard, 1994, 485 pp. This collection of articles comes in four parts: (1) Desargues and his era, (2) perspective and geometry, (3) gnomonics, stereotomy, and architecture, and (4) Desargues and the Lyonese. The contents are given in Mathematical Reviews 98b:01001. Papers by Jean Dhombres, Yves Bottineau-Fuchs, J. V. Field, and Jean-Pierre Le Goff have been abstracted in #25.4.39, #26.1.16, #26.1.56, #26.1.104, respectively. (DEZ) #26.2.35 HMAT 26 ABSTRACTS 185

Donnelly, Harold. See #26.2.23.

Donoghue, Eileen. In Search of Mathematical Treasures: David Eugene Smith and George Arthur Plimpton, Historia Mathematica 25 (1998), 359–365. An account of the role that historian of mathematics, David Eugene Smith (1860–1944), played in the assembling, displaying, and donating of the Plimpton Collection at Columbia University. Smith’s relationship with publishing mogul, George Arthur Plimpton (1855–1936), evolved from advisor to personal friend. (DEZ) #26.2.36

Dorier, Jean-Luc. On the Teaching of the Theory of VectorSpaces in the First Yearof French Science Universities, Edumath 6 (1998), 38–48. Historical analysis enables us to explain the speciﬁc meaning which formalism has in the theory, and thus the teaching, of vector spaces. Other pedagogical issues, including students’ mistakes, can be understood and acted upon better through the study of history. (JGF) #26.2.37

Doumbia, Salimata. Maths et cultures: Pythagore en Afrique, Bulletin harmonisation des programmes de mathematiques´ des pays francophones d’Afrique et de l’Ocean´ Indien 3 (1997), 6–11. Gives examples of Pythagorean ﬁgurative numbers in West Africa and presents ideas on African crafts and the Pythagorean theorem. (PG) #26.2.38

Eglash, Ron. Bamana Sand Divination, American Anthropologist 99 (1997), 112–122. The author compares the use of recursion among Bamana (or Bambara) diviners with Cantor’s recursion (the Cantor set.) He hypothesizes that an African concept of self-generated fecundity is the shared origin of both the Bamana divination and transﬁnite set theory. (PG) #26.2.39

Eglash, Ron. The African Heritage of Benjamin Banneker, Social Studies of Science 27 (1997), 307–315. “His- torical and linguistic evidence suggest that [Benjamin Banneker’s] grandfather was of Wolof origin, and that his father was from the area between what is now Ghana and Nigeria. This cultural heritage may have emerged in some of his mathematical thinking” (p. 307). (PG) #26.2.40

Ermolaeva, N. S. N. N. Luzin and the Environment of the Academy [in Russian], Istoriko-Matematicheskie Issledovaniya 37 (1997), 43–66. A detailed survey of scientiﬁc contacts between Nikolai Nikolaevich Luzin (1883– 1950) and other members of the Academy of Sciences, including some eminent academicians and mathematicians such as S. A. Chaplygin, A. N. Krylov, and N. M. Krylov. (DEZ) #26.2.41

Fauvel, John. Algorithms in the Pre-Calculus Classroom: Who Was Newton–Raphson? Mathematics in School 27 (4) (1998), 45–47. The so-called Newton–Raphson method (due in its present form to Thomas Simpson) provides insights into algorithms and iterative processes which can be useful for pupils before as well as after they learn calculus. (JGF) #26.2.42

Fava, Franco. In memoriam Franco Tricerri [in Italian], Rendiconti del Circolo matematico di Palermo 49 (1997), 13–24. An account of the life and work of the Italian differential geometer, Franco Tricerri (1947–1994), with a list of his publications. (DEZ) #26.2.43

Feffer, Loren Butler. Oswald Veblen and the Capitalization of American Mathematics: Raising Money for Research, 1923–1928, Isis 89 (1998), 474–497. A discussion of efforts led primarily by Oswald Veblen in the 1920s to solicit funds in support of mathematics, through the American Mathematical Society and Princeton University, in the context of fundraising success in physics and chemistry. “[T]he efforts of Veblen and his colleagues...insured that the mainstream of American mathematical research...would remain free and ‘pure’” (p. 497.) (DEZ) #26.2.44

Fletcher, Roger. See #26.2.19.

Folkerts, Menso. “Ein freundschaftliches Verh¨altniß”:J. L. Tiarks und C. F. Gauß, Acta Historica Leopoldina 27 (1997), 159–174. A description of the relationship between Johann Ludwig Tiarks (1789–1837) and Gauss (1777– 1865). Includes some of the letters from Tiarks to Gauss. See the review by Wilfried Schr¨oderin Mathematical Reviews 98k:01025. (JVR) #26.2.45 186 ABSTRACTS HMAT 26

Ford, Charles E. The Inﬂuence of P. A. Florensky on N. N. Luzin [in Russian], Istoriko-Matematicheskie Issle- dovaniya 37 (1997), 33–43. Correspondence between N. N. Luzin and P. A. Florensky reveals that Luzin experi- enced a profound spiritual crisis in 1905 when his materialist world-view collapsed. This crisis continued for four years and was ﬁnally resolved when Luzin had a decisive encounter with the religious philosophy of Florensky. After this time Luzin’s interest in mathematics gradually revived until he was able, by 1909, to commit himself to a career in mathematics. (DEZ) #26.2.46

Fowler, David H. The Mathematics of Plato’s Academy: A New Reconstruction, Expanded Second Edition, Oxford: Oxford Univ. Press, 1998, 496 pp., hardbound, £60. A new edition of an acclaimed study of the mathematics in Ancient Greece during the classical period. In this edition, the original text has been revised, with the chapter on Book X of Euclid’s Elements rewritten, an epilogue and a substantial appendix added, and the bibliography updated. The book is divided into three parts: interpretation, evidence, and later developments. (DEZ) #26.2.47

Fowler, David H. Wilbur Knorr Memorial Conference, BSHM Newsletter 38 (1998), 4–5. Report of a meeting held at Stanford University in March 1998 in memory of the esteemed historian of mathematics, Wilbur Knorr. See also #25.4.51. (DEZ) #26.2.48

Fowler, David H., and Robson, Eleanor. Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context, Historia Mathematica 25 (1998), 366–378. The authors reveal the detective√ work they adopted that led them to several Old Babylonian tablets which conﬁrm that the approximation of 2 on YBC 7289 (YBC = Yale Babylonian Collection) is a school exercise that makes use of a coefﬁcient list. An algorithm is derived for evaluating other such roots. (DEZ) #26.2.49

Fraser, Craig G. See #26.2.14.

Freyd, Peter. See #26.2.12.

Friedelmeyer, Jean-Pierre. What History Has To Say to Us about the Teaching of Analysis, in Evelyne Barbin and R´egineDouady, ed., Teaching Mathematics: The Relationship between Knowledge, Curriculum and Practice, Metz: Topiques Editions,´ 1996, pp. 109–122. Reforms in teaching analysis have attempted to reconcile the appar- ently irreconcilable needs for rigor and for understanding. Teaching in a historical context enables the meaning and rigor to be interactively constructed along with the student’s mathematical insight, by a process which is dynamic and living. (JGF) #26.2.50

Gerdes, Paulus. Ethnomathematik dargestellt am Beispiel der Sona Geometrie, Heidelberg/Berlin/Oxford: Spektrum Verlag, 1997, viii + 433 pp., DM 128. See #25.3.46 for an abstract of this book. An extended review of it by Bernd A. Schmeikal in Mathematical Reviews 98j:01003 describes the history of the study of ethnomathematics with Gerdes, Ubi D’Ambrosio, and the Aschers before providing an analysis of the book itself. (DEZ) #26.2.51

Gerdes, Paulus. On Culture and Mathematics Teacher Education, Journal of Mathematics Teacher Education 1 (1998), 33–53. A short history of mathematics teacher education in Mozambique since independence in 1975, highlighting the multicultural context and the role of the history of mathematics and of ethnomathematics in teacher education. (PG) #26.2.52

Gindikin, Simon G. Ramanujan the Phenomenon, Quantum 8 (March–April 1998), 4–9. A description of the mathematics discovered by Srinivasa Ramanujan that helps to understand its legacy. (DEZ) #26.2.53

Godard, Roger. History of the Least Squares Method and Its Links with Linear Algebra, in #26.2.149, pp. 133–147. Traces the history of treating observational errors from Galileo to the middle of the 20th century. (TLB) #26.2.54

Gordon, Robert J. What is the Econometric Society? History, Organization, and Basic Procedures, Econo- metrica 65 (1997), 1443–1451. A history of the Econometric Society and its ofﬁcial journal, Econometrica, which was initiated in 1933 with a primary focus on the application of mathematics and statistics to economics. (DWB) #26.2.55 HMAT 26 ABSTRACTS 187

Gowri, Aditi. Mathematical Revitalizations, in #26.2.149, pp. 117–132. An effort to address anew the question of whether Kuhnian revolutions occur in mathematics and, if not, how properly to characterize change in mathematics. (TLB) #26.2.56

Graham, Ronald L.; Korte, Bernhard; Lovász,László;Wigderson, A.; and Ziegler, GüntherM., ed. The Math- ematics of Paul Erdos,¨ I, Berlin: Springer-Verlag, 1997, 420 pp., hardbound, $105. A collection of articles that survey the life and work of Paul Erdös.Articles by BélaBollobás,Arthur H. Stone, and Cedric A. B. Smith are abstracted separately. See also #26.2.68. (DEZ) #26.2.57

Grattan-Guinness, Ivor. Karl Popper for and against Bertrand Russell, Russell 18 (1998), 25–42. A study of the personal and philosophical interactions of Russell and Popper, drawing upon personal reminiscences as well as texts. (JGF) #26.2.58

Grattan-Guinness, Ivor. Some Neglected Niches in the Understanding and Teaching of Numbers and Number Systems, Zentralblatt fur¨ Didaktik der Mathematik 30 (1998), 12–18. The author provides a selection of examples from the history of numbers and number systems that are not well understood but which could be used in teaching. (DEZ) #26.2.59

Green, H. S. See #26.2.70.

Grier, David Alan. The Math Tables Project of the Work Projects Administration: The Reluctant Start of the Computing Era, Annals of the History of Computing 20 (1998), 33–50. The success of the MTP, begun in New York in 1938, in developing and promoting methods of large-scale scientiﬁc computation, grew out of the work of strong individuals—notably Arnold Lowan and Gertrude Blanch—with a collection of motivations. (JGF) #26.2.60

Grifﬁths, Peter L. The Mathematics of the Projection Maps of Etzlaub (1511) and Mercator (1569) Including Comments from Edward Wright (1599), in #26.2.149, pp. 148–157. Argues that the conversion of degrees of latitude into secants for a map of the world was Etzlaub’s idea rather than Mercator’s. (TLB) #26.2.61

Groˇsek,Otokar; Satko, Ladislav; and Schein, Boris M. Professor Stefanˇ Schwarz (1914–1996), Semigroup Forum 56 (1998), 152–157. A photo and scientiﬁc bibliography of Stefanˇ Schwarz, known mainly for his pioneering work in the algebraic theory of semigroups. (DEZ) #26.2.62

Grosholz, Emily. Descartes’ Geometry and the Classical Tradition, in Roger Ariew and Peter Barker, ed., Essays in the History and Philosophy of Science, Indianapolis: Hackett Publishing Co., 1996, pp. 183–196. An examination of Descartes’s attempt to classify curves according to genre. See the review by William R. Shea in Mathematical Reviews 98b:01014. (DEZ) #26.2.63

Grossman, Jerrold W. See #26.2.16, #26.2.145, and #26.2.147.

Gupta, Radha Charan. False Mathematical Eponyms and Other Miscredits in Mathematics, Gan. ita-Bharat¯ ¯ı 19 (1997), 11–34. Discusses the often arguable attachment of a person’s name to some piece of mathematics. Para- graphs are devoted to 48 mathematicians, from A (witch of Agnesi) to W (Wilson’s theorem). (LH) #26.2.64

Gupta, Radha Charan. On Sungka Approximation to Pi, Gan. ita-Bharat¯ ¯ı 19 (1997), 101–106. Deals with geometrical√ approximations related to traditional Philippine mathematics—in particular, the approximation of by 6 2 2. (LH) #26.2.65

Gupta, Radha Charan. On the Approximation: = 3 1/8, Gan. ita-Bharat¯ ¯ı 19 (1997), 91–96. Overview of various claims and counterclaims made about the appearance of this approximation throughout mathematical history. (LH) #26.2.66

Gupta, Radha Charan. Primitive Area of a Quadrilateral and Averaging, Gan. ita-Bharat¯ ¯ı 19 (1997), 52–59. Explores area-approximation formulas that have appeared in various cultures. Gives particular attention to the approximation of a quadrilateral’s area by (a + c)(b + d)/4, where a, b, c, d are the side lengths in cyclic order. (LH) #26.2.67 188 ABSTRACTS HMAT 26

Gurevich, B. M. See #26.2.102.

Hajnal, András. Paul Erdös’Set Theory, in Ronald L. Graham, Bernhard Korte, LászlóLovász, A. Wigderson, and GüntherM. Ziegler, ed., The Mathematics of Paul Erdos,¨ II, Berlin: Springer-Verlag, 1997, pp. 352–393. A personal history of Paul Erdösand his collaborators. See also #26.2.57 and the review by Marion Scheepers in Mathematical Reviews 98b:01031. (DEZ) #26.2.68

Heilbron, John. Geometry Civilized: History, Culture, and Technique, Oxford: Oxford Univ. Press, 1998, 318 pp., hardbound, £35. An introduction to classical plane geometry that combines history and science and presents a cross-cultural examination. The methods of Euclid are central but approaches of the Chinese and Indians are also interwoven. (DEZ) #26.2.69

Heller, Alex. See #26.2.12.

Helou, Charles. See #26.2.119.

Hemmer, P. C.; Holden, Helge; and Kjelstrup Ratkje, S., ed. The Collected Works of Lars Onsager, River Edge, NJ: World Scientiﬁc, 1996, x + 1075 pp., hardbound, $98. Lars Onsager (1903–1976) was the Nobel-Prize-winning J. Willard Gibbs Professor of Theoretical Chemistry at Yale. Reviewer H. S. Green calls him “a worthy successor to Gibbs” in Mathematical Reviews 98b:01042. (DEZ) #26.2.70

Herring, Stewart. Archibald Sandeman and the Sandeman Library, Perth, BSHM Newsletter 38 (1998), 55–59. An account of mathematics professor, Archibald Sandeman (1822–1893), and the public library he bequeathed to the city of Perth. The Sandeman Library “has no particular interest or tradition in mathematics and Professor Sandeman did not personally leave any of his works there.” (DEZ) #26.2.71

Hill, Katherine. “Juglers or Schollers?”: Negotiating the Role of a Mathematical Practitioner, British Journal for the History of Science 31 (1998), 253–274. Tension between mathematical practitioners about the proper mixture of theory and practice in mathematical education is seen in the priority dispute (c. 1632) between William Oughtred and Richard Delamain over invention of instruments. The arguments concerned the boundaries of proper mathematical practice. (JGF) #26.2.72

Hirschfeld, James W. P. See #26.2.80, #26.2.97, and #26.2.150.

Hirzebruch, Friedrich. Kunihiko Kodaira: Mathematician, Friend, and Teacher, Notices of the American Math- ematical Society 45 (1998), 1456–1462. An appreciation of the inﬂuence that the late Japanese Fields Medalist, K. Kodaira, had on the author’s work in algebraic geometry and several complex variables. See also #25.3.114. (DEZ) #26.2.73

Hofman, Karl H., and Lawson, Jimmie D. Linearly Ordered Semigroups: Historical Origins and A. H. Clifford’s Inﬂuence, in Karl H. Hofmann and Michael Mislove, ed., Semigroup Theory and Its Applications, Cambridge, UK: Cambridge Univ. Press, 1996, pp. 15–39. An elaboration of the development of the theory of linearly ordered semigroups from 1826 to 1995, recognizing the contributions of Alfred H. Clifford (1900–1992.) See the review by Bernhard H. Neumann in Mathematical Reviews 98b:01025. (DEZ) #26.2.74

Holden, Helge. See #26.2.70.

Hoshina, Takao; Nagata, J.; Okuyama, Akihiro; and Watanabe, Tadashi. Kiiti Morita 1915–1995, Topology and its Applications 82 (1998), 3–14. A discussion of the contributions of the Japanese mathematician, K. Morita, to topology, including a list of his publications. (DEZ) #26.2.75

Hyrup, Jens. See #26.2.20.

Israel, Giorgio, and Menghini, Marta. The “Essential Tension” at Work in Qualitative Analysis: A Case Study of the Opposite Points of View of Poincar´eand Enriques on the Relationships between Analysis and Geometry, Historia Mathematica 25 (1998), 379–411. There were two diametrically opposed approaches to the qualitative theory of dynamical systems during the subject’s inception in the late 19th century. This paper analyzes how the HMAT 26 ABSTRACTS 189

“essential tension” resulting from approaches due to Henri Poincar´eand Federigo Enriques affected the relationship between geometry and analysis, and the relationship between mathematics and physics. (DEZ) #26.2.76

Jagger, Graham. Bolton Library, Cashel, BSHM Newsletter 38 (1998), 52–55. An account of the history of, and holdings in, the library founded in 1744 by Theophilus Bolton, Archbishop of Cashel. The oldest work is a 12th- century volume bound in deerskin. (DEZ) #26.2.77

Jha, V. N. Aryabhata II’s Method for Finding Cube-Root of a Number, Gan. ita-Bharat¯ ¯ı 19 (1997), 60–68. Works out several examples of computations of cube roots, following the procedure described by Aryabhata II. (LH) #26.2.78

Kanamori, Akihiro. The Mathematical Import of Zermelo’s Well-Ordering Theorem, Bulletin of Symbolic Logic 3 (1997), 281–311. Zermelo’s proof of the well-ordering theorem in 1904 is seen as central to further developments in set theory. This paper connects the theorem and its proof to Kuratowski’s ﬁxed-point theorem, Bourbaki’s results on partial orders, and applications to computer science. Includes a new, simple proof of the well-ordering theorem. See the review by Yehuda Rav in Mathematical Reviews 98k:01013. (JVR) #26.2.79

Karzel, Helmut. Survey of the Papers and Personal Memories of Giuseppe Tallini, Results in Mathematics 32 (1997), 260–271. A survey of the work of Giuseppe Tallini on structures which generalize ﬁnite projective spaces. Includes personal reminiscences. See the review by James W. P. Hirschfeld in Mathematical Reviews 98k:01032. (JVR) #26.2.80

Katz, Victor, and Voolich, Erica. HPM Conference Within a Conference at NCTM, History and Pedagogy of Mathematics Newsletter 44 (1998), 3–5. Descriptions of the talks and workshops given at a conference held within the conﬁnes of the annual NCTM meeting held in April, 1998, in Washington, DC. (DEZ) #26.2.81

Katz, Victor. See also #26.2.22.

Kjelstrup Ratkje, S. See #26.2.70.

Kleiner, Israel. From Numbers to Rings: The Early History of Ring Theory, Elemente der Mathematik 53 (1998), 417–424. An account of the development of modern ring theory from two separate paths, noncommutative ring theory and commutative ring theory. The concept of a ring is traced to M. Fraenkel and M. Sono, while ring theory is due to E. Noether and E. Artin. (DEZ) #26.2.82

Knobloch, Eberhard. See #26.2.4.

Kreyszig, Erwin. Basic Ideas in Modern Numerical Analysis and Their Origins, in #26.2.149, pp. 34–45. In- dicates the origins of the Newton–Raphson method, Newton interpolation formulas, Newton–Cotes integration formulas, Gauss’s treatment of round-off error, and least-squares approximation. (TLB) #26.2.83

Kunoff, Sharon. Advances in the Study of the Three Body Problem, in #26.2.149, pp. 46–50. Brief discussion of Poincar´e’s entry for the prize competition sponsored by King Oscar of Sweden in 1889 and the later work by G. D. Birkhoff. (TLB) #26.2.84

Kyburg, Jr., Henry E. See #26.2.24.

Lausch, Hans. Moses Mendelssohn: “Wir m¨ussen uns auf Wahrscheinlichkeiten st¨utzen,” Acta Historica Leopoldina 27 (1997), 201–213. A discussion of Mendelssohn’s papers of 1756 and 1785 on probabilities involving successive trials and coincident events. See the review by Oscar Sheynin in Mathematical Reviews 98k:01008. (JVR) #26.2.85

Lausch, Hans. See also #26.2.33.

Lawson, Jimmie D. See #26.2.24.

Lehto, Olli. Fred Gehring and Finnish Mathematics, in Peter Duren, Juha Heinonen, Brad Osgood, and Bruce Palka, ed., Quasiconformal Mappings and Analysis, New York: Springer-Verlag, 1998, pp. 19–32. The article 190 ABSTRACTS HMAT 26 begins with an overview of mathematics in Finland, then discusses the work on quasiconformal mappings by F. Gehring and his many Finnish friends and collaborators. (DEZ) #26.2.86

Li, Peter. See #26.2.23.

Loach, Judi. Le Corbusier and the Creative Use of Mathematics, British Journal for the History of Science 31 (1998), 185–215. The architect Le Corbusier (1887–1965) invoked mathematics in different ways at different stages of his career. In the late 1940s his search for mathematical legitimization of his system of proportions, the ‘Modulor,’ led to his enlisting the help Ren´eTaton. (JGF) #26.2.87

López Tascón,Carlos. Mecanica´ Newtoniana, Santaféde Bogotá,Colombia: Academia Colombiana de Ciencias Exactas, F´ısicasy Naturales, 1998, xv + 277 pp., softbound. A set of lecture notes on Newtonian mechanics from a course in physics given at the National University of Colombia. The book is neither a treatise on Newtonian mechanics nor a textbook but the author’s guide to Newton’s major concepts and ideas in kinematics, the dynamics of a system of particles, and systems of rigid bodies with six degrees of freedom. (DEZ) #26.2.88

Lumpkin, Beatrice. Cradle of Mathematics, Gan. ita-Bharat¯ ¯ı 19 (1997), 1–10. A survey of early mathematical developments in Africa, especially Egypt. (LH) #26.2.89

Lumpkin, Beatrice. From Egypt To Benjamin Banneker: African Origins of False Position Solutions, in Ronald Calinger, ed., Vita Mathematica: Historical Research and Integration with Teaching, Washington, DC: Mathe- matical Association of America, 1996, pp. 279–289. Describes the use of the rule of false position in ancient Egypt, in the work of later Alexandrian mathematicians such as Diophantus, and in Ab¯uK¯amil. Its inﬂuence on mathematicians in Europe and on Benjamin Banneker (1731–1806) is also discussed. (PG) #26.2.90

Maanen, Jan Van. Old Maths Never Dies, Mathematics in School 27 (4) (1998), 52–54. Today’s students are intrigued and inspired by 17th-century textbook problems at a number of levels, from deciphering gothic type to realizing that problems can be solved geometrically as well as algebraically. (JGF) #26.2.91

Mac Lane, Saunders. See #26.2.12.

Maligranda, Lech. Why Hölder’s Inequality Should Be Called Rogers’ Inequality, Mathematical Inequalities & Applications 1 (1998), 69–83. The author applies Boyer’s law of eponyms to the unlucky Leonard James Rogers (1862–1933), providing evidence that Rogers should receive credit for inequalities named after Otto Ludwig Hölderand Alexander Mikhailovich Lyapunov. Short biographies of Rogers and Hölderare provided. (DEZ) #26.2.92

Mancosu, Paolo, ed. From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s, Oxford: Oxford Univ. Press, 1998, 348 pp., hardbound £55, paperbound £19.50. An introduction to the debate on the foundations of mathematics during the 1920s consisting of 25 articles that had not been previously translated into English. The central ﬁgures are Hilbert, Brouwer, Bernays, and Weyl. (DEZ) #26.2.93

Maor, Eli. Trigonometric Delights, Princeton, NJ: Princeton Univ. Press, 1998, 248 pp., hardbound $24.95, £17.95. Short accounts of Ptolemy’s tables, the Hindu origin of the trigonometric functions, trigonometric identi- ties, the shape of the earth, cartography, De Moivre’s formula and its consequences, inﬁnite series, the hyperbolic trigonometric functions, complex numbers, and rigor in analysis. The author’s preface reads, “Not a comprehensive history [but] an attempt to present selected topics in trigonometry from a historical point of view.” (DEZ) #26.2.94

Masani, Pesi R. Norbert Wiener and the Future of Cybernetics, in Vidyadhar S. Mandrekar and Pesi R. Masani, ed., Proceedings of the Norbert Wiener Centenary Congress, 1994, Providence: American Mathematical Society, 1997, pp. 473–503. A discussion of the place of Wiener’s work in the history of science clarifying the meaning of “cybernetics” as Wiener used the term. Includes a discussion of the problem of evil in man and society and of retrograde trends. See the review by Dirk J. Struik in Mathematical Reviews 98k:01038. (JVR) #26.2.95

Matvievskaya, G. P. On the Paper “On Some Tasks for Establishing a University in Tashkent” by V. I. Romanovsky, Istoriko-Matematicheskie Issledovaniya 37 (1997), 66–78. A brief biography of Vsevolod Ivanovich HMAT 26 ABSTRACTS 191

Romanovsky (1876–1954), a Russian specialist in probability and statistics, followed by his 1918 paper from the magazine People’s Teacher examining the principles for organizing a university in Tashkent. The paper reﬂects the situation of education in pre-Revolutionary Russia and gives evidence of Romanovsky’s strong interest in the development of education and science in Central Asia. (DEZ) #26.2.96

Mazzocca, Francesco. Characterizations of Classical Algebraic Varieties in Some Papers of Giuseppe Tallini and Some Personal Memories of a Friend, Results in Mathematics 32 (1997), 272–275. A survey of some papers by Giuseppe Tallini on the combinatorial characterization of the Veronese surface in 5-space over a ﬁnite ﬁeld. Includes personal reminiscences. See the review by James W. P. Hirschfeld in Mathematical Reviews 98k:01033. (JVR) #26.2.97

McCleary, John. An Appreciation of the Work of Jim Stasheff, in John McCleary, ed., Higher Homotopy Struc- tures in Topology and Mathematical Physics, Washington, DC: American Mathematical Society, 1998, pp. 1–16. A survey of the work of the topologist, James Stasheff, from his thesis on the homotopy invariant way to express associativity to his work on mathematical physics. The main theme of perturbations of differential graded algebra is emphasized. Also included are a vita and lists of students, publications, and positions. (DEZ) #26.2.98

McMurran, Shawnee, and Tattersall, James. Mary Cartwright (1900–1998), Notices of the AMS 46 (1999), 214–220. Memorial tribute to Dame Mary Cartwright and her distinguished career as a researcher in the theory of functions of real and complex variables and an administrator at Girton College at Cambridge University. (DEZ) #26.2.99

Menghini, Marta. See #26.2.76.

Meskens, Ad. The Jesuit Mathematics School in Antwerp in the Early Seventeenth Century. The Seventeenth Century 12 (1997), 11–22. An important Jesuit mathematics school was founded in Antwerp in 1615. Gregory of St. Vincent helped write the curriculum and taught from 1617 to 1621. Jan-Karel della Faille and other mathematicians were trained here, but the school was moved to Leuven in 1621. (JGF) #26.2.100

Methuen, Charlotte. Kepler’s Tubingen:¨ Stimulus to a Theological Mathematics, Brookfield, VT: Ashgate, 1998, 296 pp., hardbound, £45, $76.95. An examination of the relationship among theology, astronomy, and dialectics that provides an analysis of the influences and ideas that permeated life at the University of Tübingenin the second half of the 16th century, when Johannes Kepler was a student there. The work of Kepler’s teachers is examined. See the review by June Barrow-Green in BSHM Newsletter 38 (1998), 23–25. (DEZ) #26.2.101

Minlos, Robert A.; Pechersky, E. A.; and Suhov, Yurii M. Remarks on the Life and Research of Roland L. Dobrushin, Journal of Applied Mathematics and Stochastic Analysis 9 (1996), 337–372. The contributions of the Russian mathematician, R. L. Dobrushin (1929–1995), to various ﬁelds, most prominently probability. See the review by B. M. Gurevich in Mathematical Reviews 98b:01034. (DEZ) #26.2.102

Moreno-Armella, Luis. El postulado de las paralelas, Revista de la Academia colombiana de ciencias exactas, f´ısicas y naturales 22 (1998), 393–405. A description of the road from “Euclidean truth” to “Hilbertian consistency” that resulted from attempts to prove the ﬁfth postulate. (DEZ) #26.2.103

Muroi, Kazuo. A Circular Field Problem in the Late Babylonian Metrological-Math Text W 23291-x, Gan. ita- Bharat¯ ¯ı 19 (1997), 86–90. Translates and analyzes a geometric problem and solution that appear in a Babylonian cuneiform text. (LH) #26.2.104

Mus`es,C. A Celebration of Higher-Dimensional Systems and a Man Who Notably Explored Them, Kybernetes 25 (1996), 48–52. An essay review of the book Kaleidoscopes by H. S. M. Coxeter, described as “arguably the foremost higher-dimensional geometer of the century.” See the review by Dirk J. Struik in Mathematical Reviews 98b:01035. (DEZ) #26.2.105

Myshkis, A. D., and Ovcharenko, I. E. G. E. Shilov and the Teaching of Mathematics [in Russian], Istoriko- Matematicheskie Issledovaniya 37 (1997), 155–179. Three excerpts from the pedagogical activity of G. E. Shilov. The ﬁrst, “Mathematical Analysis in the University: Current and Future,” was presented to the Commission on 192 ABSTRACTS HMAT 26

Mathematical Education (headed by A. N. Kolmogorov) of the USSR Academy of Sciences in 1965. The second concerns the teaching of common algebraic and geometrical courses in mathematics departments, while the third concerns the problem of mathematical education in other departments. (DEZ) #26.2.106

Nagata, J. See #26.2.75. √ Nahin, Paul J. An Imaginary Tale:The Story of 1 , Princeton: Princeton Univ. Press, 1998, 274 pp., hardbound, $24.95. A popular account of various properties of the complex number i, including some aspects of its history. (DEZ) #26.2.107

Nasar, Sylvia. A Beautiful Mind: Genius, Schizophrenia and Recovery in the Life of Nobel Laureate John Forbes Nash, Jr., New York: Simon & Schuster, 1998, 459 pp., hardbound, $25.00. An unauthorized biography of game theorist, John Nash, a ﬁnalist for the 1958 Fields Medal who was awarded the 1994 Nobel Prize in Economics. See the review by John Milnor in the Notices of the AMS 45 (1998), 1329–1332. See also Sylvia Nasar, The Lost Years of a Nobel Laureate, New York Times, November 13, 1994. (DEZ) #26.2.108

Nedoma, Josef. See #26.2.21.

Neumann, Bernhard H. See #26.2.24.

Neumann, Olaf. Die Entwicklung der Galois-Theorie zwischen Arithmetik und Topologie (1850 bis 1960), Archive for History of Exact Sciences 50 (1997), 291–329. A description of the development of Galois theory. The author identiﬁes two lines of development, one based on Galois groups and the other developed within the theory of functions and Riemann surfaces. See the review by Karl-Heinz Schlote in Mathematical Reviews 98k:01010. (JVR) #26.2.109

Newman, Morris. The Work of Robert C. Thompson, Linear and Multilinear Algebra 43 (1997), 1–11. The life and work, mainly in matrix theory of R. C. Thompson (1931–1995). See also #26.2.33. (DEZ) #26.2.110

Ockendon, H., and Ockendon, J. R. Alan Breach Taylor (1931–1995), Bulletin of the London Mathematical Society 30 (1998), 429–431. Alan B. Taylor was a student of George Temple whose major scientific work was in mathematical fluid dynamics. His professional legacies included initiating the mathematical Study Groups with Industry (1967, with Leslie Fox), helping to form the European Consortium for Mathematics in Industry (president in 1989), and being the first director of the Oxford Centre for Industrial and Applied Mathematics. There is a list of his 30 published papers and two books, and a photo. (JA) #26.2.111

Ockendon, J. R. See #26.2.111.

Okuyama, Akihiro. See #26.2.75.

Ovcharenko, I. E. See #26.2.106.

Pandit, M. D. Mathematicians in Ancient India, Gan. ita-Bharat¯ ¯ı 19 (1997), 69–85. Translation of, and detailed commentary on, 25 Sanskrit verses, of unknown origin, that provide lore surrounding ancient Indian mathematics. (LH) #26.2.112

Parameswaran, S. A Historical Note on Amicable Numbers, Gan. ita-Bharat¯ ¯ı 19 (1997), 97–100. Brief discussion of amicable numbers, with emphasis on a formula of Th¯abitibn Qurra. (LH) #26.2.113

Pechersky, E. A. See #26.2.102.

Pedoe, Daniel. In Love with Geometry, College Mathematics Journal 29 (1998), 170–188. Memories of studying mathematics at Cambridge with H. F. Baker, attending Wittgenstein’s lectures, collaborating with W. V. D. Hodge, teaching Freeman Dyson at Winchester, and protesting about the US system of “merit pay” which discriminates against unfashionable subjects and expository writing. (JGF) #26.2.114

Perminov, V. Ya. On the Nature of the Idea of Proof in the Development of Pre-Greek Mathematics [in Russian], Istoriko-Matematicheskie Issledovaniya 37(1997), 180–200. This article outlines a new view of pre-Greek mathematics based on a prior interpretation of initial mathematical idealizations from before 600 BC. (DEZ) #26.2.115 HMAT 26 ABSTRACTS 193

Picutti, Ettore. Sulla storia della moltiplicazione [On the History of Multiplication], Lettera Matematica 27– 28 (1998), 41–43. A description of techniques for multiplying integers from the Rhind papyrus and “lattice multiplication,” which was probably introduced by the Arabs and played a role in medieval European algebra. Also included is a brief discussion of some lesser known methods. (DCS) #26.2.116

Pieper, Herbert. Uber¨ Legendres Versuche, das quadratische Reziprozit¨atsgesetzzu beweisen, Acta Historica Leopoldina 27 (1997), 223–237. A description of the early history of the quadratic reciprocity law. Includes a discussion of the work of Euclid, Legendre, and Gauss. See the review by Charles Helou in Mathematical Reviews 98k:01011. (JVR) #26.2.117

Pier, Jean-Paul. Histoire de l’integration:´ Vingt-cinq siecles` de mathematiques´ , Paris: Masson, 1996, xii + 307 pp., 245 FF. A comprehensive history of the integral from ancient to modern times, including Chinese and Islamic contributions and the work of Newton, Leibniz, Cauchy, Fubini, and Bourbaki. Also included are discussions of the Riemann, Stieltjes, Lebesque, Denjoy, Perron, and Henstock-Kurzweil integrals. An extensive bibliography in excess of 800 citations is provided. See the review by Frank Smithies in Mathematical Reviews 98k:01003. (JVR) #26.2.118

Platonova, N. I. From Physics to General Analysis: Giovanni Giorgi (1871–1950) [in Russian], Istoriko- Matematicheskie Issledovaniya 37 (1997), 313–321. An examination of the varied works of the Italian engineer, mathematician, and physicist, G. Giorgi, in engineering and physics, electricity and magnetism, mathematical physics and functional analysis, foundations of geometry, and the theory of probability. Giorgi is little known today but was called “the great Italian” by his compatriots. (DEZ) #26.2.119

Ponza, Maria Victoria. A Role for the History of Mathematics in the Teaching and Learning of Mathematics: An Argentinian Experience, Mathematics in School 27 (4) (1998), 10–13. The experience of writing and producing a play about Galois (whose text is reproduced here) had notable effects upon the interest and enthusiasm of pupils for mathematics. (JGF) #26.2.120

Pool, Michael. Mathematical Knowledge in the Regulae, in #26.2.149, pp. 51–59. A proposal for new interpre- tations of Descartes’s discussion of universal mathematics and of perception in the Rules for the Direction of the Mind. (TLB) #26.2.121

Postnikov, Mikhail M. Pages from a Mathematics Autobiography (1942–1953) [in Russian], Istoriko- Matematicheskie Issledovaniya 37 (1997), 78–104. The eminent Russian algebraic topologist, Mikhail Mikhailovich Postnikov, describes his education at Perm University from 1942, when he was 15 years old, his move to Moscow State University under the inﬂuence of Perm professor, Sofya A. Yanovskaya, the defense of his candidate dissertation in 1949, and the defense of his doctoral dissertation in 1953. See also #22.3.70. (DEZ) #26.2.122

Pritchard, Chris. Flaming Swords and Hermaphrodite Monsters: Peter Guthrie Tait and the Promotion of Quater- nions, part II, Mathematical Gazette 82 (1998), 235–241. The main effect of P. G. Tait’s vigorous promotion of quaternions was ultimately to bolster the rival vector analysis of J. W. Gibbs. (JGF) #26.2.123

Rabinowitz, Stanley, ed. Problems and Solutions from The Mathematical Visitor 1877–1896, Westford, MA: MathPro Press, 1996, xiv + 258 pp. A collection of several hundred problems and solutions that appeared in The Mathematical Visitor, edited by the American mathematician, Artemas Martin, from 1877 to 1896. See the review by James J. Tattersall in Mathematical Reviews 98b:01044. (DEZ) #26.2.124

Rankin, Robert A. My Cambridge Years, BSHM Newsletter 38 (1998), 30–34. Eminent number theorist and De Morgan Medalist, R. A. Rankin, recalls his time at Cambridge University from his ﬁrst visit in 1933 through his work under A. E. Ingham and G. H. Hardy during 1935–1937 to his stint on the faculty during 1945–1951. (DEZ) #26.2.125

Ransom, Peter. A Day in North Wales, BSHM Newsletter 38 (1998), 43–46. Descriptions of the mathematical aspects of six sundials located in North Wales. (DEZ) #26.2.126

Rashed, Roshdi. Indian Mathematics in Arabic, in Chikara Sasaki, Mitsuo Sugiura, and Joseph W. Dauben, ed., The Intersection of History and Mathematics, Basel: Birkh¨auserVerlag, 1994, pp. 143–148. The interest of 194 ABSTRACTS HMAT 26

Arab scholars in Indian mathematics was mainly in computational methods relating to astronomy. A work of al-Samaw’al (1165) shows that al-B¯ır¯un¯ıwas fully aware of Brahmagupta’s interpolation methods, and critically compared different methods using empirical and theoretical considerations. (JGF) #26.2.127

Rav, Yehuda. See #26.2.79.

Regazzini, Eugenio. See #26.2.24.

Renteln, Michael von. Friedrich Prym (1841–1915) and His Investigations on the Dirichlet Problem, Supplemento ai Rendiconti del Circolo matematico di Palermo, Serie II 44 (1996), 43–55. Riemann’s 1861–1862 lectures on integrals of algebraic differentials greatly inﬂuenced Prym, whose future research was devoted to working out Riemann’s ideas. (JGF) #26.2.128

Rice, Adrian. The Origins and Signiﬁcance of the De Morgan Medal of the LMS, BSHM Newsletter 38 (1998), 35–36. Capsule history of the London Mathematical Society’s triennial award, the De Morgan Medal, with a list of recipients from Arthur Cayley in 1884 to Robert Rankin in 1998. See also #26.2.126. (DEZ) #26.2.129

Rice, Adrian. The Wessel Symposium, BSHM Newsletter 38 (1998), 9–11. Report of a ﬁve-day conference held in August 1998 in Copenhagen to commemorate the bicentennial of the publication of the Norwegian surveyor Caspar Wessel’s paper on complex numbers. (DEZ) #26.2.130

Robins, Gay. Proposition and Style in Ancient Egyptian Art, Austin: Univ. of Texas Press, 1994, 279 pp. From the preface (p. vii): “In this book I have attempted to base my own ideas...primarily on observations carried out on actual monuments. ...I show that the squared grid had an important inﬂuence on the composition of scenes as a whole and in helping to determine the characteristic style of a particular period. I consider the effects of the major change in the grid that occurred in the twenty-ﬁfth dynasty and persisted thereafter, and elaborate my discovery of the grid system adopted during the Amarna period.” (PG) #26.2.131

Robson, Eleanor. Counting in Cuneiform, Mathematics in School 27 (4) (1998), 2–9. Resources for teachers and suggestions for classroom activity involving Babylonian mathematics. (JGF) #26.2.132

Robson, Eleanor. See also #26.2.49.

Rodis-Lewis, Genevi`eve. Descartes: His Life and Thought, trans. Jane Marie Todd, Ithaca, NY: Cornell Univ. Press, 1998, xx + 264 pp., $39.95. This work sets out to correct many long-held errors and to paint a clearer picture of Ren´eDescartes, whose contributions to philosophy, rather than to mathematics, play a dominant role. (DEZ) #26.2.133

Roero, Clara Silvia. See #26.2.17.

Ruiz, Angel.´ See #26.2.8.

Russ, Steve. History of Mathematics in Education 1998, BSHM Newsletter 38 (1998), 60–62. Report of the three plenary talks and four workshops at the annual HIMED meeting held in April 1998 at the University of Warwick. (DEZ) #26.2.134

Sakarovitch, J. See #26.2.35.

Sasaki, Chikara. Historians of Mathematics in Modern Japan, Historia Scientiarum 6 (1996), 67–78. A collection of brief biographies of some of the principal ﬁgures in the study of the history of Japanese mathematics: Toshisada Endo (1843–1915), Yoshio Mikami (1875–1950), Tsuruichi Hayashi (1873–1935), Matsusaburo Fujiwara (1881– 1946), and Yoitsu Kondo (1911–1979). (DEZ) #26.2.135

Satko, Ladislav. See #26.2.62.

Scheepers, Marion. See #26.2.68. HMAT 26 ABSTRACTS 195

Schein, Boris M. See #26.2.62.

Schlote, Karl-Heinz. See #26.2.109.

Schmeikal, Bernd A. See #26.2.51.

Schmitz, Markus. Euklids Geometrie und ihre mathematiktheoretische Grundlegung in der neuplatonischen Philosophie des Proklos,W¨urzburg: K¨onigshausen& Neumann, 1997, viii + 449 pp., DM 98. An explanation of Proclus’s philosophy of mathematics and a philosophical interpretation of Euclid’s Elements following the ideas of Proclus. Includes an examination of the ontological status of mathematical objects, Proclus’s notion of the essence of Euclid’s propositions, and a discussion of the meaning of non-Euclidean geometries from Proclus’s viewpoint. See the review by Benno Artmann in Mathematical Reviews 98k:01005. (JVR) #26.2.136

Schr¨oder, Wilfried. See #26.2.45.

Schubring, Gert. An Unknown Part of Weierstraß’s Nachlaß, Historia Mathematica 25 (1998), 423–430. A report on the discovery in Berlin of part of Weierstrass’s Nachlass outside the Mittag-Lefﬂer Institute in Sweden. An examination of Weierstrass’s correspondence reveals the reason for the break in his relationship with Kronecker in 1883. (DEZ) #26.2.137

Schubring, Gert. Analysis of Historical Textbooks in Mathematics, Rio de Janeiro: Pontificia Universidade Católica,1997, ii + 90 pp., softbound. An edited version of a set of lectures the author presented in 1995. He examines the history and the role played by mathematics textbooks, from the oral tradition in various cultures to textbooks in the age of print. Particular examples are taken from developments surrounding the French Revolution, Lacroix’s fight for the textbook market, teacher autonomy and textbooks in Prussia, and the case of Legendre in Italy. (DEZ) #26.2.138

Scriba, Christoph. See #26.2.160.

Shea, William R. See #26.2.63.

Sheynin, Oscar B. A. A. Markov and Life Insurance [in Russian], Istoriko-Matematicheskie Issledovaniya 37 (1997), 22–33. Two newspaper articles written by Andrei A. Markov in 1906 and arguing against a proposed scheme for insuring children are reprinted and commented upon. Markov’s work in several retirement funds and his related correspondence with two colleagues are described. (DEZ) #26.2.139

Sheynin, Oscar B. See also #26.2.85.

Shulman, Bonnie. Math-Alive! Using Original Sources to Teach Mathematics in Social Context, PRIMUS 8 (1998), 1–14. The author describes her search for reasons that the logistic equation is so named and how the search itself led to numerous sources from the early history of statistical analysis. (VJK) #26.2.140

Siegmund-Schultze, Reinhard. Mathematiker auf der Flucht vor Hitler, Braunschweig/Wiesbaden: Verlag Vieweg, 1998, 368 pp., hardbound, DM 98. This book reports on the expulsion of signiﬁcant mathematicians from the sphere of inﬂuence of Hitler’s Germany after 1933. It discusses the reaction of colleagues in Germany and the helpfulness of mathematicians in the countries of sanctuary, among which the US holds a central position. About 90% of those expelled had been persecuted by the Nazis as Jews. (MK) #26.2.141

Singh, Simon. History of Cryptography, BSHM Newsletter 38 (1998), 1–4. A report on a conference of the British Society for the History of Mathematics held in June 1998 at Bletchley Park on the “rich and intriguing history” of cryptography. Items discussed include the role of mutilator tables, Polish “bombes,” public key cryptography, and the misattributed RSA system. (DEZ) #26.2.142

Singmaster, David. Coconuts: The History and Solutions of a Classic Diophantine Problem, Gan. ita-Bharat¯ ¯ı 19 (1997), 35–51. Both a historical survey and a mathematical analysis of Diophantine problems of the monkey-and- coconuts type. Traces examples as far back as an Indian text from the year 850. (LH) #26.2.143 196 ABSTRACTS HMAT 26

Singmaster, David. Mathematical Gazetteer of Britian #11: Lacock to Llanwrthwl, BSHM Newsletter 38 (1998), 38–42. Short descriptions of British mathematicians and mathematical sites (George Boole was born in Lincoln) beginning with the letter L. No guide to the pronunciation of the last entry is provided. (DEZ) #26.2.144

Siu, Man-Keung. See #26.2.10.

Skovsmose, Ole. See #26.2.158.

Smith, Cedric A. B. Did Erdo˝s Save Western Civilization? in #26.2.58, pp. 74–85. An account of Paul Erdo˝s’s enduring inﬂuence on Cambridge University during his visit there in the 1940s, including his intermediary role in helping Bill Tutte obtain a wartime job as a mathematician at Bletchley Park. See the review by Jerrold W. Grossman in Mathematical Reviews 98b:01030. (DEZ) #26.2.145

Smithies, Frank. See #26.2.118.

Solovyev, A. D. P. A. Nekrasov and the Central Limit Theorem of Probability Theory [in Russian], Istoriko- Matematicheskie Issledovaniya 37 (1997), 9–22. An objective evaluation of the eminent Russian mathematician P. A. Nekrasov’s contributions to the proof of the central limit theorem in probability theory. The author’s analysis gives a generally negative evaluation of Nekrasov’s work, which proved the theorem only for lattice random variables and under conditions that were difﬁcult to verify at that time. (DEZ) #26.2.146

Stone, Arthur H. Encounters with Paul Erdo˝s, in #26.2.58, pp. 68–73. The ﬁrst third of this work contains the author’s reminiscences of Paul Erdo˝s. See the review by Jerrold W. Grossman in Mathematical Reviews 98b:01029. (DEZ) #26.2.147

Struik, Dirk J. See #26.2.95 and #26.2.105.

Suﬁyarova I. I. See #26.2.166.

Suhov, Yurii M. See #26.2.102.

Swerdlow, Noel M. The Babylonian Theory of Planets, Princeton: Princeton Univ. Press, 1998, 264 pp., hardbound, $39.50. An examination of the omens and observations from Babylonian scribes in the second millennium BC that form the basis for the ﬁrst mathematical science, astronomy. (DEZ) #26.2.148

Tattersall, James J., ed. Proceedings of the Canadian Society for the History and Philosophy of Mathematics, Volume 10, St. Johns, Newfoundland: Memorial University, 1997 softbound, 159 pp. This volume contains most of the papers delivered at the 23rd annual meeting of the Canadian Society for the History and Philosophy of Mathematics held in June, 1997. There was a special session on science and mathematics featuring talks by R¨udiger Thiele, Rebecca Adams, Erwin Kreyszig, Sharon Kunoff, and Michael Pool. These and contributed papers by Francine F. Abeles, Amy Ackerberg-Hastings, William S. Anglin, Edward L. Cohen, Roger Godard, Aditi Gowri, and Peter L. Grifﬁths are abstracted separately. (TLB) #26.2.149

Tattersall, James. See also #26.2.99 and #26.2.124.

Thas, Joseph A. G. Tallini and His Work, Results in Mathematics 32 (1997), 276–280. A survey of the work of Giuseppe Tallini on the combinatorial characterization of quadrics and generalized quadrangles over a ﬁnite ﬁeld. Includes personal reminiscences. See the review by James W. P. Hirschfeld in Mathematical Reviews 98k:01034. (JVR) #26.2.150

Thiele, R¨udiger. Leonhard Euler (1707–1783): Episodes of his Life and Work, in #26.2.149, pp. 1–26. A survey of Euler’s mathematical career. (TLB) #26.2.151

Tian, Gang. See #26.2.23.

Tikhomirov, V. M. G. E. Shilov and Mathematical Education [in Russian], Istoriko-Matematicheskie Issle- dovaniya 37 (1997), 153–155. The early work of G. E. Shilov, a student at Moscow State University (MSU) under HMAT 26 ABSTRACTS 197

A. N. Kolmogorov, was connected with the theory of Banach algebras founded by I. M. Gelfand. After working several years at Kiev University, he was invited by I. G. Petrovsky in 1954 to return to MSU, where he investigated generalized functions, published textbooks, and wrote articles on music, poetry, philosophy, and the teaching of mathematics. See also #26.2.107. (DEZ) #26.2.152

Toomer, Gerald J. Ptolemy’s Almagest, Princeton: Princeton Univ. Press, 1998, ix + 693 pp., paperbound, $39.50. This is a paperback version of the hardbound book originally published by Duckworth in 1984. The Almagest has been translated into English and annotated by G. J. Toomer. See #11.3.53. (DEZ) #26.2.153

Trakhtenbrot, B. A. In Memory of S. A. Yanovskaya [in Russian], Istoriko-Matematicheskie Issledovaniya 37 (1997), 109–127. The author reproduces and comments on some documents sent to him by S. A. Yanovskaya in 1951 when he was accused of “bourgeois idealism.” He then describes how he overcame the danger of these accusations at a time when “idealists” and “cosmopolites” were being persecuted. The correspondence presents additional evidence of the general atmosphere surrounding mathematical logic at that time and Yanovskaya’s struggle for its legitimacy and consolidation. See also #26.2.11. (DEZ) #26.2.154

Tzanakis, Constantinos. Reversing the Customary Deductive Teaching of Mathematics by Using Its History: The Case of Abstract Algebraic Concepts, in Proceedings of the First European Summer University on the History and Epistemology in Mathematics Education, IREM de Montpellier, 1995, pp. 271–273. The customary deductive approach in mathematics teaching can be reversed by using its history as an essential ingredient, here examined in the case of complex number, rotation group, and morphisms of abstract algebraic structures. (JGF) #26.2.155

Vagliente, Victor N. See #26.2.14.

Van Brummelen, Glen. Jamshid al-Kashi: Calculating Genius, Mathematics in School 27 (4) (1998), 40–44. The remarkable and beautiful insights of al-Kashi, working in Samarkand in the 1420s, led to unprecedentedly accurate values of and the sine of 1. His method for sin(1) is essentially that of a ﬁxed-point iteration that can be done on a calculator in class. (JGF) #26.2.156

Vilkko, Risto. The Reception of Frege’s Begriffsschrift, Historia Mathematica 25 (1998), 412–422. The author argues that it is misleading to regard the public reception of Gottlob Frege’s (1848–1925) Begriffsschrift (1879) as unfavorable and tragic. He concludes that three of the six reviewers were not really interested in the book, while the others “were not so bad at all” (p. 419.) (DEZ) #26.2.157

Vithal, Renuka, and Skovsmose, Ole. The End of Innocence: A Critique of “Ethnomathematics,” Educational Studies in Mathematics 34 (1997), 131–157. A discussion of the social and political aspects of ethnomathematics, which “has achieved its purpose. It has established an understanding of mathematics and mathematics education as culturally and socially negotiated. But the concept ‘ethnomathematics’ is itself problematic. And it is not innocent” (p. 152.) (DWB) #26.2.158

Volkov, V. A., and Demidov, Sergei S. “But Your Idea Has Universal Significance and Extraordinary Value”: From the Prehistory of Nonstandard Analysis [in Russian], Istoriko-Matematicheskie Issledovaniya 37 (1997), 128–152. In 1931 Mark Yakovlevich Vygodsky published Foundations of Infinitesimal Calculus, which gave an absolutely nontraditional presentation of differential and integral calculus. He sent a copy to Nikolai N. Luzin, two of whosse responses, from 1931 and 1933, are reproduced here. They show that Luzin was the only prominent mathematician who supported Vygodsky’s approach, that he anticipated the discovery of nonstandard analysis, that he had a painful experience in searching for a subject of his future research, and that his reflections on the nature of infinitesimals served as a bridge to the theory of functions of a real variable. (DEZ) #26.2.159

Vollrath, Hans-Joachim, ed. Otto Volk: Mathematik und Erkenntnis, Litauische Aufsätze, Würzburg: Königshausen & Neumann, 1995, 120 pp., hardbound, DM 29.80. The translation from Lithuanian to German of several papers of a general mathematical nature by Otto Volk (1892–1989) that were not included in his collected works. See the review by Christoph J. Scriba in Mathematical Reviews 98b:01043. (DEZ) #26.2.160

Voolich, Erica. See #26.2.81. 198 ABSTRACTS HMAT 26

Watanabe, Tadashi. See #26.2.75.

Wood, Alastair. George Gabriel Stokes Summer School, BSHM Newsletter 38 (1998), 6–9. Report of a four-day conference held in August 1998 in Skreen, Ireland, to celebrate the diverse areas to which Sir George Gabriel Stokes made major contributions. (DEZ) #26.2.161

Yankov, V. A. The Formation of Proof in Early Greek Mathematics (A Hypothetical Reconstruction) [in Russian], Istoriko-Matematicheskie Issledovaniya 37 (1997), 200–236. An examination of the genesis of a strict method of proof in ancient Greek mathematics, contrasted with other civilizations. The author’s thesis is that the complexity and subtlety of accumulated results demanded some semblance of proof, which developed at the level of verbal transmission to students, in late Oriental civilizations as well as in ancient Egypt and Mesopotamia. (DEZ) #26.2.162

Yau,S.-T., ed. S. S. Chern: A Great Geometer of the Twentieth Century: Expanded Edition, Matawan, NJ: Interna- tional Press, 1998, 331 pp., hardbound $42. An expanded edition of a previously published work containing personal reminiscences from the geometer S. S. Chern’s friends and students. Contents: S. S. Chern, “My Mathematical Ed- ucation;” R. Palais and C. Terng, “The Life and Mathematics of Shiing-Shen Chern;” C. N. Yang, “S. S. Chern and I;” A. Weil, “S. S. Chern as Geometer and Friend;” W. Chow, “Shiing-Shen Chern, as Friend and Mathematician;” I. M. Singer, “S. S. Chern at Chicago;” P. C. W. Chu, “Professor S. S. Chern, My Father-in-law;” I. Kaplansky, “Shiing-Shen Chern, with Admiration as He Approaches his 80th Birthday;” L. Nirenberg, “Some Personal Re- marks about S. S. Chern;” F. E. Browder, “S. S. Chern;” R. Lashof, “Personal Recollection of Chern at Chicago;” R. Bott, “For the Chern Volume;” L. Auslander, “S. S. Chern as Teacher;” H. Suzuki, “Reminiscences and Acknowl- edgements;” P. A. Griffiths, “Professor S. S. Chern, 79th Birthday Celebration;” W. Stoll, “Shiing-Shen Chern’s Influence on Value Distribution: Dedicated to Shiing-Shen Chern;” W. Klingenberg, “My Encounters with S. S. Chern;” J. Simons, “My Interaction with S. S. Chern;” M. P. do Carmo, “S. S. Chern: Mathematical Influences and Reminiscences;” M. Burger, “Riemannian Manifolds: From Curvature to Topology, A Brief Historial Overview,” B. Lawson, “On Chern and Youth;” J. Cheeger, “Remarks Delivered at Chern’s 79th Birthday Celebration;” A. Weinstein, “Some Thoughts about S. S. Chern;” R. Greene, “Some Mathematical and Personal Reminiscences;” S. Y. Cheng, “My Teacher S. S. Chern;” S. M. Webster, “On Being a Chern Student;” J. P. Bourguignon, “Shiing- Shen Chern, an Optimist;” J. Wolfson, “Chern, Some Recollections;” S.-T. Yau, “S. S. Chern, as My Teacher;” C.-C. Hsiung, “My Associations with S. S. Chern;” T. Banchoff, “Becoming and Being a Chern Student.” See also #26.2.23. (DEZ) #26.2.163

Zaitsev, Evgeny A. The Calculation of Areas in the Practice of Surveying [in Russian], Istoriko-Matematicheskie Issledovaniya 37 (1997), 262–280. An examination of the notions of “geometry” and “measurement of land” in which the author interprets formulas from ancient surveyors, including the “false-area formula,” from the standpoint of agrarian practice at that time. The key point in the author’s argument is that the concept of area must be understood not in terms of mathematics but in terms of the amount of work necessary to cultivate a ﬁeld (either by plowing or by canal construction). (DEZ) #26.2.164

Zen´ˇ ıˇsek, Alexander. See #26.2.21.

Zitarelli, David E. A Collaborative Approach to Mathematics, Wyncote, PA: Raymond-Reese Book Co., 1999, vi + 360 pp., softbound. A textbook for future K–8 teachers with 31 modules on number systems, real numbers, set theory, and geometry in a historial, multicultural setting. (DEZ) #26.2.165

Zverkina, G. A., and Suﬁyarova I. I. On Methods of Approximating the Circumference by Perimeters of Regular Polygons [in Russian], Istoriko-Matematicheskie Issledovaniya 37 (1997), 237–262. A historical review of approximation methods for the circumference of a circle by perimeters of regular polygons. Methods of Archimedes and al-Kashi are compared, with an estimation of the rapidity of convergence given for both methods. The authors show that an approximation of due to Apollonius may be obtained by the method of Archimedes. (DEZ) #26.2.166