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One of the many commentators of the Chiu chang suan shu (Nine Chapters on the Mathematical Art), written approximately in the first century, is Yang Hui, who wrote Hsiang chieh chiu chang suan fa (A Detailed Analysis of the Mathematical Methods in the “Nine Chapters”) and the Chiu chang suan fa tsuan lei (Reclassification of the Mathematical Methods in the “Nine Chapters”) in 1261. The seventh chapter of the Chiu chang suan shu is called ying nu or ying pu tsu meaning “too much and not enough” or “surplus and deficiency.” It has been pointed out by historians of mathematics that the method known in Europe as the Rule of False Position is the same as the Chinese ying pu tsu method used in this chapter. Yang has made a detailed and thorough study of the twenty problems in this chapter. He has not only explained and expanded the method of ying pu tsu, but he has also suggested alternative and simpler solutions for some of the problems. This paper endeavours to translate the twenty problems, to analyse the ying pu tsu method, and above all to present a critical study of Yang's alternative methods.

The finite element method has in recent years become one of the most popular and effective numerical procedures for boundary value and eigenvalue problems. In the early 1960s it was found that the method was an independent rediscovery of a simpler idea proposed in 1943 by Richard Courant. The importance of Courant's 1943 paper has been emphasized by G. Strang in 1973 and by others. This note seeks to describe briefly Courant's finite element work which led to his publication of 1943. It is shown that Courant used the finite element ideas as early as 1922 in a proof employing Dirichlet's principle.

The paper investigates the claim that “Zorn's Lemma” is not named after its first discoverer, by carefully tracing the origins of several related maximal principles and of the name “Zorn's Lemma.” Previously unpublished information supplied by Zorn is included.

The first problem of the Seleucid mathematical cuneiform tablet BM 34 568 calculates the diagonal of a rectangle from its sides without resorting to the Pythagorean rule. For this reason, it has been a source of discussion among specialists ever since its first publication, but so far no consensus in relation to its mathematical meaning has been attained. This paper presents two new interpretations of the scribe's procedure, based on the assumption that he was able to reduce the problem to a standard Mesopotamian question about reciprocal numbers. These new interpretations are then linked to interpretations of the Old Babylonian tablet Plimpton 322 and to the presence of Pythagorean triples in the contexts of Old Babylonian and Hellenistic mathematics.

After an exploration of some of the basic categories attending mechanics, Newtonianism, and physics, a survey is given of the range of subjects covered by mechanics at the end of the 18th century, and the three main traditions operative in the subject are stressed. A few major French figures of that time are then noted, and also some views evident in certain other countries. The appendix of the paper is concerned with bibliographical questions; and an extensive bibliography, mostly of secondary literature, is appended.

This paper presents information on the life and work of Stanislaw Piątkiewicz (1849–?). HisAlgebra w logice(Algebra in Logic) of 1888 contains an exposition of the algebra of logic and its use in representing syllogisms. This was the first original Polish publication on symbolic logic. It appeared 20 years before analogous works by Łukasiewicz and Stamm. In dem Aufsatz werden Informationen zu Leben und Werk von Stanislaw Piątkiewicz (1849–?) gegeben. SeinAlgebra w logice(Algebra in der Logik) von 1888 enthalt eine Darstellung der Algebra der Logik und ihre Anwendung auf die Syllogistik. Piątkiewicz' Schrift war die erste polnische Publikation uber symbolische Logik. Sie erschien 20 Jahre vor ahnlichen Arbeiten von Łukasiewicz und Stamm. W pracy przedstawiono osobe i dzielo Stanislawa Piątkiewicza (1849–?). W szczegolności analizuje sie jego rozpraweAlgebra w logicez roku 1888, w ktorej zajmowal sie on nowymi nowczas ideami algebry logiki oraz ich zastosowaniami do sylogistyki. Praca ta jest pierwszą oryginalną polską publikacją z zakresu logiki symbolicznej. Ukazala sie 20 lat przed analogicznymi pracami Łukasiewicza i Stamma.

This paper contains a survey of the most important works on Danish mathematics and and a short description of a new archive of mathematical archival material at the Institute for Mathematical Sciences, University of Copenhagen. The archive is especially of interest for the history of Danish mathematics, but some collections contain material which might be of more general interest. Copyright 2000 Academic Press. Denne note bestar af en oversigt over de vigtigste arbejder om dansk matematik samt en kort beskrivelse af et nyt arkiv med matematisk arkivmateriale ved Institut for Matematiske Fag, Kobenhavns Universitet. Arkivet er isae r relevant for matematikkens historie i Danmark, men nogle af samlingerne indeholder dog ogsa materiale som er af mere generel interesse. Copyright 2000 Academic Press. Cette note comprend un plan l'ensemble des plus importants travaux dans de domaine des mathematiques danoises ainsi qu'une breve description d'un nouveau systeme d'archives du materiel d'archives mathematiques a l'Institut des Mathematiques de l'Universite de Copenhague. Ces archives son surtout importantes pour l'histoire des Mathematiques au Danemark, mais quelques unes des archives comportent egalement des informations d'un interet plus generel. Copyright 2000 Academic Press. AMS classification numbers: 01A55, 01A60.

Resume In this paper, we present some aspects of the investigations of the French geometer Eugene Dewulf, founding member of the Mathematical Society of France, concerning Maghrebian medieval manuscripts. L'objet de cet article est de presenter quelques elements sur les investigations du geometre francais Eugene Dewulf, membre fondateur de la Societe mathematique de France, concernant les manuscrits medievaux du Maghreb. Questa nota si propone di presentare alcuni elementi sulle ricerche del geometro francese Eugene Dewulf, membro fondatore della Societa matematica di Francia; queste ricerche hanno come argomentogli manoscritti medievali nel Maghreb.

This paper investigates the origins of the concept of mathematical truth by focusing on the development of algebra in England in the early 19th century. In particular, it investigates the reasons why the English, despite their attention to the elements of abstract algebra, never produced a system comparable to modern algebra. Special consideration is given to the works of George Peacock, Augustus DeMorgan, William Whewell, and John Herschel. It is argued that what separated the early development of English algebra from modern algebra is a fundamental difference between 19th- and 20th-century views of truth.

RésuméEn analysant un texte peu connu d'Aepinus, publié en latin en 1763, sur la démonstration de l'importante formule du binôme de Newton pour tous les exposants, entiers ou non, nous questionnons le reproche fait aux mathématiciens du XVIIIème siècle d'un manque de rigueur et d'une trop grande confiance formelle. A propos de questions mathématiques d'existence et d'unicité, nous opposons ainsi une façon de faire qui s'inscrit dans un cadre limité, même s'il est aussi imprécis que celui de l'analyse algébrique, à une volonté de rigueur irréalisable car située dans un cadre trop général.AbstractThe purpose of this paper is to test the reproach of lack of rigor and too much confidence in formal aspects made against mathematicians of the 18th century. We do so by analyzing a rather forgotten text of Aepinus, written in Latin and published in 1763, which deals with the important binomial theorem of Newton, for all exponents, integers or nonintegers. With a special emphasis on mathematical questions of existence and uniqueness, we oppose a way which is developed within a limited if imprecise frame—as in the case of algebraical analysis—to a thirst for rigor, unattainable because of its very generality.

The goal of this paper is to discuss why, how, and when nonlogical set-theoretic paradoxes were discovered. No one factor dominated the whole outlook; instead, it is argued that these paradoxes were not a simple and direct consequence of the ones discovered by Bertrand Russell and published for the first time in his The Principles of Mathematics (1903).

This paper treats Hamilton's view of algebra as the science of pure time. It contains a description of how the intuition of time, according to Hamilton, can give rise to ideas of generalized numbers that can be added and multiplied. It is also shown that the associativity of multiplication and the possibility of division are consequences of Hamilton's view. Finally it is argued that Hamilton became aware of the fact that these consequences meant a serious limitation of algebra and that this awareness made him change his view.

The Old Babylonian mathematical tablet YBC 4652 contains a series of related problems stated together with their answers. In this paper, we propose a procedure for determining the solutions to these problems and consider the pedagogy underlying the mathematics. © 2002 Elsevier Science (USA). L'ancienne tablette mathematique babylonienne YBC 4652 contient une serie de problemes avec leurs reponses. Dans cet article, nous proposons une procedure pour determiner les solutions de ces problemes et considerons la demarche pedagogique sous-jacente aux mathematiques. © 2002 Elsevier Science (USA). MSC 1991 subject classification: 01A17.

The evolution of Brouwer's ideas on logic throughout his entire scientific life is practically unknown. Still, the study of this evolution is important, on the one hand, for grasping how Brouwer's notion of intuitionism allowed for the construction of logical systems and, on the other, in order to see how some statements of intuitionistic logic which are now considered obvious were only formulated in their correct form after much rethinking. In this paper, we will thus consider Brouwer's ideas on logic in general as well as some laws in particular while tracing their development (i.e., the laws of excluded middle, testability, and reciprocity of complementarity). Die Entwicklung von Brouwers Auffassungen uber Logik wahrend seines Lebens ist nahezu unbekannt. Die Analyse dieser Entwicklung ist sehr wichtig, weil sie zeigt, erstens daβ Brouwers Intuitionismus die Grundlage fur die Konstruktion eines logischen Systems abgeben konnte; zweitens daβ einige Satze der intuitionistischen Logik, die heute als selbstverstandlich angesehen werden, erst nach einem langen ProzeB von Uberlegungen korrekt formuliert werden konnten. Deshalb wird in diesem Aufsatz Brouwers Auffassung uber Logik im allgemeinen and uber einige logische Satze (d.h. den Satz vom ausgeschlossenen Dritten, den von der Alternative von Absurditat oder Absurditat der Absurditat, and den von der Reziprozitat der Komplementaritat) im besonderen betrachten and ihre Entwicklung umrissen. L'evoluzione delle idee di Brouwer sulla logica durante tutta la sua vita scientifica e praticamente sconosciuta. Tuttavia, to studio di questa evoluzione e importante, da un lato per comprendere come il concetto di intuizionismo di Brouwer conteneva gia in se le basi per la costruzione di un sistema logico, d'altro lato per vedere come asserti della logica intuizionista, the sono ora considerati ovvi, furono formulati correttamente solo dopo molti ripensamenti. In questo articolo considereremo le idee di Brouwer sulla logica in generale e su alcune leggi in particolare (la legge del terzo escluso, di provabilita, di reciprocity della complementarita), delineandone l'evoluzione.

This paper analyzes Thomas Bayes' essay of 1763, together with the additions by Richard Price, in relation to (1) historical influences and (2) the Bayesianism of the 20th century. As regards (1), historical evidence is presented linking Price with Hume, and it is argued that Price's additions are likely to have been written as an attempt to solve Hume's problem of induction. As regards (2), it is shown that Bayes' billiard table example is so constructed as to avoid the objections to Bayesianism made, in our own time, by R. A. Fisher and J. Neyman. Bayes did, however, introduce controversial Bayesian assumptions in his scholium, and was therefore a Bayesian—though a cautious Bayesian.

This paper argues that the philosophical tradition of nominalism, as evident in the works of Pierre Gassendi, Thomas Hobbes, Isaac Barrow, and Isaac Newton, played an important role in the history of mathematics during the 17th century. I will argue that nominalist philosophy of mathematics offers new clarification of the development of a “constructivist” tradition in mathematical philosophy. This nominalist and constructivist tradition offered a way for contemporary mathematicians to discuss mathematical objects and magnitudes that did not assume these entities were real in a Platonic sense, and helped lay the groundwork for formalist and instrumentalist approaches in modern mathematics.

Diophantos in Arithmetica, without having defined previously any concept of “equality” or “equation,” employs a concept of the unknown number as a tool for solving problems and finds its value from an equality ad hoc created. In this paper we analyze Diophantos’s practices in the creation and simplification of such equalities, aiming to adduce more evidence on certain issues arising in recent historical research on the meaning of the “equation” in Diophantos’s work.

Throughout the 18th century, the notion of probabilistic expectation was a matter of controversy among mathematicians. Despite its seminal role in the earliest formulations of mathematical probability, such as that of Huygens, expectation did not remain a fixed concept but underwent several striking shifts in definition. This paper argues that the conception of expectation was altered by mathematicians in a deliberate effort to capture the salient aspects of rational decision making. As the notion of rationality successively took on legal, economic, and then psychological overtones, the definition of probabilistic expectation followed suit.

A fraction whose numerator is one is called a unit fraction . Unit fractions have been the source of one of the most intriguing mysteries about the mathematics of antiquity. Except for 2/3, the ancient Egyptians expressed all fractions as sums of unit fractions. In particular, The Rhind Mathematical Papyrus (RMP) contains the decomposition of 2 / n as the sum of unit fractions for odd n ranging from 5 to 101. The way 2 / n was decomposed has been widely debated and no general method that works for all n has ever been discovered. In this paper we provide an elementary procedure that reproduces the decompositions as found in the RMP.

This paper gives both a new interpretation of Āryabha t · a's rule for sine-differences prescribed in the second chapter of hisĀryabha t · iya(A.D. 499/510), one of the oldest astronomical texts in India, and a hypothesis about the origin of his table of sine-differences given in the first chapter of the same work.Copyright 1997 Academic Press. [FORMULA]Copyright 1997 Academic Press. [FORMULA]

The London Mathematical Society had been founded in 1865 as little more than a college club. Thanks to the support of prominent members from the British mathematical community, it had quickly grown in size and stature during its first two years; yet, while firmly established at home, it had still to secure an academic reputation overseas. This paper, a sequel to [35], examines the principal developments that occurred in the period from 1867 until the turn of the century, during which time the Society would consolidate its position as a prestigious learned body in both the national and international mathematical arenas. Copyright 1998 Academic Press. The London Mathematical Society a ete fondee en 1865 et a debute en tant que club de college. Grâce au soutien de membres eminents de la communaute mathematique britannique, le nombre d'adherents et la stature de cette Societe se sont rapidement accrus pendant ses deux premieres annees; cependant, bien que resolument etablie en Grande Bretagne elle devait encore etablir sa renommee a l'etranger. Cet expose qui fait suite a [35], examine les developpements importants qui se sont produits entre 1867 et la fin du dix-neuvieme siecle, une periode pendant laquelle la Societe a consolide sa position de Societe prestigieuse dans les arenes nationales et internationales. Copyright 1998 Academic Press. Als die London Mathematical Society 1865 gegrundet wurde, war sie kaum mehr als ein Universitatsverein. Weil prominente Mathematiker aus ganz Grosbritannien sie unterstutzten, nahm sie in den ersten zwei Jahren schnell an Grose und Gestalt zu; obwohl sie sich im eigenen Lande bestandig entwickelte, muste sie sich im Ausland zunachst noch um einen akademischen Ruf bemuhen. In dieser Arbeit werden (im Anschlus an [35]), die wesentlichen Entwicklungen in der Zeit von 1867 bis zur Jahrhundertwende untersucht, also in der Zeit, in der die “Society” ihre Stellung als anerkannte gelehrte Vereinigung sowohl auf nationaler als auch auf internationaler Ebene gefestigt hat. Copyright 1998 Academic Press. MSC 1991 subject classification: 01A55

Written in the 16th century, Cardano's Liber de Ludo Aleae was, in its time, an advanced treatment of the probability calculus. At the same time it could be viewed as a gambling manual. Several commentators on the book have concluded that it is a mishmash of several, sometimes contradictory, results and statements written over an approximate 40-year period. In the current paper the Liber de Ludo Aleae is examined as a Renaissance text written in the intellectual milieu of humanism. A close examination of the book shows that it was heavily influenced by Aristotle's Ethics, in particular Aristotle's concept of justice. By reading the Liber de Ludo Aleae in this way, it is shown that there is an internal consistency to the work with a common thread of justice (ius) and knowledge (scientia) running through it. These themes are examined in detail. It is also argued that some of Cardano's probability calculations related to dice might have been taken from a printed version of the late medieval poem De Vetula.

La demostració de la proposició 9 del llibre primer de Sobre l'esfera i el cilindre d'Arquímedes conté una afirmació sense prova, una 'llacuna', segons la denominació d'un historiador de la matemàtica. Hi ha un acord gairebé unànime entre la majoria d'editors d'Arquímedes i d'experts en la seva obra sobre l'existència d'aquest buit demostratiu. Molts, a més, han procurat omplir-lo amb demostracions variades, algunes d'incompletes o directament incorrectes. En aquest article presento arguments teòrics, històrics i textuals que permetrien justificar l'absència de prova i, per tant, tancar el debat sobre aquesta 'llacuna'. The proof of Proposition 9 in Archimedes' On the Sphere and the Cylinder, Book i, contains an unproved statement that has been referred to as a 'lacuna.' Most editors and experts in Archimedean texts have agreed on the existence of this gap and have offered different proofs for the statement, some of them with incomplete or even incorrect arguments. In this paper, I offer arguments of a mathematical, historical, and textual nature that show that it is not necessary to assume the presence of any gap in the text.

Little is known about William Wallace' work and even less about his attempts to introduce Continental calculus to Britain in the early 19th century. A letter written by him to George Peacock in 1833 reveals many interesting facts concerning his role in the reform of British mathematics. This paper presents a brief account of his life and work and the full text of the letter, and proposes that Wallace' contributions deserve to be acknowledged.

Starting from Peirce's repeated claims of priority with respect to Dedekind's definition of finite set [R. Dedekin, Was sind und was sollen die Zahlen? (Braunschweig: Vieweg, 1888), Definizione 64], this paper traces the history of Peirce's definition and its role in his research on the foundations of arithmetic. This brings to light some remarkable and neglected achievements of Peirce in this field. It also shows that his priority claims are unjustified, although understandable in terms of his desire for acknowledgment of his pioneering work on the foundations of arithmetic.RésuméCet article relate l'histoire de la définition d'ensemble fine donnée par Peirce et de son ròle dans les recherches de ce dernier sur les fondements de l'arithmétique. Il débute par les revendications répétées de Peirce quant à l'antériorité de la définition d'ensemble fini de R. Dedekind [Was sind und was sollen die Zahlen? (Braunschweig: Vieweg, 1888), Definizione 64]. Notre historique met en lumière quelques réalisations remarquables, mais méconnues, de Peirce dans ce domaine. Il montre aussi que ses revendications d'antériorité sont injustifiées, quoique compréhensibles si l'on pense à son désir de voir reconnus ses travaux de pionniers sur les fondements de l'arithmétique.

George Peacock's A Treatise on Algebra of 1830, contained in its preface the first published recognition that Algebra need not necessarily be always associated with Arithmetic, and that non-arithmetical Algebras were possible. The work also contains the first statement of Peacock's principle of the permanence of equivalent forms. It is shown in this paper that Charles Babbage had all these ideas in almost identical or superior form in an unpublished work The Philosophy of Analysis written in 1821. Peacock certainly had access to his friend Babbage's writings, and the suggestion is made of unconscious assimilation rather than deliberate plagiarism.

Evariste Galois formulated his famous theory in 1831 in the first part of his Memoire sur les conditions de resolubilite des equations par radicaux. It is titled Principes. Even though the theory is completely understood today, it is hard to follow Galois's original. The style is brief, almost aphoristic and the approach quite different from today's. The aim of this paper is to make Galois's Principes readable for contemporary mathematicians (Sections 1 and 2) and to give a survey of Galois's Applications concerning equations of prime degree, primitive equations, and the modular equation in the theory of elliptic functions (Section 3). Remarks show the relationship to the work of Lagrange and Gauss. © 2002 Elsevier Science (USA). Evariste Galois formulierte seine beruhmte Theorie 1831 im ersten Teil seines Memoire sur les conditions de resolubilite des equations par radicaux. Er tragt den Titel Principes. Auch wenn die Theorie heute vollkommen verstanden ist, ist es bisweilen schwer, Galois' ursprunglichen, auserst knappen Formulierungen zu folgen. Ziel des Artikels ist es, Galois' Principes fur heutige Mathematiker lesbar zu machen (Abschnitte 1 und 2) und einen Uberblick uber Galois' Applications zu geben, betreffend Gleichungen von Primzahlgrad, primitive Gleichungen und die Modulargleichungen aus der Theorie der elliptischen Funktionen (Abschnitt 3). Bemerkungen verweisen jeweils auf die nahe Beziehung zum Werk von Lagrange und Gauss. © 2002 Elsevier Science (USA). MSC subject classification: 12E10.

Mathematical physics played an ambiguous role during the establishment and growth of the American mathematical community. During the early years of this century, mathematical physics diminished in importance relative to newer areas of study. But by the end of World War I, some members of the American mathematical community recognized that mathematics could benefit, materially and scientifically, from increased attention to mathematical physics. This paper examines the way mathematical physics figured in the ideas and actions of key leaders of the development of the American mathematical community. Die mathematische Physik spielte eine widerspruechliche Rolle bei der Herausbildung und Entwicklung der scientific community der amerikanischen Mathematiker. Waehrend der ersten Jahre dieses Jahrhunderts verringerte sich ihre Bedeutung relativ zu neueren Forschungsgebieten. Aber gegen Ende des Ersten Weltkrieges erkannten einige amerikanische Mathematiker, dass die Mathematik materiell und wissenschaftlich von der vermehrten Aufmerksamkeit fuer mathematische Physik profitieren koennte. Dieser Artikel untersucht, auf welche Weise die mathematische Physik in den Vorstellungen und Handlungen der die Entwicklung der community bestimmenden amerikanischen Mathematiker praesent war. La physique mathematique a joue un role ambigu au sein de la jeune communaute mathematique americaine. Dans les premieres annees de ce siecle, la physique mathematique declinait relativement a d'autres specialites. Mais a la fin de la Premiere Guerre Mondiale, certains mathematiciens americains reconnurent que les mathematiques pouvaient beneficier, materiellement et scientifiquement, de la physique mathematique. Cet article analyse la maniere dont la physique mathematique fut integree, en idee et en action, par d'importants representants de la communaute mathematique des Etats-Unis.

This paper contains an analysis of the scientific climate in which the Italian school of algebraic geometry came into being and developed in a direction different from that of the German school, even though both shared a common initial matrix. Reference is made to a number of letters written by Max Noether to Luigi Cremona.

The three major encyclopedias of medieval mathematics (Pal.573, Ott.lat.3307, L.IV.21) acclaim Antonio de' Mazzinghi as the best algebraist of the 14th and 15th centuries. This paper presents Antonio's biography and an analysis of his algebraic work.

This paper treats the history of a surveying problem in which the height of a mountain or tower as well as its distance from the observer is to be determined in cases where the distance between observer and object is not passable. This problem arises in works by Chinese (Liu Hui), Indians (A¯ryabhaṫa, Brahmagupta), Arabs (al-Bi¯ru¯ni¯) and in the Christian Middle Ages (Geometria incerti auctoris, Hugo de Sancto Victore), in which the examples and methods of solution are all similar.

The challenge by Frenicle to produce a certain perfect number led Fermat to discover what is now known as Fermat's theorem. However, in this paper it is argued that the usual interpretation of this challenge, together with Fermat's response, is flawed.

The introduction of modern mathematical ideas in Spain was carried off mainly by three authors: Garcia de Galdeano, Torres Quevedo, and Rey Pastor. This paper is devoted to the study of Garcia de Galdeano's most important algebraic works, which implied great progress in the assimilation of late 19th century mathematics by the Spanish mathematical community, not only by importing the new theories but also by proposing a new synthetic and didactic approach to this discipline.

The creation of a formal mathematical language was fundamental to making mathematics algebraic. A landmark in this process was the publication of In artem analyticem isagoge by Francois Viete (1540–1603) in 1591. This work was diffused through many other algebra texts, as in the section entitled Algebra in the Cursus mathematicus (Paris, 1634, 1637, 1642; second edition 1644) by Pierre Herigone (1580–1643). The aim of this paper is to analyze several features of Herigone's Algebra. Herigone was one of the first mathematicians to consider that symbolic language might be used as a universal language for dealing with pure and mixed mathematics. We show that, although Herigone generally used Viete's statements, his notation, presentation style, and procedures in his algebraic proofs were quite different from Viete's. In addition, we emphasize how Herigone handled algebraic operations and geometrical procedures by making use of propositions from Euclid's Elements formulated in symbolic language.

Two mathematical sources, On the Geometric Constructions Necessary for the Artisan , by Abu'l-Wafā' (ca. 940–998), and the anonymous work, On Interlocks of Similar or Corresponding Figures (ca. 1300), provide us with insight into the collaboration between mathematicians and artisans in the Islamic world. In this paper I present a series of quotations from these two sources, which show that mathematicians taught geometry to artisans by means of cut-and-paste methods and of geometrical figures that had the potential of being used for ornamental purposes. Copyright 2000 Academic Press. Matematikle ilgili iki kaynak bize Islam dunyasinda matematikciler ile sanatkarlar arasindaki isbirligi konusunda aydinlatici bilgiler sunuyor. Bu kaynaklardan biri Abu'l-Vefa (ca. 940–998) tarafindan yazilan “Sanatkarin ihtiyac duydugu geometrik cizimler,” digeri anonim bir yazarin kaleme aldigi “Icice gecen benzer veya karsilikli sekiller” (ca. 1300). Bu iki kaynaktan derledigim bir dizi alintiya yer verdigim bu makalede goruyoruz ki matematikciler sanatkarlara kes-ve-yapistir yontemiyle geometri ogretirken, ayni zamanda onerdikleri geometrik sekillerin bezeme sanatlarinda kullanilabilir olmasina ozen gosteriyorlardi. Copyright 2000 Academic Press. MSC subject classifications: 01A30, 01A20.

This paper analyzes certain phases of the development of modern integration theory by emphasizing the motivation of mathematicians for their work and their own estimations of their proceedings. The following seven points are discussed: 1. Introductory sketch of the history to be covered; 2. Cauchy and Dirichlet paving the way for Riemann; 3. Riemann; 4. Darboux and Hankel continuing the work of Riemann; 5. Evaluation of Hankel and Riemann; 6. Borel's measure theory; 7. Lebesgue's idea of the integral.

Mathematical ballistics in the United States until the First World War was largely dependent on the work of European authors such as Francesco Siacci of Italy. The war brought with it a call to the American mathematical community for participation in ballistics problems. The community responded by sending mathematicians to work at newly formed ballistics research facilities at Aberdeen Proving Grounds and Washington, D.C. This paper focuses on the efforts of Forest Ray Moulton and details how he dealt with various aspects of a single problem: differential variations in the ballistic trajectory due to known factors.

In 1892, F. Engel and E. Study investigated the exponential map of classical Lie groups for the first time. They showed that the special projective Lie groups over C possess surjective exponential functions. Engel also gave a “proof” for the corresponding claims for the other projective classical groups. However, it is known that the latter claims are wrong. Nevertheless, these “results” are still cited in the literature without comments. In the present paper, the mentioned works of Engel and Study are critically discussed. © 2002 Elsevier Science (USA). Im Jahre 1892 untersuchten F. Engel und E. Study als erste die Exponentialfunktion klassischer Lie-Gruppen. Sie zeigten, das die speziellen projektiven Lie-Gruppen uber C eine surjektive Exponentialfunktion besitzen. Engel formulierte auch einen ,,Beweis“ fur die entsprechenden Aussagen uber die anderen projektiven klassischen Gruppen. Inzwischen weis man, das die letzteren Aussagen jedoch falsch sind. Dennoch werden jene ,,Ergebnisse“ immer noch unkommentiert in der Literatur zitiert. Im vorliegendein in Artikel werden die besagten Arbeiten von Engel und Study kritisch diskutiert. © 2002 Elsevier Science (USA). MSC subject classifications: 22-03; 01A55; 01A60; 22E15.

It is well known that Horner's method for the computation of a real root of a polynomial equation was anticipated in Italy by Ruffini. In the present paper it is shown that in England the method was published by Holdred, before Horner. The resulting controversy over priority is discussed, and related letters from contemporary mathematicians are reproduced. It is concluded that the dissemination of the algorithm under the inappropriate designation “Horner's method” is mainly due to De Morgan. Copyright 1999 Academic Press. Il est bien connu que la methode de Horner pour calculer une racine reelle d'une equation polynome avait ete anticipee en Italie par Ruffini. Dans ce papier il est demontre qu'en Angleterre la methode fut publiee par Holdred avant Horner. La controverse sur la priorite est discutee, et des lettres de mathe maticiens contemporains sont reproduites. Il est conclu que la dissemination de l'algorithme sous la designation inappropriee “methode de Horner” est principalement due a De Morgan. Copyright 1999 Academic Press. MSC 1991 subject classifications: 01A55; 01A70; 01A80; 65-03.

In this paper the groundsf or the positive identification of the author of “A Calculation of the Credibility of Human Testimony” are discussed.

The paper describes an approximation formula for sine (x + h) that differs from the first four terms of the Taylor expansion only by having 4 in place of 6 in the denominator of the fourth term. It appears in Sanskrit stanzas quoted in a work of about the fifteenth century and given here with translation and explanation.

The first Chinese translation of Euclid'sElementsappeared in 1607. This translation was limited to Books I-VI; the remaining books were translated in 1857. Because these translations are composed in Classical Chinese, contemporary Chinese readers cannot access them easily. This paper introduces the new translations of Euclid'sElementsinto modern Chinese and Mongolian, which have been recently made. La premiere traduction chinoise desElementsd'Euclide remonte a 1607 et se limite aux six premieres livres. Les livres restants furent traduits en 1857. Du fait que ces traductions sont redigees en Chinois classique, les lecteurs chinois actuels ne peuvent pas les apprehender facilement. Dans le present article, nous presentons deux traductions recentes desElementsen chinois et mongol modernes. [formula]

The United States Military Academy has a substantial collection of historically valuable books and materials in mathematics and mathematical education. These books are maintained in the Mathematics Department Library and the archives and special collections of the Academy Library. Both libraries are open to visiting scholars by appointment. The book collection includes the two volumes of the first mathematics textbook used at the Academy [Hutton 1800-18011; early American editions of the same volumes [Hutton 18121; and over 100 early editions of books from Charles Davies’ “Course of Mathematics,” including [Davies 1826, 18321. Davies was a graduate of the United States Military Academy in 1815 and the head of the Mathematics Department from 1823 to 1837. He wrote over 25 textbooks on elementary school to college level mathematics and was probably the premier American mathematics textbook writer in the first half of the 19th century. Over half of his textbooks are in the collection, including several first editions and many later printings by publisher A. S. Barnes. Also, a collection of Davies’ personal papers rests in the archives of the Academy Library. Since the United States Military Academy was founded in 1802 as the first engineering school in the United States, the professors at the Academy had the opportunity and responsibility to shape the direction of American education in mathematics, science, and engineering. The collection reflects the participation of the Mathematics Department in this effort by including over 30 books written in the 19th century by professors in the department. Besides Davies’ books, the