1
Historia Mathematica
ISSN:0315-0860 Volume:7 Issue:3 Page:343-365

Richards Joan L;

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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. © 1980, All rights reserved.

2
Historia Mathematica
ISSN:0315-0860 Volume:7 Issue:3 Page:234-260

Daston Lorraine J;

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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. © 1980, All rights reserved.

3
Historia Mathematica
ISSN:0315-0860 Volume:7 Issue:2 Page:156-185

Russ S.B;

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This is the first English translation of Bolzano's paper, Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege (Prague 1817). It has already appeared in French, Russian, and Czechoslovakian translations. The paper represents an important stage in the rigorous foundation of analysis and is one of the earliest occasions when the continuity of a function and the convergence of an infinite series are both defined and used correctly. © 1980.

4
Historia Mathematica
ISSN:0315-0860 Volume:7 Issue:4 Page:389-400

Becher Harvey W;

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In a recent article, J. M. Dubbey [Historia Mathematica 4 (1977), 295–302] showed that George Peacock's A Treatise on Algebra (1830) was similar to an unpublished work written by Charles Babbage in 1821. Evidently perplexed about the absence of a dispute over priority, Dubbey concluded that Peacock had unconsciously assimilated Babbage's ideas, and that Babbage was too busy with other activities to be concerned. The thesis of this article is that the innovative aspects of the work of both Babbage and Peacock are extensions of ideas put forth in 1803 by Robert Woodhouse, and that probably neither Babbage nor Peacock was overly concerned with acknowledgments because their approach to algebra was not unique at Cambridge. © 1980, All rights reserved.

5
Historia Mathematica
ISSN:0315-0860 Volume:7 Issue:3 Page:261-288

Dauben Joseph W;

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Gösta Mittag-Leffler was the founding editor of the journal Acta Mathematica. In the early 1870's it was meant, in part, to bring the mathematicians of Germany and France together in the aftermath of the Franco-Prussian War, and the political neutrality of Sweden made it possible for Mittag-Leffler to realize this goal by publishing articles in German and French, side by side. Even before the end of the First World War, Mittag-Leffler again saw his role as mediator, and began to work for a reconciliation between German and Allied mathematicians through the auspices of his journal. Similarly, G. H. Hardy was particularly concerned about the reluctance of many scientists in England to attempt any sort of rapprochement with the Central European countries and he sought to do all he could to bring English and German mathematicians together after the War. His correspondence with Mittag-Leffler survives in the Archives of the Institut Mittag-Leffler, Djursholm, Sweden, and serves as the basis for this article, which focuses upon the attempts of Mittag-Leffler to reconcile mathematicians after the War, and to renew international cooperation. © 1980, All rights reserved.

6
Historia Mathematica
ISSN:0315-0860 Volume:7 Issue:3 Page:289-342

Hawkins Thomas;

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A discussion of the manner in which discoveries in non-Euclidean geometry, combined with the Weierstrassian attitude towards mathematics, led Wilhelm Killing, one of Weierstrass' students, to initiate a research program on foundations of geometry that led to his groundbreaking investigations on the structure of Lie algebras. © 1980, All rights reserved.

7
Historia Mathematica
ISSN:0315-0860 Volume:7 Issue:4 Page:369-378

Williamson Frank;

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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. © 1980, All rights reserved.

8
Historia Mathematica
ISSN:0315-0860 Volume:7 Issue:2 Page:186-187

Lumpkin Beatrice;

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9
Historia Mathematica
ISSN:0315-0860 Volume:7 Issue:4 Page:425-443

Pier Jean-Paul;

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The Bolzano-Weierstrass and Borel-Lebesgue properties of sets constitute the fundamental ideas that led to the notion of compactness. The link between these ideas appeared for the first time to Fréchet, who formulated the first definition of a compact set. The notion developed simultaneously with that of a complete set thanks mainly to the contributions made by Hausdorff and Alexandroff. Later Moore-Smith convergence and filters enabled simplification of the language. © 1980, All rights reserved.

10
Historia Mathematica
ISSN:0315-0860 Volume:7 Issue:4 Page:407-424

Lay-Yong Lam;

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One of the significant contributions of Chinese mathematicians is the method of solving numerical equations of higher degree. A number of scholarly works have established similarities between ancient and medieval Chinese root-extraction procedures and Horner's method of solving a numerical equation of any degree. The conceptual development of the Chinese methods, which began with the procedures of extracting square and cube roots during the Han dynasty, culminated in the solution of higher numerical equations in the 13th and the beginning of the 14th centuries. This paper attempts to show the intimate role played by the triangular array of numbers (known in the West as the Pascal triangle) in the process of the development of the Chinese methods, especially when the original geometrical concept was being replaced by the more general algebraic method. © 1980, All rights reserved.

11
Historia Mathematica
ISSN:0315-0860 Volume:7 Issue:2 Page:113-117

Freudenthal Hans;

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It is generally accepted that Huygens based probability on expectation. The term “expectation,” however, stems from Van Schooten's Latin translation of Huygens' treatise. A literal translation of Huygens' Dutch text shows more clearly what Huygens actually meant and how he proceeded.

12
Historia Mathematica
ISSN:0315-0860 Volume:7 Issue:2 Page:134-140

Goldstein Bernard R;

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Although not previously noticed, the solar and lunar velocity tables in the 1483 edition of the Alfonsine Tables differ from the corresponding tables in the 1518 edition (Venice) which are identical to those that appear in the Toledan Tables and the zij of al-Battani. The velocity tables in the unpublished Hebrew version of the Alfonsine Tables, composed in 1460 (Avignon), agree with the 1483 edition, and the Hebrew translator tells us that these tables (together with some others) were added to the Alfonsine Tables by Christian scholars after the time of Alfonso. A discussion of the underlying parameters is also presented. © 1980.

13
Historia Mathematica
ISSN:0315-0860 Volume:7 Issue:1 Page:7-23

Engelsman S.B;

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In 1776, J. L. Lagrange gave a definition of the concept of a “complete solution” of a first-order partial differential equation. This definition was entirely different from the one given earlier by Euler. One of the sources for Lagrange's reformulation of this concept can be found in his attempt to explain the occurrence of singular solutions of ordinary differential equations. Another source of the new definition is contained in an earlier treatise of Lagrange [1774] in which he elaborated an approach to first-order partial differential equations briefly indicated by Euler. The method of “variation of constants,” which was fundamental to his argument, suggested to Lagrange the reformulation of the concept of a “complete solution.” In the present paper I shall discuss both sources of the new definition of “completeness.”

14
Historia Mathematica
ISSN:0315-0860 Volume:7 Issue:4 Page:457-458

Toomer G.J;

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15
Historia Mathematica
ISSN:0315-0860 Volume:7 Issue:4 Page:444-444

Gray Jeremy;

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16
Historia Mathematica
ISSN:0315-0860 Volume:7 Issue:4 Page:401-406

Yadegari Mohammad;

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An example illustrating the expansion of the binomial theorem up to the seventh power of the sum of two quantities can be found in a work by al-Zanjani (d. 1262). While the binomial theorem is presumed to have been discovered by al-Karaji (ca. 1029) and utilized by several subsequent mathematicians, the elaboration of the theorem by al-Zanjani points to the fact that it was a widespread concept in medieval Islamic mathematics. Al-Zanjani's exposition of this theorem is herewith translated from the original Arabic. © 1980, All rights reserved.

17
Historia Mathematica
ISSN:0315-0860 Volume:7 Issue:4 Page:379-388

Smith G.C;

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In 1736 Bayes wrote a tract on Newton's fluxions. The tract was one of several explanations of fluxions which appeared following Berkeley's attack upon Newton's ideas in The Analyst. Bayes began with a careful discussion of the meaning of fluxions and prime and ultimate ratios. He proceeded to develop the properties of prime and ultimate ratios in a way not unlike Cauchy's treatment of limits, using these results to prove some basic theorems concerning the calculation of fluxions. © 1980, All rights reserved.

18
Historia Mathematica
ISSN:0315-0860 Volume:7 Issue:4 Page:449-451

Van Egmond Warren;

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19
Historia Mathematica
ISSN:0315-0860 Volume:7 Issue:4 Page:454-457

Birkhoff Garrett;

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20
Historia Mathematica
ISSN:0315-0860 Volume:7 Issue:4 Page:459-467

Lewis Albert C;

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