Showing papers in "Archive for History of Exact Sciences in 1979"

The Gibbs-Wilbraham phenomenon: An episode in fourier analysis

Abstract: plays an essential r61e in computing the amount of this overshoot. While teaching a course in the theory of functions of a real variable, E. HEWITT found the value 1.71... listed for the integral (1) in HARDY & ROGOSINSKI [271, page 36. This anomaly, as well as others encountered in the literature, led us to a study of the GIBBS phenomenon and its history. In the course of this study we uncovered a maze of forgotten results, interesting and difficult generalizations, faulty constants, and some details about the GIBBS phenomenon that have escaped the attention of many writers on the subject. Despite the familiarity of our theme, we therefore entertain a hope that readers of the Archive will find some interest in a discussion of this corner of FOURIER analysis. The paper is divided into three Parts. In Part I, we examine GIBBS's phenomenon in some detail. In Part II, we take up its curious history and describe briefly some of its congeners. In Part III, we offer some conclusions. The computations given in this paper were carried out on two computers: a Hewlett-Packard 9810 and a Univac 1110. The graphs (barring the simplest) were drawn by a Hewlett-Packard 9862A plotter. All finite decimal expansions are truncated decimal expansions. It is a pleasure to record our indebtedness to GERALD B. FOLLAND, THOMAS L. HANKINS, EINAR HILLE, and STEPHEN P. KEELER, who have made valuable suggestions to us.

On the Emergence of Probability

Abstract: The modern theory of probability is usually dated from the second half of the 17th century. The famous Pascal-Fermat correspondence of 1654 began a rapid advance in the subject, and by the completion of Jacob Bernoulli's Ars Conjectandi (published posthumously in 1713, but written and discussed long before) one can say that the subject has more or less fully emerged. This of course raises an important historical question: what factors are responsible for the sudden growth of the theory of probability? Why did it happen when it did? In this paper we will examine an answer to this question recently put forward by Ian Hacking. In his book The Emergence of Probability,1 Hacking proposes that the sudden development of the theory of probability is to be explained by an important conceptual change in the way people thought about chance and evidence. The claim, in brief, is that the modern theory of probability emerged when it did because it was not until the middle of the 17th century that we possessed the modern concept of probability. We believe that Hacking is wrong. After presenting an outline of his thesis and the principal arguments that he offers for it, we will show that Hacking's explanation for the sudden activity in the theory of probability cannot be correct, since many of the concepts that Hacking believes constitute the core of our modern notion of probability were present long before the mid- 17th century. We will argue instead for a different explanation, one that accounts for the history of the theory of probability without appeal to radical conceptual revolution.

The problem of the invariance of dimension in the growth of modern topology, part II

Abstract: This work examines the historical origins of topological dimension theory with special reference to the problem of the invariance of dimension. Part I, comprising chapters 1–4, concerns problems and ideas about dimension from ancient times to about 1900. Chapter 1 deals with ancient Greek ideas about dimension and the origins of theories of hyperspaces and higher-dimensional geometries relating to the subsequent development of dimension theory. Chapter 2 treatsCantor's surprising discovery that continua of different dimension numbers can be put into one-one correspondence and his discussion withDedekind concerning the discovery. The problem of the invariance of dimension originates with this discovery. Chapter 3 deals with the early efforts of 1878–1879 to prove the invariance of dimension. Chapter 4 sketches the rise of point set topology with reference to the problem of proving dimensional invariance and the development of dimension theory. Part II, comprising chapters 5–8, concerns the development of dimension theory during the early part of the twentieth century. Chapter 5 deals with new approaches to the concept of dimension and the problem of dimensional invariance. Chapter 6 analyses the origins ofBrouwer's interest in topology and his breakthrough to the first general proof of the invariance of dimension. Chapter 7 treatsLebesgue's ideas about dimension and the invariance problem and the dispute that arose betweenBrouwer andLebesgue which led toBrouwer's further work on topology and dimension. Chapter 8 offers glimpses of the development of dimension theory afterBrouwer, especially the development of the dimension theory ofUrysohn andMenger during the twenties. Chapter 8 ends with some concluding remarks about the entire history covered.

Heaviside s Operational Calculus and the Attempts to Rigorise It

Abstract: At the end of the 19th century Oliver Heaviside developed a formal calculus of differential operators in order to solve various physical problems. The pure mathematicians of his time would not deal with this unrigorous theory, but in the 20th century several attempts were made to rigorise Heaviside's operational calculus. These attempts can be grouped in two classes. The one leading to an explanation of the operational calculus in terms of integral transformations (Bromwich, Carson, Vander Pol, Doetsch) and the other leading to an abstract algebraic formulation (Levy, Mikusinski). Also Schwartz's creation of the theory of distributions was very much inspired by problems in the operational calculus.

Absolute temperatures as a consequence of Carnot's general axiom

Abstract: 1. Program In our book, Concepts and Logic of Classical Thermodynamics,1 Mr. BHARATHA and I developed classical thermodynamics on the basis of Part I of Carnot's General Axiom, namely, the motive power of a Carnot cycle is positive and is determined by its operating temperatures and by the amount of heat it absorbs. For a given body, then, there is a function G such that for any Carnot cycle # L(%) = G(0+,0 ,C + (^j)>0. (1) The domain of G is the set of operating temperatures and heats absorbed that may appertain to Carnot cycles for the body in question. It is part of the definition of a Carnot cycle that 6+>6~ and that C+(^)>0. This definition and 1 C. Truesdell & S. Bharatha, Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines, Rigorously Constructed upon the Foundation Laid by S. Carnot and F. Reech, N.Y. etc., Springer-Verlag, 1977.

Riemann, Betti and the birth of topology

Abstract: This Archive has recently published a conscientious study ([1]) of the relations between Riemann and his Italian contemporaries. Unfortunately the writer missed what might otherwise have been the gem of his collection two letters from Betti to his colleague and friend Tardy, describing in detail his conversations with Riemann on "analysis situs". They seem to be almost unknown, in spite of having been duly quoted by Bourbaki in one of the historical notes to his Topology (cf. [2]); they were originally published by G. Loria, as one of the appendices to his obituary notice on Tardy ([3]). They are as follows (I have merely corrected obvious misprints):

Newton's “Achromatic” dispersion law: Theoretical background and experimental evidence

Abstract: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 1. Experiment VIII: The Experiments and Deduction . . . . . . . . . . . . . 93 2. Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.

Eudoxan mathematics and the Eudoxan spheres

Abstract: Liens entre les spheres d'Eudoxe de Cnide (1 moitie du 4 siecle av. J.-C.), introduites dans le modele geometrico-cinematique du mouvement des planetes, et ses autres inventions mathematiques: la theorie des proportions, la methode d'"exhaustion", la duplication du cube, etc.

Simple groups of finite order in the nineteenth century

Abstract: The history of simple groups starts with the work of Evariste Galois (1811-1832). Throughout the eighteenth century and on into the nineteenth century the all-consuming passion among algebraists was the determination of which polynomial equations could be solved by radicals. A polynomial equation of degree n, xn + alx"~1 + ... +an_lx + an = 0, where the coefficients af belong to a field F, is said to be solvable by radicals (or algebraically solvable) when it is possible to express the roots of the equation in terms of the coefficients using a finite number of algebraic operations addition, subtraction, multiplication, division, raising to powers and extraction of roots. Galois approached the problem of characterizing such equations by considering, as Lagrange had done before him, the notion of permutations of the roots of an equation. This in turn led to the concept of a group. Prior to Galois, Lagrange had worked with what is in effect the symmetric group in his studies of functions unchanged under all permutations of their variables, and GAUSS used essentially the cyclic group in the numbertheoretic setting of congruences. Galois, however, dealt not with special cases but recognized, without giving an explicit definition, a permutation group as a set of permutations having the closure property. Certain points should be clarified here. Galois was the first to use the term group in a technical sense, but he also used the word in its non-mathematical sense to refer to an arbitrary collection of objects. It is sometimes difficult to distinguish the mathematical from the colloquial meaning. Also meriting discussion is Galois' use of the terms "permutation" and "substitution". Whereas the contemporary definition of permutation is that of a one-to-one mapping of a set of objects onto itself, the word is also used more informally to denote the actual arrangement of the objects. Galois used the

Chu Shih-chieh's Suan-hsüeh ch'i-meng (introduction to mathematical studies)

Abstract: The multi-faceted content of the SHCM and its collection of rules and tables made it an important mathematical work not only in China, but also in Korea and Japan. This book clearly demonstrated Chu's predominant interest in the field of algebra and his contribution to the solution of numerical equations of higher degree, which was a prelude to his famous treatise the Ssu-yuan yu-chien.