The study of waves can be traced back to antiquity where philosophers, such as Pythagoras (c.560-480 BC), studied the relation of pitch and length of string in musical instruments. However, it was not until the work of Giovani Benedetti (1530-90), Isaac Beeckman (1588-1637) and Galileo (1564-1642) that the relationship between pitch and frequency was discovered. This started the science of acoustics, a term coined by Joseph Sauveur (1653-1716) who showed that strings can vibrate simultaneously at a fundamental frequency and at integral multiples that he called harmonics. Isaac Newton (1642-1727) was the first to calculate the speed of sound in his Principia. However, he assumed isothermal conditions so his value was too low compared with measured values. This discrepancy was resolved by Laplace (1749-1827) when he included adiabatic heating and cooling effects. The first analytical solution for a vibrating string was given by Brook Taylor (1685-1731). After this, advances were made by Daniel Bernoulli (1700-82), Leonard Euler (1707-83) and Jean d’Alembert (1717-83) who found the first solution to the linear wave equation, see section (3.2). Whilst others had shown that a wave can be represented as a sum of simple harmonic oscillations, it was Joseph Fourier (1768-1830) who conjectured that arbitrary functions can be represented by the superposition of an infinite sum of sines and cosines now known as the Fourier series. However, whilst his conjecture was controversial and not widely accepted at the time, Dirichlet subsequently provided a proof, in 1828, that all functions satisfying Dirichlet’s conditions (i.e. non-pathological piecewise continuous) could be represented by a convergent Fourier series. Finally, the subject of classical acoustics was laid down and presented as a coherent whole by John William Strutt (Lord Rayleigh, 1832-1901) in his treatise Theory of Sound. The science of modern acoustics has now moved into such diverse areas as sonar, auditoria, electronic amplifiers, etc.

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Linear and nonlinear waves

This expository paper traces the development of the Newton-Raphson method for solving nonlinear algebraic equations through the extant notes, letters, and publications of Isaac Newton, Joseph Raphson, and Thomas Simpson. It is shown how Newton's formulation differed from the iterative process of Raphson, and that Simpson was the first to give a general formulation, in terms of fluxional calculus, applicable to nonpolynomial equations. Simpson's extension of the method to systems of equations is exhibited.

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Historical development of the Newton-Raphson method

The Presocratic Philosophers

In the present edition of Allama Iqbal’s The Reconstruction of Religious Thought in Islam, an attempt has been made at providing references to many authors cited in it and, more particularly, to the passages quoted from their works. The titles of these works have not always been given by the Allama and, in a few cases, even the names of the authors have to be worked out from some such general descriptions about them as ‘the great mystic poet of Islam’, ‘a modern historian of civilization’, and the like.

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The Reconstruction of Religious Thought in Islam

This book is designed as a text for the first year of graduate algebra, but it can also serve as a reference since it contains more advanced topics as well. This second edition has a different organization than the first. It begins with a discussion of the cubic and quartic equations, which leads into permutations, group theory, and Galois theory (for finite extensions; infinite Galois theory is discussed later in the book). The study of groups continues with finite abelian groups (finitely generated groups are discussed later, in the context of module theory), Sylow theorems, simplicity of projective unimodular groups, free groups and presentations, and the Nielsen-Schreier theorem (subgroups of free groups are free). The study of commutative rings continues with prime and maximal ideals, unique factorization, noetherian rings, Zorn's lemma and applications, varieties, and Grobner bases. Next, noncommutative rings and modules are discussed, treating tensor product, projective, injective, and flat modules, categories, functors, and natural transformations, categorical constructions (including direct and inverse limits), and adjoint functors. Then follow group representations: Wedderburn-Artin theorems, character theory, theorems of Burnside and Frobenius, division rings, Brauer groups, and abelian categories. Advanced linear algebra treats canonical forms for matrices and the structure of modules over PIDs, followed by multilinear algebra. Homology is introduced, first for simplicial complexes, then as derived functors, with applications to Ext, Tor, and cohomology of groups, crossed products, and an introduction to algebraic $K$-theory. Finally, the author treats localization, Dedekind rings and algebraic number theory, and homological dimensions. The book ends with the proof that regular local rings have unique factorization. (GSM/114)

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Advanced Modern Algebra

MATHEMATICAL NOTATIONS in two volumes by FLORIAN CAJORI Volume I, Notations in Elementary Mathematics, just off the press, is sure to be an indispensable addition to every school and college library Symbols in Arithmetic, Algebra, and Geometry, as developed and used in all countries and in all ages, furnish most fascinating reading for teachers and pupils and add greatly to the appreciation and understanding of elementary mathematics This volume, beautifully printed and bound, contains 45I pages and io6 reproductions from old manuscripts and books Price $6oo It is another notable addition to the Open Court Mathemnatical Series Send for a complete list

A history of mathematical notations

Introduction The Babylonians The Egyptians The Greeks The Romans The Maya The Chinese The Japanese The Hindus The Arabs Europe During the Middle Ages: Introduction of Roman mathematics Translation of Arabic manuscripts The first awakening and its sequel Europe During the Sixteenth, Seventeenth and Eighteenth Centuries: The Renaissance Vieta to Descartes Descartes to Newton Newton to Euler Euler, Lagrange and Laplace The Nineteenth and Twentieth Centuries. Introduction: Definition of mathematics Synthetic Geometry: Elementary geometry of the triangle and circle Link-motion Parallel lines, non-Euclidean geometry and geometry of $n$ dimensions Analytic Geometry: Analysis Situs Intrinsic co-ordinates Definition of a curve Fundamental postulates Geometric models Algebra: Theory of equations and theory of groups Solution of numerical equations Magic squares and combinatory analysis Analysis: Calculus of variations Convergence of series Probability and statistics Differential equations. Difference equations Integral equations, integro-differential equations, general analysis, functional calculus Theories of irrationals and theory of aggregates Mathematical logic Theory of Functions: Elliptic functions General theory of functions Uniformization Theory of Numbers: Fermat's "Last Theorem," Waring's theorem Other recent researches. Number fields Transcendental numbers. The infinite Applied Mathematics: Celestial mechanics Problem of three bodies General mechanics Fluid motion Sound. Elasticity Light, electricity, heat, potential Relativity Nomography Mathematical tables Calculating machines, planimeters, integraphs Editor's notes Alphabetical index.

Florian Cajori

Papers

A history of mathematical notations

A history of mathematics

The Rhind Mathematical Papyrus.

The Teaching And History Of Mathematics In The United States

Mathematical principles of natural philosophy ; Optics . Treatise on light