Solving the Quintic by Iteration

Mathematic

Complex numbers from 1600 to 1840

This thesis uses primary and secondary sources to study advances in complex number theory during the 17th and 18th Centuries. Some space is also given to the early 19th Century. Six questions concerning their rules of operation, usage, symbolism, nature, representation and attitudes to them are posed in the Introduction. The main part of the thesis quotes from the works of Descartes, Newton, Wallis, Saunderson, Maclaurin, d'Alembert, Euler, Waring, Frend, Hutton, Arbogast, de Missery, Argand, Cauchy, Hamilton, de Morgan, Sylvester and others, mainly in chronological order, with comment and discussion. More attention has been given tp algebraists, the originators of most advances in complex numbers, than to writers in trigonometry, calculus and analysis, who tended to be users of them. The last chapter summarises the most important points and considers the extent to which the six questions have been resolved. The most important developments during the period are identified as follows: (i) the advance in status of complex numbers from 'useless' to 'useful'. (ii) their interpretation by Wallis, Argand and Gauss in arithmetic, geometric and algebraic ways. (iii) the discovery that they are essential for understanding polynomials and logarithmic, exponential and trigonometric functions. (iv) the extension of trigonometry, calculus and analysis into the complex number field. (v) the discovery that complex numbers are closed under exponentiation, and so under all algebraic operations. (vi) partial reform of nomenclature and symbolism. (vii) the eventual extension of complex number theory to n dimensions

Teaching Calculus with Infinitesimals: New Perspectives

The present research corroborates K. Sullivan\u27s initial results as stated in her epoch making study about the effectiveness of teaching elementary calculus using Robinson\u27s non standard approach. Our research added to her results related to the teaching of the elementary integral, with similar positive results. In this essay we propose a definition of the notion of cognitive advantage mentioned by Sullivan in expressing the dramatic differences in understanding of students of non standard calculus as opposed to those of its standard counterpart. Our proposal is based on ideas of Kitcher and Kuhn and allows us to better understand the didactics of Calculus. Formally K. Sullivan\u27s claim of an observed advantage when referring to the improved understanding of non standard calculus students (as opposed to the standard approach of Weierstrass) is a consequence to the accepted fact that mathematical truths remain the same when changes of paradigms ensue, a situation markedly different from that science. While mathematical truths remain, mathematical justifications (proofs) change dramatically and increase in complexity

The Emergence of Analysis in the Renaissance and After

Paper by Salomon Bochne

A development of some relationships between astronomy and mathematics for the secondary schools.

Thesis (Ed.M.)--Boston Universit

Methods of applied dynamics

The monograph was prepared to give the practicing engineer a clear understanding of dynamics with special consideration given to the dynamic analysis of aerospace systems. It is conceived to be both a desk-top reference and a refresher for aerospace engineers in government and industry. It could also be used as a supplement to standard texts for in-house training courses on the subject. Beginning with the basic concepts of kinematics and dynamics, the discussion proceeds to treat the dynamics of a system of particles. Both classical and modern formulations of the Lagrange equations, including constraints, are discussed and applied to the dynamic modeling of aerospace structures using the modal synthesis technique