A STUDY OF THE CONTRIBUTIONS OF EARLY NINETEENTH CENTURY BRITISH MATHEMATICIANS TO THE DEVELOPMENT OF ABSTRACT ALGEBRA AND THEIR INFLUENCE ON LATER ALGEBRAISTS AND MODERN SECONDARY CURRICULA.

摘要

The nineteenth century ushered in a period of upheaval and revolution in the history of mathematics. The non-Euclidean geometries of Gauss, Lobachensky, Bolyai, and Riemann opened new vistas to both mathematicians and physicists alike, whereas the works of Cauchy, Dedeking, Dirichlet, Weierstrass, and others placed the real number system and analysis on a firm logical foundation. Another area which experienced a radical change in both its structure and spirit was algebra. In contrast to the developments in geometry and analysis that were largely generated by Continental mathematicians, algebra received its major impetus from British mathematicians who for over a century had remained isolated from Continental mathematics after the Newton-Leibniz controversy over priority in the invention of calculus. Although the contributions of some early nineteenth century British mathematicians have been treated in summary form in various studies and histories of mathematics, no comprehensive history of this development is available for use by instructors and students in both undergraduate and graduate courses in the history of mathematics. Furthermore, the limited available literature is fragmented and does not provide a unified source from which secondary school teachers of mathematics may draw to incorporate historical materials related to the teaching of modern algebra into their classrooms.; This thesis traces the early British development of concepts associated with abstract algebra and the evolution of an expanded view of number amid much controversy. The works of Playfair, Greenfield, Woodhouse, Buee, and Gilbert during this period of controversy are examined together with the work of the two chief opponents of negative and imaginary numbers, Francis Maseres and William Frend.; The emancipation of algebra from its association with arithmetic is established and promulgated through the works of Peacock, Gregory, Babbage, and DeMorgan. However, none of these individuals succeeded in actually creating an abstract algebraic system. It was Hamilton through his discovery of quaternions, Boole through his work on the algebra of logic and classses, and Cayley for his development of matrix theory, who brought abstract algebra to its fruition. The impact of these ideas on modern secondary curricula is interwoven throughout the thesis.