Swetz - The European Mathematical Awakening

The European Mathematical Awakening

The European Mathematical Awakening

A Journey Through the History of Mathematics, 10001800

Edited by Frank J. Swetz

Dover Publications, Inc.
Mineola, New York

Copyright

Copyright 1994, 2013 by Frank J. Swetz
All rights reserved.

Bibliographical Note

This Dover edition, first published in 2013, contains a selection of thirty-two corrected chapters from Five Fingers to Infinity: A Journey Through the History of Mathematics, originally published in 1994 by Open Court Publishing Company, Chicago. The Preface, Epilogue, Historical Exhibit 9, Editors Notes, Bibliography, and introductory Perspectives have been specially prepared for this Dover edition by Frank J. Swetz.

Library of Congress Cataloging-in-Publication Data

From five fingers to infinity. Selections.

The European mathematical awakening: a journey through the history of mathematics, 10001800 / edited by Frank J. Swetz. Dover edition.

p. cm.

This Dover edition, first published in 2013, contains a selection of thirty-two corrected chapters From Five Fingers to Infinity: A Journey Through the History of Mathematics, originally published in 1994 by Open Court Publishing Company, Chicago

Summary: A global survey of the history of mathematics, this newly corrected and updated collection of 32 highly readable essays features contributions by such distinguished educators as Carl Boyer and Morris Kline. Fascinating articles explore studies by Fibonacci, Descartes, Cardano, Kepler, Galileo, Pascal, Newton, Euler, and others. Suitable for readers with no background in math Provided by publisher.

Summary: A history of mathematics in Europe from 1000 to 1800 Provided by publisher.

Includes bibliographical references and index.

eISBN-13: 978-0-486-31027-5

1. MathematicsEuropeHistory I. Swetz, Frank J., 1937 II. Title.

QA27.E85F76 2013

510.9dc23

2012031301

Manufactured in the United States by Courier Corporation
49805001
www.doverpublications.com

Table of Contents

The Growth of Mathematical Knowledge

VERA SANFORD

Bedes Finger Mathematics

G. A. MILLER

The Geometry of Gothic Church Windows

DOROTHY V. SCHRADER

E. R. SLEIGHT

Algorist versus the Abacist

CHARLES KING

R. B. M c CLENON

THOMAS M. SMITH

DOROTHY I. CARPENTER

PHILLIP S. JONES

RICHARD W. FELDMANN

Cardanos Technique for the Solution of a Reduced Cubic Equation

PHILLIP S.JONES

VERA SANFORD

JAMES KING BIDWELL

PHILLIP S. JONES

BERNARD H. TUCK

Multiplication Algorithms of the Fifteenth and Sixteenth Centuries

D. J. STRUIK

Mathematical Considerations on the Trajectory of a Cannon Ball

CARL B. BOYER

C. B. READ

The Evolution of Algebraic Symbolism

MORRIS KLINE

EDMOND R. KIELY

Torricellis Wine Glass

CARL B. BOYER

HAROLD MAILE BACON

Robervals Quadrature of the Cycloid

V. FREDERICK RICKEY

Newtons Method of Fluxions

DOROTHY V. SCHRADER

Mengolis Proof for the Divergence of the Harmonic Series

HOWARD EVES

WILLIAM DUNHAM

CARL B. BOYER

D. J. STRUIK

JERRY D. TAYLOR

LEO GAFNEY

Mathematicians of the French Revolution

CARL B. BOYER

TERI PERL

Women in Mathematics

Preface

T he following journey through the history of mathematics spans eight centuries, from 1000 CE to 1800 CE. This is a long period of time, which marks a turning point in European mathematical thinking, An inflection point in the graphical display of the Growth of Mathematical Knowledge as shown in Historical Exhibit 1 (p.4). It signifies a true societal awakening to the scope and power of mathematics and its applications. Other societies also experienced such intellectual transitions but their experiences will not be encountered directly during this journey. One route of understanding will be circuitous and challenging enough.

The landmarks of mathematical achievement and their relevant impact are dispersed over time and location. As your principal tour guide, I have tried to mark a meaningful travel agenda focusing on the development of mathematics and the conditions supporting that development. At each stop during the journey other tour guides, the authors of the individual articles, provide more experienced guidance to for the mathematical encounter. The route I have selected is chronologically organized so that in traversing it a reader follows the path of historical influences, the circumstances and discoveries, which gave rise to the particular event. However, each event in itself maybe visited separately, and still conveys a meaningful encounter. Historical Exhibits, snapshots, of some prominent features that reflect on the developments considered are included at relevant intervals. These Historical Exhibits are intended to provide further insights into the events visited. At each stop in our journey, further guidance is also provided by bibliographies and notes.

While the period of our intellectual journey covers 800 years, the developments in mathematics are not constrained or limited by particular intervals. Mathematical knowledge is, and has been, constantly growing. It is an ongoing process, an expression of human existence, one in which we all can participate.

FRANK J. SWETZ

Harrisburg, Pennsylvania

October, 2011

The European Mathematical Awakening

Perspective: The European Mathematical Awakening

B y the beginnings of the 11th century, Europe had survived a series of cataclysmic events: barbarian invasions; plagues and crop failures, to emerge from a period of intellectual and political stagnation known as the Dark Ages. Advances in agriculture and animal husbandry provided a better and larger food supply. Improved nutrition stimulated population increase. Land reclamation projects were undertaken and new towns founded. The harnessing of wind and water power made many of lifes daily chores easier. Opportunities in villages and towns attracted freemen, craftsmen and artisans. Road systems were improved. Trade and commerce increased. Cities grew and prospered. In particular, the Italian maritime city states of Pisa, Venice and Genoa benefited from the weakening Islamic domination of the Mediterranean to become trade entrepts for the goods and commodities of the Levant. Merchants from these cities established trading houses abroad, conducting business at the sources of their imports. Interacting with local merchants, they learned their customs and habits and brought much of this new knowledge back to Europe. Civic authority and structure were resurrected and strengthened throughout Europe, giving rise to regional and local identities, beginnings of a sense of nationalism. Trade guilds provided a united voice for some skilled working classes. These new institutions of identity and wealth gave rise to political power, which would challenge and modify existing sources of authority. Since the fall of the Roman Empire, the dominant encompassing political, as well as spiritual, authority in Europe was the Catholic Church. The Church was primarily concerned with otherworldly matters, and initially viewed inherited Greek scientific and mathematical theories as suspect pagan knowledge. However, the study of mathematics was formally sanctioned by St. Augustine (ca.400) as worthy of Christian involvement; still, the Catholic Churchs actual interest with mathematics was minimal, limited primarily to the determination of the Church calendar based on a correct dating of Easter. Handbooks called computi were written to assist in this task. A few churchmen pursued mathematics for its intrinsic and classical values. One such scholar was the French monk, Gerbert of Aurillac (ca. 9501003), who eventually became Pope Sylvester II in the year 999. Gerbert sought out existing Arabic sources of Euclids