!

Approximating the Roots of Numerical
Higher Equations up to the Year

1819

By

MARTIN ANDREW NORDGAARD, Ph.D.
Assistant Professor of Mathematics,
Grinnell College, Grinnell, Iowa

LB5

.C8

ho.123

HARVARD UNIVERSITY
GRADUATE SCHOOL
OF EDUCATION

Sept. 13, 1922

ACKNOWLEDGMENT

The author is deeply appreciative of the services of Professor David Eugene Smith, of Teachers College, Columbia University, whose inspiration and guidance have been a great help in carrying on this research.

CHAPTER

I. INTRODUCTION

CONTENTS

II. PRE-ALGorithmic METHODS

1. Egyptian and Babylonian Work in Finding Roots

2. Early Concepts of Higher Roots and Irrationals among the

Greeks.

3. The Approximations of Archimedes and Heron of Alexandria

4. Theon's Method of Exhaustion

5. Summary.

III. ALGORITHMIC METHODS FOR APPROXIMATING THE ROOTS OF PURE

POWERS

1. The Hindu Method of Exhaustion

2. Formulas for Irrational Roots in the Middle Ages

3. Summary.

IV. APPROXIMATIONS FOR SPECIFIC PURPOSES

PAGE

I. The Rise of Numerical Higher Equations among the Arabs

2. A Solution by an Unknown Arab Scholar

3. A Solution by Leonardo of Pisa.

4. Summary. .

V. METHODS INVOLVING DOUBLE FALSE POSITION AND PROCESSES OF

VII. NEWTON'S METHOD OF APPROXIMATION

1. Advance in Theory of Equations

2. A Description of Newton's Process

3. Raphson's Development of Newton's Method

INTERPOLATION

1. The General Nature of these Methods

2. Chuquet's Rule of Mean Numbers .

3. The Regula Aurea of Cardan

4. Stevin's Contribution

5. The Methods of Bürgi and Pitiscus

6. Summary . .

VI. VIETA'S EXTENSION OF THE HINDU METHOD OF APPROXIMATION TO

COMPLETE EQUATIONS

1. Vieta's Method of Approximation

2. The Modifications of Vieta's Method by Harriot .

3. The Improvements of Oughtred and Wallis .

4. The Contribution of Hume

5. Summary.