AN ABSTRACT OF THE THESIS OF ·.. David L. Baughman for the Master of Science in Mathematics/Computer Science presented on 1 October 1991. A large number of reference works on the history of mathematics seem to suggest either that an in-depth discussion of the ideas which led up to the simul­ taneous discovery of logarithms by Napier and Burgi would be too arcane to be of interest, or that not much did lead up to the achievement of these two discoverers and that the idea of logarithms simply spontaneously created itself in Napier and Burgi, as if of nothing. Much the same attitude seems prevalent in the area of what the discovery of logarithms meant in subsequent developments in mathematical thinking. Ease of calculations is of course mentioned, but subsequent contributions to the development of calculus and the considerable scientific importance of the discovery of the number e, which occurred as a direct result of the discovery of logarithms, most often receives scant attention. In fact, however, the history of logarithms stretches from Babylon to Newton, and a considerable number of interesting problems and ingenious solutions can be encountered along the way. The purpose of this thesis has been to explore in considerable detail the development of the idea of logarithms and of logarithms themselves from the first logarithmic-like tables of the Babylonians through the work of Isaac Newton. No known study of the history of logarithms in such detail either exists or is currently available. This work attempts to fill that void. Additionally, the history of logarithms runs parallel to, is influenced by, or is in turn influential in a number of other significant developments in the history of mathematics and science, not the least of which is the development of trigonometry. These parallels and influences are an interesting source of study in their own right. AN EARLY HISTORY OF LOGARITHMS A Thesis Presented to the Division of Mathematics and Computer Science EMPORIA STATE UNIVERSITY In Partial Fulfillment of the Requirements for the Degree Master of Science by David L. Baughman July 1992 Y:~~ Approved for the Division of Mathematics/Computer Science /: ~GUUt ln~_LLQUv~ _ ~ the Graduate Council CONTENTS CHAPTER I: INTRODUCTION 1 1.1 Introduction 1 1.2 Statement of the Problem 2 1.3 Importance of the Study 2 1.4 Sources of Information 2 1.5 Organization 2 CHAPTER II: THE IMPETUS 4 2.1 Introduction 4 2.2 The Babylonians 5 2.3 Archimedes 10 2.4 Summary and Conclusions 13 CHAPTER III: WORKING UP TO IT 15 3.1 Introduction 15 3.2 Early Significant Work in 16 Trigonometry 3.3 The Middle Ages 16 3.4 The Renaissance 17 3.5 Prosthaphaeresis and Francois 19 Viete 3.6 Summary and Conclusions 21 CHAPTER IV: FULFILLMENT .23 4.1 Introduction 23 4.2 Napier, the Man 27 4.3 The Logarithms of John Napier 29 4.4 Methods of Discovery 32 4.5 Summary and Conclusions 36 CHAPTER V: AFTERMATH 38 5.1 Introduction 38 5.2 Jobst Burgi 38 5.3 Common Logarithms 41 5.4 Natural Logarithms 49 5.5 Summary and Conclusions 52 CHAPTER VI: OVERVIEW. .53 6.1 Introduction. 53 6.2 Summary. 54 6.3 Need for Further Study. 57 6.4 Conclusions 58 ENDNOTES .60 BIBLIOGRAPHY 67 APPENDICES 70 A A Computer Simulation of Prostha­ phaeresis . 71 B A Program for Constructing Napier Logarithms . .74 C The Numerical Approximation of f(x) = l/x Using Simpson's Rule. .78 LIST OF TABLES Number Page 1. Sexagesimal Numbers and Their 7 Inverses. 2. Samples of Logarithmic Tables 40 of Jobst Burgi. 3. Logarithms of 'Whole Sines'. 43 4. Logarithms by Geometric Mean. 46 5. Partial List of Logarithms 48 by Henry Briggs. LIST OF FIGURES Number Page 1. Napier Right-Angle Triangle. 32 1 CHAPTER I INTRODUCTION 1.1 Introduction. A large number of reference works on the history of mathematics seem to assume either that an in-depth discussion of the ideas which led up to the nearly simultaneous discovery of logarithms by Napier and Burgi would be too arcane to be of interest, or that not much did lead up to the achievement of these two discoverers and that the idea of logarithms simply spontaneously created itself in them, as if of nothing. Much the same attitude seems prevalent in the area of what the discovery of logarithms meant in subsequent developments in mathematical thinking. Ease of calculations is of course mentioned, but subsequent contributions to the development of calculus and the considerable scientific importance of the discovery of the number e, which occurred as a direct result of the discovery of logarithms, most often receives scant attention. In fact, however, the history of logarithms stretches from Babylon to Newton, and a considerable number of interesting problems and ingenious solutions can be encountered along the way. 2 1.2 Statement of the Problem. The purpose of this thesis is to explore in considerable detail the development of the idea of logarithms and of logarithms themselves from the first logarithmic-like tables of the Babylonians through the work of Isaac Newton. 1.3 Importance of the Study. No known study of the history of logarithms in such detail either exists or is currently available. This work attempts to fill that void. Additionally, the history of logarithms runs parallel to, is influenced by, or is in turn influential in a number of other significant developments in the history of mathematics and science, not the least of which is the development of trigonometry. These parallels and influences are an interesting source of study in their own right. 1.4 Sources of Information. Libraries of the Kansas Regents' universities have been used to complete the study. 1.5 Organization. The thesis has been divided into six major parts. Chapter I introduces the subject. Chapter II explores early Babylonian work with tables which resemble logarithmic tables as we know them today, and additionally explores Archimedes' work on extremely large numbers and his discovery, in consequence, of logarithmic-like laws. Chapter III discusses parallel developments in trigonometry, especially during the 16th Century and most especially the work of Viete. Such 3 work influenced the staff of Tycho Brahe and their work was in turn known to Napier. Chapter IV examines the period in which logarithms were actually discovered: their nearly simultaneous discovery by John Napier and Jobst Burgi, and the introduction of the concepts of base number and exponent. Chapter V discusses the subsequent discovery of the number e, and the later work of Newton which expanded the use and mathematical interest in logarithms. Finally, Chapter VI summarizes and draws conclusions based on the study. 4 CHAPTER II THE IMPETUS The Babylonians and Archimedes 2.1 Introduction. If we might say that the problem of the computation of very large and very small numbers, often over and over again, was not truly solved until the inventions of fully-functional and efficient calculators and computers, we may also say that a major step in the improvement of such calculations was made in the Seventeenth Century with the invention of logarithms. For what made such complicated calculations far simpler as was then becoming increasingly necessary was the new-found ability to reduce multiplication to the vastly simpler act of addition. Division thus became a process of subtraction, finding powers simple multiplication, and that of finding roots equally simple diVision. But these processes, though infrequent in antiquity, were neither especially new nor unknown to those who may rightly claim for themselves the invention of logarithms. They are found upon Babylonian tablets dating as far back as 2400 B. C. I and in Archimedes' The Sand Reckoner we find the rule students of logarithms have learned for nearly four centuries, a-an = a-·n. 2 5 The proper beginning for a study of the history of logarithms, then, takes us into the past more than 4000 years. 2.2 The Babylonians. What is commonly considered or termed 'Babylonia' was a series of Mesopotamian civilizations which existed between the Tigris and Euphrates rivers from about 2000 to 600 B.C., which neither at the beginning nor at the end were entirely dominated by Babylonia itself. Indeed, Babylonia's fall to Cyrus of Persia in 538 B.C. brought an end to the Babylonian empire, but what historians continue to call Babylonian mathematics continued until nearly the birth of Christ,3 and going all the way back to the fourth millennium before the present era we find a remarkable period of cultural development and a high order of civilization, which included the use of writing, the wheel and metals.· Throughout this entire period we find a people who were highly skilled computationally, makers of sophisticated mathematical tables, and sophisticated algebraists. 6 Babylonian mathematicians skillfully developed algorithms for mathematical procedures of considerable complexity, one of which was for "a square­ root process often ascribed to later men."6 This process was, it is claimed, highly efficient, but Babylonians seemed to prefer what we may consider the much more modern approach by resorting to tables, which were seemingly in abundance. 7 Of some 300 6 mathematical tablets recovered to date by archeologists fUlly 200 of these are table tablets covering such topics as multiplication, reciprocals, squares and cubes, and exponentials. In combination with interpolation these latter tables seem to have been used in problems relating to compound interest. s In the opinion of mathematics historian Carl B. Boyer, "There is a clear instance [in the Babylonian texts] of the use of interpolation within exponential tables wherein the scribe also uses the compound interest formula a = P<l+r)n.