Enhanced descriptions from Syndetics:

The interest earned on a bank account, the arrangement of seeds in a sunflower, and the shape of the Gateway Arch in St. Louis are all intimately connected with the mysterious number e . In this informal and engaging history, Eli Maor portrays the curious characters and the elegant mathematics that lie behind the number. Designed for a reader with only a modest mathematical background, this biography brings out the central importance of e to mathematics and illuminates a golden era in the age of science.

Table of contents provided by Syndetics

• Preface
• 1 John Napier, 1614 (p. 3)
• 2 Recognition (p. 11)
• 3 Financial Matters (p. 23)
• 4 To the Limit, If It Exists (p. 28)
• 5 Forefathers of the Calculus (p. 40)
• 6 Prelude to Breakthrough (p. 49)
• 7 Squaring the Hyperbola (p. 58)
• 8 The Birth of a New Science (p. 70)
• 9 The Great Controversy (p. 83)
• 10 e[superscript x]: The Function That Equals its Own Derivative (p. 98)
• 11 e[superscript theta]: Spira Mirabilis (p. 114)
• 12 (e[superscript x] + e[superscript -x])/2: The Hanging Chain (p. 140)
• 13 e[superscript ix]: "The Most Famous of All Formulas" (p. 153)
• 14 e[superscript x + iy]: The Imaginary Becomes Real (p. 164)
• 15 But What Kind of Number Is It? (p. 183)
• App. 1 Some Additional Remarks on Napier's Logarithms (p. 195)
• App. 2 The Existence of lim (1 + 1/n)[superscript n] as n [approaches] [infinity] (p. 197)
• App. 3 A Heuristic Derivation of the Fundamental Theorem of Calculus (p. 200)
• App. 4 The Inverse Relation between lim (b[superscript h] - 1)/h = 1 and lim (1 + h)[superscript 1/h] = b as h [approaches] 0 (p. 202)
• App. 5 An Alternative Definition of the Logarithmic Function (p. 203)
• App. 6 Two Properties of the Logarithmic Spiral (p. 205)
• App. 7 Interpretation of the Parameter [phi] in the Hyperbolic Functions (p. 208)
• App. 8 e to One Hundred Decimal Places (p. 211)
• Bibliography (p. 213)
• Index (p. 217)

Excerpt provided by Syndetics

John Napier, 1614 Seeing there is nothing that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers.... I began therefore to consider in my mind by what certain and ready art I might remove those hindrances. -John Napier, Mirifici logarithmorum canonis descriptio (1614) Rarely in the history of science has an abstract mathematical idea been received more enthusiastically by the entire scientific community than the invention of logarithms. And one can hardly imagine a less likely person to have made that invention. His name was John Napier. The son of Sir Archibald Napier and his first wife, Janet Bothwell, John was born in 1550 (the exact date is unknown) at his family's estate, Merchiston Castle, near Edinburgh, Scotland. Details of his early life are sketchy. At the age of thirteen he was sent to the University of St. Andrews, where he studied religion. After a sojourn abroad he returned to his homeland in 1571 and married Elizabeth Stirling, with whom he had two children. Following his wife's death in 1579, he married Agnes Chisholm, and they had ten more children. The second son from this marriage, Robert, would later be his father's literary executor. After the death of Sir Archibald in 1608, John returned to Merchiston, where, as the eighth laird of the castle, he spent the rest of his life. Napier's early pursuits hardly hinted at future mathematical creativity. His main interests were in religion, or rather in religious activism. A fervent Protestant and staunch opponent of the papacy, he published his views in A Plaine Discovery of the whole Revelation of Saint John (1593), a book in which he bitterly attacked the Catholic church, claiming that the pope was the Antichrist and urging the Scottish king James VI (later to become King James I of England) to purge his house and court of all "Papists, Atheists, and Newtrals." He also predicted that the Day of Judgment would fall between 1688 and 1700. The book was translated into several languages and ran through twenty-one editions (ten of which appeared during his lifetime), making Napier confident that his name in history-or what little of it might be left-was secured. Napier's interests, however, were not confined to religion. As a landowner concerned to improve his crops and cattle, he experimented with various manures and salts to fertilize the soil. In 1579 he invented a hydraulic screw for controlling the water level in coal pits. He also showed a keen interest in military affairs, no doubt being caught up in the general fear that King Philip II of Spain was about to invade England. He devised plans for building huge mirrors that could set enemy ships ablaze, reminiscent of Archimedes' plans for the defense of Syracuse eighteen hundred years earlier. He envisioned an artillery piece that could "clear a field of four miles circumference of all living creatures exceeding a foot of height," a chariot with "a moving mouth of mettle" that would "scatter destruction on all sides," and even a device for "sayling under water, with divers and other stratagems for harming of the enemyes"-all forerunners of modern military technology. It is not known whether any of these machines was actually built. As often happens with men of such diverse interests, Napier became the subject of many stories. He seems to have been a quarrelsome type, often becoming involved in disputes with his neighbors and tenants. According to one story, Napier became irritated by a neighbor's pigeons, which descended on his property and ate his grain. Warned by Napier that if he would not stop the pigeons they would be caught, the neighbor contemptuously ignored the advice, saying that Napier was free to catch the pigeons if he wanted. The next day the neighbor found his pigeons lying half-dead on Napier's lawn. Napier had simply soaked his grain with a strong spirit so that the birds became drunk and could barely move. According to another story, Napier believed that one of his servants was stealing some of his belongings. He announced that his black rooster would identify the transgressor. The servants were ordered into a dark room, where each was asked to pat the rooster on its back. Unknown to the servants, Napier had coated the bird with a layer of lampblack. On leaving the room, each servant was asked to show his hands; the guilty servant, fearing to touch the rooster, turned out to have clean hands, thus betraying his guilt. All these activities, including Napier's fervent religious campaigns, have long since been forgotten. If Napier's name is secure in history, it is not because of his best-selling book or his mechanical ingenuity but because of an abstract mathematical idea that took him twenty years to develop: logarithms. * * * The sixteenth and early seventeenth centuries saw an enormous expansion of scientific knowledge in every field. Geography, physics, and astronomy, freed at last from ancient dogmas, rapidly changed man's perception of the universe. Copernicus's heliocentric system, after struggling for nearly a century against the dictums of the Church, finally began to find acceptance. Magellan's circumnavigation of the globe in 1521 heralded a new era of marine exploration that left hardly a corner of the world unvisited. In 1569 Gerhard Mercator published his celebrated new world map, an event that had a decisive impact on the art of navigation. In Italy Galileo Galilei was laying the foundations of the science of mechanics, and in Germany Johannes Kepler formulated his three laws of planetary motion, freeing astronomy once and for all from the geocentric universe of the Greeks. These developments involved an ever increasing amount of numerical data, forcing scientists to spend much of their time doing tedious numerical computations. The times called for an invention that would free scientists once and for all from this burden. Napier took up the challenge. We have no account of how Napier first stumbled upon the idea that would ultimately result in his invention. He was well versed in trigonometry and no doubt was familiar with the formula sin A · sin B = 1/2[cos( A - B ) - cos( A + B )] This formula, and similar ones for cos A · cos B and sin A · cos B , were known as the prosthaphaeretic rules , from the Greek word meaning "addition and subtraction." Their importance lay in the fact that the product of two trigonometric expressions such as sin A · sin B could be computed by finding the sum or difference of other trigonometric expressions, in this case cos( A - B ) and cos( A + B ). Since it is easier to add and subtract than to multiply and divide, these formulas provide a primitive system of reduction from one arithmetic operation to another, simpler one. It was probably this idea that put Napier on the right track. A second, more straightforward idea involved the terms of a geometric progression , a sequence of numbers with a fixed ratio between successive terms. For example, the sequence 1, 2, 4, 8, 16, ... is a geometric progression with the common ratio 2. If we denote the common ratio by q , then, starting with 1, the terms of the progression are 1, q, [q.sup.2], [q.sup.3] , and so on (note that the nth term is [q.sup.n-1] ). Long before Napier's time, it had been noticed that there exists a simple relation between the terms of a geometric progression and the corresponding exponents , or indices, of the common ratio. The German mathematician Michael Stifel (1487-1567), in his book Arithmetica integra (1544), formulated this relation as follows: if we multiply any two terms of the progression 1, q, [q.sup.2] , ..., the result would be the same as if we had added the corresponding exponents. For example, [q.sup.2] · [q.sup.3] = (q · q) · (q · q · q) = q · q · q · q · q = [q.sup.5] , a result that could have been obtained by adding the exponents 2 and 3. Similarly, dividing one term of a geometric progression by another term is equivalent to subtracting their exponents: [q.sup.5]/[q.sup.3] = (q · q · q · q · q)/(q · q · q) = q · q = [q.sup.2] = [q.sup.5-3] . We thus have the simple rules [q.sup.m] · [q.sup.n] = [q.sup.m+n] and [q.sup.m]/[q.sup.n] = [q.sup.m-n] . A problem arises, however, if the exponent of the denominator is greater than that of the numerator, as in [q.sup.3]/[q.sup.5] ; our rule would give us [q.sup.3-5] = [q.sup.-2] an expression that we have not defined. To get around this difficulty, we simply define [q.sup.-n] to be 1/ [q.sup.n] , so that [q.sup.3-5] = [q.sup.-2] = 1/[q.sup.2] , in agreement with the result obtained by dividing [q.sup.3] by [q.sup.5] directly. (Note that in order to be consistent with the rule [q.sup.m]/[q.sup.n] = [q.sup.m-n] when m = n , we must also define [q.sup.0] = 1 .) With these definitions in mind, we can now extend a geometric progression indefinitely in both directions: ..., [q.sup.-3], [q.sup.-2], [q.sup.-1], [q.sup.0] = 1, q, [q.sup.2], [q.sup.3] ,..... We see that each term is a power of the common ratio q, and that the exponents ..., -3, -2, -1, 0, 1, 2, 3, ... form an arithmetic progression (in an arithmetic progression the difference between successive terms is constant, in this case 1). This relation is the key idea behind logarithms; but whereas Stifel had in mind only integral values of the exponent, Napier's idea was to extend it to a continuous range of values. His line of thought was this: If we could write any positive number as a power of some given, fixed number (later to be called a base), then multiplication and division of numbers would be equivalent to addition and subtraction of their exponents . Furthermore, raising a number to the n th power (that is, multiplying it by itself n times) would be equivalent to adding the exponent n times to itself-that is, to multiplying it by n -and finding the n th root of a number would be equivalent to n repeated subtractions-that is, to division by n. In short, each arithmetic operation would be reduced to the one below it in the hierarchy of operations, thereby greatly reducing the drudgery of numerical computations. Let us illustrate how this idea works by choosing as our base the number 2. Table 1.1 shows the successive powers of 2, beginning with n = -3 and ending with n = 12. Suppose we wish to multiply 32 by 128. We look in the table for the exponents corresponding to 32 and 128 and find them to be 5 and 7, respectively. Adding these exponents gives us 12. We now reverse the process, looking for the number whose corresponding exponent is 12; this number is 4,096, the desired answer. As a second example, supppose we want to find [4.sup.5]. We find the exponent corresponding to 4, namely 2, and this time multiply it by 5 to get 10. We then look for the number whose exponent is 10 and find it to be 1,024. And, indeed, [4.sup.5] = ([2.sup.2])[sup.5] = [2.sup.10] = 1,024. Of course, such an elaborate scheme is unnecessary for computing strictly with integers; the method would be of practical use only if it could be used with any numbers, integers, or fractions. But for this to happen we must first fill in the large gaps between the entries of our table. We can do this in one of two ways: by using fractional exponents, or by choosing for a base a number small enough so that its powers will grow reasonably slowly. Fractional exponents, defined by [a.sup.m/n] = [sup.n][square root of [a.sup.m] (for example, [2.sup.5/3] = [sup.3][square root of [2.sup.5]] = [sup.3] [square root of 32] [approximately equals] 3.17480), were not yet fully known in Napier's time, so he had no choice but to follow the second option. But how small a base? Clearly if the base is too small its powers will grow too slowly, again making the system of little practical use. It seems that a number close to 1, but not too close, would be a reasonable compromise. After years of struggling with this problem, Napier decided on .9999999, or 1 - [10.sup.-7]. But why this particular choice? The answer seems to lie in Napier's concern to minimize the use of decimal fractions. Fractions in general, of course, had been used for thousands of years before Napier's time, but they were almost always written as common fractions, that is, as ratios of integers. Decimal fractions-the extension of our decimal numeration system to numbers less than 1-had only recently been introduced to Europe, and the public still did not feel comfortable with them. To minimize their use, Napier did essentially what we do today when dividing a dollar into one hundred cents or a kilometer into one thousand meters: he divided the unit into a large number of subunits, regarding each as a new unit. Since his main goal was to reduce the enormous labor involved in trigonometric calculations, he followed the practice then used in trigonometry of dividing the radius of a unit circle into 10,000,000 or [10.sup.7] parts. Hence, if we subtract from the full unit its [10.sup.7]th part, we get the number closest to 1 in this system, namely 1 - [10.sup.-7] or .9999999. This, then, was the common ratio ("proportion" in his words) that Napier used in constructing his table. And now he set himself to the task of finding, by tedious repeated subtraction, the successive terms of his progression. This surely must have been one of the most uninspiring tasks to face a scientist, but Napier carried it through, spending twenty years of his life (1594-1614) to complete the job. His initial table contained just 101 entries, starting with [10.sup.7] = 10,000,000 and followed by [10.sup.7] (1 - [10.sup.-7]) = 9,999,999, then [10.sup.7][(1 - [10.sup.-7]).sup.2] = 9,999,998, and so on up to [10.sup.7][(1 -[10.sup.-7]).sup.100] = 9,999,900 (ignoring the fractional part .0004950), each term being obtained by subtracting from the preceding term its 107th part. He then repeated the process all over again, starting once more with [10.sup.7], but this time taking as his proportion the ratio of the last number to the first in the original table, that is, 9,999,900 : 10,000,000 = .99999 or 1 - [10.sup.-5]. This second table contained fiftyone entries, the last being [10.sup.7][(1 - [10.sup.-5]).sup.50] or very nearly 9,995,001. A third table with twenty-one entries followed, using the ratio 9,995,001 : 10,000,000; the last entry in this table was [10.sup.7] x [.9995.sup.20], or approximately 9,900,473. Continues... Excerpted from e: The Story of a Number by Eli Maor Copyright © 1994 by Princeton University Press Excerpted by permission. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

Reviews provided by Syndetics

Library Journal Review

Everyone whose mathematical education has gone beyond elementary school is familiar with the number known as pi. Far fewer have been introduced to e, a number that is of equal importance in theoretical mathematics. Maor (mathematics, Northeastern Illinois Univ.) tries to fill this gap with this excellent book. He traces the history of mathematics from the 16th century to the present through the intriguing properties of this number. Maor says that his book is aimed at the reader with a ``modest'' mathematical background. Be warned that his definition of modest may not be yours. The text introduces and discusses logarithms, limits, calculus, differential equations, and even the theory of functions of complex variables. Not easy stuff! Nevertheless, the writing is clear and the material fascinating. Highly recommended.-- Harold D. Shane, Baruch Coll., CUNY (c) Copyright 2010. Library Journals LLC, a wholly owned subsidiary of Media Source, Inc. No redistribution permitted.

CHOICE Review

A disarming study of 17th-, 18th-, and 19th-century analysis, seen through the lens of logarithms and exponentials. Maor attempts to give the irrational number e its rightful standing alongside pi as a fundamental constant in science and nature; he succeeds very well. As a historical work, the book is quite carefully done but also clearly fanciful in places, such as the fantasy transcript of a meeting between J.S. Bach and Johann Bernoulli. The mathematical content is adroitly presented, full of intriguing tidbits and applications to music and art. The book would make excellent supplementary reading for motivated college and even high school mathematics students, and for the general reader with some mathematical background. (Although Maor explains the notions of the calculus as needed, it helps to have prior understanding.) Maor writes so that both mathematical newcomers and long-time professionals alike can thoroughly enjoy his book, learn something new, and witness the ubiquity of mathematical ideas in Western culture. Highly recommended. General; undergraduate; faculty. S. J. Colley; Oberlin College

Booklist Review

The discovery of e (the base for natural logarithms) did much to confirm the faith--central to modern science--that the empirical world is encoded in mathematics. Indeed, today scientists and engineers rely heavily on e and its derivatives when solving numerous real-world problems. Yet many who routinely employ this powerful number know little of its history. Maor recounts the rich drama surrounding the number that emerged at the center of the investigations of some of the most brilliant thinkers of all time--Fermat and Descartes, Newton and Leibniz, Laplace and the Bernoullis, Euler and Gauss. Though the exposition inevitably requires many formulas and some abbreviated derivations, the author tries to smooth the way for general readers who do not know calculus or analytical geometry, while still conveying some sense of the imaginative daring of the pioneers who opened up these fields of mathematics. Through appendixes and footnotes, mathematicians can find their way to more rigorous and exhaustive treatment of the subject; nonspecialists can easily wend their way around the more theoretical questions while still enlarging their understanding of intellectual and cultural history. ~--Bryce Christensen

Kirkus Book Review

This book that dares to use ``e'' in its title is not for mathematicians only. Adults with open minds and students just beginning to make their way through algebra and trigonometry will find much that is easily digestible and even palatable in this lively presentation of the mathematical revolution that took place between the age of Newton and the late 19th century. Maor (Math/Northeastern Illinois) begins with logarithms, the work of John Napier, a well-born Scot and fervent anti-Papist inventor who spent 20 years working out the first log tables. The author then introduces ``e'' in a thoroughly practical fashion: The number (2.718...) is the limit of a special case of the formula for compound interest. With that as a teaser, Maor goes on to demonstrate how e crops up in marvelous ways in calculus and in beautiful graphs that link it with other memorable numbers. Indeed, one of the most celebrated equations in mathematics states that e raised to the pi times i power = -1 (i= the square root of -1). This is all nicely wrought, with diagrams and informal developments of equations in the text. (Appendices supply formal treatments.) But to the bare bones of the math Maor adds descriptions of the major innovators, their quirks, and their quarrels, ranging from Newton and Leibniz fighting over the invention of calculus to the not-so-petty jealousies among the Bernoullis, from the brilliance of Leonhard Euler (who first named e) to the eccentric Georg Cantor, who established orders of infinity and demonstrated that e and pi were only two of an infinite collection of transcendental numbers. It's worth reading the book just to find out exactly what that last phrase means. Pithy, punchy, and surprisingly accessible.

Author notes provided by Syndetics

Eli Maor is a teacher of the history of mathematics who has successfully popularized his subject with the general public through a series of informative and entertaining books.

In "E: The Story of a Number," Maor uses anecdotes, excursions and essays to illustrate that number's importance to mathematics. "Trigonometric Delights" brings trigonometry to life by blending history, biography, scientific curiosities and mathematics to achieve the goal of showing how trigonometry has contributed to both science and social development. "To Infinity and Beyond: A Cultural History of the Infinite" explores the idea of infinity in mathematics and art through the use of the illustrations of the Dutch artist M.C. Escher.

Eli Maor's readable books have made the world of numbers accessible even to those with little or no background in mathematics.

(Bowker Author Biography)