


e: The Story of a Number 
Chapter 1. John Napier, 1614Seeing 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)^{1} 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.^{2} 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.^{3} 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."^{4} He also predicted that the Day of Judgment would fall between 1688 and 1700. The book was translated into several languages and ran through twentyone editions (ten of which appeared during his lifetime), making Napier confident that his name in historyor what little of it might be leftwas 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 modem military technology.^{5} 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 halfdead 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.^{6} 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 bestselling 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^{2}, q^{3}, and so on (note that the nth term is q^{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 (14871567), in his book Arithmetica integra (1544), formulated this relation as follows: if we multiply any two terms of the progression 1, q, q^{2}, . . . , the result would be the same as if we had added the corresponding exponents.^{7} For example, q^{2} · q^{3} = (q · q) · (q · q · q) = q · q · q · q · q = q^{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^{5}/q^{3} = (q · q · q · q · q)/(q · q · q) = q · q = q^{2} = q^{53}. We thus have the simple rules q^{m} · q^{n} = q^{m+n} and q^{m}/q^{n} = q^{mn}. A problem arises, however, if the exponent of the denominator is greater than that of the numerator, as in q^{3}/q^{5}; our rule would give us q^{35} = q^{2}, an expression that we have not defined. To get around this difficulty, we simply define q^{n} to be 1/q^{n}, so that q^{35} = q^{2} = 1/q^{2}, in agreement with the result obtained by dividing q^{3} by q^{5} directly.^{8} (Note that in order to be consistent with the rule q^{m}/q^{n} = q^{mn} when m = n, we must also define q^{0} = 1.) With these definitions in mind, we can now extend a geometric progression indefinitely in both directions: . . . . q^{3}, q^{2}, q^{1}, q^{0} = 1, q, q^{2}, q^{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 nth power (that is, multiplying it by itself n times) would be equivalent to adding the exponent n times to itselfthat is, to multiplying it by nand finding the nth root of a number would be equivalent to n repeated subtractionsthat 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^{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^{5} = (2^{2})^{5} = 2^{10} = 1,024. TABLE 1.1 Powers of 2
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 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 fractionsthe extension of our decimal numeration system to numbers less than 1had only recently been introduced to Europe,^{10} 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 107 parts. Hence, if we subtract from the full unit its 10^{7}th part, we get the number closest to 1 in this system, namely 1  10^{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 (15941614) to complete the job. His initial table contained just 101 entries, starting with 10^{7} = 10,000,000 and followed by 10^{7}(1  10^{7}) = 9,999,999, then 10^{7}(1  10^{7})^{2} = 9,999,998, and so on up to 10^{7}(1  10^{7})^{100} = 9,999,900 (ignoring the fractional part .0004950), each term being obtained by subtracting from the preceding term its 10^{7}th part. He then repeated the process all over again, starting once more with 10^{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^{5}. This second table contained fiftyone entries, the last being 10^{7}(1  10^{5})^{50} or very nearly 9,995,001. A third table with twentyone entries followed, using the ratio 9,995,001: 10,000,000; the last entry in this table was 10^{7} x .9995^{20}, or approximately 9,900,473. Finally, from each entry in this last table Napier created sixtyeight additional entries, using the ratio 9,900,473 : 10,000,000, or very nearly .99; the last entry then turned out to be 9,900,473 x .99^{68}, or very nearly 4,998,609roughly half the original number. Today, of course, such a task would be delegated to a computer; even with a handheld calculator the job could done in a few hours. But Napier had to do all his calculations with only paper and pen. One can therefore understand his concern to minimize the use of decimal fractions. In his own words: "In forming this progression [the entries of the second table], since the proportion between 10000000.00000, the first of the Second table, and 9995001.222927, the last of the same, is troublesome; therefore compute the twentyone numbers in the easy proportion of 10000 to 9995, which is sufficiently near to it; the last of these, if you have not erred, will be 9900473.57808."^{11} Having completed this monumental task, it remained for Napier to christen his creation. At first he called the exponent of each power its "artificial number" but later decided on the term logarithm, the word meaning "ratio number." In modern notation, this amounts to saying that if (in his first table) N = 10^{7}(1  10^{7})^{L}, then the exponent L is the (Napierian) logarithm of N. Napier's definition of logarithms differs in several respects from the modern definition (introduced in 1728 by Leonhard Euler): if N = b^{L}, where b is a fixed positive number other than 1, then L is the logarithm (to the base b) of N. Thus in Napier's system L = 0 corresponds to N = 10^{7} (that is, Nap log 10^{7} = 0), whereas in the modern system L = 0 corresponds to N = 1 (that is, log_{b}l = 0). Even more important, the basic rules of operation with logarithmsfor example, that the logarithm of a product equals the sum of the individual logarithmsdo not hold for Napier's definition. And lastly, because 1  10^{7} is less than 1, Napier's logarithms decrease with increasing numbers, whereas our common (bazse 10) logarithms increase. These differences are relatively minor, however, and are merely a result of Napier's insistence that the unit should be equal to 10^{7} subunits. Had he not been so concerned about decimal fractions, his definition might have been simpler and closer to the modern one.^{12} In hindsight, of course, this concern was an unnecessary detour. But in making it, Napier unknowingly came within a hair's breadth of discovering a number that, a century later, would be recognized as the universal base of logarithms and that would play a role in mathematics second only to the number pi. This number, e, is the limit of (1 + 1/n)^{n} as n tends to infinity.^{13}
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