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....

Napier published his invention in 1614 in a Latin treatise entitledMirifici logarithmorum canonis descriptio(Description of the wonderful canon of logarithms). A later work,Mirifici logarithmorum canonis constructio(Construction of the wonderful canon of logarithms), was published posthumously by his son Robert in 1619. Rarely in the history of science has a new idea been received more enthusiastically. Universal praise was bestowed upon its inventor, and his invention was quickly adopted by scientists all across Europe and even in faraway China. One of the first to avail himself of logarithms was the astronomer Johannes Kepler, who used them with...

For many of us—at least those who completed our college education after 1980—logarithms are a theoretical subject, taught in an introductory algebra course as part of the function concept. But until the late 1970s logarithms were still widely used as a computational device, virtually unchanged from Briggs’s common logarithms of 1624. The advent of the hand-held calculator has made their use obsolete.

Let us say it is the year 1970 and we are asked to compute the expression

$x = \sqrt[3]{{(493.8\cdot{{23.67}^2}/5.104)}}$.

For this task we need a table of four-place common logarithms (which can still be found at the back...

From time immemorial money matters have been at the center of human concerns. No other aspect of life has a more mundane character than the urge to acquire wealth and achieve financial security. So it must have been with some surprise that an anonymous mathematician—or perhaps a merchant or moneylender—in the early seventeenth century noticed a curious connection between the way money grows and the behavior of a certain mathematical expression at infinity.

Central to any consideration of money is the concept ofinterest,or money paid on a loan. The practice of charging a fee for borrowing...

At first thought, the peculiar behavior of the expression${(1 + 1/n)^n}$for large values of$n$must seem puzzling indeed. Suppose we consider only the expression inside the parentheses,$1 + 1/n$As$n$increases,$1/n$gets closer and closer to 0 and so$1 + 1/n$gets closer and closer to 1, although it will always be greater than 1. Thus we might be tempted to conclude that for “really large”$n$(whatever “really large” means),$1 + 1/n$to every purpose and extent, may be replaced by 1. Now 1 raised to any power is always equal to 1, so it seems that${(1 + 1/n)^n}$for...

${e^{ - e}} = 0.065988036$. . .

Leonhard Euler proved that the expression${x^{{x^{{x^{{x^x}}}}}}}$, as the number of exponents grows to infinity, tends to a limit if$x$is between${e^{ - e}}( = 1/{e^e})$and${e^{1/e}}{.^1}$.

${e^{ - \pi /2}} = 0.207879576...$. . .

As Euler showed in 1746, the expression${i^i}$(where$i = \sqrt { - 1} $) has infinitely many values, all of them real:${i^l} = {e^{ - (\pi /2 + 2k\pi )}}$, where\[k = 0, \pm 1, \pm 2,\]. . . . The principal value of these (the value for$k = 0$) is${e^{ - (\pi /2)}}$.

$1/e = 0.367879441...$. . .

The limit of${(1 - 1/n)^n}$as$n \to \infty $. This number is used to measure the rate of decay of the exponential function$y = {e^{ - at}}$. When$t = 1/a$we have$y = {e^{ - 1}} = 1/e$...

Great inventions generally fall into one of two categories: some are the product of a single person’s creative mind, descending on the world suddenly like a bolt out of the blue; others—by far the larger group—are the end product of a long evolution of ideas that have fermented in many minds over decades, if not centuries. The invention of logarithms belongs to the first group, that of the calculus to the second.

It is usually said that the calculus was invented by Isaac Newton (1642-1727) and Gottfried Wilhelm Leibniz (1646-1716) during the decade 1665-1675, but this is not...

About eighteen hundred years after Archimedes, a French mathematician by the name of François Viète (or Vieta, 1540-1603), in the course of his work in trigonometry, found a remarkable formula involving the number π:

$\frac{2}{\pi } = \frac{{\sqrt 2 }}{2}.\frac{{\sqrt {2 + \sqrt 2 } }}{2}.\frac{{\sqrt {2 + \sqrt {2 + \sqrt 2 } } }}{2}$.....

The discovery of thisinfinite productin 1593 marked a milestone in the history of mathematics: it was the first time an infinite process was explicitly written as a mathematical formula. Indeed, the most remarkable feature of Viète’s formula, apart from its elegant form, is the three dots at the end, telling us to go on and on . . .ad infinitum.It shows...

As an example of the method of indivisibles, let us find the area under the parabola$y = {x^2}$from$x = 0$to$x = a$. We think of the required region as made up of a large number$n$of vertical line segments (“indivisibles”) whose heights$y$vary with$x$according to the equation$y = {x^2}$(fig. 13). If these line segments are separated by a fixed horizontal distance$d$, their heights are${d^2},{\left( {2d} \right)^2},{\left( {3d} \right)^2},...,{\left( {nd} \right)^2}.$The required area is thus approximated by the sum

$\left[ {{d^2} + {{\left( {2d} \right)}^2} + {{\left( {3d} \right)}^2} + ... + {{\left( {nd} \right)}^2}} \right] \cdot d = \left[ {{1^2} + {2^2} + {3^2} + ... + {n^2}} \right] \cdot {d^3}$.

Using the well-known summation formula for the squares of the integers, this expression is equal to$[n(n\; + \;1)(2n\; + \;1){\kern 1pt} /{\kern 1pt} 6]\; \cdot \;{d^3}$, or after a slight...

The problem of finding the area of a closed planar shape is known asquadrature, or squaring. The word refers to the very nature of the problem: to express the area in terms of units of area, that is, squares. To the Greeks this meant that the given shape had to be transformed into an equivalent one whose area could be found from fundamental principles. To give a simple example, suppose we want to find the area of a rectangle of sides$a$and$b$. If this rectangle is to have the same area as a square of side$x$,...

Isaac Newton was born in Woolsthorpe in Lincolnshire, England, on Christmas Day (by the Julian calendar) 1642, the year of Galileo’s death. There is a symbolism in this coincidence, for half a century earlier Galileo had laid the foundations of mechanics on which Newton would erect his grand mathematical description of the universe. Never has the biblical verse, “One generation passeth away, and another generation cometh: but the earth abideth for ever” (Ecclesiastes 1:4), been more prophetically fulfilled.¹

Newton's early childhood was marked by family misfortunes. His father died a few months before Isaac was born; his mother soon remarried,...

Newton and Leibniz will always be mentioned together as the coinventors of the calculus. In character, however, the two men could hardly be less alike. Baron Gottfried Wilhelm von Leibniz (or Leibnitz) was born in Leipzig on 1 July 1646. The son of a philosophy professor, the young Leibniz soon showed great intellectual curiosity. His interests, in addition to mathematics, covered a wide range of topics, among them languages, literature, law, and above all, philosophy. (Newton’s interests outside mathematics and physics were theology and alchemy, subjects on which he spent at least as much time as on his more familiar...

A working knowledge of a mathematical topic requires a good system of notation. When Newton invented his “method of fluxions,” he placed a dot over the letter representing the quantity whose fluxion (derivative) he sought. This dot notation—Newton called it the “pricked letter” notation—is cumbersome. To find the derivative of$y = {x^2}$, one must first obtain a relation between the fluxions of$x$and$y$with respect to time (Newton thought of each variable as “flowing” uniformly with time, hence the termfluxion), in this case$\dot y = 2x\dot x$(see p. 75). The derivative, or rate of change, of$y$with...

When Newton and Leibniz developed their new calculus, they applied it primarily toalgebraic curves,curves whose equations are polynomials or ratios of polynomials. (Apolynomialis an expression of the form${a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ... + {a_1}x + {a_0};$the constants${a_1}$are thecoefficients,and$n$, thedegreeof the polynomial, is a non-negative integer. For example,$5{x^3} + {x^2} - 2x + 1$is a polynomial of degree 3.) The simplicity of these equations, and the fact that many of them show up in applications (the parabola$y = {x^2}$is a simple example), made them a natural choice for testing the new methods of the calculus. But in applications one also...

Among the numerous problems whose solution involves the exponential function, the following is particularly interesting. A parachutist jumps from a plane and at$t = 0$opens his chute. At what speed will he reach the ground?

For relatively small velocities, we may assume that the resisting force exerted by the air is proportional to the speed of descent. Let us denote the proportionality constant by$k$and the mass of the parachutist by$m$. Two opposing forces are acting on the parachutist: his weight$mg$(where$g$is the acceleration of gravity, about 9.8 m/sec²), and the air resistance$kv$(where...

In 1825 the German physiologist Ernst Heinrich Weber (1795-1878) formulated a mathematical law that was meant to measure the human response to various physical stimuli. Weber performed a series of experiments in which a blindfolded man holding a weight to which smaller weights were gradually added was asked to respond when he first felt the increase. Weber found that the response was proportional not to the absolute increase in weight but to therelativeincrease. That is, if the person could still feel an increase in weight from ten pounds to eleven pounds (a 10 percent increase), then, when the...

An air of mystery always surrounds the members of a dynasty. Sibling rivalries, power struggles, and family traits that pass from one generation to the next are the stuff of countless novels and historical romances. England has its royal dynasties, America its Kennedys and Rockefellers. But in the intellectual world it is rare to find a family that, generation after generation, produces creative minds of the highest rank, all in the same field. Two names come to mind: the Bach family in music and the Bernoullis in mathematics.

The ancestors of the Bernoulli family fled Holland in 1583 to escape...

Did any member of the Bach family ever meet one of the Bernoullis? It’s unlikely. Travel in the seventeenth century was an enterprise to be undertaken only for compelling reasons. Barring a chance encounter, the only imaginable reason for such a meeting would have been an intense curiosity about the activities of the other, and there is no evidence of that. Nevertheless, the thought that perhaps such an encounter did take place is compelling. Let us imagine a meeting between Johann Bernoulli (Johann I, that is) and Johann Sebastian Bach. The year is 1740. Each is at the peak of...

Probably no curve has had greater appeal for scientists, artists, and naturalists than the logarithmic spiral. Dubbedspira mirabilisby Jakob Bernoulli, the spiral possesses remarkable mathematical properties that make it unique among plane curves (see p. 121). Its graceful shape has been a favorite decorative motif since antiquity; and, with the possible exception of the circle (which itself is a special case of a logarithmic spiral), it occurs more often in nature than any other curve, sometimes with stunning accuracy, as in the nautilus shell (fig. 54).

Perhaps the most remarkable fact about the logarithmic spiral is that it...

We are not quite done with the Bernoullis yet. Among the outstanding problems that occupied the mathematical community in the decades following the invention of the calculus was the problem of thecatenary—the hanging chain (from the Latincatena,a chain). This problem, like the brachistochrone, was first proposed by one of the Bernoulli brothers, this time Jakob. In the May 1690 issue ofActa eruditorum,the journal that Leibniz had founded eight years earlier, Jakob wrote: “And now let this problem be proposed: To find the curve assumed by a loose string hung freely from two fixed points.”¹...

Consider the unit circle—the circle with center at the orgin and radius 1—whose equation in rectangular coordinates is${x^2} + \;{y^2} = 1$(fig.66). Let$P(x,\;y)$be a point on this circle, and let the angle between the positive$x$-axis and the lineOPbe$\varphi $(measured counterclockwise in radians). Thecircularortrigonometric functions“sine” and “cosine” are defined as the$x$and$y$coordinates ofP:

$x\; = \;{\rm{cos}}\;\varphi {\rm{,}}\quad y\; = \;{\rm{sin}}\;\varphi {\rm{.}}$.

The angle$\varphi $can also be interpreted as twice the area of the circular sectorOPRin figure 66, since this area is given by the formula$A = {r^2}\varphi {\kern 1pt} /{\kern 1pt} 2\; = \;\varphi {\kern 1pt} /{\kern 1pt} 2$, where$r=1$is the radius....

$e\: = \:1\: + \:\frac{1}{{1!}} + \frac{1}{{2!}} + \frac{1}{{3!}} + \frac{1}{{4!}} + \ldots $

This infinite series was discovered by Newton in 1665; it can be obtained from the binomial expansion of${(1\; + \;1{\kern 1pt} /{\kern 1pt} n)^n}$by letting$n\; \to \;\infty $. It converges very quickly, due to the rapidly increasing values of the factorials in the denominators. For example, the sum of the first eleven terms (ending with 1/10!) is 2.718281801; the true value, rounded to nine decimal places, is 2.718281828.

${e^{\pi {\kern 1pt} \iota }} + \;1\; = \;0$

This is Euler’s formula, one of the most famous in all of mathematics. It connects the five fundamental constants of mathematics,$0,\;1,\;e,\;\pi ,$and$i\; = \;\sqrt - 1$.

$\begin{array}{*{20}{c}} {e = 2\: + \:\underline {\quad \quad \quad 1\:\quad \quad \quad } } \\ {{\rm{ }}1\: + \:\underline {\quad \:\:\:\:\:1\:\:\quad \quad } } \\ {{\rm{ }}2\: + \:\underline {\quad \,\:\:\:2\quad \quad } } \\ {{\rm{ }}3\: + \:\:\underline {\quad 3\quad \:} } \\ {{\rm{ }}4\: + \:\:\underline {\:\:4\:\:} } \\ {{\rm{ }}\:\:\:\:\:\:5\: + \:\: \ldots } \\\end{array}$

This infinitecontinued fraction, and many others involving$e$and$\pi $, was discovered by Euler in...

If we compared the Bernoullis to the Bach family, then Leonhard Euler (1707-1783) is unquestionably the Mozart of mathematics, a man whose immense output—not yet published in full—is estimated to fill at least seventy volumes. Euler left hardly an area of mathematics untouched, putting his mark on such diverse fields as analysis, number theory, mechanics and hydrodynamics, cartography, topology, and the theory of lunar motion. With the possible exception of Newton, Euler’s name appears more often than any other throughout classical mathematics. Moreover, we owe to Euler many of the mathematical symbols in use today, among them$i$,...

Benjamin Peirce (1809-1880) became professor of mathematics at Harvard College at the young age of twenty-four.¹ Inspired by Euler’s formula${e^{\pi i}} = - 1$,he devised new symbols for$\pi $and$e$, reasoning that The symbols which are now used to denote the Naperian base and the ratio of the circumference of a circle to its diameter are, for many reasons, inconvenient; and the close relation between these two quantities ought to be indicated in their notation. I would propose the following characters, which I have used with success in my lectures: —

to denote ratio of circumference to diameter,

to denote Naperian base....

The introduction of expressions like${e^{ix}}$into mathematics raises the question: What, exactly, do we mean by such an expression? Since the exponent is imaginary, we cannot calculate the values of${e^{ix}}$in the same sense that we can find the value of, say,${e^{3.52}}$—unless, of course, we clarify what we mean by “calculate” in the case of imaginary numbers. This takes us back to the sixteenth century, when the quantity$\sqrt - 1$first appeared on the mathematical scene.

An aura of mysticism still surrounds the concept that has since been called “imaginary numbers,” and anyone who encounters these numbers...

Aprime numberis an integer greater than 1 that can be divided evenly only by itself and 1. The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. A positive integer > 1 that is not prime is calledcomposite.(The number 1 itself is considered neither prime nor composite.) The significance of the prime numbers in number theory—and in all of mathematics—is due to the fact that every integer > 1 can be factored into (that is, written as a product of) primes in one and only one way. For...

The history of$\pi $goes back to ancient times; that of$e$spans only about four centuries. The number$\pi $originated with a problem in geometry: how to find the circumference and area of a circle. The origins of$e$are less clear; they seem to go back to the sixteenth century, when it was noticed that the expression${({\rm{1}}\: + \:1/n)^n}$appearing in the formula for compound interest tends to a certain limit—about 2.71828—as$n$increases. Thus$e$became the first number to bedefinedby a limiting process,$e\; = \;{\rm{lim}}\,{{\rm{(1}}\; + \;1{\kern 1pt} /{\kern 1pt} n)^n}$as$n\; \to \;\infty $. For a while the new number was regarded...