Mathematics for the liberal arts (2013)

Part I. MATHEMATICS IN HISTORY

Chapter 3. MODERN MATHEMATICS

Mathematics and Poetry are… the utterance of the same power of imagination, only that in the one case it is addressed to the head, in the other, to the heart.

THOMAS HILL (1818-1891)

This chapter tells the story of mathematics in a time when the way people live begins to undergo a massive shift, a shift in which mathematics plays a major role. One difficulty of this chapter is that the level of sophistication, and the sheer volume, of the new math prevents us from providing more than a sketch of the history. Nonetheless, we can at least touch on some of the major developments of this time, and make a few stops at some scenic attractions.

Earth.¹

3.1 The 17th Century: Scientific Revolution

Introduction

The 17th century was one of the most creative periods in all of cultural history. Modem science was bom, with mathematics at its heart. This Scientific Revolution, as it is usually called, combined with the Industrial Revolution of the following century, are largely responsible for creating our modem society.

Much of the first half of the century was dominated by religious wars, including the Eighty Years’ War (1568–1648) in the Low Countries of Northwestern Europe, the English Civil Wars (1642–1651), and above all the Thirty Years War (1618–1648).

The Thirty Years War was the worst European war outside of the 20th century; millions died. It involved all the European powers, although most of the fighting took place in what is now Germany and the Czech Republic. In many of those regions, 25% or more of the population perished. The war ended with the Treaty of Westphalia (1648). Although devastated by the war, Germany did manage to produce the great mathematician and philosopher Gottfried Wilhelm Leibniz (1646–1716).

The 17th century was the Dutch Golden Age. The Dutch Republic, part of what is now the Netherlands, won its independence from the Hapsburg Empire, ruled at that time from Spain. Although not officially recognized by all powers until 1648, it was de facto independent from early in the century. It was at this time one of the great commercial centers of Europe, with Amsterdam probably the richest city in the world. The tolerant Dutch cities attracted Protestant refugees from the Southern Netherlands and Jewish refugees from Portugal and Spain. Dutch traders became dominant, displacing the Portuguese in Asia. Some historians credit the Dutch for establishing the first stock exchange, which financed the founding of the first multinational corporation, the Dutch East India Company, in 1602.

This was a period of brilliant artists such as Johannes Vermeer (1632–1675) and the great Rembrandt Harmenszoon van Rijn (1606–1669). Dutch universities, particularly the University of Leiden, were among the best. Among the prominent scientists were Anton van Leeuwenhoek (1632–1723), “the father of microbiology,” who improved the microscope and discovered red blood cells and micro-organisms, and the astronomer, mathematician, and inventor of the pendulum clock, Christiaan Huygens (1629–1695).

France emerged from the Thirty Years War the most powerful nation in Europe. This was the century of the philosopher and mathematician René Descartes (1596–1650), the dramatist Molière (1622–1673), and the heyday of the absolute monarchy under Louis XIV, the Sun King (r. 1661–1715), who ruled from his sumptuous palace at Versailles. Paris was arguably the most important cultural center of the time.

In England the 16th century began in the time of Elizabeth I (1533–1603), William Shakespeare (1564–1616), and the philosopher (and Lord Chancellor) Francis Bacon (1561–1626), who promoted the “New Science” based on observation and experimentation. The middle of the century was difficult with the civil wars, the Great Plague of 1665–66, and the Great Fire of London in 1666. Nonetheless England emerged strongly from this period. London was rebuilt, led by the architect Christopher Wren (1632–1723). The Royal Society was founded in 1660, and hired the brilliant Robert Hooke (1635–1703) as curator of experiments. This was also the time of the incomparable Isaac Newton (1642–1727).

In Italy, by contrast, the most productive scientists of the century lived in the first half. The greatest of these was the man many consider to have been the first modem scientist, Galilei Galileo (1564–1642), who died in the year of Newton’s birth. A brilliant polemicist as well as scientist, Galileo did much to promote the new astronomy of Nicolaus Copernicus (1473–1543). The Church was unwilling to accept this, and Galileo spent his last years under house arrest.

Above all, the 17th century saw the triumph of the new science, which was based on the primacy of experimentation and couched in the language of mathematics. Its first great success was in astronomy. Copernicus had published his heliocentric theory in 1543. Johannes Kepler (1571–1630), working from the careful observations of Tycho Brahe (1546–1601), studied planetary orbits, and discovered his three “planetary laws” in the first decade of the 17th century. In particular, he discovered that the planets travel around the Sun in elliptical orbits. In 1610 Galileo improved on the new telescope and was able to see through it a mini-solar system of moons orbiting Jupiter, as well as the phases of Venus, the mountains of the Moon, and sunspots, observations inconsistent with the old Ptolemaic astronomy. Finally, Isaac Newton, using the new calculus he helped to develop, formulated his laws of physics and from them derived Kepler’s planetary laws. His new physics was published in his masterwork, Philosophiœ Naturalis Principia Mathematica (1687).

In mathematics, the 17th century is remembered for the invention of calculus, the “mathematics of change.” There were other major advances as well, most notably the marriage of algebra and geometry—analytic geometry—with the invention of the Cartesian coordinate system. Also, computation was made easier with the invention of logarithms, and major work was done in probability theory and number theory.

The modem world of mathematical research is dominated by universities where most of the work is done, and by journals which disseminate the results. Neither institution was fully developed in the 17th century. Most mathematicians of the time were trained in universities, and some research was carried out there, but much was done outside them. In the early part of the century, especially, the patronage of rulers was essential. The Holy Roman Emperor Rudolph II hosted scientists including Johannes Kepler in Prague. Prince Maurice of Holland supported Simon Stevin (c. 1548–1620) and René Descartes. Prince Leopold de’ Medici supported research in Florence.

Communication of research advances was still rudimentary at this time, although the custom of hoarding instead of publicizing one’s work was beginning to break down. Most important in this respect were the new scientific societies, including the Accademia dei Lincei (1603–30) and Accademia del Cimento (Academy of experiments, 1657–67) in Italy, the Academia Parisiensis (c. 1635) and its successor, the Académie des Sciences (1666–present), in France, as well as the Royal Society in England. Mathematicians also corresponded by letter. The French monk Marin Mersenne (1588–1648), who organized the Academia Parisiensis, had a network of correspondents throughout Europe and the Middle East. He would have letters reproduced and sent to other researchers, spreading the latest discoveries.

Logarithms

The most detailed, laborious calculations at this time were done by astronomers. They used a number of techniques to simplify their work. For example, they utilized the following identity to multiply numbers.

To multiply two numbers, say x = .3295 and y = .0827, they would first use trig tables to find angles α and β such that x = sin α and y = sin β, in this case α ≈ 19.238 and β ≈ 4.744. They would then look up the cosines on the right-hand side, subtract them

and divide by 2 to get .3295 × .0827 ≈ .02725.

The last calculation may seem a complicated way to multiply numbers. The point was to replace hand multiplication by table lookups, addition, and subtraction, which were easier. In general, multiplication and division were time-consuming and prone to error. That was why logarithms were invented in the early 17th century.

Logarithms were invented twice, by a Swiss mathematician named Joost Bürgi (1552–1632), and a Scotsman named John Napier. Napier’s work was published first, in 1614 versus 1620 for Bürgi, and was much more influential.

John Napier (1550–1617)

John Napier was a Scottish aristocrat, the eighth laird of Merchiston. As a teenager, he studied at St. Andrews but did not get a degree. He probably traveled for a while in Europe, but lived most of his life as a landed aristocrat in Scotland. He had twelve children.

Napier was most famous in his time for his labors on behalf of religion. In particular, he wrote A Plaine Discovery of the Whole Revelation of Saint John: Set Downe in Two Treatises. In this work, he revealed that the Pope is the Antichrist, and that the world will probably end between 1688 and 1700. The book was quite popular, with twenty-one editions in English and numerous translations.

The following discussion uses the modem form of logarithms. Napier’s original logarithms, although using the same essential ideas, were a bit different in detail.

What is a logarithm? In a word, it is a power. For example, since 100 = 10², we call 2 the logarithm of 100, and write log 100 = 2. Note that log 10 = 1, since 10¹ = 10, and log 1000 = 3, since 10³ = 1000.

What happens to logarithms when we multiply numbers? Consider 10 · 100 = 10¹ · 10² = 10³. When we multiply numbers, their powers, that is their logarithms, are added to produce the new power. In other words, multiplication is reduced to addition. For any numbers m and n, we have

We have only talked about 10ⁿ where n is a positive integer, but the notion of powers can be extended, so that we can talk about 10^x for any positive real number x. The full explication of this is beyond this text, but it starts with the following definitions, for n a positive integer.

With this extension, we have these important identities, for real numbers x and y:

Thus we replace multiplication by addition and division by subtraction.

How can we use this? Consider the problem of multiplying two numbers, say x = 2.3456 and y = .827. We can use a table of logarithms to look up log x ≈ .3703 and log y ≈ –.0825. (Numbers between 0 and 1 have negative logarithms.) We then add the logs to get log(xy) = log x + log y ≈ .2878. Finally, we look this log up in a table to get xy ≈ 1.9400.

We also note some further properties of logarithms, for any positive numbers x and y and nonnegative integer n.

1. If x < y, then log x < log y.

2. log 1 = 0

3. If log x ≥ n and log x < n + 1, then the number of digits to the left of the decimal point of x is n + 1.

4. log(x^y) = y log x

Napier published Mirifici Logarithmorum Canonis Descriptio (Description of the Marvelous Canon of Logarithms), with the first table of logarithms, in 1614. It had taken him twenty years to calculate the tables. (This text is also responsible for our use of the decimal point.) He explained the theory behind his logarithms in the work Mirifici Logarithmorum Canonis Constructio (Construction of the Marvelous Canon of Logarithms), which appeared in 1619, two years after his death.

Napier’s logarithms spread like wildfire. Mathematicians and astronomers immediately appreciated their value in calculations. A number of people, including Napier himself, quickly refined his original ideas, and as early as 1628, Henry Briggs (1561–1630) and Adriaan Vlacq (1600–1667) published a table of logarithms of numbers from 1 to 100,000 to ten decimal places. This was the standard reference until the 20th century.

It may be difficult for someone in this age of computers to appreciate the value of logarithms. Two centuries after their introduction, however, the mathematician Pierre-Simon de Laplace (1749–1827) wrote that they “by shortening the labors, doubled the life of the astronomer.”

Logarithms are still useful. Recall the Fibonacci sequence: 1,1,2,3,5,…, where each term is the sum of the preceding two. How would we calculate the millionth Fibonacci number? Even today, a calculator (or computer) is likely to choke on the calculation. Logarithms to the rescue!

One way to approximate the nth term f_n of the sequence is by using the golden ratio,

The approximation becomes more precise as n grows. Here are the first few terms.

Using the properties above, we can compute the log of the nth Fibonacci number:

We can compute the two logs above using a table (or calculator), giving

The millionth Fibonacci number has n = 10⁶, so in this case

Thus, the millionth Fibonacci number is about

In particular, it has 208,988 digits.

Napier also invented ways of multiplying and dividing numbers using small rods that became known as Napier’s bones. These were forerunners of one of the most important computational aids in history, the slide rule.

The slide rule (Figure 3.1) used a mechanical equivalent of logarithms to approximate multiplication and division. Over time, other computations were added. For about three hundred years, the slide rule was an essential tool for the scientist and engineer. Before the arrival of the personal computer, the quintessential nerd was often pictured with a slide rule in the shirt pocket.

The Cartesian Coordinate System

In the 17th century, the mathematician and philosopher René Descartes (1596–1650) introduced a method of solving geometric problems using rectangular coordinates. Today, we call the system of coordinates the Cartesian coordinate system. The Cartesian plane is shown in Figure 3.2.

The Cartesian plane is determined by two infinite lines at right angles, called the axes. The x-axis is horizontal and the y-axis is vertical. Each axis is a real number line, with the two axes crossing at 0. The crossing point is called the origin of the coordinate system.

Each point in the plane has two coordinates. The first coordinate gives the point’s horizontal position. The second coordinate gives its vertical position. We write the coordinates as (x, y). For example, the point (2,1) is located 2 units to the right of the y-axis and 1 unit above the x-axis.

Figure 3.1 A slide rule.

Geometry based on coordinates is called analytic geometry. In analytic geometry, algebraic equations are identified with sets of points in the plane (lines, circles, etc.).

René Descartes (1596–1650)

René Descartes was bom in La Haye (later renamed La Haye-Descartes), France. He was the son of a lawyer, and himself obtained a law degree in 1616. After that, he traveled extensively throughout Europe, and participated for a while in the Thirty Years War, before settling in Paris in 1625. In 1628 Descartes moved to Holland, where he stayed for twenty years, although he moved eighteen times during this stay.

Descartes’ fame resets as much on his philosophy as his mathematics. His most important works are considered to be Discourse on the Method (1637), The Geometry (1637), and Meditations on First Philosophy (1641). He is perhaps best known for his phrase “cogito ergo sum” (“I think, therefore I am.”). Descartes believed in mind- body dualism and posited that the link between the two realms was the pineal gland at the base of the brain. He also subscribed to the vortex theory of gravitation, which has since been disproved.

Descartes was a colorful character. He said that a philosopher should not get out of bed before noon, and he claimed to have conceptualized the coordinate system while lying in bed and looking at a fly on the ceiling. He realized that the fly’s position could be specified with three coordinates. Locating a point by coordinates is an elementary observation that others had made before, but Descartes gets the credit (and the name of the coordinate system) because he showed how to use it to solve algebraic problems. The coordinate system reunified algebra and geometry after the schism that occurred in ancient Greece with the discovery of irrational numbers. Now, geometry and algebra are recognized as two aspects of the same mathematics. The coordinate system is the foundation for modem calculus and physics.

Figure 3.2 The Cartesian plane.

In 1649 Descartes was made philosopher at the court of Queen Christina in Sweden. Obligated to start work early in the morning, he died the first year of pneumonia.

Distance Formula From the Pythagorean Theorem, we obtain a formula for the distance between any pair of points in the plane (Figure 3.3). Let P(x₁,y₁) and Q(x₂,y₂) be two points in the plane. Then the line segment PQ is the hypotenuse of a right triangle with side lengths |x₁ – x₂| and |y₁ – y₂|

From the Pythagorean Theorem, the distance between P and Q is

Note that if x₁ = x₂ or y₁ = y₂, the triangle will have a leg of zero length, but the distance formula still holds.

Figure 3.3 Two points determine a right triangle.

EXAMPLE 3.1

The distance between the points (4,6) and (10,10) is

The three-dimensional Cartesian coordinate system is defined in an analogous way to the two-dimensional Cartesian coordinate system. Three mutually perpendicular number lines intersect at an origin. These lines are called the x-axis, y-axis, and z- axis. Each point in three-dimensional space is given by three coordinates, (x, y, z).

It is straightforward to extend the formula for the distance between two points in the plane to a formula for the distance between two points in three-dimensional space. Let P = (x₁,y₁,z₁) and Q = (x₂,y₂,z₂) be two points in three-dimensional space. Then the distance between P and Q is

This formula can be proved using two applications of the Pythagorean Theorem. We leave it to you to draw a diagram and show the steps.

Pierre de Fermat (1601–1665)

Pierre de Fermat’s father was a merchant, and his mother was from a family of lawyers. Pierre became a lawyer, and spent his entire career practicing law in France.

Fermat was a lawyer by vocation, but a mathematician by avocation. He tended not to publish his work, but instead disseminated his ideas through correspondence with other mathematicians. His work on analytic geometry predates Descartes’ publications, but wasn’t circulated as widely. He described a method for finding maxima and minima of functions and tangent lines to curves, which are basic problems of calculus (see Chapter 4). A number of subsequent French mathematicians gave him credit for inventing calculus, but most modem scholars disagree. Collaborating with Blaise Pascal, Fermat helped lay the foundations of probability theory.

In number theory, Fermat investigated certain Diophantine equations (equations whose solutions must be integers or rational numbers). He proved that there are no nonzero integer solutions to

Fermat thought he had a proof that there are no nonzero integer solutions to

for n ≥ 3. df n = 2 then the side lengths of Pythagorean triangles provide infinitely many integer solutions.) In 1637 he wrote of his discovery in the margin of his copy of Diophantus’ Arithmetics saying the margin was too narrow to contain the proof. However, no proof appears in his correspondence and it is likely that his proof was in error. His conjecture, which became known as Fermat’s Last Theorem, generated much mathematical research over the following three centuries, until it was finally proved in 1995 by Andrew Wiles (see Section 5.16).

Pascal’s Triangle

As described in Chapter 2, Jia Xian discovered the triangle of binomial coefficients in the 11th century. However, the triangle is named after Blaise Pascal (1623–1662), because he studied it extensively and used it to calculate odds in gambling games. He presented his findings in Treatise on the Arithmetical Triangle (published posthumously in 1665).

Recall that Pascal’s triangle has a 1 at the top. Each successive row has Is at the two ends. To find each other entry, add the two numbers above that entry. The triangle can be extended indefinitely.

Pascal’s triangle has many uses. It gives the coefficients when the binomial power (x + y)ⁿ is multiplied out. For example,

and we see that the coefficients, 1, 3, 3, 1, are the entries in the third row of Pascal’s triangle. (We say “third row” instead of “fourth row” because the top row is considered to be row 0.)

Figure 3.4 The normal distribution from probability theory.

Another use of Pascal’s triangle, related to probability, is to find the number of combinations from a given set. For example, let S be the set {a, b, c, d, e}. The two-element combinations from S (order of elements unimportant) are

We see that there are 10 two-element combinations from the five-element set S. The number 10 is the second entry of the fifth row of Pascal’s triangle (as with row entries, column entries are numbered starting with 0). How many three-element combinations from S are there?

If each entry of a row of Pascal’s triangle is divided by the sum of the entries in that row, there results the binomial distribution from probability theory. If the row number is large, this distribution approximates the normal distribution well known for its bell-shaped curve (Figure 3.4). Abraham de Moivre (1667–1754) was the first to study this probability distribution.

Blaise Pascal (1623–1662)

Blaise Pascal was bom in Claremont Ferrand, France, the son of a judge. His mother died when he was three. His father homeschooled Blaise, and in 1635 introduced him to Marin Mersenne’s circle in Paris.

Pascal was a mathematician, physicist, and philosopher. His interests were extremely varied. He invented a mechanical calculator when he was 16. He investigated conic sections, probability, and hydrodynamics, as well as theology. He corresponded with several great thinkers of his day, including Pierre de Fermat (1601–1665) and Christiaan Huygens (1629–1695). Pascal was able to work out some problems in calculus before the subject was fully developed by Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1716).

Pascal had poor health and an irascible personality. He never married and died at the age of 39. This saying is attributed to him: “The more I see of men, the better I like my dog.”

Figure 3.5 Tangent to y = at the point (4, 2).

Figure 3.6 Area under y = between 0 and 4.

Calculus

Calculus begins with two basic geometric questions. Given a curve and a point, can we find a tangent line to the curve at that point; and given a curve and an interval, can we find the area under the curve?

An example of the tangent line problem is illustrated in Figure 3.5. The curve is y = , and a tangent line is indicated through the point (x, y) = (4,2).

An example of the area problem is illustrated in Figure 3.6. Again, the curve is y = The area of interest extends from x = 0 to x = 4.

These geometric questions had been studied for millennia, but they came to the fore in this century, in part because of the invention of analytic geometry. This greatly expanded the number of curves available to the mathematician: every algebraic expression in x and y was associated to a curve in the Cartesian plane.

A number of mathematicians studied the tangent line problem in the first half of the century, most notably Fermat and Descartes. As in the invention of analytic geometry, these two were rivals in developing methods to find tangent lines. Fermat also tied this problem to that of finding maximum and minimum values of a quantity.

The area problem had been studied by Archimedes and Kepler, among others. In the early 17th century, advances were made by Fermat, and by two students of Galileo, Bonaventura Cavalieri (1598–1647) and Evangelista Torricelli (1608–1647). The latter made a remarkable discovery about what is now known as Torricelli’s trumpet, the solid obtained by rotating the hyperbola xy = k² about the x-axis, and extending it from x = a to x = ∞, where a is some positive number. Torricelli showed that this solid has finite volume but infinite surface area.

In the middle of the century, several mathematicians discovered connections between our two geometric problems. It turns out that they are in some sense inverses of one another. Important figures in this discovery were the Dutchman Hendrick van Heuraet (1634–1660?), the Scot James Gregory (1638–1675), and the Englishmen Isaac Barrow (1630–1677). The connection is now called the Fundamental Theorem of Calculus.

Although many of the major pieces of calculus were now in place, it remained to put them all together. Like logarithms, calculus can be said to have two fathers. It was first invented by Isaac Newton around 1665–66 at his country home in Grantham, England, where he retreated from an outbreak of plague that was running through Cambridge. At the same time that he was inventing calculus, he was conducting experiments into the nature of light and discovering the laws of motion that would ultimately become his paradigm-changing law of universal gravitation.

Sir Isaac Newton (1642–1727)

If I have seen further it is by standing on the shoulders of giants.

ISAAC NEWTON

Nature and nature’s laws lay hid in Night.

God said, ‘Let Newton be!’ and all was light.

Alexander Pope (1688–1744)

Isaac Newton was bom on January 4, 1643 in Woolsthorpe, England (though by the English calendar in use at the time it was December 25, 1642). His father had died two months before he was bom, and Newton spent much of his youth in the care of his grandparents. Although he was destined to become a great mathematician, he had little scholarly training in mathematics until 1661 when at age 18 he enrolled in Cambridge University’s Trinity College.

Newton did some important work on infinite series, but he is remembered primarily for three discoveries that he made during the plague years of 1665–67. He discovered the particle nature of light, invented calculus, and discerned his law of universal gravitation.

In 1672 Newton published a paper called “New Theory about Light and Colours” in the Philosophical Transactions of the Royal Society in London. His paper described light as particles in which white light is a combination of different kinds of colored light. This controversial discovery contradicted the prevailing wave theory of light, and he experienced criticism over the work, especially by the influential Robert Hooke (of Hooke’s law of elasticity).

The reaction to his theory of light left Newton reluctant to incur more criticism, and it wasn’t until 1687 that Edmond Hailey (for whom Hailey’s comet is named) finally persuaded Newton to publish his masterwork, Philosophice Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy). This contained his law of universal gravitation, which according to popular legend is said to have been inspired by an apple falling from a tree. Interestingly, although Newton used his calculus to derive his conclusions, he included only geometric proofs in the arguments of the Principia.

Newton himself did not publish his calculus until 1704, when he included it as an appendix to his book Opticks. This was nearly 40 years after he developed it, and well after calculus had been independently developed by Leibniz.

In addition to pursuing his research, Newton was twice elected to Parliament and, in 1705, was knighted by Queen Anne. He was appointed warden of the London mint in 1696 and was promoted to master of the mint in 1699, a post he held for decades. He died on March 20, 1726, and was buried, after a grand funeral ceremony, in Westminster Abbey.

Gottfried Wilhelm Leibniz (1646–1716)

Gottfried Wilhelm Leibniz was an attorney, mathematician, and philosopher from Leipzig. His father, a professor of ethics and vice chairman of the philosophy faculty at the University of Leipzig, died when he was six years old, leaving him an extensive library. By age eight, Leibniz was given access to his late father’s library, and thus began a life of independent learning.

When he failed to achieve a doctorate in October of 1666 from the University of Leipzig, he transferred to the University of Altdorf and received a doctorate in February of 1667 for a dissertation on difficult legal cases. He worked as an attorney for the remainder of his life because he felt he could do more good for people as a lawyer than as a professor.

Even as an attorney, Leibniz continued to study philosophy, science, and mathematics. He published “Discourse on Metaphysics” in 1686, which defined his philosophy. His book Theodicy came out in 1710. In it he considered questions of church doctrines that required resolution for reunification of the church (something he worked toward much of his lifetime).

As a mathematician, Leibniz was mostly self-taught. He is most famous for his calculus, which he created around 1675 (about ten years after Newton). Although Newton gets credit for creating calculus first, Leibniz was responsible for disseminating calculus throughout Europe, beginning with his 1684 publication of Nova Methodus Pro Maximis et Minimis (New Method for Maxima and Minima). Moreover, Leibniz created a lucid notation for calculus, using symbols that are still recognized and popular today, such as dx for infinitesimal changes and the elongated S symbol ∫ for integration.

Leibniz was a polymath. He could write in Latin, French, German, and other languages, though he spoke only poor English. He was a copious letter writer, and authored as many as 300 letters per year. Among his notable inventions were calculating machines that could add, subtract, multiply, and divide. One could multiply a 10-digit number by a 4-digit number with only four turns of a crank. A later model could operate on numbers up to 12 digits. He was an accomplished geologist who held that the earth had originally been molten, then covered with oceans.

Leibniz died November 14, 1716, and was buried inside the Neustäder church (something that was rare for a person without a title). Though at the time of his death his reputation had been tarnished by a dispute with Newton over the invention of calculus, by the end of the 18th century his accomplishments were again celebrated. In particular, he is recognized today as one of calculus’ true and independent creators.

If calculus were merely about solving two isolated geometrical questions, it might be important but it probably would not have had the profound impact that it has had. Tangent lines and areas, however, can be used in a great variety of interesting problems. Leibniz’ 1684 paper was about maximizing and minimizing quantities. Newton used the principles of calculus to answer physical questions about force and acceleration. It has since been used to derive statistical methods, principles of economics, and rules of engineering. It is hard to imagine an endeavor of mathematics or science that has not been enhanced somehow by the powerful tools of calculus.

Calculus will be taken up in detail in Chapter 4.

EXERCISES

Use this log table for the first five exercises.

3.1 Use the log table to approximate log 6. (Hint: 6 = 2·3.)

3.2 Use the log table to approximate log(2/3).

3.3 Use the log table to approximate log (2 · 3⁵).

3.4 Use the log table to approximate 1.211 × 5.122.

3.5 How many digits does 2¹⁰⁰⁰ have?

3.6 Plot the points (2,3) and (5,7). Find the distance between the points.

3.7 Find five points at distance 3 from (0,4).

3.8 What is the slope of the line through (0, –2) and (–3, –4)?

3.9 What is the slope of the line passing through the origin and the point (–5,4)?

3.10 Find an equation of the circle with center (3,2) and radius 5.

3.11 Find the distance between (1,2,3) and (4,5,6).

3.12 Give an equation for the sphere of radius 5 with center (0,2, –3).

3.13 A cube has vertices (0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), and (1,1,1). Let P = (3/4,0,0), Q = (0,3/4,0), R = (1,1/4,1), and S = (1/4,1,1). Show that PQRS is a square and find its side length. A cube of this side length is called Prince Rupert’s cube. It is the largest cube that can pass through a cube of side length 1 (the cube that we started with).

3.14 Use Pascal’s triangle to find the expansion of (x + y)⁵.

3.15 Use Pascal’s triangle to find the number of three-element combinations from a set of six elements. List the combinations.

3.16 Divide each entry of the eighth row of Pascal’s triangle by the sum of the entries in that row. Plot the nine values on a graph and connect the points by a curve. Do you recognize this curve?

3.17 The tangent line in Figure 3.5 has slope m = 1/4 and goes through the point (4,2).

a) Use the point-slope form of a line,

to write the equation of the line.

b) Write the equation of the tangent line in the form y = mx + b.

3.18 For the curve in Figure 3.6, the area under the curve and above an interval [0,x] is given by the formula A = .

a) Find the area under the curve and above the interval [0,1].

b) Find the area under the curve and above the interval [0,5].

c) Find the area under the curve and above the interval [1,5] by subtracting your answer for (a) from your answer for (b).

3.19 Newton’s universal law of gravitation says that the gravitational force between two point-masses m₁ and m₂ is given by

where G is a constant and r is the distance between the point-masses. Explain how this law shows that the acceleration due to gravity of an object near the Earth’s surface may be taken to be a constant (which we call g). (Hint: use Newton’s law that the force on an object equals its mass times its acceleration.)