by DK
The exponential function
Later in the 1600s, logarithms revealed something of further significance. While studying number series, Italian mathematician Pietro Mengoli showed that the alternating series 1 ˗ 1⁄2 + 1⁄3 ˗ 1⁄4 + 1⁄5 ˗… has a value of around 0.693147, which he demonstrated to be the natural logarithm of 2. A natural logarithm (ln)—so-called because it occurs naturally, revealing the time required to reach a certain level of growth—has a special base, later known as e, with an approximate value of 2.71828. This number is hugely significant in mathematics due to its links with natural growth and decay.
It was through work such as that of Mengoli that the important concept of the exponential function came to light. This function is used to represent exponential growth—where the rate of growth of a quantity is proportional to its size at any particular moment, so the bigger it is, the faster it grows—which is relevant to fields such as finance and statistics, and most areas of science. The exponential function is given in the form f(x) = bx, where b is greater than 0, but does not equal 1, and x can be any real number. In mathematical terms, logarithms are the inverse of exponentials (powers of a number) and can be to any base.
The slide rule, used here in 1941 by a member of the Women’s Auxiliary Air Force, is marked with logarithmic scales that facilitate multiplication, division, and other functions. Invented in 1622, it was a vital mathematical tool before the advent of pocket calculators.
By shortening the labors, [Napier] doubled the life of the astronomer.
Pierre-Simon Laplace
A basis for Euler’s work
The push for accurate log tables spurred mathematicians such as Nicholas Mercator to pursue further research in this area. In Logarithmo-technica, published in 1668, he set out a series formula for the natural logarithm ln(1 + x) = x - x2⁄2 + x3⁄3 - x4⁄4 +… This was an extension of Mengoli’s formulation, in which the value of x was 1. In 1744, more than 130 years after Napier produced his first logarithm table, Swiss mathematician Leonhard Euler published a full treatment of ex and its relationship to the natural logarithm.
Logarithmic scales
The pH logarithmic scale measures alkalinity and acidity. A pH of 2 is 10 times more acidic than a pH of 3 and 100 times more acidic than pH 4.
When measuring physical variables, such as sound, flow, or pressure, where values may change exponentially, rather than by regular increments, a logarithmic scale is often used. Such scales use the logarithm of a value instead of the actual value of whatever is being measured. Each step on a logarithmic scale is a multiple of the preceding step. For example, on a log10 scale, every unit up the scale represents a 10-fold increase in whatever is being measured.
In acoustics, sound intensity is measured in decibels. The decibel scale takes the hearing threshold, defined as 0 dB, as its reference level. A sound 10 times louder is assigned a decibel value of 10; a sound 100 times louder has a decibel value of 20; a sound 1,000 times louder a value of 30, and so on. This logarithmic scale fits well with the way we hear things, as a sound must become 10 times more intense to sound twice as loud to the human ear.
See also: Wheat on a chessboard • The problem of maxima • Euler’s number • The prime number theorem
IN CONTEXT
KEY FIGURE
Johannes Kepler (1571–1630)
FIELD
Geometry
BEFORE
c. 240 BCE In Method of Mechanical Theorems, Archimedes uses indivisibles to estimate the areas and volumes of curvilinear shapes.
AFTER
1638 Pierre de Fermat circulates his Method for determining Maxima and Minima and Tangents for Curved Lines.
1671 In Treatise on the Method of Series and Fluxions, Isaac Newton produces new analytical methods for solving problems such as the maxima and minima of functions.
1684 Gottfried Leibniz publishes New Method for Maximums and Minimums, his first work on calculus.
Astronomer Johannes Kepler is best known for his discovery of the elliptical shape of the planets’ orbits and his three laws of planetary motion, but he also made a major contribution to mathematics. In 1615, he devised a way of working out the maximum volumes of solids with curved shapes, such as barrels.
Kepler’s interest in this field began in 1613, when he married his second wife. He was intrigued when the wine merchant at the wedding feast measured the wine in the barrel by sticking a rod diagonally through a hole in the top and checking how far up the stick the wine went. Kepler wondered whether this worked equally well for all shapes of barrel and, concerned that he may have been cheated, decided to analyze the issue of volumes. In 1615, he published his results in Nova stereometria doliorum vinariorum (New solid geometry of wine barrels).
Kepler looked at ways of calculating the areas and volumes of curved shapes. Since ancient times, mathematicians had discussed using “indivisibles”—elements so tiny they cannot be divided. In theory these can be fitted into any shape and added up. The area of a circle could be determined, for example, by using slender pie-slice triangles.
To find the volume of a barrel or any other 3-D shape, Kepler imagined it as a stack of thin layers. The total volume is the sum of the volumes of the layers. In a barrel, for example, each layer is a shallow cylinder.
Infinitesimals
The problem with cylinders is that if they have thickness, their straight sides will not fit into the curve of a barrel, while cylinders without thickness have no volume. Kepler’s solution was to accept the notion of “infinitesimals”—the thinnest slices that can exist without vanishing. This idea had already been mooted by ancient Greeks such as Archimedes. Infinitesimals bridge the gap between continuous things and things broken into discrete units.
Kepler then used his cylinder method to find the barrel shapes with the maximum volume. He worked with triangles defined by the cylinders’ height, diameter, and a diagonal from top to bottom. He investigated how, if the diagonal was fixed, like the merchant’s rod, changing the barrel height would change its volume. It turned out that the maximum volume is held in short, squat barrels with a height just under 1.5 times the diameter—like the barrels at his wedding. In contrast, the tall barrels from Kepler’s homeland on the Rhine River held much less wine.
Kepler also noticed that the closer to the maximum the shape gets, the less the rate at which the volume increases: an observation that contributed to the birth of calculus, opening up the exploration into maxima and minima. Calculus is the mathematics of continuous change, and maxima and minima are the turning points, or limits in any change—the peak and trough of any graph.
Pierre de Fermat’s analysis of maxima and minima, which quickly followed Kepler’s, opened the way for the development of calculus by Isaac Newton and Gottfried Leibniz later in the 17th century.
The merchant’s rod is submerged to an equal extent when pushed at a diagonal into these two barrels, so he charges the same price for both. However, the elongated shape of the second barrel means it has a smaller volume, containing less wine but for the same price as the first.
JOHANNES KEPLER
Born near Stuttgart, Germany, in 1571, Johannes Kepler witnessed the “Great Comet” of 1577 and a lunar eclipse, and remained interested in astronomy throughout his life.
Kepler taught at the Protestant seminary in Graz, Austria. In 1600, non-Catholics were expelled from Graz and Kepler moved to Prague, where his friend Tycho Brahe lived. Following the death of his first wife and son, he moved to Linz in Austria, where his main job as imperial mathematician was to make astronomical tables.
Kepler was convinced that God had made the Universe according to a mathematical plan. He is best known for his work in astronomy, especially his laws of planetary motion and his astronomical tables. A year after his death in 1630, the transit of Mercury was observed as he had predicted.
Key works
1609 New Astronomy
1615 New Solid Geometry of Wine Barrels
1619 Harmon
ies of the World
1621 Epitome of Copernican Astronomy
See also: Euclid’s Elements • Calculating pi • Trigonometry • Coordinates • Calculus • Newton’s laws of motion
IN CONTEXT
KEY FIGURE
René Descartes (1596–1650)
FIELD
Geometry
BEFORE
2nd century BCE Apollonius of Perga explores positions of points within lines and curves.
c. 1370 French philosopher Nicole Oresme represents qualities and quantities as lines defined by coordinates.
1591 French mathematician François Viète introduces symbols for variables in algebraic notation.
AFTER
1806 Jean-Robert Argand uses a coordinate plane to represent complex numbers.
1843 Irish mathematician William Hamilton adds two new imaginary units, creating quaternions, which are plotted in four-dimensional space.
In geometry (the study of shapes and measurements), coordinates are employed to define a single point—an exact position—using numbers. Several different systems of coordinates are in use, but the dominant one is the Cartesian system, named after Renatus Cartesius, the Latinized name of French philosopher René Descartes. Descartes presented his coordinate geometry in La Géométrie (Geometry, 1637), one of three appendices to his philosophical work Discours de la Méthode (Discourse on the Method), in which he proposed methods for arriving at truth in the sciences. The other two appendices were on light and the weather.
Problems which can be constructed by means of circles and straight lines only.
René Descartes
describing geometry
Building blocks
Coordinate geometry transformed the study of geometry, which had barely evolved since Euclid had written Elements in ancient Greece some 2,000 years earlier. It also revolutionized algebra by turning equations into lines (and lines into equations). By using Cartesian coordinates, scholars could visualize mathematical relationships. Lines, surfaces, and shapes could also be interpreted as a series of defined points, which changed the way people thought about natural phenomena. In the case of events such as volcanic eruptions or droughts, plotting elements such as intensity, duration, and frequency could help to identify trends.
RENÉ DESCARTES
The son of a minor noble, René Descartes was born in Touraine, France, in 1596. His mother died shortly after his birth, and he was sent to live with his grandmother. He later attended a Jesuit college, then went to study law in Poitiers. In 1618, he left France for the Netherlands and joined the Dutch States Army as a mercenary.
Around this time, Descartes began to formulate philosophical ideas and mathematical theorems. Returning to France in 1623, he sold his property there in order to secure a lifelong income, then moved back to the Netherlands to study. In 1649, he was invited by Christina, Queen of Sweden, to tutor her and to launch a new academy. His weak constitution could not resist the cold winter. In February 1650, Descartes caught pneumonia and died.
Key works
1630–33 Le Monde (The World) 1630–33 L’Homme (Man)
1637 Discours de la Méthode (Discourse on the Method)
1637 La Géométrie (Geometry)
1644 Principia philosophia (Principles of Philosophy)
Finding a new method
There are two accounts of how Descartes came to develop the coordinate system. One suggests that the idea dawned on him as he watched a fly moving over the ceiling of his bedroom. He realized he could plot its position, using numbers to describe where it was in relation to the two adjacent walls. Another account relates that the idea came to him in dreams in 1619, when he was serving as a mercenary in southern Germany. It was at this time, too, that he is thought to have figured out the relationship between geometry and algebra that is the basis of the coordinate system.
The simplest Cartesian coordinate system is one-dimensional; it indicates positions along a straight line. One endpoint of the line is set as the zero point, and all other points on the line are counted from there in equal lengths, or fractions of a length. Just a single coordinate number is needed to describe an exact point on the line—as when measuring a distance with a ruler from zero to a unit of length. More commonly, coordinates are used to describe points on two-dimensional surfaces that have a length and width, or within a three-dimensional space, which also has depth. To achieve this, more than one number line is needed—each starting at the same zero point, or origin. For a point on a plane (a flat two-dimensional surface), two number lines are needed. The horizontal line, called the x-axis, and the vertical y-axis are always perpendicular to each other; the origin is the only place they will ever meet. The term for the x-axis is abscissa, while the y-axis is the ordinate. Two numbers, one from each axis, “coordinate” to pinpoint an exact position.
When taking a graph reading, these two numbers are now presented as a tuple—a strictly ordered sequence listed inside brackets. The abscissa (value of x) always precedes the ordinate (value of y) to create the tuple (x,y). Although they were conceived before negative numbers were fully accepted, coordinates now often include both negative and positive values—negative values below and to the left of the origin; positive values above and to the right of the origin. Together, the two axes create a field of points called a coordinate plane, which extends outward in two dimensions with the origin (0,0) at the center. Any point on that plane, which could stretch to infinity, can be described exactly using a pair of numbers.
I realized that it was necessary… to start again right from the foundations if I wanted to establish anything in the sciences that was stable and likely to last.
René Descartes
This edition of La Géometrie (in Latin because that was the language of scholars) was printed in 1639. Descartes originally published the book in French so it could be read by less well-educated people.
Plotting 3-D space
For a three-dimensional space, the coordinates require a third number, ordered in the tuple (x, y, z). The z refers to a third axis, which is perpendicular to the plane formed by the x and y axes (see 3-D Cartesian coordinates). Each pair of axes creates its own coordinate plane; these intersect at right angles to each other, thus dividing the space into eight zones called octants. The coordinates within each octant follow one of eight sequences of values for x, y, and z, ranging from all negative values to all positive values, with six possible negative and positive combinations in between.
Each problem that I solved became a rule which served afterwards to solve other problems.
René Descartes
Curved lines
La Géométrie sets out what soon became the foundation of the coordinate system. Descartes, however, was primarily interested in finding out how coordinates could help him use algebra to better understand lines, especially curved lines. In so doing he created a new field of mathematics, called analytic geometry, where shapes are described in terms of their coordinates and the relationships between a pair of variables, x and y. This was very different from Euclid’s “synthetic geometry,” in which shapes are defined by the way they are constructed using a ruler and pair of compasses. The ancient method was limiting; Descartes’ new method opened up all sorts of new possibilities.
La Géométrie contains much discussion about curves, which were the subject of renewed interest in the 1600s—partly because treatises by ancient Greek mathematicians had been newly translated, but also because curves featured prominently in fields of scientific exploration such as astronomy and mechanics.
Coordinates make it possible to convert curves and shapes into algebraic equations, which can be shown visually. A straight line that runs diagonally from the origin, equidistant from both axes, can be described using algebra as y = x, and has coordinates (0,0); (1,1); (2,2), and so on. The line y = 2x would follow a steeper path along a line including the coordinates (0,0); (1,2); (2,4), for instance. A line running parallel to y = 2x would pass through the y axis at a
point other than the origin, such as at (0,2). The formula for this particular line is y = 2x + 2 and that includes the points (0,2); (1,4); (2,6).
Cartesian coordinates help to reveal the great power of algebra to generalize relationships. All the straight lines described above have the same general equation: y = mx + c, where the coefficient m is the slope of the line, indicating how much bigger (or smaller) y is compared to x. The constant c, meanwhile, shows where the line meets the y axis when x is equal to zero.
With me, everything turns into mathematics.
René Descartes
A geometric shape such as the curve of a roller-coaster can be mapped on to a graph and described in relation to the x and y axes. The straight section of the curve has the equation y = x.
The circle equation
In analytic geometry, all circles centered on the origin can be defined as r = , known as the circle equation. This is because a circle can be thought of as all the points that lie at an equal distance from a central point (that distance being the radius of the circle). If that central point is (0,0) on an x, y graph, the circle equation emerges, by drawing on Pythagoras’s theorem. The circle’s radius can be conceived as the hypotenuse of a right-angled triangle with short sides x and y, so r2 = x2 + y2, which can be rewritten as r = . The circle can then be plotted on axes using different values of x and y that give the same value of r. For example, if r is 2, then the circle crosses the x axis at (2,0) and (˗2,0), and it crosses the y axis at (0,2) and (0,˗2). All the other points on the circle can be seen as one corner of a right-angled triangle moving around in a circle. As the corner moves around the circle, the short sides of the triangle vary in length, but the hypoteneuse does not because it is always the radius of the circle. The line formed by a point moving in this defined way is called a locus. This idea was developed by the Greek geometer Apollonius of Perga about 1,750 years before Descartes’ birth.