The Math Book
Page 19
An early version of the law of large numbers (LLN)—a theorem examining the results of performing the same action (such as throwing a die) a number of times—was part of Swiss mathematician Jacob Bernoulli’s Ars Conjectandi (The Art of Conjecturing, 1713). In the late 18th and early 19th century, Pierre-Simon Laplace applied probability theory to practical and scientific problems, setting out his methods in his Théorie Analytique des Probabilités (Analytic Theory of Probabilities) in 1812.
Probability theory
While ancient and medieval law graded probability in the assessment of judicial evidence, there was no theory on which to base it. Similarly, in Renaissance times, when insurance was calculated for ships, premiums were based on an intuitive estimate of risk. Odds were a feature of gaming, but Gerolamo Cardano was the first to apply mathematics to the study of probability. Games of chance were the focus of such studies even after the deaths of both Pascal and Fermat, although their letters on the subject contributed much to subsequent theory.
In the late 1700s, Pierre-Simon Laplace extended the scope of probability theory to science, and introduced his mathematical tools for predicting the probability of many incidents, including natural phenomena. He also recognized its application in statistics. Probability theory is also used in many other fields, such as psychology, economics, engineering, and sports.
See also: The law of large numbers • Bayes’ theorem • Buffon’s needle experiment • The birth of modern statistics
IN CONTEXT
KEY FIGURE
Vincenzo Viviani (1622–1703)
FIELD
Geometry
BEFORE
c. 300 BCE Euclid defines a triangle in his book Elements and proves many theorems concerning triangles.
c. 50 CE Heron of Alexandria defines a formula for finding the area of a triangle from its side lengths.
AFTER
1822 German geometer Karl Wilhelm Feuerbach publishes a proof for the nine-point circle, which passes through the midpoint of each side of a triangle.
1826 Swiss geometer Jakob Steiner describes the triangle center that has the minimum sum of distances from the triangle's three vertices.
Italian mathematician Vincenzo Viviani studied under Galileo in Florence. After Galileo’s death in 1642, Viviani gathered together his master’s work, editing the first collected edition in 1655–56.
Viviani’s research included work on the speed of sound, which he measured to within 82 ft (25 m) per second of its true value. He is best known, however, for his triangle theorem, which states that the sum of the distances between any point in an equilateral triangle and that triangle’s sides is equal to the altitude (height) of the triangle.
Proving the theorem
Starting with an equilateral triangle of base (side) a, and an altitude of h, a point is made inside the triangle. Perpendicular lines (p, q, and r) are drawn from that point to each of the three sides, meeting each side at 90°. The triangle is divided into three smaller triangles by drawing a line from the point to each corner of the main triangle. The area of a triangle is 1⁄2 × base × height, so if the lengths of the perpendiculars are p, q, and r, the areas of the triangles add up to 1⁄2 (p + q + r)a. This is also the area of the large triangle, which is 1⁄2 ha, and so h = p + q + r. If you were to break a stick of length h into three, there would always be a point in the triangle from which the pieces form the perpendiculars p, q, and r.
The altitude in an equilateral triangle, such as the above, is always equal to the combined length of lines drawn from any point in the triangle perpendicular to its three sides.
See also: Pythagoras • Euclid’s Elements • Trigonometry • Projective geometry • Non-Euclidean geometries
IN CONTEXT
KEY FIGURE
Christiaan Huygens (1629–95)
FIELD
Geometry
BEFORE
1503 French mathematician Charles de Bovelles is the first to describe a cycloid.
1602 Galileo discovers that the time taken for a pendulum to complete a swing does not depend on the swing’s width.
AFTER
1690 Swiss mathematician Jacob Bernoulli draws on Huygens’s imperfect solution to the tautochrone problem to solve the brachistochrone problem—finding a curve of the fastest descent.
Early 1700s The longitude problem is resolved by British clockmaker John Harrison and others—using springs rather than pendulums.
In 1656, Dutch physicist and mathematician Christiaan Huygens created the pendulum clock, a clock with a swinging weight that was constant. He wanted to resolve the navigational problem of determining a ship’s longitude. This was impossible without precise calculations of time, so it required an accurate clock to cope with the rolling motion of the waves, which caused wide variations in pendulum swing, leading to time discrepancies.
Seeking the right curve
The key lay in finding a curved path for the pendulum to follow (known as a tautochrone curve), whereby the time the pendulum takes to return to its lowest point is constant whatever its highest point. Huygens identified the cycloid, a curve that was steep at the top and shallow at the bottom. The curved path of any pendulum would have to be adjusted so it traveled in a cycloid. Huygens’s idea was to constrain the pendulum by adding cycloid-shaped “cheeks.” In theory, the time of each movement would now be the same from any starting point. However, friction introduced a larger error than the one Huygens was trying to resolve. It was only in the 1750s that the Italian Joseph-Louis Lagrange arrived at a solution, where the height of the curve needs to be in proportion to the square of the length of the arc traveled by the pendulum.
I was… struck by the remarkable fact that in geometry all bodies gliding along the cycloid… descend from any point in precisely the same time.
Herman Melville
Moby Dick (1851)
See also: The area under a cycloid • Pascal’s triangle • The law of large numbers
IN CONTEXT
KEY FIGURES
Isaac Newton (1642–1727), Gottfried Leibniz (1646–1716)
FIELD
Calculus
BEFORE
287–212 BCE Archimedes uses the method of exhaustion to calculate areas and volumes, introducing the concept of infinitesimals.
c. 1630 Pierre de Fermat uses a new technique for finding tangents to curves, locating their maximum and minimum points.
AFTER
1740 Leonhard Euler applies the ideas of calculus to synthesize calculus, complex algebra, and trigonometry.
1823 French mathematician Augustin-Louis Cauchy formalizes the fundamental theorem of calculus.
The development of calculus, the branch of mathematics that deals with how things change, was one of the most significant advances in the history of mathematics. Calculus can show how the position of a moving vehicle changes over time, how the brightness of a light source dims as it moves further away, or how the position of a person’s eyes alters as they follow a moving object. It can ascertain where changing phenomena reach a maximum or minimum value, and at what rate they travel between the two.
Alongside rates of change, another important aspect of calculus is summation (the process of adding things), which developed from the need to calculate areas. Eventually, the study of areas and volumes was formalized into what became known as integration, while calculating rates of change was termed differentiation.
By providing a better understanding of the behavior of phenomena, calculus can be used to predict and influence their future state. In much the same way as algebra and arithmetic are tools for working with numerical or generalized quantities, calculus has its own rules, notations, and applications, and its development between the 17th and 19th centuries led to rapid progress in fields such as engineering and physics.
Nothing takes place in the world whose meaning is not that of some maximum or minimum.
Leonhard Euler
Ancient origins
The ancient Babyl
onians and Egyptians were particularly interested in measurement. It was important for them to be able to calculate the dimensions of fields for growing and irrigating crops and to work out the volume of buildings to store grain. They developed early notions of area and volume, although these tended to be in the form of very specific examples, such as in the Rhind papyrus, where one problem involves the area of a round field with a diameter of 9 khet (a khet being an ancient Egyptian unit of length). The rules laid down in the Rhind papyrus led ultimately to what would become known more than 3,000 years later as integral calculus.
The concept of infinity is central to calculus. In ancient Greece, Zeno’s paradoxes of motion, a set of philosophical problems devised by the philosopher Zeno of Elea in the 5th century BCE, posited that motion was impossible because there are an infinite number of halfway points in any given distance. In around 370 BCE, the Greek mathematician Eudoxus of Cnidus proposed a method of calculating the area of a shape by filling it with identical polygons of known area, and then making the polygons infinitely smaller. It was thought that their combined area would eventually converge toward the true area of the shape.
This so-called “method of exhaustion” was taken up by Archimedes in around 225 BCE. He approximated the area of a circle by enclosing it within polygons with increasing numbers of sides. As the number of sides increases, the polygons (of known area) more closely resemble the circle. Taking this idea to the limit, Archimedes imagined a polygon with sides of infinitesimally smaller length. The recognition of infinitesimals was a pivotal moment in the development of calculus: previously insoluble puzzles, such as Zeno’s paradoxes of motion, could now be solved.
For by the ultimate velocity is meant that, with which the body is moved, neither before it arrives at its last place, when the motion ceases nor after but at the very instant when it arrives.
Isaac Newton
As civilizations developed, accurate measurement became essential. This ancient Egyptian tomb painting shows surveyors using rope to calculate the dimensions of a wheat field.
Fresh ideas
Mathematicians in medieval China and India made further advances in dealing with infinite sums. In the Islamic world, too, the development of algebra meant that, rather than spelling out a calculation millions of times for all possible variations, generalized symbols could be used to prove that a case is true for all numbers to infinity.
Mathematics had suffered a long period of stagnation in Europe but, as the Renaissance took hold in the 1300s, renewed interest in the subject led to fresh ideas about motion and the laws governing distance and speed. French mathematician and philosopher Nicole Oresme studied the velocity of an accelerating object against time, and he realized that the area under a graph depicting this relationship was equivalent to the distance traveled by the object. This notion would be formalized in the late 1600s by Isaac Newton and Isaac Barrow in England, Gottfried Leibniz in Germany, and Scottish mathematician James Gregory. Oresme’s work was inspired by that of the “Oxford Calculators,” a 14th-century group of scholars based at Merton College, Oxford, who developed the mean speed theorem, which Oresme later proved. It states that if one body is moving with a uniformly accelerated motion and a second body is moving with a uniform speed equal to the mean speed of the first body, and both bodies are moving for the same duration, they will cover the same distance. The Merton scholars were devoted to solving physical and philosophical problems using calculations and logic, and were interested in the quantitive analysis of phenomena such as heat, color, light, and velocity. They were inspired by the trigonometry of Arab astronomer al-Battani (858–929 CE) and the logic and physics of Aristotle.
This illustration of Kepler’s Platonic solid model of the Solar System appeared in a book published in 1596. Kepler used infinitesimally small strips to measure the distance covered in an orbit. This method was the forerunner of integration.
New developments
The incremental steps toward the development of calculus gathered pace toward the end of the 16th century. In around 1600, French mathematician François Viète promoted the use of symbols in algebra (which had previously been described in words), while Flemish mathematician Simon Stevin initiated the concept of mathematical limits, whereby the sum of amounts could converge to a limiting value, much like the area of Archimedes’ polygons converged to the area of a circle.
At much the same time, German mathematician and astronomer Johannes Kepler was researching the motion of the planets, including calculating the area enclosed by a planetary orbit, which he recognized as elliptical rather than circular. Using ancient Greek methods, he worked out the area by dividing the ellipse into strips of infinitesimal width.
A forerunner of the more formal integration to come, Kepler’s method was further developed in 1635 by Italian mathematician Bonaventura Cavalieri in Geometria indivisibilibus continuorum nova quadam ratione promota (Geometry, Advanced in a New Way by the Indivisibles of the Continua). Cavalieri worked out a “method of indivisibles,” which was a more rigorous method of determining the size of shapes. More developments followed in the 1600s with the work of English theologian and mathematician Isaac Barrow and Italian physicist Evangelista Torricelli, followed by that of Pierre de Fermat and René Descartes, whose analysis of curves advanced the new area of graphical algebra. Fermat also located maxima and minima, the greatest and least values of a curve.
Fluxion model
In 1665–66, English mathematician Isaac Newton developed his “method of fluxions,” a method for calculating variables that changed over time, which was a milestone in the history of calculus. Like Kepler and Galileo, Newton was interested in studying moving bodies and was particularly eager to unify the laws governing the motion of celestial bodies with motion on Earth.
In Newton’s fluxion model, he considered a point moving along a curve as being divided into two perpendicular components (x and y), and then considered the velocities of those components. This work laid the foundation for what became known as differential calculus (or differentiation), which together with the related field of integral calculus led to the fundamental theorem of calculus (see box, right). The idea of differential calculus is that the rate at which a variable changes at a point is equal to the gradient of a tangent at that point. This can be pictured by drawing a tangent (a straight line that touches a curve at only one point). The gradient or steepness of this line will be the rate of change of the curve at that point. Newton recognized that at the maxima and minima, the gradient of the curve was zero, because when something is at its highest or lowest point, it is momentarily not changing. Newton went on to develop his theory further by considering the converse problem—if the rate at which a variable changes is known, is it possible to calculate the shape of the variable itself? This “anti-differentiation” entailed working out areas under the curve.
Differentiation can be used to find the rate of change at a given point in time. The blue line shows the rate of change overall and the orange tangent shows the rate of change at a given point.
The fundamental theorem of calculus
James Gregory (1638–75) was the first person to formulate the fundamental theorem of calculus.
The study of calculus is underpinned by the fundamental theorem of calculus, specifying the relationship between differentiation and integration, both of which rely on the concept of infinitesimals. First articulated by James Gregory in his 1668 Geometriae Pars Universalis (The Universal Part of Geometry), it was then generalized by Isaac Barrow in 1670, and formalized in 1823 by Augustin-Louis Cauchy.
The theorem has two parts. The first states that integration and differentiation are opposites—for any continuous function (one that can be defined for all values), there exists an “anti-derivative” (or “integral”), whose derivative (a measure of the rate of change) is the function itself. The second part of the theorem states that if values are inserted into the anti-derivative F(x), the result—the definite integral of the function
f(x)—makes it possible to calculate areas under the curve of the function f(x).
Newton v. Leibniz
Around the time that Newton was developing his calculus, German mathematician Gottfried Leibniz was working on his own version, based on the consideration of infinitesimal changes in the two coordinates defining a point on a curve. Leibniz used very different notation from Newton’s, and in 1684 published a paper on what would later become known as differential calculus. Two years later, he published another paper, this time about integration, again using different notation from that of Newton. In an unpublished manuscript dated October 29, 1675, Leibniz was the first person to use the “integral” symbol ∫, which is used and recognized universally today.
There was much debate about who discovered modern calculus first: Newton or Leibniz. It led to protracted bitterness between the two rivals and across much of the mathematical community. Although Newton devised his theory of fluxions in 1665–66, he did not publish it until 1704, when it was added as an appendix to his work Opticks. Leibniz began to devise his version of calculus around 1673, and published it in 1684. Newton’s subsequent Principia is said by some to have been influenced by Leibniz’s work.
By 1712, Leibniz and Newton were openly accusing one another of plagiarism. The modern consensus is that Leibniz and Newton developed their ideas on the subject independently.