The Math Book

by DK

  Any point P, with coordinates (x, y), on the circumference of a circle can be connected to the center of the circle (0, 0) by a straight line (the circle’s radius) that forms the hypotenuse of a right-angled triangle with sides of length x and y. The equation of the circle is r2 = x2 + y2.

  Exchange of ideas

  In addition to drawing on theorems formulated by the ancient Greeks, Descartes exchanged ideas with other French mathematicians, among them Pierre de Fermat, with whom he frequently corresponded. Descartes and Fermat both made use of algebraic notation, the x and y system that François Viète had introduced at the end of the 1500s. Fermat also independently developed a coordinate system, but he did not publish it. Descartes was aware of Fermat’s ideas, no doubt using them to improve his own. Fermat also helped Dutch mathematician Frans van Schooten to understand Descartes’ ideas. Van Schooten translated La Géométrie into Latin and also popularized the use of coordinates as a mathematical technique.

  A modified form of polar coordinates that gives an aircraft’s destination in terms of angle and distance can be used as an alternative to GPS.

  New dimensions

  Van Schooten and Fermat had both suggested extending Cartesian coordinates into the third dimension. Today, mathematicians and physicists use coordinates to go much further than that and to imagine a space with any number of dimensions. Although it is almost impossible to visualize such a space, mathematicians can use these tools to describe lines moving in four, five, or as many spatial dimensions as they desire.

  Coordinates can also be used to examine the relationship between two quantities. This idea was pioneered as long ago as the 1370s, when a French monk called Nicole Oresme used rectangular coordinates and the geometric forms created by his results to understand, for instance, the relationship of elements such as speed and time, or the links between heat intensity and the degree of expansion due to heat.

  Some quantities can be represented using coordinates known as vectors, and exist in a purely mathematical “vector space.” Vectors are quantities with two values, which can be plotted as a magnitude (the length of a line) and a direction. Velocity is a vector as it has exactly those values (a quantity of speed and a direction of motion), while other vectors, such as Oresme’s heat and expansion, are visualized in this way to make it easier to add and subtract different sets of values or to manipulate them in another way.

  Mathematicians in the 1800s also found new purposes for Cartesian coordinates. They used them to represent complex numbers (sums of imaginary numbers, such as , and real numbers) or quaternions (the system that extends complex numbers) as vectors plotted in two, three, or more dimensions.

  The triumph of Cartesian ideas in mathematics… is in no small degree due to the Leiden professor Frans van Schooten.

  Dirk Struik

  Dutch mathematician

  The key coordinates

  The Cartesian coordinate system is by no means the only one. Geographic coordinates plot points on the globe as angles from preset great circles—the Equator and the Greenwich Meridian. A similar system, using celestial coordinates, describes the location of stars in an imaginary sphere centered on Earth and extended infinitely into space. Polar coordinates, determined by distance and angles from the center of Earth, are also useful for certain types of calculation.

  Cartesian coordinates remain an ubiquitous tool, however, able to plot anything from simple survey data to the movements of atoms. Without them, breakthroughs such as analytical calculus (which divides quantities into infinitesimally small amounts) and advances in space-time and non-Euclidean geometries could not have happened. Cartesian coordinates have had an immense impact in mathematics, and in many fields of science and the arts, from engineering and economics to robotics and computer animation.

  Mathematics is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency.

  René Descartes

  3-D Cartesian coordinates can be used to plot an object that has, for instance, width, depth, and height. Three axes (x, y, z) are set at right angles to each other. Where they meet is the origin (O).

  Polar coordinates

  The polar coordinate system is often used to calculate the movement of objects around, or in relation to, a central point.

  In mathematics, polar coordinates, which define points on a plane using two numbers, are the closest rivals to Descartes’ system. The first number, the radial coordinate r, is the distance from the central point—called the pole, not the origin. The second number, the angular coordinate (θ), is the angle that is defined as 0° from a single polar axis. To compare it with the Cartesian system, the polar axis would be the Cartesian x axis, and the polar coordinates (1,0°) would replace the Cartesian coordinates (1,0). The polar version of the Cartesian point (0,1) is (1,90°).

  Polar coordinates are used to help manipulate complex numbers plotted on a plane, especially for multiplication. Multiplying complex numbers is simplified when they are treated as polar coordinates, a process that involves multiplying the radial coordinates and adding the angular ones.

  See also: Pythagoras • Conic sections • Trigonometry • Rhumb lines • Viviani’s triangle theorem • The complex plane • Quaternions

  IN CONTEXT

  KEY FIGURES

  Bonaventura Cavalieri (1598–1647), Gilles Personne de Roberval (1602–75)

  FIELD

  Applied geometry

  BEFORE

  c. 240 BCE Archimedes investigates the volume and surface area of spheres in his Method Concerning Mechanical Theorems.

  1503 French mathematician Charles de Bovelles gives the first description of a cycloid in Introductio in geometriam (Introduction to Geometry).

  AFTER

  1656 Dutch mathematician Christiaan Huygens bases his invention of the pendulum clock on the curve of a cycloid.

  1693 De Roberval’s solution to the area of a cycloid is published more than 60 years after its discovery and 18 years after his death.

  The ancient Greeks puzzled over problems relating to areas and volumes of figures bounded by curves. They compared the areas of shapes by transforming each one into a square with the same area as the original shape, then compared the sizes of the squares. This was easy for shapes with straight edges, but curvilinear shapes caused problems.

  These problems remained unresolved until 1629, when Italian mathematician and Jesuit priest Bonaventura Cavalieri found a method for calculating the areas and volumes of curved shapes by slicing them into parallel pieces (Cavalieri’s principle), although he did not publish his results until six years later. In 1634, Gilles Personne de Roberval used this method to work out that the area beneath a cycloid (the arc traced by the rim of a rolling wheel) is three times the area of the circle used to generate the cycloid.

  This wheel has rolled over a piece of gum. The graph shows the path of the gum as the wheel rotates, creating a cycloid shape, which, as de Roberval discovered, has an area three times that of the wheel.

  Squaring the circle

  The ancient Greek mathematician Archimedes had used an ingenious method of exhaustion to determine the area between a parabola and a straight line. It entailed inscribing a triangle of known area to fit inside the parabola, then inscribing ever smaller triangles in all the gaps that remained. By adding together the areas of the triangles, Archimedes obtained a close approximation of the area he sought. The straight-edge-and-compass methods of his day, however, had their limitations. When he tried to calculate the surface area of a 3-D sphere using quadrature, a process which involves constructing a square of an area equal to a circle, he failed. He knew the surface area of the sphere was four times that of a circle of the same radius, but could not find a square that would give the surface area.

  A pretty result which I had not noticed before.

  René Descartes

  on de Roberval’s method for finding the area under a cycloid

  New spins on the problem

/>   The first description of a cycloid was published by Charles de Bovelles in 1503. Italian polymath Galileo gave the cycloid its name (from the Greek for “circular”) and tried to calculate its area by cutting up models of a cycloid and a circle, weighing the pieces, and comparing the results.

  Around 1628, Frenchman Marin Mersenne challenged his fellow mathematicians, including de Roberval, René Descartes, and Pierre de Fermat, to find both the area under the arch of a cycloid and a tangent to a point on the curve. When de Roberval told Descartes of his success, the latter dismissed it as “so small a result.” Descartes, in turn, discovered the tangent to a cycloid in 1638, and challenged de Roberval and Fermat to do the same. Only Fermat succeeded.

  In 1658, English architect Christopher Wren calculated the length of an arc of a cycloid as four times the diameter of the generating circle. The same year, Blaise Pascal calculated the area of any vertical slice of a cycloid. He also imagined rotating these vertical slices about a horizontal axis, and worked out the surface area and volume of the disks swept out by this rotation. Pascal’s use of infinitely small slices of shapes to solve the properties of cycloids would lead to the “fluxions” introduced by Isaac Newton as he developed early calculus.

  Since this shark-fin shape (left) and triangle (right) are the same height and the same width at equivalent points along their height, Cavalieri’s principle states that they can be sliced into parallel pieces that have similar area.

  GILLES PERSONNE DE ROBERVAL

  Born in 1602, in a field near Roberval in northern France, where his mother was bringing in the harvest, Gilles Personne de Roberval was tutored in classics and mathematics by the local priest. In 1628, he moved to Paris, where he joined Marin Mersenne’s circle of intellectuals.

  In 1632, de Roberval became professor of mathematics at the Collège Gervais, and two years later he won a competition for a highly prestigious post at the Collège Royale. He lived frugally, but managed to buy a farm for his extended family and leased out plots to generate income. He continued to practice mathematics all his life. In 1669, he invented a set of scales known as the Roberval balance. He died in 1675.

  Key work

  1693 Traité des Indivisibles (Treaty on Indivisibles)

  See also: Euclid’s Elements • Calculating pi • Mersenne primes • The problem of maxima • Pascal’s triangle • Huygens’s tautochrone curve • Calculus

  IN CONTEXT

  KEY FIGURE

  Girard Desargues (1591–1661)

  FIELD

  Applied geometry

  BEFORE

  c. 300 BCE Euclid’s Elements sets down ideas that will later constitute Euclidean geometry.

  c. 200 BCE In Conics, Apollonius describes the properties of conic sections.

  1435 Italian architect Leon Battista Alberti codifies the principles of perspective in De Pictura (On Painting).

  AFTER

  1685 In Sectiones Conicæ, French mathematician and painter Philippe de la Hire defines the hyperbola, parabola, and ellipse.

  1822 French mathematician and engineer Jean-Victor Poncelet writes a treatise on projective geometry.

  Unlike traditional Euclidean geometry, where all 2-D figures and objects belong in the same plane, projective geometry is concerned with how the apparent shape of an object is altered by the perspective from which that object is viewed. The 17th-century French mathematician Girard Desargues was a founder of such geometry.

  The idea of perspective had been addressed two centuries earlier by Renaissance artists and architects. Fillipo Brunelleschi had rediscovered the principles of linear perspective known to the ancient Greeks and Romans, and explored them in his architectural plans, sculptures, and paintings. Fellow architect Leon Battista Alberti used “vanishing points” to create a sense of 3-D perspective and wrote about the use of perspective in art.

  These two triangles are in perspective from a viewpoint called the center of perspectivity (P). Lines connecting the corresponding vertices of the triangles (X to x; Y to y, and Z to z) will always meet at P. If XYZ were a real triangular object, it would appear as the triangle xyz when viewed from P. Desargues’ theorem states that lines extending from the corresponding sides of each triangle will always meet on a line known as the axis of perspectivity.

  Perspective makes the parallel lines on sides of this flat-roofed building appear as though they will eventually meet. This meeting point is called a vanishing point.

  Good architecture should be a projection of life itself.

  Walter Gropius

  German architect

  From maps to math

  As Western explorers sailed to new lands, they needed accurate maps depicting the spherical world in two dimensions. In 1569, Flemish cartographer Gerardus Mercator devised a method now known as “cylindrical map projection.” This can be envisaged as the surface of the globe transferred onto a surrounding cylinder. When the cylinder is cut from top to bottom and rolled out, it becomes a two-dimensional map.

  In the 1630s, Desargues began investigating which properties were unchanged (invariant) when an image is projected onto a surface (perspective mapping). While its dimensions and angles may change, collinearity is preserved; this means that if three points XYZ are on a straight line, with Y between X and Z, then their images xyz are also on a straight line with y between x and z. An image of any triangle is another triangle. The corresponding sides of each triangle can be extended to meet at three points on a line (axis of perspectivity), and a line from each vertex to its corresponding vertex and beyond will meet at a point (the center of perspectivity).

  Desargues realized that all conic sections are equivalent in this way under projection. A single invariant property, such as collinearity, needs only to be proved for a single case, rather than tested on each conic. Pascal’s “mystic hexagram” theorem, for instance, states that the intersections of lines connecting pairs of six points on a conic all lie on a straight line. It can be shown by connecting six points on a circle, a proof valid for other conics, too.

  Desargues then considered what happens as the vertex of the projection cone moves further away. Parallel rays come from a point at infinity (such as the Sun). By adding these points at infinity to the Euclidean plane, each pair of lines meets at a point, including parallel lines, which meet at infinity.

  The method was developed into a full geometry by Poncelet in 1822. Today, projective geometry is used by architects and engineers in CAD technology, and in computer animation for films and gaming.

  When six arbitrary points are drawn on a circle and connected as shown (Ab, aB; Ac, Ca; Cb, cB), a straight line can be drawn through the points where lines of the same color cross. Using projection, this is true for an ellipse, too.

  GIRARD DESARGUES

  Born in 1591, Girard Desargues lived in Lyon all his life. He came from a family of wealthy lawyers who owned several properties, including a manor and a small chateau with fine vineyards. Desargues made several visits to Paris and, through Marin Mersenne, became friends with Descartes and Pascal.

  Desargues worked initially as a tutor and later as an engineer and architect. He was an excellent geometer and shared his ideas with his mathematical friends. Some of his pamphlets were later expanded into published papers. He wrote on perspective and applied mathematics to practical projects, such as designing a spiral staircase and a new form of pump. Desargues died in 1661. His work was rediscovered and republished in 1864.

  Key works

  1636 Perspective

  1639 Rough Draft of Attaining the Outcome of Intersecting a Cone with a Plane

  See also: Pythagoras • Euclid’s Elements • Conic sections • The area under a cycloid • Pascal’s triangle • Non-Euclidean geometries

  IN CONTEXT

  KEY FIGURE

  Blaise Pascal (1623–62)

  FIELDS

  Probability, number theory

  BEFORE

  975 Indian mathematician Halayudha gives the first surviving
description of numbers in Pascal’s triangle.

  c. 1050 In China, Jia Xian, describes the triangle later known as Yang Hui’s triangle.

  c. 1120 Omar Khayyam creates an early version of Pascal’s triangle.

  AFTER

  1713 Jacob Bernoulli’s Ars Conjectandi (The Art of Conjecturing) develops Pascal’s triangle.

  1915 Wacław Sierpinski describes the fractal pattern of triangles later known as Sierpinski triangles.

  Mathematics is often about the identification of number patterns, and one of the most remarkable number patterns of all is Pascal’s triangle. Pascal’s triangle is an equilateral triangle built from a very simple arrangement of numbers in ever-widening rows. Each number is the sum of the two adjacent numbers in the row above. Pascal’s triangle can be any size, ranging from just a few rows in depth to any number.

  While it might seem that such a simple rule for arranging numbers could only lead to simple patterns, Pascal’s triangle is fertile ground for several branches of higher mathematics, including algebra, number theory, probability, and combinatorics (the mathematics of counting and arranging). Many important sequences have been found in the triangle, and mathematicians believe that it may reflect some truths about relationships that we have yet to understand between numbers.