Mathematics

Around 300 BC in Alexandria, Euclid collected the geometry of his day into the Elements, thirteen books that were still used as a school text two thousand years later. Its method mattered more than any single result. It starts from a short list of definitions, five postulates and a few common notions, and then proves every proposition from those and from propositions already proved.

The fifth postulate, on parallel lines, is the odd one out: longer and less obvious than the rest. For centuries people tried to prove it from the other four and failed. In the nineteenth century Lobachevsky, Bolyai and Gauss showed why. You can replace it and still get a consistent geometry, one in which the angles of a triangle add up to less than two right angles. Riemann went on to describe curved spaces in general, and Einstein used that geometry for gravity.

Book I ends with Pythagoras' theorem and its converse. Book IX proves that there is no largest prime: multiply any finite list of primes together, add one, and the result has a prime factor missing from the list.

Calculus is the mathematics of change. The derivative measures how fast a quantity changes at an instant; the integral adds up infinitely many small pieces to give a total, such as the area under a curve or the distance covered at a varying speed.

Newton and Leibniz found the methods independently, Newton in the mid-1660s and Leibniz in the mid-1670s. Leibniz published first, in 1684 and 1686. The quarrel over priority soured relations between English and continental mathematicians for a century. Both men saw the central fact: differentiation and integration undo each other. That is the fundamental theorem of calculus.

The notation in use today is mostly Leibniz's: dy/dx for the derivative and the long S of the integral sign, from the Latin summa. It survived because it does some of the thinking for you. The chain rule, for example, looks like cancelling fractions. Cauchy and Weierstrass gave the subject a rigorous footing in the nineteenth century with limits, replacing the 'infinitely small' quantities that Bishop Berkeley had attacked in 1734.

Gottfried Wilhelm Leibniz (1646–1716) was a lawyer, diplomat and librarian to the House of Hanover, and did his mathematics alongside all of it. Besides the calculus, he worked out binary arithmetic and showed how every number can be written with 0 and 1. He built a mechanical calculator, the stepped reckoner, that could multiply and divide. He also hoped for a universal symbolic language in which disputes could be settled by calculation.

His series for π, the alternating sum of the reciprocals of the odd numbers, converges far too slowly to be useful for computing. It is still a striking result, because it links π to nothing more than the odd integers. Madhava of Sangamagrama in Kerala had found the same series about two and a half centuries earlier.

A differential equation states a relation between a quantity and its rates of change. Most laws of physics take this form. They describe how a system changes from one moment to the next, and the solution tells you where it will be later.

The simplest interesting example says that a quantity grows at a rate proportional to its size. Its solution is the exponential function. It describes unchecked population growth, compound interest and, with a negative constant, radioactive decay. A mass on a spring obeys a second-order equation, and its solution is a sine wave.

Most differential equations cannot be solved in closed form. Euler gave the first numerical method in the 1760s: take small steps, each in the direction the equation points. Poincaré, studying the three-body problem in the 1880s, changed the question from 'what is the formula?' to 'what does the solution look like?' That shift led eventually to chaos theory.

A vector is a quantity with both size and direction, such as a velocity, a force or a displacement. In coordinates it is a list of numbers, and vectors are added component by component. The dot product of two vectors measures how far they point the same way. It is zero exactly when they are perpendicular.

A matrix is a rectangular array of numbers that describes a linear map. A linear map sends straight lines to straight lines and keeps the origin fixed. Rotations, reflections, stretches and shears are all linear maps. Multiplying two matrices gives the map that does one and then the other, so the order matters.

Solving many linear equations at once is the oldest use of the subject. The Chinese text The Nine Chapters on the Mathematical Art, compiled by the second century AD, solves such systems by the method now called Gaussian elimination. Eigenvectors are the directions a map only stretches, without turning. They underlie the natural frequencies of a bridge, the principal components of a data set and the energy levels of an atom.

A parametric curve gives each coordinate as a function of a third variable, usually called t and often read as time. Instead of asking which y goes with which x, you follow a point as it moves. This handles curves that loop back on themselves or run vertically, which no single function y = f(x) can describe.

The circle is the simplest case. The cycloid is the path traced by a point on the rim of a rolling wheel. It answers two classic problems. Johann Bernoulli posed the brachistochrone problem in 1696: which curve takes a sliding bead from one point to another in the least time? The answer is the cycloid. Huygens had already shown that the cycloid is also the tautochrone. A bead released anywhere on an inverted cycloid reaches the bottom in the same time.

The velocity of the moving point is the derivative of each coordinate with respect to t, and its length gives the speed. Integrating the speed over t gives the length of the curve.