by DK
Laplace’s theory had a startling philosophical consequence. It can only work if the Universe follows a predictable mechanical path, so that everything from the spin of galaxies to the tiny atoms in nerve cells controlling thoughts could be mapped out into the future. This would mean that every aspect of a person’s life up until their death has already been predetermined; they have no free will and no agency over their thoughts and deeds.
Chance and statistics
Although mathematics helped to create such a crushing vision of reality, it also helped to dismiss it. By the 1850s, the study of heat and energy—thermodynamics—was ushering in a new model, the atomic world. To do this, it needed to describe the motion of atoms and molecules inside matter. Classical mechanics was not up to the task. Instead, physicists used a technique invented by Swiss mathematician Daniel Bernoulli in 1738, which used probability theory to model the movement of independent units within a space. Refined by Austrian physicist Ludwig Boltzmann, this technique became known as statistical mechanics. It described the atomic world in terms of random chance—something at odds with the mechanical determinism of Laplace’s demon. By the 1920s, the idea of a probabilistic Universe was solidified with the development of quantum physics, which has uncertainty at its heart.
PIERRE-SIMON LAPLACE
Born into an aristocratic family in 1749, Laplace lived through the French Revolution and the Reign of Terror, in which many of his friends were killed. In 1799, he became Minister of the Interior under Napoleon Bonaparte, but was dismissed after only six weeks for being too analytical and ineffectual. Laplace later sided with the Bourbons (the French royal family) and was rewarded with the return of his original title of marquis when the monarchy was restored.
Laplace’s demon was a side note to a career that also encompassed physics and astronomy, where Laplace was the first to postulate the concept of a black hole. His many contributions to mathematics were in classical mechanics, probability theory, and algebraic transformations. Laplace died in Paris in 1827.
Key works
1798–1828 Celestial Mechanics
1812 Analytic Theory of Probability
1814 A Philosophical Essay on Probabilities
See also: Probability • Calculus • Newton’s laws of motion • The butterfly effect
IN CONTEXT
KEY FIGURE
Siméon Poisson (1781–1840)
FIELD
Probability
BEFORE
1662 English merchant John Graunt publishes Natural and Political Observations upon the Bills of Mortality, marking the birth of statistics.
1711 Abraham de Moivre’s De Mensura Sortis (On the Measurement of Chance), describes what is later known as the Poisson distribution.
AFTER
1898 Russian statistician Ladislaus Bortkiewicz uses the Poisson distribution to study the number of Prussian soldiers killed by horse kicks.
1946 British statistician R. D. Clarke publishes a study, based on the Poisson distribution, of patterns of V-1 and V-2 flying bomb impacts on London.
In statistics, the Poisson distribution is used to model the number of times a randomly occurring event happens in a given interval of time or space. Introduced in 1837 by French mathematician Siméon Poisson, and based on the work of Abraham de Moivre, it can help to forecast a wide range of possibilities.
Take, for example, a chef who needs to forecast the number of baked potatoes that will be ordered in her café. She needs to decide how many potatoes to pre-cook each day. She knows the daily average order, and decides to prepare n potatoes where there is at least 90 percent certainty that n will match demand.
To use the Poisson distribution to calculate n, conditions must be met: orders must occur randomly, singly, and uniformly—on average, the same number of potatoes are ordered each day. If these conditions apply, the chef can find the value of n—how many potatoes to pre-bake. The average number of events per unit of space or time (lambda, or λ) is key. If λ = 4 (the average number of potatoes ordered in one day), and the number of potato orders on any one day is B, the probability that B is less than or equal to 6 is 89 percent, while the probability that B is less than or equal to 7 is 95 percent. The chef must be at least 90 percent sure that demand will be met, so n will be 7 here.
Siméon Poisson is credited with finding the Poisson distribution, but this may be an example of Stigler’s Law—no scientific discovery is credited to the true discoverer.
See also: Probability • Euler’s number • Normal distribution • The birth of modern statistics
IN CONTEXT
KEY FIGURE
Friedrich Wilhelm Bessel (1784–1846)
FIELD
Applied geometry
BEFORE
1609 Johannes Kepler shows that the orbits of the planets are ellipses.
1732 Daniel Bernoulli uses what later become known as Bessel functions to study the vibrations of a swinging chain.
1764 Leonhard Euler analyzes a vibrating membrane using what are later understood to be Bessel functions.
AFTER
1922 British mathematician George Watson writes his hugely influential A treatise on the theory of Bessel functions.
In the early 1800s, German mathematician and astronomer Friedrich Wilhelm Bessel gave solutions to a particular differential equation, the so-called Bessel equation. He systematically investigated these functions (solutions) in 1824. Now known as Bessel functions, they are useful to scientists and engineers. Central to the analysis of waves, such as electromagnetic waves moving along wires, they are also used to describe the diffraction of light, the flow of electricity or heat in a solid cylinder, and the motions of fluids.
Movement of the planets
The origins of Bessel functions lie in the pioneering work of German mathematician and astronomer Johannes Kepler in the early 1600s on the motions of the planets. His meticulous analysis of observations led him to realize that the orbits of the planets around the Sun are elliptical, not circular, and he described the three key laws of planetary motion. Mathematicians later used Bessel functions to make breakthroughs in various fields. Daniel Bernoulli found equations for the oscillations of a pendulum, and Leonhard Euler developed corresponding equations for the vibration of a stretched membrane. Euler and others also used Bessel functions to find solutions to the “three-body problem,” concerned with the motion of a body, such as a planet or moon, being acted upon by the gravitational fields of two other bodies.
Bessel’s functions are very beautiful functions, in spite of their having practical applications.
E. W. Hobson
British mathematician
See also: The problem of maxima • Calculus • The law of large numbers • Euler’s number • Fourier analysis
IN CONTEXT
KEY FIGURES
Charles Babbage (1791–1871), Ada Lovelace (1815–52)
FIELD
Computer science
BEFORE
1617 Scottish mathematician John Napier invents a manual calculating device.
1642–44 In France, Blaise Pascal creates a calculating machine.
1801 French weaver Joseph-Marie Jacquard demonstrates the first programmable machine – a loom controlled by a punchcard.
AFTER
1944 British codebreaker Max Newman builds Colossus, the first digital electronic programmable computer.
British mathematician and inventor Charles Babbage anticipated the computer age by more than a century with two ideas for mechanical calculators and “thinking” machines. The first he called the Difference Engine, a calculating machine that would work automatically, using a combination of brass cogs and rods. Babbage only managed to part-build the machine, but even this was able to process complex calculations accurately in moments.
The second, more ambitious, idea was the Analytical Engine. It was never built, but was envisaged as a machine that could respond to new problems and solve them without human intervention. The project rec
eived crucial input from Ada Lovelace, a brilliant young mathematician. Lovelace anticipated many of the key mathematical aspects of computer programming and foresaw how the machine could be used to analyze any kinds of symbol.
Charles Babbage was spurred to start his work on a mechanical calculator by the errors he found in astronomical tables produced by poorly paid and unreliable workers.
Automatic calculation
In the 17th and 18th centuries, mathematicians such as Gottfried Leibniz and Blaise Pascal had created mechanical calculating aids, but these were limited in power and also prone to error as human input was needed at every step. Babbage’s idea was to create a calculating machine that worked automatically, eliminating human error. He called his machine the Difference Engine because it allowed complex multiplications and divisions to be reduced to additions and subtractions—“differences”—that could be handled by scores of interlocking cogs. It would even print out the results.
No previous calculator had ever worked with numbers larger than four digits. Yet the Difference Engine was designed to handle numbers of up to 50 digits by means of more than 25,000 moving parts.
To set up the machine for a calculation, each number was represented by a column of cogwheels, and each cogwheel was marked with digits from 0 to 9. A number was set by turning the cogwheels in the column to show the correct digit on each. The machine would then work through the entire calculation automatically.
Babbage built several small working models with just seven number columns but remarkable calculating power. In 1823, he managed to persuade the British government to part-fund the project, with the promise that it would make producing official tables much quicker, cheaper, and more accurate. However, the full machine was hugely expensive to develop, and tested the technological capability of the day to its limits. After two decades’ work, the government canceled the project in 1842.
Meanwhile, in drawings and calculations, Babbage had also been working on his idea for an Analytical Engine. His papers suggest that the machine, if built, could have been close to what we now call a computer. His design anticipated virtually all of the key components of the modern computer, including the central processing unit (CPU), memory storage, and integrated programs.
One problem facing Babbage was what to do with numbers carried over into the next column when adding up columns of digits. At first, he used a separate mechanism for each carryover, but that proved too complicated. Then he split his machine into two parts, the “Mill” and the “Store,” which made it possible to separate the addition and carryover processes. The Mill was where the arithmetical operations were performed; the Store was where numbers were held before processing and then received back from the Mill after processing. The Mill was Babbage’s version of a computer’s CPU, while the Store acted as its memory.
The idea of telling a machine what it should do—programming—came from a French weaver, Joseph-Marie Jacquard. He developed a loom that used cards punched with holes to tell it how to weave complex patterns in silk. In 1836, Babbage realized he too could use punched cards—to control his own machine but also to record results and calculation sequences.
At each increase of knowledge, as well as on the contrivance of every new tool, human labor becomes abridged.
Charles Babbage
This replica of the demonstration model Babbage made in 1832 of Difference Engine No. 1 has three columns, each with its numbered cogwheels. Two are for calculation, one for the result.
A supporting genius
One of the greatest advocates for Babbage’s work was his fellow mathematician Ada Lovelace, who wrote of the Analytical Engine that it would “weave algebraic patterns just as the Jacquard loom weaves flowers and leaves.” As a teenager in 1832, Lovelace had seen one of the Difference Engine models working and had been instantly entranced. In 1843, she arranged the publication of her translation of a pamphlet about the Analytical Engine written by Italian engineer Luigi Menabrea, to which she added extensive explanatory notes.
Many of these notes covered systems that would become part of modern computing. In “Note G,” Lovelace described possibly the first computer algorithm, “to show an implicit function can be worked out by the engine without human head and hands first.” She also theorized that the engine could solve problems by repeating a series of instructions—a process known today as “looping.” Lovelace envisaged a program card, or set of cards, that returned repeatedly to its original position to work on the next data card or set. In this way, Lovelace argued, the machine could solve a system of linear equations or generate extensive tables of prime numbers. Perhaps the greatest insight in her notes was Lovelace’s vision of machines as mechanical brains with wide applications. “The engine can arrange and combine its numerical quantities exactly as if they were letters or any other general symbols,” she wrote, realizing that any kind of symbol, not just numbers, could be manipulated and processed by machines. This is the difference between calculation and computation—and the basis of the modern computer. Lovelace also foresaw how such machines would be limited by the quality of the input. Arguably, the first programmable computer— rather than calculator—was created by Konrad Zuse in 1938.
The object of the Analytical Engine is twofold. First, the complete manipulation of number. Second, the complete manipulation of algebraical symbols.
Charles Babbage
Delayed legacy
Lovelace’s plans to develop Babbage’s work were curtailed by her early death, by which time Babbage himself was tired, ill, and disillusioned by the lack of support for his Difference Engine. The high-precision mechanics required to build the machine were beyond what any engineer could achieve at the time. Largely forgotten until they were republished in 1953, Lovelace’s notes confirm that she and Babbage foresaw many of the features of the computer now found in every home and office.
The more I study [the Analytical Engine], the more insatiable I feel my genius for it to be.
Ada Lovelace
ADA LOVELACE
Born Augusta Byron in London in 1815, Ada, Countess of Lovelace, was the only legitimate child of the poet Lord Byron. Byron left England a few months after her birth, and Lovelace never saw her father again. Her mother, Lady Byron, was mathematically gifted—Byron called her his “Princess of Parallelograms”— and insisted Lovelace study mathematics, too.
Lovelace became renowned for her talents in mathematics and languages. She met Charles Babbage when she was 17 and was fascinated by his work. Two years later, she married William King, Earl of Lovelace, with whom she had three children, but she continued to study mathematics and follow the progress of Babbage, who called her “the Enchantress of Number.”
Lovelace wrote exhaustive notes on Babbage’s Analytical Engine. She set out many ideas about what was to become computing, earning herself a reputation as the first computer programmer. Lovelace died in 1852 of uterine cancer; in line with her wishes, she was buried next to her father.
See also: Binary numbers • Matrices • The infinite monkey theorem • The Turing machine • Information theory • The four-color theorem
IN CONTEXT
KEY FIGURE
Carl Gustav Jacob Jacobi (1804–51)
FIELDS
Number theory, geometry
BEFORE
1655 John Wallis applies calculus to the length of an elliptic curve; the elliptic integral he derives is defined by an infinite series of terms.
1799 Carl Gauss determines the key characteristics of elliptic functions, but his work is not published until 1841.
1827–28 Niels Abel independently derives and publishes the same findings as Gauss.
AFTER
1862 German mathematician Karl Weierstrass develops a general theory of elliptic functions, showing that they can be applied to problems in both algebra and geometry.
The “squashed circle” of an ellipse is one of the most recognizable curves in math. Ellipses have a long history in mathematics. They wer
e studied by the ancient Greeks as one of the conic sections. Slicing through a cone horizontally creates a circle; slicing at a steeper angle creates an ellipse (and then open curves called a parabola and a hyperbola). An ellipse is a closed curve that is defined as the set of all points in a plane, the sum of whose distances from two fixed points—each one called a focus—is always the same number. (A circle is a special ellipse with just one central focus, not two.) In 1609, German astronomer and mathematician Johannes Kepler demonstrated that the orbits of the planets were elliptical, with the Sun being located at one of the foci.
I learnt with as much astonishment as satisfaction that two young geometers…succeeded in their own individual work in considerably improving the theory of elliptic functions.
Adrien-Marie Legendre
New tools
Just as the mathematics of a circle could be used to model and predict natural phenomena that varied and repeated in a rhythmic (or periodic) way, such as the up-and-down motion of a simple sound wave, the mathematics of the ellipse can be used to do the same for phenomena that follow more complex periodic patterns, such as electromagnetic fields or the orbital motion of planets.