Gregersen E. (editor) The Britannica Guide to Statistics and Probability
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of the state in the economy and touched on the labour
theory of value.
Petty studied medicine at the Universities of Leiden,
Paris, and Oxford. He was successively a physician; pro-
fessor of anatomy at Oxford; professor of music in London;
inventor, surveyor, and landowner in Ireland; and a mem-
ber of Parliament. As a proponent of the empirical
scientific doctrines of the newly established Royal Society,
of which he was a founder, Petty was one of the originators
of political arithmetic, which he defined as the art of rea-
soning by figures upon things relating to government. His
Essays in Political Arithmetick and Political Survey or Anatomy
of Ireland (1672) presented rough but ingeniously calcu-
lated estimates of population and of social income. His
ideas on monetary theory and policy were developed in
Verbum Sapienti (1665) and in Quantulumcunque Concerning
Money, 1682 (1695).
Petty originated many of the concepts that are still
used in economics today. He coined the term full employ-
ment, for example, and stated that the price of land equals
the discounted present value of expected future rent on
the land.
siMéon-denis poisson
(b. June 21, 1781, Pithiviers, France—d. April 25, 1840, Sceaux)
The French mathematician Siméon-Denis Poisson was
known for his work on definite integrals, electromagnetic
theory, and probability.
Poisson’s family had intended him for a medical career,
but he showed little interest or aptitude and in 1798 began
studying mathematics at the École Polytechnique in Paris
under the mathematicians Pierre-Simon Laplace and
Joseph-Louis Lagrange, who became his lifelong friends.
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He became a professor at the École Polytechnique in 1802.
In 1808 he was made an astronomer at the Bureau of
Longitudes, and, when the Faculty of Sciences was insti-
tuted in 1809, he was appointed a professor of pure
mathematics.
Poisson’s most important work concerned the applica-
tion of mathematics to electricity and magnetism,
mechanics, and other areas of physics. His Traité de méca-
nique (1811 and 1833; “Treatise on Mechanics”) was the
standard work in mechanics for many years. In 1812 he
provided an extensive treatment of electrostatics, based
on Laplace’s methods from planetary theory, by postulat-
ing that electricity is made up of two fluids in which like
particles are repelled and unlike particles are attracted
with a force that is inversely proportional to the square of
the distance between them.
Poisson contributed to celestial mechanics by extend-
ing the work of Lagrange and Laplace on the stability of
planetary orbits and by calculating the gravitational
attraction exerted by spheroidal and ellipsoidal bodies.
His expression for the force of gravity in terms of the
distribution of mass within a planet was used in the late
20th century for deducing details of the shape of the
Earth from accurate measurements of the paths of orbit-
ing satellites.
Poisson’s other publications include Théorie nouvelle de
l’action capillaire (1831; “A New Theory of Capillary Action”)
and Théorie mathématique de la chaleur (1835; “Mathematical
Theory of Heat”). In Recherches sur la probabilité des juge-
ments en matière criminelle et en matière civile (1837; “Research
on the Probability of Criminal and Civil Verdicts”), an
important investigation of probability, the Poisson distri-
bution appears for the first and only time in his work.
Poisson’s contributions to the law of large numbers (for
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independent random variables with a common distribu-
tion, the average value for a sample tends to the mean as
sample size increases) also appeared therein. Although
originally derived as merely an approximation to the bino-
mial distribution (obtained by repeated, independent
trials that have only one of two possible outcomes), the
Poisson distribution is now fundamental in the analysis of
problems concerning radioactivity, traffic, and the ran-
dom occurrence of events in time or space.
In pure mathematics his most important works were a
series of papers on definite integrals and his advances in
Fourier analysis. These papers paved the way for the
research of the German mathematicians Peter Dirichlet
and Bernhard Riemann.
adolphe queTeleT
(b. Feb. 22, 1796, Ghent, Belg.—d. Feb. 17, 1874, Brussels)
The Belgian mathematician, astronomer, statistician, and
sociologist Lambert Adolphe Jacques Quetelet was known
for his application of statistics and probability theory to
social phenomena.
From 1819 Quetelet lectured at the Brussels Athenaeum,
military college, and museum. In 1823 he went to Paris
to study astronomy, meteorology, and the management
of an astronomical observatory. While there he learned
probability from Joseph Fourier and, conceivably, from
Pierre-Simon Laplace. Quetelet founded (1828) and
directed the Royal Observatory in Brussels, served as per-
petual secretary of the Belgian Royal Academy (1834–74),
and organized the first International Statistical Congress
(1853). For the Dutch and Belgian governments, he col-
lected and analyzed statistics on crime, mortality, and other
subjects and devised improvements in census taking. He
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also developed methods for simultaneous observations of
astronomical, meteorological, and geodetic phenomena
from scattered points throughout Europe.
In Sur l’homme et le développement de ses facultés, ou essai de
physique sociale (1835; A Treatise on Man and the Development
of His Faculties), he presented his conception of the homme
moyen (“average man”) as the central value about which
measurements of a human trait are grouped according to
the normal distribution. His studies of the numerical con-
stancy of such presumably voluntary acts as crimes
stimulated extensive studies in “moral statistics” and wide
discussion of free will versus social determinism. In trying
to discover through statistics the causes of antisocial acts,
Quetelet conceived of the idea of relative propensity to
crime of specific age groups. Like his homme moyen idea,
this evoked great controversy among social scientists in
the 19th century.
Jakob sTeineR
(b. March 18, 1796, Utzenstorf, Switz.—d. April 1, 1863, Bern)
The Swiss mathematician Jakob Steiner was one of the
founders of modern synthetic and projective geometry.
As the son of a small farmer, Steiner had no early
schooling and did not learn to write until he was 14. Against
the wishes of his parents, at 18 he entered the Pestalozzi
School at Yverdon, Switzerland, where his extraordinary
geometric intuition was discovered. Later, he went to the
University of Heidelberg and the University of Berlin to
study, supporting himself precariously as a tutor. By 1824
he had studied the geometric transformations that led
him to the theory of inversive geometry, but he did not
publish this work. In 1826 the first regular publication
devoted to mathematics, Crelle’s Journal, was founded,
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giving Steiner an opportunity to publish some of his other
original geometric discoveries. In 1832 he received an hon-
orary doctorate from the University of Königsberg, and
two years later he occupied the chair of geometry estab-
lished for him at Berlin, a post he held until his death.
During his lifetime some considered Steiner the great-
est geometer since Apollonius of Perga (c. 262–190 BCE),
and his works on synthetic geometry were considered
authoritative. He had an extreme dislike for the use of
algebra and analysis, and he often expressed the opinion
that calculation hampered thinking, whereas pure geom-
etry stimulated creative thought. By the end of the century,
however, it was generally recognized that Karl von Staudt
(1798–1867), who worked in relative isolation at the
University of Erlangen, had made far deeper contributions
to a systematic theory of pure geometry. Nevertheless,
Steiner contributed many basic concepts and results in
projective geometry. For example, he discovered a trans-
formation of the real projective plane (the set of lines
through the origin in ordinary three-dimensional space)
that maps each line of the projective plane to one point on
the Steiner surface (also known as the Roman surface).
His other work was primarily on the properties of alge-
braic curves and surfaces and on the solution of
isoperimetric problems. His collected writings were pub-
lished posthumously as Gesammelte Werke, 2 vol. (1881–82;
“Collected Works”).
JaMes Joseph sylvesTeR
(b. Sept. 3, 1814, London, Eng.—d. March 15, 1897, London)
British mathematician James Joseph Sylvester, with Arthur
Cayley, was a cofounder of invariant theory, the study of
properties that are unchanged (invariant) under some
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transformation, such as rotating or translating the coordi-
nate axes. He also made significant contributions to
number theory and elliptic functions.
In 1837 Sylvester came second in the mathematical tri-
pos at the University of Cambridge but, as a Jew, was
prevented from taking his degree or securing an appoint-
ment there. The following year he became a professor of
natural philosophy at University College, London (the
only nonsectarian British university). In 1841 he accepted
a professorship of mathematics at the University of
Virginia, Charlottesville, U.S., but resigned after only
three months following an altercation with a student for
which the school’s administration did not take his side.
He returned to England in 1843. The following year he
went to London, where he became an actuary for an insur-
ance company, retaining his interest in mathematics only
through tutoring (his students included Florence
Nightingale). In 1846 he became a law student at the Inner
Temple, and in 1850 he was admitted to the bar. While
working as a lawyer, Sylvester began an enthusiastic and
profitable collaboration with Cayley.
From 1855 to 1870 Sylvester was a professor of mathe-
matics at the Royal Military Academy in Woolwich. He
went to the United States once again in 1876 to become a
professor of mathematics at Johns Hopkins University in
Baltimore, Maryland. While there he founded (1878) and
became the first editor of the American Journal of
Mathematics, introduced graduate work in mathematics
into American universities, and greatly stimulated the
American mathematical scene. In 1883 he returned to
England to become the Savilian Professor of Geometry at
the University of Oxford.
Sylvester was primarily an algebraist. He did brilliant
work in the theory of numbers, particularly in partitions
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(the possible ways a number can be expressed as a sum of
positive integers) and Diophantine analysis (a means for
finding whole-number solutions to certain algebraic equa-
tions). He worked by inspiration, and frequently it is
difficult to detect a proof in his confident assertions. His
work is characterized by powerful imagination and inven-
tiveness. He was proud of his mathematical vocabulary
and coined many new terms, but few have survived. In
1839 he was elected a fellow of the Royal Society, and he
was the second president of the London Mathematical
Society (1866–68). His mathematical output includes sev-
eral hundred papers and one book, Treatise on Elliptic
Functions (1876). He also wrote poetry, although not to crit-
ical acclaim, and published Laws of Verse (1870).
John von neuMann
(b. Dec. 28, 1903, Budapest, Hung.—d. Feb. 8, 1957, Washington,
D.C., U.S.)
Hungarian-born American mathematician John von
Neumann grew from a child prodigy to one of the world’s
foremost mathematicians by his mid-twenties. He pio-
neered game theory and, along with Alan Turing and
Claude Shannon, was one of the conceptual inventors of
the stored-program digital computer.
Von Neumann showed signs of genius in early child-
hood: He could joke in Classical Greek and, for a family
stunt, he could quickly memorize a page from a telephone
book and recite its numbers and addresses. Upon comple-
tion of von Neumann’s secondary schooling in 1921, his
father discouraged him from pursuing a career in mathe-
matics, fearing that there was not enough money in the
field. As a compromise, von Neumann simultaneously
studied chemistry and mathematics. He earned a degree
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in chemical engineering (1925) from the Swiss Federal
Institute in Zürich and a doctorate in mathematics (1926)
from the University of Budapest.
Von Neumann commenced his intellectual career
when the influence of David Hilbert and his program of
establishing axiomatic foundations for mathematics was
at a peak, working under Hilbert from 1926 to 1927 at the
University of Göttingen. The goal of axiomatizing math-
ematics was defeated by Kurt Gödel’s incompleteness
theorems, a barrier that Hilbert and von Neumann imme-
diately understood. The work with Hilbert culminated in
von Neumann’s book The Mathematical Foundations of
Quantum Mechanics (1932), in which quantum states are
treated as vectors in a Hilbert space. This mathematical
synthesis reconciled the seemingly contradictory quan-
tum mechanical formulations of Erwin Schrödinger and
Werner Heisenberg.
In 1928 von Neumann published “Theory of Parlor
Games,” a key paper in the field of game theory. The
nominal inspiration was the game of poker. Game theory
focuses on the element of bluffing, a feature distinct from
the pure logic of chess or the probability theory of rou-
lette. Though von Neumann knew of the earlier work of
the French mathematician Émile Borel, he gave the sub-
ject mathematical substance by proving the mini-max
theorem. This asserts that for every finite, two-person
zero-sum game, there is a rational outcome in the
sense that two perfectly logical adversaries can arrive
at a mutual choice of game strategies, confident that
they could not expect to do better by choosing another
strategy. In games like poker, the optimal strategy incor-
porates a chance element. Poker players must bluff
occasionally—and unpredictably—to avoid exploitation
by a savvier player.
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In 1933 von Neumann became one of the first professors
at the Institute for Advanced Study (IAS), Princeton, N.J.
The same year, Adolf Hitler came to power in Germany, and
von Neumann relinquished his German academic posts.
Although Von Neumann once said he felt he had not
lived up to all that had been expected of him, he became a
Princeton legend. It was said that he played practical jokes
on Einstein and could recite verbatim books that he had
read years earlier. Von Neumann’s natural diplomacy
helped him move easily among Princeton’s intelligentsia,
where he often adopted a tactful modesty. Never much
like the stereotypical mathematician, he was known as a
wit, bon vivant, and aggressive driver—his frequent auto
accidents led to one Princeton intersection being dubbed
“von Neumann corner.”
In late 1943 von Neumann began work on the Manhattan
Project, working on Seth Neddermeyer’s implosion design
for an atomic bomb at Los Alamos, N.M. This called for a
hollow sphere containing fissionable plutonium to be sym-
metrically imploded to drive the plutonium into a critical
mass at the centre. The implosion had to be so symmetri-
cal that it was compared to crushing a beer can without
splattering any beer. Adapting an idea proposed by James
Tuck, von Neumann calculated that a “lens” of faster- and
slower-burning chemical explosives could achieve the
needed degree of symmetry. The Fat Man atomic bomb
dropped on Nagasaki used this design.
Overlapping with this work was von Neumann’s mag-
num opus of applied math, Theory of Games and Economic
Behavior (1944), cowritten with Princeton economist
Oskar Morgenstern. Game theory had been orphaned
since the 1928 publication of “Theory of Parlor Games,”
with neither von Neumann nor anyone else significantly
developing it. The collaboration with Morgernstern
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burgeoned to 641 pages, the authors arguing for game
theory as the “Newtonian science” underlying economic
decisions. The book invigorated a vogue for game theory
among economists that has partly subsided. The theory
has also had broad influence in fields ranging from evolu-
tionary biology to defense planning.
Starting in 1944, he contributed important ideas to the
U.S. Army’s hard-wired Electronic Numerical Integrator
and Computer (ENIAC) computer. Most important,
von Neumann modified the ENIAC to run as a stored-
program machine. He then lobbied to build an improved
computer at the Institute for Advanced Studies (IAS). The
IAS machine, which began operating in 1951, used binary
arithmetic (the ENIAC had used decimal numbers) and
John von Neumann pioneered game theory, contributed to the infamous
Manhattan Project, and was one of the conceptual inventors of the stored-
program digital computer. Alfred Eisenstaedt/Time & Life Pictures/
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