Gregersen E. (editor) The Britannica Guide to Statistics and Probability
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College, Cambridge (founded in 1871), during the
1880s. Despite aiding the careers of a few students who
naturally took to pure mathematics, Cayley never estab-
lished a full-fledged research school of mathematics at
Cambridge.
In mathematics Cayley was an individualist. He
handled calculations and symbolic manipulations with
formidable skill, guided by a deep intuitive understanding
of mathematical theories and their interconnections. His
ability to keep abreast of current work while seeing the
wider view enabled him to perceive important trends and
to make valuable suggestions for further investigation.
Cayley made important contributions to the algebraic
theory of curves and surfaces, group theory, linear algebra,
graph theory, combinatorics, and elliptic functions. He
formalized the theory of matrices. Among Cayley’s most
important papers were his series of 10 “Memoirs on
Quantics” (1854–78). A quantic, known today as an alge-
braic form, is a polynomial with the same total degree for
each term. For example, every term in the following poly-
nomial has a total degree of 3:
x
3
+ 7x
2
y − 5xy
2
+ y
3
.
Alongside work produced by his friend James Joseph
Sylvester, Cayley’s study of various properties of forms that
are unchanged (invariant) under some transformation, such
as rotating or translating the coordinate axes, established a
branch of algebra known as invariant theory.
In geometry Cayley concentrated his attention on
analytic geometry, for which he naturally employed invari-
ant theory. For example, he showed that the order of
points formed by intersecting lines is always invariant,
regardless of any spatial transformation. In 1859 Cayley
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outlined a notion of distance in projective geometry (a
projective metric), and he was one of the first to realize
that Euclidean geometry is a special case of projective
geometry—an insight that reversed current thinking. Ten
years later, Cayley’s projective metric provided a key for
understanding the relationship between the various types
of non-Euclidean geometries.
Cayley was essentially a pure mathematician, but he
also pursued mechanics and astronomy. He was active in
lunar studies and produced two widely praised reports on
dynamics (1857, 1862). Cayley had an extraordinarily pro-
lific career, producing almost a thousand mathematical
papers. His habit was to embark on long studies punc-
tuated by rapidly written “bulletins from the front.”
Cayley effortlessly wrote French and often published in
Continental journals. As a young graduate at Cambridge,
he was inspired by the work of the mathematician Karl
Jacobi (1804–51). In 1876 Cayley published his only book,
An Elementary Treatise on Elliptic Functions, which drew out
this widely studied subject from Jacobi’s point of view.
Cayley was awarded numerous honours, including the
Copley Medal in 1882 by the Royal Society. At various
times he was president of the Cambridge Philosophical
Society, the London Mathematical Society, the British
Association for the Advancement of Science, and the
Royal Astronomical Society.
fRancis ysidRo edgewoRTh
(b. Feb. 8, 1845, Edgeworthstown, County Longford, Ire.—d. Feb. 13,
1926, Oxford, Oxfordshire, Eng.)
Francis Ysidro Edgeworth was an Irish economist and
statistician who innovatively applied mathematics to the
fields of economics and statistics.
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Edgeworth was educated at Trinity College in Dublin
and Balliol College, Oxford, graduating in 1869. In 1877 he
qualified as a barrister. He lectured at King’s College in
London from 1880, becoming professor of political econ-
omy in 1888. From 1891 to 1922 he was Drummond Professor
of Economics at Oxford. He also played an important role
as editor of the Economic Journal (1891–1926).
Although Edgeworth was strong in mathematics, he
was weak at prose, and his publications failed to reach
a popular audience. He had hoped to use mathematics
to illuminate ethical questions, but his first work, New
and Old Methods of Ethics (1877), depended so heavily on
mathematical techniques—especially the calculus of
variations—that the book may have deterred otherwise
interested readers. His most famous work, Mathematical
Psychics (1881), presented his new ideas on the generalized
utility function, the indifference curve, and the contract
curve, all of which have become standard devices of eco-
nomic theory.
Edgeworth contributed to the pure theory of interna-
tional trade and to taxation and monopoly theory. He also
made important contributions to the theory of index
numbers and to statistical theory, in particular to proba-
bility, advocating the use of data from past experience as
the basis for estimating future probabilities. John Kenneth
Galbraith once remarked that “all races have produced
notable economists, except the Irish.” Edgeworth is a
strong counterexample to Galbraith’s claim.
pieRRe de feRMaT
(b. Aug. 17, 1601, Beaumont-de-Lomagne, France—d. Jan. 12,
1665, Castres)
The French mathematician Pierre de Fermat is often
called the founder of the modern theory of numbers.
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Co-founder of the theory of probability, Pierre de Fermat was the most pro-
ductive mathematician of his day. Photos.com
Together with René Descartes, Fermat was one of the
two leading mathematicians of the first half of the 17th
century. Through his correspondence with Blaise Pascal,
he was a co-founder of the theory of probability.
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In 1631 Fermat received the baccalaureate in law from
the University of Orléans. He served in the local parlia-
ment at Toulouse, becoming councillor in 1634. Sometime
before 1638 he became known as Pierre de Fermat, but the
authority for this designation is uncertain. In 1638 he was
named to the Criminal Court.
Through the mathematician and theologian Marin
Mersenne, who, as a friend of Descartes, often acted as
an intermediary with other scholars, Fermat in 1638 main-
tained a controversy with Descartes on the validity of their
respective methods for tangents to curves. Fermat’s views
were fully justified some 30 years later in the calculus of Sir
Isaac Newton. Recognition of the significance of Fermat’s
work in analysis was tardy, in part because he adhered to
the system of mathematical symbols devised by François
Viète, notations that Descartes’s Géométrie had rendered
largely obsolete. The handicap imposed by the awkward
notations operated less severely in Fermat’s favourite
field of study, the theory of numbers, but, unfortunately,
he found no correspondent to share his enthusiasm. In
1654 he had enjoyed an exchange of letters with his fellow
mathematician Blaise Pascal on problems in probability
concerning games of chance, the results of which were
extended and published by Huygens in his De Ratiociniis
in Ludo Aleae (1657).
Fermat vainly sought to persuade Pascal to join him in
research in number theory. Inspired by an edition in 1621
of the Arithmetic of Diophantus, the Greek mathemati-
cian of the 3rd century CE, Fermat had discovered new
results in the so-called higher arithmetic, many of which
concerned properties of prime numbers (those positive
integers that have no factors other than 1 and themselves).
One of the most elegant of these had been the theorem
that every prime of the form 4n + 1 is uniquely expressible
as the sum of two squares. A more important result, now
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known as Fermat’s lesser theorem, asserts that if p is a
prime number and if a is any positive integer, a
p
− a is divis-
ible by p. For occasional demonstrations of his theorems
Fermat used a device that he called his method of “infinite
descent,” an inverted form of reasoning by recurrence or
mathematical induction. By far the best known of Fermat’s
many theorems is a problem known as his “great,” or “last,”
theorem. This appeared in the margin of his copy of
Diophantus’ Arithmetica and states that the equation x
n
+
y
n
= z
n
, where x, y, z, and n are positive integers, has no solu-
tion if n is greater than 2. This theorem remained unsolved
until the late 20th century.
Fermat was the most productive mathematician of his
day. But his influence was circumscribed by his reluctance
to publish.
siR Ronald aylMeR fisheR
(b. Feb. 17, 1890, London, Eng.—d. July 29, 1962, Adelaide, Austl.)
The British statistician and geneticist Sir Ronald Aylmer
Fisher pioneered the application of statistical procedures
to the design of scientific experiments.
In 1909 Fisher was awarded a scholarship to study
mathematics at the University of Cambridge, from which
he graduated in 1912 with a B.A. in astronomy. He remained
at Cambridge for another year to continue course work in
astronomy and physics and to study the theory of errors.
(The connection between astronomy and statistics dates
back to Carl Friedrich Gauss, who formulated the law of
observational error and the normal distribution based on
his analysis of astronomical observations.)
Fisher taught high school mathematics and physics
from 1914 until 1919 while continuing his research in sta-
tistics and genetics. Fisher had evidenced a keen interest
in evolutionary theory during his student days—he was a
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266
founder of the Cambridge University Eugenics Society—
and he combined his training in statistics with his
avocation for genetics. In particular, he published an
important paper in 1918 in which he used powerful statis-
tical tools to reconcile what had been apparent
inconsistencies between Charles Darwin’s ideas of natural
selection and the recently rediscovered experiments of
the Austrian botanist Gregor Mendel.
In 1919 Fisher became the statistician for the
Rothamsted Experimental Station near Harpenden,
Hertfordshire, and did statistical work associated with
the plant-breeding experiments conducted there. His
Statistical Methods for Research Workers (1925) remained in
print for more than 50 years. His breeding experiments
led to theories about gene dominance and fitness, pub-
lished in The Genetical Theory of Natural Selection (1930).
In 1933 Fisher became Galton Professor of Eugenics at
University College, London. From 1943 to 1957 he was
Balfour Professor of Genetics at Cambridge. He inves-
tigated the linkage of genes for different traits and
developed methods of multivariate analysis to deal with
such questions.
At Rothamsted Fisher designed plant-breeding
experiments that provided greater information with less
investments of time, effort, and money. One major prob-
lem he encountered was avoiding biased selection of
experimental materials, which results in inaccurate or
misleading experimental data. To avoid such bias, Fisher
introduced the principle of randomization. This princi-
ple states that before an effect in an experiment can be
ascribed to a given cause or treatment independently of
other causes or treatments, the experiment must be
repeated on a number of control units of the material
and that all units of material used in the experiments must
be randomly selected samples from the whole population
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they are intended to represent. In this way, random
selection is used to diminish the effects of variability in
experimental materials.
An even more important achievement was Fisher’s
origination of the concept of analysis of variance, or
ANOVA. This statistical procedure enabled experiments
to answer several questions at once. Fisher’s principal idea
was to arrange an experiment as a set of partitioned subex-
periments that differ from each other in one or more of
the factors or treatments applied in them. By permitting
differences in their outcome to be attributed to the dif-
ferent factors or combinations of factors by means of
statistical analysis, these subexperiments constituted a
notable advance over the prevailing procedure of varying
only one factor at a time in an experiment. It was later
found that the problems of bias and multivariate analysis
that Fisher had solved in his plant-breeding research are
encountered in many other scientific fields as well.
Fisher summed up his statistical work in Statistical
Methods and Scientific Inference (1956). He was knighted in
1952 and spent the last years of his life conducting research
in Australia.
John gRaunT
(b. April 24, 1620, London, Eng.—d. April 18, 1674, London)
John Graunt was an English statistician who is generally
deemed the founder of the science of demography, the
statistical study of human populations. His analysis of the
vital statistics of the London populace influenced the pio-
neer demographic work of his friend Sir William Petty
and, even more importantly, that of Edmond Halley, the
astronomer royal.
A prosperous haberdasher until his business was
destroyed in the London fire of 1666, Graunt held
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268
municipal offices and a militia command. While still active
as a merchant, he began to study the death records that
had been kept by the London parishes since 1532. Noticing
that certain phenomena of death statistics appeared regu-
larly, he was inspired to write Natural and Political
Observations . . . Made upon the Bills of Mortality (1662). He
produced four editions of this work. The third (1665) was
published by the Royal Society, of which Graunt was a
charter member.
Graunt classified death rates according to the causes
of death, among which he included overpopulation,
observing that the urban death rate exceeded the rural.
He also found that although the male birth rate was higher
than the female, it was offset by a greater mortality rate
for males, so that the population was divided almost evenly
between the sexes. Perhaps his most important innova-
tion was the life table, which presented mortality in terms
of survivorship. Using only two rates of survivorship (to
ages 6 and 76), derived from actual observations, he pre-
dicted the percentage of persons that will live to each
successive age and their life expectancy year by year. Petty
was able to extrapolate from mortality rates an estimate of
community economic loss caused by deaths.
pieRRe-siMon,
MaRquis de laplace
(b. March 23, 1749, Beaumount-en-Auge, Normandy, France—d.
March 5, 1827, Paris)
The French mathematician, astronomer, and physicist
Pierre-Simon, marquis de Laplace is best known for his
investigations into the stability of the solar system. He
also demonstrated the usefulness of probability for inter-
preting scientific data.
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Laplace was the son of a peasant farmer. Little is
known of his early life except that he quickly showed his
mathematical ability at the military academy at Beaumont.
In 1767 he arrived in Paris with a letter of recommenda-
tion to the mathematician Jean d’Alembert, who helped
him secure a professorship at the École Militaire, where
he taught from 1769 to 1776.
In 1773 he began his major lifework—applying
Newtonian gravitation to the entire solar system—by tak-
ing up a particularly troublesome problem: why Jupiter’s
orbit appeared to be continuously shrinking while
Saturn’s continually expanded. The mutual gravitational
interactions within the solar system were so complex
that mathematical solution seemed impossible. Indeed,
Newton had concluded that divine intervention was peri-
odically required to preserve the system in equilibrium.
Laplace announced the invariability of planetary mean
motions (average angular velocity). This discovery in 1773
was the first and most important step in establishing
the stability of the solar system and the most important
advance in physical astronomy since Newton. He removed
the last apparent anomaly from the theoretical descrip-
tion of the solar system in 1787 with the announcement
that lunar acceleration depends on the eccentricity of the
Earth’s orbit.
In 1796 Laplace published Exposition du système du
monde (The System of the World), a semipopular treatment of
his work in celestial mechanics and a model of French
prose. The book included his “nebular hypothesis”—
attributing the origin of the solar system to cooling and
contracting of a gaseous nebula—which strongly influ-
enced future thought on planetary origin. His Traité de
mécanique céleste (Celestial Mechanics), appearing in five vol-
umes between 1798 and 1827, summarized the results