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CHAPTER LII
MATHEMATICS
There is no single book in the English language, save the Britannica, in
which the whole body of mathematical knowledge is examined and
classified with special reference to the inter-relation of its various
parts and to the results obtained in the neighboring domains of physics,
chemistry, and engineering. Text-books necessarily have a somewhat
narrow purpose, namely to teach the student how to solve problems in a
single given field; wide views over the surrounding country can,
therefore, seldom be afforded. The Britannica, however, does for English
readers, what the _Encyclopädie der Mathematischen Wissenchaften_ does
for German, and more, in that in the Britannica the shadowy borderlands
are illuminated and the roads cleared which connect the mathematical and
the experimental sciences. In fact if anyone possessed every
mathematical text-book that had ever been published, he would still find
the articles full of suggestion to him, for in them the whole subject
has been presented, in all its complex bearings, logically and as a
whole.
[Sidenote: History]
It is nearly 4,000 years since a mathematician was last deified in the
person of Amenophis, and as far as can be ascertained only one other of
his calling ever received this honour, and he also was an Egyptian who
had entered into his godship a full thousand years earlier (Vol. 9, p.
46). To the ancient Egyptians mathematics owes the first fragmentary
ideas of arithmetic and mensuration, but little else, for despite their
amazing mechanical achievements very little record of purely
mathematical knowledge has come down from them. It was the Greeks,
starting with Thales (600 B.C.), who really created the sciences of
geometry and numbers. To them we owe the great abstract ideas which
dominate the science. The Greek period lasted till the capture of
Alexandria by the Mohammedans, A.D. 640, at which time the Arabian
school took shape, and to it we owe the development of algebra
(_al-jebr-wa’l-muqubala_, which means the transposition and removal [of
terms of an equation]). With the Renaissance the centre of scientific
research shifted to Western Europe and from then on the boundaries of
mathematical knowledge were rapidly extended, till to-day the subject is
the common ground on which all the physical sciences meet. The student
is referred to the article MATHEMATICS (Vol. 17, p. 878), by A. N.
Whitehead, fellow and senior lecturer in mathematics, Trinity College,
Cambridge, for a brilliant exposition of the foundations of the subject.
The professed mathematician will, of course, not need any set guide to
his reading, but it may be well to point out one or two articles which
he will find especially worthy of his attention.
[Sidenote: Leading Articles]
The article PROBABILITY, (Vol. 22, p. 376), by Professor Edgeworth,
author of _Mathematical Psychics_, and numerous papers on the calculus
of probabilities, gives, to the best of our belief, the only statement
of the whole problem in the English language. That on ALGEBRAIC FORMS
(Vol. 1, p. 620), by Major Macmahon, former president of the London
Mathematical Society, includes a number of results not previously
published. The article ELASTICITY (Vol. 9, p. 141), by A. E. H. Love,
professor of natural philosophy in the University of Oxford, embodies
the experience of a distinguished mathematician who has made this
subject the object of his special study for years. Sir George Darwin
(son of Charles Darwin) in the article TIDE (Vol. 26, p. 938) summed up
the results of his life’s work. The new electrical theory of the
properties of MATTER (Vol. 17, p. 891) is discussed by Sir J. J.
Thomson, professor of physics, Cambridge, who has done more than anyone
else to develop it. There are many other valuable articles, e.g.,
GEOMETRY, _Axioms_ (Vol. 11, p. 730), and GEOMETRY, _Non-Euclidean_
(Vol. 11, p. 724), by A. N. Whitehead; UNITS, DIMENSIONS OF (Vol. 27, p.
736), by Professor J. A. Fleming; ENERGY and ENERGETICS (Vol. 9, p. 398
and p. 390), by Sir Joseph Larmor; GROUPS, by Prof. Burnside, author of
_Theory of Groups of Finite Order_. Articles which will be found highly
useful to the engineer are MENSURATION (Vol. 18, p. 134); EARTH, FIGURE
OF (Vol. 8, p. 801); GEODESY (Vol. 11, p. 607); STRENGTH OF MATERIALS
(Vol. 25, p. 1007).
[Sidenote: Leading Contributors]
The mathematician will at once recognize the peculiar fitness of the
contributors to deal with the subjects allotted to them, and this
fitness is the more noticeable in the following list, arranged in
alphabetical order, which names and briefly describes the distinguished
mathematicians who have collaborated in the Britannica, and indicates
the principal articles written by each.
H. F. Baker, Fellow and Lecturer of St. John’s College, Cambridge.
Cayley Lecturer in Mathematics in the University. Author of _Abel’s
Theory and the Allied Theory_, etc.:
DIFFERENTIAL EQUATION; _Function_, _Functions of Complex Variables_.
Ludwig Boltzmann, formerly Professor of Theoretical Physics in the
Universities of Munich, Vienna, and Leipzig. Author of _Lectures on
the Theory of Gas_; _Lectures on Maxwell’s Theory of Electricity and
Light_:
MODEL.
W. Burnside, Professor of Mathematics, Royal Naval College, Greenwich.
Hon. Fellow of Pembroke College, Cambridge. Author of the _Theory of
Groups of Finite Order_, etc.:
GROUPS, THEORY OF
Arthur Cayley, formerly Professor of Pure Mathematics in the
University of Cambridge. See the biographical article (Vol. 5, p.
589):
CURVE (in part); DETERMINANT; EQUATION; NUMBERS, PARTITION OF;
SURFACE (in part); GAUSS, K. F.; MONGE, G.
George Chrystal, Professor of Mathematics and Dean of the Faculty of
Arts, Edinburgh University, Hon. Fellow and formerly Fellow and
Lecturer, Corpus Christi College, Cambridge:
PERPETUAL MOTION; PASCAL (in part); RIEMANN, GEORG.
Col. A. R. Clarke, Royal Medal of Royal Society 1887; in charge of
trigonometrical operations of the Ordnance Survey 1854–1881:
EARTH, FIGURE OF THE (in part); GEODESY (in part); MAP,
_Projections_ (in part).
Agnes Mary Clerke, Author of _History of Astronomy in the 19th
Century_; _The System of the Stars_; _Problems in Astrophysics_; and
many other astronomical books. See the biographical article (Vol. 6,
p. 497):
ASTRONOMY, _History_: ZODIAC; BRAHE, TYCHO; COPERNICUS; FLAMSTEED;
HALLEY; HUYGENS; KEPLER, etc.
Lt. Col. C. F. Close, head of the Geographical Section, British
General Staff, formerly British Representative on the Nyasa-Tanganyika
Boundary Commission. Author of _Text-Book of Topographical Surveying_,
etc.:
MAPS, _Projections_ (in part).
W. E. Dalby, Professor of Civil and Mechanical Engineering at the City
and Guilds of London Institute, Central Technical College, South
Kensington. Author of _The Balancing of Engines_, etc.:
MECHANICS, _Applied_ (in part); and several engineering subjects.
Sir George H. Darwin, late Fellow of Trinity College, Cambridge, and
Plumian Professor of Astronomy and Experimental Philosophy in the
University. President of the British Association, 1905. Author of _The
Tides and Kindred Phenomena in the Solar System_, etc.:
TIDE.
F. Y. Edgeworth, Professor of Political Economy in the University of
Oxford, etc. Author of _Mathematical Psychics_, and numerous papers on
the Calculus of Probabilities in the _Philosophical Magazine_, etc.:
PROBABILITY.
E. B. Elliott, Waynflete Professor of Pure Mathematics, and Fellow of
Magdalen College, Oxford. Formerly Fellow of Queen’s College, Oxford.
President of the London Mathematical Society, 1896–1898. Author of
_Algebra of Quantics_, etc.:
CURVE, (in part); GEOMETRY, IV _Analytical Geometry_.
C. Everitt, Magdalen College, Oxford:
ALGEBRA, _History_: DENSITY; LIGHT, _Introduction_, _History_, etc.
J. A. Ewing, Director of (British) Naval Education. Hon. Fellow of
King’s College, Cambridge. Formerly Professor of Mechanism and Applied
Mechanics in the University of Cambridge. Author of the _Strength of
Materials_, etc.:
STRENGTH OF MATERIALS, and several engineering subjects.
J. A. Fleming, Pender Professor of Electrical Engineering in the
University of London. Fellow of University College, London. Formerly
Fellow of St. John’s College, Cambridge, and Lecturer on Applied
Mechanics in the University. Author of _Magnets and Electric
Currents_, etc.:
UNITS, PHYSICAL; and many articles on Electrical Science.
Rev. A. H. Frost:
MAGIC SQUARE.
W. Garnett, Educational Adviser to the London County Council; formerly
Fellow and Lecturer of St. John’s College, Cambridge. Principal and
Professor of Mathematics, Durham College of Science. Author of
_Elementary Dynamics_, etc.:
ENERGY (in part); HYDROMETER; KELVIN, LORD.
J. W. L. Glaisher, Fellow of Trinity College, Cambridge. Formerly
President of the Cambridge Philosophical Society and the Royal
Astronomical Society, Editor of _Messenger of Mathematics_ and the
_Quarterly Journal of Pure and Applied Mathematics_:
LOGARITHM; TABLE, MATHEMATICAL; LEGENDRE, A. M.; NAPIER, JOHN.
J. H. Grace, Lecturer in Mathematics at Peterhouse and Pembroke
College, Cambridge. Fellow of Peterhouse:
GEOMETRY, _Line Geometry_.
Sir A. G. Greenhill, formerly Professor of Mathematics in the Ordnance
College, Woolwich. Author of _Differential and Integral Calculus with
Applications_; _Hydrostatics_; _Notes on Dynamics_, etc.:
BALLISTICS; GYROSCOPE AND GYROSTAT; HYDROMECHANICS.
Sir Thomas Little Heath, Assistant-Secretary to the Treasury, London.
Fellow of Trinity College, Cambridge. Author of _Apollonius of Perga_;
_Treatise on Conic Sections_; _The Thirteen Books of Euclid’s
Elements_, etc.:
ANTHEMIUS; APOLLONIUS OF PERGA; ARCHIMEDES; HERO OF ALEXANDRIA;
PAPPUS OF ALEXANDRIA; PORISM, etc.
F. R. Helmert, Professor of Geodesy in the University of Berlin:
EARTH, FIGURE OF THE (in part); GEODESY (in part).
O. M. F. Henrici, Professor of Mechanics and Mathematics in the
Central Technical College of the City and Guilds of London Institute.
Author of _Vectors and Rotors_; _Congruent Figures_, etc.:
CALCULATING MACHINES; GEOMETRY, I. _Euclidean_; II. _Projective_;
III. _Descriptive_; PERSPECTIVE; PROJECTION.
E. W. Hobson, Fellow and Tutor in Mathematics, Christ’s College,
Cambridge. Stokes Lecturer in Mathematics in the University:
FOURIER’S SERIES; SPHERICAL HARMONICS; TRIGONOMETRY.
A. E. Jolliffe, Fellow, Tutor and Mathematical Lecturer, Corpus
Christi College, Oxford. Senior Mathematical Scholar, 1892:
CONTINUED FRACTIONS; MAXIMA AND MINIMA; SERIES.
H. Lamb, Professor of Mathematics, University of Manchester, formerly
Fellow and Assistant Tutor of Trinity College, Cambridge; Member of
Council of Royal Society, 1894–1896. Royal Medallist, 1902. President
of London Mathematical Society 1902–1904. Author of HYDRODYNAMICS,
etc.:
DYNAMICS; HARMONIC ANALYSIS; MECHANICS, I. _Theoretical_; VECTOR
ANALYSIS; WAVE.
A. E. H. Love, Sedleian Professor of Natural Philosophy in the
University of Oxford. Hon. Fellow of Queen’s College; formerly Fellow
of St. John’s College, Cambridge; Secretary to the London Mathematical
Society:
ELASTICITY; VARIATIONS, CALCULUS OF; FUNCTION, _Functions of Real
Variables_; INFINITESIMAL CALCULUS.
W. H. Macaulay, Fellow and Tutor of King’s College, Cambridge:
MOTION, LAWS OF.
Major P. A. Macmahon, Deputy Warden of the Standards, Board of Trade.
Joint General Secretary, British Association. Formerly Professor of
Physics, Ordnance College. President of London Mathematical Society,
1894–1896:
ALGEBRAIC FORMS; COMBINATORIAL ANALYSIS; CAYLEY, ARTHUR.
G. B. Mathews, formerly Professor of Mathematics, University College
of N. Wales, sometime Fellow of St. John’s College, Cambridge:
ALGEBRA, _Special Kinds of Algebra_; NUMBER.
J. Clerk Maxwell, former Professor of Experimental Physics in the
University of Cambridge. See biographical article (Vol. 17, p. 929):
CAPILLARY ACTION (in part); DIAGRAM.
Simon Newcomb, former Professor of Mathematics and Astronomy, Johns
Hopkins University, etc. See the biographical article (Vol. 19, p.
474):
ASTRONOMY, _Descriptive_; and many other astronomical subjects.
J. H. Poynting, Professor of Physics and Dean of the Faculty of
Science in the University of Birmingham. Formerly Fellow of Trinity
College, Cambridge. Joint-author of _Text-Book of Physics_:
ACOUSTICS; GRAVITATION (in part); SOUND.
F. Purser, formerly Fellow of Trinity College, Dublin; Professor of
Natural Philosophy in the University of Dublin; Member of the Royal
Irish Academy:
SURFACE (in part).
J. Purser, formerly Professor of Mathematics in Queen’s College,
Belfast. Member of the Royal Irish Academy:
SURFACE (in part).
W. J. M. Rankine, former Professor of Civil Engineering at Glasgow
University. See the biographical article (Vol. 22, p. 894):
MECHANICS, _Applied_ (in part).
Hon. B. A. W. Russell, formerly Fellow of Trinity College, Cambridge.
Author of _Foundations of Geometry_; _Principles of Mathematics_,
etc.:
GEOMETRY, VI. _Non-Euclidean_ (in part).
W. F. Sheppard, Senior Examiner in the Board of Education; formerly
Fellow of Trinity College, Cambridge; Senior Wrangler, 1884:
ALGEBRA, _Principles of Ordinary Algebra_; ARITHMETIC; DIFFERENCES,
CALCULUS OF; INTERPOLATION; MENSURATION.
P. G. Tait, late professor of Natural Philosophy, Edinburgh
University. Author of _Elementary Treatise on Quaternions_. Joint
author with Lord Kelvin of _Treatise on Natural Philosophy_:
KNOT; QUATERNIONS (in part); HAMILTON, SIR WILLIAM; MAXWELL, JAMES
CLERK.
Rev. Charles Taylor, formerly Master of St. John’s College, Cambridge.
Vice-Chancellor, Cambridge University, 1887–1888. Author of
_Geometrical Conics_, etc.:
GEOMETRICAL CONTINUITY.
H. M. Taylor, Fellow of Trinity College, Cambridge; formerly Tutor and
Lecturer. Smith’s Prizeman, 1865. Editor of the Pitt Press _Euclid_:
NEWTON, SIR ISAAC.
Sir J. J. Thomson, Cavendish Professor of Experimental Physics and
Fellow of Trinity College, Cambridge. President of the British
Association, 1909–1910. Author of _A Treatise on the Motion of Vortex
Rings_; _Application of Dynamics to Physics and Chemistry_:
MATTER; and several articles on Electrical Science.
J. Walker, Christ Church, Oxford. Demonstrator in the Clarendon
laboratory. Formerly Vice-President of the Physical Society. Author of
_The Analytical Theory of Light_, etc.:
POLARIZATION OF LIGHT; REFRACTION, _Double Refraction_.
A. N. Whitehead, Fellow and Lecturer in Mathematics, Trinity College,
Cambridge. Author of _A Treatise on Universal Algebra_, etc.:
GEOMETRY VI. _Non-Euclidean Geometry_ (in part); GEOMETRY VII.
_Axioms on Geometry_; MATHEMATICS.
These are the men who are responsible for the mathematical sections of
the Britannica. A fuller list of articles on mathematical subjects is
given below.
Abel, Niels Henrik
Abscissa
Acceleration
Agnesi, Maria Gaetana
Aguillon, F. D.
Algebra
Algebraic Forms
Aliquot
Allen, or Alleyn T.
Amicable Numbers
Anderson, Alexander
Angle
Anthemius
Apollonius of Perga
Archimedes
Argument
Arithmetic
Autolycus of Pitane
Axis
Babbage, Charles
Baldi, Bernardino
Ballistics
Barlow, Peter
Barrow, Isaac
Bernoulli (family)
Bessel Function
Binomial
Biquadratic
Bisectrix
Boole, George
Borda, Jean Charles
Boscovich, Roger J.
Bouguer, Pierre
Bowditch, Nathaniel
Brachistochrone
Briggs, Henry
Buxton, Jedediah
Calculating Machines
Camus, Charles E. L.
Cardan, Girolamo
Cardioid
Castel, Louis Bertrand
Catenary
Cauchy, A. L., baron
Cayley, Arthur
Charles, J. A. C.
Chebichev, P. L.
Circle
Cissoid
Clairault, A. C.
Clifford, William K.
Cocker, Edward
Colburn, Z.
Combinatorial Analysis
Conchoid
Cone
Conic Section
Conoid
Continued Fractions
Cotes, Roger
Cremona, Luigi
Cube
Curve
Cycloid
Cylinder
Demoivre, Abraham
De Morgan, Augustus
Determinant
Diagonal
Diagram
Diameter
Differences, Calculus of
Differential Equation
Dimension
Diophantus of Alexandria
Ditton, Humphry
Dodecahedron
Dynamics
Earth, Figure of the
Elasticity
Ellipse
Ellipsoid
Emerson, William
Energetics
Energy
Epicycloid
Equation
Euclid
Euler, Leonhard
Fermat, Pierre de
Figurate Numbers
Focus
Folium
Fourier, J. B. J.
Fourier’s Series
Frisi, Paolo
Frustum
Function
Galloway, Thomas
Galois, Evariste
Gauss, Karl Friedrich
Geodesy
Geometrical Continuity
Geometry
Gnomon
Graphical Methods
Gravitation
Greaves, John
Gregory (family)
Gregory, Olinthus G.
Groups, Theory of
Gunter, Edmund
Gyroscope and Gyrostat
Hachette, J. N. P.
Hamilton, Sir W. R.
Harmonic
Harmonic Analysis
Harriot, T.
Hero of Alexandria
Hodograph
Hutton, Charles
Huygens, Christiaan
Hydrodynamics
Hydromechanics
Hydrostatics
Hyperbola
Icosahedron
Inaudi, Jacques
Infinite
Infinitesimal Calculus
Interpolation
Inversion
Involution
Ivory, Sir James
Jacobi, Karl G. J.
Kelvin, William Thomson, 1st baron
Kinematics
Kinetics
Kircher, Athanasius
Knot
Kovalevsky, Sophie
Lagrange, Joseph L.
Landen, John
Laplace, P. S., de
Lardner, Dionysius
Legendre, Adrien Marie
Lemniscate
Leonardo of Pisa
Leslie, Sir John
Lever
Lie, Marius Sophus
Limaçon
Line
Lobachevskiy, N. I.
Locus
Logarithm
Logocyclic Curve, Strophoid or Foliate
MacCullagh, James
Maclaurin, Colin
Magic Square
Map
Mascheroni, Lorenzo
Mathematics
Matter
Maupertuis, Pierre de
Maxima and Minima
Maxwell, J. Clerk
Mechanics
Mensuration
Mersenne, Marin
Monge, Gaspard
Montucia, Jean-Étienne
Motion, Laws of
Murphy, Robert
Napier, John
Newton, Sir Isaac
Nicomachus of Gerasa
Number
Numbers, Partition of
Numeral
Octahedron
Ordinate
Oughtred, William
Oval
Pantograph
Pappus of Alexandria
Parabola
Peacock, George
Peirce, Benjamin
Pell, John
Perpetual Motion
Perspective
Pfaff, J. F.
Playfair, John
Plücker, Julius
Poinsot, Louis
Poisson, Siméon Denis
Polygon
Polygonal Numbers
Polyhedral Numbers
Polyhedron
Poncelet, Jean Victor
Porism
Price, Bartholomew
Prism
Probability
Projection
Quadratrix
Quaternions
Recorde, Robert
Riccati, J. F., count
Riemann, G. F. B.
Roberval, G. P. de
Robins, Benjamin
Roulette
Routh, Edward John
Russell, John Scott
Salmon, George
Saunderson, N.
Serenus of Antissa
Series
Serpentine
Simpson, Thomas
Simson, Robert
Smith, H. J. S.
Smith, Robert
Snell, Willebrord
Sphere
Spherical Harmonics
Spheroid
Spiral
Spottiswoode, W.
Statics
Steiner, Jakob
Stevinus, Simon
Stirling, James
Stokes, Sir George G.
Strength of Materials
Sturm, J. C. F.
Surface
Sylvester, J. J.
Table, Mathematical
Tait, Peter G.
Tartaglia, Niccolo
Taylor, Brook
Tetrahedron
Theodosius of Tripolis
Thompson, T. P.
Tide
Todhunter, Isaac
Triangle
Trigonometry
Trisectrix
Units, Dimensions of
Units, Physical
Variations, Calculus of
Vector Analysis
Vernier, Pierre
Vieta (or Viète), F.
Wallace, William
Wallis, John
Wave
Witch of Agnesi
Zero