Mathematics

Prideof Mathematics 1
Definition of
Mathematics and
the Main
Branches

Mathematics
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”)
is the study of topics such as quantity(numbers), structure, space,
and change. There is a range of views among mathematicians and
philosophers as to the exact scope and definition of mathematics.
Calculus
Calculus is the mathematical study of change, in the same way
that geometry is the study of shape and algebra is the study of
operations and their application to solving equations. It has two major
branches, differential calculus(concerning rates of change and slopes of
curves), and integral calculus (concerning accumulation of quantities and
the areas under and between curves); these two branches are related to
each other by the fundamental theorem of calculus. Both branches make
use of the fundamental notions of convergence of infinite
sequences and infinite series to a well-defined limit. Generally, modern
calculus is considered to have been developed in the 17th century by Isaac
Newton and Gottfried Leibniz. Today, calculus has widespread uses
in science, engineering and economics and can solve many problems
that algebra alone cannot.

Trigonometry
Trigonometry (from Greek trigōnon, "triangle" and metron,
"measure"[
) is a branch of mathematics that studies relationships involving
lengths and angles of triangles. The field emerged in the Hellenistic
world during the 3rd century BC from applications of geometry to
astronomical studies.
Algebra
Algebra (from Arabic "al-jabr" meaning "reunion of broken
parts") is one of the broad parts of mathematics, together with
number, geometry and analysis. In its most general form, algebra is
the study of mathematical symbols and the rules for manipulating
these symbols; it is a unifying thread of almost all of mathematics.
Geometry
Geometry deals with spatial relationships, using fundamental qualities
or axioms. Such axioms can be used in conjunction with mathematical

definitions for points, straight lines, curves, surfaces, and solids to draw
logical conclusions.
Statistic
The science of making effective use of numerical data from
experiments or from populations of individuals. Statistics includes not only
the collection, analysis and interpretation of such data, but also the
planning of the collection of data, in terms of the design
of surveys and experiments.
Arithmetic
Arithmetic or arithmetics (from the Greek ἀριθμός arithmos,
"number") is the oldest and most elementary branch ofmathematics. It
consists of the study of numbers, especially the properties of the
traditional operations between them—
addition, subtraction, multiplication and division. Arithmetic is an
elementary part of number theory, and number theory is considered to be
one of the top-level divisions of modern mathematics, along
with algebra, geometry, and analysis. The terms arithmetic and higher
arithmetic were used until the beginning of the 20th century as synonyms
for number theory and are sometimes still used to refer to a wider part
of number theory

Branches of
Mathematics:
Mathematician

And
Their Contribution
Algebra
Algebra (from Arabic "al-jabr" meaning "reunion of broken parts") is
one of the broad parts of mathematics, together with
number, geometry and analysis. In its most general form, algebra is
the study of mathematical symbols and the rules for manipulating
these symbols; it is a unifying thread of almost all of mathematics.
 Elementaryalgebra, the part of algebra that is
usually taught in elementary courses of
mathematics.
 Abstractalgebra, in which algebraic structures such
as groups, rings and fields are axiomatically de
fined and investigated.
 Linear algebra, in which the specific properties

Al-Khwarizmi wrote one of the first Arabic algebras, a systematic exposé of the
basic theory of equations, with both examples and proofs. By the end of the 9th
century, the Egyptian mathematician Abu Kamil had stated and proved the basic laws

and identities of algebra and solved such complicated problems as finding x,
y, and z such that x + y + z = 10, x2 + y2 = z2, and xz = y2.
Omar Khayyam showed how to express roots of cubic equations by line segments
obtained by intersecting conic sections, but he could not find a formula for the roots.
Leonardo Fibonacci achieved a close approximation to the solution of the cubic
equation x3 + 2x2 + cx = d. Because Fibonacci had traveled in Islamic lands, he probably
used an Arabic method of successive approximations.
Ludovico Ferrari, soon found an exact solution to equations of the fourth degree
(see quartic equation), and as a result, mathematicians for the next several centuries
tried to find a formula for the roots of equations of degree five, or higher
Niccolò Fontana Tartaglia (1499/1500 - 1557) Tartaglia was an Italian
mathematician. The name "Tartaglia" is actually a nickname meaning
"stammerer", a reference to his injury-induced speech impediment. He was
largely self-taught, and was the first person to translate Euclid's Elementsinto a
modern European language. He is best remembered for his contributions to
algebra, namely his discovery of a formula for the solutions to a cubic equation.
Such a formula was also found by Gerolamo Cardano at roughly the same time,
and the modern formula is known as the Cardano-Tartaglia formula. Cardano also
found a solution to the general quartic equation.
Joseph-Louis Lagrange (1736 - 1813) Despite his French-sounding name,
Lagrange was an Italian mathematician. Like many of the great mathematicians
of his time, he made contributions to many different areas of mathematics. In
particular, he did some early work in abstract algebra. We will learn about
Lagrange's Theorem fairly soon, which is one of the most fundamental results in
group theory.
Évariste Galois (1811 - 1832) Galois was a very gifted young French
mathematician, and his story is one of the most tragic in the history of
mathematics. He was killed at the age of 20 in a duel that is still veiled in

mystery. Before that, he made huge contributions to abstract algebra. He
helped to found group theory as we know it today, and he was the first to use
the term "group". Perhaps most importantly, he proved that it is impossible to
solve a fifth-degree polynomial (or a polynomial of any higher degree) using
radicals by studying permutation groups associated to polynomials. This area of
algebra is still important today, and it is known as Galois theory in his honor.
Carl Friedrich Gauss (1777 - 1855) Along with Leonhard Euler, Gauss is
considered to be one of the greatest and most prolific mathematicians of all
time. He made significant contributions to algebra, number theory, geometry,
and physics, just to name a few areas. In algebra, there are several results in
ring theory (specifically regarding rings of polynomials) bearing his name.
Niels Henrik Abel (1802 - 1829) Abel was a Norwegian mathematician
who, like Galois, did seminal work in algebra before dying at a very young age.
Strangely enough, he proved similar results regarding the insolvability of the
quintic independently from Galois. In honor of his work in group theory, abelian
groups are named after him. The Abel Prize in mathematics, sometimes thought
of as the "Nobel Prize in Mathematics," is also named for him.
Emmy Noether (1882 - 1935) Noether is widely considered to be the
greatest female mathematician of all time, and in fact one of the greatest
mathematicians ever. Her most important work was related to abstract algebra,
specifically the theory of rings and fields. The concept of a Noetherian ring, as
well as several theorems in algebra, are named in her honor. She became a
lecturer at the University of Göttingen in 1915, at the invitation of David
Hilbert. She was forced to leave in 1933, when Adolf Hitler expelled Jewish
faculty members from Göttingen. She emigrated to the United States, where
she took up a position at Bryn Mawr, which she held until her death in 1935.
Arthur Cayley (1821 - 1895) Cayley was a British mathematician whose
work is known to students of abstract algebra and linear algebra. The Cayley-

Hamilton Theorem for matrices is named after him and William Rowan Hamilton,
and a fundamental theorem in group theory, Cayley's Theorem, is due to him.
Camille Jordan (1838 - 1922) Like Cayley, Jordan made contributions to
both abstract algebra and linear algebra. He is known for developing the Jordan
normal form of a matrix, and for originating the Jordan-Hölder Theorem in
group theory.

Geometry
Geometry deals with spatial relationships, using fundamental qualities
or axioms. Such axioms can be used in conjunction with mathematical
definitions for points, straight lines, curves, surfaces, and solids to draw
logical conclusions.
Euclidean geometry, elementary geometry of two and three
dimensions (plane and solid geometry), is based
largely on the Elements of the Greek mathematician
Euclid (fl. c.300 B.C.). In 1637, René Descartes
showed how numbers can be used to describe points
in a plane or in space and to express geometric
relations in algebraic form, thus founding analytic
geometry, of which algebraic geometry is a further
development (see Cartesian coordinates). The problem
of representing three-dimensional objects on a two-
dimensional surface was solved by Gaspard Monge,
who invented descriptive geometry for this purpose in the
late 18th cent. differential geometry, in which the concepts
of the calculus are applied to curves, surfaces, and
other geometrical objects, was founded by Monge and
C. F. Gauss in the late 18th and early 19th cent. The
modern period in geometry begins with the
formulations of projectivegeometry by J. V. Poncelet (1822)
and of non-Euclidean geometry by N. I. Lobachevsky (1826)
and János Bolyai (1832). Another type of non-
Euclidean geometry was discovered by Bernhard
Riemann (1854), who also showed how the various
geometries could be generalized to any number of
dimensions.

Babylon (2000 BC - 500 BC)
The Babylonians replaced the older (4000 BC - 2000 BC) Sumerian
civilization around 2000 BC. The Sumerians had already developed writing
(cuniform on clay tablets) and arithmetic (using a base 60 number system).
The Babylonians adopted both of these. But, Babylonian math went beyond
arithmetic, and devloped basic ideas in number theory, algebra, and geometry.
The problems they wanted to solve usually involved construction and land
estimation, such as areas and volumes of rectangular objects. Some of their
methods were rules that solved specialized quadratic, and even some cubic,
equations. But, they didn’t have algebraic notation, and there is no indication
that they had logical proofs for the correctness of their rule-based methods.
Nevertheless, they knew some special cases of the "Pythagorean Theorem"
more than 1000 years before the Greeks (see: Pythagorean Knowledge
In Ancient Babylonia and Pythagorus’ theorem in Babylonian mathematics).
Their durable clay tablets have preserved some of their knowledge (better
than the fragile Eygptian papyri). Four specific tablets (all from the period
1900 BC - 1600 BC) give a good indication of Babylonian mathematical
knowledge:

Yale tablet YBC 7289
- shows how to compute the diagonal of a square.
Plimpton 322
- has a table with a list of Pythagorean integer triples.

Susa tablet -
shows how to find the radius of the circle through the three vertices of an
isoceles triangle.
Tell Dhibayi tablet –
shows how to find the sides of a rectangle with a given area and diagonal.
There is no direct evidence that the Greeks had access to this knowledge. But,
some Babylonian mathematics was known to the Eygptians; and probably through
them, passed on to the Greeks (Thales and Pythagorus were known to have
traveled to Egypt).
Egypt (3000 BC - 500 BC)
The geometry of Egypt was mostly experimentally derived rules used by the
engineers of those civilizations. They developed these rules to estimate and

divide land areas, and estimate volumes of objects. Some of this was to
estimate taxes for landowners. They also used these rules for construction of
buildings, most notably the pyramids. They had methods (using ropes to
measure lengths) to compute areas and volumes for various types of objects,
various triangles, quadrilaterals, circles, and truncated pyramids. Some of their
rule-based methods were correct, but others gave approximations. However,
there is no evidence that the Egyptians logically deduced geometric facts and
methods from basic principles. And there is no evidence that they knew a
form of the "Pythagorean Theorem", though it is likely that they had some
methods for constructing right angles. Nevertheless, they inspired early Greek
geometers like Thales and Pythagorus. Perhaps they knew more than has been
recorded, since most ancient Eygptian knowledge and documents have been
lost. The only surviving documents are the Rhind and Moscow papyri.
Ahmes (1680-1620 BC)
wrote the Rhind Papyrus (aka the “Ahmes Papyrus”). In it, he claims
to be the scribe and annotator of an earlier document from about 1850
BC. It contains rules for division, and has 87 problems including the
solution of equations, progressions, areas of geometric regions, volumes
of granaries, etc.
Anon (1750 BC)
The scribe who wrote the Moscow Papyrus did not record his name.
This papyrus has 25 problems with solutions, some of which are
geometric. One, problem 14, describes how to calculate the volume of a
truncated pyramid (a frustrum), using a numerical method equivalent
to the modern formula: , where a and b are the
sides of the base and top squares, and h is the height.
The book Mathematics in the Time of the Pharaohs gives a more detailed
analysis of Egyptian mathematics.

India (1500 BC - 200 BC)
Everything that we know about ancient Indian (Vedic) mathematics is
contained in:
The Sulbasutras
These are appendices to the Vedas, and
give rules for constructing sacrificial
altars. To please the gods, an altar's
measurements had to conform to very
precise formula, and mathematical
accuracy was very important. It is not
historically clear whether this
mathematics was developed by the Indian
Vedic culture, or whether it was
borrowed from the Babylonians. Like the
Babylonians, results in the Sulbasutras are stated in terms of ropes; and
"sutra" eventually came to mean a rope for measuring an altar. Ultimately,
the Sulbasutras are simply construction manuals for some basic geometric
shapes. It is noteworthy, though, that all the Sulbasutras contain a method
to square the circle (one of the infamous Greek problems) as well as the
converse problem of finding a circle equal in area to a given square. The main
Sulbasutras, named after their authors, are:
Baudhayana (800 BC)
Baudhayana was the author of the earliest known Sulbasutra. Although he was
a priest interested in constructing altars, and not a mathematician, his
Sulbasutra contains geometric constructions for solving linear and quadratic
equations, plus approximations of (to construct circles) and . It also
gives, often approximate, geometric area-preserving transformations from one
geometric shape to another. These include transforming a square into a
rectangle, an isosceles trapezium, an isosceles triangle, a rhombus, and a circle,

and finally transforming a circle into a square. Further, he gives the special
case of the “Pythagorean theorem” for the diagonal of a square, and also a
method to derive “Pythagorian triples”. But he also has a construction (for a
square with the same area as a rectangle) that implies knowin g the more
general “Pythagorian theorem”. Some historians consider the Baudhayana as
the discovery of the “Pythagorian theorem”. However, the Baudhayana
descriptions are all empirical methods, with no proofs, and were likely
predated by the Babylonians.
Manava (750-690 BC)
contains approximate constructions of circles from rectangles, and squares from
circles, which give an approximation of = 25/8 = 3.125.
Apastamba (600-540 BC)
considers the problems of squaring the circle, and of dividing a segment into 7
equal parts. It also gives an accurate approximation of = 577 / 408 =
1.414215686, correct to 5 decimal places.
Katyayana (200-140 BC)
states the general case of the Pythagorean theorem for the diagonal of any
rectangle.
Greek Geometry (600 BC - 400 AD)
Thales of Miletus (624-547 BC)
was one of the Seven pre-Socratic Sages, and brought the science
of geometry from Egypt to Greece. He is credited with the

discovery of five facts of elementary geometry, including that an angle in a
semicircle is a right angle (referred to as “Thales Theorem ”). But some
historians dispute this and give the credit to Pythagorus. There is no evidence
that Thales used logical deduction to prove geometric facts.
Pythagorus of Samos (569-475 BC)
is regarded as the first pure mathematician to logically deduce
geometric facts from basic principles. He is credited with proving
many theorems such as the angles of a triangle summing to 180
deg, and the infamous "Pythagorean Theorem" for a right-angled
triangle (which had been known experimentally in Babylon and Egypt
for over 1000 years). The Pythagorean school is considered as the (first
documented) source of logic and deductive thought, and may be regarded as the
birthplace of reason itself. As philosophers, they speculated about the structure
and nature of the universe: matter, music, numbers, and geometry. Their legacy
is described in Pythagorus and the Pythagoreans : A Brief History
Hippocrates of Chios (470-410 BC)
wrote the first "Elements of Geometry" which Euclid may have
used as a model for his own Books I and II more than a hundred
years later. In this first "Elements", Hippocrates included geometric
solutions to quadratic equations and early methods of integration.
He studied the classic problem of squaring the circle showing how to
square a "lune". He worked on duplicating the cube which he showed to be
equivalent to constructing two mean proportionals between a number and its
double. Hippocrates was also the first to show that the ratio of the areas of
two circles was equal to the ratio of the squares of their radii.
Plato (427-347 BC)

founded "The Academy" in 387 BC which flourished until 529 AD.
He developed a theory of Forms, in his book "Phaedo", which
considers mathematical objects as perfect forms (such as a line
having length but no breadth). He emphasized the idea of 'proof'
and insisted on accurate definitions and clear hypotheses, paving the
way to Euclid, but he made no major mathematical discoveries himself. The
state of mathematical knowledge in Plato's time is reconstructed in the scholarly
book: The Mathematics of Plato's Academy.
Theaetetus of Athens (417-369 BC)
was a student of Plato's, and the creator of solid geometry. He was the first
to study the octahedron and the icosahedron, and to construct all five regular
solids. His work formed Book XIII of Euclid's Elements. His work about rational
and irrational quantities also formed Book X of Euclid.
Eudoxus of Cnidus (408-355 BC)
foreshadowed algebra by developing a theory of proportion which is presented in
Book V of Euclid's Elements in which Definitions 4 and 5 es tablish Eudoxus'
landmark concept of proportion. In 1872, Dedekind stated that his work on
"cuts" for the real number system was inspired by the ideas of
Eudoxus. Eudoxus also did early work on integration using his method of
exhaustion by which he determined the area of circles and the volumes of
pyramids and cones. This was the first seed from which the calculus grew two
thousand years later.
Euclid of Alexandria (325-265 BC)
is best known for his 13 Book treatise "The Elements" (~300 BC),
collecting the theorems of Pythagorus, Hippocrates, Theaetetus,

Eudoxus and other predecessors into a logically connected whole. A good modern
translation of this historic work is The Thirteen Books of Euclid's Elements by
Thomas Heath
Archimedes of Syracuse (287-212 BC)
is regarded as the greatest of Greek mathematicians, and was also
the inventor of many mechanical devices (including the screw,
pulley, and lever). He perfected integration using Eudoxus' method
of exhaustion, and found the areas and volumes of many objects. A
famous result of his is that the volume of a sphere is two-thirds
the volume of its circumscribed cylinder, a picture of which was inscribed on h is
tomb. He gave accurate approximations to and square roots. In his treatise
"On Plane Equilibriums", he set out the fundamental principles of mechanics,
using the methods of geometry, and proved many fundamental theorems
concerning the center of gravity of plane figures. In "On Spirals", he defined and
gave fundamental properties of a spiral connecting radius lengths with angles as
well as results about tangents and the area of portions of the curve. He also
investigated surfaces of revolution, and discovered the 13 semi-regular (or
"Archimedian") polyhedra whose faces are all regular polygons. Translations of his
surviving manuscripts are now available as The Works of Archimedes. A good
biography of his life and discoveries is also available in the book Archimedes:
What Did He Do Beside Cry Eureka?. He was killed by a Roman soldier in 212
BC.
Apollonius of Perga (262-190 BC)
was called 'The Great Geometer'. His famous work was "Conics"
consisting of 8 Books. In Books 5 to 7, he studied normals to
conics, and determined the center of curvature and the evolute of
the ellipse, parabola, and hyperbola. In another work "Tangencies",
he showed how to construct the circle which is tangent to three
objects (points, lines or circles). He also computed an approximation for

better than the one of Archimedes. English translations of his Conics Books I -
III, Conics Book IV, and Conics Books V to VII are now available.
Heron of Alexandria (10-75 AD)
wrote "Metrica" (3 Books) which gives methods for computing areas
and volumes. Book I considers areas of plane figures and surfaces of
3D objects, and contains his now-famous formula for the area of a
triangle = where s=(a+b+c)/2 [note: some
historians attribute this result to Archimedes]. Book II considers volumes of 3D
solids. Book III deals with dividing areas and volumes according to a given ratio,
and gives a method to find the cube root of a number. He wrote in a practical
manner, and has other books, notably in Mechanics
Menelaus of Alexandria (70-130 AD)
developed spherical geometry in his only surviving work "Sphaerica" (3 Books). In
Book I, he defines spherical triangles using arcs of great circles which marks a
turning point in the development of spherical trigonometry. Book 2 applies
spherical geometry to astronomy; and Book 3 deals with spherical trigonometry
including "Menelaus's theorem" about how a straight line cuts the three sides of
a triangle in proportions whose product is ( -1).
Claudius Ptolemy (85-165 AD)
wrote the "Almagest" (13 Books) giving the mathematics for the
geocentric theory of planetary motion. Considered a masterpiece
with few peers, the Almagest remained the major work in
astronomy for 1400 years until it was superceded by the
heliocentric theory of Copernicus. Nevertheless, in Books 1 and 2,
Ptolemy refined the foundations of trigonometry based on the chords of a circle
established by Hipparchus. One infamous result that he used, known as

"Ptolemy's Theorem", states that for a quadrilateral inscribed in a circle, the
product of its diagonals is equal to the sum of the products of its opposite
sides. From this, he derived the (chord) formulas for sin(a+b), sin(a -b), and
sin(a/2), and used these to compute detailed trigonometric tables.
Pappus of Alexandria (290-350 AD)
was the last of the great Greek geometers. His major work in geometry is
"Synagoge" or the "Collection" (in 8 Books), a handbook on a wide variety of
topics: arithmetic, mean proportionals, geometrical paradoxes, regular polyhedra,
the spiral and quadratrix, trisection, honeycombs, semiregular solids, minimal
surfaces, astronomy, and mechanics. In Book VII, he proved "Pappus' Theorem"
which forms the basis of modern projective geometry; and also proved "Guldin's
Theorem" (rediscovered in 1640 by Guldin) to compute a volume of revolution.
Hypatia of Alexandria (370-415 AD)
was the first woman to make a substantial contribution to the
development of mathematics. She learned mathematics and
philosophy from her father Theon of Alexandria, and assisted him in
writing an eleven part commentary on Ptolemy's Almagest, and a
new version of Euclid's Elements. Hypatia also wrote commentaries
on Diophantus's “Arithmetica”, Apollonius's “Conics” and Ptolemy's astronomical
works. About 400 AD, Hypatia became head of the Platonist school at
Alexandria, and lectured there on mathematics and philosophy. Although she had
many prominent Christians as students, she ended up being brutally murdered by
a fanatical Christian sect that regarded science and mathematics to be pagan.
Nevertheless, she is the first woman in history recognized as a professional
geometer and mathematician
Rene Descartes (1596-1650)

in an appendix "La Geometrie" of his 1637 manuscript "Discours de
la method ...", he applied algebra to geometry and created analytic
geometry. A complete modern English translation of this appendix is
available in the book “The Geometry of Rene Descartes“. Also, the
recent book “Descartes's Mathematical Thought” reconstructs his
intellectual career, both mathematical and philosophical.
Girard Desargues (1591-1661)
invented perspective geometry in his most important work titled
"Rough draft for an essay on the results of taking plane sections
of a cone" (1639). In 1648, he published.
Pierre de Fermat (1601-1665)
is also recognized as an independent co-creator of analytic geometry
which he first published in his 1636 paper "Ad Locos Planos et
Solidos Isagoge". He also developed a method for determining
maxima, minima and tangents to curved lines foreshadowing calculus.
Descartes first attacked this method, but later admitted it was
correct. The story of his life and work is described in the book “ The
Mathematical Career of Pierre de Fermat;.
Blaise Pascal (1623-1662)
was the co-inventor of modern projective geometry, published in his
"Essay on Conic Sections" (1640). He later wrote "The Generation
of Conic Sections" (1648-1654
Giovanni Saccheri (1667-1733)

was an Italian Jesuit who did important early work on non-euclidean geometry.
In 1733, the same year he died, Saccheri published his important e arly work on
non-euclidean geometry, “Euclides ab Omni Naevo Vindicatus”. Although he saw
it as an attempt to prove the 5th parallel axiom of Euclid. His attempt tried
to find a contradiction to a consequence of the 5th axiom, which he failed to
do, but instead developed many theorems of non-Euclidean geometry. It was 170
years later that the significance of the work realised. However, the discovery of
non-Euclidean geometry by Nikolai Lobachevsky and Janos Bolyai was not due to
this masterpiece by Saccheri, since neither ever heard of him.
Leonhard Euler (1707-1783)
was extremely prolific in a vast range of subjects, and is the
greatest modern mathematician. He founded mathematical analysis,
and invented mathematical functions, differential equations, and
the calculus of variations. He used them to transform analytic into
differential geometry investigating surfaces, curvature, and geodesics. Euler,
Monge, and Gauss are considered the three fathers of differential geometry. In
classical geometry, he discovered the “Euler line” of a triangle; and in analytic
geometry, the “Euler angles” of a vector. He also discovered that the "Euler
characteristic" (V-E+F) of a surface triangulation depends only on it’s genus,
which was the genesis of topology. Euler made other breakthrough contributions
to many branches of math. Famous formulas he discovered include “ Euler’s
formula” (eix
= cos x + i sin x), “Euler’s identity” (eiπ
+ 1 = 0), and many
formulas with infinite series. The list of his discoveries goes on and on. A
representative selection of his work (in 8 different fields) is given in the popular
book “Euler: The Master of Us All”. In 1766, Euler became almost totally blind,
after which he produced nearly half of all his work, dictating his papers to
assistants. He published over 800 papers and books, and his collected works fill
25,000 pages in 79 volumes. A large repository of his work is now available
online at The Euler Archive.

Gaspard Monge (1746-1818)
is considered the father of both descriptive geometry in "Geometrie
descriptive" (1799); and differential geometry in "Application de
l'Analyse a la Geometrie" (1800) where he introduced the concept
of lines of curvature on a surface in 3-space.
Adrien-Marie Legendre (1752-1833)
made important contributions to many fields of math: differential
equations, ballistics, celestial mechanics, elliptic functions, number
theory, and (of course) geometry. In 1794 Legendre published
“Elements de Geometrie” which was the leading elementary text on
the topic for around 100 years. In his "Elements" Legendre greatly
rearranged and simplified many of the propositions from Euclid's "Elements" to
create a more effective textbook. His work replaced Euclid's "Elements" as a
textbook in most of Europe and, in succeeding translations, in the United
States, and became the prototype of later geometry texts, including those being
used today. Although he was born into a wealthy family, in the 1793 French
Revolution he lost his capital, and became dependent on his academic salary.
Then in 1824, Legendre refused to vote for the government's candidate for the
French Institut National; and as a result, his academic pension was stopped. In
1833 he died in poverty.
Carl Friedrich Gauss (1777-1855)
invented non-Euclidean geometry prior to the independent work of
Janos Bolyai (1833) and Nikolai Lobachevsky (1829), although
Gauss' work on this topic was unpublished until after he died. With
Euler and Monge, he is considered a founder of differential geometry. He
published "Disquisitiones generales circa superficies curva" (1828) which contained
"Gaussian curvature" and his famous "Theorema Egregrium" that Gaussian

curvature is an intrinsic isometric invariant of a surface embedded in 3-space.
The story of his life and work is given in the popular book “ The Prince of
Mathematics: Carl Friedrich Gauss.
Nikolai Lobachevsky (1792-1856)
published the first account of non-Euclidean geometry to appear in
print. Instead of trying to prove Euclid’s 5th axiom (about a unique
line through a point that is parallel to another line), he studied the concept of
a geometry in which that axiom may not be true. He completed his major work
Geometriya in 1823, but it was not published until 1909. In 1829, he published
a paper on hyperbolic geometry, the first paper to appear in print on non-
Euclidean geometry, in a Kazan University journal. But his papers were rejected
by the more prestigious journals. Finally in 1840, a paper of his was published in
Berlin; and it greatly impressed Gauss. There has been some speculati on that
Gauss influenced Lobachevsky’s work, but those claims have been refuted. In any
case, his great mathematical achievements were not recognized in his lifetime,
and he died without a notion of the importance that his work would achieve.
Janos Bolyai (1802-1860)
was a pioneer of non-Euclidean geometry. His father, Farkas, taught
mathematics, and raised his son to be a mathematician. His father
knew Gauss, whom he asked to take Janos as a student; but Gauss
rejected the idea. Around 1820, Janos began to follow his father’s path to
replace Euclid's parallel axiom, but he gave up this approach within a year, since
he was starting to develop the basic ideas of hyperbolic and absolute geometry.
In 1825, he explained his discoveries to his father, who was clearly disappointed.
But by 1831, his father’s opinion had changed, and he encour aged Janos to
publish his work as the Appendix of another work. This Appendix came to the
attention of Gauss, who both praised it, and also claimed that it coincided with
his own thoughts for over 30 years. Janos took this as a severe blow, became
irritable and difficult with others, and his health deteriorated. After this he did

little serious mathematics. Later, in 1848, Janos discovered Lobachevsky’s 1829
work, which greatly upset him. He accused Gauss of spiteful machinations
through the fictitious Lobachevsky. He then gave up any further work on math.
He had never published more than the few pages of the Appendix, but he left
more than 20000 pages of mathematical manuscripts, which are now in a
Hungarian library.
Jean-Victor Poncelet (1788-1867)
was one of the founders of modern projective geometry. He had
studied under Monge and Carnot, but after school, he joined
Napoleon’s army. In 1812, he was left for dead after a battle with
the Russians, who then imprisoned him for several years. During this time, he
tried to remember his math classes as a distraction from the hardship, and
started to develop the projective properties of conics, including the pole, polar
lines, the principle of duality, and circular points at infinity. After being freed
(1814), he got a teaching job, and finally published his ideas in “Traite des
proprietes projectives des figures” (1822), from which the term “projective
geometry” was coined. He was then in a priority dispute about the duality
principle that lasted until 1829. This pushed Poncelet away from projective
geometry and towards mechanics, which then became his career. Fifty years
later, he incorporated his innovative geometric ideas into his 2 -volume treatise
on analytic geometry “Applications d'analyse et de geometrie” (1862, 1864). He
had other unpublished manuscripts, which survived until World War I, when they
vanished.
Hermann Grassmann (1809-1877)
was the creator of vector analysis and the vector interior (dot)
and exterior (cross) products in his books "Theorie der Ebbe and
Flut" studying tides (1840, but 1st published in 1911), and

"Ausdehnungslehre" (1844, revised 1862). In them, he invented what is now
called the n-dimensional exterior algebra in differential geometry, but it was not
recognized or adopted in his lifetime. Professional mathematicians regarded him
as an obscure amateur (who had never attended a university math lecture), and
mostly ignored his work. He gained some notoriety when Cauchy purportedly
plagiarized his work in 1853 (see the web page Abstract linear spaces for a
short account). A more extensive description of Grassmann's life and work is
given in the interesting book “A History of Vector Analysis”.
Arthur Cayley (1821-1895)
was an amateur mathematician (he was a lawyer by profession)
who unified Euclidean, non-Euclidean, projective, and metrical
geometry. He introduced algebraic invariance, and the abstract groups of
matrices and quaternions which form the foundation
Bernhard Riemann (1826-1866)
was the next great developer of differential geometry, and
investigated the geometry of "Riemann surfaces" in his PhD thesis
(1851) supervised by Gauss. In later work he also developed geodesic
coordinate systems and curvature tensors in n-dimensions. An engaging and
readable account of Riemann’s life and work is given in the book “ Bernhard
Riemann 1826-1866: Turning Points in the Conception of Mathematics ”
Felix Klein (1849-1925)
is best known for his work on the connections between geometry
and group theory. He is best known for his "Erlanger Programm"
(1872) that synthesized geometry as the study of invariants under
groups of transformations, which is now the standard accepted

view. He is also famous for inventing the well-known "Klein bottle" as an
example of a one-sided closed surface.
David Hilbert (1862-1943)
first worked on invariant theory and proved his famous "Basis
Theorem" (1888). He later did the most influential work in
geometry since Euclid, publishing "Grundlagen der Geometrie"
(1899) which put geometry in a formal axiomatic setting based on 21 axioms. In
his famous Paris speech (1900), he gave a list of 23 open problems, some in
geometry, which provided an agenda for 20th century mathematics. The story
of his life and mathematics are now in the acclaimed biography “ Hilbert”.
Oswald Veblen (1880-1960)
developed "A System of Axioms for Geometry" (1903) as his
doctoral thesis. Continuing work in the foundations of geometry led
to axiom systems of projective geometry, and with John Young he
published the definitive "Projective geometry" in 2 volumes (1910-
18). He then worked in topology and differential geometry, and published "The
Foundations of Differential Geometry" (1933) with his student Henry
Whitehead, in which they give the first definition of a differentiable manifold.
Donald Coxeter (1907-2003)
is regarded as the major synthetic geometer of the 20th century,
and made important contributions to the theory of polytopes, non-
Euclidean geometry, group theory and combinatorics. Coxeter is
noted for the completion of Euclid's work by giving the complete classification of
regular polytopes in n-dimensions using his "Coxeter groups". He published many
important books, including Regular Polytopes (1947, 1963, 1973) and
Introduction to Geometry (1961, 1989). He was a Professor of Math at Univ.

of Toronto from 1936 until his death at the age of 96. When asked about how
he achieved a long life, he replied: "I am never bored". Recently, a biography of
his remarkable life has been published in the interesting book “ King of Infinite
Space: Donald Coxeter, the Man Who Saved Geometry”.
Trigonometry
Trigonometry (from Greek trigōnon, "triangle" and metron, "measure"[
) is
a branch of mathematics that studies relationships involving lengths
and angles of triangles. The field emerged in the Hellenistic world during
the 3rd century BC from applications of geometry to astronomical studies.
Core Trigonometry
 This type of trigonometry is used for triangles that have
one 90 degree angle. Mathematicians use sine and
cosine variables within a formula (as well as data from
trigonometry tables such as decimal values) to
determine the height and distance of the other two
angles. A scientific calculator has the trigonometry
tables programmed within, making the formulations
easier to equate than through using long division. Core
trigonometry is taught in high schools, and studied in
depth by mathematic majors in college.

Spherical Trigonometry
 Spherical trigonometry deals with triangles that are
drawn on a sphere, and this type is often used by
astronomers and scientists to determine distances
within the universe. Unlike core or plane
trigonometry, the sum of all angles in a triangle is
greater than 180 degrees. Sine and cosine tables

Ahmes
 was the Egyptian scribe who wrote the Rhind Papyrus - one of the
oldest known mathematical documents.
Thales
 was the first known Greek philosopher, scientist and
mathematician. He is credited with five theorems of elementary
geometry.
Pythagoras
 was a Greek philosopher who made important developments in
mathematics, astronomy, and the theory of music. The theorem
now known as Pythagoras's theorem was known to the Babylonians
1000 years earlier but he may have been the first to prove
Euclid
 was a Greek mathematician best known for his treatise on
geometry: The Elements . This influenced the development of
Western mathematics for more than 2000 years. http://www-
groups.dcs.st-and.ac.uk/~hist...
Heron or Hero of Alexandria
 was an important geometer and worker in mechanics who invented
many machines ncluding a steam turbine. His best known
mathematical work is the formula for the area of a triangle in terms
of the lengths of its sides. A is the area of a triangle with sides a, b
and c and s = (a + b + c)/2 then A^2 = s (s - a)(s - b)(s - c).
Menelaus
 was one of the later Greek geometers who applied spherical
geometry to astronomy. He is best known for the so-called
Menelaus's theorem.
François Viète
 was a French amateur mathematician and astronomer who
introduced the first systematic algebraic notation in his book In
artem analyticam isagoge . He was also involved in deciphering
codes. he calculated π to 10 places using a polygon of 6 216=
393216 sides. He also represented π as an infinite product which, as

far as is known, is the earliest infinite representation of π....
Johannes Kepler
 was a German mathematician and astronomer who postulated that
the Earth and planets travel about the sun in elliptical orbits. He
gave three fundamental laws of planetary motion. He also did
important work in optics and geometry.
René Descartes
 was a French philosopher whose work, La géométrie, includes his
application of algebra to geometry from which we now have
Cartesian geometry. His work had a great influence on both
mathematicians and philosophers.
Leonhard Euler
 was a Swiss mathematician who made enormous contibutions to a
wide range of mathematics and physics including analytic
geometry, trigonometry, geometry, calculus and number theory.
Firstly his work in number theory seems to have been stimulated
by Goldbach but probably originally came from the interest that
the Bernoullis had in that topic. Goldbach asked Euler, in 1729, if
he knew of Fermat's conjecture that the numbers 2^n + 1 were
always prime if n is a power of 2. Euler verified this for n = 1, 2, 4,
8 and 16 and, by 1732 at the latest, showed that the next case
2^(32) + 1 = 4294967297 is divisible by 641 and so is not prime.
Euler also studied other unproved results of Fermat and in so
doing introduced the Euler phi function (n), the number of integers
k with 1 k n and k coprime to n. He proved another of Fermat's
assertions, namely that if a and b are coprime then a^2 + b^2 has
no divisor of the form 4n - 1, in 1749. Other work done by Euler on
infinite series included the introduction of his famous Euler's
constant , in 1735, which he showed to be the limit of
1/1 + 1/2 + 1/3 + ... + 1/n - log(e) n
Lagrange
 excelled in all fields of analysis and number theory and analytical
and celestial mechanics. He also worked on number theory proving
in 1770 that every positive integer is the sum of four squares. In
1771 he proved Wilson's theorem (first stated without proof by
Waring) that n is prime if and only if (n -1)! + 1 is divisible by n.
Giovanni Ceva

 was an Italian mathematician who rediscovered Menelaus's
theorem and proved his own well-known theorem.
Pitiscus
Although Pitiscus worked much in the theological field, his proper
abilities concerned mathematics, and particularly trigonometry.
The word 'trigonometry' is due to Pitiscus and first occurs in the title of
his work Trigonometria: sive de solutione triangulorum tractatus brevis
et perspicuus first published in Heidelberg in 1595 as the final section of
A Scultetus's Sphaericorum libri tres methodice conscripti et utilibus
scholiis expositi.
The first section, divided into five books, covers plane and spherical
trigonometry.
In the first book he introduced the main definitions and theorems of
plane and spherical trigonometry.
The third of the five books is devoted to plane trigonometry and it
consists of six fundamental theorems.
The fourth book consists of four fundamental theorems on spherical
trigonometry, while the fifth book proves a number of propositions on
the trigonometric functions.
Trigonometry: or, the doctrine of triangles.
Hipparchus
He made an early contribution to trigonometry producing a table of
chords, an early example of a trigonometric table; indeed some
historians go so far as to say that trigonometry was invented by him.
Finally let us examine the contributions which Hipparchus made to
trigonometry.
Even if he did not invent it, Hipparchus is the first person whose
systematic use of trigonometry we have documentary evidence.
If this is so, Hipparchus was not only the founder of trigonometry but
also the man who transformed Greek astronomy from a purely
theoretical into a practical predictive science.
Aryabhata I
The mathematical part of the Aryabhatiya covers arithmetic, algebra,
plane trigonometry and spherical trigonometry.
We now look at the trigonometry contained in Aryabhata's treatise.
He also introduced the versine (versin = 1 - cosine) into trigonometry.
Regiomontanus
Regiomontanus made important contributions to trigonometry and

astronomy.
In the Epitome Regiomontanus, realising the need for a systematic
account of trigonometry to support astronomy, promised to write such a
treatise.
With Book II the study of trigonometry gets under way in earnest.
Books III, IV and V treat spherical trigonometry which, of course, is of
major importance in astronomy.
Guo Shoujing
Making sense of the data gathered from the instruments required a
knowledge of spherical trigonometry and Guo devised some remarkable
formulae.
We should now look at the rather remarkable work which Guo did on
spherical trigonometry and solving equations.
The first column is the value of x using Guo's formula taking an accurate
modern approximation to π, the second column is the result given by the
formula with π = 3, while the third column is the correct answer
calculated using trigonometry (in fact the cosine).
Theodosius
Sphaerics contains no trigonometry although it is likely that Hipparchus
introduced spherical trigonometry before Sphaerics was written (although,
one has to assume, after the book on which Sphaerics is based, which
would certainly be the case if this earlier book was written by Eudoxus).
Perhaps it is worth remarking that despite our comment above that the
work contains no trigonometry, there are some results which we could
easily interpret in trigonometrical terms.
Peirce Benjamin
For example An Elementary Treatise on Plane Trigonometry (1835), First
Part of an Elementary Treatise on Spherical Trigonometry (1936), An
Elementary Treatise on Sound (1936), An Elementary Treatise on Algebra
: To which are added Exponential Equations and Logarithms (1937), An
Elementary Treatise on Plane and Solid Geometry (1937), An Elementary
Treatise on Plane and Spherical Trigonometry (1940), and An Elementary
Treatise on Curves, Functions, and Forces Vol 1 (1841), Vol 2 (1846).
Girard Albert
Albert Girard worked on algebra, trigonometry and arithmetic.
In 1626 he published a treatise on trigonometry containing the first use
of the abbreviations sin, cos, tan.
It appears that Girard spent some time as an engineer in the Dutch army

although this was probably after he published his work on trigonometry.
Durell
Among the books he wrote around this time were: Readable relativity
(1926), A Concise Geometry (1928), Matriculation Algebra (1929),
Arithmetic (1929), Advanced Trigonometry (1930), A shorter geometry
(1931), The Teaching of Elementary Algebra (1931), Elementary Calculus
(1934), A School Mechanics (1935), and General Arithmetic (1936).
For the second of our more detailed looks at one of Durell's texts let us
consider Advanced Trigonometry which was also originally published by G
Bell & Sons.
This volume will provide a welcome resource for teachers seeking an
undergraduate text on advanced trigonometry, when few are readily
available.
Casey
1893); A treatise on elementary trigonometry (1886); A treatise on plane
trigonometry (1888); A treatise on spherical trigonometry (1889).
Briggs
Gellibrand was professor of astronomy at Gresham College and was
particularly interested in applications of logarithms to trigonometry.
He therefore added a preface of his own on applications of logarithms to
both plane trigonometry and to spherical trigonometry.
Bhaskara II
It covers topics such as: praise of study of the sphere; nature of the
sphere; cosmography and geography; planetary mean motion; eccentric
epicyclic model of the planets; the armillary sphere; spherical
trigonometry; ellipse calculations; first visibilities of the planets;
calculating the lunar crescent; astronomical instruments; the seasons;
and problems of astronomical calculations.
There are interesting results on trigonometry in this work.
In particular Bhaskaracharya seems more interested in trigonometry for
its own sake than his predecessors who saw it only as a tool for
calculation.
Al-Tusi Nasir
One of al-Tusi's most important mathematical contributions was the
creation of trigonometry as a mathematical discipline in its own right
rather than as just a tool for astronomical applications.

In Treatise on the quadrilateral al-Tusi gave the first extant exposition of
the whole system of plane and spherical trigonometry.
This work is really the first in history on trigonometry as an independent
branch of pure mathematics and the first in which all six cases for a right-
angled spherical triangle are set forth.
Klugel
Klugel made an exceptional contribution to trigonometry, unifying
formulae and introducing the concept of trigonometric function, in his
Analytische Trigonometrie.
Klugel's trigonometry was very modern for its time and was exceptional
among the contemporary textbooks.
Doppelmayr
Doppelmayr wrote on astronomy, spherical trigonometry, sundials and
mathematical instruments.
He also wrote several mathematics texts himself, including one on
spherical trigonometry and Summa geometricae practicae.
Puissant
The map was produced with considerable detail, the projection used
spherical trigonometry, truncated power series and differential geometry.
Puissant wrote on geodesy, the shape of the earth and spherical
trigonometry.
Herschel Caroline
Slowly Caroline turned more and more towards helping William with his
astronomical activities while he continued to teach her algebra, geometry
and trigonometry.
In particular Caroline studied spherical trigonometry which would be
important for reducing astronomical observations.
Viete
The Canon Mathematicus covers trigonometry; it contains trigonometric
tables, it also gives the mathematics behind the construction of the
tables, and it details how to solve both plane and spherical triangles.
Viete also wrote books on trigonometry and geometry such as
Supplementum geometriae (1593).
Fuss

Most of Fuss's papers are solutions to problems posed by Euler on
spherical geometry, trigonometry, series, differential geometry and
differential equations.
His best papers are in spherical trigonometry, a topic he worked on with A
J Lexell and F T Schubert.
Al-Jayyani
Another work of great importance is al-Jayyani's The book of unknown
arcs of a sphere, the first treatise on spherical trigonometry.
Although it is certain that Regiomontanus based his treatise on Arabic
works on spherical trigonometry it may well be that al-Jayyani's work was
only one of many such sources.
Ulugh Beg
This excellent book records the main achievements which include the
following: methods for giving accurate approximate solutions of cubic
equations; work with the binomial theorem; Ulugh Beg's accurate tables of
sines and tangents correct to eight decimal places; formulae of spherical
trigonometry; and of particular importance, Ulugh Beg's Catalogue of the
stars, the first comprehensive stellar catalogue since that of Ptolemy.
As well as tables of observations made at the Observatory, the work
contained calendar calculations and results in trigonometry.
Stevin
The author of 11 books, Simon Stevin made significant contributions to
trigonometry, mechanics, architecture, musical theory, geography,
fortification, and navigation.
The collection included De Driehouckhandel (Trigonometry), De Meetdaet
(Practice of measuring), and De Deursichtighe (Perspective).
Calculus
Calculus is the mathematical study of change, in the same way
that geometry is the study of shape and algebra is the study of

operations and their application to solving equations. It has two major
branches, differential calculus(concerning rates of change and slopes of
curves), and integral calculus (concerning accumulation of quantities and
the areas under and between curves); these two branches are related to
each other by the fundamental theorem of calculus. Both branches make
use of the fundamental notions of convergence of infinite
sequences and infinite series to a well-defined limit. Generally, modern
calculus is considered to have been developed in the 17th century by Isaac
Newton and Gottfried Leibniz. Today, calculus has widespread uses
in science, engineering and economics and can solve many problems
that algebra alone cannot.
Differential calculus
Divides things into small (different) pieces and tells
us how they change from one moment to the next.
Integral calculus
Joins (integrates) the small pieces together and tells
us how much of something is made, overall, by a
series of changes.

Descartes was educated in the Jesuit
preparatory school of La Flèche and the
University of Poitiers, taking a degree in
law. He then spent two years in Paris
where, outwardly living the life of a
frivolous young gentleman, he began a
serious study of mathematics. To see more
of the world Descartes joined several
armies as an unpaid volunteer; the brief
intervals of tranquility during nine years of
service provided him time to develop his
mathematical and philosophic ideas. In
1628, Descartes decided to settle in
Holland, where he remained for the next
twenty years. There he wrote his great
philosophic treatise on the scientific
method, the Discours de la méthode (1637).
(The still-quoted sentence, "I think,
therefore I am," comes from the Discours.)
In 1649, after much hesitation, Descartes
accepted the invitation of the 22-year-old
Queen Christina to come to Sweden as her
private tutor. After only four months of
winter tutoring sessions, always held at

5:00 in the morning in the ice-cold library,
Descartes died of pneumonia.
The last of the three appendices to
Descartes’s Discours was a 106-page essay
entitled La géométrie. It provides the first
printed account of what is now called
analytic or coordinate geometry. The work
exerted great influence after being
published in a Latin translation along with
explanatory notes. The Géométrie
introduced many innovations in
mathematical notation, most of which are
still in use. With Descartes, small letters
near the beginning of the alphabet indicate
constants and those near the end stand for
variables. He initiated the use of numerical
superscripts to denote powers of a
quantity, while occasionally writing aa for
the second power, a2. The familiar symbols
+, -, and are also encountered in
Descartes’s writing.
Descartes "algebrized" the study of
geometry by shifting the focus from curves
to their equations, allowing the tools of
algebra, rather than diagrams, to be applied
to the solution of various geometric
problems. The Géométrie also treated one
of the most important problems of the day,
that of finding tangents to curves, by
describing a procedure for constructing the
normal to a curve at any point (the tangent
is perpendicular to the normal). Another
part of the work deals with matters in the
theory of equations: Descartes states that x
- a is a factor of a polynomial if and only if
a is a root. He also notes that the
maximum number of roots is equal to the
degree of the polynomial.

Fermat received a Bachelor of Civil Laws from
the University of Law at Orleans in 1631. Fermat
considered mathematics to be a hobby, never
publishing his work. Most of his theories and
formulations were recovered from his
correspondence with Pierre de Carcavi and
Father Mersenne. Upon his death his son Samuel
oversaw the publications of Fermats work in
Observations on Diophantus, and Mathematical
Works.
Pierre de Fermat explored such mathematical
areas as analytical geometry, pre-evolved
Calculus, and infinite descent. However his work
with Number Theory is what he is best known for.
A few of his well known theorems include
Every non-negative integer can be represented as
the sum of four or fewer squares A prime of the
form 4n + 1 can be represented as the sum of two
squares The equation Nx2 + 1 = y2 has infinitely
many integer solutions if N is not a square
Fermat was in the habit of presenting his
theorems as fact, letting others perform the task
of presenting the proofs and verifications of his
work. Perhaps his most infamous work is what is
commonly known as Fermat's Last Theorem,
named such as it was the last of his theorems to
be proven. This theorem states that xn + yn = zn
has no non-zero integer solutions for x, y and z
when n > 2. To further add to the mystery,
Fermat's last words on this were found in the
margin of a popular mathematics book, simply
stating that he had found a "remarkable proof"
but that the margin was too small in which to
explain. In 1995, over 300 years later, this
theorem was finally proven by the British
mathematician, Andrew Wiles.
Although once mistakenly declared deceased
during the plague of 1653, he continued to live
out his life in Toulouse with his wife and four
children until his death in 1665.

As a youth, Torricelli took courses in
mathematics and philosophy with the Jesuits in
Faenza, Italy. They noticed his outstanding
promise and sent him for further education to a
school in Rome run by a former student of
Galileo’s. Torricelli himself may be viewed as
Galileo’s last pupil, for he came to live with the
blind and ill Galileo in 1641. They had only a
little time to work together, for the aged scholar
died within three months.
Appointed to the chair of mathematics in
Florence, the position left vacant by Galileo,
Torricelli’s own career was cut short when he
died suddenly, probably of typhoid fever, five
years later at the age of 39. He is often
remembered today for his demonstration of the
weight of air. The demonstration consisted of
taking a long tube filled with mercury and sealed
at one end, and inverting it into a basin of
mercury; the changing pressure of air on the free
surface of mercury in the basin made the level in
the tube stand higher on some occasions than on
others.
Torricelli was a mathematician of considerable
accomplishment. Using Cavalieri’s method of
indivisibles, he solved the famous problem of
finding the area under one arch of the cycloid;
later, he determined the length of the infinitely
many revolutions of the logarithmic spiral (in
polar coordinates, In 1641, he
established a result so astonishing that
mathematicians of the day thought it to be
impossible: there is a geometric solid which is
infinitely long, but nonetheless has a finite
volume. The body, which he called "the acute
hyperbolic solid," is generated when the region
bounded by a branch of the hyperbola y = 1/x,
the line x = 1 and the x-axis is revolved around
the x-axis. Its finite volume is given in modern
notation by the integral

When he communicated his discovery to the
French geometers in 1644, Torricelli’s status
changed from being a virtual unknown to one of
the most acclaimed mathematicians in Europe.
The proof itself constituted the high point in the
Opera geometrica (1644), the only work of
Torricelli to be published in his lifetime.
Wallis entered Cambridge University in 1632,
studied theology, and received a master’s degree
in 1640, the same year in which he took Holy
Orders. He held a faculty position at Cambridge
for about a year, but vacated it upon deciding to
marry. During England’s Civil War of 1642-1648,
Wallis aided the Puritan cause by deciphering
captured coded Royalist dispatches. As a reward
for this service (and although he was yet to show
any mathematical promise), Wallis was appointed
professor of geometry at Oxford in 1649. Because
the position required him to give public lectures
on theoretical mathematics, Wallis embarked at
the age of 32 on a systematic and productive
study of the subject. He retained his post at
Oxford until his death, over 50 years later.
Wallis’s Tractus de sectionibus conicis of 1656 is
the first elementary textbook to treat conics
using Descartes’s new coordinate geometry. In it,
the ellipse, hyperbola and parabola are each
identified with an equation of second degree. In
1655, he had published the Arithmetica
infinitorum, the work on which his reputation is
grounded. The Arithmetica contains a formula
equivalent to
for the area under the curve y = xn. This is often
regarded as the first general theorem to appear in
the calculus. After giving a somewhat rigorous
demonstration for several integral powers of x,
Wallis inferred it to be true for every positive
integer; then, relying on "permanence of form,"
he asserted that the formula held even when n is
negative (but not equal to -1) or fractional. The
result was not new, having been anticipated by

Cavalieri. Where Cavalieri relied almost entirely
on geometric reasoning, Wallis held to an
arithmetic argument whenever possible. With the
advent of his "arithmetic integration," the
geometric method of indivisibles virtually ceased
to appear in the calculus.
The familiar knot symbol for infinity makes its
first appearance in print in the Arithmetica. As
does Wallis’s famous infinite product expansion
for p ,
Blaise Pascal was born in the French province of
Auvergne on June 19, 1623. Early on in his life,
Pascal's father wanted to restrict his son's
education primarily to languages. However, at a
young age Pascal became increasingly curious
about mathematics. Through his tutor, he gained
knowledge about geometry and decided to pursue
his own studies. Pascal discovered many
properties of geometric figures, such as the sum
of the angles of a triangle is equal to two right
angles. Pascal's father was so impressed by his
son's abilities that he gave him a copy of Euclid's
"Elements" (which he soon mastered). By the age
of fourteen, Pascal was attending the weekly
meetings of other French geometricians, which
later formed the basis of the French Academy.
In 1640, Pascal published an essay on conic
sections, and during the next few years, he
invented and built a mechanical calculating
machine, which was called a Pascaline. When he
became twenty-one, Pascal gained interest in
Torricelli's work on atmospheric pressure, which
led him to study hydrostatics.
In 1650, Pascal took an abrupt hiatus from his
research to pursue religion. He joined the
Jansenist monastery at Port-Royal in 1654 after
he had a religious experience that changed his
life. He broke away from the Jansenists in 1658
and returned once again to his studies in
mathematics. He worked primarily on calculus
and on probability theory with Pierre de Fermat

up until his death at the age of 39.
In 1661, Newton entered Cambridge University,
where he was awarded a master’s degree in 1668.
He was for the most part self-taught, learning his
mathematics from books, especially from
Descartes’s Géométrie and Wallis’s Arithmetica
infinitorum. During the two years 1664-1665,
when an outbreak of the Great Plague closed the
university, Newton remained in seclusion at
home. In these "wonderful years," he began to do
his own original research. Beginning in 1664 he
laid the foundations of the differential calculus,
which he described as the "method of fluxions";
and, in 1665, he began investigating the "inverse
method of fluxions," or the integral calculus.
Newton formulated his principle of universal
gravitation in the same period. This idea
culminated in his masterwork, the Principia
Mathematica (1687), which explains the motions
of the heavenly bodies in the language of
mathematics. In 1669 Newton’s former teacher
resigned his professorship in favor of his pupil,
who by that time was considered the most
promising mathematician in England. Newton
remained at Cambridge until 1696.
If Newton had overcome his "wariness to impart,"
there might never have been a controversy over
who discovered the calculus. For many years his
methods remained unknown, except to a few
friends. He wrote De Analysi per Aequationes
Infinitas in 1669 but did not publish it until
1711; while the Tractus de quadratura curvarum,
composed in 1671, did not appear until 1704.
In Newton’s terminology, a variable quantity x,
depending on time, is called a fluent; and its rate
of change with time is said to be the fluxion of
the fluent, denoted by (dx/dt in modern
notation). He chose the letter o to represent an
infinitely small quantity, with xo indicating the
corresponding change in . For an illustration of
his fluxional methods, Newton provides the
equation xy - a = 0. He substitutes x + o for x,

and y + o for y, then expands to get
After using the original equation xy - a = 0 and
dividing by o, the equation is reduced to
The term involving o is neglected, since "o is
supposed to be infinitely small," leaving
(modern: x dy/dt + y dx/dt = 0)
In 1665, Newton generalized the familiar
binomial theorem for expanding expressions of
the form (1 + a)n, n being a positive integer, to
the case where n is a fractional exponent,
positive or negative; the result is an infinite
(binomial) series, rather than a polynomial. By
means of the expansion of (1 - x2)1/2, he arrived
at what today would be written as
Leibniz received a doctorate of laws in 1667, a
step towards entering the diplomatic service of
one of the small states which then made up
Germany. Traveling extensively on political
missions to France, Holland and England, he was
brought into contact with most of the leading
mathematicians of the day. Leibniz’s real
mathematical education began in the years 1672
to 1676, in Paris, when time between
assignments allowed him to study the subject in
depth. His version of the calculus seems to have
been invented in 1673, but the first account was
not formally published until 1684. (This was
twenty years prior to the appearance of Newton’s
presentation of the calculus in De quadratura
curvarum.) Leibniz’s diplomatic career came to
an end in 1676 when he reluctantly accepted the
position of librarian in the court of Hanover, a
post which he held for the remainder of his life.
He helped to organize the Berlin Academy of
Science in 1700, and became its first President.
The most important aspect of Leibniz’s calculus
was a suitable symbolism that allowed the
geometric arguments of his predecessors to be
translated into operational rules. He proposed

the symbol for the sum of areas of infinitely
small rectangles; it is the script form of s, the
initial letter in summa (sum).. In his new
formalism, Leibniz expresses relations such as
He also originated the notation dy/dx, treating it
as a quotient of differentials (infinitely small
increments of the variable); and used the letter d,
standing alone, for differentiation. His led to
useful algorithms, such as the product rule:
d(xy) = x dy + y dx.
His formula
indicated the inverse relationship of
differentiation and integration.
One of Leibniz’s early contributions is an elegant
series for p which is now named after him:
p /4 = 1 - 1/3 + 1/5 - 1/7 + . . .
When challenged, as a test of his ability, to
calculate the sum of the series
1/1 . 2 + 1/2 . 3 + 1/3 . 4 + 1/4 . 5 + . . .,
he found that the terms could be transformed
into differences by the identity 1/n(n+1) = 1/n -
1/(n+1); the series then became
(1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + (1/4 - 1/5) +
. . .
and, when adjacent terms are canceled, had sum
1.
Michel Rolle was born at Ambert on April 21,
1652. Since he did not receive formal training as
a child, Rolle had to educate himself in
mathematics.
In 1689, he wrote a paper on algebra, which
contains the theorem on the position of the roots
of an equation. In 1675, he relocated to Paris and
worked as an arithmetical expert. Rolle primarily
worked on Diophantine analysis, algebra, and
geometry. In 1685, he was elected to the Royal
Academy of the Sciences. In 1691, Rolle
published "Rolle's Theorem", for which he is best
remembered. His theorem, which is a specialized

case of the Mean Value Theorem, guaranteed the
existance of a horizontal tangent line (f'(x)=0)
between points a and b given that f(a) = f(b) = 0.
Rolle also gained some notariety by solving a
problem posed by Jacques Ozanam in 1682.
Impressed by Rolle's achievement, Jean-Baptiste
Colbert, controller general of finance under King
Louis XIV of France, rewarded Rolle with a
pension for his diligent work.
The Swiss Bernoulli brothers, James and John,
were the first to achieve a full understanding of
Leibniz’s presentation of the calculus. Their
subsequent publications did much to make the
subject widely known to the rest of the
continent.
James Bernoulli, the elder of the two, entered
the University of Basel in 1671, receiving a
master’s degree in theology two years later and a
licentiate (a degree just below the doctorate) in
theology in 1676. Meanwhile, he was teaching
himself mathematics, much against the wishes of
his merchant father. Bernoulli spent two years in
France familiarizing himself with Descartes’
Géométrie and the work of his followers. By
1687, he had sufficient mathematical reputation
to be appointed to a vacant post at Basel. He also
wrote to Leibniz in the same year, asking to be
shown his new methods. This proved difficult
because Leibniz’s abbreviated explanations were
full of errors. Still, Bernoulli mastered the
material within several years and went on to
make contributions to the calculus equal to
those of Leibniz himself.
The Bernoulli brothers used the techniques of
Leibniz’s calculus as a means for handling a wide
range of astronomical and physical problems,
sometimes working independently to solve the
same problem. In 1690, James Bernoulli
challenged the mathematicians of Europe to
determine the shape (that is, to find the
equation) of a hanging flexible cable suspended in
equilibrium at two points. The correct solution

was presented a year later by his brother John in
his first published paper. The desired curve was
not a parabola, as some expected, but a curve
known as the catenary -- from the Latin word
catena, chain.
Bernoulli was more adapt at treating infinite
series than most mathematicians of the day. He
showed that
diverges, and that
1/12 + 1/22 + 1/32 + 1/42 + . . .
converges; but he confessed his inability to find
the sum of the latter series. (Euler succeeded in
finding its sum.) In 1690 he established what is
known as the "Bernoullian inequality,"
(1 + x)n > 1 + nx, x > -1, n > 1, n an integer.
We also owe to him the word "integral" in its
technical sense.
The Marquis de l’Hôpital, a French nobleman
living by private means, is known for the first
printed book on the newcalculus. He served
briefly as a cavalry officer, but resigned because
of his extreme nearsightedness to devote his
energies entirely to mathematics. In his time the
recently invented calculus was fully understood
only by Newton, Leibniz and the Bernoulli
brothers. In 1691-92, when John Bernoulli spent
over half a year in Paris, he was generously
compensated for giving the young Marquis
private lessons on this powerful new method. In
return for a monthly allowance, Bernoulli was
induced to continue the instruction by letter; the
agreement was that he would communicate his
future mathematical discoveries exclusively to
l’Hôpital to be used as the Marquis saw fit.
L’Hôpital eventually felt that he understood the
material well enough to compose a proper
textbook on it.
L’Hôpital’s Analyse des infiniment petits,
published in 1696, contains an account of the
differential calculus as conceived by Leibniz and
learned from Bernoulli. In its preface l’Hôpital

freely acknowledges his debt to the two
mathematicians, saying, "I have made free use of
their discoveries." The successive reprintings of
the Analyse (1716, 1720 and 1768) made the
calculus known throughout Europe. In 1730 it
was translated into English, supplemented by the
translator with work on the integral calculus; in
tribute to Newton, the book’s derivative notation
was changed to the fluxional "dottage" of their
English hero. L’Hôpital is nowadays remembered
in the name of his "0/0 rule," a rule for finding
the limiting value of a quotient whose numerator
and denominator both tend to zero. His
statement of the rule is not entirely in accord
with modern use. Making use of limit notation,
which was unavailable to l’Hôpital, a reasonable
rendition of his statement would be:
If f(x) and g(x) are differentiable functions with
f(a) = g(a) = 0, then
whenever
The Analyse dominated the field for the next 50
years, finally to find a worthy rival in Euler’s
great treatises of the 1750’s.
John Bernoulli earned a master’s degree in
philosophy and, in 1690, a medical licentiate
from the University of Basel, where his brother
James was teaching. At the same time, he was
secretly studying the publications of Leibniz with
James’s help. Shortly thereafter, Bernoulli
visited Paris where he contracted to teach the
material to the young marquis de l’Hopital. Many
of his own discoveries in calculus appeared in
l’Hopital’s textbook. In 1695, supported by a
recommendation from l’Hopital, Bernoulli
obtained a position at Gröningen in Holland.
Upon his brother’s death in 1705, he succeeded
him as professor of mathematics at Basel, to
remain there for 43 years. Bernoulli was a
zealous defender of Leibniz against charges that

he had plagiarized Newton’s discovery of the
calculus.
In 1696, John Bernoulli published a
mathematical challenge, a popular device in the
early days of the calculus. The problem he posed
was to determine the shape of the curve down
which a bead will slide, from one point to another
not directly beneath it, in the shortest possible
time. This is the famous brachistochrone
problem, which Bernoulli named from the Greek
words for "quickest time." Five prominent
mathematicians found a solution; namely, the
two Bernoullis, Leibniz, l’Hopital and Newton.
When Newton’s solution arrived, unsigned,
Bernoulli is said to have exclaimed, "I recognize
the lion by his paw." Not surprisingly, the sought-
after curve is not a straight line, but an upside-
down cycloid.
One of Bernoulli’s more notable achievements is
the expansion of a function in series through
repeated integration by parts:
This leads to interesting identities such as
Brook Taylor was born in Edmonton, England on
Aug. 18, 1685. Since Taylor's family were
wealthy, his parents could afford to have private
tutors available. Taylor entered St John's
College, in Cambridge, on April 3, 1703 where he
pursued mathematics as his field of study. In
1708, he developed a solution to the center of
oscillation of a body based on differential
calculus. Taylor's solution eventually led to a
dispute with John Bernoulli.
In 1709, Taylor graduated from St. John's
College. In 1712, Taylor joined the Royal Society.
After two years, he was elected to the position of
Secretary to the Royal Society. During this time,
he produced two very important books. The first
book, "Methodus Incrementorum Directa Et
Inversa", developed the "calculus of finite
differences", integration by parts, and the

infamous "Taylor Series". The second book,
Linear Perspective", created the foundations of
projective geometry.
Colin MacLaurin was born in Kilmodan, Scotland
in 1698. His father, John Maclaurin, was the
town's minister. Colin, the youngest of three
sons, was extremely talented from an early age.
Considered a child prodigy, he enrolled at the
University of Glasgow when he was only 11.
About one year later, he became exposed to
advanced mathematics when he discovered a
copy of Euclid's "Elements". MacLaurin quickly
mastered six of the thirteen books that
comprised "Elements". At 14, he earned his M.A.
degree. His thesis was on the power of gravity, in
which he further developed Newton's theories. By
the time he turned 19, he became a professor of
mathematics in Aberdeen. A few years later, he
became a fellow of the Royal Society of London.
During the time of his fellowship, MacLaurin met
with Sir Issac Newton in 1725. Impressed by
MacLaurin's intellect, Newton recommended that
MacLaurin be made the professor of mathematics
at the University of Edinburgh. In 1740,
MacLaurin shared a prize from the Academy of
Sciences with fellow mathematicians Leonhard
Euler and Daniel Bernoulli for an essay on tides.
In 1742, he published the first systematic
formulation of Newton's methods, where he
developed a method for expanding functions
about the origin in terms of series (now known as
a MacLaurin Series). This method was adapted
from Brook Taylor's case of an expansion about
an arbitrary point (known as a Taylor Series).
Maclaurin also made astronomical observations,
developed several theorems similar to Newton's
theorems in "Principia", improved maps of the
Scottish isles, and developed the method of
generating conics.

At the age of 14, Euler entered the University of
Basel where its most famous professor, John
Bernoulli, aroused his interest in mathematics;
he graduated three years later with a master’s
degree. Unsuccessful in obtaining a position at
Basel (partly due to his youth), Euler went to the
fledgling St. Petersburg Academy in Russia, there
to become its chief mathematician by 1730. In
1741, at the invitation of Frederick the Great, he
joined the Berlin Academy as head of its
mathematics section. Euler’s quarter-century
stay was not altogether happy and so, in 1766,
he readily accepted the generous offer of
Catherine I to return to St. Petersburg. Euler had
previously lost the sight in one eye, to all
appearances from overwork; in 1771, a clumsy
cataract operation on his other eye left him
entirely blind. Aided by a phenomenal memory
Euler remained productive until the end of his
life, dictating his thoughts to a servant who knew
no mathematics.
Euler’s enormous output of 886 papers and books
made him the most prolific of all
mathematicians. His landmark textbooks, the
Introductio in analysin infinitorum of 1748
followed by the Introductiones calculi
differentialis (1775) and the Institutiones calculi
integralis (1768-1770), brought together
everything that was then known of the calculus.
These comprehensive works divorced the subject
from its geometrical origins and shaped its
direction for the next 50 years. They also
popularized the use of the mathematical symbols
At a time when the notion of convergence was
not well-understood, Euler’s work was
conspicuous for its treatment of infinite series.
His most famous result in this regard involves an
unexpected appearance of p : namely,
In the Introductio, he expanded the
trigonometric functions sin x and cos x as power

series to obtain the relationship now known as
Euler’s Identity:
eix = cos x + i sin x (x real)
A consequence of taking x = p in Euler’s Identity
is an equation connecting five of the most
important constants in mathematics: eip + 1 = 0.
Euler’s investigations also led to the well-known
formula
(cos x + i sin x)n = cos nx + i sin nx.
Thomas Simpson was born in Leicestershire,
England on August 20, 1710. Simpson's first job
was as a weaver, the chosen profession of his
father. However, he gave up weaving to pursue a
study of mathematics. He improved his own
mathematical skills through hard work and
effort. By 1735, Simpson was able to solve
several questions that involved infinitesimal
calculus. In 1743, he was appointed Professor of
Mathematics at Woolwich in London (which he
held until his death).
Simpson is best known for his work on numerical
methods of integration, probability theory, and
interpolation. He worked on the "Theory of
Errors" and aimed to prove that the arithmetic
mean was better than a single observation. He
also taught privately and wrote several textbooks
on mathematics.
Joseph-Louis Lagrange was born in Turin, Italy
on Jan. 25, 1736 -- the oldest of 11 children. His
father planned for him to become a lawyer.
However, while at the College of Turin, Lagrange
read a paper published by the astronomer
Edmond Halley on the use of algebra in optics.
Halley's paper and Lagrange's interest in physics
eventually led him to pursue a career in
mathematics.
Lagrange is best remembered for the Lagrangian
function and Lagrange multiplier, which bear his
name. Lagrange multipliers are used to locate

multivariable maximum and minimum points
subject to a constraint of the form g(x,y) = 0 or
g(x,y,z) = 0.
He also made numerous contributions to the
calculus of variations (which include
optimization problems), calculus of probabilities,
analytical mechanics, the theory of functions,
and in differential and integral calculus.
Pierre-Simon Laplace was born in Beaumont-en-
Auge, France on Mar. 23, 1749. Very little is
known of his early childhood. He attended Caen
University, majoring in theology. Laplace
intended to join the church upon graduation.
However, he became aware of his mathematical
talents and decided to leave the university.
Laplace traveled to Paris where he studied
mathematics under Jean le Rond d'Alembert, a
brillant mathematician and scientist who
pioneered the use differential equations in
physics and studied equilibrium and fluid
motion. d'Alembert was so impressed with
Laplace that he appointed him professor of
mathematics at the Ecole Militaire at the young
age of 19. In 1773, he joined the Paris Academy
of Sciences. In 1785, Laplace was an examiner at
the Royal Artillery Corps. One of his students
was Napoleon Bonaparte who was sixteen at the
time.
Among his many contributions, Laplace is best
remembered for introducing the potential
function and Laplace coefficients and Laplace
transforms. The Laplacian, which represents the
divergence of the gradient of a scalar function, is
used to help simplify the time-independent
Schrodinger equation.
Some of his other noteworthy accomplishments
include proving the stability of the solar system,
deriving the least squares rule, contributing to
the study of electricity and magnetism,
solidifying the theory of mathematical
probability, and performing experiments on
capillary action and specific heat with Antoine

Lavoisier.
Jean-Baptiste Joseph Fourier was born in
Auxerre, France on March 21, 1768 - the ninth of
twelve children. He attended the Ecole Royal
Militaire of Auxerre in 1780 where he first
studied literature and then mathematics. He
continued to study mathematics, even while
training to become a priest in a Benedictine
abbey in 1787. However, Fourier desired to make
an impact in mathematics like Newton and
Pascal. In 1794, he went to Paris to study at the
Ecole Normale under other famous
mathematicians such as Lagrange, Laplace, and
Monge. By 1797, Fourier was an instructor and
researcher at the College de France. In 1798, he
became a scientific adviser to Napoleon's army
during France's invasion of Egypt. Fourier did
not return to Paris until 1801 when he resumed
teaching. By 1817, he was elected to the
Academy of Sciences. Five years later Fourier
became the Secretary of the mathematics section
at the Academy.
Fourier is best remembered for the Fourier
Transform, which involves the Fourier Series,
and for his theorem on the position of roots in an
algebraic equation. The Fourier Transform makes
it possible to take any periodic function of time
and equate it into an equivalent infinite
summation of sine waves and cosine waves.
Johann Carl Frederich Gauss was born on Apr.
30, 1777 in Brunswick, Germay. Many consider
him to have been a child prodigy since he taught
himself reading and arithmetic by the age of
three. In 1792, his talent caught the attention of
the Duke of Brunswick who later gave Gauss a
stipend to pursue his education. He attended
Caroline College from 1792 to 1795. While there,
he formulated the least-squares method and dealt
with the concept of congruence in number

theory. By 1799, Gauss was awarded a Ph.D for
giving the first proof of the fundamental theorem
of algebra during a doctoral dissertation. In 1801,
he published "Disquisitiones Arithmeticae",
which contained solutions to several problems in
number theory. Gauss also predicted the orbit of
a newly discovered asteroid, Ceres, using his
least squares approximation method. This
discovery eventually led to a position as
astronomer at the Gottingen Observatory.
The intricate research Gauss performed
contributed to the fields of differential geometry,
theoretical astronomy, statisics, magnetism,
mechanics, acoustics, and optics.
Cauchy attended France’s great Ecole
Polytechnique from 1805 until 1807 and worked
briefly as a military engineer. In 1813 he
abandoned his chosen career, apparently for
reasons of health, and devoted himself
exclusively to mathematics. Cauchy secured an
instructorship at the Polytechnique, where he
rose to be professor of mechanics in 1816.
During this period he undertook a thorough
reorganization of the foundations of the calculus,
infusing the subject, as he put it, with the same
rigor that was to be found in geometry. Because
of the changing political situation in 1830
Cauchy went into voluntary exile in Turin, where
he obtained an appointment at the university. In
1838 he returned to Paris and resumed his
teaching, although not at the Polytechnique.
Cauchy was the foremost French mathematician
of the nineteenth century; his 789 papers and
seven books rank him second only to Euler in
terms of productivity.
Cauchy’s celebrated Cours d’analyse de l’Ecole
Royale Polytechnique, based on his lectures at
that school, stamped elementary calculus with
the character it has today. It recognizes the limit
concept as the cornerstone of a firm logical
explanation of continuity, convergence, the
derivative and the integral. In defining "limit," he

says:
When the values successivly attributed to a
particular variable approach indefinitely a fixed
value so as to differ from it by as little as one
wishes, this latter value is called the limit of the
others.
Suffice it to say, the reliance on such phrases as
"as little as one wishes" denies precision to the
notion. The Cours describes the derivative of y =
f(x) as the limit ("when it exists") of a difference
quotient
as h goes to zero. Another aspect of Cauchy’s
work is a careful treatment of sequences and
series. One of the basic tests for sequential
convergence is a result that is today called the
"Cauchy convergence criterion"; specifically, a
sequence s1, s2, s3, ... converges to a limit if the
difference sm - sn can be made less than any
assigned value by taking m and n sufficiently
large.
August Ferdinand Mobius was born on Nov. 17,
1790 in Schulpforta, Germany. In 1809 Mobius
graduated from College and went to the
University of Leipzig. Athough his family
suggested Mobius study law, he preferred
mathematics, astronomy, and physics. In 1813,
Mobius studied astronomy under Gauss at the
Gottingen Observatory. By 1815, he started his
doctoral thesis on the occultation of fixed stars.
Shortly, he began his Habilitation thesis on
trigonometric equations. Mobius was appointed
to the chair of astronomy and higher mechanics
at the University of Leipzig in 1816. He did not
gain full professorship in astronomy until 1844.
Mobius is known for his work in analytic
geometry and topology. Specifically, he was one
of the discoverers of the M obius Strip. Mobius
also made numerous contributions in astronomy.
He wrote papers on the occultations of the

planets, astronomical principals, and on celestial
mechanics.
George Green was the son of a baker and left
school at the tender age of 9 to follow in his
father's footsteps. Even at this age he exhibitted
an interest in mathematics. Being of lower social
standing, he was not able to afford the costs of a
university. Green instead, took upon himself the
responsibility of self-education. With his basic
education, he began reading and studying
mathematical papers as well as other documents.
In 1828 at the age of 35, he published possibly
his greatest work, entitled "An Essay on the
Application of Mathematical Analysis to the
Theories of Electridcity and Magnetism." In this
publication, he made his first attempts to apply
mathematical theory to electrical phenomena.
Many of its subscribers were not able to really
understand the contents, importance, or
significance of this work. Two years later, one of
the exceptions to this, Sir Edward Bromheadm,
met with George and encouraged him to publish
two other recognized 'memoirs', "Mathematical
Investigations Concerning the Laws of
Equilibrium of Fluids Analogous to the Electric
Fluid" and "On the Determination of Exterior and
Interior Attractions of Ellipsoids fo Variable
Densities." He also published a paper entitled
"Researches on the Vibrations of Pendulum in
Fluid Media."
In 1833, at the age of 40, he turned down an
invitation from Cambrige University and
admitted himself to Caius College. He gained
recognition and went on to publish papers on
wave theory dealing with the hydrodynamics of
wave motion and reflection and refraction of
light and sound.

Pierre Verhulst was born in Brussels, Belgium on
Oct. 28, 1804. He attended the University of
Ghent where he earned a doctoral degree in 1825
within three years. He eventually came back to
Brussels and worked on number theory. He also
gained an interest in social statictics from
Adolphe Quetelet, another famous
mathematician from Belgium who studied the
theory of probability under Pierre Laplace and
Joseph Fourier. As his interest grew, Verhulst
spent more time with social statistics and less
time trying to publish the complete works of
Euler.
In 1829 Verhulst translated John Herschel's
Theory of light and published the paper. In 1835,
Verhulst was appointed professor of mathematics
at the University of Brussels where he offered
courses on geometry, trigonometry, celestial
mechanics, astronomy, differential and integral
calculus, and the theory of probability. In 1841,
Verhulst was elected to the Belgium Academy. By
1848, he became the Academy's president.
Verhulst's research on the law of population
growth showed that forces, which tend to
obstruct population growth, increase in
proportion to the ratio of the excess population
to the overall population. He proposed a
population growth model which takes into
account the possible limitation of population size
due to limited resources. Verhulst's model is
often called the "Logistic Growth Equation", or
"Verhulst Equation". His model is considered an
improvement over the Malthusian model, which
assumes human population grows exponentially
when plagues or other disasters do not occur.

Karl Gustav Jacob Jacobi, although born to a
Jewish family, was born in Germany and given
the French name Jacques Simon. Jacobi was
taught by his uncle until he was 12 years of age,
afterwhich he was enrolled in the Gymnasium in
Potsdam. While still in his first year of school he
was moved to the final year class. Jacobi, still
age 12, passed all the necessary classes to enter
the university, but was could not continue
because of the age restriction of age of the
University of 16. Jacobi continued studying
independently and was finally admitted to the
University of Berlin in the spring of 1821.
By 1824, Jacobi began teaching and in 1825
presented a paper concerning iterated functions
to the Academy of Sciences in Berlin. The
Academy did not find his results impressive and
it was not published until 1961.
In 1829, Jacobi published his paper "Fundamenta
nova theoria functions ellipticarum," translated
to New Foundations of the Theory of Elliptic
Functions, which made significant contributions
to the field of elliptic functions.
Jacobi was promoted to full professor in 1832
while at the University of Konigsberg, and
pursued his study of partial differential equations
of the first order, which led to the publishing of
"Structure and Properties of Determinants." He
applied these theories to differential equations in
Dynamics which again led to another publishing,
Lectures in Dynamics. It was also here at the
University of University of Konigsberg, that he
worked on functional determinants now called
Jacobian determinants.
In 1842, Jacobi became ill with diabetes and was
assited with grants to move to Itay where he
published many more works.
Jacobi moved to a small town called Gotha in
1848. Two years later, January of 1851, he
developed influenza and smallpox and died
shortly after.

Weierstrass came late to mathematics. He
entered the University of Bonn at his father’s
insistence to study law and public finance, to
prepare for entering the civil service; after four
years of carousing he left the university without
a degree. Weierstrass eventually obtained a
teaching license and spent the years from 1841
to 1854 at obscure secondary schools in Prussia.
A series of brilliant mathematical papers written
during this time, however, resulted in an
honorary doctoral degree from the University of
Königsberg in 1854. Then, at the age of 40,
Weierstrass was appointed to an academic
position at the University of Berlin. He exerted
great influence there through his teaching of
advanced mathematics, attracting gifted
students from around the world. Although
Weierstrass never published these lectures, his
contributions were widely disseminated by the
listeners.
Weierstrass provided a completely rigorous
treatment of calculus by using the arithmetic of
inequalities to replace the vague words in
Cauchy’s definitions and theorems. The result
was a clear-cut formulation of the notion of limit,
our now-standard one in terms of epsilon and
delta:
Mathematicians erroneously believed that a
continuous function must be differentiable at
most points. Weierstrass surprised his
contemporaries by providing an example of a
continuous function that has a derivative at no
point x of the real line; namely,
He is also known for extending the comparison
test for a series of constants to a series of
functions all defined on the same interval I. The
so-called Weierstrass M-test says:

The son of a Lutheran pastor, Riemann forsook
an initial interest in theology to study
mathematics in Berlin and then in Gottingen. He
completed his training for the doctorate in 1851
at the latter university, under the guidance of
the legendary Carl Gauss. Riemann returned to
Gottingen three years later as a lowly unpaid
tutor, working his way up the academic ladder to
a full professorship in 1859. Yet his teaching
career was tragically brief. He fell ill with
tuberculosis and spent his last years in Italy,
where he died in 1866, only 39 years of age.
Although he published only a few papers, his
name is attached to a variety of topics in several
branches of mathematics: Riemann surfaces,
Cauchy-Riemann equations, the Riemann zeta
function, Riemannian (that is, non-Euclidean)
geometry, and the still-unproven Riemann
Hypothesis.
The view that integration was simply a process
reverse to differentiation prevailed until the
nineteenth century. The familiar conception of
the definite integral as the limit of
approximating sums was given by Riemann in a
paper he submitted upon joining the faculty at
Göttingen in 1854. It was not published until 13
years later, and then only after his untimely
death. His formulation of what today is known as
the "Riemann integral" runs thus:
If f(x) is a continuous function on the interval
[a,b] and
a = xo, x1, x2, ... ,xn= b are a finite set of points
in [a,b], then
where xk * is an arbitrary point in the subinterval
[xk-1,xk] and d is the maximum of the lengths of
the subintervals.
(This is a modification of Cauchy’s definition, in

which the xk were taken to be the left-endpoints
of the subintervals [xk-1,xk].) Riemann
subsequently applied his version of the integral
to discontinuous functions, producing a
remarkable example of an integrable function
having infinitely many discontinuities. With this,
the study of discontinuous functions gained
mathematical legitimacy.
Considered one of the greatest American
scientists during the 19th century, Josiah
Willard Gibbs was born in New Haven,
Connecticut on Feb. 11, 1839. In 1854, he
enrolled at Yale University where he won prizes
for excellence in Latin and Mathematics. By
1863, Gibbs earned a Ph.D in engineering (the
first in the U.S.) from the Sheffield Scientific
School at Yale. He tutored at Yale for three years,
teaching Latin and Natural Philosophy. Gibbs also
attended several lectures in Europe before
becoming a Professor of Mathematical Physics at
Yale in 1871.
From 1871 to 1878, Gibbs worked on
thermodynamics, introducing geometrical
methods, thermodynamic surfaces, and criteria
for equilibrium. He developed the concept of
Gibbs free energy and other thermodynamic
potentials in the analysis of equilibrium. Gibbs
also built the foundation of modern vector
calculus and studied the electromagnetic theory
of light.
After writing several papers, Gibbs changed the
focus of his research from thermodynamics to
statistical methods. In 1884, he introduced the
"Gibbs principle" for statistical entropy,
canonical, and microcanonical statistical
distributions. In 1898, he studied the "Gibbs
phenomenon" in the convergence of Fourier
series. By 1902, Gibbs published "Elementary
Principles of Statistical Mechanics", from which
the foundation of statistical mechanics was built.

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