list of mathematician

mathematician in different fields

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Definition of
Mathematics and
the Main
Branches

2. Pride of Mathematics 2

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, calculu s 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.

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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

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Branches of
Mathematics:
Mathematician

5. Pride of Mathematics 5
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.











 Elementary algebra, the part of algebra that is usually
taught in elementary courses of mathematics.
 Abstract algebra, in which algebraic structures such
as groups, rings and fields are axiomatically defined and
investigated.
 Linear algebra, in which the specific properties of linear
equations, vector spaces and matrices are studied.
 Commutative algebra, the study of commutative rings.
 Computer algebra, the implementation of algebraic
methods as algorithms and computer programs.
 Homological algebra, the study of algebraic structures

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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.

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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)

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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 aNoetherian 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.

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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 projective geometry 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.

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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.

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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)

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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(1680-1620BC)
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(1750BC)
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:

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TheSulbasutras
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 knowing 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(750-690BC)
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.

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Katyayana(200-140BC)
states the general case of the Pythagorean theorem for the diagonal of any rectangle.
Greek Geometry (600 BC - 400 AD)

ThalesofMiletus(624-547BC)
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(569-475BC)

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

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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 establish 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

16. Pride of Mathematics 16
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 his 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)

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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,

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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)

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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 early 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

20. Pride of Mathematics 20
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” (e
ix
= cos x + i sin x),
“Euler’s identity” (e
iπ
+ 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)

21. Pride of Mathematics 21

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 speculation 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 encouraged 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

22. Pride of Mathematics 22
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”.

23. Pride of Mathematics 23
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)

24. Pride of Mathematics 24

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”.

25. Pride of Mathematics 25
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.

Plane Trigonometry
 Plane trigonometry is used for determining the height and distances of the
angles in a plane triangle. This type of triangle has three vertices (points of
intersection) on the surface, and the sides of the triangle are straight lines.
Values for plane trigonometry are different than for core, as the sum of
the plane must equal 180 degrees as opposed to 90 degrees. Mechanical
engineers, architects, physicists and chemists use this type of
trigonometry.

26. Pride of Mathematics 26
























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 are used, as
well as latitude and longitude variables for determining the
distance between two points. Once used to determine the
position of sunrises and sunsets, this type of trigonometry
originated in the 8th century. Mapmakers and navigation
enthusiasts continue to use spherical trigonometry today.

Analytic Trigonometry
 A subtype of core trigonometry, analytic seeks to determine
values based upon the x-y plane of a triangle. The sine (and cosine)
of the sum of two angles is used to obtain the sine (and cosine) of a
double angle. Formulas for double angles are also used to
determine the values of half angles, by using division and square
roots. Analytic trigonometry is used in engineering and science.

27. Pride of Mathematics 27












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.

28. Pride of Mathematics 28
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

29. Pride of Mathematics 29
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

30. Pride of Mathematics 30
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.

31. Pride of Mathematics 31
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

32. Pride of Mathematics 32
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.

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

33. Pride of Mathematics 33
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.

34. Pride of Mathematics 34
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.

35. Pride of Mathematics 35
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.

36. Pride of Mathematics 36


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, a
2. 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

37. Pride of Mathematics 37
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

38. Pride of Mathematics 38
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 remember ed 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, wh ich 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

39. Pride of Mathematics 39

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,

40. Pride of Mathematics 40
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.

41. Pride of Mathematics 41


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 a t 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

42. Pride of Mathematics 42
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

43. Pride of Mathematics 43
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, w hich
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

44. Pride of Mathematics 44
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 d ay. He showed
that

45. Pride of Mathematics 45
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

46. Pride of Mathematics 46
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 yo ung
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

47. Pride of Mathematics 47
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

48. Pride of Mathematics 48
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). Thi s
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

49. Pride of Mathematics 49
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

50. Pride of Mathematics 50
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

51. Pride of Mathematics 51
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 ins tructor 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

52. Pride of Mathematics 52
sine waves and cosine waves.


Johann Carl Frederich Gauss was born on Apr. 30,
1777 in Brunswick, Germay. Many conside r 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

53. Pride of Mathematics 53
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, astro nomy, 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

54. Pride of Mathematics 54
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 Mobius 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.

55. Pride of Mathematics 55


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.

56. Pride of Mathematics 56


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.

57. Pride of Mathematics 57


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:

58. Pride of Mathematics 58


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: Riem ann 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

59. Pride of Mathematics 59
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.

60. Pride of Mathematics 60


Srinivasa Ramanujan was born in Erode, India on
Dec. 22, 1887. He first developed an attraction for
mathematics at the age of 15 when he read a copy of
George Shoobridge Carr's "Synopsis of Elementary
Results in Pure and Applied Mathematics". In no
time, he was deriving his own theorems.
Unfortunately, most of the papers in Carr's collection
of theorems were about 40 years old. By 1902,
Ramanujan used what he learned about cubic
equations to create his own method of solving
quartic equations. He tried to do the same with
quintic equations but couldn't make it work.
In 1903, Ramanujan's talent earned him a
scholarship to the University of Madras, but he lost
it the following year because he concentrated solely
on mathematics and neglected his other courses.
However, he continued his work in mathematics
outside of the university, even while enduring
poverty for several years. Eventually, he became a
clerk at the Madras Port Trust.
In 1911, Ramanujan published his first paper in the
Journal of the Indian Mathematical Society. In 1913,
he became associated with Godfrey H. Hardy, a
British mathematician. This led to a scholarship
from the University of Madras and from Trinity
College in Cambridge. In 1914, he traveled to
England. Hardy tutored Ramanujan in mathematics
and also collaborated with him on research. During
his stay in London, his health became a major issue
and he did not pubslish anything for five months. Of
the work he did get to publish, they appeared in
European and British journals. By 1918, he was
elected a fellow of the Cambridge Philosophical
Society. Shortly thereafter, he was considered for a
fellowship in the Royal Society of London. Three
months later, Ramanujan became the first Indian to
join the Royal Society.
Ramanujan worked in the field of continued
fractions, which he skillfully mastered. He also did
extensive work on hypergeomteric series,
independently discovering the results of Carl
Frederich Gauss and Ernst Ed uard Kummer.

61. Pride of Mathematics 61
Ramanujan derived the elliptical integrals, the
Riemann Series, functional equations of the zeta
function and his own theory of divergent series.















Statistic

62. Pride of Mathematics 62
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.























(1) Descriptive Statistics:
In descriptive statistics, it deals with collection of data, its
presentation in various forms, such as tables, graphs and
diagrams and findings averages and other measures which
would describe the data.
For Example: Industrial statistics, population statistics,
trade statistics etc… Such as businessman make to use
descriptive statistics in presenting their annual reports,
final accounts, bank statements.
(2) Inferential Statistics:
In inferential statistics, it deals with techniques used for
analysis of data, making the estimates and drawing
conclusions from limited information taken on sample basis
and testing the reliability of the estimates.
For Example: Suppose we want to have an idea about the
percentage of illiterates in our country. We take
a sample from the population and find the proportion of
illiterates in the sample. This sample proportion with the
help of probability enables us to make some inferences
about the population proportion. This study belongs to
inferential statistics.

63. Pride of Mathematics 63

Name
National
ity
Birt
h
Deat
h
Contribution
Al-Kindi Iraqi 801 873 Developed the first code breaking
algorithm based on frequency analysis.
He wrote a book entitled "Manuscript
on Deciphering Cryptographic
Messages", containing detailed
discussions on statistics
Graunt, John English 1620 1674 Pioneer of demography who produced
the first life table
Bayes,
Thomas
English 1702 1761 Developed the interpretation
of probability now known as Bayes
theorem
Laplace,
Pierre-Simon
French 1749 1827 Co-invented Bayesian statistics .
Invented exponential families (Laplace
transform), conjugate
prior distributions, asymptotic
analysis of estimators (including
negligibility of regular priors).
Used maximum-likelihood and posterior-
mode estimation and considered
(robust) loss functions
Playfair,
William
Scottish 1759 1823 Pioneer of statistical graphics
Gauss, Carl
Friedrich
German 1777 1855 Invented least squares estimation
methods (with Legendre). Used loss
functionsand maximum-
likelihood estimation
Quetelet, Belgian 1796 1874 Pioneered the use of probability and

64. Pride of Mathematics 64
Name
National
ity
Birt
h
Deat
h
Contribution
Adolphe statistics in the social sciences
Nightingale,
Florence
English 1820 1910 Applied statistical analysis to health
problems, contributing to the
establishment of epidemiology and
public health practice.
Developed statistical graphicsespecially
for mobilizing public opinion. First
female member of the Royal Statistical
Society.
Galton,
Francis
English 1822 1911 Invented the concepts of standard
deviation, correlation, regression
Thiele,
Thorvald N.
Danish 1838 1910 Introduced cumulants and the term
"likelihood". Introduced a Kalman
filter intime-series
Peirce,
Charles
Sanders
American 1839 1914 Formulated modern statistics in
"Illustrations of the Logic of Science"
(1877–1878) and "A Theory of
Probable Inference" (1883). With
a repeated measures design ,
introduced blinded, controlled
randomized
experiments (before Fisher).
Invented optimal design for
experiments on gravity, in which he
"corrected the means ". He
used correlation, smoothing, and
improved the treatment of outliers.
Introduced terms "confidence" and
"likelihood"

65. Pride of Mathematics 65
Name
National
ity
Birt
h
Deat
h
Contribution
(before Neyman and Fisher). While
largely a frequentist, Peirce's possible
world semantics introduced the
"propensity" theory of probability. See
the historical books of Stephen Stigler
Edgeworth,
Francis Ysidro
Irish 1845 1926 Revived exponential families (Laplace
transforms) in statistics.
ExtendedLaplace's (asymptotic) theory
of maximum-likelihood estimation.
Introduced basic results on information,
which were extended and popularized
by R. A. Fisher
Pearson, Karl English 1857 1936 Numerous innovations, including the
development of the Pearson chi-
squared test and the Pearson
correlation. Founded the Biometrical
Society andBiometrika, the first journal
of mathematical
statistics and biometry
Spearman,
Charles
English 1863 1945 Extended the Pearson correlation
coefficient to the Spearman's rank
correlation coefficient
Gosset,
William
Sealy(known
as "Student")
English 1876 1937 Discovered the Student t
distribution and invented the Student's
t-test
Fisher, Ronald English 1890 1962 Wrote the textbooks and articles that
defined the academic discipline of
statistics, inspiring the creation of

66. Pride of Mathematics 66
Name
National
ity
Birt
h
Deat
h
Contribution
statistics departments at universities
throughout the world. Systematized
previous results with informative
terminology, substantially improving
previous results with mathematical
analysis (and claims). Developed
the analysis of variance, clarified the
method ofmaximum likelihood (without
the uniform priors appearing in some
previous versions), invented the
concept of sufficient statistics,
developed Edgeworth's use
of exponential families and information,
introducing observed Fisher
information, and many theoretical
concepts and practical methods,
particularly for the design of
experiments
Bonferroni,
Carlo Emilio
Italian 1892 1960 Invented the Bonferroni
correction for multiple comparisons
Wilcoxon,
Frank
Irish-
American
1892 1965 Invented two statistical
tests: Wilcoxon rank-sum test and
the Wilcoxon signed-rank test
Neyman,
Jerzy
Polish-
American
1894 1981 Discovered the confidence interval and
co-developed the Neyman–Pearson
lemma
Pearson, Egon English 1895 1980 Co-developed the Neyman –Pearson
lemma of statistical hypothesis testing
Deming, W. American 1900 1993 Developed methods for

67. Pride of Mathematics 67
Name
National
ity
Birt
h
Deat
h
Contribution
Edwards statistical quality control
Finetti,
Bruno de
Italian 1906 1985 Pioneer of the "operational subjective"
conception of probability. Used this as
the basis for exposition of the Bayesian
method of sta tistical analysis.
Developed the representation theorem
for exchangeable random
variablesshowing that they are the
basis of the IID model in statistics.
Kendall,
Maurice
English 1907 1983 Co-developed methods for
assessing statistical randomness ;
invented Kendall tau rank correlation
coefficient
Tukey, John American 1915 2000 Jointly popularized Fast Fourier
transformation, pioneer of exploratory
data analysis and graphical
presentation of data, developed
the jackknife for variance estimation,
invented the box plot.
Blackwell,
David
American 1919 2010 Co-developed Rao-Blackwell
theorem and wrote one of the
first Bayesiantextbooks, Basic
Statistics.
Rao,
Calyampudi
Radhakrishna
Indian 1920 Co-developed Cramér–Rao
bound and Rao–Blackwell theorem ,
inventedMINQUE method of variance
component estimation.
Cox, David English 1924 Developed the proportional hazards
model for the analysis of survival data

68. Pride of Mathematics 68
Name
National
ity
Birt
h
Deat
h
Contribution
Efron,
Bradley
American 1938 Invented the bootstrap resampling
technique for deriving an empirical
distribution of an estimate of a model
parameter









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. Arithme tic 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


number theory was a synonym of "arithmetic". The
addressed problems were directly related to the
basic operations and

69. Pride of Mathematics 69

70. Pride of Mathematics 70
Math
Theorem
And
Formula

71. Pride of Mathematics 71


Geometry Formulas
Areas:
 Area of a Square = a
2


Here a is the side of the square which is equal for 4sides
 Area of a Rectangle = a*b
Here a is the length measure and b is the breadth measure of rectangle
 Area of a Parallelogram = bh
Here b is the breadth of the parallelogram and h is its height measure
 Area of a Trapezoid = h/2 (b1 + b2)
Here h is the height measure of the trapezoid and b1 and b2 are its unequal
breadths measures

72. Pride of Mathematics 72


Here r is the radius of the circle, and pi=3.14
 Area of a Ellipse = pi r1 r2

Here r1 is the shorter radius and r2 is the longer radius measures, and pi = 3.14.

 Area of a triangle=1/2(bh)

Here b is the breadth measure and h is the height measure of the rectangle.
 Area of equilateral triangle=

Here a is the side measure of the triangle which is equal on the 3 sides

73. Pride of Mathematics 73
 Area of a triangle according to SAS (two sides and the opposite angle)
=

Here a and b are the 2 adjacent sides measure and C is the included angle between
them.
 Triangle given a, b, c are the sides =

where s = (a+b+c)/2

Volumes Formulas

 Volume of cube = a
3

Here a is the side measure of the cube which is equal on every side
 Volume of rectangular prism = a b c
Here a, b and c are the length. Breadth and height measure of the cuboid.
 Volume of irregular prism = b h

b and h are the breadth and height measure of the irregular prism

74. Pride of Mathematics 74
 Volume of cylinder = b h = pi r
2
h
b and h are the breadth and height measure of the cylinder
 Volume of pyramid = (1/3) b h
b and h are the breadth and height measure of the pyramid.
 Volume of cone = (1/3) b h = 1/3 pi r
2
h
and h are the breadth and height measure of the cone
 Volume of sphere = (4/3) pi r
3


 Volume of ellipsoid = (4/3) pi r1 r2 r3
Formulas to find perimeters:
 square = 4a
Here a is the side of the square which is equal for 4sides.
 Rectangle = 2a + 2b

Here a is the length measure and b is the breadth measure of rectangle
 Triangle = a + b + c

75. Pride of Mathematics 75
a , b and c are the sides
 Circle = 2pi r

circle = pi d (where d is the diameter) d is the diameter
Mathematics Formulas
 Simple interest:
Simple interest can be calculated
as:
Simple Interest, SI = PRT / 100
Here P = principal, R = rate per
annum, T = time in years.
Therefore Amount, A = P +
PRT/100 = P [1 + ( RT / 100 )]
 Compound interest
Compound interest is defined as
the interest earned on the
principal amount and on interest
accumulated on it.
P = C (1 + r/n) n*t
where
P = future value
C = initial deposit
r = interest rate (expressed as a
fraction: eg. 0.06)
n = # of times per year interest
is compounded
t = number of years invested
 Equation for simplified compound
interest:
When interest is only compounded
once per year (n=1), the equation
simplifies to:
P = C (1 + r) t
 Equation for Continuous
Compound Interest
When interest is compounded
continually (i.e. n –> ), the
compound interest will be
calculated as:
P = C e r*t
Algebra:
 Closure Property of Addition

76. Pride of Mathematics 76
Sum (or difference) of 2 real
numbers equals a real number
 Additive Identity
a + 0 = a
 Additive Inverse
a + (-a) = 0
 Associative of Addition
(a + b) + c = a + (b + c)
 Commutative of Addition
a + b = b + a
 Definition of Subtraction
a – b = a + (-b)
 Multiplicative Identity
a * 1 = a

 Multiplicative Inverse
a * (1/a) = 1 (a 0)
 (Multiplication times 0)
a * 0 = 0
 Associative of Multiplication
(a * b) * c = a * (b * c)
Commutative of Multiplication
a * b = b * a
 Distributive Law
a(b + c) = ab + ac
 Definition of Division
a / b = a(1/b)
x a x b = x (a + b)
x a y a = (xy) a
(x a) b = x (ab)
x (a/b) = bth root of (x a) = (
bth (x) ) a
x (-a) = 1 / x a
x (a – b) = x a / x b
 simple equation in two variables:
: – y=ax+b
Where’ y and x are variables and a
and b are constants.
 Algebraic equation:
: -ax2+bx+c=0;
 Polynomial equations:
(a+b) 2 = a 2 + 2ab + b 2
(a+b)(c+d) = ac + ad + bc + bd
a 2 – b 2 = (a+b)(a-b)
a 3 b 3 = (a b)(a 2 ab + b 2) x
2 + (a+b)x + AB = (x + a)(x +
b)
if ax 2 + bx + c = 0 then x = (
-b (b 2 – 4ac) ) / 2a
 Logarithms:
y = logb(x) if and only if x=b y
logb(1) = 0
logb(b) = 1
logb(x*y) = logb(x) + logb(y)
logb(x/y) = logb(x) – logb(y)
logb(x n) = n logb(x)
logb(x) = logb(c) * logc(x) =
logc(x) / logc(b)

77. Pride of Mathematics 77
1 The Irrationality of the Square Root
of 2
Pythagoras and his
school
500
B.C.
2 Fundamental Theorem of Algebra Karl Frederich Gauss 1799
3 The Denumerability of the Rational
Numbers
Georg Cantor 1867
4 Pythagorean Theorem Pythagoras and his
school
500
B.C.
5 Prime Number Theorem Jacques
Hadamard and Charles-
Jean de la Vallee
Poussin (separately)
1896
6 Godel’s Incompleteness Theorem Kurt Godel 1931
7 Law of Quadratic Reciprocity Karl Frederich Gauss 1801
8 The Impossibility of Trisecting the
Angle and Doubling the Cube
Pierre Wantzel 1837
9 The Area of a Circle Archimedes 225
B.C.
10 Euler’s Generalization of Fermat’s
Little Theorem
(Fermat’s Little Theorem)
Leonhard Euler
(Pierre de Fermat)
1760
(1640)
11 The Infinitude of Primes Euclid 300
B.C.
12 The Independence of the Parallel
Postulate
Karl Frederich
Gauss, Janos
Bolyai, Nikolai
Lobachevsky, G.F.
Bernhard
Riemanncollectively
1870-
1880
13 Polyhedron Formula Leonhard Euler 1751
14 Euler’s Summation of 1 + (1/2)^2 +
(1/3)^2 + ‘ (the Basel Problem).
Leonhard Euler 1734

78. Pride of Mathematics 78
15 Fundamental Theorem of Integ ral
Calculus
Gottfried Wilhelm von
Leibniz
1686
16 Insolvability of General Higher Degree
Equations
Niels Henrik Abel 1824
17 DeMoivre’s Theorem Abraham DeMoivre 1730
18 Liouville’s Theorem and the
Construction of Trancendental
Numbers
Joseph Liouville 1844
19 Four Squares Theorem Joseph-Louis Lagrange 1770
20 Primes that Equal to the Sum of
Two Squares (Genus theorem)
? ?
21 Green’s Theorem George Green 1828
22 The Non-Denumerability of the
Continuum
Georg Cantor 1874
23 Formula for Pythagorean Triples Euclid 300
B.C.
24 The Undecidability of the Continuum
Hypothesis
Paul Cohen 1963
25 Schroeder-Bernstein Theorem ? ?
26 Leibnitz’s Series for Pi Gottfried Wilhelm von
Leibniz
1674
27 Sum of the Angles of a Triangle Euclid 300
B.C.
28 Pascal’s Hexagon Theorem Blaise Pascal 1640
29 Feuerbach’s Theorem Karl Wilhelm Feuerbach 1822
30 The Ballot Problem J.L.F. Bertrand 1887
31 Ramsey’s Theorem F.P. Ramsey 1930
32 The Four Color Problem Kenneth Appel and
Wolfgang Haken
1976
33 Fermat’s Last Theorem Andrew Wiles 1993
34 Divergence of the Harmonic Series Nicole Oresme 1350
35 Taylor’s Theorem Brook Taylor 1715

79. Pride of Mathematics 79
36 Brouwer Fixed Point Theorem L.E.J. Brouwer 1910
37 The Solution of a Cubic Scipione Del Ferro 1500
38 Arithmetic Mean/Geometric Mean
(Proof by Backward Induction)
(Polya Proof)
Augustin-Louis Cauchy
George Polya
?
?
39 Solutions to Pell’s Equation Leonhard Euler 1759
40 Minkowski’s Fundamental Theorem Hermann Minkowski 1896
41 Puiseux’s Theorem Victor Puiseux (based
on a discovery of Isaac
Newton of 1671)
1850
42 Sum of the Reciprocals of the
Triangular Numbers
Gottfried Wilhelm von
Leibniz
1672
43 The Isoperimetric Theorem Jacob Steiner 1838
44 The Binomial Theorem Isaac Newton 1665
45 The Partition Theorem Leonhard Euler 1740
46 The Solution of the General Quartic
Equation
Lodovico Ferrari 1545
47 The Central Limit Theorem ? ?
48 Dirichlet’’s Theorem Peter Lejune Dirichlet 1837
49 The Cayley-Hamilton Thoerem Arthur Cayley 1858
50 The Number of Platonic Solids Theaetetus 400
B.C.
51 Wilson’s Theorem Joseph-Louis Lagrange 1773
52 The Number of Subsets of a Set ? ?
53 Pi is Trancendental Ferdinand Lindemann 1882
54 Konigsberg Bridges Problem Leonhard Euler 1736
55 Product of Segments of Chords Euclid 300
B.C.
56 The Hermi te-Lindemann
Transcendence Theorem
Ferdinand Lindemann 1882
57 Heron’s Formula Heron of Alexandria 75
58 Formula for the Number of ? ?

80. Pride of Mathematics 80
Combinations
59 The Laws of Large Numbers <many> <many>
60 Bezout’s Theorem Etienne Bezout ?
61 Theorem of Ceva Giovanni Ceva 1678
62 Fair Games Theorem ? ?
63 Cantor’s Theorem Georg Cantor 1891
64 L.Hopital’s Rule John Bernoulli 1696?
65 Isosceles Triangle Theorem Euclid 300
B.C.
66 Sum of a Geometric Series Archimedes 260
B.C.?
67 e is Transcendental Charles Hermite 1873
68 Sum of an arithmetic series Babylonians 1700
B.C.
69 Greatest Common Divisor Algorithm Euclid 300
B.C.
70 The Perfect Number Theorem Euclid 300
B.C.
71 Order of a Subgroup Joseph-Louis Lagrange 1802
72 Sylow’s Theorem Ludwig Sylow 1870
73 Ascending or Descending Sequences Paul Erdos and G.
Szekeres
1935
74 The Principle of Mathematical
Induction
Levi ben Gerson 1321
75 The Mean Value Theorem Augustine-Louis Cauchy 1823
76 Fourier Series Joseph Fourier 1811
77 Sum of kth powers Jakob Bernouilli 1713
78 The Cauchy-Schwarz Inequality Augustine-Louis Cauchy 1814?
79 The Intermediate Value Theorem Augustine-Louis Cauchy 1821
80 The Fundamental Theorem of
Arithmetic
Euclid 300
B.C.
81 Divergence of the Prime Reciprocal Leonhard Euler 1734?

81. Pride of Mathematics 81







Series
82 Dissection of Cubes (J.E.
Littlewood’s elegant proof)
R.L. Brooks 1940
83 The Friendship Theorem Paul Erdos, Alfred
Renyi, Vera Sos
1966
84 Morley’s Theorem Frank Morley 1899
85 Divisibility by 3 Rule ? ?
86 Lebesgue Measure and Integration Henri Lebesgue 1902
87 Desargues’s Theorem Gerard Desargues 1650
88 Derangements Formula ? ?
89 The Factor and Remainder Theorems ? ?
90 Stirling’s Formula James Stirling 1730
91 The Triangle Inequality ? ?
92 Pick’s Theorem George Pick 1899
93 The Birthday Problem ? ?
94 The Law of Cosines Francois Viete 1579
95 Ptolemy’s Theorem Ptolemy 120?
96 Principle of Inclusion/Exclusion ? ?
97 Cramer’s Rule Gabriel Cramer 1750
98 Bertrand’s Postulate J.L.F. Bertrand 1860?
99 Buffon Needle Problem Comte de Buffon 1733
100 Descartes Rule of Signs Rene Descartes 1637

82. Pride of Mathematics 82





Mathemati
cs
Quotation

83. Pride of Mathematics 83


















Quotes
If people do not believe that mathematics is simple, it is only
because they do not realize how complicated life is. ~John Louis von
Neumann

Arithmetic is where the answer is right and everything is nice and
you can look out of the window and see the blue sky - or the
answer is wrong and you have to start over and try again and see
how it comes out this time. ~Carl Sandburg

Pure mathematics is, in its way, the poetry of logical ideas. ~Albert
Einstein

Mathematics are well and good but nature keeps dragging us around

84. Pride of Mathematics 84
by the nose. ~Albert Einstein

Black holes result from God dividing the universe by zero. ~Author
Unknown

Mathematics - the unshaken Foundation of Sciences, and the
plentiful Fountain of Advantage to human affairs. ~Isaac Barrow

I never did very well in math - I could never seem to persuade the
teacher that I hadn't meant my answers literally. ~Calvin Trillin

I don't agree with mathematics; the sum total of zeros is a
frightening figure. ~Stanislaw J. Lec, More Unkempt Thoughts

If you think dogs can't count, try putting three dog biscuits in your
pocket and then giving Fido only two of them. ~Phil Pastoret

[A mathematician is a] scientist who can figure out anything except
such simple things as squaring the circle and trisecting an
angle. ~Evan Esar, Esar's Comic Dictionary

"Every minute dies a man, Every minute one is born;" I need hardly
point out to you that this calculation would tend to keep the sum
total of the world's population in a state of perpetual equipoise,
whereas it is a well-known fact that the said sum total is constantly
on the increase. I would therefore take the liberty of suggesting
that in the next edition of your excellent poem the erroneous
calculation to which I refer should be corrected as follows: "Every

85. Pride of Mathematics 85
moment dies a man, And one and a sixteenth is born." I may add
that the exact figures are 1.067, but something must, of course, be
conceded to the laws of metre. ~Charles Babbage, letter to Alfred,
Lord Tennyson, about a couplet in his "The Vision of Sin"

Math is radical! ~Bumper Sticker

There was a blithe certainty that came from first comprehending the
full Einstein field equations, arabesques of Greek letters clinging
tenuously to the page, a gossamer web. They seemed insubstantial
when you first saw them, a string of squiggles. Yet to follow the
delicate tensors as they contracted, as the superscripts paired with
subscripts, collapsing mathematically into concrete classical entities -
potential; mass; forces vectoring in a curved geometry - that was a
sublime experience. The iron fist of the real, inside the velvet glove
of airy mathematics. ~Gregory Benford, Timescape

It is a mathematical fact that fifty percent of all doctors graduate
in the bottom half of their class. ~Author Unknown

If two wrongs don't make a right, try three. ~Author Unknown

Arithmetic is where numbers fly like pigeons in and out of your
head. ~Carl Sandburg, "Arithmetic"

Arithmetic is numbers you squeeze from your head to your hand to
your pencil to your paper till you get the answer. ~Carl Sandburg,
"Arithmetic"

86. Pride of Mathematics 86

If equations are trains threading the landscape of numbers, then no
train stops at pi. ~Richard Preston

Even stranger things have happened; and perhaps the strangest of all
is the marvel that mathematics should be possible to a race akin to
the apes. ~Eric T. Bell, The Development of Mathematics

So if a man's wit be wandering, let him study the mathematics; for
in demonstrations, if his wit be called away never so little, he must
begin again. ~Francis Bacon, "Of Studies"

The essence of mathematics is not to make simple things
complicated, but to make complicated things simple. ~S. Gudder

The human mind has never invented a labor-saving machine equal to
algebra. ~Author Unknown

The mathematics are distinguished by a particular privilege, that is,
in the course of ages, they may always advance and can never
recede. ~Edward Gibbon, Decline and Fall of the Roman Empire

Go down deep enough into anything and you will find
mathematics. ~Dean Schlicter

It is not the job of mathematicians... to do correct arithmetical
operations. It is the job of bank accountants. ~Samuil
Shchatunovski

87. Pride of Mathematics 87

Trigonometry is a sine of the times. ~Author Unknown

Mathematics is not a careful march down a well-cleared highway, but
a journey into a strange wilderness, where the explorers often get
lost. Rigour should be a signal to the historian that the maps have
been made, and the real explorers have gone elsewhere. ~W.S.
Anglin

A mathematician is a device for turning coffee into theorems. ~Paul
Erdos

Let us grant that the pursuit of mathematics is a divine madness of
the human spirit, a refuge from the goading urgency of contingent
happenings. ~Alfred North Whitehead

The tantalizing and compelling pursuit of mathematical problems
offers mental absorption, peace of mind amid endless challenges,
repose in activity, battle without conflict, "refuge from the goading
urgency of contingent happenings," and the sort of beauty changeless
mountains present to sense tried by the present-day kaleidoscope of
events. ~Morris Kline, Mathematics in Western Culture

Mathematics is as much an aspect of culture as it is a collection of
algorithms. ~Carl Boyer, 1949, calculus textbook

The cowboys have a way of trussing up a steer or a pugnacious
bronco which fixes the brute so that it can neither move nor

88. Pride of Mathematics 88
think. This is the hog-tie, and it is what Euclid did to
geometry. ~Eric Bell, The Search for Truth

Sometimes it is useful to know how large your zero is. ~Author
Unknown

The hardest arithmetic to master is that which enables us to count
our blessings. ~Eric Hoffer, Reflections On The Human Condition

Mathematics is the only good metaphysics. ~William Thomson Baron
Kelvin of Largs


The most savage controversies are those about matters as to which
there is no good evidence either way. Persecution is used in
theology, not in arithmetic. ~Bertrand Russell

I prefer zeroes on the loose
to those lined up behind a cipher.
~Wisława Szymborska (1923–2012), "Possibilities," 1997, translated
from the Polish by Clare Cavanagh and Stanisław Barańczak

Music is the pleasure the human mind experiences from counting
without being aware that it is counting. ~Gottfried Leibniz

How many times can you subtract 7 from 83, and what is left
afterwards? You can subtract it as many times as you want, and it
leaves 76 every time. ~Author Unknown

89. Pride of Mathematics 89

To most outsiders, modern mathematics is unknown territory. Its
borders are protected by dense thickets of technical terms; its
landscapes are a mass of indecipherable equations and incomprehensible
concepts. Few realize that the world of modern mathematics is rich
with vivid images and provocative ideas. ~Ivars Peterson

With my full philosophical rucksack I can only climb slowly up the
mountain of mathematics. ~Ludwig Wittgenstein, Culture and Value

But mathematics is the sister, as well as the servant, of the arts
and is touched with the same madness and genius. ~Harold Marston
Morse

Still more astonishing is that world of rigorous fantasy we call
mathematics. ~Gregory Bateson

Everybody who knew anything about ciphering was called in to
consider it. A young man from a high school near here, who made a
specialty of mathematics and pimples, and who could readily tell how
long a shadow a nine-pound ground-hog would cast at 2 o'clock and
37 minutes P.M., on ground-hog day, if sunny, at the town of
Fungus, Dak., provided latitude and longitude and an irregular mass
of red chalk be given to him, was secured to jerk a few logarithms in
the interests of trade. He came and tried it for a few days, covered
the interior of the Exposition Building with figures and then went
away. ~Bill Nye, "Seeking to be Identified," Nye and Riley's Railway
Guide by Edgar W. Nye and James Whitcomb Riley, 1888

90. Pride of Mathematics 90


Anyone who cannot cope with mathematics is not fully human. At
best he is a tolerable subhuman who has learned to wear shoes,
bathe, and not make messes in the house. ~Robert Heinlein, Time
Enough for Love

You may be an engineer if your idea of good interpersonal
communication means getting the decimal point in the right
place. ~Author Unknown

The trouble with integers is that we have examined only the very
small ones. Maybe all the exciting stuff happens at really big
numbers, ones we can't even begin to think about in any very
definite way. Our brains have evolved to get us out of the rain,
find where the berries are, and keep us from getting killed. Our
brains did not evolve to help us grasp really large numbers or to look
at things in a hundred thousand dimensions. ~Ronald L. Graham

We could use up two Eternities in learning all that is to be learned
about our own world and the thousands of nations that have arisen
and flourished and vanished from it. Mathematics alone would
occupy me eight million years. ~Mark Twain

Can you do Division? Divide a loaf by a knife - what's the answer
to that? ~Lewis Carroll, Through the Looking Glass

Mathematics is the supreme judge; from its decisions there is no

91. Pride of Mathematics 91
appeal. ~Tobias Dantzig

Although he may not always recognize his bondage, modern man lives
under a tyranny of numbers. ~Nicholas Eberstadt, The Tyranny of
Numbers: Mismeasurement and Misrule

The mathematics are usually considered as being the very antipodes
of Poesy. Yet Mathesis and Poesy are of the closest kindred, for
they are both works of the imagination. ~Thomas Hill

I used to love mathematics for its own sake, and I still do, because
it allows for no hypocrisy and no vagueness.... ~Stendhal (Henri
Beyle), The Life of Henri Brulard

As far as the laws of mathematics refer to reality, they are not
certain; and as far as they are certain, they do not refer to
reality. ~Albert Einstein, Sidelights on Relativity

God does not care about our mathematical difficulties; He integrates
empirically. ~Albert Einstein

If there is a God, he's a great mathematician. ~Paul Dirac

To all of us who hold the Christian belief that God is truth,
anything that is true is a fact about God, and mathematics is a
branch of theology. ~Hilda Phoebe Hudson

The different branches of Arithmetic - Ambition, Distraction,

92. Pride of Mathematics 92
Uglification, and Derision. ~Lewis Carroll

geometry is not true, it is advantageous. ~Henri Poincaré

Infinity is a floorless room without walls or ceiling. ~Author
Unknown

Physics is mathematical not because we know so much about the
physical world, but because we know so little; it is only its
mathematical properties that we can discover. ~Bertrand Russell

God is real, unless declared integer. ~Author Unknown

If a healthy minded person takes an interest in science, he gets busy
with his mathematics and haunts the laboratory. ~W.S. Franklin
Proof is an idol before whom the pure mathematician tortures
himself. ~Arthur Stanley Eddington, The Nature of the Physical
World

There was a young man from Trinity,
Who solved the square root of infinity.
While counting the digits,
He was seized by the fidgets,
Dropped science, and took up divinity.
~Author Unknown

One cannot escape the feeling that these mathematical formulas have
an independent existence and an intelligence of their own, that they

93. Pride of Mathematics 93
are wiser than we are, wiser even than their discoverers... ~Heinrich
Hertz

In the binary system we count on our fists instead of on our
fingers. ~Author Unknown

There are 10 types of people in this world: those who understand
binary and those who don't. ~Author Unknown

The laws of nature are but the mathematical thoughts of
God. ~Euclid

In most sciences one generation tears down what another has built
and what one has established another undoes. In mathematics alone
each generations adds a new story to the old structure. ~Hermann
Hankel

Twice two makes four seems to me simply a piece of
insolence. Twice two makes four is a pert coxcomb who stands with
arms akimbo barring your path and spitting. I admit that twice two
makes four is an excellent thing, but if we are to give everything its
due, twice two makes five is sometimes a very charming thing
too. ~Fyodor Mikhailovich Dostoevsky

I know that two and two make four - & should be glad to prove it
too if I could - though I must say if by any sort of process I could
convert 2 & 2 into five it would give me much greater
pleasure. ~George Gordon, Lord Byron

94. Pride of Mathematics 94

Mathematics may be defined as the economy of counting. There is
no problem in the whole of mathematics which cannot be solved by
direct counting. ~Ernst Mach

Nature does not count nor do integers occur in nature. Man made
them all, integers and all the rest, Kronecker to the contrary
notwithstanding. ~Percy William Bridgman, The Way Things Are

A man has one hundred dollars and you leave him with two
dollars. That's subtraction. ~Mae West

I've dealt with numbers all my life, of course, and after a while you
begin to feel that each number has a personality of its own. A
twelve is very different from a thirteen, for example. Twelve is
upright, conscientious, intelligent, whereas thirteen is a loner, a
shady character who won't think twice about breaking the law to get
what he wants. Eleven is tough, an outdoorsman who likes tramping
through woods and scaling mountains; ten is rather simpleminded, a
bland figure who always does what he's told; nine is deep and
mystical, a Buddha of contemplation.... ~Paul Auster, The Music of
Chance

Why do we believe that in all matters the odd numbers are more
powerful? ~Pliny the Elder, Natural History

Uneven numbers are the gods' delight. ~Virgil, The Eclogues

95. Pride of Mathematics 95
One of the endlessly alluring aspects of mathematics is that its
thorniest paradoxes have a way of blooming into beautiful
theories. ~Philip J. Davis

Pure mathematics is the world's best game. It is more absorbing
than chess, more of a gamble than poker, and lasts longer than
Monopoly. It's free. It can be played anywhere - Archimedes did it
in a bathtub. ~Richard J. Trudeau, Dots and Lines

I like mathematics because it is not human and has nothing particular
to do with this planet or with the whole accidental universe -
because like Spinoza's God, it won't love us in return. ~Bertrand
Russell, 1912

The man ignorant of mathematics will be increasingly limited in his
grasp of the main forces of civilization. ~John Kemeny

Although I am almost illiterate mathematically, I grasped very early
in life that any one who can count to ten can count upward
indefinitely if he is fool enough to do so. ~Robertson Davies, "Of
the Conservation of Youth," The Table Talk of Samuel Marchbanks

96. Pride of Mathematics 96





References

Famous-mathematicians.org/

fabpedigree.com/james/mathmen.htm

listverse.com/2010/12/07/top-10-greatest-mathematicians/

listverse.com/2010/12/07/top-10-greatest-mathematicians/

www.famous-mathematicians.com/

www.thefamouspeople.com/mathematicians.php

www.mnn.com › Tech › Research & Innovations

www.storyofmathematics.com/mathematicians.html

fabpedigree.com/james/mathmen.htm

www.biography.com/people/groups/academics-mathematicians

www.businessinsider.com/important-mathematicians-modern-world-201...

97. Pride of Mathematics 97

www.thefamouspeople.com/mathematicians.php

www.ask.com › Math › Geometry › Trigonometry

https://au.answers.yahoo.com/question/index?qid...

https://www.scribd.com/.../Mathematicians-Who-Contributed-in-Trigono...