2. 

The Ishango bone, found near the headwaters of the Nile river
(northeastern Congo), may be as much as 20,000 years old and consists of a
series of tally marks carved in three columns running the length of the
bone. Common interpretations are that the Ishango bone shows either the
earliest known demonstration of sequences of prime numbers or a sixmonth lunar calendar. In the book How Mathematics Happened: The First
50,000 Years, Peter Rudman argues that the development of the concept of
prime numbers could only have come about after the concept of division,
which he dates to after 10,000 BC, with prime numbers probably not being
understood until about 500 BC. He also writes that "no attempt has been
made to explain why a tally of something should exhibit multiples of two,
prime numbers between 10 and 20, and some numbers that are almost
multiples of 10.“ The Ishango bone, according to scholar
Alexander Marshack, may have influenced the later development of
mathematics in Egypt as, like some entries on the Ishango bone, Egyptian
arithmetic also made use of multiplication by 2; this, however, is disputed.
Predynastic Egyptians of the 5th millennium BC pictorially represented
geometric designs. It has been claimed that megalithic monuments in
England and Scotland, dating from the 3rd millennium BC, incorporate
geometric ideas such as circles, ellipses, and Pythagorean triples in their
design.

3. 

The study of mathematics as a subject in its own right begins in the 6th
century BC with the Pythagoreans, who coined the term "mathematics"
from the ancient Greek (mahatma), meaning "subject of instruction".
Greek mathematics greatly refined the methods (especially through the
introduction of deductive reasoning and mathematical rigor in proofs)
and expanded the subject matter of mathematics.Chinese mathematics
made early contributions, including a place value system.The
Hindu-Arabic numeral system and the rules for the use of its operations,
in use throughout the world today, likely evolved over the course of the
first millennium AD in Indiaand was transmitted to the west via Islamic
mathematics. Islamic mathematics, in turn, developed and expanded the
mathematics known to these civilizations. Many Greek and Arabic texts
on mathematics were then translated into Latin, which led to further
development of mathematics in medieval Europe.
From ancient times through the Middle Ages, bursts of mathematical
creativity were often followed by centuries of stagnation. Beginning in
Renaissance Italy in the 16th century, new mathematical developments,
interacting with new scientific discoveries, were made at an
increasing pacethat continues through the present day.

5. *************************
*************************




Archimedes was a Greek mathematician, philosopher
and inventor who wrote important works on geometry,
arithmetic and mechanics.
Archimedes was born in Syracuse on the eastern coast
of Sicily and educated in Alexandria in Egypt. He then
returned to Syracuse, where he spent most of the rest of
his life, devoting his time to research and
experimentation in many fields.
In mechanics he defined the principle of the lever and
is credited with inventing the compound pulley and the
hydraulic screw for raising water from a lower to
higher level. He is most famous for discovering the law
of hydrostatics, sometimes known as 'Archimedes'
principle', stating that a body immersed in fluid loses
weight equal to the weight of the amount of fluid it
displaces. Archimedes is supposed to have made this
discovery when stepping into his bath, causing him to
exclaim 'Eureka!'
During the Roman conquest of Sicily in 214 BC
Archimedes worked for the state, and several of his
mechanical devices were employed in the defence of
Syracuse. Among the war machines attributed to him
are the catapult and - perhaps legendary - a mirror
system for focusing the sun's rays on the invaders'
boats and igniting them. After Syracuse was captured,
Archimedes was killed by a Roman soldier. It is said
that he was so absorbed in his calculations he told his
killer not to disturb him.

6. ***********************

Leonhard Euler 15 April
1707 – 18 September 1783)
was a pioneering Swiss
mathematician and physicist.
He made important
discoveries in fields as
diverse as
infinitesimal calculus and
graph theory. He also
introduced much of the
modern mathematical
terminology and notation,
particularly for
mathematical analysis, such as
the notion of a
mathematical function.[3] He is
also renowned for his work in
mechanics, fluid dynamics,
optics, and astronomy.]

7. 
Euler spent most of his adult life in St. Petersburg, Russia, and inBerlin, Prussia. He is considered to be
the pre-eminent mathematician of the 18th century, and one of the greatest mathematicians ever. He is
also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto volumes.[4] A
statement attributed to Pierre-Simon Laplaceexpresses Euler's influence on mathematics: "Read Euler,
read Euler, he is the master of us all."[5] Euler was born on April 15, 1707, in Basel to Paul Euler, a pastor
of the Reformed Church, and Marguerite Brucker, a pastor's daughter. He had two younger sisters
named Anna Maria and Maria Magdalena. Soon after the birth of Leonhard, the Eulers moved from
Basel to the town of Riehen, where Euler spent most of his childhood. Paul Euler was a friend of the
Bernoulli family—Johann Bernoulli, who was then regarded as Europe's foremost mathematician, would
eventually be the most important influence on young Leonhard. Euler's early formal education started
in Basel, where he was sent to live with his maternal grandmother. At the age of thirteen he enrolled at
the University of Basel, and in 1723, received his Master of Philosophy with a dissertation that
compared the philosophies ofDescartes and Newton. At this time, he was receiving Saturday afternoon
lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for
mathematics.[6] Euler was at this point studying theology, Greek, and Hebrew at his father's urging, in
order to become a pastor, but Bernoulli convinced Paul Euler that Leonhard was destined to become a
great mathematician. In 1726, Euler completed a dissertation on the propagation of sound with the
title De Sono.[7] At that time, he was pursuing an (ultimately unsuccessful) attempt to obtain a position at
the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition; the
problem that year was to find the best way to place the masts on a ship. Pierre Bouguer, a man who
became known as "the father of naval architecture" won, and Euler took second place. Euler later won
this annual prize twelve times.[8

8. 



Johann Carl Friedrich Gauss
(30 April 1777 – 23 February 1855)
was a German mathematician and
physical scientist who contributed
significantly to many fields,
including number theory, algebra,
statistics,analysis,
differential geometry, geodesy,
geophysics, electrostatics,
astronomy and optics.
Sometimes referred to as
the Princeps mathematicorum("the
Prince of Mathematicians" or "the
foremost of mathematicians") and
"greatest mathematician since
antiquity", Gauss had a remarkable
influence in many fields of
mathematics and science and is
ranked as one of history's most
influential mathematicians.[2]

9. 



Carl Friedrich Gauss was born on 30 April 1777 in Lower Saxony, Germany, as the son of poor working-class
parents. Indeed, his mother was illiterate and never recorded the date of his birth, remembering only that he
had been born on a Wednesday, eight days before the Feast of the Ascension, which itself occurs 40 days after
Easter. Gauss would later solve this puzzle about his birthdate in the context of finding the date of Easter,
deriving methods to compute the date in both past and future years. He was christened and confirmed in a
church near the school he attended as a child.
Gauss was a child prodigy. There are many anecdotes about his precocity while a toddler, and he made his first
ground-breaking mathematical discoveries while still a teenager. He completedDisquisitiones Arithmeticae, his
magnum opus, in 1798 at the age of 21, though it was not published until 1801. This work was fundamental in
consolidating number theory as a discipline and has shaped the field to the present day.
Gauss's intellectual abilities attracted the attention of the Duke of Brunswick,who sent him to the Collegium
Carolinum (now Braunschweig University of Technology), which he attended from 1792 to 1795, and to the
University of Göttingen from 1795 to 1798. While at university, Gauss independently rediscovered several
important theorems; his breakthrough occurred in 1796 when he showed that any regular polygon with a
number of sides which is a Fermat prime (and, consequently, those polygons with any number of sides which is
the product of distinct Fermat primes and a power of 2) can be constructed by compass and straightedge. This
was a major discovery in an important field of mathematics; construction problems had occupied
mathematicians since the days of the Ancient Greeks, and the discovery ultimately led Gauss to choose
mathematics instead of philology as a career. Gauss was so pleased by this result that he requested that a
regular heptadecagon be inscribed on his tombstone. The stonemason declined, stating that the difficult
construction would essentially look like a circle.
The year 1796 was most productive for both Gauss and number theory. He discovered a construction of the
heptadecagon on 30 March.He further advanced modular arithmetic, greatly simplifying manipulations in
number theory. On 8 April he became the first to prove the quadratic reciprocity law. This remarkably general
law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic. The
prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbersare
distributed among the integers. Gauss also discovered that every positive integer is representable as a sum of
at most three triangular numberson 10 July and then jotted down in his diary the famous note: "ΕΥΡΗΚΑ!
num = Δ + Δ + Δ". On October 1 he published a result on the number of solutions of polynomials with
coefficients in finite fields, which 150 years later led to the Weil conjectures.

10. 

French mathematician who did important work in
many different branches of mathematics. However,
he did not stay in any one field long enough to
round out his work. He had an amazing memory and
could state the page and line of any item in a text he
had read. He retained this memory all his life. He
also remembered verbatim by ear. His normal work
habit was to solve a problem completely in his head,
then commit the completed problem to paper.
Despite his keen mathematical ability, he was
physically clumsy and artistically inept. In fact, he
received a score of 0 on his Polytechnique entrance
exam. He was always in a rush and disliked going
back for changes or corrections. He was also a
popularizer of mathematics. Poincaré's brother
Raymond was president of the French Republic
during World War I.
Poincaré is quoted as saying, "It is the simple
hypotheses of which one must be most wary;
because these are the ones that have the most
chances of passing unnoticed" (Boyer and Merzbach
1991, p. 599). In 1880, he created generalized
elliptic functions called automorphic functions. He
discovered that automorphic functions invariant
under the same group are connected by an
algebraic equation. Conversely, he found that the
coordinates of a point on any algebraic curve can
be expressed in terms of automorphic functions. He
showed they could be used to solve second order
linear differential equation with algebraic
coefficients.

11. 


As a mathematician and physicist, After receiving his degree, Poincaré
began teaching at the University of Caen in Normandy (in December
1879). At the same time he published his first major article – they are
devoted to treatment with a class of automorphic functions. There, in
Caen, he met his future wife, Louise Poulin d'Andesi and on April 20,
1881, they held their wedding. They had a son and three daughters
Poincaré immediately established himself among the greatest
mathematicians of Europe, attracting the attention of many prominent
mathematicians. In 1881 Poincaré was invited to take a teaching position
at the Faculty of Sciences of the University of Paris; he accepted the
invitation. During the years of 1883 to 1897, he taught mathematical
analysis in École Polytechnique.
In 1881–1882, Poincaré created a new branch of mathematics: the
qualitative theory of differential equations. He showed how it is possible
to derive the most important information about the behavior of a family of
solutions without having to solve the equation (since this may not always
be possible). He successfully used this approach to problems in
celestial mechanics and mathematical physics.

12. 


David Hilbert ( January 23, 1862 – February
14, 1943) was aGerman mathematician. He is
recognized as one of the most influential and
universal mathematicians of the 19th and early
20th centuries. Hilbert discovered and
developed a broad range of fundamental ideas
in many areas, including invariant theory and
the axiomatization of geometry. He also
formulated the theory of Hilbert spaces,[3] one
of the foundations of functional analysis.
Hilbert adopted and warmly defended
Georg Cantor's set theory and
transfinite numbers. A famous example of his
leadership in mathematics is his 1900
presentation of a collection of problems that
set the course for much of the mathematical
research of the 20th century.
Hilbert and his students contributed
significantly to establishing rigor and
developed important tools used in modern
mathematical physics. Hilbert is known as one
of the founders of proof theory and
mathematical logic, as well as for being among
the first to distinguish between mathematics
and metamathematics.

13. DAVID HILBERT


Hilbert, the first of two children of Otto and Maria Therese (Erdtmann) Hilbert, was born in the Province of Prussia either in Königsberg (according to Hilbert's own statement) or in Wehlau (known since 1946 as Znamensk) near
Königsberg where his father worked at the time of his birth. In the fall of 1872, he entered the Friedrichskolleg
Gymnasium (Collegium fridericianum, the same school thatImmanuel Kant had attended 140 years before), but after
an unhappy period he transferred to (fall 1879) and graduated from (spring 1880) the more science-oriented Wilhelm
Gymnasium. Upon graduation he enrolled (autumn 1880) at the University of Königsberg, the "Albertina". In the
spring of 1882, Hermann Minkowski (two years younger than Hilbert and also a native of Königsberg but so talented
he had graduated early from his gymnasium and gone to Berlin for three semesters),returned to Königsberg and
entered the university. "Hilbert knew his luck when he saw it. In spite of his father's disapproval, he soon became
friends with the shy, gifted Minkowski."[ In 1884, Adolf Hurwitzarrived from Göttingen as an Extraordinarius, i.e. an
associate professor. An intense and fruitful scientific exchange among the three began, and Minkowski and Hilbert
especially would exercise a reciprocal influence over each other at various times in their scientific careers. Hilbert
obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled Über invariante
Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen ("On the invariant properties of special
binary forms, in particular the spherical harmonic functions").
Hilbert remained at the University of Königsberg as a Privatdozent (senior lecturer) from 1886 to 1895. In 1892, Hilbert
married Käthe Jerosch (1864–1945), "the daughter of a Konigsberg merchant, an outspoken young lady with an
independence of mind that matched his own“. at Königsberg they had their one child, Franz Hilbert (1893–1969). In
1895, as a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at
the University of Göttingen, at that time the best research center for mathematics in the world. He remained there for
the rest of his life.

14. 



His son Franz suffered throughout his life from an undiagnosed mental illness: his inferior
intellect was a terrible disappointment to his father and this misfortune was a matter of
distress to the mathematicians and students at Göttingen.Minkowski — Hilbert's "best and
truest friend"— died prematurely of a ruptured appendix in 1909.
The Mathematical Institute in Göttingen. Its new building, constructed with funds from the
Rockefeller Foundation, was opened by Hilbert and Courant in 1930.
The Göttingen school among the students of Hilbert were Hermann Weyl, chess champion
Emanuel Lasker,Ernst Zermelo, and Carl Gustav Hempel. John von Neumann was his assistant.
At the University of Göttingen, Hilbert was surrounded by a social circle of some of the most
important mathematicians of the 20th century, such as Emmy Noether and Alonzo Church.
Among his 69 Ph.D. students in Göttingen were many who later became famous
mathematicians, including (with date of thesis): Otto Blumenthal (1898), Felix Bernstein
(1901), Hermann Weyl (1908), Richard Courant (1910), Erich Hecke (1910), Hugo Steinhaus
(1911), and Wilhelm Ackermann (1925).Between 1902 and 1939 Hilbert was editor of the
Mathematische Annalen, the leading mathematical journal of the time.

15. *************************
************************


Euclid published a great number of works on a variety
of topics, but is often referred to as the father of
geometry and is most remembered for the thirteenvolume textbook Elements. Primarily composed of the
accumulated knowledge of other mathematicians,
Elements contains a few original theorems that are
directly attributed to Euclid and are the foundation of
his fame as a brilliant mathematician. Euclid is also
generally praised for the clarity and logic with which
he systemically presents geometric principles,
theorems, and proofs, within Elements, causing it to
remain a standard mathematical text for over two
thousand years.
Though often overshadowed by his mathematical
reputation, Euclid is a central figure in the history of
optics. He wrote an in-depth study of the phenomenon
of visible light inOptica, the earliest surviving treatise
concerning optics and light in the western world.
Within the work, Euclid maintains the Platonic tradition
that vision is caused by rays that emanate from the eye,
but also offers an analysis of the eye's perception of
distant objects and defines the laws of reflection of
light from smooth surfaces. Optica was considered to
be of particular importance to astronomy and was often
included as part of a compendium of early Greek
works in the field. Translated into Latin by a number of
writers during the medieval period, the work gained
renewed relevance in the fifteenth century when it
underpinned the principles of linear perspective.

16. *************************
************************


In mathematics, the Pythagorean
theorem—or Pythagoras'
theorem—is a relation in
Euclidean geometry among the
three sides of a right triangle. It
states that the square of the
hypotenuse (the side opposite the
right angle) is equal to the sum of
the squares of the other two sides.
The theorem can be written as an
equation relating the lengths of the
sides a, b and c, often called
the Pythagorean equation:[1]
where c represents the length of the
hypotenuse, and a and b represent
the lengths of the other two sides.

17. ************************
*************************



John Napier of Merchiston (1550 – 4
April 1617) – also signed as Neper,
Nepair – namedMarvellous
Merchiston, was a Scottish landowner
known as a mathematician, physicist,
andastronomer. He was the 8th Laird of
Merchistoun.
John Napier is best known as the
inventor of logarithms. He also invented
the so-called "Napier's bones" and made
common the use of the decimal point in
arithmetic and mathematics.
Napier's birthplace, Merchiston Tower
in Edinburgh, Scotland, is now part of
the facilities of
Edinburgh Napier University. After his
death from the effects of gout, Napier's
remains were buried in
St Cuthbert's Church, Edinburgh.

18. *************************
*************************





Aryabhatta is the author of several treatises on mathematics
and astronomy, some of which are lost.
His major work, Aryabhatiya, a compendium of mathematics
and astronomy, was extensively referred to in the Indian
mathematical literature and has survived to modern times.
The mathematical part of the Aryabhatiya covers arithmetic,
algebra, plane trigonometry, and spherical trigonometry. It
also contains continued fractions, quadratic equations,
sums-of-power series, and a table of sines.
The Arya-siddhanta, a lot work on astronomical
computations, is known through the writings of
Aryabhat ta's contemporary, Varahamihira, and later
mathematicians and commentators, including Brahmagupta
and Bhaskara I. This work appears to be based on the
older Surya Siddhanta and uses the midnight-day
reckoning, as opposed to sunrise in Aryabhatiya. It also
contained a description of several astronomical
instruments: the gnomon, a shadow , possibly anglemeasuring devices, semicircular and circular cylindrical
stick yasti-yantra, an umbrella-shaped device called
the chhatra-yantra, and water clocks of at least two types,
bow-shaped and cylindrical.
A third text, which may have survived in the Arabic
translation, is Al ntf or Al-nanf. It claims that it is a translation
by Aryabhata, but the Sanskrit name of this work is not
known.
Probably dating from the 9th century, it is mentioned by the
Persian scholar and chronicler of India,

19. *************************
************************

Varahamihir's main work is the
book Pañcasiddhāntikā on the Five
[Astronomical] Canons) dated ca. 575
CE gives us information about older
Indian texts which are now lost. The work
is a treatise on
mathematical
astronomy and it summarises five
earlier astronomical treatises, namely
the Surya Siddhanta, Romaka Siddhanta,
Paulisa Siddhanta, Vasishtha Siddhanta
and Paitamaha Siddhantas. It is a
compendium of Vedanga Jyotisha as well
as Hellenistic astronomy(including
Greek, Egyptian and Roman
elements).He was the first one to
mention in his work Pancha Siddhantika
that the ayanamsa, or the shifting of the
equinox is 50.32 seconds.6655

20. *************************** *************************


Brahmagupta (Sanskrit: ब्रह्मगुप; listen (
help·info)) (597–668 AD) was an Indian
mathematician andastronomer who
wrote two important works on
mathematics and astronomy:
theBrāhmasphuṭasiddhānta (Correctly
Established Doctrine of Brahma) (628), a
theoretical treatise, and
the Khaṇḍakhādyaka, a more practical
text. There are reasons to believe that
Brahmagupta originated from Bhinmal.
Brahmagupta was the first to give rules
to compute with zero. The texts
composed by Brahmagupta were
composed in elliptic verse, as was
common practice in Indian mathematics,
and consequently has a poetic ring to it.
As no proofs are given, it is not known
how Brahmagupta's mathematics was
derived.

21. 



Brahmagupta gave the solution of the general linear equation in
chapter eighteen of Brahmasphutasiddhanta,
The difference between rupas, when inverted and divided by the
difference of the unknowns, is the unknown in the equation.
Therupas are [subtracted on the side] below that from which the
square and the unknown are to be subtracted
which is a solution for the equation equivalent to ,
where rupas refers to the constants c and e. He further gave two
equivalent solutions to the general quadratic equation
18.44. Diminish by the middle [number] the square-root of
the rupas multiplied by four times the square and increased by the
square of the middle [number]; divide the remainder by twice the
square. [The result is] the middle [number].
18.45. Whatever is the square-root of the rupas multiplied by the
square [and] increased by the square of half the unknown,
diminish that by half the unknown [and] divide [the remainder] by
its square. [The result is] the unknown.