Euclid was an influential Greek mathematician known as the "Father of Geometry". His work Elements is one of the most important works in the history of mathematics, where he deduced the principles of Euclidean geometry from a small set of axioms. Euclid also wrote works on other areas of mathematics and may have been a student of Aristotle, founding the school of mathematics in Alexandria.

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MATHEMATICS

4. Euclid also known as Euclid of Alexandria, was a Greek
mathematician, often referred to as the "Father of
Geometry“. His Elements is one of the most influential
works in the history of mathematics. Euclid deduced the
principles of what is now called Euclidean geometry from a
small set of axioms. Euclid also wrote works on
perspective, conic sections, spherical geometry, number
theory and rigor. Euclid may have been a student of
Aristotle. He founded the school of mathematics at the
great university of Alexandria. He was the first to prove that
there are infinitely many prime numbers; he stated and
proved the unique factorization theorem; and he
devised Euclid's algorithm for computing gcd. He introduced
the Mersenne primes and observed that (M2+M)/2 is always
perfect (in the sense of Pythagoras) if M is Mersenne.
Among several books attributed to Euclid are The Division
of the Scale, The Optics, The Cartoptrics. Several of his
masterpieces have been lost, including works on conic
sections and other advanced geometric topics. Apparently
Desargues' Homology Theorem was proved in one of these
lost works; this is the fundamental theorem which initiated
the study of projective geometry

5. Carl Friedrich Gauss, the "Prince of Mathematics," exhibited
his calculative powers when he corrected his father's
arithmetic before the age of three.His genius was confirmed at
the age of nineteen when he proved that the regular n-gon
was constructible, for odd n, if and only if n is the product of
distinct prime Fermat numbers. At age 24 he
published Disquisitiones Arithmeticae, probably the greatest
book of pure mathematics ever. Gauss may be the greatest
theorem prover ever. Several important theorems and lemmas
bear his name; he was first to produce a complete proof of
Euclid's Fundamental Theorem of Arithmetic and first to
produce a rigorous proof of the Fundamental Theorem of
Algebra. Gauss himself used "Fundamental Theorem" to
refer to Euler's Law of Quadratic Reciprocity; Gauss was first
to provide a proof for this, and provided eight distinct proofs
for it over the years. Gauss proved the n=3 case of Fermat's
Last Theorem for a class of complex integers; though more
general, the proof was simpler than the real integer proof, a
discovery which revolutionized algebra. Other work by Gauss
led to fundamental theorems in statistics, vector analysis,
function theory, and generalizations of the Fundamental
Theorem of Calculus.

6. Euler may be the most influential mathematician who ever
lived he ranks #77 on Michael Hart's famous list of the Most
Influential Persons in History. His notations and methods in
many areas are in use to this day. Just as Archimedes
extended Euclid's geometry to marvelous heights, so Euler
took marvelous advantage of the analysis of Newton and
Leibniz. He gave the world modern trigonometry.He
invented graph theory.Euler was also a major figure in
number theory, proving that the sum of the reciprocals of
primes less than x is approx. (ln ln x). Euler was also first to
prove several interesting theorems of geometry, including
facts about the 9-point Feuerbach circle; relationships
among a triangle's altitudes, medians, and circumscribing
and inscribing circles; and an expression for a tetrahedron's
area in terms of its sides. Euler was first to explore
topology, proving theorems about the Euler characteristic.
he settled an arithmetic dispute involving 50 decimal places
of a long convergent series. Four of the most important
constant symbols in mathematics (π, e, i = √-1, and γ =
0.57721566...) were all introduced or popularized by Euler.

8. Place value system and zero
The place-value system, first seen in the 3rd
century Bakhshali Manuscript, was clearly in place in
his work. While he did not use a symbol for zero, the
French mathematician Georges If rah explains that
knowledge of zero was implicit in Aryabhata's place-
value system as a place holder for the powers of ten
with null coefficients
Approximation of π
Aryabhata worked on the approximation
for pi (π), and may have come to the conclusion
that π is irrational. In the second part of
the Aryabhatiyam (gaṇitapāda 10), he writes:
Trigonometry
In Ganitapada 6, Aryabhata gives the area of a
triangle as "for a triangle, the result of a
perpendicular with the half-side is the area."
Algebra
In Aryabhatiya Aryabhata provided elegant
results for the summation of series of squares
and cubes:

9. Fermat practically founded Number
Theory, and also played key roles in the
discoveries of Analytic Geometry and
Calculus. He was also an excellent
geometer and discovered probability theory
Fermat's most famous discoveries in
number theory include the ubiquitously-
used Fermat's Little Theorem; the n =
4 case of his conjectured Fermat's Last
Theorem the fact that every natural
number is the sum of three triangle
numbers
Fermat developed a system of analytic
geometry which both preceded and
surpassed that of Déscartes; he developed
methods of differential and integral
calculus which Newton acknowledged as
an inspiration.

10. 1. “Do not say a little in many words but a great deal in a few.”
2. “Friends are as companions on a journey, who ought to aid each other
to persevere in the road to a happier life.”
3. “Above the cloud with its shadow is the star with its light. Above all
things reverence thyself.”
4. “Strength of mind rests in sobriety; for this keeps your reason
unclouded by passion.”
5. “There is geometry in the humming of the strings, there is music in the
spacing of the spheres.”
6. “Concern should drive us into action and not into a depression. No
man is free who cannot control himself.”