# A Brief History & Development of Geometry

Enedina Maya MAT 410 Spring '14 The College of Saint Rose Dr. Powers

## Era

### Ancient Times

3100 B.C. - 3099 B.C.

1000 B.C - 1000

650 - 850

1500 - 2014

## Geometer

### Thales of Miletus

624 B.C - 547 B.C.

Born in current day Turkey. He is credited for the proof that "for any chord AB in a cirlce, all of the angels subtended by points anywhere on the same semiarc of the circle will be equal."
He founded the geometry of lines and is given credit for introducing abstract geometry.

### Pythagoras of Samos

569 B.C. - 475 B.C.

Born in Greece. He believed that geometry is the highest form of mathematical study. There was a Pythagoras' society, and since they shared idea it is not known if everything accredited to Pythagoras was actually just his own thought.
Accredited for the Pythagorean Theorem, discovery of irrational numbers, the Platonic solids, and that the sum of the angles of a triangle is equal to 180 degrees.

### Hippocrates of Chios

470 B.C. - 410 B.C.

Born in Ancient Greece. He was a mathematician and astronomer. His major contribution was the first organized textbook on geometry-The Elements. (Stoichia)
He worked on squaring the circle and duplicating the cube.

### Mozi

470 B.C. - 390 B.C.

Chinese philosopher who wrote the oldest surviving book of Chinese geometry.

### Theaetetus of Athens

417 B.C. - 347 B.C.

Founder of solid geometry. The first person to study the octahedron and the icosahedron, and to construct all 5 regular solids. His biggest contribution was his work on irrational lengths described in Book 10 of Euclid's Elements.

### Eudoxus of Cnidus

408 B.C. - 355 B.C.

Greek mathematician and astronomer whose major contribution is to the theory of proportion that is in Euclid's Elements.

### Menaechmus

380 B.C. - 320 B.C

Greek mathematician who studied conic sections.

### Euclid

325 B.C. - 265 B.C.

Greek mathematician, often referred to as the "Father of Geometry." Best known for his book "The Elements" that collected the theorems of previous mathematicians. He also had works on conic sections, spherical geometry, and number theory.

### Archimdes

287 B.C. - 212 B.C.

Greek Mathematician who analyzed the area of a circle and discovered how to calculate volumes and surface areas of spheres and cylinders.A famous result that is inscribed on his tomb stone is that "the volume of a sphere is two/thirds the volume of its circumscribed cylinders."

### Nicomedes

280 B.C. - 210 B.C.

Greek mathematician who discovered the Conchoid Curve, which can be used to trisect angles.

### Eratosthenes of Cyrene

276 B.C. - 194 B.C.

Greek mathematician, geographer, astronomer, and historian. His major accomplishment was his creation of a sieve that determines prime numbers up to any given limit.

### Apollonius of Perga

262 B.C. - 190 B.C.

Greek mathematician known as 'The Great Geometer'. Famous book, Conics that introduced terms such as ellipse, parabola, and hyperbola.

### Hipparchus of Rhodes

190 B.C. - 120 B.C.

Greek astronomer, geographer,and mathematician considered to be one of the greatest astronomers if his time. He made advances in geography and math, specifically trigonometry.One of his major contribution, included the division of a circle into 360 degrees and the creation of one of the first trigonometric tables for solving triangles.

### Heron of Alexandria

Greek mathematician, scientist, inventor and engineer. The vending machine was one of his ideas mentioned in his book. He was also the first to make a machine that operated with a windwheel. To geometry he approximated square roots and formulated the area of a triangle.
A = √(s(s−a)(s−b)(s−c))

### Menelaus of Alexandria

Greek mathematician and astronomer who first defined a spherical triangle. His most important work was on the geometry of the sphere.

### Pappus of Alexandria

Greek mathematician known for Papus's Theorem in projective geometry. "Projective geometry is the branch of geometry dealing with the properties and invariants of geometric figures under projection from a point."

### Hypatia of Alexandria

First woman to make a substantial contribution to the development of math. Her father was a professor at a University. She would later teach "Arithmetica". She wrote a commentary on Apllonius' "Conics".

### Proclus Lycaeus

412 - 485

A Greek philosopher who wrote a commentary on The Elements where he commented on attempting to have a proof to the 5th postulate using the other four.

### Zu Geng

Chinese mathematician. His greatest accomplishment was to calculate the diameter of a sphere of a given volume.

### Brahmagupta

598 - 670

Brahmagupta was the foremost Indian mathematician of his time. He made advances in astronomy and most importantly in number systems including algorithms for square roots and the solution of quadratic equations.

### Thabit ibn Qurra

826 - 901

An Iraqi Muslim mathematician, physician, astronomer, and translator who lived in Baghdad. He made important discoveries in algebra, geometry, and astronomy. In math, he discoverd an equation for determining amicable numbers. He also wrote on the theory of numbers, and extended their use to describe the ratios between geometrical quantities. He is also known for having calculated the solution to a chessboard problem involving an exponential series.

### Abu al-Wafa' al-Buzjani

940 - 998

Persian mathematician and astronomer who worked in Baghdad. He made important innovations in spherical trignometry. He was the first to use negative numbers in an Islamic text. He is also credited with compiling the tables of sines and tangents at 15 degree intervals. He also introduced the secant and cosecant functions.He established several trigonometric identities in their modern form.He discovered the law of sines for spherical triangles.

### Leonardo Pisano Fibonacci

1170 - 1250

Italian mathematician considered by some "the most talented western mathematician of the Middle Ages." He is best known for the spreading of the Hindu-Arabic numeral system in Europe, and for the Fibonacci numbers.

### Johannes Kepler

1571 - 1630

German mathematician, astronomer,and astrologer. A key figure in the 17th century scientific revolution, best known for his laws of planetary motion. He published the first description of the hexagonal symmetry of snowflakes, what would later become known as the Kepler conjecture. He was one of the pioneers of the mathematical applications of infinitesimals.

### Girard Desargues

1591 - 1661

French mathematician and engineer, who is considered one of the founders of projective geometry. Desargues' famous 'perspective theorem' - that when two triangles are in perspective the meets of corresponding sides are collinear.

### Rene Descartes

1596 - 1650

French philosopher, writer, and mathematician. Best known for the Cartesian coordinate system. He is credited as the father of analytical geometry.

### Pierre de Fermat

1601 - 1665

French lawyer and and mathematician who is given credit for early developments that led to calculus. He is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines. He made notable contributions to analytic geometry, probability, and optics. Best known for Fermat's Last Theorem.

### Blaise Pascal

1623 - 1662

French mathematician, physicist, inventor, writer, and philosopher. Known for Pascal's triangle.

### Gottfried Wilhelm Leibniz

1646 - 1716

German mathematician and philosopher. He developed calculus independently of Newton. Leibniz was the first to denote any of several geometric concepts derived from a curve, such as abscissa, ordinate, tangent, chord, and the perpendicular.

### Giovanni Girolamo Saccheri

1667 - 1733

Italian priest, philosopher, and mathematician. He is known for his work in non-Euclidean geometry. He wanted to develop a proof for Euclid's 5th postulate by contradiction, but failed.

### Leonhard Euler

1707 - 1783

Swiss mathematician and physicist who made important discoveries in calculus and graph theory. He introduced the concept of a function and was the fist to write f(x) to denote the function f applied to the argument x. He also introduced the modern notation for the trigonometric functions, the letter e for the base of the natural logarithm, the Greek letter Σ for summations and the letter i to denote the imaginary unit.The use of the Greek letter pi to denote the ratio of a circle's circumference was also popularized by Euler.

### Gaspard Monge

1746 - 1818

French mathematician, the inventor of descriptive geometry and the father of differential geometry.

### John Playfair

1748 - 1819

Scottish scientist and mathematician, and a professor of philosophy. He wrote a famous commentary on Euclid's 5th postulate which he proposed replacing it by an axiom, "Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line".

### Carl Friedrich Gauss

1777 - 1855

Gauss developed the Gauss method for adding large amounts of consecutive numbers when he was six. However, his most important creation is that of non-Euclidean geometry.

### Jean-Victor Poncelet

1788 - 1867

French mathematician whose most notable work was in projective geometry, in particular, his work on Feuerbach's theorem. He also made discoveries about projective harmonic conjugates; among these were the poles and polar lines associated with conic sections. These discoveries led to the principle of duality, and also aided in the development of complex numbers and projective geometry.

### Nikolai Ivanovich Lobachevsky

1792 - 1856

Russian mathematician and geometer, known for his work on hyperbolic geometry, also known as Lobachevskian geometry. He was the founder of non-Euclidean geometry, which he developed independently of János Bolyai and Carl Gauss.

### János Bolyai

1802 - 1860

Hungarian mathematician, one of the founders of non-Euclidean geometry. He also developed a geometric concept of complex numbers as ordered pairs of real numbers.

### Hermann Grassmann

1809 - 1877

German polymath who is remembered for his development of the basic calculus for vectors. Grassmann also initiated the representation of subspaces of a given space by coordinates; this leads to a point mapping of an algebraic manifold, called the Grassmannian

### Arthur Cayley

1821 - 1895

British mathematician. He defined the concept of a group-as a set with a binary operation satisfying certain laws. In group theory, Cayley's theorem,states that every group G is isomorphic to a subgroup of the symmetric group acting on G.

### Georg Friderich Bernhard Riemann

1826 - 1866

Riemann was one of the foremost geometers in the development of Non-Euclidean Geometry. He also made contributions to analysis, number theory, and differential geometry.

### Felix Klein

1849 - 1925

German mathematician known for his work in group theory, complex analysis, non-Euclidean geometry, and on the connections between geometry and group theory. At the Erlangen Program Klein proposed a new solution to the problem how to classify and characterize geometries on the basis of projective geometry and group theory.

### David Hilbert

1862 - 1943

German mathematician who developed many ideas in many areas, such as invariant theory and the axiomatization of geometry. Hilbert is known as one of the founders of proof theory and mathematical logic.

### Niels Fabian Helge von Koch

1870 - 1924

Swedish mathematician who is most famous for the Koch snowflake, one of the earliest fractal curves to be described.

### Albert Einstein

1879 - 1955

German physicist who made many contributions to the field of calculus and geometry. Einstein contributed many equations to geometry and calculus, ten of them known as Einstein Field Equations.

### Oswald Veblen

1880 - 1960

American mathematician, geometer and topologist. His work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905. There is an award named after him granted by the American Mathematical Society. He made important contributions to differential geometry and the early development of topology.

### M.C. Escher

1898 - 1972

A Dutch graphic artist. He is known for intertwining art and mathematics. In his art he used repeating patterns of interlocking motifs, tessellations of the Euclidean and the hyperbolic plane and his drawings represented impossible figures.

### Juliusz Schauder

1899 - 1943

Polish mathematician of Jewish origin known for his work in functional analysis, partial differential equations, and mathematical physics. Most of his work is in the field of functional analysis. He is best known for the Schauder fixed point theorem which is a major tool to prove the existence of solutions in various problems, the Schauder bases, and the Leray-Schauder principle.

### Jean Leray

1906 - 1998

French mathematician who worked on algebraic topology and differential equations. Together with Juliussz Schauder, discovered a topological invarient, known as the Leray-Schauder degree. His main work in toplogy was done with he was a prisoner of war at Edelbach, Austria from 1940-1945. He hid the fact that he was an expert in differential equations, because he feared that would led him to be asked to do war work.

### André Weil

1906 - 1998

French mathematician known for his foundational work was discovering the connections between algebraic geometry and number theory. This began his doctoral work that led to the Mordell-Weil theorem. Another accomplishment was Weil conjectures.

### Donald Coxeter

1907 - 2003

British-born Canadian geometer. He is known as the "Man who saved Geometry". He made contributions of major importance in the theory of Coxeter groups which gave rise to tesselations, non-euclidean geometry, group theory and combinatorics.

### Edward Lorenz

1917 - 2008

American mathematician and meteorologist. He is a pioneer of chaos theory. He introduced the butterfly effect.

### Bernard Dwork

1923 - 1998

An American mathematician known for the first general results on the Weil conjectures.

### Benoit Mandelbrot

1924 - 2010

Polish-born, French and American mathematician who is largely responsible for the present interest in fractal geometry. He showed how fractals can occur in many different places in both math and elsewhere. He discovered the Mandelbrot set, of intricate, never-ending fractal shapes.

### Jean-Pierre Serre

1926

French mathematician. Serre applied spectral sequences to the study of the relations between the homology groups of fibre, total space and base space in a fibration. This allowed him to discover fundamental connections between the homology groups and homotopy groups of a space and to prove important results on the homotopy groups of spheres. His theorem led to increased progress in algebraic topology and homological algebra.

### Alexander Grothendieck

1928

French mathematician, born in Germany. He is the central figure behind the creation of the modern theory of algebraic geometry. In algebraic geometry, he worked on sheaf theory and homological algebra, where he introduced Abelian categories.

### Michael Artin

1934

American mathematician known for contributions to algebraic geometry.

### Gerhard Frey

1944

German mathematician known for his work in number theory. His Frey curve, was central to Wiles' proof of Fermat's Last Theorem. His research is applied to coding theory and cryptography.

### Kenneth Alan Ribet

1948

American mathematician who paved the way towards Andrew Wiles's proof of Fermat's last theorem. He proved the epsilon conjecture by Jean-Pierre Serre was true and the Herbrand-Ribel theorem helped proof Fermat's Last Theorem.

### Andrew John Wiles

1953

British mathematician who specializes in number theory. He is most known for proving Fermat's Last Theorem.

### Wacław Sierpiński

1969

was a Polish mathematician. He was known for outstanding contributions to set theory, number theory, theory of functions and topology. hree well-known fractals are named after him (the Sierpinski triangle, the Sierpinski carpet, and the Sierpinski curve), as are Sierpinski numbers and the associated Sierpiński problem.

### Joanne Powers

2014

She "believes that Mathematics is the most elegant branch of learning in which I could engage and I intend to pass that attitude to my students."