THE HISTORY OF GEOMETRY ANCIENT GREEK MIDDLE AGES 20th CENTURY - PRESENT Most of the Egyptian geometry we know comes from the Rhind (written in 1550 BCE by Ahmes) and Moscow papyrus (wriiten in 1700 BCE by Egyptologist Vladimir Golenischev). EGYPT Egyptian geometry was experimentally derived from the rules used by engineers of the civilization. The Egyptians also knew about Pythagorean Theorem (a^2 + b^2= c^2) 1,000 years before Pythagoras discovered it. They also used geometry to build the pyramids. BABYLON The Babylonians developed a sexagesimal number system (based on 60). They drew cuneiform symbols (wedge-shaped) on wet clay tablets as their writing system. The Babylonians divided the day into 24hrs., each hour into 60 min., and each minute into 60 sec. The Babylonians used the formulas: ab=[(a+b)^2-a^2-b^2]/2 and ab=[(a+b)^2-(a-b)^2]/4 to make multiplication easier. They used the formula a/b=a x(1/b) to divide. The Babylonians (like the Egyptians) had an understanding of Pythagorean Theorem. CHINA Had one symbol for each of 1,2,3,4,5,6,7,8,9,10,100,1000,10000. The Chinese made calculations on counting rods. A Chinese book (The Nine Chapters on the Mathematical Art) collects mathematics to the beginning of the Han Dynasty. This is what the chapters cover: Ch 1, Field measurement: systematic discussion of algorithms using counting rods for common fractions including alg. for GCD, LCM; areas of plane figures, square, rectangle, triangle, trapezoid, circle, circle segment, sphere segment, annulus -- some accurate, some approximations.

Ch 2,3,6 on proportions, Cereals, Proportional distribution, Fair taxes.

Ch 4, What width?: given area or volume find sides. Describes usual algorithms for square and cube roots but takes advantage of computations with counting rods

Ch 5, Construction consultations: volumes of cube, rectangular parallelepiped, prism frustums, pyramid, triangular pyramid, tetrahedron, cylinder, cone, and conic frustum, sphere -- some approximations, some use pi=3

Ch 7, Excess and deficients: false position and double false position

Ch 8, Rectangular arrays: Gives elimination algorithm for solving systems of three or more simultaneous linear equations. Involves use of negative numbers (red for positive numbers, black for neg numbers). Rules for signed numbers.

Ch 9, Right triangles: applications of Pythagorean theorem and similar triangles, solves quadratic equations with modification of square root algorithm, only equations of the form x^2 + a x = b, with a and b positive. INDIA The Indians made major contributions to math but, the biggest is the number system they developed. It helped develop the one we use today. This invention goes beyond the discoveries of Archimedes and Apollonius. The Indians had an understanding of Pythagorean Theorem. Brahmagupta's theorem: If a cyclic quadrilateral has diagonals that are perpendicular to each other, then the perpendicular line drawn from the point of intersection of the diagonals to any side of the quadrilateral always bisects the opposite side.

Brahmagupta's formula: The area, A, of a cyclic quadrilateral with sides of lengths a, b, c, d, respectively, is given by: A=square root of: (s-a) (s-b) (s-c) (s-d)

where s, the semi-perimeter, given by: s=a+b+c+d/2

Brahmagupta's Theorem on rational triangles: A triangle with rational sides and rational area is of the form:

a=u^2/v+v, b=u^2/w+w, c=u^2/v + u^2/w-(v+w) CLASSICAL Thales of Miletus wrote deductive proofs for five geometric propositions, they didn't survive. Pythagoras of Ionia wasn't the first to discover Pythagorean Theorem though he was the first to give a proof for it. He also gathered a group of students and together they discovered what most of high schoolers learn in geometry today. They also discovered incommensurable lenghts and irrational numbers. Plato, a philosopher, had a great influence on geometry. Mathematicians thus believed that geometry should use no tools but compass and straight-edge. This dictum led to a deep study of compass and straightedge constuctions. HELLENISTIC Euclid of Alexandria wrote "The Elements of Geometry", in which he presented geometry in an axiomatic form (became known as Euclidean Geometry). The following are his 5 axioms: 1.Any two points can be joined by a straight line.

2.Any finite straight line can be extended in a straight line.

3.A circle can be drawn with any center and any radius.

4.All right angles are equal to each other.

5.If two straight lines in a plane are crossed by another straight line (called the transversal), and the interior angles between the two lines and the transversal lying on one side of the transversal add up to less than two right angles, then on that side of the transversal, the two lines extended will intersect (also called the parallel postulate). Archimedes of Syracuse, Sicily, developed methods very similar to the coordinate systems of analytic geometry and the limiting process of integral calculus Islamic Muslim mathematicians were responsible for algebraic geometry. The successor of Muhammad ibn Musa al-Kwarizmi was a Persian scholar who invented the Algorithm in Mathematics which is the base for Computer Science (born 780) undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra. This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose.

Thabit ibn Qurra contributed to a number of areas in mathematics, the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-Euclidean geometry. Another contribution Thabit made was the generalization on Pythagorean Theorem, he extended it from special right triangles to all triangles, along with a proof. 17th Century The rigorous methods of geometry found in Euclid's were relearned. Further development in Euclidean and Algebraic geometry continued.

There two major developments in the early 17th century: Analytic geometry (coordinates and equaations) by Rene Descartes and Pierre de Fermat. This was necessary for calculus and a precise quantitative science of physics. The second was the systematic study of projective geometry by Girard Desargues. Projective geometry is the study of geometry without measurement, just the study of how points align with each other.

In the late 17th century, calculus was developed independently and almost simultaneously by Isaac Newton and Gottfried Wilhelm von Leibniz. This was the beginning of a new field of mathematics now called analysis 18th Century Omar Khayyam attempted to prove Euclid's Parallel Postulate but, failed. His attempt led to the developement of non-Euclidean geometry. By 1854, Bernhard Riemann, a student of Gauss, had applied methods of calculus in a ground-breaking study of the intrinsic (self-contained) geometry of all smooth surfaces, and thereby found a different non-Euclidean geometry. This work of Riemann later became fundamental for Einstein's theory of relativity. Non-Euclidean geometry was proven to be as consistent as Euclidean geometry by Beltrami in 1868. In 1894 David Hilbert refined Euclid's axioms.

19th Century In the early 19th century, Gauss, Johann Bolyai, and Lobatchewsky, each independently, took a different approach. Beginning to suspect that it was impossible to prove the Parallel Postulate, they set out to develop a self-consistent geometry in which that postulate was false. In this they were successful, thus creating the first non-Euclidean geometry Developments in algebraic geometry included the study of curves and surfaces over finite fields as demonstrated by the works of among others André Weil, Alexander Grothendieck, and Jean-Pierre Serre as well as over the real or complex numbers. Finite geometry itself, the study of spaces with only finitely many points, found applications in coding theory and cryptography. With the advent of the computer, new disciplines such as computational geometry or digital geometry deal with geometric algorithms, discrete representations of geometric data, and so forth.

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# The History of Geometry:)

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