Geometry "measuring the earth“

is the branch of math that has to do with spatial relationships.

Fact:Fact: No one has been able take No one has been able take a tape measure around the earth.a tape measure around the earth.

The The circumference circumference of the planet at of the planet at the equator is the equator is 24,901.473 miles

HOW DO WE KNOW THAT?

The first known case of calculating the distance around the earth was done by Eratosthenes around 240 BCE.

Eratosthenes

was a Greek mathematician, poet, athlete, geographer and astronomer.

Eratosthenes

His contemporaries nicknamed him "beta" (Greek for "number two") because he supposedly proved himself to be the second in the ancient Mediterranean region in many fields.

Eratosthenes

He is noted for devising a system of latitude and longitude, and for being the first known person to have calculated the circumference of the Earth.

Eratosthenes

He also created a map of the world based on the available geographical knowledge of the era. Eratosthenes was also the founder of scientific chronology; he endeavored to fix the dates of the chief literary and political events from the conquest of Troy.

This is a short outline of geometry's history, exemplified by major geometers responsible for it's

evolution.

* ANCIENT GEOMETRY* GREEK GEOMETRY* MEDIEVAL GEOMETRY* MODERN GEOMETRY

Babylon (4000 BC - 500 BC)Egypt (5000 BC - 500 BC)

A BIT OF HISTORYAmong the mathematical sciences, Geometry is the earliest and historically most influential. The Babylonians and Egyptians knew many geometric facts more than a thousand years before Christ.

Their geometrical knowledge was PRACTICAL in nature as exemplified by their temples and pyramids that needed very accurate plans and models

Great SphinxMore than 4000 years old, the Great Sphinx of Giza is the most famous emblem of ancient Egypt.

A BIT OF HISTORYA BIT OF HISTORY

GeometryGeometry first became first became associated with associated with land land measurement in Egyptmeasurement in Egypt. .

The Egyptians were obliged to The Egyptians were obliged to invent it in order to restore the invent it in order to restore the landmarks that were destroyed landmarks that were destroyed by the periodic inundation of by the periodic inundation of Nile River. Nile River.

Herodotus, the Greek historian, gave the following account of the origin of Geometry.

“King Sesostres divided the land among all Egyptians so as to give each a four-sided piece of equal size and draw from each his revenue by imposing a tax to be levied annually.

Whenever the river tore part of the land due to inundation, the citizen concerned had notify the king. The king then sent his overseers to measure how much the land was reduced so that the citizen would pay the tax imposed proportionally to the land lost”.

The geometry of Babylon (in Mesopotamia) and Egypt was mostly experimentally derived rules used by the engineers of those civilizations. They knew how to compute areas, and even knew the "Pythagorean Theorem" 1000 years before the Greeks (see: Pythagoras's theorem in Babylonian mathematics). But there is no evidence that they logically deduced geometric facts from basic principles. Nevertheless, they established the framework that inspired Greek geometry. A detailed analysis of Egyptian mathematics is given in the book: Mathematics in the Time of the Pharaohs.

His work called ““Directions for Directions for Knowing All Dark ThingsKnowing All Dark Things””, consisted of a collection of problems in Geometry and Arithmetic. In the problems concerning the area of a circular field, Ahmes used the value of pi ( ) equal to 3.1604. at that time the value of pi was taken to equal 3.

India (1500 BC - 200 BC)Everything that we know about ancient Indian

(Vedic) mathematics is contained in:The Sulbasutraswhich are appendices to the Vedas giving 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:

The 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 pi (to construct circles) and Ö2 = 577 / 408 (which is accurate to 5 decimal places). It also gives the special case of the Pythagorean

theorem for the diagonal of a square.

The Manava (750 BC)contains approximate constructions

of circles from rectangles, and squares from circles, which give

approximations of p.The Apastamba (600 BC)

considers the problems of squaring the circle, and of dividing a segment into 7 equal parts. It also gives an

accurate approximation of Ö2 .The Katyayana (200 BC)

gives the general case of the Pythagorean theorem for the

diagonal of any rectangle.

Major Greek Geometers (listed chronologically):

Thales of Miletus Thales of Miletus (about (about 624-547 B.C) 624-547 B.C)

who was engaged both in who was engaged both in commerce and public commerce and public

affairs visited Egypt and affairs visited Egypt and brought his acquired brought his acquired

knowledge of Geometry knowledge of Geometry from Egypt to Greece.from Egypt to Greece.

Thales Thales Name Thales of Miletos Name Thales of Miletos

(Θαλής ο Μιλήσιος) (Θαλής ο Μιλήσιος) Birth ca. 624–625 BC Birth ca. 624–625 BC Death ca. 547–546 BC Death ca. 547–546 BC School/tradition Ionian School/tradition Ionian

Philosophy, Milesian Philosophy, Milesian school, Naturalism school, Naturalism

Main interests: Ethics, Main interests: Ethics, Metaphysics, Metaphysics, Mathematics, Astronomy Mathematics, Astronomy

Notable ideas : Water is Notable ideas : Water is the physis, Thales' the physis, Thales' theorem theorem

Influenced Pythagoras, Influenced Pythagoras, Anaximander, Anaximenes Anaximander, Anaximenes

Pythagoras of Samos (569-Pythagoras of Samos (569-475 BC)475 BC) was one of his was one of his pupils who through his pupils who through his advice went to Egypt and advice went to Egypt and gained extensive gained extensive experience. Having experience. Having become more famous than become more famous than his teacher, Thales, his teacher, Thales, Pythagoras laid the Pythagoras laid the foundation of Geometry.foundation of Geometry.

is regarded as the first pure is regarded as the first pure mathematician to logically mathematician to logically deduce geometric facts from deduce geometric facts from basic principles. He is credited basic principles. He is credited with proving many theorems with proving many theorems such as the angles of a triangle such as the angles of a triangle summing to 180 deg, and the summing to 180 deg, and the infamous "Pythagorean infamous "Pythagorean Theorem" for a right-angled Theorem" for a right-angled triangle (which had been known triangle (which had been known experimentally in Egypt for over experimentally in Egypt for over 1000 years). 1000 years).

The Pythagorean school The Pythagorean school is is considered as the considered as the (first (first documented)documented) source of logic and source of logic and deductive thought, and may be deductive thought, and may be regarded as the birthplace of regarded as the birthplace of reason itself. reason itself.

As philosophers, they speculated As philosophers, they speculated about the structure and nature of about the structure and nature of the universe: matter, music, the universe: matter, music, numbers, and geometry. Their numbers, and geometry. Their legacy is described in legacy is described in Pythagoras and the Pythagoreans : A Brief History..

Hippocrates of Chios about 470 BC - about 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 equivalent to constructing two mean proportional 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 BC - 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 thus construct all five regular solids. This work of his 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.

Menaechmus (380-320 BC)was a pupil of Eudoxus, and discovered the conic sections. He was the first to show that ellipses, parabolas, and hyperbolas are obtained by cutting a cone in a plane not parallel to the base.

Euclid

Born fl. 300 BCResidence Alexandria, Egypt

Nationality GreekFields MathematicsKnown for Euclid's Elements

Archimedes of Syracuse (287-212 BC)is regarded as the greatest of Greek

mathematicians, and was also an inventor of many mechanical devices (including the screw, the pulley, and the 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.

Archimedes of Syracuse (287-212 BC)He gave accurate approximations to p 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.

Archimedes of Syracuse (287-212 BC)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 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 p better than the one of Archimedes. English translations of his Conics: Books I - III and Conics Books V to VII are now available.

Hipparchus of Rhodes (190-120 BC)is the first to systematically use and document the foundations of trigonometry, and may have invented it. He published several books of trigonometric tables and the methods for calculating them. He based his tables on dividing a circle into 360 degrees with each degree divided into 60 minutes. This is the first recorded use of this subdivision. In other work, he applied trigonometry to astronomy making it a practical predictive science.

Heron of Alexandria (10-75 AD) 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 = sqrt[s(s-a)(s-b)(s-c)] where s=(a+b+c)/2 [but 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 "Almagest" (13 Books) giving the mathematics for the geocentric

theory of planetary motion. Considered a masterpiece with few

peers, Almagest remained the major work in astronomy for 1400 years until it was superceded by 

the heliocentric theory of Copernicus. 

Claudius Ptolemy (85-165 AD)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, 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 of Alexandria (370-415 AD) She 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.

Hypatia of Alexandria (370-415 AD) 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.

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

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.

invented modern projective geometry in his most important work titled "Rough draft for an essay on the results of taking plane sections of a cone" (1639). His famous 'perspective theorem' for two triangles was published in 1648.

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). He proved many projective geometry theorems, the earliest including "Pascal's mystic hexagon" (1639).

was extremely prolific in a vast range of subjects, and founded mathematical analysis. He invented the idea of functions and used them to transform analytic into differential geometry investigating surfaces, curvature, and geodesics.

He discovered (1752) that the well-known "Euler characteristic" (V-E+F) of a polyhedron depends only on the surface topology. Euler, Monge, and Gauss are considered the three fathers of differential geometry. He also made breakthroughs contributions to many other branches of math. A representative selection of his discoveries is given in Euler: The Master of Us All.

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.

invented non-Euclidean geometry prior to the independent work of Janos Bolyai (1833) and Nikolai Lobachevsky (1829), although Gauss' work 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.

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. The professionals regarded him as an obscure amateur mathematician (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 A more extensive description of Grassmann's life and work is given in the interesting book : A History of Vector Analysis.

was an amateur mathematician (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 for quantum mechanics.

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.

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.

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.

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" (1910-18).

He then worked in topology and differential geometry, and published with his student Henry Whitehead "The Foundations of Differential Geometry" (1933) which gives the first definition of a differentiable manifold.

is regarded as the major synthetic geometer of the 20th century, and has made important contributions to the theory of polytopes, non-Euclidean geometry, group theory and combinatorics.

His "Coxeter groups" give the complete classification of regular polytopes in n-dimensions. He has published many important books, including Regular Polytopes (1947, 1963, 1973) and Introduction to Geometry (1961, 1989).

WHY DO WE NEED TO STUDY GEOMETR

Y?

WE HAVE TO STUDY GEOMETRY TO:

ENHANCE OUR ANALYTICAL SKILLS TO ENABLE US TO EXPRESS OUR THOUGHTS ACCURATELY AND TRAIN US TO REASON LOGICALLY.

WE HAVE TO STUDY GEOMETRY TO:

PROVIDE US WITH MANY IMPORTANT FACTS OF PRACTICAL VALUE.

WE HAVE TO STUDY GEOMETRY TO:

UNDERSTAND AND APPRECIATE OUR NATURAL AND MAN-MADE ENVIRONMENT.

REMEMBER:

A KNOWLEDGE OF BASIC GEOMETRY IS USEFUL TO EVERYDAY LIFE, PARTICULARLY IN MEASURING AND DESIGNING SUCH ITEMS AS

ROOM CARPETING

PAINTING OF A HOUSE

CONSTRUCTION OF A PICTURE FRAME