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Unformatted text preview: Euclid and his Elements Euclid and his Elements
300 B.C. Euclid Euclid Very little is known about the life of Euclid. Trained at Plato’s Academy. Professor of mathematics at the University of Alexandria. Founder of the Alexandrian School of Mathematics. Intellectual centre of the Mediterranean world through the Greek and Roman times Elements Elements Euclid is well known for his elements. Over 2000 editions of Euclid’s Elements have appeared since the first one printed in 1482. For more than 2000 years, this work has dominated all teaching of geometry. The Elements consists of 13 books, 465 propositions from plane and solid geometry and number theory. Elements Elements Out of 465 theorems, only a few were Euclid’s own invention. Collected and edited all of the well known results. Studied by many mathematicians including Archimedes, Newton, Liebniz. Euclid’s great genius was in presenting old mathematics in a thoroughly clear, organized and logical fashion. Axiomatic Development Axiomatic Development 1. 2. 3. He began the elements with a few basics: 23 definitions 5 postulates (assumptions) 5 axioms (results that do not need to be proved). From these basics, he proved his first proposition. Use first proposition to prove the second and so on. Axiomatics Axiomatics Axiomatic development of the Elements was of major importance. Served as a model for all subsequent mathematical work. This postulational thinking has today penetrated into almost every field of mathematics. A relatively modern outcome has been the creation of axiomatics, the study of the general properties of sets of postulates and postulational thinking. Elements Book II Elements Book II A short book of 14 propositions Deals with the transformation of areas and the geometric algebra of the Pythagoreans Of special interest are propositions 12 and 13. Stated together as follows: “In an obtuseangled (acuteangled) triangle, the square of the side opposite the obtuse (acute) angle is equal to the sum of the squares of the other two sides increased (decreased) by twice the product of one of these sides and the projection of the other on it” Element Book II Element Book II This is the generalization of the Pythagorean Theorem – Law of Cosines. Concluded Book II with proposition 14 in which he examined the quadrature of the general polygon. B
c A b a C Element Book III Element Book III Book III contains 39 propositions, many of the familiar theorems about circles, chords, secants, tangents. In proposition 1, he showed how to find the centre of a given circle. In proposition 18, he proved that a tangent to a circle and the radius drawn to that point of tangency meet at right angles. Book III Book III Later, he proved that in a circle, angles in the same segment are equal to one another. Proposition 22 – The opposite angles of quadrilaterals in circles are equal to two right angles. Proposition 31 – Angle inscribed in a semicircle is right. Book IV Book IV Dealt with the construction of inscribing and circumscribing certain geometric figures using the compass and the ruler ( 3sided, 4 sided, 5 sided, 6sided, 15 sided) Proposition 4 – inscription of circle within a given triangle, circle centre is the intersection point of the bisector of the angles. Book IV Book IV Proposition 5 – circumscription of a circle about a given triangle, centre is the intersection point of the perpendicular bisectors of the sides. The final construction in this book was of the regular pentadecagon (15sided polygon). Starting with an equilateral triangle, Greek geometers produced regular polygons of 6, 12, 24, 48,…sides. Polygons Polygons Starting with a square, they could generate 8, 16, 32, 64,…sides. From pentagon, 10, 20, 40 … sides From decapentagon, 15, 30, 60, 120 … Rich list but not all regular polygons are on it. Euclid did not mentioned anywhere the construction of 7gon, 9gon, 17gon. Most mathematician after Euclid assumed that other polygons may not be constructible using a compass and straightedge. Regular Polygons Regular Polygons In 1796, Gauss discovered how to construct a heptadecagon (17sided). Gauss showed that a regular polygon having a prime number of sides can be constructed with Euclidean tools if this number is of the form f ( n) = 2
2n + 1 Regular Polygons Regular Polygons For n=0, 1, 2, 3, 4 then f(n) =3, 5, 17, 257, 65, 537 all prime numbers. Unknown to the Greeks, regular polygons of 17, 257, 65, 537 sides can be constructed with Euclidean tools. It is said that Gauss discovery, at the age of 19, of a regular polygon of 17 sides, convinced him to devote his life to mathematics. Books V, VI, VII and IX Books V, VI, VII and IX This book is devoted to the development of Eudoxus idea of the theory of proportion. Book VI applies the Eudoxian theory of proportion to plane geometry. Books VII, VIII and IX, which contains a total of 102 propositions, deals with elementary number theory. Book VII Book VII This book begins with a list of 22 new definitions specific to the properties of whole numbers. Euclid defined an even number to be one that is divisible into two equal parts and an odd number to be one that is not. He also defined a prime number to be greater than 1 that is divisible (Euclid said “measured”) by 1 and itself. Euclidean Algorithm Euclidean Algorithm Nonprime numbers greater than 1 are called composite. A perfect number is the sum of its “part”, that is, its proper divisors. In the first two propositions of Book VII, Euclid established the Euclidean Algorithm a technique for finding the greatest common divisor of two or more integers. Euclidean Algorithm Euclidean Algorithm Ex: Find the greatest common divisor of 4575 and 1647. Divide the smaller into the larger and keep track of the remainder. 4575 = (1647 X 2) + 1281 1647 = (1281 X 1) + 366 1281 = (366 X 3) + 183 366 = (183 X 2) Previous remainder 183 is the GCD. Significant Theorems Significant Theorems A number of significant theorems are found in Book IX. Proposition 14 is equivalent to the important “Fundamental Theorem of Arithmetic” – any integer greater than 1 can be expressed as a product of primes in one and only one way. Proposition 35 gives a geometric derivation of the formula for the sum of the first n terms of a geometric series. Theorems of Book IX Theorems of Book IX Proposition 20 – the number of prime numbers is infinite. 2 −1 Proposition 36 – if is a prime number, then is a perfect number. 2 (2 −1) Book X contains 115 propositions of irrational numbers. This is the longest and the most sophisticated. n
n− 1 n Books XI to XIII Books XI to XIII These books cover the fundamental of solid geometry or 3D geometry. Book XI has 39 propositions examining the solid geometry of intersecting planes, plane angles etc. Book XII deals with much deeper concepts of solid geometry. Here Euclid employed Eudoxus’ method of exhaustion to the volume of a cone etc. Books XII and XIII Books XII and XIII Proposition 10 – any cone is a third part of the cylinder which has the same base with it and equal height. Last proposition of Book XII established, via exhaustion, that “Spheres are to one another in triplicate ratio of their respective diameters”.
V 1 V2 D3 =1 3 D2 Other works of Euclid Other works of Euclid Besides the Elements, Euclid other works includes Data – deals with the first 6 books of the Elements On Divisions – work on geometry from an Arabic translation. Lost work – Pseudaria, Porisms, Conics, Surface Loci. Greek Mathematics after Euclid Greek Mathematics after Euclid Next great mathematician is Archimedes. Born in the Greek city of Syracuse in Sicily about 287 B.C. and died in 212 B.C. Studied at the Alexandria Library, trained in the Euclidean tradition. Among his friends were Conan, Dositheus and Eratosthenes. The first two were successors to Euclid and the last was a librarian at the University. Archimedes Archimedes Many of Archimedes mathematical discoveries were communicated to these men. While in Egypt, invented the “Archimedian Screw”, a device for raising water from a low level to a higher level. He then returned to Syracuse and spent the rest of his life there. Archimedes Archimedes He had the ability to devote himself single mindedly to any problem in extraordinary periods of intense concentration. Discovered the first law of hydrodynamics, ”When immersed in a fluid, a body is buoyed up by a force equal to the weight of the displaced fluid”. He discovered the fundamental principles of hydrostatics found in one of his work entitled “On Floating Bodies”. Archimedes Archimedes He advanced the science of optics and did pioneering work in mechanics – water pump, pulleys and compound pulleys. No one had ever thought to square the parabola. Archimedes attempted the problem and finally solved it. For all of his practical inventions, his true love was pure mathematics. Archimedes Archimedes His mathematical results, which survive in a dozen books, are of high quality. He was a master of Eudoxus’ method of exhaustion. Three of Archimedes’ works are devoted to plane geometry. They are “Measurement of a circle”, “Quadrature of a parabola” and “On Spirals”. Measurement of a circle Measurement of a circle In this work, Archimedes began the classical π method of computing . In the first proposition of this work, he gave a complete analysis of circular area. We first need to examine what was known about circular areas before Archimedes. Geometers knew, at that time, that “The ratio of a circle’s circumference to its diameter is constant”. Area of a Circle Area of a Circle Modern mathematicians define this ratio to C π = , ⇒ C =πD be . D Proposition XII.2 of the Elements established that “two circular areas are to each other as the squares on their diameters”.
2 A D1 1 = 2=, k A2 D2 ⇒ A= kD 2 Area of a Circle Area of a Circle How do these two constant relate to one another? π Archimedes found the link between and k. His proof required two fairly direct preliminary results plus a rather sophisticated logical strategy called reductio ad absurdum (reduction to absurdity). Area of a Circle Area of a Circle Definition: An apothem is the length of the line drawn from the regular polygon’s centre perpendicular to any of the sides. Theorem: The area of a regular polygon is (1/2)hQ where Q is the perimeter of the polygon and h is its apothem. Archimedes Archimedes He succeded in giving the first scientific estimate of the critical constant pi. His work entitled “Quadrature of the Parabola” contains 24 propositions. His work entitled ”On Spirals” contains 28 propositions devoted to properties of the curve today known as the “spiral of Archimedes”. Archimedes’ other works Archimedes’ other works Two of his other works are devoted to geometry of three dimensional figures. One, entitled “On the sphere and cylinder” contains volumes and surface areas of spheres and related bodies. This work consists of 2 books and contains 53 propositions. He started with a list of definitions and derived very sophisticated theorems. This work was written in the Euclidean way. On sphere and cylinder On sphere and cylinder Proposition 1 – If a polygon be circumscribed about a circle, the perimeter of the circumscribed polygon is greater than the circumference of the circle. Proposition 13 – The surface of any right circular cylinder excluding the bases is equal to a circle whose radius is a mean proportional between the side of the cylinder and the diameter of the base. On sphere and cylinder On sphere and cylinder Proposition 33 – The surface of any sphere is equal to four times the greatest circle in it. For the proof of this result, he used the method of exhaustion. He exhausted the sphere by approximating it from within and without by cones whose surface areas were previously determined On sphere and cylinder On sphere and cylinder Proposition 34 – Any sphere is equal to four times the cone which has its base equal to the greatest circle in the sphere and its height to the radius of the sphere. Archimedes expressed the volume of the sphere not as a simple algebraic formula but in terms of the volume of a simpler solid, the cone. Archimedes’ sphere in a Archimedes’ sphere in a cylinder After the proofs of Propositions 33 and 34, he considered a cylinder circumscribed about the sphere. He then claimed that “The cylinder is half again (3/2) as large as the sphere in both surface area and volume”. He expressed the complicated spherical surface and volume in terms of the simple surface and volume of a related cylinder. Archimedes Archimedes Archimedes showed that total cylinder surface = (3/2) sphere surface and total cylinder volume = (3/2) sphere’s volume. He took special pride in this discovery. He requested a sphere contained in a cylinder inscribed with the ratio 3:2 be placed on his tomb. Archimedes Archimedes The treatise ”On Conoids and Spheroids” contains 32 propositions concerned with the volumes of quadrics of revolution. Archimedes also wrote two related essays on arithmetic, one of which is lost. The essay that is not lost entitled “The Sand Reckoner” applies an arithmetic system for the representation of large numbers to the finding of an upper limit to the number of grains of sand that would fill a sphere with centre at the earth and radius reaching to the sun. Archimedes Archimedes It was here, among related remarks to astronomy, hat we find that Aristarchus (310 – 230 B.C.) had put forward the Copernican theory of the solar system. “Cattle Problems” – communicated by Archimedes to Eratosthenes. A difficult problem involving seven linear equations and eight unknowns. Archimedes Archimedes Recent discovery of the longlost treatise entitled “Method” by Heiberge in Constantinople. This work is in the form of a letter addressed to Eratosthenes. It contains information concerning a “method” that Archimedes used in discovering many of his theorems. Closely connected with the ideas of integral calculus. Eratosthenes (284 – 192 B.C.) Eratosthenes (284 – 192 B.C.) A native of Cyrene, only a few years younger than Archimedes. A widely read scholar with works on pure mathematics, philosophy, geography and astronomy. Known as the second Plato – gifted in many fields but always second best. He wrote many papers on astronomy. Most of his writings are lost. Eratosthenes Eratosthenes Among his contributions was his famous “sieve” – a simple technique for finding prime numbers less than a given number. To use the sieve, write down all the odd numbers starting with 2 and less than n. Noting 2 as the first prime, cross off all multiples of 2. The next integer that has not been eliminated is 3, which must be the next prime. All of its multiples can be eliminated. At the end, primes will be the only remaining numbers on the list. Eratosthenes Eratosthenes Eratosthenes’ best known scientific achievement may be his reported determination of the circumference of the earth. Uncertain to what he did since the original manuscript entitled “On the Measurement of the Earth” has not been found. Found the earth’s circumference to be 24,466 miles. Exact circumference is 24, Apollonius (262 – 190 B.C.) Apollonius (262 – 190 B.C.) Born in Prega. He went to Alexandria to study. He was an astronomer. One of his masterpieces was “The Conics” – an extensive treatment of conic sections – the ellipse, parabola and hyperbola. This work earned him the name of “The Great Geometer”. Conic sections had been extensively studied before Appollonius but he organized the previous work as Euclid had done. Apollonius Apollonius “The Conics” were written in 8 books containing about 400 propositions. First four books deal with the general elementary theory of conics while the remaining ones are more specialized. Even in classical times, this work was recognized as the authoritative source on conics and it was held in high regard when rediscovered during the Renaissance. When Kepler (15711630) discovered that planets travel in elliptical paths, the importance of conics was clear. Apollonius Apollonius A century later, Edmund Halley devoted years of his life to prepare the definite edition of the “Conics”, since he highly regarded this piece of classical mathematics. Today it stands, along with Euclid’s Elements and the works of Archimedes, as one of the genuine landmarks of Greek mathematics. Apollonius Apollonius Prior to Apollonius, the Greeks derived the conic sections from three types of cones, according as the vertex angle of the cone is less than, equal to, or greater than a right angle. By cutting each of three such cones with a plane perpendicular to an element of the cone, an ellipse, parabola and hyperbola results respectively. Only one branch of the hyperbola was considered. In Book I, Apollonius obtains all the conic sections in the now familiar way from one right or oblique circular double cone. Apollonius Apollonius 1. 2. 3. The names ellipse, parabola and hyperbola were supplied by Apollonius and were borrowed from the early Pythagorean terminology of applications of areas. Apollonius wrote many other books. Among those were “On Proportional Sections” – 181 propositions. “On Spatial section” – 124 propositions. “On Determinate Section” – 83 propositions. Apollonius Apollonius
1. 2. 3. “Tangencies” – 124 propositions – dealt with the problem of constructing a circle tangent to three given circles, where the given circles are permitted to degenerate independently into straight line or points. Known as the problem of Applonius, attracted many mathematicians, among them Viete, Euler and Newton. “Vergings” 125 propositions “Plane Loci” – 147 propositions Trigonometry Trigonometry The origins of trigonometry are not clear. There are some problems in the Rhind papyrus and the Babylonian tablet Plimpton 322 contains a table of secants. Babylonian astronomers of the 4th and 5th centuries B.C. had accumulated a large amount of observational data and much of this passed to the Greeks. It was this early astronomy that lead to spherical trigonometry. Hipparchus Hipparchus Considered the most eminent astronomer around 140 B.C. He was the first to carry out numerous observations of planetary positions, introduced a coordinate system for their analysis and began the tabulation of trigonometric ratios necessary to enable astronomers to solve right triangles. Hipparchus Hipparchus The Babylonians sometime before 300 B.C. initiated the division of the circumference of the circle into 360 parts, called degrees, and within the next 2 centuries this measurement was adopted in the Greek world. Hipparchus was one of the first to use this measure. He also used arcs of 1/24 of the circle and 1/48 of a circle, called “steps” and “half steps” in some of his work. Hipparchus Hipparchus Babylonians were also the first to introduce coordinates into the sky. The system they used is known as the ecliptic system. Hipparchus was the first to attempt the detailed tabulation of lengths which enable plane triangles to be solved. The basic element in Hipparchus’ trigonometry was the chord subtending to given arc in a circle of fixed radius. He gave a table listing alpha and chord(alpha) for various values of the arc alpha. Chord(alpha) is simply a length. Claudius Ptolemy Claudius Ptolemy The definitive Greek work on astronomy was written by Ptolemy about 150 A.D. This very influential treatise, called “Syntaxis Mathematica” was based on the writings of Hipparchus and is noted for its remarkable completeness and elegance. This work became known as the “Almagest”. It is in 13 books. In the final book, Ptolemy developed his table of chords, an extension of Hipparchus table of chords. Ptolemy Ptolemy Ptolemy began with a circle whose diameter was divided into 120 equal parts. If each part has length p, then the diameter is 120p. For any central angle alpha, Ptolemy wanted to find the length of chord AB subtended by this angle. He generated such a table for all angles from ½ deg to 180 deg in ½ deg increments. He found that the chord of 1 deg is 1.0472p. Ptolemy Ptolemy A table of chords is equivalent to a table of trigonometric sines. Book I of the Almagest also contains a geometrical proposition now known as Ptolemy’s Theorem. “In a cyclic quadrilateral, the product of the diagonals is equal to the sum of the products of the two pairs of opposite sides”. Ptolemy Ptolemy From this, we can derive the modern difference formulas for sin(AB) and cos(A+B). In Book VI of the Almagest is found the four place value of pi. Books VII, VIII are devoted to a catalogue of 1028 fixed stars. The remaining books are devoted to the planets. The Almagest remained the standard work on astronomy until the time of Copernicus and Kepler. Heron of Alexander Heron of Alexander An Egyptian with Greek training. We know very little about his life but we know a lot about his mathematics. His wok may be divided into two classes, the geometric and the mechanical. In Heron’s Pneumatica appear descriptions of about 100 machines. In Catoptrica, one finds the elementary properties of mirrors and problems concerning the construction of mirrors to satisfy certain requirements. Heron Heron The most important of Heron’s geometrical works is his Metrica, written in 3 books and discovered in Constantinople by R. Schone in 1896. Book I deals with the areas of squares, rectangles, triangles etc. It is in this book that we find heron’s clever derivation of the famous formula for the area of a triangle in terms of its sides. Algebra Algebra In 1842, Nesselmann characterized three stages in the historical development of algebraic notation. First, we have rhetorical algebra, in which the solution of a problem is written without symbols as a pure argument. Then comes, syncopated algebra, in which abbreviations are adopted for some of the more frequently recurring quantities and operations. At the last stage, we have symbolic algebra, in which solutions appear in symbols having little apparent connections with the entities they represent. Diophantus Diophantus Another mathematician of uncertain date and nationality. All algebra before Diophantus was rhetorical. Diophantus’ outstanding contributions to mathematics was the syncopation of Greek algebra. Rhetorical algebra persisted in the rest of the world, except India, for many hundreds of years. In western Europe, most algebra remained rhetorical until the 15th century. Symbolic algebra made its first appearance in western Europe in the 16th century and did not become prevalent until the middle of the 17th century. Much of the symbolism of our elementary algebra textbooks is less than 400 years old. Diophantus Diophantus Diophantus wrote 3 works, Arithmetica, his most important, of which 6 of 13 books are in existence. The Arithmetica is an analytic treatment of algebraic number theory and marks the author as a genius in this field. The existing portion of the work is devoted to the solution of 130 problems leading to equations of first and second degree. Diophantus Diophantus There are some number theorems stated in the Arithmetica. This results were later investigated by Viete and Fermat. They are many propositions on representation of numbers as the sum of two, three or four squares. This was later completed by Fermat, Euler and Lagrange. Diophantus had abbreviations for the unknowns, powers of the unknowns up to the sixth, subtraction, equality and reciprocals. Other Greek Mathematicians Other Greek Mathematicians 1. 2. Pappus of Alexandria Wrote commentaries on Euclid’s Elements and Data and Ptolemy’s Almagest. Wrote “Mathematical Collection” a combined commentary and guidebook of the existing geometrical works of his time. Of the 8 books, the first is lost. Other Greek Mathematicians Other Greek Mathematicians 1. 2. 3. Theon of Alexandria Author of a commentary, in 11 books, on Ptolemy’s Almagest. The modern editions of Euclid’s Elements are based upon Theon’s revision of the original work. Held an administrative post at the University of Alexandria. Hypatia of Alexandria Hypatia of Alexandria Daughter of Theon Wrote commentaries on Diophantus’ Arithmetica and Applonius’ Conic Sections First woman mathematician to be mentioned in the history of mathematics. Lectured on mathematics and philosophy. The End The End In 529 A.D., the Athenian school closed for ever under Emperor Justinian. Mathematicians and philosophers fled to Persia. In 641 A.D., Arabs burned the great Alexandria Library. The long and glorious history of Greek mathematics came to an end. ...
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