Greek Mathematics after Euclid Archimedes Eratosthenes Appollonius Hipparchus Menelaus Ptolemy Heron Diophantus

Famous Greek Mathematicians after Euclid ›Archimedes: 287–212 B.C – inventor, mathematician, his treatise on calculus was lost until 15 years ago ›Eratosthenes: B.C – astronomer, mathematician, Sieve of Eratosthenes ›Apollonius: 262–190 B.C – astronomer, mathematician, Developed conic sections ›Hipparchus: Astronomer, mathematician, Trigonometry and table of lengths of chords with given central angle

Famous Greek Mathematicians after Euclid ›Menelaus: mathematician, spherical triangles, Menelaus’ theorem ›Ptolemy: astronomer, mathematician, three point problems, Ptolemy’s theorem ›Heron: Heron’s formula, approximation for square roots, Metrica books ›Diophantus: Arithmetica, - beginning of number theory, rational solutions to curves

Archimedes KING HIERON’S CROWN ›Goldsmith was supposed to fashion crown from given weight in gold ›Archimedes tasked to devise method to test if crown was pure gold ›Law of Hydrostatics – Archimedes Principle ›Eureka! Suspend wreath and equal mass of gold suspended from the other end. immerse into a container of water. if the scale tilts in the direction of the gold, then the wreath has a greater volume than the gold its density is less than that of gold and must then be a alloy of gold and some lighter material.

Archimedes Defense Inventions ›helped Syracuse resisted Romans for 3 years ›Marcellus had great respect for Archimedes ›Ordered soldiers not to harm him

Archimedes Death ›was working on a problem in geometry ›soldiers entered his home ›he asked them to wait while he finished his proof ›they ran him through with a spear.

ARCHIMEDIAN MATHEMATICS Method of Exhaustion in approximating π Properties of spiral of Archimedes Found that sphere inscribed in cylinder is 2/3 its volume

ARCHIMEDIAN MATHEMATICS Archimedes work wiped off by a monk and used as a prayer book. Research shows that it is earliest known calculus Dealt with “potential infinities” and “The Method” which dealt with areas of curved regions The Archimedes Codex – William Noel (2007)

ERATOSTHENES BC Born in Cyrene Solved the Doubling Cube Problem Mechanically Mesolabium Three sliding rectangular plates. Height = width of the frame slide in three grooves. First plate remains fixed. Second slides under first and third under second. Drawing line through points of intersection

ERASTOTHENES Primes Determining primes Check if primes less than square root are factors Sieve Cross 1 out Circle 2 – cross out multiples of 2 Circle 3 – cross out multiples of 3 Twin Primes differ by 2 How many twin primes are there less than 100 Symmetrical primes are primes whose reverse is also prime How many symmetrical primes are there between 1 and 100

ERASTOTHENES Primes What is the 21 st prime number? Is it a symmetrical prime? What number primes is its reverse? What are the factors of 21? Write 73, 37, 7 and 3 in binary. Are they all palindromes? We call 37 and 73 a “Sheldon Prime”

ERASTOTHENES Best remembered for calculating earth’s circumference Estimated arc of great circle through Alexandria and Syene (Aswan) He also determined the angle between the two cities from the center of the earth based on the shadows cast. He calculated it to be 24,622 miles which is only 245 miles less than correct value

APPOLLONIUS Appollonius of Perga Approx 262 BC to 190 BC Rival to Archimedes Calculated more accurately than Archimedes The Comparison of a Dodecahedron and the Icosahedron The distance from the pentagonal faces of a dodecahedron to center The distance from the triangular faces of a icosahedron to center Same

›Conic Sections still have many uses today. Parabolic Reflectors Satellites Microphones Flashlights Appollonius

Elliptical Rooms Billiards Whispering Room

APPOLLONIUS On Conics Best work on Conic Sections for thousands of years Proved that parabolas, hyperbolas, and ellipses were all planar intersections of a conic (not necessarily right) He actually coined these names for the curves Ellipsis – deficiency Hyperbola – a throwing beyond Parabola – placing beside or comparison

HIPPARCHUS Hipparchus of Nicaea 180 BC – 125 BC Father of Trigonometry Aristarchus had discovered ratio arc – chord  1 as central angle  0 Hipparchus created tables of angle measures, chord measures and arc measures Used this table for astronomical calculations

The construction of the table is starts on

He also calculated chord of ½ Place F so that CB = CF Place D so that DOA = ½ Place E so that DE is perpendicular to AC What is the relationship between

He also calculated chord of ½ Place F so that CB = CF Place D so that DOA = ½ Place E so that DE is perpendicular to AC What is the relationship between

He also calculated chord of ½ Place F so that CB = CF Place D so that DOA = ½ Place E so that DE is perpendicular to AC What is the relationship between

He also calculated chord of ½ Place F so that CB = CF Place D so that DOA = ½ Place E so that DE is perpendicular to AC What is the relationship between

He also calculated chord of ½ Place F so that CB = CF Place D so that DOA = ½ Place E so that DE is perpendicular to AC Why is

He also calculated chord of ½ Why is

He also calculated chord of ½ Find the ratio which gives us AD.

MENALAUS 70 AD – 130 AD Spherical Triangles A+B+C > 180 degrees January 14 th, 98 AD made observations occulation of the Beta Scorpii by moon Menelaus realized that light follows angle incidence equals of angle of reflection

MENELAUS THEOREM If an arbitrary line (not parallel to existing triangle edges) cuts an arbitrary triangle, and we extend any side to create more triangles, the following ratio holds.

PTOLEMY 85 AD – 165 AD Most influential of Greek Astronomers Propounded geocentric theory that prevailed 1400 years Heaven is spherical in form and rotates as a sphere Earth is spherical in form Earth is situated in the middle of heaven The earth does not move in any way

PTOLEMY Cyclic Quadrilaterals Aquadrilateral is inscribed in a circle The sum of the products of opposite sides = product of diagonals. These are equivalent properties

HERON Heron of Alexandria, Egypt 10 AD – 75 AD Geometer and Mechanical Engineer Taught at Museum in Alexandria Physics, Math, Pneumatics and Mechanics

HERON Apply Pythagorean Theorem to Subtract equations and solve for d Let and substitute d into earliers equation.