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: 275-194 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 Hiero’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. if the scale tilts in the direction of the gold, then the wreath has a greater volume than the gold immerse into a container of water. 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 resist 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 Found 22 7 < 𝜋< 223 71 Properties of Spiral of Archimedes This lead to polar coordinates Found that sphere inscribed in cylinder is 2/3 its volume Found the “Quadrature of the Parabola”

Archimede’s Palimpsest 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 www.sciencenews.org

Other works by Archimedes On the Equilibrium of Planes About the mathematical theory of levers including moments and center of gravity of objects On Spirals On Floating Bodies On Conoids and Spheroids His work on conic sections and some solids obtained from them.

Contemporaries of Archimedes Conics: gave us the names “Ellipse” and “Hyperbola” and we think “Parabola” comes from his as well. Ellipse: He described as the locus of points with a fixed total distance from two points.

Erastothenes

Eratosthenes 276-194 BC Born in Cyrene Important astronomer, geographer, poet, and musical theorist 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

Eratosthenes 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

What is the 21st prime number? Is it a symmetrical prime? Eratosthenes Primes What is the 21st 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” https://www.youtube.com/watch?v=TIYMmbHik08

Apollonius

ApolLonius 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

Apollonius Parabolic Reflectors Satellites Microphones Flashlights

Apollonius Elliptical Rooms Billiards Whispering Room

ApolLonius 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 Hyperbola Parabola

Hipparchus

Hipparchus Hipparchus of Nicaea 180 BC – 125 BC Father of Trigonometry Hipparchus created tables of angle measures, chord measures and arc measures Used this table for astronomical calculations

Menalaus

Menalaus 70 AD – 130 AD Spherical Triangles A+B+C > 180 degrees January 14th, 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

Ptolemy 85 AD – 165 AD Most influential of Greek Astronomers Epicylces 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 Algamest Contained an important mathematical table. Perhaps the first trigonometry table Cyclic Quadrilaterals A quadrilateral is inscribed in a circle The sum of the products of opposite sides = product of diagonals. These are equivalent properties

Heron

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 earlier equation.

Diophantus

Diophantus of Alexandria 201 or 215 AD to between 285 and 299 AD Greek geometer whose interest was in Egyptian Mathematics Rational Ratios

Diophantine OG "This tomb holds Diophantus. Ah, what a marvel! And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. Alas! late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life."

Diophantine age problem Simplified In simpler English it says: Diophantus's youth lasted 1/6 of his life. He had the first beard in the next 1/12 of his life. At the end of the following 1/7 of his life Diophantus got married. Five years from then his son was born. His son lived exactly 1/2 of Diophantus's life. Diophantus died 4 years after the death of his son. How long did Diophantus live?

Diophantus Gave proof of the classification of Pythagorean triples. “Rhetorical Algebra” Arithmetica

Pappus Circa. 290 - 350

Pappus Final Greek Geometer to produce results that had an impact on future mathematics. May have been the best geometer of his time, but his work was mostly ignored by his contemporaries. Librarian of Alexandria Synagoge

Theon and Hypathia of Alexandria

Theon (circa 335 – 405) Theon: Extensive rewriting of Euclid’s Elements

Hypatia First woman to make substantial contributions to the development of mathematics Head of the Platonist School in Alexandria Wrote commentaries on Arithmetica Historians mark her death as the hands of a Christian mob as the end of Alexandrian intellectualism and the eventual final destruction of the library.

Works Cited https://en.wikipedia.org/wiki/Apollonius_of_Tyana History of Math, University of Arizona https://www.mathsisfun.com/puzzles/diophantus- solution.html http://www.ancient.eu/article/309/ http://www-history.mcs.st- and.ac.uk/Biographies/Hypatia.html http://aetherforce.com/who-really-burned-the-library-of- alexandria-by-preston-chesser/