Archimedes Created a number system to deal (~250BCE) with arbitrary large numbers

Archimedes wrote an essay on the number of grains of sand that would fill a sphere whose diameter was equal to the distance from earth to the fixes stars. Since he had to work with numbers larger than in myriad MAT 336 Hellenic myriads, he imagined a “doubled class” of numbers of eight numerals (instead of the four the the Greek ciphered Mathematics system) After Euclid 1. 1 to 99,999,999 2. 100,000,000 to 1016-1 3. etc

Archimedes Approximation of π Archimedes (~250BCE) (~250BCE) Computed the Ideas of calculus perimeter of a regular BD : CD = BA : AC. Derived and proved Infinitesimals polygon of 96 formulae for Method of exhaustion sides!!!!!! Area of the circle Dealt with infinity!!! Surface area and volume 3 10/71< π < 3 1/7 of the sphere Approximation of π Area of an ellipse. Created a number system to deal with Throughout this proof, Archimedes uses Area under a parabola arbitrary large several rational approximations to numbers various square roots. Nowhere does he Applied mathematics to say how he got those approximations. physics Explanation of principle of ? Images and text from: the lever https://itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.html Innovative machines Archimedes (~250BCE) Archimedes Archimedes (~250BCE) (~250BCE)

http://abel.math.harvard.edu/~knill//3dprinter/documents/paper.pdf

Proposition 33: The surface of any sphere is equal to four times the circle in it. 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 equal to the radius of the sphere

And although he made many excellent Several of Archimedes treatises discoveries, he is said to have asked his (which had been copied in 10th- The Archimedes kinsmen and friends to place over the grave century Constantinople) were found where he should be buried a cylinder enclosing a on a Byzantine prayer book from the Palimpsest sphere, with an inscription giving the proportion 13th century by which the containing solid exceeds the Discovered in 1906 by the Danish contained. scholar Johan Ludwig Heiberg. The text had been scraped away to Plutarch (AD 45-120), Parallel Lives: Marcellus make room for the prayer book. The book then went missing until it was auctioned – in a much more damaged state – at Christie's in New York in 1998. Bought by an anonymous American collector for $2m (£1.25m), it was deposited at Baltimore's Walters Art Museum, where scientists, conservators, classicists and historians have been working on A tomb in Syracuse in the Necropolis of Grotticelli referred to affectionately (or deceptively) as “Archimedes’ uncovering the secrets of oldest Tomb ”, but known to be of Roman origin dating at least two centuries after the death of Archimedes. surviving copy of Archimedes' works https://www.math.nyu.edu/~crorres/Archimedes/Tomb/TombIllus.html http://www.archimedespalimpsest.org/about/ Archimedes Archimedes Law of The Method Euclid’s Elements the Lever Proposition XII.10 Any cone is a third part of the cylinder with the same Magnitudes are in equilibrium at base and equal height. distances reciprocally proportional to their weights.

https://www.math.nyu.edu/~crorres/Archimedes/Lever/LeverLaw.html

Apollonius of Perga (~200BCE) Claudius Ptolemy ~100AD Conic sections In Alexandria, studied and later taught under the followers of Euclid Mathematical Collection (Almagest) - Conics - 8 books Mathematical Astronomy worked out more fully and Astronomer and physicist generally than in the writings of others. Trigonometry - Table of chords “the most and prettiest of these theorems are new, and it Solve plane triangles was their discovery which made me aware that Euclid did not work out the syntheses of Solved spherical triangles the locus with respect to three and four lines, but only a chance portion of it, and that not successfully; for it was not possible for the said synthesis to be completed without the aid of the additional theorems discovered by me.” Diophantus - (possibly ~250AD) - Claudius Ptolemy ~100AD Alexandria

Ptolemy's world map, reconstituted from Ptolemy's Geography (circa 150) in the 15th century, indicating Arithmetica "Sinae" (China) at the extreme right, beyond the island of "Taprobane" (Ceylon or Sri Lanka, oversized) and the "Aurea Chersonesus" (Southeast Asian peninsula). Major advance in the solution of equation: Introduction to symbolism. Linear and quadratic equations Higher degree equations The method of false position

one of the first scholars to look at the problem of representing large portions of the Earth's surface on a flat map.

“The Greeks drew upon reasoning, proof and definition in their mathematics; and, to a degree unique among ancient cultures,….” The Rainbow of Mathematics: A History of the Mathematical Science- Ivor Grattan-Guinnes