St. Mary High School

FA :-1 Mathematics Task No.1 Topic:- Biography of Great

Mathematician

EratosthenesThe Great Mathematician 

Eratosthenes• Born276 BC

Cyrene• Died194 BC

Alexandria• Ethnicity :- Greek• Occupation:-• Scholar• Librarian• Poet• Inventor

*The son of Aglaos, Eratosthenes was born in 276 BC, in Cyrene. Now part of modern-day Libya, Cyrene had been founded by the Greeks centuries earlier, and became the capital of Pentapolis (North Africa), a country of five cities: Cyrene, Arsinde, Berenice, Ptolemias, and Apollonian, Cyrenaica. Alexander the Great conquered Cyrene in 332 BC, and following his death in 323 BC its rule was given to one of his generals, Ptolemy I Soter, the founder of the Ptolemaic Kingdom

life

Continue*Eratosthenes went to Athens to further his studies. There he

was taught Stoicism by its founder, Zeno of Citium, in philosophical lectures on living a virtuous life.[9] He then studied under Ariston of Chios, who led a more cynical school of philosophy. He also studied under the head of the Platonic Academy, who was Arcesilaus of Pitane. His interest in Plato led him to write his very first work at a scholarly level, Platonikos, inquiring into the mathematical foundation of Plato's philosophies.[10] Eratosthenes was a man of many perspectives and investigated the art of poetry under Callimachus.[11] He had talent as a most imaginative poet.

Continue*He wrote poems: one in hexameters called Hermes illustrating

the god's life history; and another, in elegiacs, called Erigone, describing the suicide of the Athenian maiden Erigone (daughter of Icarius).[12] He wrote Chronographies, a text that scientifically depicted dates of importance, beginning with the Trojan War. This work was highly esteemed for its accuracy: George Syncellus was later able to preserve from Chronographies a list of 38 kings of the Egyptian Thebes. Eratosthenes also wrote Olympic Victors, a chronology of the winners of the Olympic Games. It is not known when he wrote his works, but they highlighted his abilities.

Continue*These works and his great poetic abilities led the pharaoh

Ptolemy III Euergetes to seek to place him as a librarian at the Library of Alexandria in the year 245 BC. Eratosthenes, then thirty years old, accepted Ptolemy's invitation and travelled to Alexandria, where he lived for the rest of his life. Within about five years he became Chief Librarian, a position that the poet Apollonius Rhodius had previously held. As head of the library Eratosthenes tutored the children of Ptolemy, including Ptolemy IV Philopator who became the fourth Ptolemaic pharaoh.

Continue*Eratosthenes made several important contributions to

mathematics and science, and was a friend of Archimedes. Around 255 BC, he invented the armillary sphere. In On the Circular Motions of the Celestial Bodies, Cleomedes credited him with having calculated the Earth's circumference around 240 BC, using knowledge of the angle of elevation of the Sun at noon on the summer solstice in Alexandria and on Elephantine Island near Syene (modern Aswan, Egypt).

Continue *Eratosthenes believed there was good and bad in

every nation and criticized Aristotle for arguing that humanity was divided into Greeks and barbarians, and that the Greeks should keep themselves racially pure.[14] As he aged he contracted ophthalmia, becoming blind around 195 BC. Losing the ability to read and to observe nature plagued and depressed him, leading him to voluntarily starve himself to death. He died in 194 BC at the age of 82 in his beloved Alexandria.[15]

Measurement of the Earth's circumference*Eratosthenes calculated the circumference of the Earth without

leaving Egypt. He knew that at local noon on the summer solstice in Syene (modern Aswan, Egypt), the Sun was directly overhead. He knew this because the shadow of someone looking down a deep well at that time in Syene blocked the reflection of the Sun on the water. He measured the Sun's angle of elevation at noon on the same day in Alexandria. The method of measurement was to make a scale drawing of that triangle which included a right angle between a vertical rod and its shadow. This turned out to be 1/50th of a circle. Taking the Earth as spherical, and knowing both the distance and direction of Syene, he concluded that the Earth's circumference was fifty times that distance.

Continue

Continue* Some claim Eratosthenes used the Olympic stade of 176.4

m, which would imply a circumference of 44,100 km, an error of 10%,[16] but the 184.8 m Italian stade became (300 years later) the most commonly accepted value for the length of the stade,[16] which implies a circumference of 46,100 km, an error of 15%.[16] It was unlikely, even accounting for his extremely primitive measuring tools, that Eratosthenes could have calculated an accurate measurement for the circumference of the Earth for three important assumptions he made (none of which are perfectly accurate):[17][16]

Continue• That Alexandria and Syene lay on the same

meridian,• That the distance between Alexandria and Syene

was 5000 stades,• That the Earth was a perfect sphere.• Eratosthenes later rounded the result to a final

value of 700 stadia per degree, which implies a circumference of 252,000 stadia, likely for reasons of calculation simplicity as the larger number is evenly divisible by 60.[16] Repeating Eratosthenes' calculation with more accurate data, the result is 40,074 km, which is 66 km different (0.16 %) from the currently accepted polar circumference of the Earth.[17]

Continue• Seventeen hundred years after Eratosthenes'

death, while Christopher Columbus studied what Eratosthenes had written about the size of the Earth, he chose to believe, based on a map by Toscanelli, that the Earth's circumference was one-third smaller. Had Columbus set sail knowing that Eratosthenes' larger circumference value was more accurate, he would have known that the place that he made landfall was not Asia, but rather a New World.[18]

Achievements• Eusebius of Caesarea in his Preparation Evangelica

 includes a brief chapter of three sentences on celestial distances (Book XV, Chapter 53). He states simply that Eratosthenes found the distance to the Sun to be "σταδίων μυριάδας τετρακοσίας καὶ ὀκτωκισμυρίας" (literally "of stadia myriads 400 and 80,000") and the distance to the Moon to be 780,000 stadia. The expression for the distance to the Sun has been translated either as 4,080,000 stadia (1903 translation by E. H. Gifford), or as 804,000,000 stadia (edition of Edouard des Places, dated 1974–1991). The meaning depends on whether Eusebius meant 400 myriad plus 80,000 or "400 and 80,000" myriad. With a stade of 185 m, 804,000,000 stadia is 149,000,000 km, approximately the distance from the Earth to the Sun.

Continue• Eratosthenes also calculated the Sun's diameter.

According to Macrobious, Eratosthenes made the diameter of the Sun to be about 27 times that of the Earth.[23] The actual figure is approximately 109 times.[25]

• During his time at the Library of Alexandria, Eratosthenes devised a calendar using his predictions about the ecliptic of the Earth. He calculated that there are 365 days in a year and that every fourth year there would be 366 days.[26]

Continue• He was also very proud of his solution for

Doubling the Cube. His motivation was that he wanted to produce catapults. Eratosthenes constructed a mechanical line drawing device to calculate the cube, called the mesolabio. He dedicated his solution to King Ptolemy, presenting a model in bronze with it a letter and an epigram.[27] Archimedes was Eratosthenes' friend and he, too, worked on the war instrument with mathematics. Archimedes dedicated his book The Method to Eratosthenes, knowing his love for learning and mathematics.[28]

Prime numbers

Continue • Eratosthenes proposed a simple algorithm for finding

prime numbers. This algorithm is known in mathematics as the Sieve of Eratosthenes.

• In mathematics, the sieve of Eratosthenes (Greek: κόσκινον Ἐρατοσθένους), one of a number of prime number sieves, is a simple, ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite, i.e., not prime, the multiples of each prime, starting with the multiples of 2. The multiples of a given prime are generated starting from that prime, as a sequence of numbers with the same difference, equal to that prime, between consecutive numbers. This is the sieve's key distinction from using trial division to sequentially test each candidate number for divisibility by each prime.

Works• Eratosthenes was one of the most pre-eminent

scholarly figures of his time, and produced works covering a vast area of knowledge before and during his time at the Library. He wrote on many topics — geography, mathematics, philosophy, chronology, literary criticism, grammar, poetry, and even old comedies. Unfortunately, there are only fragments left of his works after the Destruction of the Library of Alexandria.

Titles• Platonikos• Hermes• Erigone• Chronographies• Olympic Victors• Περὶ τῆς ἀναμετρήσεως τῆς γῆς (On the Measurement of the

Earth)[30] (lost, summarized by Cleomedes)• Гεωγραϕικά (Geographika)[31] (lost, criticized by Strabo)• Arsinoe (a memoir of queen Arsinoe; lost; quoted by Athenaeus in

the Deipnosophistae)• Ariston (concerning Aristo of Chios' addiction to luxury); lost;

quoted by Athenaeus in the Deipnosophistae)[32]

• A fragmentary collection of Hellenistic myths about the constellations, called Catasterismi (Katasterismoi), was attributed to Eratosthenes, perhaps to add to its credibility.

Modern Implications of Eratosthenes’ Math• The most actively used discovery of Eratosthenes’ is the leap

day, which adds an extra day onto every fourth year in order to keep the calendar system accurate.

• We also know the circumference of the earth due to his math and have been able to calculate how long trips may take around the world based on his findings.

• Another way we use Eratosthenes’ math is to know how far away the sun is from Earth. Due to this it is possible to, say, know how far a planet must be from a star to sustain life, and other associated equations.

• We also know the tilt of Earth’s axis thanks to Eratosthenes and thus have been able to figure out how the day and night cycle works based on his math.

Made By

• Preshit . Pegadpalliwar• Ritik . Atram• Aniket . Fale• Sanket . Raut• Pooja . Jaiswal• Amisha . Mankar• Priya . Tapkire• Shreya .Atram