Eratosthenes

Life The son of Aglaos, Eratosthenes was born in 276 BC in Cyrene. Now part of modern-day Libya, Cyrene had been founded by Greeks during the second half of the 7th century BCE, Cyrene came under the rule of Alexander the Great in 332 BC,

By the late 260s BCE, Eratosthenes went to Athens to further his studies. and the eclectic Bion of Borysthenes. and he sent Eratosthenes the famous Cattle Problem to be presented to the mathematicians of Alexandria. Eratosthenes subsequently wrote compositions on geography, philosophy, rhetoric, literary criticism, grammar, poetry and star lore. However, a simplified version of the method as described by Cleomedes was preserved.

The simplified method works by considering two cities along the same meridian, and the difference in angles of the shadows cast by the sun on a vertical rod (a gnomon). The two cities used by Eratosthenes were Alexandria and Syene (modern Aswan). At noon on the summer solstice, there were still shadows in Alexandria. However, in Syene, rods cast no shadows, and the Sun's rays shone straight down into the city-center well.

According to Cleomedes, Eratosthenes then measured the shadow's angle to be about 7.2 degrees, which is 1/50 of a full circle, and reasoned using alternate interior angles that this angle represented the portion of Earth's curvature between the two cities. The distance between Alexandria and Syene was reported to be about 5,000 stadia, as measured by professional bematists. Eratosthenes multiplied this number by 50 and arrived at a total of roughly 250,000 stadia for the Earth's circumference.

While Eratosthenes' method was sound he made two false assumptions which fortuitously cancelled each other out. The first was that Syene was on the Tropic of Cancer and the second was that it lay on the same meridian of longitude (directly south) of Alexandria. In fact Syene is 1° north of the Tropic and 3° east of Alexandria.

Sun measurements

Eusebius of Caesarea in his Preparatio Evangelica includes a brief chapter of three sentences on celestial distances. He states simply that Eratosthenes found the distance to the Sun to be "" (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 The meaning depends on whether Eusebius meant 400 myriad plus 80,000 or "400 and 80,000" myriad. With a stade of , 804,000,000 stadia is , approximately the distance from the Earth to the Sun.

Eratosthenes also calculated the Sun's diameter. According to Macrobius, Eratosthenes made the diameter of the Sun to be about 27 times that of the Earth. The ecliptic is the apparent circular orbit of the sun projected onto the imaginary celestial sphere over the course of a year; its obliquity is the inclination of its plane relative to the plane of the equator. a model of objects in the sky (on the celestial sphere), consisting of a spherical framework of rings, centered on Earth or the Sun, that represent lines of celestial longitude and latitude and other astronomically important features, such as the ecliptic.

Geography

Eratosthenes continued to study the Earth, and began to sketch it. In the Library of Alexandria he had access to travel books, which contained information and representations of the world that needed to be pieced together in some organized format. He placed grids of overlapping lines over the surface of the Earth. He used parallels and meridians to link together every place in the world. It was then possible to estimate the distance from remote locations with this network over the surface of the Earth. In the Geography he recorded the names of over 400 cities and their locations were shown, a feat without precedent.

Mathematics, music theory and metaphysics

In Platonikos, primarily mathematical questions were dealt with; the concepts discussed included distance, ratio, continuous and discontinuous proportion, mathematical mean, prime number and point. The focus was on the theory of proportions, in which Eratosthenes saw the key to Platonic philosophy. He applied the tool of the ratio equation ("a is to b as c is to d"), which he called "analogy", to both mathematics and philosophy. Friedrich Solmsen states that in proportion, he believed he had found the unifying bond of the "mathematical" sciences (arithmetic, geometry, astronomy, music theory), since all statements of these sciences could ultimately be traced back to statements about proportions.

According to Theon of Smyrna, he perceived ratio as the foundational principle which underlies proportion, as well as the "primary cause of the creation of all orderly things", while he saw the number one as the starting point (archḗ) and the primary element (stoicheíon) of numbers and quantity.

For Eratosthenes, numbers are unproblematic; but lines, on the other hand, are curious, as they cannot be produced by the combination of individual points, since the individual point has no extension. Eratosthenes contends rather it arises from the continuous movement of a point. This view was later criticized by the skeptic Sextus Empiricus. He dedicated his solution to King Ptolemy, presenting a model in bronze, with a letter and an epigram.

Eratosthenes' sieve is one of a number of prime number sieves, and 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.

A secondary subject of Platonikos was music theory, in which Eratosthenes applied the theory of proportions to music, Since antiquity, he is considered an authority in the field of music. Eratosthenes knew and considered the system of the music theorist Aristoxenus. This is based on the idea that the soul can only grasp sensible objects if it has a corresponding disposition in its own structure. Accordingly, it is a mixture of two components, one incorporeal and one corporeal. Its contents were lost, but it can be said that it consisted of two books, and was of enough importance to be included in what Pappus called the "Treasury of Analysis" together with the writings of Euclid, Apollonius, and Aristaeus, thus implying that it was a systematic geometrical composition. The extant work by this name in its current form cannot be attributed to Eratosthenes, however it is rooted in a genuine work by him with the same name. It has been pointed out, that Eratosthenes did not invent the myths, which had been transmitted over centuries through Greek traditions, rather he connected these tales to the constellations and attributed the different mythical characters to them. It is now lost, but 155 fragments survive, 105 in the writings of Strabo, 16 in the writings of Pliny the Elder, and the rest scattered in Byzantine sources.