Aristarchus of Samos

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Aristarchus of Samos
Statue of Aristarchus of Samos at the Aristotle University of Thessaloniki
Born	c. 310 BC Samos
Died	c. 230 BC (age c. 80) Alexandria
Nationality	Greek
Occupation	Scholar Mathematician Astronomer

Aristarchus of Samos (/ˌærəˈstɑːrkəs/; Greek: Ἀρίσταρχος ὁ Σάμιος, Aristarkhos ho Samios; c. 310 – c. 230 BC) was an ancient Greek astronomer and mathematician who presented the first known heliocentric model that placed the Sun at the center of the known universe, with the Earth revolving around the Sun once a year and rotating about its axis once a day. He was a student of Strato of Lampsacus, who was the third head of the Peripatetic School in Greece. According to Ptolemy, during his time there, he observed the summer solstice of 280 B.C. Along with his contributions to the heliocentric model, as reported by Vitruvius, he created two separate sundials; one that is a flat disc and one that is hemispherical. He was influenced by the concept presented by Philolaus of Croton (c. 470 – 385 BC) of a fire at the center of the universe, but Aristarchus identified the "central fire" with the Sun and he put the other planets in their correct order of distance around the Sun.

Like Anaxagoras before him, Aristarchus suspected that the stars were just other bodies like the Sun, albeit farther away from Earth. Often, his astronomical ideas were rejected in favor of the geocentric theories of Aristotle and Ptolemy. However, Nicolaus Copernicus knew about the possibility that Aristarchus had a 'moving Earth' theory, although it is unlikely that Copernicus was aware that it was a heliocentric theory.

Aristarchus estimated the sizes of the Sun and Moon as compared to Earth's size. He also estimated the distances from the Earth to the Sun and Moon. He is considered one of the greatest astronomers of antiquity along with Hipparchus, and one of the greatest thinkers in human history.

Heliocentrism

See also: Heliocentrism

The original text has been lost, but a reference in book by Archimedes, entitled The Sand Reckoner (Archimedis Syracusani Arenarius & Dimensio Circuli), describes a work in which Aristarchus advanced the heliocentric model as an alternative hypothesis to geocentrism:

You are now aware ['you' being King Gelon] that the "universe" is the name given by most astronomers to the sphere the centre of which is the centre of the earth, while its radius is equal to the straight line between the centre of the sun and the centre of the earth. This is the common account (τὰ γραφόμενα) as you have heard from astronomers. But Aristarchus has brought out a book consisting of certain hypotheses, wherein it appears, as a consequence of the assumptions made, that the universe is many times greater than the "universe" just mentioned. His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun on the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of the fixed stars, situated about the same centre as the sun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed stars as the centre of the sphere bears to its surface.

Aristarchus suspected the stars were other suns that are very far away, and that in consequence there was no observable parallax, that is, a movement of the stars relative to each other as the Earth moves around the Sun. Since stellar parallax is only detectable with telescopes, his accurate speculation was unprovable at the time.

It is a common misconception that the heliocentric view was held as sacrilegious by the contemporaries of Aristarchus. Lucio Russo traces this to Gilles Ménage's printing of a passage from Plutarch's On the Apparent Face in the Orb of the Moon, in which Aristarchus jokes with Cleanthes, who is head of the Stoics, a sun worshipper, and opposed to heliocentrism. In the manuscript of Plutarch's text, Aristarchus says Cleanthes should be charged with impiety. Ménage's version, published shortly after the trials of Galileo and Giordano Bruno, transposes an accusative and nominative so that it is Aristarchus who is purported to be impious. The resulting misconception of an isolated and persecuted Aristarchus is still transmitted today.

According to Plutarch, while Aristarchus postulated heliocentrism only as a hypothesis, Seleucus of Seleucia, a Hellenistic astronomer who lived a century after Aristarchus, maintained it as a definite opinion and gave a demonstration of it, but no full record of the demonstration has been found. In his Naturalis Historia, Pliny the Elder later wondered whether errors in the predictions about the heavens could be attributed to a displacement of the Earth from its central position. Pliny and Seneca referred to the retrograde motion of some planets as an apparent (and not real) phenomenon, which is an implication of heliocentrism rather than geocentrism. Still, no stellar parallax was observed, and Plato, Aristotle, and Ptolemy preferred the geocentric model that was held as true throughout the Middle Ages.

The heliocentric theory was revived by Copernicus, after which Johannes Kepler described planetary motions with greater accuracy with his three laws. Isaac Newton later gave a theoretical explanation based on laws of gravitational attraction and dynamics.

After realizing that the Sun was much larger than the Earth and the other planets, Aristarchus concluded that planets revolved around the Sun. But this brilliant insight, it turned out, "was too much for the philosophers of the time to swallow and astronomy had to wait 2000 years more to find the right path."

Distance to the Sun

Main article: Aristarchus On the Sizes and Distances

Aristarchus's third-century BC calculations on the relative sizes of (from left) the Sun, Earth, and Moon, from a tenth-century AD Greek copy

The only known surviving work usually attributed to Aristarchus, On the Sizes and Distances of the Sun and Moon, is based on a geocentric world view. Historically, it has been read as stating that the angle subtended by the Sun's diameter is two degrees, but Archimedes states in The Sand Reckoner that Aristarchus had a value of half a degree, which is much closer to the average value of 32' or 0.53 degrees. The discrepancy may come from a misinterpretation of what unit of measure was meant by a certain Greek term in the text of Aristarchus.

Aristarchus claimed that at half moon (first or last quarter moon), the angle between the Sun and Moon was 87°. He might have proposed 87° as a lower bound, since gauging the lunar terminator's deviation from linearity to one degree of accuracy is beyond the unaided human ocular limit (with that limit being about three arcminutes of accuracy). Aristarchus is known to have studied light and vision as well.

Using correct geometry, but the insufficiently accurate 87° datum, Aristarchus concluded that the Sun was between 18 and 20 times farther away from the Earth than the Moon. (The true value of this angle is close to 89° 50', and the Sun's distance is approximately 400 times that of the Moon.) The implicit false solar parallax of slightly under three degrees was used by astronomers up to and including Tycho Brahe, c. AD 1600. Aristarchus pointed out that the Moon and Sun have nearly equal apparent angular sizes, and therefore their diameters must be in proportion to their distances from Earth.

Size of the Moon and Sun

In On the Sizes and Distances of the Sun and Moon, Aristarchus discusses the size of the Moon and Sun in relation to the Earth. In order to achieve these measurements and subsequent calculations, he used several key notes made while observing a lunar eclipse. The first of these consisted of the time that it took for the Earth's shadow to fully encompass the Moon, along with how long the Moon remained within the shadow. This was used to estimate the angular radius of the shadow. From there, using the width of the cone that was created by the shadow in relation to the Moon, he determined that it was twice the diameter of the Moon at the full, non-central eclipse. In addition to this, Aristarchus estimated that the length of the shadow extended around 2.4 times the distance of the Moon from the Earth.

Using these calculations, along with the aforementioned estimated distances of the Sun from the Earth and Moon from the Earth, he created a triangle. Employing a similar method of geometry that he previously used for the distances, he was able to determine that the diameter of the Moon is roughly one-third that of the Earth's diameter. In order to estimate the size of the Sun, Aristarchus considered the proportion of the Sun's distance to Earth in comparison to the Moon's distance from Earth, which was found to be roughly 18 to 20 times the length. Therefore, the size of the Sun is around 19 times wider than the Moon, making it approximately six times wider than the Earth's diameter.

Legacy

Aristarchus (center) and Herodotus (right) from Apollo 15, NASA photograph

The lunar crater Aristarchus, the minor planet 3999 Aristarchus, and the telescope Aristarchos are named after him.

Primality test

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Primality_test 

A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy (its running time is polynomial in the size of the input). Some primality tests prove that a number is prime, while others like Miller–Rabin prove that a number is composite. Therefore, the latter might more accurately be called compositeness tests instead of primality tests.

Simple methods

The simplest primality test is trial division: given an input number, n, check whether it is evenly divisible by any prime number between 2 and √n (i.e. that the division leaves no remainder). If so, then n is composite. Otherwise, it is prime.^[1]

For example, consider the number 100, which is evenly divisible by these numbers:

2, 4, 5, 10, 20, 25, 50

Note that the largest factor, 50, is half of 100. This holds true for all n: all divisors are less than or equal to n/2.

Actually, when we test all possible divisors up to n/2, we will discover some factors twice. To observe this, rewrite the list of divisors as a list of products, each equal to 100:

2 × 50, 4 × 25, 5 × 20, 10 × 10, 20 × 5, 25 × 4, 50 × 2

Notice that products past 10 × 10 merely repeat numbers which appeared in earlier products. For example, 5 × 20 and 20 × 5 consist of the same numbers. This holds true for all n: all unique divisors of n are numbers less than or equal to √n, so we need not search past that. (In this example, √n = √100 = 10.)

All even numbers greater than 2 can also be eliminated since, if an even number can divide n, so can 2.

Let's use trial division to test the primality of 17. We need only test for divisors up to √n, i.e. integers less than or equal to ${\displaystyle \scriptstyle {\sqrt {17}}\approx 4.12}$, namely 2, 3, and 4. We can skip 4 because it is an even number: if 4 could evenly divide 17, 2 would too, and 2 is already in the list. That leaves 2 and 3. We divide 17 with each of these numbers, and we find that neither divides 17 evenly -- both divisions leave a remainder. So, 17 is prime.

We can improve this method further. Observe that all primes greater than 3 are of the form 6k ± 1, where k is any integer greater than 0. This is because all integers can be expressed as (6k + i), where i = −1, 0, 1, 2, 3, or 4. Note that 2 divides (6k + 0), (6k + 2), and (6k + 4) and 3 divides (6k + 3). So, a more efficient method is to test whether n is divisible by 2 or 3, then to check through all numbers of the form ${\displaystyle \scriptstyle 6k\ \pm \ 1\leq {\sqrt {n}}}$. This is 3 times faster than testing all numbers up to √n.

Generalising further, all primes greater than c# (c primorial) are of the form c# · k + i, for i < c#, where c and k are integers and i represents the numbers that are coprime to c#. For example, let c = 6. Then c# = 2 · 3 · 5 = 30. All integers are of the form 30k + i for i = 0, 1, 2,...,29 and k an integer. However, 2 divides 0, 2, 4,..., 28; 3 divides 0, 3, 6,..., 27; and 5 divides 0, 5, 10,..., 25. So all prime numbers greater than 30 are of the form 30k + i for i = 1, 7, 11, 13, 17, 19, 23, 29 (i.e. for i < 30 such that gcd(i,30) = 1). Note that if i and 30 were not coprime, then 30k + i would be divisible by a prime divisor of 30, namely 2, 3 or 5, and would therefore not be prime. (Note: Not all numbers which meet the above conditions are prime. For example: 437 is in the form of c#k + i for c#(7)=210, k=2, i=17. However, 437 is a composite number equal to 19*23).

As c → ∞, the number of values that c#k + i can take over a certain range decreases, and so the time to test n decreases. For this method, it is also necessary to check for divisibility by all primes that are less than c. Observations analogous to the preceding can be applied recursively, giving the Sieve of Eratosthenes.

A good way to speed up these methods (and all the others mentioned below) is to pre-compute and store a list of all primes up to a certain bound, say all primes up to 200. (Such a list can be computed with the Sieve of Eratosthenes or by an algorithm that tests each incremental m against all known primes < √m). Then, before testing n for primality with a serious method, n can first be checked for divisibility by any prime from the list. If it is divisible by any of those numbers then it is composite, and any further tests can be skipped.

A simple, but very inefficient primality test uses Wilson's theorem, which states that p is prime if and only if:

${\displaystyle (p-1)!\equiv -1{\pmod {p}}\,}$

Although this method requires about p modular multiplications, rendering it impractical, theorems about primes and modular residues form the basis of many more practical methods.

Example code

Python

The following is a simple primality test in Python using the simple 6k ± 1 optimization mentioned earlier. More sophisticated methods described below are much faster for large n.

def is_prime(n: int) -> bool:
    """Primality test using 6k+-1 optimization."""
    if n <= 3:
        return n > 1
    if n % 2 == 0 or n % 3 == 0:
        return False
    i = 5
    while i ** 2 <= n:
        if n % i == 0 or n % (i + 2) == 0:
            return False
        i += 6
    return True

C#

The following is a primality test in C# using the same optimization as above.

bool IsPrime(int n)
{
    if (n == 2 || n == 3)
        return true;

    if (n <= 1 || n % 2 == 0 || n % 3 == 0)
        return false;

    for (int i = 5; i * i <= n; i += 6)
    {
        if (n % i == 0 || n % (i + 2) == 0)
            return false;
    }

    return true;
}

JavaScript

The following is a primality test in JavaScript using the same optimization as above.

function isPrime(num) {
  if (num <= 3) return num > 1;
  
  if ((num % 2 === 0) || (num % 3 === 0)) return false;
  
  let count = 5;
  
  while (Math.pow(count, 2) <= num) {
    if (num % count === 0 || num % (count + 2) === 0) return false;
    
    count += 6;
  }
  
  return true;
}

R

The following is a primality test in R (programming language) using the same optimization as above.

is.prime <- function(number) {
  if(number <= 1 || number %% 2 ==0 || number %% 3 == 0) {
    return (FALSE)
  } else if (number == 2 || number == 3 ) {
    return(TRUE)
  }
  i <- 5
  while (i*i <= number) {
    if(number %% i == 0 || number %% (i+2)==0) {
      return(FALSE)
    } else {
      return(TRUE)
    }
    i = i + 6
  }
  return(TRUE)
}

Heuristic tests

These are tests that seem to work well in practice, but are unproven and therefore are not, technically speaking, algorithms at all. The Fermat test and the Fibonacci test are simple examples, and they are very effective when combined. John Selfridge has conjectured that if p is an odd number, and p ≡ ±2 (mod 5), then p will be prime if both of the following hold:

• 2^p−1 ≡ 1 (mod p),
• f_p+1 ≡ 0 (mod p),

where f_k is the k-th Fibonacci number. The first condition is the Fermat primality test using base 2.

In general, if p ≡ a (mod x²+4), where a is a quadratic non-residue (mod x²+4) then p should be prime if the following conditions hold:

• 2^p−1 ≡ 1 (mod p),
• f(1)_p+1 ≡ 0 (mod p),

f(x)_k is the k-th Fibonacci polynomial at x.

Selfridge, Carl Pomerance, and Samuel Wagstaff together offer $620 for a counterexample. The problem is still open as of September 11, 2015.

Probabilistic tests

Probabilistic tests are more rigorous than heuristics in that they provide provable bounds on the probability of being fooled by a composite number. Many popular primality tests are probabilistic tests. These tests use, apart from the tested number n, some other numbers a which are chosen at random from some sample space; the usual randomized primality tests never report a prime number as composite, but it is possible for a composite number to be reported as prime. The probability of error can be reduced by repeating the test with several independently chosen values of a; for two commonly used tests, for any composite n at least half the a's detect n's compositeness, so k repetitions reduce the error probability to at most 2^−k, which can be made arbitrarily small by increasing k.

The basic structure of randomized primality tests is as follows:

1. Randomly pick a number a.
2. Check equality (corresponding to the chosen test) involving a and the given number n. If the equality fails to hold true, then n is a composite number and a is a witness for the compositeness, and the test stops.
3. Get back to the step one until the required accuracy is reached.

After one or more iterations, if n is not found to be a composite number, then it can be declared probably prime.

Fermat primality test

The simplest probabilistic primality test is the Fermat primality test (actually a compositeness test). It works as follows:

Given an integer n, choose some integer a coprime to n and calculate a^{n − 1} modulo n. If the result is different from 1, then n is composite. If it is 1, then n may be prime.

If aⁿ⁻¹ (modulo n) is 1 but n is not prime, then n is called a pseudoprime to base a. In practice, we observe that, if aⁿ⁻¹ (modulo n) is 1, then n is usually prime. But here is a counterexample: if n = 341 and a = 2, then

${\displaystyle 2^{340}\equiv 1{\pmod {341}}}$

even though 341 = 11·31 is composite. In fact, 341 is the smallest pseudoprime base 2.

There are only 21853 pseudoprimes base 2 that are less than 2.5×10¹⁰. This means that, for n up to 2.5×10¹⁰, if 2ⁿ⁻¹ (modulo n) equals 1, then n is prime, unless n is one of these 21853 pseudoprimes.

Some composite numbers (Carmichael numbers) have the property that a^{n − 1} is 1 (modulo n) for every a that is coprime to n. The smallest example is n = 561 = 3·11·17, for which a⁵⁶⁰ is 1 (modulo 561) for all a coprime to 561. Nevertheless, the Fermat test is often used if a rapid screening of numbers is needed, for instance in the key generation phase of the RSA public key cryptographic algorithm.

Miller–Rabin and Solovay–Strassen primality test

The Miller–Rabin primality test and Solovay–Strassen primality test are more sophisticated variants, which detect all composites (once again, this means: for every composite number n, at least 3/4 (Miller–Rabin) or 1/2 (Solovay–Strassen) of numbers a are witnesses of compositeness of n). These are also compositeness tests.

The Miller–Rabin primality test works as follows: Given an integer n, choose some positive integer a < n. Let 2^sd = n − 1, where d is odd. If

${\displaystyle a^{d}\not \equiv \pm 1{\pmod {n}}}$

and

${\displaystyle a^{2^{r}d}\not \equiv -1{\pmod {n}}}$ for all ${\displaystyle 0\leq r\leq s-1,}$

then n is composite and a is a witness for the compositeness. Otherwise, n may or may not be prime. The Miller–Rabin test is a strong pseudoprime test (see PSW page 1004).

The Solovay–Strassen primality test uses another equality: Given an odd number n, choose some integer a < n, if

${\displaystyle a^{(n-1)/2}\not \equiv \left({\frac {a}{n}}\right){\pmod {n}}}$, where ${\displaystyle \left({\frac {a}{n}}\right)}$ is the Jacobi symbol,

then n is composite and a is a witness for the compositeness. Otherwise, n may or may not be prime. The Solovay–Strassen test is an Euler pseudoprime test (see PSW page 1003).

For each individual value of a, the Solovay–Strassen test is weaker than the Miller–Rabin test. For example, if n = 1905 and a = 2, then the Miller-Rabin test shows that n is composite, but the Solovay–Strassen test does not. This is because 1905 is an Euler pseudoprime base 2 but not a strong pseudoprime base 2 (this is illustrated in Figure 1 of PSW).

Frobenius primality test

The Miller–Rabin and the Solovay–Strassen primality tests are simple and are much faster than other general primality tests. One method of improving efficiency further in some cases is the Frobenius pseudoprimality test; a round of this test takes about three times as long as a round of Miller–Rabin, but achieves a probability bound comparable to seven rounds of Miller–Rabin.

The Frobenius test is a generalization of the Lucas pseudoprime test.

Baillie–PSW primality test

The Baillie–PSW primality test is a probabilistic primality test that combines a Fermat or Miller–Rabin test with a Lucas probable prime test to get a primality test that has no known counterexamples. That is, there are no known composite n for which this test reports that n is probably prime. It has been shown that there are no counterexamples for n ${\displaystyle <2^{64}}$.

Other tests

Leonard Adleman and Ming-Deh Huang presented an errorless (but expected polynomial-time) variant of the elliptic curve primality test. Unlike the other probabilistic tests, this algorithm produces a primality certificate, and thus can be used to prove that a number is prime. The algorithm is prohibitively slow in practice.

If quantum computers were available, primality could be tested asymptotically faster than by using classical computers. A combination of Shor's algorithm, an integer factorization method, with the Pocklington primality test could solve the problem in ${\displaystyle O((\log n)^{3}(\log \log n)^{2}\log \log \log n)}$.

Fast deterministic tests

Near the beginning of the 20th century, it was shown that a corollary of Fermat's little theorem could be used to test for primality. This resulted in the Pocklington primality test. However, as this test requires a partial factorization of n − 1 the running time was still quite slow in the worst case. The first deterministic primality test significantly faster than the naive methods was the cyclotomy test; its runtime can be proven to be O((log n)^{c log log log n}), where n is the number to test for primality and c is a constant independent of n. Many further improvements were made, but none could be proven to have polynomial running time. (Note that running time is measured in terms of the size of the input, which in this case is ~ log n, that being the number of bits needed to represent the number n.) The elliptic curve primality test can be proven to run in O((log n)⁶), if some conjectures on analytic number theory are true. Similarly, under the generalized Riemann hypothesis, the deterministic Miller's test, which forms the basis of the probabilistic Miller–Rabin test, can be proved to run in Õ((log n)⁴). In practice, this algorithm is slower than the other two for sizes of numbers that can be dealt with at all. Because the implementation of these two methods is rather difficult and creates a risk of programming errors, slower but simpler tests are often preferred.

In 2002, the first provably unconditional deterministic polynomial time test for primality was invented by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena. The AKS primality test runs in Õ((log n)¹²) (improved to Õ((log n)^7.5) in the published revision of their paper), which can be further reduced to Õ((log n)⁶) if the Sophie Germain conjecture is true. Subsequently, Lenstra and Pomerance presented a version of the test which runs in time Õ((log n)⁶) unconditionally.

Agrawal, Kayal and Saxena suggest a variant of their algorithm which would run in Õ((log n)³) if Agrawal's conjecture is true; however, a heuristic argument by Hendrik Lenstra and Carl Pomerance suggests that it is probably false. A modified version of the Agrawal's conjecture, the Agrawal–Popovych conjecture, may still be true.

Complexity

In computational complexity theory, the formal language corresponding to the prime numbers is denoted as PRIMES. It is easy to show that PRIMES is in Co-NP: its complement COMPOSITES is in NP because one can decide compositeness by nondeterministically guessing a factor.

In 1975, Vaughan Pratt showed that there existed a certificate for primality that was checkable in polynomial time, and thus that PRIMES was in NP, and therefore in NP ∩ coNP. See primality certificate for details.

The subsequent discovery of the Solovay–Strassen and Miller–Rabin algorithms put PRIMES in coRP. In 1992, the Adleman–Huang algorithm reduced the complexity to ZPP = RP ∩ coRP, which superseded Pratt's result.

The Adleman–Pomerance–Rumely primality test from 1983 put PRIMES in QP (quasi-polynomial time), which is not known to be comparable with the classes mentioned above.

Because of its tractability in practice, polynomial-time algorithms assuming the Riemann hypothesis, and other similar evidence, it was long suspected but not proven that primality could be solved in polynomial time. The existence of the AKS primality test finally settled this long-standing question and placed PRIMES in P. However, PRIMES is not known to be P-complete, and it is not known whether it lies in classes lying inside P such as NC or L. It is known that PRIMES is not in AC⁰.

Number-theoretic methods

Certain number-theoretic methods exist for testing whether a number is prime, such as the Lucas test and Proth's test. These tests typically require factorization of n + 1, n − 1, or a similar quantity, which means that they are not useful for general-purpose primality testing, but they are often quite powerful when the tested number n is known to have a special form.

The Lucas test relies on the fact that the multiplicative order of a number a modulo n is n − 1 for a prime n when a is a primitive root modulo n. If we can show a is primitive for n, we can show n is prime.

Eratosthenes

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Eratosthenes 

Eratosthenes
Born	276 BC Cyrene (in modern Libya)
Died	194 BC (around age 82) Alexandria
Occupation	Scholar Librarian Poet Inventor
Known for	Sieve of Eratosthenes Founder of Geography

Eratosthenes of Cyrene (/ɛrəˈtɒsθəniːz/; Greek: Ἐρατοσθένης [eratostʰénɛːs]; c. 276 BC – c. 195/194 BC) was a Greek polymath: a mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria. His work is comparable to what is now known as the study of geography, and he introduced some of the terminology still used today.

He is best known for being the first person known to calculate the circumference of the Earth, which he did by using the extensive survey results he could access in his role at the Library; his calculation was remarkably accurate. He was also the first to calculate Earth's axial tilt, which also proved to have remarkable accuracy. He created the first global projection of the world, incorporating parallels and meridians based on the available geographic knowledge of his era.

Eratosthenes was the founder of scientific chronology; he endeavoured to revise the dates of the main events of the semi-mythological Trojan War, dating the Sack of Troy to 1183 BC. In number theory, he introduced the sieve of Eratosthenes, an efficient method of identifying prime numbers.

He was a figure of influence in many fields. According to an entry in the Suda (a 10th-century encyclopedia), his critics scorned him, calling him Beta (the second letter of the Greek alphabet) because he always came in second in all his endeavours. Nonetheless, his devotees nicknamed him Pentathlos after the Olympians who were well rounded competitors, for he had proven himself to be knowledgeable in every area of learning. Eratosthenes yearned to understand the complexities of the entire world.

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 centuries earlier and became the capital of Pentapolis (North Africa), a country of five cities: Cyrene, Arsinoe, Berenice, Ptolemias, and Apollonia. 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. Under Ptolemaic rule the economy prospered, based largely on the export of horses and silphium, a plant used for rich seasoning and medicine. Cyrene became a place of cultivation, where knowledge blossomed. Like any young Greek at the time, Eratosthenes would have studied in the local gymnasium, where he would have learned physical skills and social discourse as well as reading, writing, arithmetic, poetry, and music.

Eratosthenes teaching in Alexandria by Bernardo Strozzi (1635)

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. He then studied under Aristo 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. Eratosthenes was a man of many perspectives and investigated the art of poetry under Callimachus. 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). 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.

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 traveled 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. He expanded the library's holdings: in Alexandria all books had to be surrendered for duplication. It was said that these were copied so accurately that it was impossible to tell if the library had returned the original or the copy. He sought to maintain the reputation of the Library of Alexandria against competition from the Library of Pergamum. Eratosthenes created a whole section devoted to the examination of Homer, and acquired original works of great tragic dramas of Aeschylus, Sophocles and Euripides.

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, with a high precision.

Eratosthenes believed there was both good and bad in every nation and criticized Aristotle for arguing that humanity was divided into Greeks and barbarians, as well as for arguing that the Greeks should keep themselves racially pure.^[12] 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 82 in Alexandria.^[9]

Scholarly career

Measurement of Earth's circumference

Main article: Earth's circumference § Eratosthenes

Measure of Earth's circumference according to Cleomedes' simplified version, based on the approximation that Syene is on the Tropic of Cancer and on the same meridian as Alexandria

The measurement of Earth's circumference is the most famous among the results obtained by Eratosthenes, who estimated that the meridian has a length of 252,000 stadia (39,060-40,320 km), with an error on the real value between −2.4% and +0.8% (assuming a value for the stadion between 155 and 160 metres). Eratosthenes described his arc measurement technique, in a book entitled On the measure of the Earth, which has not been preserved. However, a simplified version of the method has been preserved, as described by Cleomedes.

The simplified method works by considering two cities along the same meridian and measuring both the distance between them and the difference in angles of the shadows cast by the sun on a vertical rod (a gnomon) in each city at noon on the summer solstice. The two cities used were Alexandria and Syene (modern Aswan), and the distance between the cities was measured by professional bematists. A geometric calculation reveals that the circumference of the Earth is the distance between the two cities divided by the difference in shadow angles expressed as a fraction of one turn.

Geography

19th-century reconstruction of Eratosthenes' map of the (for the Greeks) known world, c. 194 BC

See also: History of geodesy and history of longitude

Eratosthenes now continued from his knowledge about the Earth. Using his discoveries and knowledge of its size and shape, he began to sketch it. In the Library of Alexandria he had access to various travel books, which contained various items of information and representations of the world that needed to be pieced together in some organized format. In his three-volume work Geography (Greek: Geographika), he described and mapped his entire known world, even dividing the Earth into five climate zones: two freezing zones around the poles, two temperate zones, and a zone encompassing the equator and the tropics. He had invented geography. He created terminology that is still used today. 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 now possible to estimate one's distance from remote locations with this network over the surface of the Earth. In the Geography the names of over 400 cities and their locations were shown, which had never been achieved before. However, his Geography has been lost to history, although fragments of the work can be pieced together from other great historians like Pliny, Polybius, Strabo, and Marcianus.

• The first book was something of an introduction and gave a review of his predecessors, recognizing their contributions that he compiled in the library. In this book Eratosthenes denounced Homer as not providing any insight into what he now described as geography. His disapproval of Homer's topography angered many who believed the world depicted in the Odyssey to be legitimate. He also commented on the ideas of the nature and origin of the Earth: he thought of Earth as an immovable globe while its surface was changing. He hypothesized that at one time the Mediterranean had been a vast lake that covered the countries that surrounded it and that it only became connected to the ocean to the west when a passage opened up sometime in its history.
• The second book contains his calculation of the circumference of the Earth. This is where, according to Pliny, "The world was grasped." Here Eratosthenes described his famous story of the well in Syene, wherein at noon each summer solstice, the Sun's rays shone straight down into the city-center well. This book would now be considered a text on mathematical geography.
• His third book of the Geography contained political geography. He cited countries and used parallel lines to divide the map into sections, to give accurate descriptions of the realms. This was a breakthrough and can be considered the beginning of geography. For this, Eratosthenes was named the "Father of Modern Geography."

Achievements

Eratosthenes was described by the Suda Lexicon as a Πένταθλος (Pentathlos) which can be translated as "All-Rounder", for he was skilled in a variety of things: He was a true polymath. He was nicknamed Beta because he was great at many things and tried to get his hands on every bit of information but never achieved the highest rank in anything; Strabo accounts Eratosthenes as a mathematician among geographers and a geographer among mathematicians.

• Eusebius of Caesarea in his Preparatio 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 (607 ft), 804,000,000 stadia is 149,000,000 km (93,000,000 mi), approximately the distance from the Earth to the Sun.
• 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. The actual figure is approximately 109 times.
• 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.
• 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. 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.

Number theory

Sieve of Eratosthenes: algorithm steps for primes below 121 (including optimization of starting from the prime's square).

Main articles: Sieve of Eratosthenes and primality test

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 no documents left of his work 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) (lost, summarized by Cleomedes)
• Гεωγραϕικά (Geographika) (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)
• The Catasterismi (Katasterismoi), a collection of Hellenistic myths about the constellations, was attributed to Eratosthenes.