The Secret Teachings of All Ages · 78 of 236

The Sieve of Eratosthenes

Redrawn from Taylor's Theoretic Arithmetic.

This sieve is a mathematical device originated by Eratosthenes about 230 B.C. far the purpose of segregating the composite and incomposite odd numbers. Its use is extremely simple after the theory has once been mastered. All the odd numbers are first arranged in their natural order as shown in the second panel from the bottom, designated Odd Numbers. It will then be seen that every third number (beginning with 3) is divisible by 3, every fifth number (beginning with 5;) is divisible by 5, every seventh number (beginning with 7) is divisible by 7, every ninth number (beginning with 9) is divisible by 9, every eleventh number (beginning with 11) is divisible by 11, and so on to infinity.

This system finally sifts out what the Pythagoreans called the "incomposite" numbers, or those having no divisor other than themselves and unity. These will be found in the lowest panel, designated Primary and Incomposite Numbers. In his History of Mathematics, David Eugene Smith states that Eratosthenes was one of the greatest scholars of Alexandria and was called by his admirers "the second Plato." Eratosthenes was educated at Athens, and is renowned not only for his sieve but for having computed, by a very ingenious method, the circumference and diameter of the earth. His estimate of the earth's diameter was only 50 miles less than the polar diameter accepted by modern scientists.

Context

Eratosthenes of Cyrene (c. 276-194 BCE) was head librarian of the Library of Alexandria, a serious mathematician, geographer, and astronomer. His sieve for finding prime numbers is still taught today as the foundational algorithm for prime generation. The 'second Plato' epithet Hall reports is from David Eugene Smith's History of Mathematics; Eratosthenes was also nicknamed 'Beta' in antiquity, second-best in everything — a slightly unfair characterization given his actual achievements, which include the first accurate measurement of the earth's circumference. He measured the angle of the sun at Syene (Aswan) and Alexandria on the same day at noon, used the distance between the cities and the angular difference, and arrived at about 250,000 stadia for the earth's circumference, accurate to within a few percent of the modern value.Smith, History of Mathematics; van der Waerden, Science Awakening.

This and other mathematical achievements of Eratosthenes, are indisputable evidence that in the third century before Christ the Greeks not only knew the earth to be spherical in farm but could also approximate, with amazing accuracy, its actual size and distance from both the sun and the moon. Aristarchus of Samos, another great Greek astronomer and mathematician, who lived about 250 B.C., established by philosophical deduction and a few simple scientific instruments that the earth revolved around the sun. While Copernicus actually believed himself to be the discoverer of this fact, he but restated the findings advanced by Aristarchus seventeen hundred years earlier.

Insight

Aristarchus of Samos (c. 310-230 BCE) is the genuinely scandalous figure of Hellenistic astronomy. He proposed a heliocentric system with the earth orbiting the sun seventeen centuries before Copernicus, as Hall correctly notes. The proposal is preserved by Archimedes in The Sand Reckoner — Archimedes had to record it in order to refute it, and the system was rejected on the (in 250 BCE) reasonable ground that if the earth moves, the fixed stars should show parallax, and they did not. Aristarchus replied that the stars must be vastly farther away than was supposed; his hypothesis was correct on this point too, but unverifiable by the instruments of the time. Copernicus knew of Aristarchus through citations in classical sources and acknowledged the precedent in early drafts of De Revolutionibus, though he removed the acknowledgment from the published version.Archimedes, The Sand Reckoner; Heath, Aristarchus of Samos (1913).

10 is one-half of the sum of 6 and 14; 18 is one-half the sum of 14 and 22; and 6 is one-half the sum of 2 and 10.

The oddly-odd, or unevenly-even, numbers are a compromise between the evenly-even and the evenly-odd numbers. Unlike the evenly-even, they cannot be halved back to unity; and unlike the evenly-odd, they are capable of more than one division by halving. The oddly-odd numbers are formed by multiplying the evenly-even numbers above 2 by the odd numbers above one. The odd numbers above one are 3, 5, 7, 9, 11, and so forth. The evenly-even numbers above 2 are 4, 8, 16, 32, 64, and soon. The first odd number of the series (3) multiplied by 4 (the first evenly-even number of the series) gives 12, the first oddly-odd number. By multiplying 5, 7, 9, 11, and so forth, by 4, oddly-odd numbers are found. The other oddly-odd numbers are produced by multiplying 3, 5, 7, 9, 11, and so forth, in turn, by the other evenly-even numbers (8, 16, 32, 64, and so forth). An example of the halving of the oddly-odd number is as follows: 1/2 of 12 = 6; 1/2 of 6 = 3, which cannot be halved further because the Pythagoreans did not divide unity.

Even numbers are also divided into three other classes: superperfect, deficient, and perfect.

Superperfect or superabundant numbers are such as have the sum of their fractional parts greater than themselves. For example: 1/2 of 24 = 12; 1/4 = 6; 1/3 = 8; 1/6 = 4; 1/12 = 2; and 1/24 = 1. The sum of these parts (12+6+8+4+2+1) is 33, which is in excess of 24, the original number.

Deficient numbers are such as have the sum of their fractional parts less than themselves. For example: 1/2 of 14 = 7; 1/7 = 2; and 1/14 = 1. The sum of these parts (7+2+1) is 10, which is less than 14, the original number.

Perfect numbers are such as have the sum of their fractional parts equal to themselves. For example: 1/2 of 28 = 14; 1/4 = 7; 1/7 = 4; 1/14 = 2; and 1/28 = 1. The sum of these parts (14+7+4+2+1) is equal to 28.

Insight

Perfect numbers — those equal to the sum of their proper divisors (6 = 1+2+3, 28 = 1+2+4+7+14, 496, 8128) — are one of the cleaner pieces of mathematics inherited from the Pythagorean school. Euclid proved (Elements IX.36) that if 2^n - 1 is prime (now called a Mersenne prime), then 2^(n-1) times (2^n - 1) is perfect, which generates the formula Hall describes here. Euler in the eighteenth century proved that every even perfect number has this form. Whether any odd perfect number exists is still unknown — one of the oldest unsolved problems in mathematics. The Pythagoreans treated perfect numbers as mathematical reflections of moral and theological perfection, an attitude Boethius and Cassiodorus preserved into the medieval quadrivium.Euclid, Elements IX.36; Dickson, History of the Theory of Numbers.

The perfect numbers are extremely rare. There is only one between 1 and 10, namely, 6; one between 10 and 100, namely, 28; one between 100 and 1,000, namely, 496; and one between 1,000 and 10,000, namely, 8,128. The perfect numbers are found by the following rule: The first number of the evenly-even series of numbers (1, 2, 4, 8, 16, 32, and so forth) is added to the second number of the series, and if an incomposite number results it is multiplied by the last number of the series of evenly-even numbers whose sum produced it. The product is the first perfect number. For example: the first and second evenly-even numbers are 1 and 2. Their sum is 3, an incomposite number. If 3 be multiplied by 2, the last number of the series of evenly-even numbers used to produce it, the product is 6, the first perfect number. If the addition of the evenly-even numbers does not result in an incomposite number, the next evenly-even number of the series must be added until an incomposite number results. The second perfect number is found in the following manner: The sum of the evenly-even numbers 1, 2, and 4 is 7, an incomposite number. If 7 be multiplied by 4 (the last of the series of evenly-even numbers used to produce it) the product is 28, the second perfect number. This method of calculation may be continued to infinity.

Perfect numbers when multiplied by 2 produce superabundant numbers, and when divided by 2 produce deficient numbers.

The Pythagoreans evolved their philosophy from the science of numbers. The following quotation from Theoretic Arithmetic is an excellent example of this practice:

"Perfect numbers, therefore, are beautiful images of the virtues which are certain media between excess and defect, and are not summits, as by some of the ancients they were supposed to be. And evil indeed is opposed to evil, but both are opposed to one good. Good, however, is never opposed to good, but to two evils at one and the same time. Thus timidity is opposed to audacity, to both [of] which the want of true courage is common; but both timidity and audacity are opposed to fortitude. Craft also is opposed to fatuity, to both [of] which the want of intellect is common; and both these are opposed to prudence. Thus, too, profusion is opposed to avarice, to both [of] which illiberality is common; and both these are opposed to liberality. And in a similar manner in the other virtues; by all [of] which it is evident that perfect numbers have a great similitude to the virtues. But they also resemble the virtues on another account; for they are rarely found, as being few, and they are generated in a very constant order. On the contrary, an infinite multitude of superabundant and diminished numbers may be found, nor are they disposed in any orderly series, nor generated from any certain end; and hence they have a great similitude to the vices, which are numerous, inordinate, and indefinite."