The evenly-even numbers possess certain unique properties. The sum of any number of terms but the last term is always equal to the last term minus one. For example: the sum of the first and second terms (1+2) equals the third term (4) minus one; or, the sum of the first, second, third, and fourth terms (1+2+4+8) equals the fifth term (16) minus one.

In a series of evenly-even numbers, the first multiplied by the last equals the last, the second multiplied by the second from the last equals the last, and so on until in an odd series one number remains, which multiplied by itself equals the last number of the series; or, in an even series two numbers remain, which multiplied by each other give the last number of the series. For example: 1, 2, 4, 8, 16 is an odd series. The first number (1) multiplied by the last number (16) equals the last number (16). The second number (2) multiplied by the second from the last number (8) equals the last number (16). Being an odd series, the 4 is left in the center, and this multiplied by itself also equals the last number (16).

The evenly-odd numbers are those which, when halved, are incapable of further division by halving. They are formed by taking the odd numbers in sequential order and multiplying them by 2. By this process the odd numbers 1, 3, 5, 7, 9, 11 produce the evenly-odd numbers, 2, 6, 10, 14, 18, 22. Thus, every fourth number is evenly-odd. Each of the even-odd numbers may be divided once, as 2, which becomes two 1’s and cannot be divided further; or 6, which becomes two 3’s and cannot be divided further.

Another peculiarity of the evenly-odd numbers is that if the divisor be odd the quotient is always even, and if the divisor be even the quotient is always odd. For example: if 18 be divided by 2 (an even divisor) the quotient is 9 (an odd number); if 18 be divided by 3 (an odd divisor) the quotient is 6 (an even number).

The evenly-odd numbers are also remarkable in that each term is one-half of the sum of the terms on either side of it. For example: [paragraph continues]

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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. 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.

__________________________ [paragraph continues] 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.

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