(525) how steadily the masters of the art repel and ridicule any one who attempts to divide absolute unity when he is calculating, and if you divide, they multiply 4 , taking care that one shall continue one and not become lost in fractions. That is very true. Now, suppose a person were to say to them: O my friends, what are these wonderful numbers about which you are reasoning, in which, as you say, there is a unity such as you demand, and each unit is equal, invariable, indivisible,—what would they answer? They would answer, as I should conceive, that they were speaking of those numbers which can only be realized in thought. Then you see that this knowledge may be truly called necessary, necessitating as it clearly does the use of the pure intelligence in the attainment of pure truth? Yes; that is a marked characteristic of it. And have you further observed, that those who have a natural talent for calculation are generally quick at every other kind of knowledge; and even the dull, if they have had an arithmetical training, although they may derive no other advantage from it, always become much quicker than they would otherwise have been. Very true, he said.

(1) CONCERNING the secret significance of numbers there has been much speculation. Though many interesting discoveries have been made, it may be safely said that with the death of Pythagoras the great key to this science was lost. For nearly 2500 years philosophers of all nations have attempted to unravel the Pythagorean skein, but apparently none has been successful. Notwithstanding attempts made to obliterate all records of the teachings of Pythagoras, fragments have survived which give clues to some of the simpler parts of his philosophy. The major secrets were never committed to writing, but were communicated orally to a few chosen disciples. These apparently dated not divulge their secrets to the profane, the result being that when death sealed their lips the arcana died with diem.

(89) 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.