Pythagoras

Theory of Numbers.—The scientific doctrines of the Pythag orean school have no apparent connection with the mysticism of the society or their rules of Their discourses and speculations all connect themselves with the idea of number, and the school holds an important place in the history of mathe matical and astronomical science. Aristotle tells us that the "applied themselves to the study of mathematics and were the first to advance that science ; insomuch that, having been brought up in it, they thought that its principles must be the principles of all existing Pythagoras is said to have at tached supreme importance to arithmetic, which he advanced and took out of the of commercial utility. He also made etry a part of a liberal education, examining the principles of the science and treating the theorems from an immaterial and in tellectual standpoint.

discovery was, perhaps, that of th

e de pendence of the musical intervals on certain arithmetical ratios of of string at the same tension, 2:1 the octave, 3 :2 the fifth and 4 :3 the fourth. This discovery could not but have powerfully contributed to the idea that "all are num bers." According to Aristotle, the theory in its form re numbers, not as relations predicable of but as actually constituting their essence or substance. Numbers, he says, seemed to the Pythagoreans to be the first in the whole of nature, and they supposed the elements of numbers to be the elements of all and the whole heaven to be a musical scale and a number (Metaph. A. a). Later, e.g., in the of "Philolaus," things are spoken of, not as being numbers, but as having number and thereby knowable.

The development of these ideas into a comprehensive meta physical system was probably the work of Philolaus. The "ele ments of numbers" referred to by Aristotle were, to the the Odd and the Even, which they identified with the Limit and the Unlimited. The Unlimited, and therefore the Limit also, was conceived as spatial. Numbers were thus spatially and "one" was identified with the point, which was thus a unit having position and "two" was simi larly identified with the line, "three" with surface, and "four" with solid. The Odd and Even and the Limit and Unlimited were the first two of a set of ten fundamental oppositions postulated by the the remaining being : one and many, right and left. male and female, rest and motion, straight and curved, the Universe was in a sense the realization of these opposites. The further speculations of the on the subject of number rest mainly on fanciful analogies. Thus "seven" is called rapOevos and because within the decade it has neither factors nor products; "five," on the other hand, is marriage because it is the union of the first masculine and the first femi nine number 2, unity not considered a number) ; "one" is identified with reason because it is unchangeable; "two" with opinion because it is unlimited and indeterminate; "four" with justice because it is the first square number, the product of Aristotle has the remark that Eurytus, who was a disciple of Philolaus, used to assign numbers to all sorts of such as horses or men, and imitated their shapes by ar pebbles after the manner of those who bring numbers into the forms of triangles or squares (Metaph. N. 5, 1092 b io).

This us to figured numbers and the connection between numbers and The "holy tetractys," by which the later Pythagoreans used to swear, was a of this kind _ • _ • _ rep the number io and at a its composition as 1+ 2+3+4. To add a row of five dots gives the next number with 5 as the side, and so on, that the sum of any number of the series of natural numbers be with 1 is a number. The sum of any number of the series of odd numbers with 1 is similarly seen to be a square ; thus 3 and 5 added successively to I a of The successive odd numbers after I were called because the addition of each of them to the sum of the ones with 1) makes a number into the next If the added to a square is itself a square number, e.g., 9, we have a square number which is the sum of two squares; thus 1+3+5+7 =16 or 42, and the addition of 9 ( = 25 or 52, that is, 32+42=52. himself is credited with a formula for two numbers the sum of which is also a square, namely (if m is any odd number), \ This connects itself with the theorem of the square on the hypotenuse of a which tradition universally associates with the name of Pythagoras. This being a property of all consideration would show that, while some such triangles have their sides in the ratio of rational numbers, some, and in particular the isosceles, have not. In the case of the isosceles right-angled the ratio of the hypotenuse to either of the other sides is what we write as V2, which is "irrational" in the sense that its value can not be expressed exactly as a ratio between numbers. It was the who discovered the irrationality of V2, and not only so, but they discovered the law of formation of the series of "side"—and numbers (as they were called), which sat isfy the equations = +I and so enable us to find, as x and y increase, closer and closer approximations to the value of V2. The law depends on the proposition in Eucl. ii. 1 o to the effect that (2x+y)2— 2 (x+y)2= 2x2--y2, whence it follows that if x, y satisfy one of the above then 2x+y, x+y is a solution in numbers of the other equation. From this we derive +, and so on ad infinitum as approximations to V2.