geometry, or science, without any scientific precedent or a “scientific method.” How
would
one begin? The Pythagoreans turned to the world itself and followed up on the suspicion that there was something special about the numbers 1, 2, 3, and 4 that appeared in the musical ratios.Those numbers were popping up in another line of investigation they were pursuing.
They had at their fingertips a simple but productive way of working with numbers. Maybe at first it was a game, setting out pebbles in pleasing arrangements. Most of the information about “pebble figures” and the connections with the cosmos and music that the Pythagoreans found in them comes from Aristotle. He knew about Pythagorean ideas of “triangular numbers,” the “perfect” number 10, and the
tetractus
.
The dots that still appear on dice and dominoes are a vestige of an ancient way of representing natural numbers, the positive integers with which everyone normally counts. Dots and strokes stood for numbers in Linear B, the script the Mycenaeans used for the economic management of their palaces a thousand years before Pythagoras, and also in cuneiform, an even older script. Pebble figures were a related way of visualizing arithmetic and numbers, but they seem to have been unique to the Pythagoreans.
By tradition, Pythagoras himself first recognized links between the pebble arrangements and the numbers he and his colleagues had discovered in the ratios of musical harmony. Two of the most basic arrangements worked as follows: Begin with one pebble, then place three, then five, then seven, etc.—all odd numbers—in carpenter’s angles or “gnomons,” to form a square arrangement. *
Or, begin with two pebbles and then set out four, then six, then eight, etc.—all even numbers—and the result is a rectangle.
That is easier to understand visually than verbally, one reason to use pebbles.
Pythagoras and his associates were alert for hidden connections. The pebble figures of the square and rectangle dictated a division of the world of numbers into two categories, odd and even, and this struckthem as significant. It was a link with what they were thinking of as the two basic principles of the universe, “limiting” and “limitless.” “Odd” they associated with “limiting”; “even” with “limitless.”
Another way of manipulating the pebbles was to cut a triangle from either the square or the rectangular figure.
In the line of pebbles that then forms the diagonal or hypotenuse of the triangle, the pebbles are not the same distances from one another as they are in the other two sides, nor are they touching one another. Having all the pebbles in all three sides of a triangle at equal distances from their immediate neighbors, or all touching one another, requires a new figure: Set down one pebble, then two, then three, then four, with all the pebbles touching their neighbors. The result is a triangle in which all three sides have the same length, an equilateral triangle. Notice that the four numbers in this triangle are the same as the numbers in the basic musical ratios, 1, 2, 3, and 4, and the ratios themselves are all here: Beginning at a corner, 2:1 (second line as compared with first), then 3:2, then 4:3. The numbers in these ratios add up to 10. The Pythagoreans decided 10 was the perfect number. They also concluded that there was something extraordinary about this equilateral triangle, which they called the
tetractus
, meaning “fourness.” The
tetractus
was, in a nutshell, the musicalnumerical order of the cosmos, so significant that when a Pythagorean took an oath, he or she swore “by him who gave to our soul the
tetractus
.”
Most scholars think it was after Pythagoras’ death that the Pythagoreans found they could construct a tetrahedron (or pyramid)—a four-sided solid—out of four equilateral triangles, and they probably knew this by the time Philolaus wrote the first Pythagorean book in the second half of
would
one begin? The Pythagoreans turned to the world itself and followed up on the suspicion that there was something special about the numbers 1, 2, 3, and 4 that appeared in the musical ratios.Those numbers were popping up in another line of investigation they were pursuing.
They had at their fingertips a simple but productive way of working with numbers. Maybe at first it was a game, setting out pebbles in pleasing arrangements. Most of the information about “pebble figures” and the connections with the cosmos and music that the Pythagoreans found in them comes from Aristotle. He knew about Pythagorean ideas of “triangular numbers,” the “perfect” number 10, and the
tetractus
.
The dots that still appear on dice and dominoes are a vestige of an ancient way of representing natural numbers, the positive integers with which everyone normally counts. Dots and strokes stood for numbers in Linear B, the script the Mycenaeans used for the economic management of their palaces a thousand years before Pythagoras, and also in cuneiform, an even older script. Pebble figures were a related way of visualizing arithmetic and numbers, but they seem to have been unique to the Pythagoreans.
By tradition, Pythagoras himself first recognized links between the pebble arrangements and the numbers he and his colleagues had discovered in the ratios of musical harmony. Two of the most basic arrangements worked as follows: Begin with one pebble, then place three, then five, then seven, etc.—all odd numbers—in carpenter’s angles or “gnomons,” to form a square arrangement. *
Or, begin with two pebbles and then set out four, then six, then eight, etc.—all even numbers—and the result is a rectangle.
That is easier to understand visually than verbally, one reason to use pebbles.
Pythagoras and his associates were alert for hidden connections. The pebble figures of the square and rectangle dictated a division of the world of numbers into two categories, odd and even, and this struckthem as significant. It was a link with what they were thinking of as the two basic principles of the universe, “limiting” and “limitless.” “Odd” they associated with “limiting”; “even” with “limitless.”
Another way of manipulating the pebbles was to cut a triangle from either the square or the rectangular figure.
In the line of pebbles that then forms the diagonal or hypotenuse of the triangle, the pebbles are not the same distances from one another as they are in the other two sides, nor are they touching one another. Having all the pebbles in all three sides of a triangle at equal distances from their immediate neighbors, or all touching one another, requires a new figure: Set down one pebble, then two, then three, then four, with all the pebbles touching their neighbors. The result is a triangle in which all three sides have the same length, an equilateral triangle. Notice that the four numbers in this triangle are the same as the numbers in the basic musical ratios, 1, 2, 3, and 4, and the ratios themselves are all here: Beginning at a corner, 2:1 (second line as compared with first), then 3:2, then 4:3. The numbers in these ratios add up to 10. The Pythagoreans decided 10 was the perfect number. They also concluded that there was something extraordinary about this equilateral triangle, which they called the
tetractus
, meaning “fourness.” The
tetractus
was, in a nutshell, the musicalnumerical order of the cosmos, so significant that when a Pythagorean took an oath, he or she swore “by him who gave to our soul the
tetractus
.”
Most scholars think it was after Pythagoras’ death that the Pythagoreans found they could construct a tetrahedron (or pyramid)—a four-sided solid—out of four equilateral triangles, and they probably knew this by the time Philolaus wrote the first Pythagorean book in the second half of
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