An interesting snippet transcribed from a 1980 recording by Dr. Leslie Ralph, a well known expert on Pythagoras in his day.
“The concept of number was an assemblage of discrete units. The quantity of units determined its particular value, and the abstract ideas of numerical quantitative relationships did not occur to the practical minds of the early Greeks.
Pythagoras experimented with beads and stones and came upon the morphological structure of number.
He found that the natural sequence of numbers such as one, two, three or four and onwards always formed a triangle.
Once again, if we take the sequence one, three, five, seven and onwards, it always formed a square.
And yet again, if we take one, four, seven, 10 and onwards, they form the pentagon and so on until one reached a polygon of 10000 sides, which would bring the value of PI to a four figure accuracy.
The significance of discovering was clearly displayed in showing the units in their natural sequence. One two three four, which revealed the triangle. This assemblage became a symbol for the morphology of number, the quality essential for man's cognizance and understanding of the universe.
Pythagoras extended this discovery into a real doctrine by linking it with the concept of harmony so dear to his heart. The discrete unit, which formed the basis of number, was, in his mind, the ultimate particle of substance.”
*A quadratic morphology*
Pythagoras here considers the series 1, 1+d, 1+2d, etc., for the cases d=1, 2, & 3. The nth term is 1+(n-1)d. For d=1 the sum of the first n terms is (n^2)/2 + n/2, for d=2 it is n^2 and for d=3 it is (3/2)n^2 -n/2, for d=4: 2(n^2)-n. Note that the coefficient of n^2 in each case is d/2. In fact the general formula for the sum is (d/2)n^2 – ((d-2)/2)n or (d/2)(n^2-n)+n, a general formula for figurate numbers as we call them. The second derivative w.r.t. n is just d, the constant difference in the arithmetical series we started with. Perhaps the pentagram gains its "occult potency" from pentagons being "generated" by a constant difference "engine" of step 3- a surprising idea which presents itself naturally here. Again the sequence 3,5,7 suggests a square, the sequence 2,3 a triangle and 1,4 a pentagon/gram. Pythagoras doesn't consider the case where d=0 - then 1,1,1 would form a straight line.
*I have not been able to find out much about Leslie Ralph, of Wimbledon. His old house is still a doctor's surgery. If anyone has a biographical info I'd love to see it (email link on home page here).