(BEING CONTINUED FROM 9/10/2015)
The Pentagonal and Hexagonal numbers are shown in the graphs below.
Note that the sequences have sums given by
Similarly, polygonal numbers of all orders are dsignated; this process can be extended to three dimensional space, where there results the polyhedral numbers. Philolaus is reported to have said:
All things which can be known have number; for it is not possible that without number anything can be either conceived or known.
The Pythagorean Theorem
Whether Pythagoras learned about the 3, 4, 5 right triangle while he studied in Egypt or not, he was certainly aware of it. This fact though could not but strengthen his conviction that all is number. It would also have led to his attempt to find other forms. How might he have done this?
One place to start would be with the square numbers, and arrange that three consecutive numbers be a Pythagorean triple! Consider for any odd number m,
which is the same as
Put the gnomon around . The next number is 2n+1, which we suppose to be a square.
It follows that
This idea evolved over the years and took other forms.
Did Pythagoras or the Pythagorean s actually prove the Pythagoras Theorem? Probably not. Although a proof is simple to give, the Pythagorean s had only a limited theory of similarity. And perhaps the reason was that rigor had not yet advanced to that level, at least in the early and middle period. The late Pythagorean s ( 400 B.C) however probably did supply a proof of this most famous of theorems.
Theorem I-47. In right-angled triangles, the square upon the hypotenuse is equal to the sum of the squares upon the legs.
This figure is modeled after the original figure used by Euclid to prove the result. It is known to the French as pon asinorum and to the Arabs as the Figure of the Bride.
More Pythagorean Geometry
- Various theorems about triangles, parallel lines, polygons, circles, spheres and regular polyhedra.
- Work on a class of problems in the applications of areas. (e.g. to construct a polygon of given area and similar to another polygon.)
- Given a line segment, construct on part of it or on the line segment extended a parallelogram equal to a given rectilinear figure in area and falling short or exceeding by a parallelogram similar to a given one. (In modern terms, solve .)
From Kepler we have the quote
“Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.”
A line AC divided into extreme and mean ratio is defined to mean that it is divided into two parts, AP and PC so that AP:AC=PC:AP, where AP is the longer part.
Let AP=x and AC= a. Then the golden section is
and this gives the quadratic equation
The solution is
And this was all connected with the construction of a pentagon.
(TO BE CONTINUED)