ythagorean 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 modelled 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
Contributions by the Pythagorean s include
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
The golden section is the positive root:
And this was all connected with the construction of a pentagon.
The Pythagorean Pentagram
First we need to construct the golden section. The geometric construction, the only kind accepted, is illustrated below.
Assume the square ABCE has side length a. Bisecting DC at E construct the diagonal AE, and extend the segment ED to EF, so that EF=AE. Construct the square DFGH. The line AHD is divided into extrema and mean ratio.
The key to the compass and ruler construction of the pentagon is the construction of the isoceles triangle with angles and . We begin this construction from the line AC. Note the figure
Divide a line AC into the `section' with respect to both endpoints. So PC:AC=AP:PC; also AQ:AC=QC:AQ. Bisect the line AC and construct on the perpendicular at this midpoint the point B so that AP=PB=QB=QC.
Define PAB and QPB. Then . This implies , and hence . Solving for we, get . Since PBQ is isoceles, the angle QBP . Now complete the line BE=AC and the line BD=AC and connect edges AE, ED and DC. Apply similarity of triangles to show that all edges have the same length. This completes the proof.
The only regular polygons known to the Greeks were the equilaterial triangle and the pentagon. It was not until about 1800 that C. F. Guass added to the list of constructable regular polyons by showing that there are three more, of 17, 257, and 65,537 sides respectively. Precisely, he showed that the constructable regular polygons must have
sides where the are distinct Fermat primes. Recall, a Fermat prime is a prime having the form
Fermat ( 1630) conjectured that all numbers of this kind are prime.
Pierre Fermat (1601-1665), was a court attorney in Toulouse (France). He was an avid mathematician and even participated in the fashion of the day which was to reconstruct the masterpieces of Greek mathematics. He generally refused to publish, but communicated his results by letter.
Are there any other Fermat primes? Here's what's known to date.
By the theorem of Gauss, there are constructions of regular polygons of only 3, 5 ,15 , 257, and 65537 sides, plus multiples,
sides where the are distinct Fermat primes.
The Pythagorean Theory of Proportion
Besides discovering the five regular solids, Pythagoras also discovered the theory of proportion. Pythagoras had probably learned in Babylon the three basic means, the arithmetic, the geometric, and the subcontrary (later to be called the harmonic).
Beginning with a>b>c and denoting b as the --mean of a and c, they are:
The Pythagorean Theory of Proportion
In fact, Pythagoras