Arturo Reghini here extends the Pythagorean arithmetic to those properties which actually exist. The end of third chapter of “Sacred Numbers” presents some Pythagora classics: egyptian triangles and a hypotenuse theorem extension.
My translation from “I numeri Sacri nella Tradizione Pitagorica Massonica” or sacred numbers in masonic pythagorean tradition. The second part of the third chapter on the driad of odd prime numbers within the decade.
A second way to reach the number five is instead suggested by a Plutarch’s consideration. The mention of Plutarch is located in De Iside et Osiris and reconnects to the “Egyptian” triangle rectangle, the simplest of integers 3, 4, 5 triangles. Geometrically the Pythagorean theorem, which applies to every right triangle, says that in a right triangle the sum of the squares created on catheti is equal to the square on the hypotenuse, arithmetically, when the sides of the triangle are integers, it happens that the sum of the squares of these integers is equal to the square that has side to the hypotenuse. In the case of the Egyptian triangle where the short sides are the three and four and five is the hypotenuse, Plutarch presents an analogical interpretation of the Pythagorean theorem to the effect that the five would be the result or fruit of the spiritual action of the three vertically set and symbolizing the male above the horizontal base of the four that symbolizes the female. Thus, the five would come not from integers but from the polygonal numbers and precisely from the square numbers.
The triad of consecutive numbers 3, 4, 5 therefore displays the property that the sum of the squares of the first two is equal to the square of the third. Indeed, it is easy to recognize that this is the only triad of three consecutive integers to have this property; in fact denoting by x – 1, x and x + the three consecutive numbers in the equation
(x -1)² + x² = ( x + 1)²
that is x² – 4× = 0 has only the solutions x = 0 and x = 4. If then instead of squares are considered three consecutive triangular or pentagonal or pentagonal of a same gender r, and one tries, when it happens, that the sum of the first two polygons is equal to the third, is that this fact occurs only when the polygons are squares, one finds that this occur only when polygons are square and namely in the case of the third, fourth and fifth square.
In fact the polygonal x ° gender r is expressed by the formula
and consider the indeterminate equation in the unknowns r and x
P (r , x – 1) + P (r , x) = P (r , x + 1).
It admits no solution other than the solution r = 4, x = 4.
In fact, replacing to symbols their expressions, the equation becomes:
that developing and reducing becomes:
(r – 2) x² – 4 (r – 2) x + r – 4 = 0
and applying the well-known terminate formula of the quadratic equation is obtained
where the discriminant is equal to 5 for r = 3, is equal to 4 for r = 4 and is always between 3 and 4 for any other value of r. We have only a rational entire and positive value of x in r = 4 and is x = 4. Therefore, as in the case of the linear numbers, the only triad of three linear consecutive numbers for which happens that the sum of the first two is equal to the third consists of the triad 1, 2, 3, so in the case of polygonal numbers the same triad of consecutive polygonal numbers of the same kind for which happens that the sum of the first two is equal to the third is constituted by the third, fourth and fifth square that’s to say by the sides of the Egyptian triangle. The Egyptian triangle is presented in this respect as a hypostasis of the fundamental triad 1, 2, 3. With the triad of the numbers 3, 4, 5 takes place in the surface manifestation or epiphany what happens in the linear irradiation for the triad 1, 2, 3. The number five takes third place and replaces the three, as the pentagram or blazing star takes the place of the Delta, or luminous triangle, passing from the first degree Chamber to that of second degree.
The three numbers from this triad 3, 4, 5 are the numbers of the sides of the Egyptian triangle. But it can be shown more generally the following properties: In a right triangle coprime integers always happens: 1) a cathetus is even and the other two sides are odd; 2) The even cathetus is always a multiple of four; 3) The hypotenuse is always the sum of two squares, one even and the other odd, and therefore is of the form 4 n + 1; 4) The hypotenuse is never multiple of three; 5) One of the short sides is always a multiple of three. 6) One of the catheti is always a multiple of five. 7) The perimeter and the area is equal to a multiple of six.
These simple and interesting properties of right triangles in integers can be demonstrated in various ways, but since it is not easy to find these demonstrations together we will give a demonstration that the reader less demanding and mistrustful can jump.
Let’s give first of all the general formulas for right triangles in integers prime among them. Denoting by x, y and z the catheti of the hypotenuse, we have:
y2 =z2 –x2 =(z–x)(z+x)
and the two factors of the second member must both be squares or contain a common factor α. In this second case their sum 2 z and their difference 2 x must have in common this factor α, and as x and z are for the hypothesis prime between them, so 2 x and z 2 can not have other common factor than the one or the two.
It will therefore be:
z + x = α m2 z – x = α n2 con α = 1 , 2
y = α m n
and canceling the common factor α and multiplying by two the three last equalities are obtained for x, y, z the formulas
y=2mn z=m² +n² x=m² –n²
where m and n are numbers of different parity, that is, one even and the other odd, otherwise the three sides would be equal and therefore not prime between them.
The even cathetus y resulta therefore multiple of 4, the hypotenuse and the other cathetus are odd, and so the perimeter is even and the area is even because it is given by the semi-product of catheti.