This Pythagorean classification of numbers in tern of even numbers and tern of odd numbers should not be confused with the modern classification of odd and even numbers in four classes depending on whether the remainder of the division of a number four is 0, 1, 2, 3 ; attention all the more necessary because the same terminology with the same word means different things in the two classifications, and for instance, the Pythagorean equally odd numbers have the form 2 (2m + 1), while modern have the form 4m + 1. (31)
This classification in terns and this ternary custom, in agreement with the archaic numbering based three that makes the four to be a new unit, leads to a classification in terns of all natural numbers. And in fact there is in Theon the following set of the first nine numbers:
1 4 7 α δ ζ
2 5 8 that is β ε η
3 6 9 γ ς θ
represented in the Theon text by the first nine letters of the greek alphabet, which in their turn were used just as numeral signs of the first nine numbers. In this ennead or tern theme the individual numbers of the first line, divided by three, give as rest the unit, those of the second give as rest two and those of the third give no rest. It may be noted that in this arrangement the only internal number is five, which is always the case for the five, in the disposition of the ten numbers of the decade according to the Tetractys.
Continuing to arrange the numbers in terns, one gets a set of enneads with 27 numbers represented by the 24 letters of the greek alphabet and three episemi, or signs added to the greek alphabetic system of written numeration. The second ennead starts with 10 and the third ends with 27, which is the third power of the three and is therefore a perfect number because it ends the tern of enneads. Continuing again one gets an ennead of enneads whose last number is 81. If we stop to this quadruplet 3, 9, 27, 81 of powers of the three it is composed of perfect numbers in the Aristotelian greek sense of the word.
We found the number 27 in Porphyrius who insists that Pythagoras spent three times nine days in the sanctuary of Zeus in Crete; and that reappears as an object of particular attention by the Egyptian Freemasonry of Cagliostro. In a letter sent to Cagliostro by the Master of the Lodge “Sagesse Triomphante” (32) to give an account of the inauguration of the temple work, there is this passage: ” L’adoration et les travaux ont durés trois jours et par un concours remarcable de circonstances nous étions réunis au nombre de 27, et il y a eu 54 heures d’adoration” “the Worship and work have lasted for three days and by a remarcable combination of circumstances we gathered 27 in number, and there were 54 worship hours.
As for the number eighty one we see it appear in Dante and this time without the usual screen of hierarchies and principalities. According to Dante, the natural life of a perfect man should have a duration of 81 years, and he observes (33) that “Plato lived eighty-one years as testifies by Tullius”; and adds that if Christ had not been crucified would have lived 81 years. As you can see, Dante knew a lot.
Dante has divided his Comedia in three parts each of 33 songs written in triplets each of 33 syllables. He manifests in “De Vulgari Eloquio” the aesthetic reasons which he honors the hendecasyllable, but it may be that the choice of hendecasyllable was due to other reasons as well. The 99, last two-digit number, is a perfect number multiple of three and nine; it is the number of songs of the three parts if you do not assign the first to a particular song and one hundred is the total number of songs. Each song contains 33 as each triplet contains 33 syllables. 33 is the product of 3 to 11, the 99 is the product of 9 to 11; and if it is the sum of the first four powers of the three and the unit, one gets the square of 11, which is the fourth prime odd number.
In this numeration based three, that is in this arrangement of numbers in terns and enneads, the new units are the numbers connective to the powers of the three, namely 4, 10, 28, 82. Of the 4 and 10 we’ve been already dealt with. As for the number 28, firstly it is a perfect number in the sense technically modern and strict to the word because its divisors are 1, 2, 4, 7 and 14 whose sum is exactly 28. For these reasons it was especially considered by the Pythagoreans and we know it in two ways.
The Antologia Palatina (34) retained under the name of the epigrammatist Socrates a dialogue between Polycrates and Pythagoras in which Polycrates asks Pythagoras how many athletes is leading towards wisdom. Pythagoras replies, I’ll tell you, Polycrates: half are studying the admirable science of mathematics, the eternal nature is the subject of the studies of a quarter, the seventh part is practicing meditation and silence, there are also three women whose Theano is the most distinct. That’s the number of my students who are still those of the Muses. The solution to this problem, and the corresponding equation of the first degree is just the number 28; and the way the problem is set shows how to Pythagoras interested precisely to demonstrate that this number was a perfect number.
The other documentation about the number 28 is provided by the Pythagorean underground to the Basilica di Porta Maggiore in Rome. Carcopino in his study on the Pythagorean basilica shows (35) how the members of the Pythagorean brotherhood to which the basilica belonged were in the number of 28, that is based on the observation already made by Mrs. Strong (36) that the funeral stucco of the basilica cell were indeed 28. Without the purely fortuitous discovery of this underground Pythagorean basilica we can not assert with confidence that the 28 is a sacred number in the Pythagorean sacred architecture. Neither Carcopino nor epigrammatist Socrates indicate the reason for the choice of the number 28, it is clearly due to its perfection, and to being equal to the sum of their own dividers, as well as being a new unit in the terns and enneads system.
Other less immediate relations intercede between the quadruplet numbers .: 4, 10, 28, 82. The 28-sided polygon has 350 diagonals that’s to say ten times the number of diagonals of the decagon that has 35 diagonals. Furthermore, the 28th tetrahedral number is ten times the 28th triangular number; and the tenth triangular number is 55 which is at a time harmonic mean and the ratio between the tenth pyramid with a square base which is 1540 and the fourth hexagonal which is 28. Similarly the 82, following the perfect number 81 as the 28 follows the 27, is such that the tetrahedral 82 ° is equal to 28 times the triangular 82 °: So one has the two relations: F (3, 28) = 10 P (3, 28): F (3, 82) = 28 P (3, 82).
These are some of the relations between the numbers: 4, 10, 28, 81. Among the multiple of three six is the perfect number; and its square, the 36, is the only triangular number that is square of another triangular. In fact
and taking into account that the two factors to the first member are two prime numbers together and that the same should happen to the second member, and examining the four possible cases depending on whether x and y are odd or even, is easily found that the only integer and positive solutions are x = y = 1 and x = 8, y = 3. The six is also the only number for which happens that its cube is equal to the sum of the cubes of the three consecutive numbers preceding it. In fact indicating with x – 1, x, x + 1 and x + 2 the four consecutive numbers must be:
which does not admit other real solution than x = 4, and then we have:
If we consider the right triangles in integers, the only one whose sides have for measures three consecutive integers is, as we know, the Egyptian triangle (3, 4, 5) which area has to fit 6; then there are two classes of triangles in integers which it belongs as a first triangle, the Egyptian triangle, 1 – those in which the hypotenuse exceeds of one their larger cathetus, 2 – those in which the larger cathetus exceeds of one the lesser. The first of these two classes is given by the formula
which for each odd value of n provides a right triangle in integer numbers (37).
This resolution is the same as that, in the words of Proclus, was given by Pythagoras. The first triangle given by this formula, is for n = 3 and the Egyptian triangle; the second is for n = 5 and is the triangle (5, l2, 13) whose area is 30; the sum of the areas of the two triangles is 36. The problem of determining a right triangle in which the difference among catheti is equal to one, is a little more difficult and it was solved by the mathematician Girard; the first of these triangles is the Egyptian triangle, the second is the triangle (20, 21, 29) whose area is 210; the sum of the two areas is 216 = 63.
We observe now that the 36 is the eighth triangular number and, at the same time, the sixth square that’s to say we have:
P ( 3, 8 ) = P ( 4, 6 ) = 36
and we observe that the two numbers 3 and 8 are respectively the number of sides of the octahedron face and the number of faces, while the numbers 4 and 6 are similarly the number of sides of the face of the cube or hexahedron and the number of faces; and that these two cosmic figures, cube and octahedron are mutually polar. Similarly, the twentieth triangle, which is 210, is equal to 12ve pentagonal, we have that:
P ( 3, 20 ) = P ( 5, 12 ) = 210
and also this time the number 20 is the number of triangular faces of the icosahedron and 12 the number of the pentagonal faces of the polar polyhedron that’s to say the dodecahedron. In the end for the tetrahedron, that is auto-polar, happens that P (3, 4) = 10.
So these three numbers 10, 36 and 210 are obtained by considering the five cosmic figures, namely the three pairs of polar polyhedra: the auto-polar tetrahedron, the octahedron and cube, the icosahedron and dodecahedron. As for the existing five cosmic figures, that so much takes part in Pythagorean and Platonic geometry and cosmology, does exist then the admirable property: the triangular numbers that have to order the number of faces of a triangular faces polyhedron are equal to the polygonal numbers that have typically the number of sides 3, 4, 5 of the polar polyhedron and to order the number of faces of this polyhedron. That is, the polygonal number that has to gender the gender of the face of the polyhedron and to order n the number of faces a polyhedron remains unchanged, going from the polyhedron to polar polyhedron.
One sees immediately that the sum of these three numbers 10, 36 and 210 is equal to 256 that’s to say to the fourth power of the 4.
while the product of these three numbers decomposed into prime factors contains the four factors 7, 5, 3, 2 high respectively to the first, second, third and fourth power. The bases form the quadruplet of two and the first three odd primes and the exponents are the numbers of Tetractys.