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 28, which is based on the observation already made by Mrs. Strong (36) that the funeral stucco of the basilica cell was indeed 28. Without the purely fortuitous discovery of this underground Pythagorean basilica, we can not assert with confidence that 28 is a sacred number in the Pythagorean sacred architecture. Neither Carcopino nor epigrammatist Socrates indicates the reason for the choice of the number 28, it is clearly due to its perfection, and to be 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 are 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; the tenth triangular number is 55 which is at a time-harmonic mean, and the ratio between the tenth pyramid with a square base 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, and 81. Among the multiple of three six is the perfect number; and its square, 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) whose 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, 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 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 the 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 is the number of the pentagonal faces of the polar polyhedron that’s to say the dodecahedron. In the end for the tetrahedron, which 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 take 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. The triangular who have to order the number of faces of the tetrahedron, octahedron, and icosahedron respectively:
P ( 3, 20 ) = P ( 5, 12 ) = 210 = 3 P ( 5, 7 ) = 2 . 3. 7 . = product of 2 and the first three primes of the decade.
It has also:
We do not know if these properties have already been observed by others, sive Deus, sive Dea (either god or goddess).
The numbers of Tetractys appear in some formulas that express the cosmic figures as sums of tetrahedra and also appear in atomic physics in connection with the number of electrons that form the nuclear covering of the rare gas atoms.
We have observed that a pyramidal number can always be expressed as the sum of the tetrahedral numbers. Similarly, it can be shown that the same thing occurs for octahedral, cubic, icosahedral, and dodecahedral numbers; namely, that a polyhedral of order n is always equal to an additive combination of the three consecutive tetrahedral to order n – 2, n – 1 and n, and are the following identities:
formulas that are easy to verify bearing in mind that the first members are given by the following general formulas of polyhedral numbers:
tetrahedral n, 
cubical n 
octahedral n 
n icosahedral 
n dodecahedral 
In the four preceding identities the coefficient of the average term that’s to say of the tetrahedral (n – 1) ° is respectively 2, 4, 8, 16, that’s to say the power of the two which has as exponent the numbers 1, 2, 3, 4 of the Tetractys.
This would happen according to the Platonic constitution of matter. In atomic physics instead, appear the squares of the numbers of Tetractys. And here’s how: If you order the chemical elements according to the laws of Moseley and Mendelejev according to the similarity of their chemical behavior, the first column is occupied by the so-called rare gases, namely helium, neon, argon, krypton, xenon, the emanation of radium. It is found that the number of electrons that constitute their atomic nucleus covering, in the above-written order, which is the natural one according to their atomic weight and the atomic number, is respectively:
2 10 18 36 54 86
The corresponding finite difference or gnomons are then respectively and neatly
2 8 8 18 18 32 that’s to say 2. 12, 2. 22, 2. 32, 2. 42
which is double the squares of the numbers of Tetractys.
We observe that the first four triangles given from the Pythagorean formula (see footnote 39) are: ( 3, 4, 5 ), ( 5, 12, 13 ), ( 7, 24, 25 ), ( 9, 40, 41 ), and in them, the difference, between the hypotenuse and the odd cathetus has precisely the values 2, 8, 18, 32. These triangles have in fact sides
and the difference between the hypotenuse and the odd cathetus n is
- Gino Loria, Le scienze esatte, second publ., Milano, 1914, page 821;
- Federici Cardinalis Borromaei Archiepis. Mediolani, De Pythagoricis Numeris, Libri tres, Mediolani 1627. See lib. II. chap. XXVI, page 116;
- Theonis Smyrnaei Platonici, Expositio rerum mathematicarum ad legendum Platonem utilium, publ. Hiller, Lipsia, 1878, page 4 and page 100;
- Lidus, De mensibus; publ. Lipsia, 1898; IV, 64;
- Porphyrius, Vita Pythagorae, 51;
- Verg., Aen. I, 94;
- See Delatte, Etudes …, 112. Other passages containing the same association terque quaterque are: Verg., Aen., IV. 589; XII, .155; G. I. 411; G. n. 399; Oratius Car. XXXI, 23; Tibullus, 3, 3. 26;
- Dante, Purg. VII, 2;
- Gomperz, Les penseurs de la Grèce, I, 116;
- Erwin Rohde, Psiche, italian version, Bari, 1914; I, 255, note 11;
- Anatolius, περί δέκαδος , 9; Delatte, Etudes …, 122, nota 1;
- Ps. Plutarch, Vita Homeri, 145;
- See Delatte, Etudes …, 120 e 122;
- Porphyrius, Life of Pythagoras, ed. Carabba, Lanciano, 1913, page 57;
- Eduard Zeller, Sibyllinische Blättern, Berlin, 1890, page 40 and follow;
- Adolf Kaegi, Die Neunzahl hei den Ostarien. Separatdruck aus den philologischen Abhandlungen;
- Dante, Par. XV, 97-98;
- See A. Meillet, Aperçu d’une histoire de la langue grecque, Paris, 1913, page 98.
- Homer, Odyssey, XIX, 175;
- Fabre d’Olivet, Les vers dorés de Pythagore expliqués, Paris, 1813, 24;
- Servius, Comm. a Vergil. – Egloga VIII, 75;
- Bungo, Numer. Mysteria, 1591, second ed., page 96;
- Bungo, Numer. Mysteria, 18.5;
- V. Pareto, Trattato di Sociologia generale, I, 499;
- See Abel Rey, La jeunesse de la science grecque, 119:
- Delatte, Etudes … 19 e cfr. Iamblichus, Vita Pithagorae, 114;
- See Proclus, ap. Taylor, I, 148;
- See Apollonius Conicals, ediz. Helberg, Lipsia, 1893, II, 170;
- Nicomachus, Introduction to Arithmetic, II, 8 page 294.
- Taylor, The Theoretic Arithmetic of the Pythagoreans, Los Angeles, 1934, page 243;
- Finally we note as an example of the archaic grouping in terns that in the Greek spoken numeration applies, as in the Italian one, the Handel law, that’s to say the words that express large numbers are formed by dividing the number in groups of three units, the class of units, the class of the thousands, the class of millions, etc;
- See Marc Haven, Le maître inconnu, 154;
- Dante, Conv. IV, 24;
- Anthol. Palatina, XIV, 1;
- Jerome Carcopino, La basilique pythagoricienne de la Porte Majeure, Paris, 1927, pag. 255;
- Eugenie Strong, The stuccoes of the underground basilica near the Porta Maggiore nel Journal of Hellenic Studies, XLIV, 1924, pag. 65;
- This Pythagorean formula is an immediate consequence of the fundamental property that has the square of growing preserving the similarity of the shape. When the gnomon is a square, the two consecutive squares have to difference a square. Now the quadratic gnomons are nothing more than odd numbers, if np odd number is a square, that’s to say if you have 2 n – 1 = m2, the sum of the first odd numbers that precede it is (n – 1) 2 and we have:
and substituting it has
Since m is odd then that is of the form m = 2 p + 1 equal the catheter can be written:
which is four times the p triangular number. This formula expresses then the Pythagorean theorem: the quadruplet of p triangular number and the (p + 1) odd number are the two catheti of a right triangle in integers in which the hypotenuse exceeds of one the even cathetus. It has namely:
for example, for p = 5 has the triangle (60, 11, 61). This formula can be deduced as a special case of the general formulas page 40 and 41 ( Arturo Reghini Sacred Pythagorean Numbers 9) by placing them in m = p + 1 and n = p; In fact, the x becomes:
.
38. (40 in the formula image) The two identities
P ( 3, 8 ) = P ( 4, 6 ) = 36 P ( 3 , 20 ) = P ( 5, 12 ) = 210
say that the 8th triangular is equal to the sixth square and the twentieth triangle is equal to the twelfth pentagonal. The problem of determining a triangle which is also a square was solved by Euler, the indeterminate equation
admits infinite integer solutions given by the double series
x 1 8 49 288 y 1 6 35 204 for which applicants apply formulas
Similarly, Euler (Algebra, ed. Leipzig, page 391) has solved the problem of determining the triangular which is also pentagonal, that’s to say he solved the equation
whose infinite solutions are given by the double series x 1 20 285 3976 55385 … y 1 12 165 2296 31977. . .
for which formulas apply applicants
The triangular corresponding to odd values to the order x are also diagonal numbers, that’s to say they have the form z (z 2 – 1) with z = 1, 143, 27693. . .
The first solution, after the unity, of the two problems is that given by the two identities, that is that connected to the cube and octahedron, and the icosahedron and dodecahedron, and expresses the property that we set out on the cosmic figures.
Previous Chapter Arturo Reghini Sacred Pythagorean Numbers 11 .