The number three and how deeply affected the Greek mentality. And not only. Arturo Reghini’s Sacred Numbers in Masonic Pythagorean Tradition. Chapter five: The number and its powers.

“*Pythagoreans assigned to the supreme God the perfect ternary number where there is a start, middle, end*“. Servius, Commentary to Virgil – Eclogue VIII, 75.

What we have so far exposed reconnects undoubtedly to the Pythagorean school, and only to the Pythagorean school. But there are also other more archaic elements, that the Pythagoreans found, accepted, assimilated, and even exalted, although they are independent of the development of arithmetic multiplication which focuses on the consideration of tetraktys. These items relate to three, its multiple, its powers, and immediately consecutive numbers.

We have already said that the spoken Greek numeration was decimal, such as ours, in which the powers of ten and ten represent the units of a higher order. But the spoken Greek numeration, and so the Sanskrit and Latin, to be confined to these, shows traces of a spoken numeration based on three-putting a powerful seal to the very mentality of people and justified, if not determined, the Pythagorean predilection for the number three.

The echo of this predilection has come down to us; three is universally regarded as the perfect number for excellence, the popular mottos: omne Trinum est perfectum (in the three is perfection), there is no two without three, are inspired by this concept. In the Pythagorean and Neo-Pythagorean school, writes Loria (1), there existed the general motto that any collection of things had to admit a division into three categories “; Aristotle gives the Pythagorean judgment that everything concludes with the ternary number which is inherent in all things, and found that the three s frequently “inter sacra” ( among sacred) ; and Cardinal Borromeo, in a rare and little-known work of his (2), quotes that Aristotle’s note. As for Plato, he begins the Timaeus, the Pythagorean dialogue par excellence, with the words: “one, two, three.” The three, writes the neo-Pythagorean Theon from Smyrna (3) in his “statement of the mathematical things useful to the reading of Plato,” is the first (number) to have a beginning, middle, end; and Lidus (4) wrote nearly the same thing. And the Alexandrian Pythagorean Porphyrius (5) says that “there is something in nature that has a beginning, middle, end, and to indicate that form and nature the Pythagoreans suggested the number three.”

The number three is the end of this triplet or triad and the Indo-European nomenclature numbers show that anciently when counting it was the last number, and then started again. In fact, the Latin Quator or Quater means etymologically “et tres” (and three) because “qua” was the Latin enclitic; The Sanskrit catur has exactly the same origin. In Greek one of the enclitics is τε which appears in Aeolian τέτορες and Doric τέττορες and this structure is also found in Italian in the words caterva, quaterna, and quaderna. Proof of this connection between the three and four is supplied from the frequency of the expression τρὶς ϰᾳὶ τετρᾲϰΊς in Greek and in Latin of the corresponding expression “terque quaterque”, for example in Virgil’s passage “O terque quaterque beati (blessed)” (6), which according to Macrobius is imitated from a passage of Homer in which the author of Theologumena Arithmetica (7) finds a mystical sense. Even Dante continues the tradition by saying (8): “They were repeated three and four times”; and this archaic combination of the three and four agrees with that of the Pythagorean Tetractys which has representation for the equilateral triangle, that is the Delta which is the fourth letter of the alphabet. Three is in a certain way the last number, and so the perfect number for excellence, and thus in the spoken numbering system ternary-based the four is a new unit, as it is ten in the decimal system; and the two Number Four and Ten, of which we have seen the connection in Tetractys, are also associated because of the establishing of a new unit respectively in the two numbering systems.

Even grammar helps to give the three special importance because are many grammatical ternary distinctions although some of them may be intentional work of grammarians and therefore more consequence than a cause of the number three excellence. However, language precedes grammar and the distinction of the three grammatical numbers, the three genera, and people, it is an artificial distinction wanted by grammarians. We observe that the three also serve the greek language in the superlative formation: τρισμάκαρες means most blessed, and τρισμέγιστος which means great are formed like the French très grand.

Of course, the tern of triads, that’s to say the number nine, the product of three times three, is for this reason, as noted by Dante, a most perfect number: and it is not surprising that the three and nine have great importance in worship and magic. According to Gomperz (9), the number three sanctity is already encountered in Homer every time a trinity of Gods gets unified in the same invocation, for example, Zeus, Athena, and Apollo. Ancestor worship honors especially under the name of Tritopatores, or fathers trinity, the father, grandfather, and great-grandfather. “Nine, writes Rohde (10), as it is easy to see, is especially in Homer a round figure; in ancient times it was very common and normal a division of time periods according to groups of nine. ‘The Pythagorean Anatolius (11) cites the verse of Homer (Il. V, 160) to prove that Homer recognized special value to the number nine. And the pseudo-Plutarch notes Homer seems to show a special predilection for the number three (12) and recognizes a special value to the number nine, and points out the fact about the verse IL. XV, 169, verse that for the same reason it is also detected by Lidus (13) and by the anonymous author of Theologumena Arithmetica.

Porphyrius (14), narrating Pythagoras’ trip to Crete, says that Pythagoras climbed up to the antrum called the veiled by black wool, and there according to the rite passed three times nine days and saw the throne which was annually set up to that God. “So we see appearing the twenty-seven as well, the third power of the three; and Porphyrius, who well knew of course that three nine is equal to twenty-seven, insisted on the ritual and sacred behavior of this period of time, doubly sacred because composed of three enneads.

Zeller (15) lingers over the continued use of the number three in Greek funeral ceremonies; and Adolf Kaegi (16) long disserts on the three and nine in mortuary ceremonies in India, Iran, Greece, and Rome. Many of these customs have come down to us going from paganism to Christianity, and the Catholic liturgy offers us an example in the Triduum, the Novena, and ceremonies for the thirtieth day after death. The Roman calendar has the day of the reference in Nonae, which is the ninth day before the Ides. In the Middle Ages still existed the temporal hours, and Dante speaks in the Vita Nova and recalls them (17) in the verses: “Florence, within the ancient circle so she took still her tierce and nones.” The ninth was noon; and it is a voice that still lives in English noon and in some Italian dialects, for example, Barbarani uses it in his poems in the Veronese vernacular.

This veneration for the number three and number nine, so deeply rooted in language, customs, and the Greek mentality, has contributed to the Pythagorean tradition of distinguishing a theme in every collection of things; especially as Cotrone, home of the Pythagorean school founded by Pythagoras, was a Dorian colony, and the oldest institutions common to Dorians is the division into three tribes (18). The ships of Dorians were counted by multiples of three and Dorians were qualified as τριχάιρες, that’s to say three tribes; the qualification was old because Homer too speaks (19) of the triple Dorians.

Of course, this veneration of the number three is not a Pythagorean peculiarity; the Far East tradition, for instance, exposes the Tao-te-king with the formula: “One has produced two, two produced three, three produced all numbers”; and Fabre d’Olivet (20) observes that this doctrine is elegantly exposed in the so-called Zoroaster Oracles: The ternary shines everywhere in the universe and the monad is its principle. But this veneration is accentuated in by Pythagorean in correspondence to its arithmetic character; “The Pythagoreans writes Servius (21), assigned to the supreme God the perfect ternary number which is beginning, middle, end.” According to Bungo (22), the ternary is almost a return to the One and the origin. Bungo notes (23) the ancient theologians primarily worshiped three Gods, Jupiter, Neptune, and Pluto, sons of Saturn and Rhea. After the unit Saturnia, Bungo says, that’s to say the union of the intelligible world where all things are implied, they divided the sense world into three regions; Heavenly ruled by Jupiter, media by Neptune, underground by Pluto; So you have three brothers, three kingdoms, three scepters and all three tripartite. We have then the three Furies: Alecto, Tisiphone, Megaera; the three Harpies: Aello, Ocypeta, Celano; the three Fates: Clotho, Lachesis, Atropos. Pareto (24) recognizes the application of this tradition in the Capitoline triad and in the triple sign by which almost every deity flaunts her/his power, Jupiter’s triple thunderbolt, Neptune’s trident of, Pluto’s three-headed dog. Christianity has the Holy Trinity, the three Magi, and their threefold offering, the three crosses of Calvary. But by Pythagorean, the veneration of three assumes a very special significance, because the Pythagorean characteristic is precisely the key function recognized to the number. As for Freemasonry, we have already seen what significance has the number three among the Freemasonry sacred numbers.

A distinction in three categories, which dates back to Pythagoras himself, is the three lives one (25) which Aristotle uses in Ethics, namely, the theoretical life, practice, and apolaustic. Heraclid, shortly posterior to Plato, says that Pythagoras was the first to make this distinction; as those who came to the Olympic games could be divided into three classes: those who were going to buy and sell, the ones who were going to participate in races and those who went simply to observe, so men could be divided into corresponding three classes. Rey, who is usually inclined to attribute the posterior Pythagoreans back to what the ancients attributed to Pythagoras himself, speaking of this distinction, recognizes “how it is impossible to doubt the coming from the very beginnings of the school.”

I decided to publish this paragraph alone because of the importance I have recognized in it. Reghini translates and comments the last two Pythagoras golden verses to demonstrate how he meant for them to be somewhat threefold practiced. Not only, but Reghini takes the opportunity to point out how both incorrect and “sanitized” the version, of the original Greek, of many contemporary scholars and hermetists.

Another important distinction between the three categories is as follows: “The Pythagoreans writes Delatte (26), divide the reasonable beings into three categories: the man, the deity, and a being of an essence intermediate as was Pythagoras.” Three classes have also divided the members of the Pythagorean association; which, according to Iamblichus, were those of novices, mathematicians, and physicists. Other names are given by other writers, but the division is always threefold.

In geometry, the Pythagoreans distinguished three kinds of angles: acute, straight and obtuse which ascribed to three species of divinity (27), and three species of triangles (28): equilateral, isosceles, and scalene. They knew that a plane filled with regular polygons is only possible with three species of polygons: triangle, square, and hexagon; and they knew that there are three regular polygons that make up the faces of the five regular polyhedra or cosmic figures. And, although no Pythagorean geometry text has survived, it is symptomatic that Euclid’s Elements could suddenly begin with the consideration of the equilateral triangle; one may suspect that this traditionally happened also before the geometry of the Pythagoreans. And in music we have seen the importance of the three progressions mentioned by Archita, the arithmetic, geometric and harmonic progression with their three medium; and as the whole octave or harmony is an extension of the Philolaos tetrachord, which is constituted by three strings do, fa, sol ( C, F, G) and the harmonic of the first.

In arithmetic, we have already seen that the Pythagoreans divided the numbers into perfect, elliptic, and hyperbolic numbers. Likewise, the rectangular, or epiped, numbers were divided into square, heteromech, and promech and so did the Pythagoreans distinguishing three classes of even numbers and of three classes of odd numbers.

Nicomachus (29) distinguishes between even numbers: 1st – numbers equally even, that’s to say the powers of two; 2nd – the equally odd numbers, that’s to say the form 2 (2m + 1) numbers; 3rd – the numbers not equally even, that’s to say the numbers of form 2n (2m + 1) with n ≥ 2. The three categories are composed of the numbers:

numbers equally even: 4, 8, 16, 32… numbers equally odd: 6, 10, 14, 18… numbers not equally even: 12, 20, 24, 28…

The classification exhausts all the possibilities, and the even numbers of the third class are those not belonging to the other two. The classification of the even numbers resembles that of rectangular numbers like the heteromech are distinguished from promech because the difference between the sides, in the case of heteromech, is only one point and more points in the case of promech, so the numbers are equally odd contain only a factor two, while the not equally equal contain in addition to the odd factor times the factor of two.

We note that Euclid in Book VII called equally even the product of two even factors, but this does not conform to the Pythagorean tradition and Iamblichus blames Euclid for this definition and, as reported by Taylor (30), Asclepius in his manuscript commentary on the first book of Nicomachus, says this Euclid’s definition to be incorrect because with it one get even numbers and not numbers equally even.

To this ternary classification of even numbers another also ternary did correspond for odd numbers according to Nicomachus, Iamblichus, and Theon testimonies; but it has been badly transmitted. Nicomachus distinguished: 1 – odd primes; 2 ° – secondary and synthetic numbers as the 9, 15, 21, 25, 27, 33 … which are products of two or more prime factors even not distinct; 3rd – the numbers that are secondary and compounds in themselves but primes respect to another number such as 25 and 9. It is obvious that the second class contains all the numbers that do not belong to the first, and in the example reported by Nicomachus, both 25 and 9 belong simultaneously to the two classes. It is, therefore, necessary to return the Pythagorean ternary classification of the odd numbers; and we seem to be able to do this by the following: Noting that, in the Pythagorean ternary classification between the unit and the number there are only the two; that, similarly in the classification of rectangular numbers, between the square and the promech there is only the heteromech (which has only one more point in one of its sides) and similarly in the case of ternary classification of even numbers between the number equally even 2n and the number not equally even 2n (2 m + 1), where with n ≥ 2 there is only the equally odd 2 (2 m + 1) where the factor two is unique, the ternary classification of odd numbers was likely to be the following: 1 odd prime; 2nd the prime factors powers of which at least two distinct; 3rd powers of a single odd prime with exponent at least equal to two. That’s to say: odd prime numbers a; powers of a single prime and with n at least equal to two; other cases of odd numbers in which there are at least two distinct prime factors. 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 has the form 4m + 1. (31)

This classification in terns and this ternary custom, in agreement with the archaic numbering based on three that makes the four to be a new unit, leads to a classification in triplets 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 triplet 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 episemes, 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 at 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 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 testified 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 into three parts each of 33 songs written in triplets each of 33 syllables. He manifests in “De Vulgari Eloquio” the aesthetic reasons for which he honors the hendecasyllable, but it may be that the choice of hendecasyllable was due to other reasons as well. The 99, the 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, 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 are 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 Pythagoras is interested precisely to demonstrate that this number was a perfect number.