Pythagorean Tetraktys and Masonic Delta by Arturo Reghini: “Every whole, or sum of four things, is called Tetractys, after a Pythagorean word, and there are several Tetractys.
But that we now have dealt with is the Tetractys par excellence, which Pythagoreans took their oath under. A Speusippus fragment notes that ten contains within itself the linear variety, plane, and solid number because one is a point, 2 a line, 3 a triangle, and 4 a pyramid (1).
We are continuing our translation from Arturo Reghini “ Sacred numbers in masonic Pythagorean Tradition” 1947. First chapter.
Jew Philo (2), repeating Pythagorean concepts said that four are the limits of things: point, line, surface, and solid, and Geminus says that arithmetic is divided into the linear theory of numbers, the numbers theory, the plane numbers theory, and the solid numbers theory.
Perfection, or the completion of universal manifestation, is achieved with ten which is the sum of the numbers up to four. The decade has it all, like the unit, which contains all potential. The name δεκάς is just for that receptive property δεχάς.
This finding is the result of the limit set for the development of numbers from the three-dimensional quality of space and would lead to the recognition of the same properties of four and ten even if the spoken numbering, instead of being decimal numbering, was, for example, a duodecimal based or based ternary numbering.
As a matter of fact, we note the coincidence. The reason why Greek, Latin, and Italian spoken numbers and so on is decimal, is that humankind has ten fingers, which are of great help in the account (with your finger) so that in writing Latin and ancient Greek unit was represented by a finger identified by the letter I. The last finger is the tenth, and then 10 is perfect. Five has special representation in the two writings, in greek by the initial word of pente, in latin by the palm or open hand span later identified with the letter V, as in the latin world the writing of numbers had been ahead of the knowledge and use of the alphabet; and 10 is represented in greek by the letter Δ, decade initial, and which has the shape of an equilateral triangle. While in Latin is represented by two open and opposite hands, that’s to say by the sign later identified with the letter X. These signs are enough in the greek and latin writing of numbers to write or representation of numbers up to one hundred, which is carried in H of the original greek word Hecaton, and in Latin a sign then identifiable with the initial of centum.
Both Pythagorean Tetractys and spoken numbering highlight the importance of the number ten in completely independent ways. And this is not the only correlation between four and ten because the greek language forms the names of the numbers from ten to 99 using the names of the first ten numbers, introduces a new name to indicate 100, and then a new name to indicate the thousand, finally a new and final name for the tens of thousands. This very word μύριοι, μυρίοι if differently marked, indicates an indefinitely great number. In short, the greek language has only four names, after nine, to designate the first four powers of ten and stops at the fourth power, as the sum of the integers ends with four in Tetractys.The third finding concerning decade (and therefore Tetractys) is as follows: After the unit, that is potentially polygonal, pyramidal, and versatile of any kind, the first number which is simultaneously linear, triangular, and tetrahedral, and then appears in the irradiation of unit and in the simplest form of manifestation and realization of unity, is the number ten. It is the first number that appears simultaneously in the three sequences of linear, triangular, and tetrahedral numbers:
We know only five numbers that have this property, they are 1, 10, 120, 1540, and 7140. The determination of the other numbers that are simultaneously triangular and tetrahedral depends on the solution of the equation obtained by equating triangular x to tetrahedral y, that is on the solution of the indeterminate third degree
with two unknown equations.
equation which we know the five solutions:
but modern mathematics can not determine the other possible integer solutions.
A fourth finding is provided by the observation that the letter delta is the fourth letter of the greek alphabet and takes the shape of an equilateral triangle. Letter D = delta is the fourth letter also in Etruscan, Phoenician and various latin and greek alphabets, and, although the good order of alphabet letters is not an order established by a law of nature, we must not neglect this observation for the value Pythagoreans could annex to it or any part thereof. The decade is therefore the fourth triangular number and tetrahedral third and it is represented, in the writing of numbers, by its initial which is the fourth letter of the alphabet and forms the shape of a triangle.
If you take the fourth triangular its representation is:
which is found in Theon of Smyrna and Nicomachus of Jerash. This representation is a decade symbol, in the etymological sense of the word, that is a meeting of more than one sense. There is a symbol that is triangular or a triangle; it is the fourth triangular, composed of ten points arranged in four lines containing respectively one, two, three, and four points. “Look,” says Lucianus, what you believe are four instead are ten, and they represent the perfect triangle, and our oath” (3).
A fifth finding that is very important, certainly for the Pythagoreans, is obtained from consideration of the musical scale. Modern music uses the tempered scale, which is approximately the natural scale based on the simple ratios principle. While Greeks made use of the Pythagorean scale based on the principle of the fifth. We shall see later, the genesis of this scale, for the moment let’s just note that all three of these scales are made up of seven fundamental notes arranged in the well-known order. Greeks called the octave harmony.
Fundamental notes in this range or octave, whereof law of the other fifth are deduced, are the first, fourth, fifth, and octave, that is the four strings of the Philolaus tetrachord: the first, fourth, or syllable, the fifth or diapente, and diapason. According to tradition, Pythagoras, by observation and experiment, had found that the relationship between the length of these strings and the length of the first was expressed by numerical ratios 4:3, 3:2, 2:1 that’s by the relations between the Tetractys numbers, which are not only simple ratios but also the simplest possible. Philolaus tetrachord shows that, in harmony, appear the same foundation numbers 1, 2, 3, and 4, which appear in Tetractys. “This discovery writes Delatte (4), produced on all minds, especially on the Pythagoreans, an extraordinary effect, that today we can no longer appreciate. Tetractys gave them the key to acoustics mysteries, and they extended to the whole domain of physics the same conclusions. It became a cornerstone of their philosophy of rhythmicity, and we understand how they may have considered Tetractys as the source and root of eternal nature. “
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- Philo, De Mundi opificio, 10, 16, 34.
- LUCIANUS, Vita. auct., 4.
- A. DELATTE, Etudes sur la litterature pithagoriciénne, page 259.
- Ibidem, page 250.