A third finding concerning decade (and therefore tetractys) is as follows: After 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, 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.
but which modern mathematics can not determine the other possible integer solutions.
A fourth finding is provided by the observation that 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 an 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. 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.
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 which is triangular or a triangle; it is the fourth triangular, is composed by 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 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 were 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, 4, which appears 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 of 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.