The fourth and last number of the decade which is obtained through the multiplication is the number nine which is the square of number three, or a number that was considered perfect for reasons we shall see, and that even today is by definition taking into account to be a perfect number for example by puzzlers. Nine is the last monadic number, that’s to say in modern terms it is the last single-digit number; with this number ends the numbers ennead, the number ten being a new unity, so it is perfect according to pythagoreans. An ancient writer, Pseudo Plutarch, enumerates the reasons for the perfection of number nine (8), saying, “The number nine is perfect because it is the first odd square, and it is odd since it is divided into three triads which again are divided into three units”.

Dante in his Vita Nova, that’s to say a life which is after the initiation life, undertakes a persistent use of this “perfect number nine ” (9), noting that this number was a friend of Beatrice, and that he was in ninth grade when he saw Beatrice for the first time, and he arrives to identificate Beatrice with this number. He explains the reasons which, partly at least, justify its consideration of number nine, and why (10) “this number was a friend of the glorious woman of my mind, who was called Beatrice by many”, and says that “the more subtle thinking, and according to the infallible Truth, this number nine used to be she herself, by way of similitude”. The number three is the root of nine, however, that is without any other number, it’s nine for itself, as we see clearly that three times three is nine”. According to Dante the criterion of perfection is the traditional one, pythagorean criterion, and he himself specifies the description quoting (11) Aristotle when saying in the seventh book of physics on perfection: “Each thing is almost perfect when it touches and adds its own virtues, and then it is almost perfect according to its nature”. Of course, he ends up to explain the division of spiritual beings into three hierarchies and principalities, each composed of three tiers, in order to give the chrism of orthodoxy to its design, but the fact is that the nine heavens idea do not belong to the Jewish tradition and Christianity adopted it here, as for other things, a pagan idea and more especially a pythagorean conception. In short, for Dante, the number nine is a power of three, does not include other factors that the number three, it is a direct Trinity manifestation, its power, which is the glorious woman that dominates the mind of Dante, or Domina, and no doubt those of all the other “true lovers” who called her with the same name.

To better understand this Dante’s numerical symbolism one should expose the interpretation of the writings on Dante and medieval literature in general, according to Gabriel Rossetti, Ugo Foscolo, Giovanni Pascoli and in particular Luigi Valli, and we can not do much more than referring back the reader. In Masonry the number nine has special significance when determining the age of initiation in various degrees of Scottish Rite, and it is the basis for calculating any architectural design, for the plotting table on which such calculations are carried out is divided into nine boxes and it was precisely for this reason called tiercel boad or tripartite board by the ancient Freemasons. In the third grade legend nine brothers go in search of Hiram, exploring three by three of the east, the south and west, to find themselves on the ninth day of searches at a given place in the north.

We will see later, when dealing with ternary numbers, the archaic reasons that have made the number three a perfect number in the classical Aristotelian sense of the word. The number nine, the power of three, it is also perfect for the same reasons, also it is perfect for being the last digit of a number, and because it is the last number of the quaternion of composed numbers contained within a decade.

We have thus found a second tetractys, that composed by numbers 4, 6, 8, 9. We can observe that the sum of the first four numbers composing the first tetractys was ten, while the sum of this second tetractys components is 27, that’s to say the cube of three, another perfect number which we should deal. Of course the first tetraktys is the tetractys par excellence. It can identify the decade, symbolized by the equilateral triangle, or the letter Delta, which was in the sanctuary of Delphi. In fact, the Catechism of Acousmatics, the archaic traditional pythagorean sect, asks: (12) What is there in the sanctuary of Delphi?” and the answer is, “The tetractys because in it is harmony, inside are Sirens”. We will see later the significance of this somewhat curious response; for the moment be content to see how inside the sanctuary of Delphi, from which takes origin the motto “Know thyself”, there was the same symbol of tetractys and Delta Masonic appearing in Freemason temples. Evidently, awareness of self and knowledge of tetractys have some relationship to each other.

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- A number was named promeco if the difference between points on one side and consecutive exceeded one. For example 15 was a promeco;
- A number having a rectangular shape was called eteròmeco if containing only one point more than the consecutive, for example 20 was an eteromeco;
- Matila C. Ghyka, “L’esthétique dans les arts et des proportions” see by the same author, “Le Nombre d’Or”, 1931;
- See A. Delatte, Etudes, page 159;
- See Delatte, Etudes, 152, 156;
- See Delatte, Etudes, p.. 196, 202, 212, 216;
- Four is also the only number that is composed by decade and at the same time is factor of a number of the decade, in fact 2 . 2 = 4 and 4 . 2 = 8;
- Ps. Plutarch, De vita et Homer’s poems, 145;
- DANTE, Conv II, 6, and Vita Nova XXIX;
- DANTE, Vita Nova, XXIX;
- DANTE, Conv , IV, 16;
- See Delatte, Etudes, 276.