From four on all numbers can be obtained by addition of all the terms others than unit. The geometric representation of numbers in a three-dimensional space is perfect and end with number four, and since the sum 1 + 2 + 3 + 4 = 10 is the new unit of the decimal numbering system, and consequently the perfection of four and ten and the tetractys symbol. Thus Pythagoreans did not specially deal in numbers larger than ten that were expressed in language and writing by ten and previous numbers and for this reason, perhaps, they reduced to first nine numbers the numbers greater than ten by the consideration of their pitmene or bottom, substituting to them the remaining of their division for nine or the very nine when the number was a multiple of nine: remaining easily obtained through the well known rule of the division remaining for nine.

Since the development of the numbers by addition ends with four, one must now consider the development or generation of numbers by multiplication. It seems sure that pythagoreans did actually appeal to this canon of distinction, because number seven was sacred to Minerva since as Minerva it was a virgin and not generated, that is it was not a factor of any number (within the decade) and was not a factors product. Thus numbers are divided into numbers that are not products of other numbers, or prime numbers, and numbers that are products. Taking into account only the numbers within decade, numbers are divided into four classes: the class of prime numbers within decade, that are factors of decade numbers: and they are the two (which really is not a number), but appears as a factor of 4 of 6 of the 8 and 10, three that is a factor of 6 and 9; and 5 which is a factor of 10. Second class consists of the prime numbers less than 10 which are not factors of numbers less than 10, and is represented by only seven. Third class consists of the composed numbers, less than ten, and that factors of numbers less than 10, and is made only by the number four, which is at the same time the square of the two and a factor of 8; the fourth class is formed by composite numbers less than 10 which are compounds of other numbers without being factors within the decade, it is represented by six, eight and nine, because 2 · 3 = 6, 2 · 2 · 2 = 2 · 4 = 8 and 3 · 3 = 9. Not counting 10 and taking into account two we have four prime numbers: 2, 3, 5, 7 whose only one does not produce other numbers, and four composite numbers: 4, 6, 8, 9 of which only one is factor.

It is worth to note that this pythagorean canon of distinction for numbers classification within decade coincides perfectly with the traditional canon of distinction compliant with Vedanta for the fourfold classification of twenty-five principles or tattwa, precisely the first principle (Prakriti) which is not a production but it is productive, seven principles (Mahat, Ahamkara and the 5 tanmatras) which are both products and productive, 16 principles (the 11 indriya, including Manas and the 5 bhuta) which are unproductive productions, and finally Purushawhich is not neither production nor productive. We refer the reader about the exposure that makes René Guenon in his “The Man and his becoming according to the Vedanta”, Bari, Laterza, 1937. This same principle of distinction affects, as noted by the Colebrooke (Essais sur la Philosophie des Hindous, trans. Pauthier), the division of nature, made in Scotus Erigena De divisione Naturae, who says: “The division of Nature I believed to be established in four different species, of which the first is what creates and is created, the second is what is created and in turn creates: the third that is created and does not create, and finally the fourth what is not created nor creates. “Of course it is not appropriate speaking of derivation, however, Pythagoras, chronologically precedes, not only Scotus Erigena but Shankaracharya too. So that leaves the established traditional character of the pythagorean doctrine of numbers.

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