In chapter two of Pythagorean Tetraktys by Arturo Reghini we are discovering how number six is Aphrodite’s and why, four, six, eight and nine are perfect numbers.

This second chapter of “ Sacred numbers in masonic Pythagorean Tradition” 1947 is entitled ” The quaternion of composed and synthetic numbers”. Arturo Reghini starts it presenting an excerpt by Apuleio’s De Magia : *Non ex omni Ligno, ut Pythagoras dicebat, debet Mercurius Exculpi*“

or not from any wood, as Pythagoras said, Mercurius has to be carved.

In the development of numbers from the unity and passing through the line segment, triangle and tetrahedron, we had to stop at number four because the addition of a new point outside the three-dimensional space is not possible for the human insight. We can continue to consecutively add units, but we can only do so while remaining in the field of linear, plane and solid numbers; and, considering for example linear numbers, there is lack of any criterion to distinguish the various properties of numbers.

Moreover, instead of generating numbers by addition, one may then considers the generation of numbers through the operation of multiplication. Indeed, the same terminology, traditionally remained unchanged, is clearly indicating that numbers were regarded as produced by this operation. Following this method the distinction immediately presents itself in two and four classes of numbers: the class of prime numbers or *protoi* or asynthetic that cannot be obtained by multiplication, and the class of synthetic or secondary numbers, or compounds that are the product of other numbers called their factors. Since two is not a number and even numbers are always the product of two for another factor, prime numbers are always odd numbers; and the odd numbers which are not *promechi* (1) or squares are prime numbers. The first decade, not counting the two, contains only three prime numbers: three, five and seven, which means the sacred numbers of apprentice, teacher and fellow freemason.

The numbers in the decade that can be obtained by multiplication are four: four, six, eight and nine. Four is the product of two factors equal to two, which is a square, and since the square is obtained by a consecutive addition of gnomons, which are nothing more than the odd consecutive numbers, and since the squares in their growing keep the similarity of form, so according to Pythagoreans in a certain way, they retain their character of odd numbers, even if their basis is not odd, because they differ from other rectangular numbers which in geometric development do not preserve the similarity of form. For example the *eteromechi* (2) numbers are obtained by adding the number two after a gnomon square shaped like a square where one side contains a point more

and one has the numbers: 2, 2 · 3 = 6, 3 · 4 = 12, 4 · 5 = 20. . . . . . n (n + 1). These numbers have always different form, because two *eteromechi* numbers as n (n + 1) and m (m + 1) can not be balanced by the same sides of one and the other, that’s to say can not happen to be n: m = (n + 1) (m + 1) because it should be m (n + 1) = n (m + 1) and then n = m. And the same goes for rectangular or oblong numbers called *promechi* which are products of two factors that differ by a number r any different from one as 8, 15, 24. . . m (m + 2); for it can never happen that the number m (m + r) is similar to the number n (n + r), that’s to say m: n = (m + r) (n + r) being in that case mr = nr and so m = n. And a number can be Promeco and eteromeco in various forms and all of in a dissimilar way, for example 45 = 5 · 9 = 3 · 15; and 16 = 4 · 4 = 2 · 8. Therefore, the square numbers are the only growing rectangulars retaining the similarity of form. The number four, after the one, is the first of these numbers. In this way it escapes the imperfection of even numbers: We know indeed that it is perfect because with four it ends the development of the number one in space and one obtains the decade.

As for the number six is an *eteromeco* number, but it is also a perfect number, and precisely it is the first perfect number in the modern sense. Perfect numbers are those numbers which are equal to the sum of their divisors (except the number itself). In fact, the divisors of six are: 1, 2, 3 and we have: 1 + 2 + 3 = 6. From this point of view Pythagoreans distinguished three kinds of numbers: the perfect numbers, elliptic numbers and hyperbolic numbers. The perfect numbers were those for which the sum of divisors was exactly equal to the number itself, the elliptical numbers or deficient those for which the sum of divisors was less than the number itself, and the hyperbolic numbers or abundant those for which the sum of divisors exceeded the number itself. For example, the number 15 is an elliptical one because the sum of its divisors is 1 + 3 + 5, which is less than 15, while 12 is a hyperbolic number because the sum of its divisors is 1 + 2 + 3 + 4 + 6 = 16 which exceed 12. Of course, the perfect numbers are relatively rare and the ancients only knew four, and namely the first four: 6, 28, 496, 8128. There are no known odd perfect numbers, those that are must end for 6 or 8, and are given by Euclid’s formula 2r – 1 (2r – 1) if the exponent r is prime and so also is the first factor 2r – 1; and since we do not know a general rule for recognizing the prime numbers, the study of perfect numbers presents formidable difficulties and we know a little more than a dozen of them. The number six is perfect in this, even modern, sense, of the mathematical term.

Another property of the number six is this: the sum of its factors is equal to their product, for example 1 + 2 + 3 = 1 · 2 · 3 = 6. Indeed this is the only case where this happens for three consecutive positive numbers. In fact, denoting by x the medium term, it should be (x – 1) + x + (x + 1) = (x – 1) x (x + 1) or 3 x = x³ – x, and dividing by x, it must be 3 = x² – 1 or x² = 4 and x = 2. The number two is therefore the only solution: and so the only set of three consecutive numbers whose sum is equal to the product and consists of the triad 1, 2, 3 or from the monad, the dyad and triad.

As the number six results from the multiplication of the number two, the female principle, for the first odd number that is male, it was called *gamos* and was the Pythagorean number sacred to Aphrodite. Note here that, according to Matila G. Ghyka, a rumanian author of two very valuable works, widely used and of great interest especially for the study of sacred architecture, the pythagorean number sacred to Aphrodite would be the five and not the six (3). Matila G. Ghyka bases his belief on a single excerpt by Nicomachus, who also points at an analogy (4) between the number five and Aphrodite, because five is the sum of two and three, but that is a simple analogy recognized only by this late Pythagorean, while the number six is identified with Aphrodite not only by the same Nicomachus (5), but it corresponds to Aphrodite also according to other ancient writers, like Lidus, Moderatus and San Clemente (6). Matila G. Ghyka takes no account of this fact, rather it does not even brings notice, and says that number five was according to Pythagoreans the symbol of procreation and marriage, and this false allegation is used to assign a Pythagorean character to a theory of his, as a matter of fact to his central theory, whereas the five is the symbol and the number of organic life and the six is the symbol and the number of inorganic life. He presents this theory as a pythagorean theory that Pythagoras had overshadowed by just taking number five as the number of Aphrodite. This is not true, and, whatever may be the philosophical and scientific value of Ghyka theory, this is a theory of this modern writer, not a Pythagorean theory. The pythagorean number for Aphrodite is six and not five.

The third issue of the decade, which is obtained by multiplication is the number eight. It is a *promeco* number because 8 = 2 · 4, but is the first cubic number in space, such as the four was the first square in the plane. Now the cubic numbers have a representation in space similar to that of squares in plane as well as growing while preserving the similarity of form. So even the cubes whose edge is an even number, and in particular the number two, are beyond the general character of the even numbers (7).