Was an Italian mason, Arturo Reghini (1878-1946), to write one of the most comprehensive documents on masonic Delta and Pythagorean tetraktys.

And that’s despite British masonry having kept its connection with Pythagoreans much more alive than Latin’s. In fact, Old Charges clearly mentioned Pythagoras as the founder of European masonry.

Sacred Numbers, Sacred Geometry and Acoustics. All of them play an important role in the interdicted part of Alchemy keys. While sharing common sources on these topics, nowadays Alchemy and Masonry are revealing different and sometimes conflicting aims. Alchemy is a discipline without brotherhood, Masonry is a brotherhood without discipline. Most misleading beliefs on Alchemy have been spread around by Masonry. Reghini, educated in maths, nom de plume Pietro Negri, although a highly ranked Mason was one of the most Italian prominent hermetic researchers. His aversion to fascism, after an early trusting endorsement, was finally overtly ended. I thought that a serial translated divulgation of his “ I Numeri Sacri nella Tradizione Pitagorica Massonica” or “ Sacred Numbers in Traditional Pythagorean Masonry) posthumously published in Roma 1947, could be of some interest. My translation is as follows:

Chapter 1

The Pythagorean Tetractys and Masonic Delta.

*“Bless us, divine number generating gods and humans, sacred tetraktys, containing the root and source of creation that eternally regenerates”. *Golden Verses.

Restoring the ancient Pythagorean arithmetic work is very difficult, for reports are still scarce and not all reliable. We should at every step and statement cite sources and discuss their value, but this would make our exposure long and heavy and difficult to understand. Therefore, in general, we will refrain from any philology, we will stick only to what has been resulted less controversial and always declare that it is just our opinion or our work.

The Pythagorean ancient and modern literature is very extensive, and so we decide to waive the listing of hundreds of books, studies, articles, and authors of ancient and modern steps constituting it. According to some critics, historians and philosophers, Pythagoras was a simple moralist and would never deal with mathematics, according to some hypercritical Pythagoras would not have existed, but we have for sure the existence of Pythagoras, and, accepting the witness of almost contemporary philosopher Empedocles, we do believe that his knowledge in every field of knowledge was very great. Pythagoras lived in the sixth century before Christ, he founded a school in Calabria and an Order that Aristotle called the Italian school, and taught among other things, arithmetic and geometry. According to Proclus, head of the School of Athens in the fifth century of our era, Pythagoras was the first who raised geometry to the dignity of science and, according to Tannery, geometry comes from the brain of Pythagoras as Athena comes armed with all points from the brain of Jupiter.

But no writings of Pythagoras or attributed to him have come down to us, and it is very possible he did not write anything. If it were otherwise, in addition to remote antiquity, it should be kept in mind that the Pythagoreans kept secret their teachings, or at least some of them. A Belgian philologist, Armand Delatte, in his first work: “ *Études sur la littérature pythagoricienne*, Paris, 1915”, made a most learned criticism of the Pythagorean literature sources, and he has made clear, among other things that the famous “*Golden Verses*”, although a compilation by a neo-Pythagorean of the second and fourth centuries of our era, would allow us to go back almost to the top of the Pythagorean school, for they were transmitting archaic knowledge. This work by Delatte will be our principal source. There are other ancient witnesses in the writings of Philolaus, Plato, Aristotle, and Timaeus Tauromenium. Philolaus was, along with Tarentum Archytas, one of the most prominent contemporary close to Pythagoras, Timaeus was a Pythagorean historian. And the great philosopher Plato is strongly affected by the influence of the Pythagorean and we consider him a Pythagorean, although not belonging to the sect. Much less ancient are Pythagoras’ biographers, that’s to say Iamblichus, Porphyry, and Diogenes Laertius, who were neo-Pythagoreans in the first centuries of our era, along with writers and mathematicians such as Theon of Smyrna and Nicomachus of Jerash. The mathematical writings of the two latter authors are the source transmitting Pythagorean arithmetic. Even Boethius has fulfilled this task. A lot of information we owe to Plutarch.

Among the moderns, besides Delatte and the old work by Chaignet on “*Pythagore et la philosophie pythagoricienne*”, Paris, 2nd ed. 1874, and “ the Word of Pythagoras” by Augusto Rostagno, Turin, 1924, we will make use of the work “ *The Theoretic Arithmetic of the Pythagoreans*”, London 1816, 2nd ed., Los Angeles, 1934, Dr. Thomas Taylor English Greek scholar who was a neo-Plato and a neo-Pythagorean, and among historians of mathematics we will make use of “*Maths in ancient Greece*”, Milano, Hoepli, 1914, 2nd ed., Gino Loria, and his work “ A History of Greek Mathematics” T . Heath, 1921.

In modern mathematics, the unit is the first issue of the natural series of whole numbers. They are obtained starting from the unit and adding another unit later. The same is not true for arithmetic multiplication. In fact, the same word, monad, indicated the arithmetic unit and the monad in the sense that today we would call metaphysical monad, and the shift from universal to duality is not as simple as switching from one to two by the addition of two units. In arithmetic, including multiplication, there are three direct operations: addition, multiplication, and raising to a power, accompanied by the three inverse operations. Now the product of the unit itself is still the unit, and a power unit is still the unit, then only the addition allows the passage from unity to duality. This means that to obtain two we must admit that there may be two units, which means having the concept of two, namely that the monad can lose its unique character, that it can stand out, and that there may be a duplication of units or multiple units. Philosophically there is the question of monism and dualism, the metaphysical question of Being and its representation, and the question of the biological cell and its reproduction. Now if we admit the intrinsic and essential oneness of the Unit, one must admit that another unit can be only an appearance and that its appearance is an alteration of uniqueness that comes from the Monad operating a distinction in itself. Consciousness operates in a similar manner a distinction between self and not self. According to Advaita Vedanta this is an illusion, and indeed is a great illusion, and there isn’t any choice but to get rid of it. It is not an illusion that there is this illusion, even if it can be overcome. The Pythagoreans said that the dyad was generated by moving away or separated from itself, which divided by two: and indicated that differentiation or polarization with different words: diaeresis, tolma.

In math, a multiplication unit was not a number, but it was the principle, ἀρχή’s of all numbers, we say the beginning, not the start. Once admitted the strength of another unit and several units, then from the unit, by addition, the two and then all the numbers result. Pythagoreans conceived numbers as drawn up or composed or variously depicted as points. A point was defined by Pythagoreans as a unit having position, while, in Euclid’s opinion, the point has no parts. The unit was represented by the point (σημεῖον semeion sign or seed) or even, when a written alphabetic system of numerals came in use, from the letter A or α, which was used to write the unit.

Once having admitted the possibility of the unit addition resulting in two, represented by the two endpoints of a segment line, we can continue to add units, and subsequently obtain all the numbers represented by two, three, and four … points aligned. In that way, we thus have the linear development of the numbers. Except the two that can be achieved only by the addition of two units, all integers can be considered either as the sum of other numbers, for example, five is 5 = 1 + 1 + 1 + 1 + 1, or also 5 = 1 + 4 and 5 = 2 + 3. One and two did not have this general property of numbers: and therefore, to the ancient Pythagoreans, as well as unit two can’t be considered a number, but the principle of even numbers. This view was lost over time because Plato ( Parmenides, 143 d) speaks of the two as equal, and Aristotle (Ta Topikà, 2, 137) speaks of the two as the only even prime. The three in turn can only be considered as the sum of one and two, while all other numbers as well as being the sum of several units are also both different from the sum of parts, some of them can be regarded as a sum of two equal parts in the same way that the two is the sum of two units, and are called even numbers because of their similarity with the pair, so for example 4 = 2 + 2, 6 = 3 + 3, etc.. are even numbers, while others, such as three and five which are not the sum of two parts or two equal addends, are called odd numbers. So the triad 1, 2, 3 has the mentioned properties which number greater than 3 don’t.

In the natural series of numbers, even and odd numbers take alternately place one after another; even numbers have in common with two of the peculiarity above mentioned and then they can always be represented in the form of a rectangle (epiped) in which one side contains two points, while odd numbers, like unity, don’t, and when they may be rectangular in shape, it happens that the base and height contain a number of points which in turn is an odd number. Nicomachus also reports a more ancient definition: excluding fundamental dyad, even is a number that is divisible into two equal parts or unequal parts that are either even or odd, or, as we should say, that have the same parity, while odd number can be divided only into two unequal parts, one even and the other odd, i.e. parts that have different parity.

According to Heath (A History of Greek Mathematics, I, 70 ) the distinction between odd and even goes back to Pythagoras of course, which is not hard to believe, and Reidemeister (Die arithmetic der Griechen, 1939, pag. 21.) says that the theory of even and odd is Pythagorean, and this concept makes explicit the mathematical logic of Pythagoreans as well as being the foundation of Pythagorean metaphysics. In Unequal Numbers, says Virgil, Deus Gaudet, God takes delight.

The Masonic tradition conforms to this recognition of the sacred or divine odd numbers, as indicated by the numbers expressing initiation ages, the number of lights, jewelry, brothers members of a factory, etc. Wherever there is a distinction, a polarity, there is an analogy with even and odd pair, and one can establish a correspondence between the two poles as well as equal and odd, so for Pythagoreans, the male was an odd and female peer, the right was odd and the left was equal …

to be continued at Arturo Reghini Sacred Pythagorean Numbers. Part 2 .