An appendix of Clavicule de la Science Hermétique displays a symbolical, though not arithmetical, rendition of Mercurial partitions and transmutations.
This second part is entitled “Cabala of the Pythagorean Table”; in fact, a kind of Pythagorean Tetraktys is presented. I found two slightly different versions of this document. Let’s have a look at both.
You can find an introduction and the first part of Clavicule de la Science Hermétique at Clavicule Science Hermétique and the Know Yourself Search.
“Cabala of the Pythagorean Table
or I
The Chaos or the Matter, the number I first. & the circle O through which the Soul
I I
of the World O or the Sun through R the Moon its Matrix B gives
(the second version: of the World O or the Sun through 
the Moon its Matrix 
gives)
I I I
The Spirit, Soul, and Body
These three principles of nature, being yet intellectual, give a perfect construction of the four Elements after adding the first number.
I I I I
PONM
That’s to say + or our Earth. (the second version adds 
)
Hence this divine number, inscrutable magic ternary, after having won the two by means of four, moves gloriously towards the first, where it becomes perfect.
Hence the ineffable.
The Tetragram of Greeks & the Schemhammerphorasch of Jews.
Of which every line expresses the name of the Lord, and since it is composed of pure vowels, and no term in the world can be pronounced without a vowel, in the same way, nothing in the world can neither exist nor subsist without God; from which comes this Egyptian motto: Jupiter fills all ( the author uses remplit).
The same Pythagoras wanted to signify the same thing through this divine table Which he had brought from Egypt or Palestine.
These unities together give the number X. Perfect, which is anatomically dissected in the middle, and the left horn being erected, give the letter L, but the two horns together give the letter V, and in total, the letter X, whence comes LVX, which is a single word by which it has pleased God to leave some idea of him to man.
Corollaries
The Geometer learns from there, the squaring of the circle, and the perpetual motion, knowledge through the movement of the four elements.
The Arithmetician uses the production of substantial numbers by means of purely intellectual things.
The Grammarian uses the origin of letters.
The End “
It is clear enough that this interpretation of the Pythagorean Tetraktys is purely symbolical and not arithmetical. That’s to say the author does take the drawing of Tetraktys to represent the alchemical principles of Mercurius-Sulphur-Salt within the four Elements and not points in the space. You can see an arithmetical, as long as mathematical, rendition of the Tetraktys at Arturo Reghini & Pythagorean Tetraktys 2, where I have translated in the English language the essay of the Italian mathematician, as well as freemason and researcher in esotericism, Arturo Reghini on the sacred numbers in the Pythagorean tradition ( at least what has survived till nowadays). Also, the mention of Schemhammerphorasch seems functional for alchemical purposes.
The author of Clavicule de la Science Hermétique did glamorize his concepts by adding some pieces of “esoteric” knowledge, as it was commonly accepted and also rewarded in the early eighteenth century. He employs this syncretism to explain better, or better hide, the “cabala,” as it was often called the secret language of an alchemist (1). But let’s try to give a sense to his diagrams: giving to the basic letter I a Mercurius valence, or our basic alchemical unit of Secret Fire from Stars and Suns and still dwelling in everybody, you have our foundation, our Chaos, our Spirit of Life, our first volatile extraction from matter or coming from the sky.
I I is, as in arithmetics, equal to 1 + 1 = 2. That’s to say, one Mercurius plus another Mercurius ( or a Mercurius partitioned) is symbolically equal to Sulphur. Or a Mercurius, once fixed in the first meaning of Sulphur ( since for the rule of three, every symbol in Alchemy has at least three meanings) or Mercurius Philosophorum, is ready to double and get married to a companion Mercurius Philosophorum to give a Sulphur or Soul.
I I I is the rendition of the product of this union: a new life. In this case, 1 + 1 = 3, because they together become a different thing, a child. The Mercurius/Secret Fire, after having mated with itself, is ready to give birth to a new itself. The single Mercurius Spirit of Life has extracted the Soul Sulfur and given birth to Salt, that’s to say Mercurius plus Sulphur, that’s to say the same Mercurius become Sulphur, is now Spirit of Life plus a Soul. They are a ” ready to be” new body; they are Salt, but not a new body yet.
I I I I is the “fixation” of the child/Salt in a new “earthy” body. A real new body is being born ( since a Spirit of Life plus a Soul cannot but give life to a new Body). The density stairs of elements have been achieved from the more ineffable Fire to the denser Earth. We have dissolved the raw matter in the preparatory work, and now we have to solidify the extracted part in a fixed-embodied one in the last work ( see an Opus magnum scheme). Sun is here synonymous with Sulphur/Soul, Moon of Mercurius/Spirit of Life (2). The Cross is a symbol of fixation (3).
This is the squaring of the circle (fixation), and the perpetual motion (extraction of Secret Fire/Mercurius), knowledge through the movement of the four elements ( the new Body, the new Earth). This is also the production of substantial numbers by means of purely intellectual things. Concerning the origin of letters, it is clear to be a secret I doubt the same author may have the knowledge of, but simply to evoke it.
Concerning the LVX letter formation, it is clear that the Tetraktys was in no way conceived to make a latin word ( LVX is, in fact another way to express the term LUX or light). Try instead to give numerical values to the letters: L is the latin for 50, V for 5, and X for 10. So 50 + 5 – 10 ( since a smaller number before a greater one makes a subtraction). The result is superior to 50 but inferior to 60. very often, the days it takes the light to come out of the dark during a putrefactive phase.
is the symbol of the Sun + Moon, or Mercurius becomes Sulfur.
is the symbol of our Red Earth or “Adam”.
is the symbol of the last perfect Sulphur.