On the Geomantic Parts of Fortune and Spirit

Whether it’s Tarot, geomancy, runes, or any other kind of art, I consider divination in general to be a process of three basic steps:

  1. Hash out, refine, and formally ask the query.
  2. Perform the divination to manipulate the symbols into a readable format.
  3. Interpret the reading.

In geomancy, that second step is the whole process of developing the four Mothers and the rest of the chart from them.  After the querent and I refine the query sufficiently and settle on the final form of the question to be asked, and once I manipulate my tools (cards, dice, or whatever) to come up with the four Mother figures, I then proceed to draw out the entire geomantic chart with all the relevant information I’d need to start with.  Once that’s done, this is what my scribbling and scratching typically ends up like:

The exact process I follow to arrive at this mess of lines and symbols from which I divine the fates and facts of the world is this:

  1. Draw out the four Mothers, then the Daughters, Nieces, and Court.
  2. Label the terminals for the Via Puncti with the elemental glyphs above the Mothers and Daughters, where possible.
  3. Draw out a simple square house chart, and populate it with the first twelve figures of the Shield Chart.
  4. Count the number of odd points in the House Chart to find the Part of Spirit, and label it (I use a circle with two diagonal lines coming out of the bottom like legs, for which I can’t find a compatible Unicode glyph that looks similar enough, but Chris Brennan suggests using an uppercase Greek letter phi Φ, for which I like using the specific glyph U+233D “APL Functional Symbol Circle Style” ⌽).
  5. Based on the Part of Spirit, label the coordinating house for the Part of Fortune (⊕).
  6. Based on the sum of odd points from calculating the Part of Spirit, add the odd points of the Court to find the odd point sum of the Shield Chart.
  7. Find the difference between the odd point sum of the Shield Chart and 64, double it, and add that to the odd point sum to find the Sum of the Chart.

You can see the different steps I took broken down by the above list fairly clearly as I did them (orange, red, green, yellow, pink, blue, cyan):

Making the Shield and House Charts is nothing special for us at this point, and I’ve discussed the Via Puncti before on my blog.  The Sum of the Chart is also fairly common knowledge, whereby you sum up all the points of the sixteen figures in the Shield Chart and compare it to 96 to determine how fast or slow the situation will resolve; again, it’s something I’ve discussed before.  Still, it might surprise you that I don’t actually calculate it directly, but base it on my calculations of the Part of Spirit (due to the mathematics of geomancy, the method works out to the same result).  Likewise, I don’t calculate the Part of Fortune directly, but also base it on the Part of Spirit.  So what gives?  What are these Parts, how are they calculated, and how are they used in geomancy?

First, let’s go with the more well-known of the two Parts, the Part of Fortune.  How do we find this indication?  From Christopher Cattan’s book The Geomancie (book III, chapter 21):

The question being made, after that we have judged by the houses, figures, angles, companions, aspects, the way of point, and by all the other sorts and manners before said, now resteth it to judge by the Part of Fortune.  The Part of Fortune figures, which afterwards ye must divide into twelve parts, and that which remaineth give unto the figures.  As if there rest two ye must give into unto the second figure, if there do remain four to the fourth figure, if there be six to the sixth figure, if there be eight to the eighth figure, if there be ten to the tenth figure, if there be twelve to the twelfth figure.  As by example, if the figure be of 72 points, or 84 or 96 or 108 points, then the part of fortune shall go into the twelfth.  But if the said points of the figure made, being divided by twelve, there do remain but two, as if there remain seventy and four where there remaineth but two, then (as before we have said) ye must give that unto the second house, and there shall be the Part of Fortune.  The which if the figure and house be good (for both the one and the other must be looked upon) you shall judge good, and if it be evil ye shall also judge evil; and so likewise shall ye do of all the other figures.  But if the figure be good, and the house ill, or contrary, the house good and the figure ill, you shall judge the said Part of Fortune to be mean.  And, to end ye may the more easier know the place where the figure falleth, which is called the Part of Fortune, ye shall mark it with this mark, 🌞, and thereafter ye shall judge all the question by the example that followeth. …

Many do use another manner to find Part of Fortune, in taking all the points as well of the twelve houses as the two Witnesses, and the Judge, which they do part by twelve (as is aforesaid) but because I have found no truth therein I will speak no more thereof.

If the mark Cattan proposes shows up as an embarrassingly incongruous sun emoji (like it does for me), then that’s just how it appears on your browser.  I’m using the Unicode character U+1F31E “Sun with Face” glyph as the closest approximation without overlapping with the usual glyph for the Sun (☉) for the symbol from the original text (fourth line, first character):

From Robert Fludd’s Fasciculus Geomanticus (book II, chapter 2):

Of the discovery of the part of fortune, and its placement in schemata.

Now the part of fortune ⊕ is to be discussed.  The part of fortune is of great importance in the view of the Geomancers just as in the view of the Astrologers, and is of great consideration: for in their view the sign of ⊕ and the steps to discover the Hyleg are chiefly considered, and through them the house, into which [the part of fortune] falls into as a result of the projection, truly seizes great life and energy by itself.  …

This part of fortune is to be considered with the utmost exactness, for if it falls into a good house and figure, it is of no small weight for bringing about judgment; if truly in an evil [house and figure], it brings about no meager impediment to judging [the schema].

Fludd then goes on to give other methods of calculating similar things “if the above method is seen to be obscure”, but the phrase “Part of Fortune” doesn’t appear, and he mostly focuses on ways of constructing entirely new charts for the purpose of a clearer judgment.

Lastly, the description of the Part of Fortune from John Michael Greer in his Art and Practice of Geomancy (chapter 6) on the Part of Fortune:

… The Part of Fortune, as the name implies, indicates a house from which the querent can expect good fortune to come in the situation.  In financial divinations it usually refers to a source of ready cash.

What about the Part of Spirit?  To start with, calling it that is my own innovation.  In the extant geomantic literature, it’s more commonly called the Index.  JMG discusses it since it appears in Fludd and Cattan, and though I’m unsure if it appears any earlier, Cattan is the one who (as far as I’m aware) introduced it (book III, chapter 18) by calling it one of the ways to find “the point of instruction”:

Another rule [to know for what intent a chart was made for] is to take all the uneven points of all the twelve figures, and give one to the first, one to the second, one to the third, and so consequently unto all the others, until that all the points be bestowed, and then if the last point remain on the first house, it signifieth thereby that the person hath desired to have that figured be made upon some of the demands which be of the first house; if it rest upon the second, it signifieth that the question or demand of the movable goods, or other things contained in the second house; and so shall you judge of the other houses where the point doth stay.  And if it do happen that the point of the intent do stay in the house of the thing demanded, or in the fifth, ye must judge according to the significations that the Judge doth show unto you; and when ye will judge by the same Judge, you must also take the uneven points of the Witness and the Judge, and bestow them amongst them; but that rule which is only by the 12 houses, is the better, more sure and certain. …

Fludd basically says the same thing (book II, chapter 3) and even with the same name in the chapter header (“De punctis instructionis…”), so I won’t translate it here.  As for JMG, he calls it the method the “projection of points”  and the resulting figure the “Index” (chapter 6):

… This can ferret out hidden factors in the chart.  Projection of points is done by counting up the number of single points in the first twelve figures of the chart, leaving the double points uncounted.  Take the total number of single points and subtract 12; if the result is more than 12, subtract 12 again, and repeat until you have a number less than 12.  If the final number is 0, this stands for the twelfth house.

The house identified by the projection of points is called the Index, and represents the hidden factor at work in the situation. …

Okay, enough reciting from resources.  Based on all the above, the methodology for finding the Part of Fortune goes like this:

  1. Add up the number of all points in the twelve houses of the House Chart.
  2. Divide by twelve.
  3. The remainder points to the house of the Part of Fortune.  If the remainder is 0, then it points to the twelfth house.

The Part of Spirit’s method is nearly identical, except instead of counting all the points, we count just the single points.  For example, given the figure Acquisitio, if we’re counting all the points in it, we have six points, but if we’re just counting single points, then we only have two.  Thus, if (for either sum) we get 88, we divide that by 12.  That gets us 7.333…, so our remainder is 4 because 12 × (7.333… – 7) = 4; phrased another way, 88 ÷ 12 = 7 + 4/12.  Thus, we look at the fourth house for the given Part for which we’re doing a calculation.

Before continuing on with how we use these indications in geomancy, it’s probably best to talk about what a Part even is.  The Parts (also sometimes called Arabic Parts or Lots) are an old doctrine in astrology, dating back to at least the time of Ptolemy’s Tetrabiblos and seen in both Arabic and European astrological treatises since.  At least 97 were in use in the ninth century according to the Arabic astrologer Albumassar, over a hundred listed by the Italian astrologer Bonatti in his works, and more were developed since then, even in our modern era incorporating the outer planets past Saturn.   The Parts are constructed points in a horoscope based on the sums and differences of other observable points (e.g. Ascendant or Midheaven) or physical objects (e.g. planets or luminaries).  In essence, a Part is a mathematical harmonic between different astrological notes that describes certain in-depth areas in a querent’s life or situation that could, in theory, be sussed out by looking at the planets and their aspects alone, but are more explicitly specified by their corresponding Part.

For instance, if we’re looking at indications of someone’s mother, we could look at the ruler of the fourth house in a chart, or we could look at the Part of the Mother, which is calculated as follows:

Mother = Asc + Moon – Saturn

In other words, we start from the Ascendant, add the ecliptic longitude (the position in the Zodiac) of the Moon, then subtract the ecliptic longitude of Saturn.  Thus, in a horoscope where we have the Ascendant at 25° Scorpio, the Moon at 19° Gemini, and Saturn at 3° Taurus, then our calculation would look like this:

(25° Sco) + (19° Gem) – (3° Tau)
205° + 79° – 33°
251°
(11° Cap)

With those points as above, we end up with 251° on the ecliptic, which in zodiacal notation is 11° Capricorn, which is the degree of the Part of the Mother.  This is strictly a mathematical point, much like midpoints are in modern astrology, but used specifically to determine the presence, state, and effects of one’s mother (or all mothers) in a horoscope, and can then be interpreted like any other planet in the horoscope, except that they only receive aspects instead of making them.

While the technique isn’t as popular as it once was, even today many modern astrologers take note of the Part of Fortune.  From Bonatti’s Liber astronomiae (translated by Robert Zoller in The Arabic Parts in Astrology):

This part signifies the life, the body, and also its soul, its strength, fortune, substance, and profit, i.e. wealth and poverty, gold and silver, heaviness or lightness of things bought in the marketplace, praise and good reputation, and honors and recognition, good and evil, present and future, hidden and manifest, and it has signification over everything.  It serves more for rich men and magnates than for others.  Nevertheless, it signifies for every man according to the condition of each of those things.  And if this part and the luminaries are well disposed in nativities or revolutions, it will be notably good.  This part is called the part of the Moon or the ascendant of the Moon, and it signifies good fortune.

The Part of Fortune is a weird part, because it actually has two formulas to calculate it, only one of which is used depending on whether the horoscope is that of a day chart (Sun above the horizon) or a night chart (Sun below the horizon):

Day Fortune: Ascendant + Moon – Sun
Night Fortune: Ascendant + Sun – Moon

Later in Liber astronomiae, Bonatti describes the Part of Spirit, which he also calls the Part of the Sun or the Part of Things to Come, as follows:

The pars futurorum signifies the soul and the body after the pars fortunae and the quality of these, and faith, prophecy, religion, and the culture of God and secrets, cogitations, intentions, hidden things and everything which is absent, and courtesy and liberality, praise, good reputation, heat, and cold. …

In other words, if the Part of Fortune describes the material well-being (or lack thereof) of a horoscope, then the Part of Spirit describes the spiritual well-being; just as the Part of Fortune describes our connections to the world outside us, the Part of Spirit describes the connections of the world inside us.  Fittingly enough, the calculation for the Part of Spirit is the reverse of the Part of Fortune: while the Part of Spirit also uses two formulas, one for day and one for night, the formulas themselves are switched from the Part of Fortune:

Day Spirit: Ascendant + Sun – Moon
Night Spirit: Ascendant + Moon – Sun

Thus, the Part of Fortune and Part of Spirit are intimately connected by how they’re calculated; if you know the location of one, you know the location of the other.

Bringing the notion of the Part of Fortune into geomancy from astrology necessitated an obvious conceptual change in how it’s calculated; without degrees or the ability for certain things to fall among them, it would normally have been impossible to calculate any Part.  However, Cattan either invented or learned a way to find an equally-significant sign in geomancy by adapting the methods available to us in geomancy by counting the points and divvying the sum of the House Chart among the houses.  What none of the older geomancers seem to have noticed is that there’s an intimate relationship between the Part of Fortune and the Index in geomancy: if you know the location of one, you know the location of the other.

First, note that the Part of Fortune and the Index can only fall in even-numbered houses (e.g. house II, house IV, house VI, etc.) due to the mathematical intricacies of geomancy; this is true for similar reasons and with similar logic for why the Judge of a geomantic chart must always be an even figure.  (Why Cattan makes this explicit for the Part of Fortune but suggests wrongly that the Index can be in odd houses is a mystery to me; perhaps he simply didn’t anticipate that a calculation based on odd points could result in only even numbers.)  Thus, by performing the calculations of the Part of Fortune and Index, we can get only one of six numerical results: 2, 4, 6, 8, 10, and 0 (with 0 signifying that the sum in the calculation was evenly divisible by 12, and thus indicates the twelfth house).

After many charts of calculating the Part of Fortune and Index separately, I noticed a pattern emerging: the sums of the two separate calculations for the Part of Fortune and Index always add up to 12 (2 + 10, 4 + 8, 6 + 6, 8 + 4, or 10 + 2) or 24 (12 + 12).  Thus, if the Part of Fortune were in the eighth house, then because 12 – 8 = 4, I knew immediately that the Index would be in the fourth house; if the Index were in the sixth house, then the Part of Fortune would also need to be in the sixth house; if either indication was in the twelfth house, so would the other indication.  Again, if you know the location of one, you know the location of the other.

The mathematics behind this relationship can be described like this: if there are four rows in each figure and we’re looking at a collection of twelve figures, then there are 4 × 12 = 48 total rows.  Each row must be odd or even, and the number of odd rows plus the number of even rows must equal 48.  Plus, we know that since the houses of the Part of Fortune and Part of Spirit must both add up to 12 or 24, both of which are evenly divisible by 12, then we know that the sum of all the odd points plus all the points total must also be evenly divisible by 12.  We can check this mathematically as follows.  First, in mathematical notation, let us use the % sign to represent the modulo function, which is “the remainder after dividing by a number”.  Thus,

x = number of odd rows in the House Chart
x = number of points in the odd rows of the House Chart
x % 12 = remainder of x divided by 12 = Part of Spirit

y = number of even rows in the House Chart
y + x = 48
y = 48 – x

2y = number of points in the even rows of the House Chart
2y + x = number of all points in the House Chart
2 × (48 – x) + x
96 – 2x + x
96 – x
(96 – x) % 12 = Part of Fortune

((2y + x) + x) % 12
(96 – 2x + x + x) % 12
96 % 12
0
Q.E.D.

It was this interesting relationship between these two indications that reminded me of the relationship between the astrological Parts of Fortune and Spirit, and thus what led me to start calling the Index the Part of Spirit and reanalyzing it in that light.  Even though there’s a huge difference between how the astrologers calculate these two Parts in astrology versus how we would in geomancy and where they might be found in their separate House Charts, I find that the relationship between them is identical and, for that purpose, hugely useful in geomantic interpretation.

To briefly describe my own personal view of these Parts based on all the foregoing, the geomantic Part of Fortune indicates the source, manner, and condition of the material life of the querent: bodily health, material wealth, worldly means, and so forth.  Likewise, the geomantic Part of Spirit indicates the same but for the spiritual life of the querent: mental and spiritual well-being, divine gifts, aid from spirits or gods, and so on.  I also read notions of resources and capabilities for the querent (to answer “what can I count on to accomplish it?”) in the Part of Fortune and notions of fate and destiny of the querent (“what should I be focusing on or having faith in?”) into the Part of Spirit.

Going beyond the basic interpretation of the Parts themselves, I’ve also found a trend in charts when the two Parts are both in the sixth house or both in the twelfth house:

  • If the Part of Fortune and Part of Spirit are both in house VI, then the matter is completely in the hands of the querent.  The querent has the ultimate say and ability to determine how the situation will proceed, and can change the reality of it as they need to depending on the course of action they take.  Their actions or lack thereof will be the crucial determiner in whether and how the situation will proceed.
  • If the Part of Fortune and Part of Spirit are both in house XII, then the matter is completely out of the querent’s hands.  All the querent can do in the situation is react accordingly and adjust their conceptions and perceptions of the situation, because the reality of the situation will proceed without their input regardless of their attempts.  No matter what the querent might attempt, the situation will continue unfolding as it will.

Also, as one other use, I often use the Part of Spirit in readings about magical, occult, or divine ritual for the sake of figuring out what particular courses of action might be best, or determining what path one ought to take, whether in a specific ritual or in a general direction.  It’s a small extra thing, but for a practicing magician like myself who consults with and is consulted by other magicians, it’s a useful thing to know.  I touched on this very briefly in my old post on geomancy and magic, but now the reasoning behind it all becomes clear.

All that said, remember that the Parts can only fall in even-numbered houses.  In a sense, this is similar to the idea that figures that are even can be considered objective because only even figures can be Judges (as I wrote at length before).  In this case, the even-numbered houses deal with, in order: material goods, land and family, health and servants, death and spirits, work and office, mystery and restriction.  We exclude the odd-numbered houses, which deal with: the querent themselves, communication, creation/procreation/recreation, relationships and rivalries, religion and faith, friendships and patronage.  There’s a similar “inherent to my personal life and relationships” versus “external to my personal life and relationships” difference between the even and odd houses as there is between the objective versus subjective qualities between the even and odd figures.  It is because these things are more external to us that they can be things pointed to help us or focus on, because they’re things that we’re not necessarily in full control or knowledge of.

As a side note, I only read the Parts in a radical (unrotated) chart.  When the chart is rotated for a third-party reading, I don’t bother looking at or interpreting the Parts of Fortune and Spirit, because they’re house-based calculations and not figure-based, so they don’t get rotated with the chart and (to my mind) have no importance or meaning in such a rotated chart.  I find that the Parts work best (if at all) when applied to the querent themselves in a situation, and I haven’t found it useful to rotate the Parts with the rest of the chart for a third party.

Similarly, I don’t swap my calculations of the Parts of Fortune and Spirit around based on whether it’s daytime or nighttime, because the notion of a diurnal or nocturnal geomantic chart doesn’t make sense; after all, a solar figure might never even appear in a given chart, or it might appear both above and below the horizon in a geomantic House Chart.  Instead, it makes more sense for the Part of Spirit to only rely on odd points (the points that represent active elements, excised and above the world of passive matter) and the Part of Fortune to rely on both odd and even points (the co-mingling of active Spirit and passive Matter that results in the world around us).

Further, although there are over a hundred possible Arabic Parts (depending on tradition, era, and author you’re looking at), I’m disinclined to say that there are more than these two Parts in geomancy.  After all, the logic for the Parts in astrology is easily extensible, but in geomancy we’re far more limited based on the techniques and tools that we use, but at the same time, we have other techniques that can fill in just as easily (such as adding the figures of two houses together, the triads in the Shield Chart, and so forth).  That we call them “Parts” in geomancy is more due to conceptual parallel in what they mean more than how they’re calculated than anything else.

The only other way I can think of to extend the technique of geomantic Parts would be to calculate a new Part based on tallying only the even points in a House Chart and taking the remainder after dividing by 12, which could be worth exploring, but I’m unsure what it might indicate; perhaps using my own tripartite view of the world, if the Part of Spirit (odd points only) indicates the influence of the spiritual Cosmos and the Part of Fortune (odd and even points) indicates the influence of the humane World, then this third unnamed Part (even points only) might indicate the influence of the material Universe.  Who knows?  It might show something of good use in divination, if a pattern can be detected.

Ah, and one final thing, just to finish off the intro to the post regarding the Sum of the Chart.  Instead of tallying up all the individual points of the 16 figures in the Shield Chart, I take a shortcut method: find the odd sum of the chart (odd sum of the House Chart, already calculated for the Part of Spirit, plus the number of odd rows in the four Court figures), find the difference between that and 64, double it, and add it to the odd sum to come up with the total Sum of the Chart.  The reason why this works is much like some of the logic in why the Parts of Fortune and Spirit have to add up to 12 or 24: because each figure has four rows and there are 16 figures, then there are 4 × 16 = 64 total rows of points in the Shield Chart.  Since every row must be even or odd, the number of odd rows added to the number of even rows must add to 64.  Since it’s easiest to find the number of odd rows in the chart after we calculate the Part of Spirit (we just need to take into account four more figures), once we have that number we just subtract it from 64 to get the number of even rows.  Remembering that an even row has two points in it, we double that to get the number of points in the even rows, add to it the number of odd rows (which have only one point in each), and voilà, the Sum of the Chart is yours.

More Thoughts on Shield Chart Company

Last time, I posted my collected thoughts on the rule of company in interpreting geomantic charts.  The rule, as taught nowadays, seems to have originated with the French geomancer Christopher Cattan, but after a bit of discussion with a student, seems to have pointed more towards something like the rule of triads like what Robert Fludd used in his interpretation of the Shield Chart rather than an extra way to get more significators out of the House Chart in case the significators themselves don’t perfect, like what John Michael Greer proposes in his Art and Practice of Geomancy.  I offered my thoughts there on how we might apply those same rules of company (company simple, company demi-simple, company compound, and company capitular) to the parents in a given triad, but I think we could offer more variations based on what we know of the figures, as well.

First, let’s talk about company capitular.  This rule has bugged me in the past, where we say that two figures are in company if they share the same Fire line (so Albus and Populus would be in company, but not Albus and Puer).  Why don’t we care about the other lines?  When it comes to company capitular, much like the case with the Via Puncti being limited in the literature to just the Fire line, we can also expand this rule a bit to focus on the similarity of the figures based on which of their lines are in agreement.  Using the above framework, I would normally say that c.  However, if we were to go to a more elemental way of looking at the figures, we can then rename and refine “company capitular” into “elemental company” and offer a new set of analytical rules:

  • Elemental company can be made multiple ways at once, and can be seen as a separate system beyond the methods of company simple, demi-simple, and compound.
  • A shared active line indicates an overwhelming desire or power in the method indicated by the elemental line.
  • A shared passive line indicates a complete apathy or powerlessness in the method indicated by the elemental line.
  • Company by Fire (same Fire line) shows that both parents want the same thing out of the situation.
  • Company by Air (same Air line) shows that both parents are thinking and saying the same things about the situation.
  • Company by Water (same Water line) shows that both parents feel the same way about the situation.
  • Company by Earth (same Earth line) shows that both parents have the same material means and physical basis to attain the outcome.

So, let’s say we have a First Triad (describing the nature and condition of the querent) where we have Coniunctio and Rubeus as the parents; the resulting child is Albus.  Thus, we can see that the parents of this triad are in passive company by Fire and Earth, in active company by Air, and not in company by Water.  While we know that the overall condition of the querent is placid and calm and not very active (Albus), we can also say that this is because they’re only constantly thinking about something intently (active company by Air) without having much to act (passive company by Fire) nor having much to act upon (passive company by Earth).  Through the querent’s reflection and mulling things over, they lose their intense and active feelings on the matter and let it go (not in company by Water).

That said, I suppose that this particular example isn’t particularly helpful, as it’s more a description of how the figures are interacting based on their elemental composition rather than an interaction between people or whether there’s support involved for the querent or other people involved in a given matter.  We know that we have passive company by Fire and Earth and active company by Air, so if we were interpreting this as a normal rule of company, we could say that there’s lots of concerted talk with others and lots of talking to people, but not much else going on, and that talk isn’t helpful when it comes to communicating feelings or helping sympathize or empathize with others, leading to solitude and loneliness on the parts of individual people.

Maybe elemental company isn’t the best approach.  However, there’s another way we could expand on the rule of company when implemented in the triads, and that’s based on the rule of company compound, where two figures are in company if they’re reverses of each other (e.g. Albus and Rubeus, or Caput Draconis and Cauda Draconis).  With company compound, the parent and their allies are approaching the same matter from different directions and have different results in mind, looking for their own ends, but find a common thing to strive for and will help each other out where they themselves lack the power they get from the other.  The thing is, however, that the reversion of a figure is essentially a mathematical transformation of a figure, not elemental or otherwise occult, and there are other mathematical transformations we could use instead to obtain other forms of company.

Although I haven’t discussed it explicitly on my blog much, I have briefly gone over the mathematical transformations of the figures in an earlier post, and I’ve also explicitly stated what the given transformation is of each figure in the relevant posts in my De Geomanteia series.  For our purposes here, there are three types of mathematical transformations of the figures:

  • Inversion: replacing all the single dots with double dots and vice versa (e.g. Puer inverted becomes Albus).  Everything a figure is not, but on an external level.
  • Reversion: rotating a figure upside down (e.g. Puer reverted becomes Puella).  The same qualities of a figure taken to its opposite, internal extreme.
  • Conversion: inversion with reversion (e.g. Puer converted becomes Rubeus).  The same qualities of a figure expressed in a similar, contraparallel manner.

So, if we were to make separate rules of company for these transformations, we might end up with four types of company, were we to keep company simple around as well.  Company compound would be renamed company reverse, and we’d add in “company inverse” and “company converse” into the mix as well, for a total of four “mathematical company” methods:

  • Company simple: both parents are the same figure (e.g. Albus and Albus).  The significator and their allies are completely in line with each other, from approach to energy, and are identical in all regards.  Complete harmony and support.
  • Company inverse: the parents are inverses of each other (e.g. Albus and Puer).  The significator and their allies fulfill each other’s deficit of power or means, yet mesh together to form one complete and total force that will conquer and achieve everything that alone they could not.
  • Company reverse: the parents are reverses of each other (e.g. Albus and Rubeus).  The significator and their allies are approaching the same matter from different directions and have different results in mind, looking for their own ends, but find a common thing to strive for and will each benefit from the whole.
  • Company converse: the parents are converses of each other (e.g. Albus and Puella).  The significator and their allies are similar enough to act along the same lines of power and types of action, but express it in completely different ways from the outside.  Internally, the action and thoughts are the same, but externally, they are distinct.  Think bizarro-world reflections of each other.

Interestingly, because these are mathematical operations performed on the figures, if we know what the operation is, we nearly always already know what the child will be if we know the parents and type of company they’re in.  For instance, we know that when two figures are added to each other, if those figures are inversions, the result will always be Via (e.g. Populus and Via, Albus and Puer, Laetitia and Caput Draconis).  Likewise, if two figures are in company simple, we’re adding the same figure to itself, so the result will always be Populus.  However, the other types of company give us a bit more interesting stuff to chew on:

  • Company reverse
    • Cannot be formed if parents are both Via, both Populus, both Coniunctio, or both Carcer.  These figures are reversions of themselves, the so-called “axial” figures.  In these cases, we have company simple.
    • Cannot be formed if parents are Fortuna Major and Fortuna Minor (or vice versa), or Acquisitio and Amissio.  These figures are inversions of themselves, and so we have company inverse.
    • Child will be Carcer if parents are Laetitia and Tristitia, or Caput Draconis or Cauda Draconis.
    • Child will be Coniunctio if parents are Albus and Rubeus, or Puer and Puella.
  • Company converse
    • Cannot be formed if parents are Populus and Via, or Carcer and Coniunctio.  The axial figures have a converse that is their inverse, and so we have company inverse.
    • Cannot be formed if parents are both Fortuna Maior, both Fortuna Minor, both Acquisitio, or both Amissio.  These figures are converses of themselves, and so we have company simple.
    • Child will be Carcer if parents are Laetitia and Cauda Draconis, or Tristitia and Caput Draconis.
    • Child will be Coniunctio if parents are both Albus and Puella, or Rubeus and Puer.

Note that, in all cases where we use these company rules for parents in a triad, we always have a child that will be an axial figure: always Populus if company simple, always Via if company inverse, and either Carcer or Coniunctio if company reverse or company converse.  Thus, if we see any child figure in the Shield Chart as an axial figure, we know immediately that its parents will be in company.  Further, based on this child figure, we could see at a glance whether a triad is referring to a single person developing over time with the help or assistance of others (if Via or Carcer), or whether the triad is referring to multiple people interacting and dealing amongst themselves (if Populus or Coniunctio); additionally, we can see whether there is progress and change involved (if Via or Coniunctio) or whether things stagnate and become fixed (if Populus or Carcer).  However, this is a very naïve way of reading a triad, and may not always hold up depending on the specific triad being interpreted as well as the query and intuition of the diviner.

As an example, let’s consider a First Triad where the First Mother is Albus.  Again, we’re considering what the condition and overall state of the querent is, so let’s see what the four possibilities of company would be and their resulting triads:

  • Company simple (Second Mother Albus, First Niece Populus):  Not much to speak of, really.  As in all cases where the child is Populus, what has been is what will be.  However, the querent is likely not alone and has at least one other friend who shares their same state of mind and condition, and are coming together in harmony and unison to help each other out or facilitate their actions together.
  • Company inverse (Second Mother Puer, First Niece Via):  On its own, we could say that the state of the querent will be turned completely on its head, with all this passive contemplation turning into daring, heedless action.  If the chart or intuition of the diviner suggests that the querent is with someone else, this is someone who’s constantly playing devil’s advocate and goading the querent onto radical change, and together they complete and fulfill each other in many ways.
  • Company reverse (Second Mother Rubeus, First Niece Coniunctio):  Fun times, except ew.  This is a weird combination of people, and I’d hardly call them “allies” in any sense; they’re both arguing with each other to the point of talking past each other, yet in their harsh and loud words, they eventually come to a concordance and progress together.  Strange bedfellows, indeed.
  • Company converse (Second Mother Puella, First Niece Carcer): This is probably the most pleasing of all companies possible, as it provides the querent with someone sufficiently different yet operating on the same principles to reinforce the condition and state of the querent.  In this case, this would be good to solidify the nature of the querent and give them some stability, but with the risk of codependency and a potential for getting locked into their current state without trying to actively change things.

All these rules of company so far discussed are based on something structural about the figures, either the elemental structure in the first set (originally based on an expansion of company capitular) or the mathematical structure in the second set (expanding off company compound).  What about company demi-simple?  In that rule, both figures in company are ruled by the same planet, and indicates that the significator and their allies are different, but share enough characteristics for them to complement each other and understand each other enough to accomplish the same thing.  If we use a more occult basis for establishing company, I can think of two more ways to find these out, forming a set of four “magical company” rules:

  • Company simple: both parents are the same figure (e.g. Albus and Albus).  The significator and their allies are completely in line with each other, from approach to energy, and are identical in all regards.  Complete harmony and support.
  • Company zodiacal: both figures are ruled by the same zodiacal sign (e.g. Caput Draconis and Coniunctio).  The significator and their allies are put together by fate and must contend with the same matter together, though not perhaps in the same way.  The zodiacal rulership of the figures can be found in this post.  Not all signs have two figures, so company zodiacal can only be formed when both figures are ruled by the signs Taurus, Gemini, Virgo, and Scorpio, the only signs using Gerard of Cremona’s method that have two figures assigned to them.  Otherwise, using Agrippa’s method, company zodiacal can only be formed when both figures are ruled by the signs Cancer, Leo, and Virgo.
  • Company planetary: both figures are ruled by the same planet (e.g. Albus and Coniunctio).  This would have been company demi-simple in the original rules of company given by Cattan, but here, we can say that the inner drive of the significator and their allies are the same, though their external expression is different but aimed at the same overall goal.
  • Company elementary:  both figures are ruled by the same element (e.g. Albus and Populus).  The outer expression and actions of the figures are similar and get along well enough for the time being, although their inner drives and ultimate goals differ.  The elemental rulership of the figures can be found in this post.

These methods of company do not rely on anything structural in the figures (with the exception of company simple), but rely on the higher meanings of element, planet, and sign attributed to the figures to see how close the figures are to each other and whether they can form enough of a relationship to work together.  Additionally, unlike the other sets of company rules, I think it’s best that two figures can be in company multiple ways at the same time (like Carcer and Tristitia, which would be in company both planetary and elemental) rather than having one form of company “overwrite” the others.  Still, if an overwriting rule were put in place, I think it would go company simple (sameness), then company zodiacal (fated), company planetary (inner drive the same), and company elementary (outer expression the same).  It is a little frustrating that so few figures can enter into company zodiacal with each other, however, but I think that might also be for the best.

So, to recap, we have four sets of rules of company:

  1. Canonical company (given by Cattan): company simple, company demi-simple, company compound, company capitular
  2. Elemental company (based on the elemental structure of the figures): company by Fire, company by Air, company by Water, company by Earth
  3. Mathematical company (based on the mathematical relationships of the figures): company simple, company inverse, company reverse, company converse
  4. Magical company (based on the occult associations of the figures): company simple, company zodiacal, company planetary, company elementary

Of these, I think elemental company can be thrown out as a viable technique, as it doesn’t really tell us anything we didn’t already know, but instead is another way to look at the simple addition of figures, which isn’t a great way of telling whether someone has allies or external support, and strongly differs from the other methods entirely.  Mathematical company and magical company, however, bear much more possibility because they explore actual relationships among the figures, one by means of their structure and one by means of their correspondences.  When applied to the parents in a triad, I think we can definitely use these in addition to or instead of Cattan’s canonical company rules to understand whether a person in a reading has allies and, if so, of what type and means.

All this hasn’t really touched on the role of the child in a triad, however, when it comes to rules of company.  That said, these rules are all about pairs of figures, and with the exception of the Sentence, all figures are parents and can enter into company with at least one other figure.  I think it might be best to leave it at Cattan’s barely-explained way of seeing which parent the child agrees with most, whether it be by ruling planet or element or whatever, and judge a triad much as we might judge the Court with the added clarity of seeing who helps who attain what in a given triad.

Internumeric Relationships by Addition on the Tetractys

It’d be rude and vulgar of me to leave the Tetractys as some simple geometric diagram used for plotting paths or meditations.  I mean, the Tetractys is a meditation tool, yes, but to use it merely for working with the Greek alphabet with in a mathetic framework is to ignore the deeper meaning of the Tetractys.  For the Pythagoreans, especially, the Tetractys was more than a set of ten dots; it was the key to all creation and all cosmos.  There’s no evidence that anybody’s used it to plot paths on like I did, which is probably because this is an innovative use for an already heavily used tool based purely on number.  As we’re all aware by now, the Tetractys is a representation of the Monad, Dyad, Triad, and Tetrad to yield the Decad: 1 + 2 + 3 + 4 = 10.  All these numbers are holy to the Pythagoreans and to Western occultists generally, but there’s so much more to the Tetractys than this.

One of the traditional ways of understanding the mysteries of the Tetractys was to take the different ranks of numbers present and add them together to yield a particular number.  For instance, the Monad plus Tetrad yields the Pentad (1 + 4 = 5), while the Monad, Dyad, and Triad together yield the Hexad (1 + 2 + 3 = 6).  All these numbers have their own meaning, all of which are based ultimately on the Monad and, in succession, the meanings given to the other numbers built upon the Monad.  I’d thought I’d investigate what some of these properties are and see what the Tetractys represents in building the numbers of the Decad together based on these relationships between the ranks of the Tetractys.  Specifically, these relationships are based on the arithmetical operation of addition, the straightforward aggregation of two numbers by combining their distinct magnitudes into a single one.  Other operations exist, but those are for another time.

So, to start off with, we have four basic numbers, starting with the Monad and ending with the Tetrad.  We can say that, with the exception of the Monad, all numbers are just collections of Monads in a particular relationship:

  1. Monad = individuation, undifferentiated, undifferentiatable
  2. Dyad = two Monads in relation
  3. Triad = three Monads in harmony
  4. Tetrad = four Monads in form

Note that some of these can be broken down further into simpler groups.  Without repeating any particular number (such as saying that the Dyad is two Monads or the Tetrad is two Dyads), we end up with two extra identities:

  1. Triad = Monad + Dyad
  2. Tetrad = Monad + Triad

It’s crucially important to note that the Dyad, Triad, and Tetrad are more than just a collection of monads.  Number in the esoteric sense is more than just a magnitude or amount, but also a relationship formed between the individuals in the collection.  The only number in this set that has no relationship is the Monad itself, since it exists as a unity unto itself without anything to relate to.  The Dyad is the first number that has a relationship, but can be said to be relationship itself; without the Dyad, relationship cannot exist.  In a more arithmetic sense that the Pythagoreans preferred, all numbers can be divided into two partially overlapping groups of odd (able to be divided into unequal parts only) and even (able to be divided into two equal and unequal parts).  Four, for instance, is even because it can be split up into groups of 1/3 and 2/2.  Five, however, is odd, because it can be split into 1/4 or 2/3, and neither of those are equal splits.  However, the Monad cannot be split at all into anything, and the Dyad can not be split into unequal parts, so neither the Monad nor Dyad are even nor odd, and are thus not true number, though they are sources of number.

Thus, based on the individuation of the Monad and relation of the Dyad, all other numbers can be made, such as the Triad.  It is because of this that the Triad is considered by the Pythagoreans to be the first true number, since the Monad and Dyad are something rarer and rawer.  All amounts can be formed from the Monad, but it’s the relationship (Dyad) between individual Monads that produce a number.  Thus, as the Triad is the first true number, it is also the first odd number, and the Tetrad is the first even number.

So, based on the six above identities, we can form the rest of the numbers from the Pentad (5) to the Decad (10).  If we omit the identities from above and reduce all things to a collection of Monads, Dyads, Triads, and Tetrads, we end up with two ways to form the Pentad, and one way each to form the Hexad, Heptad, Octad, Ennead, and Decad:

  1. Pentad = (Monad + Tetrad) or (Dyad + Triad)
  2. Hexad = Dyad + Tetrad
  3. Heptad = Triad + Tetrad
  4. Octad = Monad + Triad + Tetrad
  5. Ennead = Dyad + Triad + Tetrad
  6. Decad = Monad + Dyad + Triad + Tetrad

Yes, this is all basic arithmetic that we’ve been able to do since kindergarten.  Of course, it’s always the simplest things that hide some of the more profound secrets.  I won’t go over all the associations and theologies behind the numbers for that; you can get a copy of the Theology of Arithmetic by Iamblichus for cheap (or even, dare I say it, for free), and you can read about what the Pythagoreans thought about the numbers of the Decad way back when.  What I want to point out is, at a high level, what these additions of the numbers mean based on the four concepts of monadic individuation, dyadic relation, triadic harmony, and tetradic form.

Monad
The Monad is an individual, unchanging, static, and stable.  It is the only thing that exists, and thus cannot be differentiated from anything (since there’s nothing to differentiate it from).  While we can say that it contains all opposites and extremities within itself, it’d be more proper to say that no concept of opposition or extremity exists within the Monad.  While the Monad exists, nothing exists within the Monad; it can become all and any qualities, but it itself has no qualities.  It is the source of all nature, but is itself beyond nature.  It cannot be divided since it is a unit, an atom, the core of existence itself.  The Monad cannot move, as there is nothing within which it can move (which would imply something that is Monad and something that is not-Monad).  The Monad has no shape, consisting only of a single point that indicates both all sizes and all angles but without anything else to connect to.

Dyad
The Dyad is relation and difference.  Between two Monads, we now know of two things that can be compared as equals, but as different equals.  The Dyad is representative of differentiation, distinction, opposition, and motion, all of which can be thought of as different types of relation.  The Dyad represents a line defined by two points, but is still without shape; it can possess direction and magnitude, but is as yet without definition.  The Dyad allows for things to exist within, around, and outside of other things, since it creates space between and among other things.  While the Monad is pure potential for creation (and all other things), the Dyad is the act of creation itself, since it distinguishes a Creator from the Creature, or the Acted from the Actor.  The Dyad is space, change, action, and relativity.

Triad
The Triad is harmony and proportion, formed from a combination of individuation and relation.  It is the first odd number, and the first number that can be added from other distinct numbers.  The Triad gives the first shape of something, as three points can define an enclosed space.  The Triad indicates actuality, the Creature made through Creation (Dyad) from the Creator (Monad).  However, it is also indicates harmony, since two distinct and different things are linked to and joined by a third.  With the Triad, there is real existence as opposed to potential existence or becoming existence.  Quoth Iamblichus, “‘this’ belongs to the Monad, ‘either’ to the Dyad, and ‘each’/’every’ to the Triad”.  With Triad, there is time: beginning, middle, end; there is communication: speaker, listener, message; there is work: actor, action, acted upon. However, like the Monad, the Triad is static, since it provides for space and size but not change, since it is construction and creation that brought a static shape to being.

Tetrad
The Tetrad is the root of form, formed from a combination of individuation and harmony.  With three points we can define a two-dimensional shape, but with four we can define a solid three-dimensional object.  Moreover, the Tetrad is dynamic, since it is even; while the Triad measures static quantity, the Tetrad measures dynamic quantity, since it provides for motion and change while the Tetrad does not.  Further, the Tetrad allows for forms present in relationship to each other; while the Triad offers a two-dimensional form, the Tetrad allows for two-dimensional forms next to each other as the Dyad allows for Monads to be next to each other.  With both individuation and harmony, one can choose to be part of a harmony or break away from it, acting either inside or outside a given group, and allows for distinct existence apart from, aggregated with, or in conjunction with others.

Pentad
Alone among the numbers, the Pentad is the only one that can be formed in two distinct ways: from the Monad and Tetrad (a combination of individuation and form) and from the Dyad and Triad (a combination of relation and harmony).  In a way, it’s fitting; between all the numbers of the Decad, the Pentad is the middle of them.  Consider that any two numbers that add up to 10 have 5 as the mean (9 + 1, 8 + 2, 7 + 3, etc.); the Pentad is halfway to the Decad, and itself is vital to life.  It is the combination of pure potential and discrete aggregation (Monad and Tetrad), as well as of relation and harmony (Dyad and Triad); it is the combination of an even and odd number in either case, and considered to unify opposites in a dynamic way that allows for growth and change as opposed to the static way of the Triad.  If we consider the Pentad as the sum of Monad and Tetrad, we obtain a view of eternality and potentiality combined with and suspended among temporality and discretion (the four changeable elements acting under unchanging Spirit); if we consider the Pentad as the sum of Dyad and Triad, we obtain a view of motion and action mixed with and changing stasis and relationship.  In either case, the Pentad is where life and concrete reality itself begins, since in the Pentad there is balance, reciprocity, distribution, and especially of growth.

Hexad
The Hexad is the combination of relation and form, producing a dynamic harmony.  Unlike the Pentad, which is dynamic growth, the Hexad is a balance between things in motion.  The presence of distinct qualities bestowed by the Tetrad in relation of the Dyad allows for various dynamic forces to exist dynamically, moving with and acting, co-acting, or reacting together without destruction.  As the Tetrad represents a body and the Dyad represents motion, the Hexad represents a body in motion and can move in six ways, or three sets of two ways: up/down, left/right, forward/backward.  Seen the other way, as the Tetrad represents qualities and the Dyad represents opposition, the Hexad represents an ordering and balance of opposites.  Further, as two Tetrads, the Hexad represents what we commonly see as “Merkava stones”, two interlocked tetrahedrons that represent a combination of bodies and opposites that together unite to form a whole.  While the Pentad is the number of life, the Hexad is the number of order.

Heptad
The Heptad is the combination of harmony and form, producing foundation.  This is hard to describe in a single word, but within the Heptad there are all things finally present to create everything, yet is short of actively creating everything; all manifest sources are present in the Heptad (seven planets of astrology, seven vowels of Greek speech, etc.), though they are as yet too unmanifest on their own.  As a combination of Triad and Tetrad, the Heptad represents the four elements and three reagents, or the three processes that transform the four elements so as to create all things.  As an odd number that cannot be divided, the Heptad is similar to the Monad in that it provides for potential creation, but unlike the Monad, the Heptad is a collection of seven entities that provide the foundation of all manifest things, while the Monad is an undifferentiatable source from which all manifest and unmanifest things come.  If the Hexad represents order, then the Heptad are the things that are ordered within the cosmos provided for by the Hexad, the meat to fill out the Hexad’s bones.  The Heptad is that which essentially exists; the Heptad is essence.

Octad
The Octad is the first addition that involves three numbers: the Monad, Triad, and Tetrad.  Thus, the Octad combines individuation, harmony, and form.  As the Heptad is the combination of the Triad and Tetrad, we can say that the Octad is that which results from the essences of creation into which they flow.  However, as we saw with the Pentad, we can also say that the Monad and Heptad combine such that the Heptad is mixed in within the Monad, as the seven planets are within the eighth sphere of the fixed stars, as the four elements are within the Quintessence.  However, we can also say that the Octad is the combination of two Tetrads, allowing for mixtures and combinations of that which otherwise could only relate to each other by processes; although Sulfur combines and transforms Air into Fire and vice versa if we use the Tetrad + Triad view, we end up with dry air or cool fire between Air and Fire if we use the Tetrad + Tetrad view.  The Octad represents solution and combination of qualities, a single entity produced from essences or qualities and their interquality transformations.  The Octad is mixture.

Ennead
The Ennead is the combination of relation, harmony, and form.  Based on how we might conceive of this, we can say that the Ennead combines the Tetrad and Pentad, the Triad and Hexad, the Dyad and Heptad, or the Monad and Octad, but at its root it combines the Dyad, Triad, and Tetrad.  At its core, it lacks the Monad and possesses the Dyad, indicating that the Ennead is an active number related to creating but not as creator or creature.  In the Ennead is all creating of manifest things, combining tetradic body, triadic intermediation, and dyadic motion.  In the number nine are all the other numbers brought together, the final single-digit whole number.  As there were nine Muses who lead to all Art and nine Curetes who watched over the infant Zeus, the Ennead brings things to completion and perfection without itself being perfect.  The Ennead is realization.

Decad
At long last, we finally reach the Decad, the combination of the Monad, Dyad, Triad, and Tetrad; of individuation, relation, harmony, and form.  In the Decad are all the basic numbers of the Tetractys, and there are many ways to add to the Decad using the lesser numbers, but at its core it is the number formed from 1, 2, 3, and 4 summed together.  Just as in the Ennead there is the process of realization and completion but without something to realize or complete, the Decad augments this with the Monad, allowing for something to be filled with the Ennead.  The Decad represents a discrete entity (Monad) that is distinct from other things (Dyad) that is stable unto itself (Triad) given physical a body (Tetrad).  Moreover, it is also something that can grow (Pentad) while maintaining itself in an order (Hexad) that combines all ethereal essences (Heptad) and concrete mixtures (Octad) being brought together (Ennead).  Without any other number preceding it, the entity represented by the Decad would be lacking and could not be fully realized.  Whether it is the universe we live in or the individual people we live as, we are all representative of the Decad and the journey it has taken to get here.  The Decad is the Whole.

I think it goes without saying that this Pythagorean analysis of the ten numbers of the Decad can easily be mapped onto the Tree of Life in Jewish kabbalah or Hermetic qabbalah, and indeed, I recall seeing many of these things present in the explanations given in works like Alan Moore’s Promethea series.  It makes sense, too, since Pythagoreanism is one of the fundamental philosophies underlying Western occult thought, deep enough to not clearly be distinguished as Pythagorean but also profound enough to affect everything that’s built upon it.  While numerology has never quite been my strong suit, this little exploration of the basic numbers has considerably helped.

Pythagorean Correspondences to the Tetractys

As many of my readers know, as well as those in Western occulture generally, correspondences are a big thing for us.  Based on our shared philosophical and educational lineages, we like to say that “A is like B”; we understand that the light of the Sun is much like the heat of fire, which itself is like the luster of gold based on certain shared properties.  In recognizing these shared properties, we immediately come to a system of symbols, where one thing can stand in for another, as well as to a system of harmonic relationships, where two things can be used compatibly with each other because they share the same ideas.  On a large scale, we call this system of symbolism one of correspondence, where something corresponds to something else.  This is often used in emanationist frameworks, where these correspondences cross levels of manifestation.  For instance, the Sun being an astrological planet is on a higher level than the element of Fire, which is itself on a higher element than actual fire or gold.  However, we can use any of these things to represent or produce a harmony with the other since they’re all corresponded to each other.

Probably one of the most valuable resources for this comes from the Second Book of Occult Philosophy by Cornelius Agrippa, where Agrippa presents a set of correspondences that link various names of God, planets, choirs of angels, ranks of the blessed, elements, prophets, and the like to each other based on certain shared properties.  Crucially, however, Agrippa organizes this by number.  Thus, he has a Scale of Four (book II, chapter 7) to correspond things that are easily divisible into one of four groups, a Scale of Seven (chapter 10) for things grouped into sevens, a Scale of Ten (chapter 13), and so forth.  Each of these are immensely useful for magicians, since they provide us with symbols and ritual ideas at a glance.  Aleister Crowley’s famous Liber 777 and, more recently, Stephen Skinner’s Complete Magician’s Tables offer these but on a much grander scale, corresponding far more things together on a qabbalistic basis than Agrippa does in his Scale of Ten.

Of course, finding systems of correspondence is an old thing, and even back in classical and antique times do we see the foundations of these systems of correspondence set up and used.  And, well, you can see where I’m taking this, aren’t you?  The Tetractys, that venerable Pythagorean symbol, was seen to contain within itself the foundations of all life and existence in every conceivable form, and not just in a strictly emanationist way.  Each rank of the tetractys, based on whether it related to the Monad, Dyad, Triad, or Tetrad, was associated to something else that formed part of the cosmos.

One good source for this comes from Iamblichus’ Life of Pythagoras, where he gives a good overview of the life of Pythagoras (duh) as well as a number of his teachings (though nowhere in depth as I’d like).  The Taylor translation linked above, however, also contains an extensive collection of other Pythagoreans who followed Pythagoras and wrote down what the Teacher (ostensibly) said, as well as a set of notes where Taylor inspects the things Iamblichus says and expands on them where the original author was annoyingly terse to our modern readers.  Part of this expansion is where Taylor talks about how the Tetractys wasn’t just a number but a graphical mnemonic, if you will, of various things

Monad Dyad Triad Tetrad
Number 1 2 3 4
Doubling Progression 1 2 4 8
Tripling Progression 1 3 9 27
Even Geometry Point Line Polygon Solid
Odd Geometry Point Open curve Closed curve (circle) Cylinder
Element Fire Air Water Earth
Platonic Solid Tetrahedron Octahedron Icosahedron Cube
Growth of Vegetation Seed Length Breadth Depth
Communities Individual Family Town State
Power of Judgment Intellect Science Opinion Sense
Parts of an Animal Rational Irascible Epithymetic Body
Seasons Spring Summer Autumn Winter
Ages of Man Infancy Youth Adulthood Old Age

Well, would you look at that, it’s a table of correspondence along the same path as Agrippa’s Scale of Four.  It’s not quite the same (Agrippa gives Summer, Spring, Winter, and Autumn instead of Pythagoras’ Spring, Summer, Autumn, Winter, and I’m personally in favor of using Agrippa’s associations or a variation thereof, especially considering how Athenians started their year at the summer solstice), and there are a few hard-to-understand terms and progressions, but for the most part it’s definitely something useful in seeing how emanation works in everything.

I mean, sure, the can of Monster energy drink next to me is something that emanated from the Source just as I did, but it has a different body and different contents than I do.  Consider the body of the can, the metallic mostly-cylindrical shape the drink comes in.  The can wasn’t born, so it can’t age in the way a human ages, but consider how soft drink cans are made for a bit.  The cylindrical can was stretched out from a circular cut from a flat sheet of aluminum; from this, we got the tetrad-corresponded cylinder from the triad-corresponded circle.  Of course, this circle itself has depth, since it’s a cutout from an aluminum sheet which is a body; all bodies have three dimensions (length, breadth, and depth), without any one of which it’d only be a two-dimensional shape.  So, whence the circle itself?  The circle itself is a form, not a body, an idea that can interact with others.  Whence the form of a circle?  The form of a circle is made from a curved line traveling around a point.  After all, all circles only need two points for a definition: a center and a boundary.  The curved line demonstrates motion and direction, both of which are relative concepts (in order to move, you need something to move from both in terms of location, speed, orientation, etc.).  The curved line, then, comes from the single point, the Monad of all shapes and forms and bodies.

So why is the tetradic form of a circle a cylinder and not a sphere?  After all, isn’t the sphere the thing most like a circle in the third dimension?  Sorta, yeah, but a sphere is (according to Pythagoras and other Pythagoreans) a perfect body, and there is nothing we can make in the cosmos that is perfect due to the constant actions of Difference, Existence, and Sameness as well as the upheaval and drama in the four elements.  Rather, the tetradic form of a circle is a circle with depth, the most straightforward of which is a stack of circles, forming a cylinder.  It makes sense, though a little counterintuitive.

Between Agrippa and Taylor’s exposition of the correspondences of fourfold things to the Tetractys, a lot of intellectual work has already been cut out for us in studying how the Tetractys can relate to individual things.  Then again, that’s just it; this kind of analysis is good for understanding individual things, and it’s the relationships of those things that are just as important, if not moreso.  In fact, one of the more famous divisions of things is the Quadrivium, literally “four ways”: four types of mathematics used throughout the classical, medieval, and Renaissance worlds.  In this, arithmetic is an understanding of bare number (Monad), followed by music (in the broad sense) as an understanding of relationship and modulation (Dyad), followed by geometry as an understanding of static form (Triad), followed by astronomy which is an understanding of moving bodies (Tetrad).  Just as one can’t study astronomy without a knowledge of geometry, and geometry of music (for the study of proportions and ratios is a type of music in the classical, ideal sense!), and music of arithmetic, the Tetractys itself indicates that the relationships between things are where the real action lies in the cosmos.

After all, wasn’t that the whole point of my developing mathesis, anyway?  To discover relationships more than units?  To understand the changes between the different methods of manifestation rather than the methods themselves?  Something is still missing, and that’s where mathesis becomes mathematic, in our modern sense of numbers and relationships.  After all, if we’re still trying to analyze stuff as individual units, then we’re dealing with things as individual monads.  A Dyad is more than just two monads put next to each other; it is a relationship between the two that makes two monads into a Dyad.  That relationship is often called “music” in Pythagorean literature, but it’s not necessarily the music of instruments or sounds.  Music, in this case, is the means of progression, movement, and patterns.  It is not enough to study sheer quantity in the arithmetic sense, and it is yet too much to study harmony in the geometric sense.  Another type of analysis-and-synthesis is needed for the Dyad.