Number in Concept – Arithmetic

Number in concept – Arithmetic.

References

Guides 

            The principle guides followed in this chapter are listed below: 

            Plato‘s work, written about 400 – 350 BC, forms the background of discussion here;  particularly the Timaeus, and the Republic. 

            Theon of Smyrna, who lived about 75 – 150 AD, wrote an introduction to Plato called ‘Mathematics Useful for Understanding Plato’. 

            Theology of Arithmetic, is an anonymous work, attributed to Iamblichus, tentativly dated 350 AD.  It is a compilation of other philosophical works and perhaps lecture notes on the numbers of the decad.

            Thomas Taylor, in 1816 edited and structured a compilation of ancient writers on arithmetic, called the ‘Theoretic Arithmetic of the Pythagoreans’.  He translated Plato and other major Pythagorean writers.  His style is difficult for a modern reader, but his understanding is great.

            The principle present day guides are:  Keith Critchlow and Robert Lawlor.  Both have written on the subjects at hand in deep sympathy with the ancient authors;  and in a mannor accessable to the contemporary reader.    

Character of the Four Subjects

            Nicomachus provides the following introduction to the origin and nature of the quadrivium:  [IA, 1.1.1]  “The ancients, who under the leadership of Pythagoras first made science systematic, defined philosophy as love ofof wisdom. … Pythagaras restricted the title to apply to the knowledge and comprehension of reality. … This wisdom he difined as the science or knowledge  of the truth in real things, conceiving science to be a steadfast and firm apprehrnsion of the underling substance, and real things to be those which continue uniformly the same and never depart even briefly from their existance;  these real things would be immaterial … for bodily material things are … forever involved in continuoud flow and change … .  The bodiless things however, of which we conceive in connection with or together with matter, such as … quantity, equality, relation, disposition … those things whereby the qualities found in bodies are comprehrnded – all these are of themselves immovable and interchangeable, but accidentally they partake of and and share in the body to which they belong.  Now it is with suvh things that wisdom is particulary concerned, but accidentally also with things that share in them, that is, bodies”.

            [1.2.3]  “Therefore if we crave for happiness, a goal that is worthy and fitting, this is accomplishrd by philosophy alone : and philosophy means for us desire for wisdom, and wisdom the science of truth of things, … it is reasonable and most necessary to disttinguish and systematize the accidental qualities of things.”

            [1.2.4]  “Things, then, both those properly so called [real  things] and those that simply have the name [material things], are some of them unified and continuoud, for example, an animal, … a tree and the like, which are called … magnitudes;  others are discontinuous, in a side-by-side arrangement ]or] in heaps, which are called multitudes, for example, a flock, … a chorus ann the like.”

            [1.2.5] “Wisdom then must be considered to be the knowledge of these two forms.  Since however all multitude snd magnitude are their own nature infinite [multiplication and division of a unit can be continued indefinitely] … and since sciences are always of limited things and never of infinates, it is evident that a science of either of them could ever be formulated, for each of them is limitless in itself … .   A science, however, would arise to deal with something separated from each of them, with quantity, set off from multitude and magnitude.”

            [1.3.1]  “Since quantity may be viewed bt itself, having no relation to anything rlse … or together with its relation to another thing … it is clear that two scientific methods will deal with the investigation of quantity;  srithmetic, absolute quantity and music, relative quantity.”

            [1.3.2]  “… inasmuch as part of size [or quantity] is in a state of rest snd stsbility and another part in motion and revolution two other sciences will accurately reat of size, geometry of the part that abides at rest and astronomy of that which moves and revolves.”

            [1.3.3]  “Without the aid of these then it is not poaaible to deal accurately with the forms of being nor to discover the truth in things, knowledge of which is wisdom …”

            [1.3.5]  Nicomachus closes this section by quotlng Plato’s dialog of the Laws, book 13 “… Every diagram, system of numbers, every scheme of harmony and every law of the movement of the stars, ought to appear as one to those who study rightly, and what we say will properly apear if you study all things looking to one principal, for there will be seen to be one bond for all these things, and if you attempt to study philosophy in any other way you must call no fortune to assist him.  For there is never a path without these …”

Precedence of the Subjects

            Nicomachus asks:  [IA 1.4 – 1.5]  Which of theese four methods must we learn first?  … the one which naturally exists before them all and takes the place of origin and root and as it were, mother and nurse of the rest.  And this is arithmetic, not solely because we said that it existed before all the others in the mind of the creating God like some universal and exemplary plan, relying upon which as a design and archetypal example the creator of the universe sets in order material creations … ;  but also because it is naturally prior in birth, inasmuch as it abolishes other sciences within itself, but is not abolished with them.  … cince it has the property of abolishing the others within itself, it is the older.

            So it is with the foregoing sciences;  if geometry exists arithmetic must needs be implied, for it is with the help of this latter that we can speak of triangle, quadrilateral,octahedron, icosahedron, double, eightfold, or one-and-a-half times, or anything else of the sort which is used as a term by geometry, and such things cannot be conceived of without the numbers that are implied with each one …

            [1.5.1-2]  And once more this is true in the case of music;  not only because the absolute is prior to the relative … but also because the musical harmonies, diatessaron,diapente and dispason, are named for numbers;  similarly all of their harmonic ratios are arithmrtical ones …  More evidently still astronomy attains through arithmrtic  the investigations that pertain to it, not alone because it is later than geometry in origin – for motion naturally comes after rest – nor because the motions of the stars have a perfectly memodius harmony, but also because risings, settings, progrssions, retrogressions, increases and all dorts of phases are governed by numerical cycles and quantities.”

Definition and Nature of Number

Definition

            Number is considered to be the basic unit of counting;  and this elemental unit, thought of as the source of number, is called the Monad.  Multitude, as extension may thus be thought of as the repitition of the Monad. 

            Theon writes (MUUP, p12), “Number is a collection of monads, or a progression of the multitude beginning from and returning to the monad (through the successive addition or subtraction of one unit).”              

                        Taylor says (TA, p 3), “Thales defined number conformably to the doctrine of the Egyptians by whom he was instructed, to be a collection of monads.  But Pythagoras defined it to be the extension and energy of the spermatic reasons contained in the monad.  Or otherwise, to be that which prior to all things subsists in a divine intellect, by which and from which all things are coordinated, and remain connumerated in an indissoluble order.  Others, however of his followers defined it to be a progression from unity through the magnitude of it.  But Eudoxus the Pythagorean says, that number is definate multitude, in which defination he distinguishes the species from the genus.  The followers of Hippassus who were called ‘Acousmatici’ said that number is the first paradigm of the fabrication of the world;  and again, that it is the judicial instrument of the god who is the demiurgus of the universe.”             

Critchlow writes (KW #5, GIN), “Arithimos which we currently translate as number was generaly consideredto be a plurality of units (plethos monadon)…”.

Realms of Number. 

            There are three realms or worlds of number;  1, concrete number, that is, things which we can count;  2, intellectual or mathematical number, what we might call the science of counting;  and 3, philosophical or qualitative number, that is, the essential differentiating aspect of each integer;  a concept associated with the Platonic Forms. 

            As Critchlow puts it (KW #5, GIN, p 2),  “…number has at least three principle aspects, all fundamentally different in quality yet each subject to a unifying principle.  These are firstly ideal number or the arithmos eidetikos that Plato indicated was the archtypical or the eide, the governing intelligable principle, they are singular, incomparable (that is not able to be added or subtracted from each other) and at this level, although they are still refractions of unity, they are self identical and complete between one and ten.  This first level is principle or the ‘first principles’ of number after the unitary generating principle.  The second level or secondary aspect is mathematical number or arithmos mathematicos which Plato indicated was an intermediary function (metaxe) between the eide (ideal) and the aistheta or sensable expression in things.  This secondary domain , which we might typify as pure mathematics today, deals with the number aspect of things which clearly goes indefinately beyond ten.  Plotinus, the later Platonist philosopher, said that this kind of mathematical number could be philosophically understood as an image (eidolon) of the Ideal numbers which are their governing principle and source.  The value of this class on mathematics was seen by Plato as a means, like the soul, to raise the contemplation ‘from earth to heaven’, to remember our source and goal.  Finally the third aspect of number was the numbered aspect of all things, an amalgamation and inseparable relation between number and all physically experienced things.  Every existent or created thing has a necessary number aspect.” 

            In this quote Lawlor considers philosophical and intellectual number together as qualitative;  and material numbers as quantitative.  He writes (SG, p 10), “In addition to simple quantity, numbers on the ideal level are possessed of quality, so that ‘twoness’, ‘threeness’, or ‘fourness’, for example are not merely composed of 2, 3, or 4 units, but are wholes or unities in themselves, each having related powers.”  I

Number and Form

            Number in its qualitative aspect can be considered the source of perceptable forms, that is, of the ‘convenient objects’ of the world.              Lawlor writes (SG, p 14), “‘Numbers are the sources of form and energy in the world.  They are dynamic and active even among themselves…and almost human in their capacity for mutual influence.’ (Theon of Smyrna).  Numbers in the Pythagorean view can be androgynous or sexual, procreators or progeny, active or passive, hetrogeneous or promiscuous, generous or miserly, undefined or indivualized.  They have their attractions, repulusions, families, friends; they make marriage contracts.  They are in fact the very elements of nature.”

Qualities of the First Ten Numbers

Oneness  

            Philosophically, oneness may be taken as representing the quality of Unity.  As such it is called ‘the One’, the all encompasing, the totality, the One which is indivisible because, if divided or multiplied, it would result in more than one;  and the new multitude would be of a different quality.  The One is associated with the mysterious point of origin, at which the percievable world comes into being; the point which may be said to have location without dimension.  At the other end of the scale, so to speak, the One symbolizes the equally mysterious All, the total unity of the entire percievable multitude;  for if the point is the begining of all it  must contain already within it some aspect of all that is to be.  So, the One or quality of Oneness may be thought of as a kind of Alpha – Omega;  a cyclic begining and end;  like the symbol of the Uroburos, the serpent biting its own tail. 

            Geometrically, the One as potentiality may be represented as a point or dot;  and the One as All, may be represented as a circle.  Mathematically, one can be taken as the integer, the unit of addition.  Applied to the material world, the One is the scale unit or a convenient unit of observation, the inch, foot, etc.  The concepts of ‘atom’ and ‘element’ are derived from this idea of the indivisible unit.

            Lawlor says (SG, p12) that the One “perfectly represents the principal of absolute unity and as such has been used as the symbol to represent God.  As a statement of form it can in one sense represent the point -the BINDU or seed in the Hindu mandala – or in another sense it can represent the circle.” 

            Critchlow writes (IP, p9), “The manifestation of an action, object or thought (if it can be defined) necessitates a point of origin or departure, in relation both to the manifestation itself and to the person who is conscious of its emergence.  The point of emergence does not necessarily reveal its causation either in the field of its emergence or in the mind of the viewer.  In the mind the point represents a unitary focus of conscious awareness; in the physical world it represents a focal event in a field which was previously uninterrupted.”   Critchlow, quoting the Ikhwan al-Safa (the medieval, Islamic ‘Brotherhood of Purity’), writes “…everything indivisible is one:  it is so by reason of its oneness that it is indivisible.”   In comments to this author Critchlow added, “In this way  the Ikhwan give oneness a power, not mearly a defination.  Thus it can be considered the binding power of a singular cosmos.”

The Monad.

            The idea or form (in the Platonic sense), or qualitative aspect of the One is called the Monad.  As Theon puts it (MUUP, p12), “As number differs from that which is numbered, in the same way the monad differs from that which is one”.  The monad is indivisible.  It represents the primary entity;  that which cannot be divided.  It includes every-thing in potential.  The monad is not considered by the Pythagoreans to be a number but rather, the principle of number. 

            Theon tells us (MUUP, p 13), “The Pythagoreans claim that the monad is the principal of all numbers and that the one is free of all variety”. 

            Lawlor writes (PN, p 34), “The monad in order to be the ARCHAI of the unit, must be indivisible, that is, its singularity is not made up of any component parts.”  

            Taylor writes (TA, p4), ” …the monad is in discrete quantity that which is the least, or it is the first and common part of discrete quantity, or the principle of it.  According to Thymaridas indeed it is bounding quantity; since the beginning and end of everything is called a bound.  Euclid, in Book seven of the Elements defines ‘the monad to be that according to which every thing that exists is called one.”   

            Taylor adds (TA, p5), ” …it is called the monad either from remaining immutable, and not departing from its own nature;  for as often as the monad is multiplied into itself, it remains the monad;  since once one is one;  and if we multiply the monad to infinity, it still continues to be the monad.  Or it is called the monad, because it is separated, and remains by itself alone apart from the remaining multitude of numbers. (Footnote) Archytas and Philolaus , as we are informed by Theon, called indiscriminantly ‘the one’, the monad, and the monad, ‘the one’.  But according to the best of the Platonists, Proclus, Damascius, & in divine natures, ‘the monad’ is that which contains distinct, but at the same time profoundly united multitude;  and the one is the summit of the many, so that the one is more simple than the monad.  Observe too, that in the sensible universe, the first monad is the world itself, which comprehends in itself all the multitude of which it is the cause (in conjunction with the cause of all).”   

            The One has been associated by Taylor (TA, p 169 & 171) with the male or Light power and with the ‘enlightened’ sun god Apollo. 

            In the TOA (p 37) we read, “Just as without the monad there is in general no composition of anything, so also without it there is no knowledge of anything whatsoever, since it is a pure light, most authoritative over everything in general, and it is sun-like and ruling, so that in each of these respects it resembles the Divinity, and especially because it has the power of making things cohere and combine, even when they are composed of many ingredients and are very different from one another, just as the Divinity made this universe harmonious and unified out of things which are likewise opposed.”   This is another way of saying that the monad, if it contains and bounds multiplicity, also must contain its own opposite.

             The TOA continues (p 37), “So they  [the Pythagoreans] say that the monad is not only Divinity, but also ‘intellect’ and ‘androgyne’.  It is called Divinity because of that aspect of Divinity which is most authoritative both in the creation of the universe and … in all skill and reason:  even if this aspect of the Divine were not manifest itself … in particular matters, yet in respect of its activity it is Intellect, since in respect of its knowledge it is sameness and unvarying.  … the monad, which even if differentiated … is as it were  a creative principle … and does not alter from its own principle, and forbids anything else to alter, but is truly unchanging…”   This passage links the monad to the Divine Intellect which is the source of the Platonic Forms.  It also suggests that the monad is related to the form of the ‘Same’ (as opposed to the ‘Different’);  forms used by Plato in the dialogue of Timaeus to construct the ‘World Soul’.             As to the androgony of the monad, the TOA says (p 38), “So, in short, they consider  [the monad]to be the seed of all, both male and female at once – not only because they think that what is odd is male in so far as it [cannot be divided equally]and what is even is female in so far as it [can be equally divided], and  [the Monad] alone is both even and odd, but it is also taken to be father and mother, since it contains the principles of both matter and form, craftsman and what is crafted …”  

Cyclic Number

             Monad, and number (considered as multitude), can be thought of as having a linked cyclic nature;  the cycle going from the One into the multitude and back to the One. 

            Taylor writes (TA, p 4), “The Monad is therefore, the principal and element of numbers, which while multitude is diminished by subtraction, is itself deprived of every number, and remains stable and firm;  since it is not possible for division to proceed beyond the Monad.  For if we divide the one which is in sensibles into parts, again the one becomes multitude and many; and by a subtraction of each of the parts we end in one.  And if we again divide this one into parts, the parts will become multitude;  and by an ablation of each of the parts, we shall at length arive at unity.” 

            Theon says (MUUP, p12), “If we divide a tangible body in several parts, what was one becomes several, and if each part is subtracted, it will terminate at one; and if we again divide this one into several parts, the multitude will arise out of it, and in taking away each of these parts, we again come to one in such a way that that which is one, as one, is without parts and indivisible.”  This may be thought of as a ‘cycle of creation’. 

            Taylor adds (TA, p6, footnote), “Observe too, that in the sensible universe, the first monad is the world itself, which comprehends in itself all the multitude of which it is the cause (in conjunction with the cause of all).  The second monad is the inerratic sphere.  In the third place, the spheres of the planets succeed, each of which is also a monad, comprehending an appropriate multitude.  And in the fourth and last place are the spheres of the elements, which in a similar manner are monads.  All these monads likewise, are denominated …’wholeness’, and have a perpetual subsistence.”   The ‘inneratic’ sphere is probably that of the fixed stars.  See the Astronomy chapter for a more complete description.

Zero and Number

            The classical system starts with one, not zero.  Zero is not considered a number.  Rather it is the absence of number.  Zero can be taken to represent or symbolize ‘no-thing’.  This concept is relativly recent in intellectual history, dating from the Indian Buddist tradition of emptiness or non-being, of approximately the seventh century AD.  The concept of zero appearantly played no role in the Pythagorean system.

            As Lawlor puts it (SG, p 17), “The origins of this symbol date back to sometime before the eighth century AD, when we have a record of its first written appearance in a mathematical text from India.  It is interesting to note that during the century just prior to this time a particular line of thought had begun to develop in India which found its expression in both Hinduism (through Shankhara) and Buddhism (through Narayana).  This school laid exclusive emphasis on the goal of obtaining personal transcendence and escape from ‘karma’ through renunciation of the natural world…“

            Zero was not used in the ancient world;  nor were any of our current number symbols;  nor was the modern decimal place system.  In early sacred alphabets, like Hebrew or Greek, numbers were represented by letters.  Number science was taught,through the manipulation of physical objects, pebbles or stones (similar to modern glass ‘marbles’), or even with folded sheets of fabric.  Each of the subjects of the Quadrivium has both a mental and an active, physical aspect.  Zero has no place (literally) in such a method.

Twoness, the Dyad. 

            The Monad, as the All, contains everything;  so it must also contain its own ‘opposite’.  This oppositness represents the quality of the Dyad.  The Dyad is created, in one sense, by changing our viewpoint of the One;  or we might say, by reminding ourselves of ourselves as viewers of the One, as distinct from being one with the One.  In another sense, the Dyad is created by the movement or internal division of the One.   With the Dyad appear all relative qualities such as, good-bad, up-down, day-night, etc., which are contained within, and make up the One.  Absolute Oneness, it may thus be said, is made up of the perfect balance of opposites.  This balance is an ideal relationship of 1:1, equality, which may be considered the beginning of proportion. 

            Schwaller writes (SN, p 30), “This first Unity is always, although under different expressions, the idea of the absolute, of the eternal…  This idea contains contraries (i.e., the same nature twice, but opposed in its tendencies), because the idea of the Absolute can only exist as the perfect stabilization of two essentially contrary natures.” 

            Theon says (MUUP, p 66), “The first increase, the first change from unity is made by the doubling of unity which becomes two, in which are seen matter and all that is perceptable, the generation of motion, multiplication and addition, composition and the relationship of one thing to another.” 

            The Dyad is symbolicly related to the Earthly or dark realm, the female power, and the form of the Different.  It is associated (by Taylor, TA, p 174) with the goddess Artemis, Apollo’s twin sister;  goddess of the Moon, whose appearance is different every night. 

            Critchlow writes (Intro. to Adam, NNP, p 11),  “TWO can also be said to be the principle of repetition, division and multiplicity, it is often associated with a negativity because it is the ‘departure from oneness’.  Duality, the first polarization and the beginning of difference.”  

            In the TOA (p45) we read that the Dyad, “…is also called ‘deficiency and excess’ and ‘matter’ (for which, in fact, another term is the ‘indefinate dyad’) because it is in itself devoid of shape and form and any limitation, but is capable of being limited and made definate by reason and skill.”   Here, the Dyad is related to the form of the Different, as the Monad is related to the form of the Same.  The quality of the Different, the indefinate, is that it is not limited by a boundry;  the quality of the Same, the definate, is limited by boundry.

              As Lawlor puts it (SG, p 23), the Dyad comes about from an internal division of the One, not by adding two ones,“From both the metaphysical and natural points of view it is false to say that in order to arrive at two, you take two ones and put them together.  One only need look at the way in which a living cell becomes two.  For one by defination is singular, it is Unity, therefore all inclusive.  There cannot be two Ones.  Unity as the perfect symbol for God, divides itself from within itself, thus creating Two:  the ‘self’ and the ‘me’ of God, so to speak; the creator unity and the created multiplicity.”   This differentiation of two from one can only exist on an ideal level because it must remain in perfect balance or perfect stillness;  a condition not found in the material, perceptible world, indeed, as we have seen, a condition not perceptable at all. 

            Geometrically, the division and separation of the original one point implies the creation of the line or axis between them. 

            The TOA says (p 45), “The Dyad is clearly formless because the infinate sequence of polygons arise, in actuality from triangularity and the triad, while as a result of the monad everything is together in potential, and no rectilinear figure consistes of two straight lines or two angles.  So what is indefinate and formless falls under the dyad alone.”  

            As Lawlor puts it (SG, p 12),  “Two is a quantity, but symbolicly it represents, as we have seen, the principle of Duality, the power of multiplicity.  At the same time it has its formal sense in the representation of the line, in that two points define a line.”   Note that the Monad, as the Same, is represented as a circle which is a completely contained or bounded figure;  while the Dyad, as the Different, is represented as a line, which can be extended indefinately. 

            Neither two nor one, are considered, by the Pythagoreans to be ‘actual’ numbers, but rather the principle of all number;  and two is not necessarily considered an even number, but rather, the cause or form of Evenness. 

   The TOA tells us (p 45), “The Dyad is not number, nor  is it an even number, because it is not actual;  …every even number is divisible into both equal and unequal parts, but the Dyad alone cannot be divided into unequal parts;  and also when it is divided into equal parts, it is completely unclear to which class its parts belong.”  Waterfield adds in a footnote, “It is unclear whether its parts are odd or even, since the monad is both odd and even.”  

Threeness the Triad

            The One, divided from within itself, gives rise to the Two   The relationship of Oneness to Twoness gives rise to Threeness.  Or, we may say that two polar opposites plus the uniting quality of their relationship consitiutes a threeness;  or that Unity plus its two divisions within are a threeness.  Yet another way of expressing this is that two polar opposites must have a relationship, in this case equality.  This connective, or transformative third element creates the triad.  Perceptable structures and events embody the quality of a Triadic cycle in that they all have as their minimal expression, a beginning, a middle, and an end. 

            Mathematically, for Pythagoreans, three is the first ‘actual’ number;  that is, the smallest number whose qualitative state can be realized in the material realm.  With three we can, for the first time, establish the basic proportional number relationship of two extremes and a mean.  Geometrically, three is represented by the triangle;  the prime polygon, three being the fewest number of sides needed to enclose an area.  In Pythagorean music, three ratios establish the structure of the octave;  1:2, 2:3, and 3:4 (as we will see in the Music session). 

            In astronomy, Taylor reminds us (TA, p 180), “With respect to the triad possessing authority in whatever pertains to astronomy … this will be evident from considering that there are three quaternions of the celestial signs, viz. the fixed, the moveable, and the common.”   [This will be further explained in the Astronomy session.]  The TOA (p52) adds, “There are three configurations of the moon – waxing, full moon and waning;  there are three types of irregular motion of the planets – direct motion, retrogression and, between these, the stationary mode;  there are three circles which define the zodiacal plane – that of summer, that of winter, and the one midway between these, which is called the ecliptic …”  So, we see that each of the four subjects of the Quadrivium is, in its own way, founded on a threeness or triad. 

            As Theon describes the triad (MUUP, p 66), “The number two added to unity produces three, which is the first number having a beginning, a middle and an end.  This is why this number is the first to which the name ‘multitude’ applies, for all the numbers less than this are not called multitude (or many) but one, or one and other;  while three is called multitude.  We make three libations to show that we ask everything which is good.  We call thrice woeful those who are at the limit of misfortune, and thrice favored those who are at the limit of success.  The ternary number also represents the first nature of the plane because it is in a sense its image, the first form of the plane being the triangle.  It is for this reason that there are three types of triangles, the equalateral, the isosceles, and the scalene, and also that there are three types of angles, the right angle, whose property it is to be unique, well defined and composed of the equal and the similar, and which causes all right angles to be equal to one another, being in the middle between the acute and the obtuse angles, larger than one and smaller than the other.  As for all the other angles, they are infinate  [we might be permitted to substitute ‘indefinate’] in number and indetermined, since they are either greater or smaller.”

            Critchlow says (Intro. to NNP, p 12), “Thus with the advent of THREE we might characterise that which has a beginning, a middle and an end as the ‘first number’.  That is the first essential set of conditions of the unified principle containing the essential ingredients of number, numbering and the numbered.  It is at just this point in the process of the departure from ONE that we find our first recapitulation of the three essential aspects of ONENESS, which are unity, the unified and the unique.  Also, if we follow Plato and observe the objective laws of geometry, which can be considered to be Number in space, we find that threeness as the triangle (or trigon) is the first shape: hence another way of appreciating its possible role as being the first ‘expression’ of number.  Three points, three lines, three angles: a triangle is a single threeness of three.  Plotinus, in the Platonic succession, places particular emphasis on the three hypostases of the simultaineously proceeding, maintaining and returning modes of all things.” 

            Lawlor writes (SG, p 12), “With three a qualitative transition is made from the pure abstract elements of point and line to the tangable, measurable state which is called a surface.  In India the triangle was called the Mother, for it is the membrane or birth channel through which all the trancendant powers of Unity and its initial division into polarity must pass in order to enter into the manifest realm of surface.  The triangle acts as the mother of form.”               According to Taylor [TA, p 178], citing Nicomachus“in the extracts from him preserved by Photius”  three is associated with the goddess Leto [Latona], mother of the twins, Apollo and Artemis.  As we have seen, Apollo is associated with the Same, and Artemis is associated with the Different.  We may see Leto as the third element necessary to bring the two twin opposites into being;  or we may see a viseral example of how the One may divide from within itself to create the Two, and the relation of the Two to the One create the Three.  The Triad may be seen here to have a birth giving quality, which will be discussed further in regard to the Vesics Pisces, in the Geometry chapter.

Fourness,  the Tetrad

            The Tetrad, may be thought of as composed of two Dyads.  Four is the first number to embody and balance growth by both multiplication and addition, as 2+2 = 2×2 = 4.  In plane geometry, four is the first ‘square’ number, and it can also be represented by the cross, +.   As three represents the cycle of creation;  four represents the created thing.  It is the symbol of the material realm.  Four is the least number that can describe a three dimensional object, i.e. that which is necessary for senseable perception.  The primary solid geometrical form, the tetrahedron, uniquely, has four faces and four vertices.  The tetrahedron represents the first, or ‘highest’ element, fire or light. 

            Four is associated with equality through the square, and with coming into being through the cross.  The balanced intersection of two Dyads keeps their four polar qualities equidistant from each other, creating the ‘right’ angle at the meeting point, figure 01.  The angle is ‘right’ because it is equidistant from either extreme of the duality.  A just individual is sometimes called ‘upright’ or ‘righteous’;  and a right angle of 90o divides a circle into four equal parts. 

Figure 1 – The Tetrad as double Dyad generates the square and the right angle cross

            In astronomy, the rising and setting points of the sun give us the east – west axis, while the noon and midnight positions provide the north – south axis.  These cross axies are the basis of ‘orientation’ [as we will see in the Astronomy session] and of the latitude – longitude system for determining position on the Earth’s surface;  as in the expression ‘the four corners of the earth’.  There are four material elements in the traditional system, fire, air, water and earth;  in terms of modern Physics these are similar to the four states of matter;  radiation or plasma, gas, liquid, and solid.

            Four is also associated by Taylor (TA, p 181) with Hermes, perhaps in his role as protector of the square deal in the market square;  and also with regard to Hermes being the son of Maia, “For the Tetrad is the son of Maia, i.e. of the Duad.”   Although this makes symbolic sense, Graves (GM, p 399) does not substantiate it, translating Maia’s name as ‘grandmother’, he notes that she is the daughter of Atlas. 

            The TOA (p 61) puts the relationship of Hermes to the Tetrad this way, “Squares are, so to speak, not easy to shake from their distain, just like Hermes, who is fashioned in this way.”  In a footnote Waterfield adds, “Square cut statues of Hermes looked down (in distain, our author suggests) on many road junctions.”   See figure 02.  Hermes is more fully discussed in the Astronomy chapter.            Taylor (TA, p182) writes that since the tetrad, ” …causally contains all mundane natures, it may very properly be called the fountain of natural effects.  Because likewise it opens and shuts the recesses of generation, it is denominated … the key bearer of nature, as is also the mother of the gods who is represented with a key.”  This also may be related to the symbol of St. Peter, whose two keys of the kingdom are crossed;  figure 03.

            Any description of the tetrad would not be complete without mention of the Pythagorean Tetraktys.  This, in brief, is a triangular arrangement of ten dots, based on the fact that the sum of the first four numbers is ten;  that is, 1+2+3+4 = 10.  But a proper description of this powerful symbol is best defered until each of the individual numbers in the Decade has been described.

Fiveness, or the Pentad

            The Pentad is considered the middle or median number because it is the Arithmetic mean [the average] between any two numbers that add up to ten, as shown in this so called ‘square of Saturn’ arrangement:

In this same sense, the pentad is likened, by Taylor (TA, p191) and by the TOA (p 71) to the balancing point on a weighing scale.                             

            Five symbolizes the human realm in that humanity stands above earth and below heaven, as five stands midway in the cycle from one to nine.  The human body projects nicely over the pentagon, the two arms, two legs and the head picking up the five verticies, figure 04.  The pentagonal symmetry in humans is continued out to our five fingers and toes.  Within pentagonal geometry, the Ø (PHI) ratio, 1:1.618, is generated.  The √5 = 2.236 = (.618 or Ø) x 2 + 1; figure 05.  This ratio, as we shall see in the Geometry session, plays a key role in the geometry and dimensions of organic growth.  Certain flowers, for example, utilize fivefold symmetry.            Lawlor notes (SG, p 58) “…because of its appearance in the pentagon the golden section can be found in all flowers having five petals or any multiple of five …  The rose family is one of those based on five, as are all the flowers of the edible fruit bearing plants.  Thus five signals to man his proper foods  Five is dominant in the substructure of living forms, while 6 and 8 are most characteristic of the geometry of mineral or inanimate forms.  The plants displaying a sixfold structure, such as the tulip, the lily and the poppy, are very often poisonous or only medicinal for man.  Traditional medicine considered seven petalled plants to be poisonous.  Among these are the tomato and other plants of the belladonna or nightshade family.  The very exotic flowers, on the other hand, the flowers of love such as the orchid, the azalea and the passion flower, are all governed by pentagonal symmetry.”  

            Five is associated with the additive aspect of marriage as it is composed of the female even number two, plus the male odd number three.  As the TOA puts it (p 65), “The pentad is the first number to encompass the specific identity of all number, since it encompasses two, the first even number, and three, the first odd number. Hence it is called ‘marriage’, since it is formed of male and female.“

            Five is also associated with Aphrodite, or Venus, goddess of beauty, through its relationship to the Phi [Ø] or ‘Golden’ ratio, 1:.618;  or as the TOA says (p 73) “…because it binds to each other a male and female number.  Likewise it is called ‘nuptual’ and ‘androgyny’ …”

            There are five ‘Platonic solids’, defined in the TOA (p 65) as “…solid figures with equal sides and equal angles –  the tetrahedron, octahedron,  icosahedron, cube, and dodecahedron.  And Plato says (Timaeus 53-56) that the first is the figure of fire, the second of air, the third of water, the fourth of earth, and the fifth of the universe.”   

            The TOA continues (p 68), “They called the Pentad ‘lack of strife’, not only because aether, the fifth element, which is set apart on its own, remains unchanging, while there is strife and change among the things under it, from the Moon to the Earth, but also because the primary two different and dissimilar kinds of number, even and odd, are  … reconciled and knitted together by the pentad, which is a composite whole consisting of their conjunction …”    

Sixness, the Hexad

            The Hexad is produced from two multiplied by three, or we might say, female x male, as in the multiplicative aspect of marriage.  Two and three are the first numbers that can multiply in their roles as male and female.  One, as noted above, cannot increase by multiplication.  Six is the first ‘perfect’ number in that it is the sum of its whole number fractional parts (numerated by one).  Six has three such parts;  1 is 1/6th of six, 2 is its 1/3rd, and 3 is its 1/2;  1+2+3=6 and 1x2x3=6.  For a more complete discussion see the section on Perfect numbers in the Appendix to these notes. 

            Six may be associated with the comprehending or structuring of physical space.  There are three spacial dimensions depth, width, and height;  each characterized by two Dyadicly opposite directions, forward-back, right-left, and up-down.  Each of these directions can be seen as perpendicular to one of the six faces of a cube, figure 06. 

            As the TOA points out (p 80), “They also called it  [the hexad]‘health’ and ‘anvil’ (as it were, the unwearying one), because it is reasonable to think that the most fundamental triangle of the elements of the universe partake in it, since each triangle is six, if it is divided by three perpendiculars:  for it would be divided altogether into six parts.  That is why there are as many edges to a pyramid (tetrahedron) as there are (6), and as many faces of a cube, and as many vertices in an octahedron (6), and faces of a dodecahedron (12=6×2), and edges to a cube and an octahedron (12) and an icosahedron (30=6×5) and nothing pertaining to their faces vertices or edges is altogether free of the hexad.”   (In a footnote, Waterfield adds “The Greek words for ‘anvil’ and ‘unwearing’ are superficially similar.“) 

            The faces of the Platonic solids are all either equalateral triangles, square or pentagonal.  Figure 07 shows how each of these kinds of faces are formed of six triangles or groups of six triangles.  For further discussion see the Geometry session.  Multiples of six are also used to measure linear, circular or spherical space;  for example, 360 degrees of a circle, 12 signs of the zodiac, 24 hours of the day, 60 minutes of the hour, 12 inches of the foot, the 180 degrees of a triangle.

            The perfection of the hexad also has another dimension;  that of the union of opposites in the soul.  In Plato’s Timaeus (35) we read, “And  [the Divinity] composed  [soul] in the following way and out of the following constituents.  From the indivisible, eternally unchanging Being and the divisible, changing Being of the physical world he mixed a third kind of Being intermediate between them:  again with the Same and the Different he made, in the same way, compounds intermediate between their indivisible element and their physical, divisible element:  and taking these three compounds he mixed them into a single unity …”   There are, then, three components of soul, Same, Different and Being, but each of them is composed of the divisible, and its dyadic opposite, the indivisible;  for a total of six parts. 

            The TOA adds (p 77), “For when soul is present, the opposites which have been admitted by the living creature are reconciled and ordered and tuned as well as possible, as they yield and correspond to each other and hence cause health in the compound … which would not exist together without some harmony;  assuredly as far as soul is present, they can congregate, but when soul departs, then dissolution of all the components of the creature occur.   … 6 is the first number to subsume both a half and a third at once, since it is made up of different and contrary factors – the root of things which are divisible by two, and the root of things which are divisible by three, the dyad and the triad – so that, just as there occurs in it the association of things which are altogether at variance, so the hexad is constituted to bring together and into unison things which are altogether different.”   The mention of harmony, and of 6 ‘subsuming’ the half and third may be references to the structure of the octave;  the 1/2 point of which marks the fifth note and the 1/3 point of which marks the fourth note.  See figure 08 and fthere will be a more complete discussion of the octave in the Mussic session.

            The TOA continues (p77)“…it is also necessary for the largest class of soul to be a solid number – spherical in fact, and not solid only in a male or only in a female way, but in both ways … and since, in this context, 6 is the first to contain the principle of even-odd nature  [6 is an even number whose half is odd], then what is spherical in accordance with it, and not with the pentad, is considered to be more suited to the soul, in as much as it is androgynous, while five has only one or the other identity.”   A spherical number is a cube whose last digit is the same as the last digit of the base;  in this case, 63 = 216.  The spherical nature of the soul is discussed further at various points in the preceeding and following sessions.  The point here is that the arithmetic, harmonic and geometric aspects of the soul can be united through the hexad.

            There is a further geometric perfection of six;  as the circumference of a circle can be laid off perfectly into six parts by its own radius [As we will see in the Geometry session].  Six can also be represented geometrically as the double triangle and in this way generates the star hexagon, figure 09.  The double triangle symbolizes the ‘perfect’ balance between the upward movement or aspiration of earthly forces and the downward movement or grace of heavenly forces, as in the Hebrew ‘seal of Solomon’. 

            As Solomon may be considered a paradgime of wisdom, and as Plotinus reminds us (On Beauty, 6.2), “… every virtue is a purification, and so is even wisdom itself.” , Solomon’s perfectly balanced, double triad seal makes a connection, or draws a thread, between wisdom, virtue, perfection and the hexad.

Seveness, the Heptad

            The Heptad, as a prime, is not created by the multiplication of any other numbers, it is only created by addition of ones.  Nor do multiples of it create any other number within the decade;  as 7×2 = 14.  The other primes below ten do multiply within the Decade;  3×2=6, 5×2=10.  Theon (MUUP, p 68) gives a more complete picture, “Among the numbers contained in the decad, some create and some are created, for example, 4 multiplied by 2 creates 8, and is created by 2.  Others are created but do not create, like 6 which is the product of 2 by 3, but which does not create any other numbers in the decad.  Others create but are not created, such as 3 and 5 which are not created by any combination of numbers, but which create:  3 produces 9, and multiplied by 2 produces 6, and 5 multiplied by 2 produces 10.”

            Seven may be considered a virginal number in the sense that it does not engage in multiplication.  Theon (MUUP, p68) also associates this virginal arithmetic quality of seven with the goddess Athena “because this goddess was not born out of a mother and gave birth to none.”   

            As Graves recounts the myth (GM, p 46), “… Athena’s own priests tell the following story of her birth.  Zeus lusted after Metis the Titaness, who turned into many shapes to escape him until she was caught at last and got with child.  An oracle of Mother Earth then declared that this would be a girl child and that, if Metis concieved again, she would bear a son who was fated to depose Zeus …  Therefore, having coaxed Metis to a couch with honeyed words, Zeus suddenly opened his mouth and swallowed her …  In due process of time, he was siezed by a raging headache … so that his skull seemed about to burst …  Up ran Hermes, who at once divined the cause of Zeus’s discomfort.  He persuaded Hephaestus, or some say Prometheus, to fetch his wedge and beetle and made a breech in Zeus’s skull, from which Athene sprang, fully armed, with a mighty shout.”   A connection of this story to the arithmetic character of seven is made in the TOA (p 99), “They called the Heptad Athena … because it is a virgin and unwed, just like Athena in myth, and is born neither of mother (i.e. of even number) nor of father (i.e. odd number), but from the head of the father of all (i.e. from the monad, the head of number);  and like Athena it is not womanish, but divisible number is female.” 

            Seven is associated with the astronomical heavens.  There are seven ‘planets’ or wandering stars in the ancient cosmology.  There are seven days in the week and seven times the material Tetrad is twenty eight, the number of days in both the lunar and human menstrual cycles.  It may also be noted here that the sum of the numbers 1+2+3+4+5+6+7 = 28.

            There are seven intervals between the eight notes of the musical octave.  The musical fourth is positioned by the ratio of 4:3.  The two right angle sides of the 3-4-5 triangle add up to 7.

            The TOA (p87) quotes Hippocrates as saying, “Seven are the seasons, which we call ages – child, boy, adolescent, youth, man, elder, old man.  One is a child up to the shedding of teeth, untill 7 years;  a boy up to puberty, untill twice 7;  an adolescent up to the growth of the beard, until three times 7;  a youth during the general growth of the body, until four times 7;  a man up to one short of fifty years, until seven times 7;  an elder up to 56 years, until eight times 7;  from then on one  is an old man.”   Traditionally, the human lifespan is 70 years, until ten times 7.            Seven as 4+3 is related to the triadic cycle of proceeding, maintaining and returning (or we might see it here as the three axes of measurement);   plus the four stages of coming into being point, line, plane, and solid.  As Taylor puts it (TA, p197), “The forms or parts of the soul being three, viz. the intellective, the irasible and the epithymetic, four most perfect virtues are produced belonging to these parts, just as of the three intervals (length, breadth and depth) there are four boundries in corporeal increase, (viz. a point, a line, a superficies or surface, and a solid).”   And of course there are seven Liberal Arts.

Eightness, or the Octad

The Octad is the first cubic number, as 2x2x2 = 8;  that is to say,  a cube with length, width and height of two units has a volume of eight units. 

            On the arithmetic of the Octad, the TOA tells us (p 101), “All the ways in which it is put together are excellent and equilibrated tunings.  First it results from the only two numbers within the decad which are neither engenderers or engendered (1 and 7);  then, it results from the two which are even-odd, one potentially, the other actually (2 and 6);  then, it results from the first two odd numbers (3 and 5);  …and fourthly, it is 4 taken twice, and 4 is the only number which both engenders and is engendered  [within the decad].  The consequence is that 8 is completed by means of the first two unengendered numbers, and from their opposites (numbers which engender) and from the number which contains both characteristics.”  An even-odd number is an even number whose half is odd.  There is a section on this topic in the Appendix to this chapter. 

            The TOA next reminds us that the numbers needed to form the musical consonances are complete at 4, and we might add, there are 8 notes in the octave.  The TOA continues (TA, p 102), “Hence they used to call the ogodad ’embracer of all harmonies’ because of this marvellous attunement, or because it is the first to have been attuned and multiplied so as to be equal times equal times equal, which is a most lawful generation.  So when they call it ‘Cadmean’, they should be understood to be referring to the fact that, as all historians tell us, Harmonia was the wife of Cadmus.” 

            Eight is also associated with other female aspects of deity.  Taylor writes (TA. p 199). “According to the extracts from Nicomachus, the names given to this number by the Pythagoreans are as follows:  ‘Universal harmony and Cadmeia, mother and Rhea, the producing cause of females and Cybele, Cybebe and Dindymene, …  With respect to the first of these epithets universal harmony, … the Pythagoreans distinguished musical ratios by this number …”  Taylor then points out that there are eight heavenly spheres, one for each of the seven planets and one for the fixed stars.  Each is assigned a harmonic ratio, as though it were a note in an octave.  He continues (p 200), “Hence the ogodad was called by the Pythagoreans Cadmeia, because Harmony is said to have been the wife of Cadmus.  And Cadmus, as we learn from Olympiodorus, is the sublunary world.”   Sublunary, here means ‘pertaining to the Earth’, as the Moon occupies the lowest heavenly sphere [see Astronomy session for further discussion].

            Taylor continues,  “With respect to the epithets mother, Rhea, Cybele, and Dindymene, the ogodad was doubtless thus denominated from being the first cube, and a cube is the element of earth which is participated by Rhea who is the same divinity as Ceres, and was called Cybele and Dindymene.  As Rhea too is the vivific goddess, the life of which she is the source must be of a feminine characteristic;  and hence the reason is obvious why the ogodad is celebrated as the producing cause of females.”   Rhea is the earth mother, wife of Uranus, the sky father.  In the above passage, Taylor has linked Rhea with Ceres by virtue of their birth giving qualities.  Ceres is the Roman name for the Greek Demeter, the grain goddess whose Mysteries were celebrated at Eleusis.

            Cadmus, whose name according to Graves (GM, p 197) means ‘eastern’ or ‘from the east’, comes to Greece seeking his sister, Europe.  She had been kidnapped by Zeus, who had taken the form of a bull.  During the search Cadmus is forced to kill a serpent belonging to Ares.  Athene then advises Cadmus to plant, or sow, the serpents teeth, from which spring up armed men.

            But, Graves tells us (GM, p196), “… Ares demanded vengeance for the death of the serpent, and Cadmus was sentanced by a divine court to be his bondsman for a Great Year.”

            “When Cadmus had served eight years in bondage to Ares … Athene secured him the land of Boeotia.  With the help of his Sown Men he built the Theban acropolis, named ‘the Cadmea’ in his honor and, after being initiated into the mysteries which Zeus had taught to Iasion, married Harmonia, the daughter of Aphrodite and Ares …”

            “This was the first mortal wedding ever attended by the Olympians … and they all brought gifts.”   The gifts include flutes from Athene, a lyre from Hermes, “and Electra, Iasion’s mother taught Harmonia the rites of the Great Goddess;  while Demeter assured her a prosperous barley harvest by lying with Iasion in a thrice plowed field during the celebrations.  The Thebans still show the place where the Muses played the flute and sang on this occasion, and where Apollo performed on the lyre.”   The Mysteries refered to are, according to Graves, the Samothracian.  The octad is linked to the Mysteries through the earthly, female, birth giving powers.  The Earthly, or sub-lunary world is symbolized by the cube;  and, arithmetically, the octad is the first cubic number.

            Geometricly, the octagon may be produced by the double square, 4 + 4, as shown in figure 10a-d.  The Monad, 10a differentiates into the Dyad,10b.  If the Dyad is considered a dynamic balance of polar opposites, this creates a new center of movement, here called point CM.  Doubling the Dyad while maintaining the balance of opposites creates the right angle cross and the square, 10c.  Doubling this tetrad creates the octagon, 10d.  

            The Octad is associated with the harmony of the elemental forces of the world;  the cube, symbol of the earthly realm, has eight vertices.  The Orphics, according to Theon (MUUP, p69), named eight elemental principles in the oath “By the creator of things forever immortal: fire and water, earth and heaven, moon and sun, the great torch and the black night.”   A possible arrangement of these is shown in figures 10e-f.

Nineness, the Ennead

            The Ennead, is the last number, or the completion of number in the sense that ten may be thought of as a new Monad, the begining of the next cycle. 

            As the TOA puts it (p 105), “That number admits nothing beyond the Ennead , but rather everything circles around within it, is clear from the so called recurrences:  there is a natural progression up to it, but after it there is repetition.  For 10 becomes a monad by the subtraction of one … ennead, and again 11 and 20 become a dyad by the subtraction of either one or two enneads, and 12 and 30 become a triad, and again 100 becomes a monad when 11 enneads are subtracted …“. 

            Taylor (TA, p 202) says, “Thus if from each of the numbers 10, 11, 12, 13, 14, 15, 16, 17, 18, and 19, the number 9 is subtracted, the numbers that remain will be 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.  And vice versa :… For if to each of the numbers 1, 2, 3, 4, 5, &c., 9 is added, the numbers produced will be 10, 11, 12, 13, 14, &c. …  Hence it is not possible there should be any elementary number beyond the ennead.  Hence the Pythagoreans called it Ocean and horizon, because all numbers are comprehended by and revolve within it.”   Waterfield reminds us, in a footnote (TOA, 105), “Oceanus was envisaged as an expanse of water encircling the outer limits of the world.”

            Nine is the first odd square;  3×3=9, and is also the sum of the first two cubic numbers 1 and 8 (the monad being a cube in potential, or we might say, 13 = 1).  As three is the symbol of the cycle of beginning, middle, and end;  three times three is symbolicly the cycle brought to completion or to perfection.  The human gestation period, for example, is nine months. 

            The TOA (p 106) tells us that the Pythagoreans,  “call  [the Ennead] Prometheus because it prevents any number from proceeding further than itself, as is reasonable, since it is thrice perfect and does not lack the advantage of multiplication, but in fact is both the combination of two cubes (1 and 8) and, since it is a square, it alone of the numbers up to it has a triangular number as its square root.”               Waterfield translates Prometheus as ‘not outrunning’.  Graves translates it as ‘forethought’.  Both suggest an awareness of the completion of cycles. 

            As to the ennead being thrice perfect, the TOA (p 61) tells us that the triad  “is special in being both equal to and successive to the monad and dyad, while it is perfect in another way and in itself, because it in particular contains beginning, middle and end.”   In this sense, the ennead is the cycle completed, perfected, or made manifest in three dimensions. 

            Taylor (TA, p204) adds, “The ennead may likewise be said to be the perfector as we learn from Proclus in Tim. p. 298, ‘because it gives completion to the fabric of generation.’  For he says ‘this number is adapted to generation (i.e. the sub-lunary region);  since it proceeds from the monad as far as to the last numbers without retrogression;  and this is the peculiarity of generation.'”    

            The ennead is also associated by Taylor (TA, p 203) with Ares, as the dissolver or slayer, saying, “With respect to the appellation … Mars, we are informed by Martianus Capella, that the ennead was thus called because it is the extremity of the first series of numbers (i.e. of the numbers within ten);  and that the end of all things is from Mars.”  

            The TOA (p 106) gives these associations, “They used to call it ‘Hephaestus’, because the way up to it is, as it were, by smelting and evaporation …” in the sense that Hephaestus is the diety of the fiery hearth, and fire, being the least dense element, rises to the limit;  “… and Hera …’sister-consort of Zeus’ because of its being paired with the monad.”   Hera as 9 stands opposite to Zeus as 1, the monad as beginning and end of the cycle of number;  and Hera as 9 stands next to Zeus as 10, the next cycle of the monad.

Tenness, or the Decad

            The Decade, embodies the completed cycle, and the begining of the new cycle;  it becomes a new monad.  Taylor (TA, p 207) says, “We have already observed that all numbers revolve within the number 9, and that on this account it was called by the Pythagoreans the ocean and the horizon;  so that it is in reality paronymous to the monad.  This being the case, it may seem wonderful that the decad should also be considered analogous to the monad.  The reason however of this is, that the first offspring of a monad is likewise a monad, which contains in a more expanded mannor all that consists more contractedly, and as it were spermatically in the prior monad.  Hence both the ennead and the decad are monads; but in the former all numbers subsist more unitedly, and in the latter with more abundant diffusion  and separation.’

            The decad is associated with the idea of Cosmos, the All, or the all encompassing.  The decad is said to contain all the other numbers and implicitly all possible numbers;  just as the monad contains the possibility of all other numbers, but in an undifferentiated state. 

            In the TOA (p 110), we read, “They called it ‘universe’ because all things are arranged by it both in general and in particular …”  Waterfield adds a footnote: “Kosmos (universe) means literally ‘order’ or ‘arrangemment’.”    The decad contains five even and five odd numbers and the TOA (p 114) quotes Anatolius as saying, “The decad is potentially generated by even and odd: for ten is five times two. …  It is called ‘power’ and ‘all fulfiller’, because it limits all number by encompassing within itself the whole nature of even and odd, moving and unmoving, good and bad. ”  Speusippus is quoted to add that the decad contains, “… an equal amount of prime and incomposite numbers  [1, 2, 3, 5, and 7], and secondary and composite numbers  [4, 6, 8, 9, and 10].”  These terms are discussed later in the Appendix to these notes.

            The TOA continues (p 102), “They called it ‘All’ because there is no natural number greater than it, but, if one thinks about it, number recures and circles back, in a sense, to the decad;  for 100 is 10 decades, and 1,000 is 10 hundreds, and 10,000 is10 thousands, and similarly any other number recurs and retrogresses either to the decad or to some number within the decad.  Anyway, the reduction and returning of all number to it is manifold.”

            The TOA (p 111) relates the decade to the structure of the uinverse, “Given that theologians claim in unison that Necessity occupies the most remote rim of the whole heaven, and perpetually drives and urges on the whole rotation with an adamantine and indefagitable whip, then the decad would be ‘Necessity’ since it circumscribes everything and, by mingling things with one another and again separating them, it imbues things with change and continuity.”   This description seems to be related to the Myth of ER, in Plato’s Republic (616c).  This model will be more fully described in the Astronomy session.

            The TOA continues (p 111), “The spheres of the universe are ten and fall under the decad.  It is called ‘Atlas’ because in myth the Titan carries heaven on his shoulders;  as Homer says: ‘He holds the great pillers which keep Earth and heaven apart’;  and the decad holds together the principles of the spheres, as if it were a diameter of all of them, which both turns them around and limits them, so that they can be best maintained.”    The 10 spheres may be understood to be:  the earth, the 7 planets, the the zodiac or fixed stars,  and the sphere of Divinity.  Waterfield suggests the 10th sphere is a ‘counter-Earth’ and he adds in a footnote (TOA, p90); “The Counter-Earth is an invisible counterpart to Earth.  It is the body closest to the central fire or hearth, but is invisible to us because the inhabited part of the Earth always faces away from it.”   Your author knows nothing else of this notion.

The Tetraktys  

            The decad is the sum of the first four numbers, 1+2+3+4=10.  When each integer is taken as a dot and arranged in a trianglular pattern, as in figures 11 & 12, the Decade is known as the Tetraktys. 

            Taylor (TA, p 205) quotes Proclus on the relationship of the tetrad to the decad, “Proclus likewise informs us … ‘that divine number proceeds from the undecaying retreats of the monad, till it arrives at the divine tetrad which generated the mother of all things, the universal recipient, venerable, the boundry of all things, immutable and unwearied, and which both the immortal gods and earth born men denominate the sacred decad.’  Proclus adds, ‘that by the monad and its undecaying retreats in these verses, that occult cause, the one being, … is signified;  but that divine tetrad is the evolution into light of intelligible multitude, which the dyad that is between the monad and tetrad unfolds;  and that the decad is the world, which recieves the images of all the divine numbers, which are supernally imparted to it’.” 

            In the TOA (p 55) we read, “Everything in the universe turns out to be completed in the natural progression up to the tetrad, in general and in particular, as does everything numerical – in short, everything whatever its nature.  The fact that the decad … is consumated by the tetrad along with the numbers that preceed it, is special and particularly important for the harmony which completion brings;  so is the fact that it provides the limit of corporeality and three dimensionality.” 

            Theon says (MUUP, p62), “The importance of the quaternity obtained by addition (that is to say 1+2+3+4) is great in music because all the consonances are found in it.  But it is not only for this reason that all Pythagoreans hold it in high esteem:  it is also because it seems to outline the entire nature of the universe.  It is for this reason that the formula of their oath was:  ‘I swear by the one who has bestowed the tetraktys to the coming generations, source of eternal nature, into our souls’.  The one who bestowed it was Pythagoras, and it has been said that the tetractys appears to have been discovered by him.” 

            In his Forward to the TOA Critchlow writes (p 9), “That ten is ‘complete at four’ is a well known Pythagorean paradox based on the simple cumulative progression of 1+2+3+4 = 10;  or in the direct manner of those who had no separate number symbols”   [as in figure 11].

            “However there is obviously an allegorical reasoning which allows paradox to play a free part in the ‘generation’ or simultaneous ‘being’ of the first nine (or ten) ideal numbers.  … we have a most interesting Pythagorean idea to contend with – that ideal number is not necessarily subject to a sequential or causal progression from one through to ten, but is rather a unity with ten essential or potential qualities, simultaineously  present in the Decade or Tetraktys.”

            The tetraktys can be seen as a model of ‘coming into being’.  Continuing in his forward to the TOA (p10) Critchlow reminds us, “… there is the possibility of understanding the shadow allegory of Plato’s cave in terms of the number of points, which we might call ‘dots’, representing ‘incomparable sets’ of one dot for oneness, two dots for duality, three dots for triplicity, four dots for quaternity, and so on.  In this case the ‘shadows’ would work in reverse, as it were.  From the pure, indescribable light of unity, the first visual evidence, we might say, is for the units to become surface ‘dots’, which eventually become matter as spheres, i.e. projecting into dimensions.”

            “In this way we can see a progression that embraces Plato’s doctrine that ‘images’ in the human mind reflect the universal source in the mind of the Divine Artificer.  So we get one reflecting as a single dot or shadow on a surface for us to ‘imagine’ or remember oneness from.  If we should take the Pythagorean tradition literally, the first and  comprehensive evidence of ‘ideal’ or archtypal number is an array of dots in triangular form  [see figure 12].  This pattern might be called, in current usage, a refraction pattern of a single triangle. … This Decade or pattern can be taken both as a graphic (differentiated) form of ten dots in four rows and/or as one triangular form outlined in nine dots with a center point.  From the above pattern we can see how the Tetraktys of the Pythagoreans may represent both a minimal oneness and a maximal tenness simultaineously.”

            Some correspondences to the tetraktys are indicated in the table below:

The Pythagorean Lamda

            The Pythagorean Lamda is a geometric arrangement of the double and triple number series.  Lambda, the Greek letter  ‘L’, is shaped like an inverted ‘V’.:         This is a model of coming into being used by Plato in the Timaeus (35).  He starts at the first stage, with unity, at the apex, and proceeds downwards with the double series, the powers of 2, on one side and the triple series, the powers of 3, on the other.  The second stage is the principle of odd and even, expressed in the first odd and even numbers, 2 and 3.  In the third stage, that of the 2 dimensional plane, the odd and even principles are ‘squared’, expressed in the numbers 4 and 9.  In the fourth stage the still idealized 2 dimensional forms enter the solid, 3 dimensional, perceivable world, expressed in the first two cubic numbers, 8 and 27.

            The chart below shows the Lambda and the four stages of coming into being.  Also shown are some of the cycles of four mentioned by Theon (MUUP, p 63).  The geometry of the elements are more fully discussed in the Geometry chapter.

            Critchlow, in the introduction to the TOA (p 13) points out that the Lambda is a tetractys with three positions missing.  Comparing the Lambda and the Tetractys, he points out that the Lambda consists of seven numbers, “…we have seven positions representing two progressions, which leave us with the impression that there could be three more ‘points’ or positions within the open triangle, so that if filled the model would become like the sacred Tetraktys of the Pythagoreans or the triangular ‘four’ representing the full decad.”  He then fills in the three empty positions, figure 13.  “If we start from the monad … and follow down the right hand slope or angle the fourfold progression is 1×3=3, x3=9, x3=27.  If we now shift down from 1 to the position of 2 and follow a similar angle down through the pattern, we pass through the center point, which becomes 2×3=6, and arrive at the point at the base, which becomes 6×3=18.  If we do the same with the diagonal that moves down from 4, we have 4×3=12.”

            “In summary, we might say that the point positions, one by one, represent the decade; … the allocated number progressions represent ‘states of being’ of the refraction of one, we might say, and particularly of two and three respectivly as they move from plane to cubic form and thereby provide a pattern that has within it the essential numerical harmonies.  In particular, this decad contains those numerical harmonies proposed by Plato for such a vital story as the generation of the ‘world soul’.”   Critchlow goes on to show, in figure 14, how the three means and the musical ratios are present in the ten allocated numbers.  Finally, he shows, in figure 15, that by adding 1 to each number in the Lambda decade, a new set of ten numbers is produced, each of which has astronomical significance.  The Lambda- Tetraktys may thus be linked to the four subjects of the Quadrivium;  arithmetic, in the three means;  geometry, in the triangular structure;  music, in the octaval ratios;  and astronomy, through a kind of ‘inflation’ to a new monadic level, just as the Heavenly may be thought of as being in a ‘higher’ sphere than the Earthly.  

A Mythic Journey of Number.

            The chart that follows illustrates a cyclic, spiritual or mythic journey from unity, through multiplicity, and the return to a new unity.

            The journey begins in undifferentiated unity (1).  The first step is separation, or recognition of polarity, self and other (2).  Coming into being requires structure (3) and the substance of materiality (4).  Once in the material world we seek the balance of the mean or median (5), leading to the perfection of virtue (6), ascendance to higher realms (7) and experience of the harmony of the spheres (8).  When the experience is completed (9), the new integration is achieved (10).

            The above presentation may be seen as a way of viewing the qualitative aspect of number that is quite different from the contemporary view of number as limited to quantity or mathematics.  The above chart should not be considered as definitive or exclusive in any way;  but rather might be useful a seed for thoughtful meditative attention by those wishing to approach the qualitative states described.

Elements of Arithmetic

            The following is a synthesis of Nicomachus, ‘On Arithmetic’ written c. 100 AD;   Theon of Symerna’s ‘Math Useful for Understanding Plato’, written in the 4th century BC and Thomas Taylor’s ‘Theoretic Arithmetic written in the late 18th century’.

Generation of Number

            Generation of number is either by addition or multiplication.  Addition means increase by units of one.  All numbers can be created by continuous adding of one, that is 1+1+1+1+1…  The process is one dimensional or linear.  Some numbers called ‘primes’ can only be generated by addition of monads.  They are not ‘produced’ by the multiplication of other numbers.  Multiplication is the process of increase in two or more dimensions.  That is, two groups of three units produces six units, whereas three units plus two units produces five units.

Odd & Even  

            Two is the first even number.  Three is the first odd number.  Even numbers can be separated into two equal parts and odd numbers can be separated into two equal parts with a remainder of one.  Odd and even numbers thus alternate.    One is both even and odd because as The One, the Monad,  it contains the polar opposites which ‘even and odd’ are. If one is added to an odd number it becomes even. If one is then added to that even number, it becomes odd.  A progression of numbers differing by two will be entirely composed of odd numbers or enitrely of even numbers.  The duality of odd and even arises from the bi-polar principal of the Dyad.

            [I.6.3 -4]  “Everything that is harmoniously constituted is knit together out of opposites a,d of course, out of real things:  for neither can non-existant things be set in harmony, nor can things that exist, but are like one another, nor yet things that are different but have no relation to one another.  It remains accordingly, that those things out of which harmony is made are both real, different and have some relation to one another.”

            “Of such things therefore, scientific number consists:  for the most fundamental species in it are two, embracing the essence of quantity, different from oneanother and not of a wholly different genus, odd and even, and they are reciprocally woven into harmony with each other, inseparably and uniformly by a wonderful and divine Nature … .”

            Theon  writes (MUUP, p 14), “Initally the numbers are divided into two kinds: those which are called even and those which are called odd.  The even numbers are those which can be divided into two equal parts, such as two and four;  odd numbers on the contrary, are those which  can only be divided into unequal parts, such as five and seven.  Some have said that the first odd number is the Monad (unit) for even is the opposite of odd, and the Monad is necessarily either odd or even.  Now it cannot be even since, not only does it not divide into two equal parts, but it does not divide at all; therefore unity is odd.  If you add an even number to another even number, the total will be even; now unity added to an even number gives an odd number; therefore again, unity is not even, but odd.  However, Aristotle says, in ‘The Pythagorean’ that ONE participates in both natures.  Indeed, added to an odd number it gives an even number, which it would not be able to do if it did not participate in both natures.  This is why it is called ‘odd-even’.  Archytas also appears to have had this feeling.  The first idea of the odd is the Monad, as also in the world, the quality of odd is attributed to that which is defined and well ordered.  On the other hand, the first idea of the even is the indefinate Dyad, which causes that which is indefinate, unknown and disorderly to be attributed to the quality of even, in the world also.  This is why the dyad is called indefinate, because it is not defined, as is the Unit (or monad).  As for the terms which follow in a continued series, beginning with unity, they always increase by an equal quantity, each surpassing the preceeding one by a unit.  But in the measure of the terms 1, 2, 3, 4, 5, 6,  the ratio of the number 2 to unity is double;  that of 3 to 2 is sesquialter (1+1/2) finally that of 6 to 5 is sesquiquintan (1+1/5).  Now the relationship 1+1/5 is smaller than 1+1/4; 1+1/4 is smaller than 1+1/3;  1+1/3 is smaller than 1+1/2; and finally 1+1/2 is smaller than 2.  And one would find that the relationship would decrease in like manner for the other numbers.  It can thus be seen that the sucessive numbers are alternately odd and even.”

Even Numbers 

            There are three kinds of even numbers.  They are known as the EVENLY-EVEN, the EVENLY-ODD and the UNEVENLY-EVEN.

            EVENLY-EVEN numbers are those that can be divided into two other smaller even numbers all the way to 1.  For example, 64;  64/2=32, 32/2=16, 16/2=8, 8/2=4, 4/2=2, 2/2=1. The evenly even numbers are generated by the double progression, 1, 2, 4, 8, 16, 32, 64, 128 …  In any segment of this progression, if the number of terms is even, the product of the means equals the product of the extremes.  For example, 1, 2, 4, 8, 16, 32.  8 x 4 = 32, 16 x 2 = 32 and the extremes, 1 x 32 = 32.  If the number of terms in the progression is odd then the square of the median and the products of the means equal the product of the extremes.  For example, 1, 2, 4, 8, 16.  42 = 16, 2 x 8 = 16 and the product of the extrems, 1 x 16 = 16.  The sum of the terms in an evenly-even progression is equal to the last term minus one, for example, 1, 2, 4, 8;  1+ 2 = 4 – 1,  1 + 2 + 4 = 8 – 1.

            EVENLY-ODD numbers can be equally divided only once, then an odd number is reached, preventing further even division, for example, 18/2 = 9.  The progression that generates them starts from 2 and increases by increments of 4;  2, 6, 10, 14, 18, 22…  This progression can also be generated by multiplying the odd sequence by two.  Thus, 1, 3, 5, 7, 9, x 2 = 2, 6, 10, 14, 18.  When any given term is divided, if the divisor is even the quotient will be odd.  If the divisor is odd, the quotient will be even.  For example, 18/2 = 9 and 18/3 = 6.  In progressions with an even number of terms, the sum of the means is equal to the sum of the extremes, for example, 6 + 10 = 2 + 14.  In progressions with an odd number of terms, the median is equal to one half the sum of the extremes, for example, 6 = [10 + 2] ÷ 2, 10 = [6 + 14] ÷ 2.  In the progression of the evenly even numbers only the number one recieves no division, but in the evenly odd progression each term can be evenly divided only once.  These kinds of numbers can thus be thought of as opposites or polarities.

            UNEVENLY-EVEN numbers mediate between them.  These can be equally divided more than once but not as far as to the number one. For example, 28. 28 ÷ 2 = 14, 14 ÷ 2 = 7.  Their progression is generated by multiplying the progression of odd numbers by the progression of evenly-even numbers starting with four.  3, 5, 7, 9, 11, 13, 15…x 4, 8, 16, 32, 64, 128, 256…  This yields the series 12, 20, 24, 28, 36, 40, 44…  Divisors and quotients may or may not be of the same polarity.  For example 24 ÷ 6 = 4, 24 ÷ 8 = 3.

            The three types of even numbers can be graphically represented in this chart, figure 16:

            In the first vertical row (1V) are the odd numbers.  In the second vertical row (2V) are the evenly-odd numbers.  In the first horizontal row (1H) are the evenly-even numbers.  The remaing numbers in rows 3V and higher are unevenly-even numbers.

            Odd numbers in row 1V can not be divided into an equal number of parts.  Evenly-odd numbers in row 2V can be divided into equal parts only once, 10/2=5.  Numbers in row 3V can be equally divided three times and numbers in row 4V can be equally divided four times;  and they will yield the odd numbers in row 1V, for example, 20/4=5.  In row 8V;  into eight equal parts, 56/8=7.

            In the horizontal rows of even numbers:  If the number of terms in a sequence is odd,  12, 24, 48,  then, 12×48=242.  24 is thusthe geometric mean between 12 and 48.  It is expressed mathematically as A:B=B:C, B=√AC, AC=B2.  If the number of terms is even, then the product of the means  equals the product of the extremes.  For example, 12, 24, 48, 96; 12×96=24×48.  In the vertical rows of even numbers if the number of terms in a sequence is odd, 12, 20, 28,  then, 12+28/2=20.  20 is thus the arithmetic mean between 12 and 28.  It is expressed mathematicly as B=AC/2.  If the number of terms is even, 12, 20, 28, 36, then, 12+36=20+28.

            Even numbers have the property of being PERFECT, DEFICIENT, or ABUNDANT. 

            PERFECT NUMBERS are equal to the sum of their equally divisible whole number parts;  or said another way, perfect numbers are equal to the sum of their factors.  For example, 6 is equally divisible into six parts of one unit each, or two parts of three units each, or three parts of two units each;  6/6=1,  6/2=3,  6/3=2,  1+2+3=6.  Note that 6/4 and 6/5 do not yield whole numbers.  In the case of 28 we have,  28/28=1, 28/14=2, 28/7=4, 28/4=7, 28/2=14.  1+2+4+7+14=28. 

            ABUNDANT NUMBERS are greater than the sum of their equally divisible parts.  For example, 12.  12/12=1, 12/6=2, 12/4=3, 12/3=4, 12/2=6.  1+2+3+4+6=16 which is greater than 12.  Or 24;  24/24=1, 24/12=2, 24/8=3, 24/6=4, 24/4=6, 24/3=8, 24/2=12.  1+2+3+4+6+8+12 = 36 which is greater than 24. 

            DEFICIENT NUMBERS are less than the sum of their equally divisible parts.  For example 8;  8/8=1, 8/4=2, 8/2=4.  1+2+4=7 which is less than 8.  Or 14; 14/14=1, 14/7=2, 14/2=7;  1+2+7=10 which is less than 14.  Multiples of perfect numbers are abundant.  Divisions of perfect numbers are deficient.  For example, 3  6  12;  3 is deficient, 12 is abundant.  Or 2  6  18;  2 is deficient, 18 abundant.  6 is the Geometric mean between the multiplied and divided extremes.

            Perfect numbers are generated starting with the evenly-even (EE) series.  From this, generate a series of odd numbers created by the sum of the EE terms up to that term.  1+2=3, 1+2+4=7.  If the odd number is a prime multiply it by its corresponding term in the EE series.

            EE #              1   2    4     8    16    32      64   128    256     512    1024    2048   4096

            Odd #            1   3    7   15    31    63    127   255    511   1023    2047    4095   8191

              Perfect #        1   6   28    *   496     *   8128      *       *        *         *          *         *

            Asterisks indicate numbers that are not perfect.

            Taylor (TA, p28, in paraphrase) likens the abundant numbers to Briarus, the hundred handed giant and to Gerion, the triple bodied.  The deficient numbers he says, are like the one eyed Cyclops.  The perfect numbers are like the mean between polar extremes and in the sense they are like virtue.  Like virtues, Taylor tells us, perfect numbers are rare, few and generated by strict order.  Abundant and deficient numbers are like vices;  they are very numerous and not limited.

Odd Numbers

            There are three types of odd number, PRIME or INCOMPOSITE, COMPOSITE and RELATIVE INCOMPOSITE. 

            PRIME NUMBERS can be divided into equal parts only by one.  They are generated only from simple addition by adding units of one.  They are not multiples of any other numbers.  The progression of primes is,  1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, … 

            COMPOSITE NUMBERS are multiples of primes.  The progression is 9, 15, 21, 25, 27, 33, 33, 35, 39, … 

            A number may be RELATIVLY INCOMPOSITE in relation to another number if they have no common divisor, for example 9 is relatively incomposite when compared to 25.  The squares of the primes have no common divisors,  9, 25, 49, 121, 169 … To find if two odd numbers have a common divisor, subtract the smaller from the larger untill one or some odd number is reached.  If one is reached then the numbers have no common divisor.  If an odd number is reached, then that number is the common factor.  For example, 9 & 29.  29 – 9 = 20,  20 – 9 = 11, 11 – 9 = 2,  9- 2 = 7,  7 – 2 = 5,  5 – 2 = 3,  3 – 2 = 1, there is no common factor.  Or 9 & 21.  21 – 9 = 12,  12 – 9 = 3,  9 – 3 = 6,  6 – 3 = 3,  3 is the common factor. 

The Sieve 

            In the odd progression, multiples of three occur every third term, multiples of five occur every fifth term, and so on.  Primes and composites can be graphically related by the following chart which Taylor (TA, p23) tells us “Eratosthenes denominates a sieve”.   A number that has only one entry in its vertical row, opposite itself, is a prime, figure 17.

Ratio & Proportion

 Equality & Inequality 

            When compared to something else, that which is equal is neither greater nor less; but is the same magnitude.  That which is unequal when compared to something else, is either greater or less.  These qualities of greater and lesser may be thought of as Dyadic opposites.  There are five types of greater inequality;  Multiple, Superparticular, Superpartient, Multiple Superparticular and Multiple Superpartient.  Lesser inequality is denominated by the same terms, using the prefix ‘Sub’.  In the sections that follow, the number being compared is called the ‘leader’ and the number to which it is compared is called the ‘attendant’.

            MULTIPLE RATIO.  The leader, when compared to the attendant, contains it in some whole number multiple.  Multiple enequality is first seen in the natural sequence 1, 2, 3, 4, 5, 6,… All the terms are multiples of one.  Thus all numbers are multiple inequalities of One.  For example, 2:1, two is the leader and it contains one, the attendant, twice.  This is a double ratio.  3:1 is a triple ratio, etc.  In the ratio 1:2, one, the leader, is contained by 2, the attendant, twice.  This is a sub-double ratio.  The number of multiple inequalities is unlimited.

            The table of multiple ratios is called by Taylor (TA p 38), “…subtle, …scientific,  and, …delightful” ,  adding, “This table is called Pythagorean, from Pythagoras, who is said to be the author of it.”    It is used by Taylor to show that multiple inequality generates the other types of inequality.  The reader may recognise it as the ‘Multiplication Table’ studied in Primary school, figure 18.

In the first vertical and horizontal rows are the natural series.  All the terms in the double series are even.  Terms in the triple series are alternatly odd and even.  Each row differs by one addition of the natural series;  that is, the four series minus the three series equals the natural series.  The ratios of terms in the 3rd row to terms in the second row is 3:2 and so on for all rows.  All the underlined numbers are squares.  Numbers adjacent to the squares are unequalateral in that they have factors that differ by one.  6 = 2 x 3,  12 = 4 x 3.  The squares and unequalateral numbers form a  ‘stepped’ pattern in the chart.  They form the series,  2  4  6  9  12  16  20  25  30  36…  Between each square is an unequilateral number.  The numbers on either side of the square are equal to twice the square.  2 + 6 = 4 x 2.  Of the four corners of the chart, 1 and 100 are squares;  and 10 is the root of 100.  10×10=100×1.

            SUPERPARTICULAR RATIO.   When the leader contains the attendant and a whole number part of it once, the ratio is superparticular.  For example: 4:3, four contains three and one third part of three.  The initial terms of superparticular ratio series’ always differ by one.  The two smallest numbers that can be compared this way are 3:2 (2:1 is a multiple ratio).  3 contains 2 and 1, which is half of 2.  This 1 &1/2 to 1 comparison is called a SESQUIALTER ratio.  The series of such ratios is generated by comparing  the triple (the natural series x 3) and double series (natural x 2).

            Triple            3    6    9   12   15   18   21   24   27    30    

            Double          2    4    6     8   10   12   14   16   18    20    

The ratio 4:3, or 1 &1/3 to 1 is called SESQUITERTIAN.  The series is produced by comparing the quadruple and triple series..

            Quadruple       4    8   12   16   20   24   28   32

            Triple              3    6     9   12   15   18   21   24

The ratio of 5:4, or 1 & 1/4 to 1,is called Sesquiquartan. The ratio of 6:5, or 1 & 1/6 to 1, is called Sesquiquintan.  Note that the first terms of a superparticular ratio series always differ by one.  The second terms in the series differ by two, the third terms by three, etc.

            SUPERPARTIENT RATIO.   When the leader contains the attendant and a whole number part of it more than once, it is called a superpartient.  The initial ratio of this type is 5:3 because, 3:2 is sesquialter, 2:3 is sub sesquialter,  3:3 is equality, and  4:3 is sesquitertian.  The series is:

            Natural series from 3         3   4   5    6    7    8    9   10         

            Odd series from 5              5   7   9  11  13  15  17   19        

5:3 is called super bi-partient, as  5-3=2.  5:3 is also called super bipartient – tertian or super bitertian. This term signifies that the leader equals one and two thirds of the attendant.  7:4 is called super tripartient.  7 – 4 = 3.  And so on.  2 x (5:3) = 6:10 and this is also super bipartient.

            MULTIPLE Superparticular & Superpartiant ratios are produced when the leader contains the attendant more than once plus an additional portion.  These ratios begin with 5:2.  1:2 is sub double.  2:2 is equality.  3:2 is sesqualter, 4:2 is double.  The series of initial ratios is:

            Odd series from 5               5     7     9    11    13    15    17    19     21    

            Natural series from 2          2     3     4      5      6      7      8      9     10    

The double sesqualter (2 & 1/2 to 1)series is:

            Multiples of 5 from 5          5     10     15     20     25     30

            Multiples of 2 from 2          2       4      6        8     10     12

The double sesquitertian (2 & 1/3 to 1) series is:

            Multiples of 7 from 7          7     14     21     28     35     42

            Multiples of 3 from 3          3       6       9     12    15      18

Triple sesquialter (3 & 1/2 to 1):

            Multiples of 7 from 7          7     14     21     28      35     42

            Multiples of 2 from 2          2       4       6       8      10     12

Triple sesquitertian (3 & 1/3 to 1):

            Multiples of 10 from 10    10      20    30      40      50     60

            Multiples of 3 from 3          3        6      9      12      15     18

All of the series used in the the above sections are found in the table of multiple ratios.  That table is indefinately extendable.

Inequality Proceeds From Equality 

            All types of inequality originate from equality.  Equality is the “mother and root”  of them and “pours them forth”  says Taylor (TA, p47).  The following, he says, is a “profound dicipline”.   The nature of Good is definate.  The nature of Bad is indefinate.  It is Goodness “which by the impression of itself”  bounds the indefinate.  In the Soul, the vestage of  Divine Goodness moderates the inequality of the Soul’s motion.  This is the rational power which restrains the emotional powers, which are like inequality in that they are indefinate.

            Taylor’s arithmetic analogy proceeds thus:  Let there be three equal terms.  First, multiple ratios are produced.  Then multiples are converted to superparticulars.  These are then converted to superpartients.  There are three rules for performing the conversions.  l, The first term remains equal to the first term.  2, The second term is equal to the first plus the second.  3, the third term is equal to the first term plus twice the second term plus the third term.

            We have seen how inequality proceeds from equality.  Inequality may also be reduced to equality.  All ‘things’ says Taylor are composed of ‘elements’ and all things can be resolved into such elements.  Letters are elements of speech, as they create and bound sylables.  Sounds are the elements of music.  The percievable world is composed of the four elements. Inequality is bounded by equality so it should be resolvable into equality.  The rules for reduction are, l, first term remains the same, 2, the second term is equal to the second term minus the first, and 3, the third term is equal to the third term minus the first minus twice the second.  For example:

            Reduces to                             4                4                    4               This is equality.

            Taylor concludes: ‘Unity is the principle of quantity.’

Tables of Ratios 

            Each multiple number will be the first of as many other ratios as it has numbers appearing under it in the tables that follow.  In each of the tables, if the first horizontal series is double, all horizontal terms are doubles.  If triples, then all horizontal terms are triples, etc.  The numbers along the angular lower edge of each table are the next higher order of multiple.  If the series is double, the angular numbers are triples.

            Extension of the double series:

The first double, 2, has one sesqualter ratio possible, 3:2.  Three is odd;  it has no whole number half;  so 3 does not have a whole number sesquialter ratio.  Four is the second double.  There are two sesqualter ratios possible, 6:4 and 9:6.  Nine has no whole number half.

            The extension of the triple series:

            The first triple has only one sesquitertian ratio, 4:3.  Four has no whole number third.  Nine is the second triple.  There are two sesquitertian ratios possible, 12:9 and 16:12.  Sixteen has no whole number third.

            Extension of the quadruple series:

Number into Music 

            The additive aspects of ratio bring us to the place where number merges into music.  Intervals produce multiples.  The interval of the octave, the double ratio, is composed of a sesqualter and a sesquitertian ratio, figure 19.

Number into Geometry 

marbles or dice.  Relationships between numbers was visually and tactily constructed.  When represented this way, number relationship generates geometric form. In the following paragraphs the Monadic unit will be represented as 1

            PLANER NUMBERS have two factors, as planes can be described by two numbers, length and width for rectangles, or base and altitude for triangles, etc.  Planer numbers are arrainged as polygons such as triangles, squares, rectangles, pentagons, hexagons, etc.  The Monadic unit contains within itself all the other intervals and forms, so it is not properly considered as a planar number but may be thought of as the seed of all of them.

            RECTILINEAR NUMBER   One may be considered a square with its side equal to 1.  If another unit is added we have 1 1.  This is considered as a rectangle whose area is two units and whose sides are 1 x 2.  A rectangle whose sides differ by one is called UNEQUALATERAL and its area is an unequilateral number.  If another Monadic unit is added this can be arrainged 1 1 1.  The side ratio of 1:3 is called an OBLONG number.  The sides of an oblong number always differ by two or more.  But three units can also be arranged:

                    TRIANGULAR number is the first type of polygonal number;  as triangles may be considered the primary geometric shape in that they have the fewest number of sides and angles.  Succesive terms in the natural series 1, 2, 3, 4, 5, when added together produce triangular numbers.  Adding the terms in the natural series to one yields triangular numbers.  Each successive layer of number is called a GNOMON.  The side of the triangle is equal to the last gnomon.  One as we have said, is not a triangular number but is the ‘seed’ of all numbers and forms.  When two is added to one, the triangle arises.  The first gnomon is 1.  The second gnomon is 2.  1 + 2 = 3, the third gnomon.  The next gnomon is equal to 1 plus the last gnomon figure 20.

SQUARE numbers are produced by multiplying any number by itself.  2 x 2 = 4, 3 x 3 = 9,  4 x 4 = 16,  5 x 5 = 25. The Geometric mean between any two square numbers is an unequilateral number.  For example, the mean between 1 and 4 is found by √1 x 4.  The mean is 2.  2 x 2 = 1 x 4.  Or in 4:9, the mean is equal to √4 x 9 = √36.  The mean is 6.  The square numbers however, are the Arithmetic means between unequilateral numbers.  For example, 2, 6.  The Arithmetic mean is equal to the sum of the extremes divided by two.  6 + 2 = 8,  8 ÷ 2 = 4.  The square number 4 is the mean.  Square numbers are alternately even and odd.  1, 4, 9, 16, 25, 36, 49, 64, …  The difference between terms increases by two, 3, 5, 7, 9, 11, 13, …  This series when added to the square preceeding it gives the next square,  1+ 3 = 4,  5 + 4 = 9.  The odd series is also the series of square gnomons.  The square gnomons differ by two as the triangular gnomons differ by one, figure 21. 

A square number is also formed by the sum of sucessive triangular numbers for example 10 + 6 = 16 = 4 x 4, figure 22.

            CIRCULAR numbers, when squared or cubed, produce numbers that end in the original number.  For example, 5 x 5 = 25,  25 x 5 = 125,  125 x 5 =  625.  Or 6 x 6 = 36,  36 x 6 = 216.  Or 10 x 10 = 100.  When cubed, they may be thought of as SPHERICAL numbers.

            UNEQUILATERAL numbers have sides that differ by one.  Two is the first unequilateral number because two and one differ by one.  The sides of an unequilateral number are always even and odd because even and odd are always one number apart.  Numbers are created by addition and by multiplication.  If an even number is added to the sum of the even numbers that precede it an unequilateral number will be produced.  The 1st even term is 2.  2 + 4 (the 2nd even term) = 6,  6 + 6 (the 3rd even term) = 12,  12 + 8 (the 4th term) = 20.  Unequilateral numbers are also produced by multiplying successive terms of the natural series.  1 x 2 = 2,  2 x 3 = 6,  3 x 4 = 12,  4 x 5 = 20…  The unequilateral series is 2, 6, 12, 20, 30, 42, 56, 72, 90… 

            Sucessive even numbers added together and successive natural numbers multiplied together produce unequilateral numbers.  Thus:

                2 + 4 = 6         1 1 1        or        1 1 1 1      2 + 4 + 6 = 12         

                2 x 3 = 6         1 1 1                    1 1 1 1      4 x 3 = 12         

                                                                   1 1 1 1                     

            SIMILAR square, oblong and unequalateral numbers have their sides in the same proportion.  All square numbers are similar.  Their sides are in the ratio of 1:1.  Example, 3 x 2 = 6,  6 x 4 = 24,  3:6 = 2:4 so 6 and 24 are similar.

            PENTAGONAL numbers are generated by Gnomons separated by three.  The gnomons are 1, 4, 7, 10, 13, 16, 19…  figure 23.

          HEXAGONAL numbers are generated by Gnomons seperated by four.

            Gnomons                    1, 5,  9, 13, 17, 21, 25…..

            Hexagonal numbers    1, 6, 15, 28, 45, 66, 91….

            OTHER polygonal numbers follow in a similar manner.

End of Ariåthmetic Notes