The μonaΔ ακα τηε (-)ΝΣ (Θ)ΓπφιομΣΜΣ ΤΗΣ (-/+)ΝΣ ΤΗΣ ΨΓΣΑΤΟΓ οφ ΙΤ ΙτΣΣΛφ τηε ΣΥΣ οφ Ιπππ

Ψαλψθλοστ οΝΣ  ιν μορε ωαγΣ τηαν

 ΟΝΕ

1

|

-

ΘΣΙΝΓ Α ΨΑΛΨΘΛΑΤΟΓ

Τηθ Σ 

ΛΥ 

ΤηΘΣ

Τηε ΨΑΛΨΘΛΟΣΤ

ωασ ΨΓΣΑΤΣΔ

....."There follow the seven portions

 multiplying unity

 by two

 and three

 to get

 the sequence

 1, 2, 3, 4, 9, 8, 27.

 Having made this division

 and related it to

 the sevenness of

 the planetary system,

 Plato goes on to

 describe the filling in of

 the intervals.

 This is done by

 placing two means

 between each of

 the powers of

 2 and 

powers of

 3 .

 These are 

the arithmetic

 and harmonic

 means

 which,

 with

 the geometric mean

 complete

 the triad of

 means.

 The means set up proportional

 unions between extremes

 and are therefore in themselves

 the epitome, in mathematical terms, 

of the mediating principle —

 in common with the definition 

of psyche ('soul') 

as mediating 

between 

the metaphysical 

domain

(τηοτη αβλε ΨΣΣΝ ΛΕΣΣ ΦΣΛΛΤ Μ(Ο)ΓΣ ) 

 and 

the physical 

domain:

 (FELT MORE OBServeΔ Λεσσ) 

 The

 means:

...themselves 

The means aka :themselves

 are 

set in a hierarchical tendency,

 as one might call

 it.

 The geometrical

 is

 the most SKY ward 

(Forces unseen and observed

orign = ζηατ moΣΤ ΣΠΗΕΓΕ)

 the harmonic 

the most central 

(psychic/ΝοερόΣ+ανθρωπολογική επιστήμη),

 and 

the ari th:metic 

Τηατ βε Λοω

Ω

the most earth-

Τηε ΓΣΣΙΔΘΑΛ

ΓΣΣΙΔΙΝΓ

ΙΝ

ΙΤ

ΑΚΑ

Τ

Π

ΤΗΙΣ

ΤΗΑΤ

ΤΗΕ ΟΤΗΕΓ ΤΗΙΝΓ

TA THEOLOGOUMENA ARIthMITIKIS

The the ology of the ology

might go around in a logo ish

way more than logo phishing

 Moreover, 

it is demonstrably

 both point 

and angle 

(with all forms of angle)

 and 

beginning

 middle

 and end

 of all things

...:...:...

 since, if you 

decrease it, it limits

 the infinite dissection of what is

aka... continuous 

and if you increase

 it,

 it defines 

the increase 

aka the multiplic ands

as being the same as the divided ands

 and this 

is due to the di Σposit 

ion of

EVERY 

ΔIVIΣοΓ

 over

human nature

There is a certain plausibility in their also calling it 'matter' and even 'receptacle of all,' since it is productive even of the dyad (which is matter, strictly speaking) and since it is capable of containing all principles; for it is in fact productive and disposed to share itself with everything.

Likewise, they call it 'Chaos/ which is Hesiod's first generator, 13 because Chaos gives rise to everything else, as the monad does. It is also thought to be both 'mixture' and 'blending,' 'obscurity' and 'darkness,' thanks to the lack of articulation and distinction of everything which ensues from it.

Anatolius says that it is called 'matrix' and 'matter,' on the grounds that without it there is no number.

The mark which signifies the monad is a symbol of the source of all things. 14 [6] And it reveals its kinship with the sun in the summation of its name: for the word 'monad' when added up yields 361, which are the degrees of the zodiacal circle. 15

The Pythagoreans called the monad 'intellect' because they thought that intellect was akin to the One; for among the virtues, they likened the monad to moral wisdom,- for what is correct is one. And they called it 'being,' 'cause of truth,' 'simple/ 'paradigm/ 'order/ 'concord/ 'what is equal among greater and lesser/ 'the mean between intensity and slackness/ 'moderation in plurality/ 'the instant now in time,' and moreover they called it 'ship,'

'chariot/ 'friend/ 'life/ 'happiness.'

Furthermore, they say that in the middle of the four elements there lies a certain monadic fiery cube, whose central position they say Homer was aware of when he said: "As far beneath as is Hades, so far above the Earth are the heavens." 16 In this context, it looks as though the disciples of Empedocles and Parmenides and just about the majority of the sages of old followed the Pythagoreans and declared that the principle of the monad is situated in the middle in the manner of the Hearth, and keeps its location because of being equilibrated; and Euripides too, who was a disciple of Anaxagoras, mentions the Earth as follows: "Those among mortals who are wise consider you to be the Hearth." 17

Moreover, [7] the Pythagoreans say that the right-angled triangle too was formed by Pythagoras when he regarded the numbers in the triangle monad by monad. 

Calling the monad 'Proteus/ as they do, is not implausible, since he was the demigod in Egypt who could assume any form and contained the properties of everything, as the monad is the factor of each number.

And we say that the mean between what is greater and what is smaller is what is equal. Therefore equality lies in this number alone. Therefore the product of its multiplication will be equal to the sum of its addition: [1 1] for 2+2=2x2. Hence they used to call it 'equal.'

That it also causes everything which directly relates to it to have the same property of being equal is clear not only (and this is why it is the first to express equality in a plane and solid fashion — 

equality of length and breadth in the plane number 4 

and in the solid number 

eight equality 

of depth and height

 as well

 in its very divisibility into two 

monads which are equal to

 each other

 also in the number 

which is said to be 

'evolved' from it 

(that is,

 16, 

which is 2x2x2x2)

 which is a plane number 

of the so-called 

'color' on base 2: for 16

 is 4 x 4

is 4 ^2

is 16/4

is 4^2 /4

is 16/2^3

is 64/4^4

 5. 'Color' is a traditional

 Pythagorean term for

 surface area Thus

 16 is a

 'plane number'

 IS A

 'square'

 of the 

'color'

 'on base 2' 

i.e. 

2 

squared

 or

 4

6. A square whose area is 16 has four sides each 4 in length: the sum of the sides is also 16.

 Smaller squares have areas less than

 the sum of their sides

 larger squares have areas greater than

 the sum of their sides

7. the equality shared by the area 

and the sum of the sides 

See Plato, Theaetetus 147d.

[147c] for we give in our answer something that knowledge belongs to, when that was not what we were asked.

Theaetetus
So it seems.

Socrates
Secondly, when we might have given a short, everyday answer, we go an interminable distance round; for instance, in the question about clay, the everyday, simple thing would be to say “clay is earth mixed with moisture” without regard to whose clay it is.

Theaetetus
It seems easy just now, Socrates, as you put it; but you are probably asking the kind of thing that came up among us lately when

[147d] your namesake, Socrates here, and I were talking together.

Socrates
What kind of thing was that, Theaetetus?

Theaetetus
Theodorus here was drawing some figures for us in illustration of roots, showing that squares containing three square feet and five square feet are not commensurable in length with the unit of the foot, and so, selecting each one in its turn up to the square containing seventeen square feet and at that he stopped. Now it occurred to us, since the number of roots appeared to be infinite, to try to collect them under one name,

 by which we could henceforth call all the roots.¹

Socrates
And did you find such a name?

Theaetetus
I think we did. But see if you agree.

Socrates
Speak on.

Theaetetus
We divided all number into two classes. The one, the numbers which can be formed by multiplying equal factors, we represented by the shape of the square and called square or equilateral numbers.

Socrates
Well done!

Theaetetus
The numbers between these, such as three

[148a] and five and all numbers which cannot be formed by multiplying equal factors, but only by multiplying a greater by a less or a less by a greater, and are therefore always contained in unequal sides, we represented by the shape of the oblong rectangle and called oblong numbers.

Socrates
Very good; and what next?

Theaetetus
All the lines which form the four sides of the equilateral or square numbers we called lengths, and those which form the oblong numbers we called surds, because they are not commensurable with the others

[148b] in length, but only in the areas of the planes which they have the power to form. And similarly in the case of solids.¹

Socrates
Most excellent, my boys! I think Theodorus will not be found liable to an action for false witness.

Theaetetus
But really, Socrates, I cannot answer that question of yours about knowledge, as we answered the question about length and square roots. And yet you seem to me to want something of that kind. So Theodorus appears to be a false witness after all.

Socrates
And do you think that the discovery of knowledge, as I was just now saying, is a small matter and not a task for the very ablest men?

Theaetetus
By Zeus, I think it is a task for the very ablest.

Socrates
Then you must have confidence in yourself, and believe that Theodorus is right,

[148d] and try earnestly in every way to gain an understanding of the nature of knowledge as well as of other things.

Theaetetus
If it is a question of earnestness, Socrates, the truth will come to light.

Socrates
Well then—for you pointed out the way admirably just now—take your answer about the roots as a model, and just as you embraced them all in one class, though they were many, try to designate the many forms of knowledge by one definition.

Theaetetus

If it is a question of earnestness

 Socrates, the truth will come to light.

 And this number is obviously in a sense a sort of mean between greater and lesser in the same way that the dyad is. For the squares before it have perimeters which are greater than their surface areas, while the squares after it, on the other hand, have perimeters which are less than their surface areas, but this square alone has perimeter equal to surface area. 6 This is apparently why Plato in Theaetetus went up to 16, but stopped 'for some reason' at the square whose area is 1 7 feet, when he was faced with the manifestation of the specific property of 16 and the appearance of a certain shared equality. 

It is also called 'deficiency and excess' and 'matter' (for which, in fact, another term is the 'indefinite dyad') because it is in itself devoid of shape and form and any limitation, but is capable of being limited and made definite by reason and skill.

The dyad is clearly formless, because the infinite 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 consists of two straight lines or two angles. So what is indefinite and formless falls under the dyad alone.

It also turns out to be 'infinity,' since it is difference, and difference starts from its being set against 1 and extends to infinity. And it can be described as productive of infinity, since the first manifestation of length is in the dyad, based on the monad as a point, and length is both infinitely divisible and infinitely extensible. Moreover, the nature of inequality proceeds in an infinite sequence whose source is the dyad [13] in opposition to the monad. For the primary distinction between them is that one is greater, the other smaller.

The dyad is 

not number

 nor even

 because it is not actual

 at any rate

 every even number is

 divisible into both equal

 and unequal parts

 the dyad alone 

cannot be divided into unequal parts

 when it is divided into equal parts,

 it is completely unclear 

to which class its parts belong

 as any part

 is the source. 

  Say the monad were half the dyad

Then the existence of the dyad is necessarily prior.

their mutual relations are to be preserved for them,

as they necessarily co-exist,

because double is double what is half,

 and half is half what is double,

 and they are

 neither prior

 nor posterior,

 because they 

generate 

and are 

generated 

by each other

destroy 

and are 

destroyed

 by each other.

They also name it 'Diometor', the mother of Zeus (they said that the monad was 'Zeus'), and 'Rhea', after its flux and extension, 16 which are the properties both of the dyad and of Nature, which is in all respects coming into being. And they say that the name 'dyad' is suited to the moon, both because it admits of more settings than any of the other planets , 17 and because the moon is halved or divided into two: for it is said to be cut into half or into two.

17. Here duas (dyad) is linked with duseis (settings).

The triad is called 'prudence' and 'wisdom' — that is, when people act correctly as regards the present, look ahead to the future, and gain experience from what has already happened in the past: so wisdom surveys the three parts of time, and consequently knowledge falls under the triad.

[17] They call the triad 'piety': hence the name 'triad' is derived from 'terror' — that is, fear and caution. 5

From Anatolius

The triad, the first odd number

 is called perfect

 because it is the first number to signify the totality

 — beginning, middle and end.

 When people exalt extraordinary events, they derive words from the triad and talk of 'thrice blessed,' 'thrice fortunate.' Prayers and libations are performed three times.

 Triangles both 

reflect 

and are 

the first substantiation of plane BEING

 there are three kinds of triangle —

 equilateral

 isosceles

 scalene

 Moreover, there are three rectilinear angles — 

acute

 obtuse

 right

 And there are three parts of time 

Among the virtues

 they likened it to moderation: for it is 

commensurability between excess and deficiency

the triad makes 6 by the addition of the monad

 dyad and itself

1+2+3 = 6

 and 6 is the first perfect number.

One could relate to all this the words of Homer, "All was divided into three ," 10 given that we also find that the virtues are means between two vicious states which are opposed both to each other and to virtue ; 11 and there is no disagreement with the notion that the virtues fall under the monad and are something definite and knowable and are wisdom — for the mean is one — while the vices fall under the dyad and are indefinite, unknowable and senseless.

They call it 'friendship' and 'peace,' and further 'harmony' and 'unanimity': for these are all cohesive and unificatory of opposites and dissimilars. Hence they also call it 'marriage.' And there are also three ages in life.

1. Here 'gnomon' is being used primarily in its original sense as a carpenter's tool (see Glossary). As defined by Heron, however, a mathematical gnomon is that which when added to any figure or number makes the result similar to the original to which it was added; and the decad does this (see pp. 61 and 77, n. 6). The similarity between this passage and p. 109 makes me sure that a 'joiner' is also some unknown tool.

2. 1+2+3+4=10.

3. Perhaps the four Aristotelian causes, mentioned on p. 58; or perhaps the four elements (cf. Plato, Timaeus 32b-c, and pp. 95-6).

VII. Now that which came into being must be material and such as can be seen and touched. Apart from fire nothing could ever become visible, nor without something solid could it be tangible, and solid cannot exist without earth: therefore did God when he set about to frame the body of the universe form it of fire and of earth. But it is not possible for two things to be fairly united without a third; for they need a bond between them which shall join them both. The best of bonds is that which makes itself and those which it binds as complete a unity as possible ; and the nature of proportion is to accomplish this most perfectly. For when of any three numbers, whether expressing three or two dimensions, one is a mean term, so that as the first is to the middle, so is the middle to the last ; and conversely as the last is to the middle, so is the middle to the first ; then since the middle becomes first and last, and the last and the first both become middle, of necessity all will come to be the same, and being the same with one another all will be a unity. Now if the

body of the universe were to have been made a plane surface having no thickness, one mean would have sufficed to unify itself and the extremes; but now since it behoved it to be solid, and since solids can never be united by one mean, but require two —God accordingly set air and water betwixt fire and earth, and making them as far as possible exactly proportional, so that fire is to air as air to water,:and as air is to water water is to earth, thus he compacted and constructed a universe visible and tangible. For these reasons and out of elements of this kind, four in number, the body of the universe was created, being brought into concord through proportion; and from these it derived friendship, so that coming to unity with itself it became in-

dissoluble by any force save the will of him who joined it.

For since all things in general 

are subject to quantity 

when they are juxtaposed and heaped together

 as discrete things

 and are subject to size 

when they are combined and continuous

  in terms of quantity

 things are conceived as either

 absolute or relative

 in terms of size

 as either 

at rest or in motion

 accordingly the four mathematical systems or sciences will make their respective apprehensions in a manner appropriate to each thing:

 arithmetic 

apprehends quantity in general

 especially absolute quantity

 music apprehends

quantity when it is relative

- and geometry apprehends

 size in general

 especially [21] static size

 astronomy apprehends 

size when 

it is in motion and

 undergoing orderly 

change.

If number is the form of things,

 and the terms up to the tetrad are

 the roots and elements, as it were, 

of number,

 then these terms would

 contain 

the aforementioned properties and the manifestations

 of the four mathematical sciences — 

the monad of arithmetic

 the dyad of music

 the triad of geometry

  the tetrad of astronomy

 just as in the text entitled 

On the Gods Pythagoras distinguishes them as follows: "Four are the foundations of wisdom — arithmetic, music, geometry, astronomy — ordered 1, 2, 3, 4." 

And geometry falls under the triad, not only because it is concerned with three-dimensionality and its parts and kinds, but also because it was characteristic of this teacher 4 always to call surfaces (which they used to term 'colors') the limiters of geometry, on the grounds that geometry concerns itself primarily with planes,but the most elementary plane is contained by a triad, either of angles or of lines; and when depth is added, from this as a base to

4. Pythagoras.

a single point, then in turn the most elementary of solids, the pyramid, is formed, which (even though in itself it is encompassed by at least four angles or surfaces ) is fitted together by virtue of three equal dimensions, and these dimensions form the limits of anything subsisting in Nature as a solid.

And astronomy — the science of the heavenly spheres — falls under the tetrad, because of all solids the most perfect and the one which particularly embraces the rest by nature, and is outstanding in thousands of other respects, is the sphere, which is a body consisting of four things — center, diameter, circumference and area (i.e. surface).

Because the tetrad is like this, people used to swear by Pythagoras on account of it, obviously because they were astounded at his discovery and addressed him with devotion for it; so Empedocles says somewhere, "No, by him who handed down to our generation the tetraktys, the fount which holds the roots of ever-flowing Nature." 5 [23] For they used "ever-flowing Nature" as a metaphor for the decad, since it is, as it were, the eternal and everlasting nature of all things and kinds of thing, and in accordance with it the things of the universe are completed and have a harmonious and most beautiful limit. And its 'roots' are the numbers up to the tetrad — 1, 2, 3, 4. For these are the limits and, as it were, the sources of the properties of number — the monad of sameness which is regarded as absolute, the dyad of difference and what is already relative, the triad of particularity and of actual oddness, the tetrad of actual evenness. (For the dyad is often viewed by us as being oddlike on account of being like a source, since it is not yet receptive of the pure properties of evenness and is not capable of being subdivided.) 6

The tetrad is the first to encompass minimal and most seminal embodiment, since the most elementary body and the one with the smallest particles is fire, and this is the body whose shape as a solids a pyramid (hence the name), 7 which alone is enclosed by four bases and four angles.

7. Puramis (pyramid) is linked with pur (fire).

They even distinguish four kinds of planetary movements — progression, retrogression and two modes of being stationary, primary and secondary. 10

There are four distinct senses in living creatures; for touch is a common background to the other four, which is why it alone does not have a location or a regular organ. There are four kinds of plants — trees, shrubs, vegetables and herbs. There are four kinds of virtues: first, wisdom in the soul and, corresponding to it, keen sensibility of the body and good fortune in external matters,second, moderation in the soul and health in the body and good repute in external matters; third, on the same arrangement, courage, strength and power; and fourth, justice, beauty and friendship.

Moreover, just as there are four seasons such as summer, so there are four seasons for man — childhood, youth, adulthood and old age.

The most elementary numerical properties are four — sameness in the monad, difference in the dyad, color in the triad 11 and solidity in the tetrad.

The pentad is the first to exhibit the best and most natural mediacy, when, in conjunction with the dyad, it is taken in disjunct proportion 6 to both the limits of natural number — to the monad as source and the decad as end: [32] for as 1 is to 2, so 5 is to 10, and again as 10 is to 5, so 2 is to 1; and alternately, as 10 is to 2, 5 is to 1, and as 2 is to 10, 1 is to 5. And the product of the limits is equal to the product of the means, as is the way with geometrical proportion: for 2x5=1x10. Reciprocally, we are able to see first in the pentad, compared with the greater limit, the principle of half, just as we see this principle first in the dyad, compared with the smaller limit: for 2 is double 1, and 5 is half 10.

Hence the pentad is particularly comprehensive of the natural phenomena of the universe: it is a frequent assertion of ours that the whole universe is manifestly completed and enclosed by the decad, and seeded by the monad, and it gains movement thanks to the dyad and life thanks to the pentad, which is particularly and most appropriately and only a division of the decad, since the pentad necessarily entails equivalence, while the dyad entails ambivalence.

there are five general elements of the universe — 

earth

 water

 air

 fire

  aether

 and their figures are five — 

tetrahedron

 hexahedron

 octahedron

 dodecahedron

 icosahedron

 the sum of the bases

 of these figures is

 ten times the principle of

 the pentad.