777,600,000,000 α Νυμβεγ πθ Σπανδ ΥνΔΣγ
八九七
波浪線 直線 界
Bā
bōlàng xiàn
Ξiǔ
zhíxiàn
Qī
jiè
βα
βΩθλαηγ χιαη
Ξιυ
ζψηιαη :ι... :...Ξηι
ξιΣ
Short
story shorter
everything that exists
can be expressed as an
expression of triangle
the way
the writing was done in back
in the ΔΑΥ
 |
| d/1HbXmZOA3ixG5aIBl4tcyhvRL8CyFDm3ooI0j9GtrJq0/=sharing |
They say numbers have something to say
the significance of the word is proportional to the word of significance contained in the word
based on this there is some stretch of some argument made by someone that the people who put together for instance the
so called Plimption 322 tablet knew something about scratching
lines in dirt as the students studying these ideas seem to know and
what is going on in the background is left to languish in the lofty lairs of lords and ladies leaving loose ands to the end menders
But I must make an effort to explain this anew, even more clearly. Indeed if someone having
moulded all manner of shapes out of gold kept on remoulding each shape into all the others
without stopping, and someone were to point to one of them 50B and ask what it is; in truth, the
safest answer is gold. But the triangle and all the other shapes arising in it we should never refer
to as “these”, as if they were things that are, for they change even as we make the assertion; rather
we should be content if they deign to accept the cautious designation “such as this”. We must
certainly give the same account also of the nature of the recipient of all these bodies. We must
Copyright © 2021 by David Horan 19
assert that it is always the same for it never departs at all from its own power. For, it is always
receiving everything, but it never adopts a shape 50C like the things which enter into it, in any way
at all. There it lies, the natural recipient of all impressions, moved and shaped by the things that
enter it, seeming to change from time to time because of these. What enters it and leaves it are
imitations of the things that are always, modelled upon them in some manner which is hard to
describe and extraordinary, which we shall deal with on another occasion.
Now in the first place, fire and earth, water and air are bodies, as is obvious to everyone, I
presume, and the form of every body also possesses depth.
And it is absolutely necessary that
depth,
in turn,
be contained by a flat surface,
and any straight-sided flat surface
is constructed
from triangles.
All triangles originate from
two triangles 53D
each of which possesses
a single right
angle
while the others are acute.
One of them has half a right angle on either side,
divided by
equal sides,25
while in the other case
the portions of the right angle
are unequal,
apportioned by
unequal sides.26
So proceeding on the basis of the likely account accompanied by necessity,
we
propose this
as the cause
of fire and the other bodies,
but any causes still higher than these,
ΓθΔ knows, and so do any men who are dear to him.
So we
need to state what
the most beautiful 53E
bodies would be,
four in number,
dissimilar to one another,
and yet,
capable of arising from one
25 This is a right angled isosceles triangle.
.svg.png)
By lengths of sides
Ancient Greek mathematician Euclid defined three types of triangle according to the lengths of their sides:[1][2]
n mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.
The theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation:[1]
25 This is a right angled isosceles triangle.
26 This is a right angled scalene triangle.
Copyright © 2021 by David Horan 21
another in some cases when they are broken apart. For once we accomplish this, we are in
possession of the truth concerning the generation of earth and fire and whatever lies between
them, in a proportion. Indeed we shall not concede to anyone that there are visible bodies more
beautiful than these anywhere, each based upon a single type.
So we must be eager to construct the four types of body that excel in beauty, and to declare that
we have sufficiently understood their nature. 54A In the case of the pair of triangles, there is only
one type of isosceles triangle but an unlimited number of scalenes, therefore we must proceed to
select the most beautiful from this unlimited number if we intend to begin methodically. Yet if
anyone can declare that he has chosen something more beautiful for the construction of these
bodies, he will be victorious, not as an enemy but as a friend. And so, passing over the others, we
propose one of the numerous triangles as the most beautiful, the one from which the equilateral
triangle is constructed as a third. The explanation 54B is a longer story but if anyone refutes this
and discovers that it is not the case, the prize is his, as a friend.
So we shall select two triangles from which the bodies of fire and of the others have been devised:
one is the isosceles while the other has a long side that is always triple the short side when they
are squared. However something stated earlier in an imprecise manner should now be made more
definite. For when it appeared that the four kinds all underwent generation from one another and
into one another, the appearances were incorrect. The four kinds do indeed arise from the
triangles 54C we selected, and while three are constructed from the one that has unequal sides, the
fourth one alone is constructed from the isosceles. So it is not possible for them all to be
generated from one another when they are broken apart, so that a few large kinds are produced
from numerous smaller kinds and vice versa: only three of them can do this because they arise
naturally from a single triangle, and by breaking up the larger bodies, numerous smaller bodies
can be formed from the same triangles as the bodies take on the shapes appropriate to themselves.
Then again the small bodies when dispersed into their constituent triangles 54D would give rise to
another large single form once they constitute a single number belonging to a single volume.
Now that is enough said about their intergeneration; next we should explain what the form of each
of them is like and the number that come together to constitute it. Now we begin with the
construction of the first and smallest form; its element of composition is the triangle having its
hypotenuse double the shortest side in length. If a pair of such triangles is placed along a
diagonal, and this is done 54E three times, with the diagonals and the short sides fixed to the same
point as though it were a centre, then from these, six in number, a single equilateral triangle is
produced. And when four equilateral triangles are combined based on three flat angles coming
together, a single solid angle 55A is produced, the one that comes next after the most obtuse of flat
angles;27 and when four such solid angles have been completed, the first solid form that
apportions the entire surface of a sphere into equal and similar parts is constructed.28
The second is composed of the same triangles but arranged into eight equilateral triangles which
produce a single solid angle from four flat angles, and when six solid angles like this have been
generated the second solid form is, in turn, brought to completion. The third is composed of one
hundred and twenty of the elemental triangles combined together; 55B it has twelve solid angles
each bounded by five flat equilateral triangles and twenty faces consisting of equilateral triangles.
Now this elemental triangle was released from duty once it had produced these three forms, and
the isosceles triangle then produced the fourth kind when it was assembled in groups of four with
their right angles joining at the centre to form a single equilateral quadrangle;29 and six such
surfaces when brought together 55C produced eight solid angles each consisting of three flat right
angles, and the shape of the assembled body was the cube with its six flat equilateral quadrangular
27 This simply means that the three sixty degree angles that meet at the corner of a tetrahedron add up to 180o which is
not an obtuse angle but next after the most obtuse angle. 28 This refers to a tetrahedron inscribed within a sphere. 29 A square.
Copyright © 2021 by David Horan 22 faces. One construction still remained, the fifth, and god used this in order to embroider shapes upon the universe. Now suppose someone considering all this with due measure were to become perplexed over whether we should say there are an unlimited number of worlds or that there is some limit. He would regard the opinion that the number is unlimited 55D as belonging to someone who is actually inexperienced on issues whereon he should be experienced but, on the other hand, the question of whether it is, in truth, ever appropriate to say that there is naturally one world or five, if he were to stop there, would provide more reasonable grounds for perplexity. Now our approach, based upon the likely account, reveals that it is, by nature, a single god, but someone else, in view of some other considerations may form different opinions. Now we should dismiss this fellow and allocate the four types that have been generated to fire, earth, water and air. So let’s assign the cubic form to earth since earth is the most immobile 55E of the four types and it is the most malleable of bodies, so it is especially necessary that this be the sort that has the most stable bases; and of the two triangles we proposed at the outset, the one with equal sides is naturally more stable than the one with unequal sides; and in the case of the equal sided surfaces constructed from either triangle, the square is necessarily more resistant to motion than the equilateral triangle, both as a whole and on the basis of its parts. Therefore we shall preserve the likely account by assigning this form to earth, 56A and of those that remain, the hardest to move is assigned to water, the easiest to move to fire, and the intermediate kind to air. Again the smallest body is assigned to fire, the largest, for its part, to water, and the intermediate to air; and the sharpest again to fire, the second sharpest to air, and the third sharpest to water. Now in all these cases the one that has the fewest bases is necessarily, by nature, the most mobile since it is the most incisive and the sharpest 56B of them all in every respect, and also the lightest since it is composed of the fewest identical parts. The second lies in second place based on these same factors while the third lies in third
So in accord with the correct and likely account, let the solid form that constitutes a pyramid be
the element30 and seed of fire, and we may say that the second in order of generation belongs to
air, and the third to water. Now we need to realise that these are all so small that an individual
particular 56C of each kind is invisible to us due to its extreme smallness, and yet, when a large
number are aggregated together, their bulks can be seen. As for the proportions associated with
their multiplicity, movements, and their other capacities, insofar as the nature of necessity
allowed under willing persuasion, we must realise that god fitted them together entirely in due
proportion, once he had precisely perfected them in this comprehensive manner.
Now from all that we have said earlier about the kinds, they would be most likely 56D to behave as
follows. Earth, on encountering fire and being broken apart by its sharpness, regardless of
whether it is in fire itself, when it gets broken up or in a mass of air or water, would travel along
until such time as it somehow meets up with its own parts which, once fitted together with
themselves, become earth again; for they could never have adopted any other form. Water, when
broken apart either by fire or air, is capable of combining to constitute one body of fire and two of
air, 56E while the components of air, when a single portion is broken up would become two bodies
of fire. And again, when a little fire is surrounded by a lot of air or water or even by some earth
and is in motion within them, does battle, gets defeated and is shattered, two bodies of fire
combine into a single figure of air. And if air is defeated and broken in pieces, one whole figure
of water will be compounded from two and a half of air.
In fact we may consider them again as follows: whenever one of the other 57A types is surrounded
by fire and is cut by its sharp corners and edges, it stops being cut once it recombines into the
nature of fire. For in each case, a kind that is the same and similar is unable to effect any change
in, or be affected by, anything that is the same as itself and has similar characteristics. However as
long as it transforms into something else, as a lesser contesting against a stronger, its dissolution
30 Literally “letter”.place.
Copyright © 2021 by David Horan 23
is unceasing. And furthermore whenever a few smaller units, being surrounded 57B by many that
are larger, are broken apart and quenched, then, if they recombine and adopt the form of the
dominant units their quenching ceases and air arises from fire, or water from air. But if as they are
going in this direction, one of the other kinds comes upon them and attacks them, there is no end
to their dissolution until they are either completely broken up by the pressure and take flight
towards their kindred, or they are overcome and, out of the multiplicity, constitute a unit like unto
their conqueror and abide thereafter as its neighbour. And indeed, 57C during these processes they
all exchange their locations, for although the vast bulk of each kind is set apart in its own place
because of the motion of the receptacle, yet those that are in the process of becoming unlike
themselves but like unto others, are borne by the shaking towards the place belonging to the kinds
they are becoming like.
Now it is through such causes as these that the unmixed and primary bodies have arisen, and yet,
the different types that exist within the forms of these four must be attributed to the construction
of the two elementals. This initially produced a triangle, in each case, that was not of one size
only, 57D rather, some were smaller, some larger and the number of different sizes corresponded to
the number of different types among the four kinds. And so, as they combine with themselves and
one another, their diversity is unlimited, a diversity that must be contemplated by anyone who
intends to have recourse to the likely account in relation to natural phenomena.
Now if there is no agreement on the origin of motion and rest, and the manner in which they arise,
our subsequent reasoning would be much impeded. 57E And although something has been said
about them already, the following should still be added: where there is uniformity motion is never
inclined to arise. For it is difficult, impossible really, that there be something that is to be moved
without that which is to move it, or for something to be a mover without something that is to be
moved. In the absence of these there is no movement, yet it is impossible for them ever to be
uniform, so accordingly, we should always propose 58A that there is rest in uniformity, and motion
in non-uniformity, and furthermore, the cause of the non-uniform nature is inequality.
Copyright © 2021 by David Horan 24
penetrated and broken apart by fire it loses its uniformity and once this is undone it is more
inclined towards motion and once it becomes mobile it is extended out upon the earth under the
pressure of the surrounding air. Each of these processes has acquired a name: “melting” is the
disintegration of the bulk and “flowing” is the spreading upon the earth. But when the fire is
expelled from there again 59A it does not emerge into a void; the surrounding air is pressurised,
and it compresses the still mobile liquid mass into the seats that belonged to fire thus mixing it
together with itself. And once this is compressed it acquires its uniformity again and since fire,
the artificer of its non-uniformity has withdrawn, it reverts to sameness with itself. Being quit of
fire is termed “cooling” while the contraction that follows its departure is termed “being in a solid
state”.
Now of all these 59B liquefiable kinds of water, as we have called them, the most dense arises
from the finest and most uniform components. It is of one form only, glistening and yellow in
colour, a most revered possession; gold, filtered through rock and solidified; while the offspring
of gold which is very hard and black due to its density is called adamant. The kind whose parts
bear the closest resemblance to gold has more forms than one. In density it is, in a way, more
dense than gold and it contains a small fine portion of earth and so it is harder, yet in another way
it is lighter as it contains larger 59C interstices within itself. This compound is copper, one form of
the bright and solid types of water. But as the two components of the mixture age, they separate
from one another again and the earth that is in the mixture appears on the surface and is called
verdigris.
It should no longer be a complex matter to work out the features of the other substances of this kind whilst adhering to the structure of likely stories. In this regard, whenever someone, for the sake of relaxation, laying aside discussions concerning things that are always, gives consideration to likely 59D accounts concerned with becoming, and derives a pleasure he will not later regret, he would be introducing a measured and sensible pastime into his life. So having indulged in this just now, dealing with the same issues, we shall proceed with the likely accounts that follow this in due order, as follows. Water that is mixed with fire and is fine and liquid is called liquid due to its motion and the course it takes as it rolls along the ground. Now it is also soft because its bases are yielding, being less stable than those of earth, and whenever it separates off from fire and air, and is on its own, it becomes 59E more uniform, and is compressed into itself by the emerging substances and it is solidified in this way. When this happens to it well above the earth it is hail, when on the earth it is ice, and when it happens to a lesser extent and it is still half-solidified it is snow when above the earth, and when it is compacted from dew upon the earth it is called frost. Now most types of water are mixed with one another and because this category, in general, is filtered through 60A the plants of the earth they are called juices. Although each involves dissimilarity on account of the intermixtures, and while the differences give rise to numerous kinds that are nameless, yet four types containing fire are particularly evident and these have received names: wine, that can heat the soul as well as the body; an oily sort that is smooth and splits the ray of sight and so appears glittering and bright to behold, resin, castor oil, oil itself and any others with the same capacity; there is one that can relax 60B the passageways of the mouth back to their natural condition and by this capacity furnishes sweetness, it is generally referred to as honey; the last kind corrodes the flesh by burning, it is a foamy sort secreted from all other juices and it is called “sour-juice”. As for the forms of earth, that which is filtered through water becomes a strong material in the following manner: whenever the water that is mixed in with it is broken up in the commingling process, it transforms into the figure of air, and once it has become air 60C it rises up to its own region, but there isn’t a void above them so the air in the vicinity is pressurised. And since it is heavy it presses down and pours around the mass of earth, squeezes it intensely and compresses it into the seats from which the newly formed air had arisen. And earth, when compressed by air constitutes stone that is indissoluble in water. The superior kind composed of equal and uniform parts is transparent, while the opposite applies to the inferior kind. The one from which all 60D the
Copyright © 2021 by David Horan 25
moisture has been forced out by the rapid action of fire and is a more brittle compound than the
other, is the kind to which we have given the name ceramic-clay. And on occasion, when some
moisture is left behind, and the earth has been melted by fire and then cooled down, the black
coloured substance constitutes (lava)31stone. Similarly there are two more that are extracted from
the mixture containing a great deal of water. They are composed of finer particles of earth and are
salty, and being semi-solid they are soluble once more in water. One is soda, which can cleanse us
of oil and earth while the other is salt which blends well in the interactions 60E associated with
sense perception in the mouth, and according to tradition is beloved of the gods. The compounds
of both earth and water that can be dissolved by fire but not by water, are compacted in that way
for the following reason: a mass of earth is not broken up by fire or air, since their particles are
naturally smaller than the structure of its interstices so they pass through the considerable open
spaces without exertion leaving the earth undissolved and rendering it indissoluble. However
since the particles of water are naturally larger, they make their way through by force, loosening
the earth 61A and breaking it apart. So earth that is not forcibly solidified in this way is dissolved
only by water, yet when it has been so solidified it is dissolved by nothing but fire, for no way is
left for anything to enter it except fire.
Water, for its part, when compressed with extreme force is dispersed by fire alone, but if the force
of compression is weaker it is dispersed by both fire and air; air acts upon the interstices, fire
upon the elemental triangles. And when air is forcibly solidified nothing dissolves it except by
acting upon its elemental triangles, and when the solidification is not forceful, only fire can break
it apart.
In the case of the mixed bodies composed of earth and water, as long as water 61B occupies its
interstices and these are forcibly pressed together, the particles of water arriving from outside
having no way of entering, flow around it leaving the entire mass intact. Yet the particles of fire
entering the interstices of the water and acting upon it, fire acting on water as water acted upon
earth, constitute the sole causes whereby the compound body is broken apart and flows. Some of
these happen to have less water than earth; the entire category associated with glass and any 61C
stones that we call liquefiable. Others, for their part, have more water; all materials that are waxlike or useful as incense.
In relation to bone, flesh and any substances of this kind the following account applies: the origin of all these is the production of marrow; for whilst the soul is conjoined with the body, the bonds of life, bound fast in marrow, are the secure roots of the mortal race; and yet, the marrow itself has come into being from other materials. For the god separated out those primary triangles that were unwarped and smooth and best able to produce fire, water, air and earth in a precise manner, and set each of these apart 73C from their own kinds. Having mixed these with one another in due proportion, he devised a universal seed-mixture for the entire mortal race by fashioning marrow from them, and after that he planted therein the various kinds of soul and bound them fast. In his initial allocation, he divided the marrow immediately into shapes corresponding in number and type to the shapes that were to belong to the particular forms of soul. He moulded into a complete sphere the portion of marrow that was about to receive the divine seed into itself, as if it were ploughed land, and this he called the brain, implying that the head34was going to become the container for the brain once the construction of each living creature had been completed. Furthermore, he divided the portion of marrow that was to retain the remaining part of soul, the mortal part, into shapes that were both round and elongated at the same time; and yet, he referred to them all as marrow. He also cast out fetters of the entire soul from these as if they were being thrown from anchors, and then fashioned all of this body of ours around the marrow after he had constructed a complete 73E covering of bone to surround it. He constructed bone as follows: he sifted out earth that was pure and smooth, kneaded it, soaked it in marrow, then placed it in fire, and after that he plunged it into water, again into fire, and then into water once more. So by constantly transferring it in this way from one medium to the other, it was rendered indissoluble by either of them. Making use of this he turned a sphere of bone around the brain of the creature and he left a narrow opening in the sphere. He also moulded vertebrae out of bone to surround the marrow 74A that runs through the neck and down the spine, arranging them vertically like pivots beginning from the head and running down the whole trunk. And so they preserved the entire seed by fencing it about with a stone-like enclosure, creating joints and making use of the quality of difference within these as an intermediate power, introduced for the sake of motion and flexibility.
Now whenever more is going out than is coming in, all things decay, but when less is going out
they grow. So when the entire structure is young and the elemental triangles of its constituents are
still new, like fresh wood, the bond holding them together is strong, and yet the overall
construction is soft 81C since it has been newly formed from marrow and nourished with milk. So
when the surrounding triangles come into it from outside, the ones that the food and drink are
made from, these are older and weaker than its own triangles, so it overpowers these older
triangles with the new ones, cuts them up and makes the creature grow large by nourishing it with
an abundance of substances similar to its own. However when the root of the triangles is
slackened because of numerous such contests fought 81D against many opponents over a long
period of time, they are no longer able to cut the entering triangles and assimilate them to
themselves, on the other hand, they themselves are easily split apart by those triangles that are
coming in from outside. Every creature in this situation is overpowered and decays and the
condition is called old age. And in the end, once the conjoining bonds of the triangles around the
marrow endure no longer under the strain and are split apart, they in turn release the bonds of the
soul, and once she is set free in accordance with nature she flies away with pleasure. 81E Indeed
everything contrary to nature is painful, while anything that arises naturally is pleasant. The same
goes for death; when it happens as a result of disease or injury it is painful and violent but when it
Copyright © 2021 by David Horan 37
comes with old age, an ending that accords with nature, it is the least painful of deaths, attended
more by pleasure than by pain.
The origin of diseases is presumably obvious to everyone. For since the body has been
constructed from four 82A constituents, earth, fire, water and air, any unnatural excess or
deficiency or any change that occurs, removing them from their own place to somewhere alien,
causes disorder and disease. Furthermore since there is more than one kind of fire and of the other
constituents, the acquisition by the body of an inappropriate variety, and all phenomena of this
sort, have the same effect. For when any of these arises or changes its location unnaturally,
anything that was previously being cooled is warmed, what was dry 82B becomes moist; the same
goes for light and heavy, and everything changes in all sorts of ways. In fact I maintain that only
the addition or removal of same from same, on the same basis and in due proportion, will allow
something to remain the same as itself, sound and healthy, while anything entering or leaving that
strikes a discordant note by transgressing any of these requirements, will engender a whole
variety of alterations and countless diseases and corruptions.
What’s more, because of the density of bone, whatever is filtered through it consists of the purest kind of triangles, and being extremely smooth and oily it pours, drop by drop, from the bones and waters the marrow. 82E And when each of these substances comes into being based on these processes, the consequence, for the most part, is health, but when they are reversed the result is disease. For whenever flesh is decomposed and releases decomposed matter back into the veins once more, then a lot of variegated blood, mixed with air, adopts multifarious colours and bitterness, as well as acidic and saline properties, and retains all sorts of bile, serum and phlegm. Once these substances have become perverted and corrupted, they first destroy the blood itself, and although they themselves do not provide any 83A nourishment at all to the body, they are borne in all directions through the veins, no longer holding to the natural arrangement of the cycles, at enmity towards themselves because they afford no advantage to themselves, and hostile to any part of the body that stands firm and holds its position; this they destroy and corrupt.
Now when it comes to movements, the best movement is in oneself, by oneself, for this has most
kinship with the movement of thought and of the universe. Being moved by another is an inferior
form of movement but the worst form involves moving the body, part by part, by means of
various agents, while it is lying still. Accordingly the best way of purging or restoring the body is
through gymnastic exercises, second is through the swaying motion of ships and any other means
of carriage that do not induce tiredness. There is a third 89B form of motion, useful in cases of
extreme necessity, which should not be accepted under any other circumstances by anyone
possessed of intelligence; the use of purgative drugs for medical purposes. For diseases that do
not pose a huge threat should not be incited with drugs. Indeed, in a way, the structure of any
disease resembles the nature of a living being. In fact, the constitution of these beings involves an
assigned span of life applicable to the species as a whole, and each particular creature is born with
its own allotted life span, in the absence of 89C the intervention of necessity. For the triangles
belonging to each individual are constructed, from the very outset, with the capacity to endure for
a certain period of time, and no creature could ever live beyond that limit. Now the same tendency
applies to the constitution of diseases; whenever someone ignores its allotted time span and
destroys it with drugs, minor diseases are inclined to become major; few to become many.
Therefore we should manage everything of this sort through our lifestyle insofar as leisure allows,
and 89D not incite a troublesome malady through the use of drugs.
That’s enough said about the living creature as a whole and the bodily parts thereof, and how a
person may live in accord with reason, guiding and being guided by himself. But first and most
importantly we must somehow arrange for the part that will guide us to be as beautiful and
excellent as possible for its guiding role. Now to 89E elaborate upon these matters in detail would
constitute a task in its own right, yet we would not go much astray if we were to conclude our
account by adding a secondary topic in accord with what has gone before, by considering the
issue as follows. As we have stated on numerous occasions, there are three forms of soul dwelling
in three regions within us each possessing movements of its own. Accordingly, we should now
declare as briefly as possible, and on the same basis as before, that any form which continues in
idleness, and keeps quiet its own movements, necessarily becomes weaker, while any that are
exercised become stronger. 90A Therefore we should be careful that these movements retain a due
proportion with respect to one another.
As for the most lordly form of our soul, we should think of it as follows: it is a daimon given to
each of us by god, and it is said to dwell at the very pinnacle of our bodies to raise us up from
earth, to our kindred in heaven, we who are, properly speaking, a plant not of the earth but of
heaven. Indeed it is from there, the place from which soul first had her birth, that the divine
agency suspends 90B our head and root, making straight the entire body.
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