THE LEXARITHMIC THEORY
AND ITS APPLICATIONS (Theory of the Hellenic Language)
by Kosmas Milt. Markatos
Prof. of the Technological Education Institute (T.E.I) of Larisa
Introduction
A. What it is?
“Lexarithmic theory” is the cosmic system which – via the Hellenic language – not only interprets the Cosmos the Universe and every of its activity but also produces knowledge (in the plural) and yet in the limit the knowledge itself through axiomatic way and independently of time.
It is the fruit of more than 30 years research (19732008) of the structure of the Greek language.
Many parts of the theory were presented in several lectures and were published in the daily press magazines and other publications.
The theory – in very synoptic form – is published in the 1^{st} issue of my quarterly magazine ΠΑΜΜΕΓΑΣ (PAMMEGAS:
ΠΑγκόσμιος Μαθηματική Μελέτη Ελληνικής ΓλώσσΑΣ (1980) or PAngosmios Mathematical Melete (study) of the Ellenic (Greek) GlossAS (of language)).
In the present pamphlet the theory is formulated with some completions and theoretical explanations.
B. Historical going back
The letterwordnumber relation as a practice and knowledge goes back to what we call “prehistory of man”. Our sources referring to the existence and the practice of the relation in historical times are few and scattered apparent and insinuated. We mention some of them:
References of historical author to former ones (scientists, philosophers, artists, theologists, poets).
· Ομηρος or Homer [Ιλιάς ΗΤ orIlias HT) Ισόψηφα Επη or Isopsephe verses (equal in number).
· Πλάτων (Platon) (in his book Κρατύλος or Cratylos] the first glossological book of mankind.
· Ιάμβλιχος or Iamvlichos (in his book “Τα Θεολογούμενατης Αριθμητικής” or The Theologoumena of Arithmetic)
· Χρήστου Χρ. or Chrestou Chr. (in his book Αρχαία Σπάρτη or Ancient Sparta) (1960) Isopsephe verses on graves and tombs and PAMMEGAS issue Γ΄.
· Σταμάτης Ευάγγ. Σ. (Stamates Euagg. S) in his book “Ιστορία των Ελληνικών Μαθηματικών” or “History of the Hellenic Mathematics” (1980).
· References and comments of Christian authors.
In the twentieth century this relation is studied from its beginning:
· Χαλάς Αντών. Φ. (Chalas Anton. F) in his books “Το εις το ελληνικόν αλφάβητον και το σύμπαν υπολανθάνον μυστήριον” or “The concealed mystery found in Greek alphabet and in the universe” (1921) και “ Η απόκρυφος διάταξις του ελληνικού αλφαβήτου” or “The hidden arrangement of the Greek alphabet” (1930).
· Μανιάς Θεοφ. Ν. (Manias Theof. Ν.) mainly in his book “Αγνωστα μεγαλουργήματα των αρχαίων Ελλήνων” or “Unknown masterpieces of ancient Greeks” [1970].
· Michel Tohn in his book “City of revelation” (1973).
· Blich Bond and Lee in the book “Gematria” (1977).
· Μαρκάτος Κοσμάς Μ. (Markatos Kosm. M.) The magazine PAMMEGAS (see above) (1980).
C. Formulation of the theory
The lexarithmic theory supports reduces (goes back) and enrolls all the preexistent knowledge beliefs traditions and all the parallel studies on the letterwordnumber relation to an axiomatic system of study and interpretation of the Cosmos through the mathematical study of the Hellenic glossa (language).
The theory is based on the Hellenic Alphabetical System on which mathematics was founded through which in turn – by adopting the notion of proof – each and every science was established .
The study starts with the idendification of the alphabetic and arithmetic system of the Hellenes (Greeks) which is called alpharithmetic system*.
Αα= 1

Ββ= 2

Γγ= 3

Δδ= 4

Εε= 5

F=6

Ζζ= 7

Ηη= 8

Θθ= 9

Ιι = 10

Κκ = 20

Λλ =30

Μμ= 40

Νν= 50

Ξξ= 60

Οο= 70

Ππ= 80

Ϟ = 90

Ρρ=100

Σςσ=200

Ττ=300

Υυ=400

Φφ=500

Χχ=600

Ψψ=700

Ωω=800

Ϡ=900

* The elements (letters) Ϝ (digamma) Ϟ (koppa) and Ϡ (sampi) are no longer used in the Greek alphabet.
Postulations axioms and definitions
The lexarithmic theory as axiomatic is constructed on postulates axioms and definitions but is built (unfolded) on conditions:
Aitema (postulate) 1^{st} (nature of the element αβγ … ω)
Every element [γράμμα letter] αβγ … ω is both quantity (αριθμός arithmos number) and quality clear and distinct sound voice):
Quantity number arithmos alpharithmos
Α, α }
Quality sound voice
Aitema (postulate) 2^{d} (nature of the name)
Onoma (name) is the eggrammatos form of being [eggrammatos: form in letters from gramma (letter)].
Note
In the Hellenic language the notion λέξις (lexis or word) and όνομα (onoma or name) are distinguished:
· “Word” is a general notion and is referred or belongs to γένος (genos genus or gender).
· “Name” is a special meaning of those which are produced from the corresponding word (lexis) and is referred or belongs to είδος [eidos or kind (species)].
· The word name (όνομα) etymologically goes back to the verb γι{γνω}σκω [gignosko k{no}w (root: GNO or KNO or NO)]:
From the alpharithms (element letters) avon 24 words (anagrams transpositions of the letters of a name so as to form another one) are produced without repetitions but only two are names namely the names avon and nova:
avon→ (avno anvo avon anov aonv) (vnoa vona vano vnao voan vaon) (noav naov nvoa novanavo nvao) (oavn ovan onav oanv ovna onva).
· The name avon (ηβόν evon) is Doric type (avos avee avon) of the Ionian one (ηβός ηβή ηβόν: evos eve evon) and means “νέος” (neos new).
· The name nova is new Latin word and means (astronomy) “the suddenly appeared new star which gradually becomes faint or disappeared”.
· Model:
The being ΚΟΣΜΟΣ or COSMOS (represented by the sphere) the notion namely itself is reflected in the eggrammatos form ΚΟΣΜΟΣ consisted from the letters Κ,Ο,Σ,Μ,Ο,Σ under this arrangement:
Definition of lexarithmos (in plural: lexarithmoi)
To every eggrammatos form corresponds the sum of its alpharithms (elements letters) according to the alpharithmetic system of Hellenes which is called lexarithmos:
(eggrammatos form word in general or lexis) lexarithmos
Examples
Name (lexarithmos)
ΚΟΣΜΟΣ (Cosmos) = Κ+Ο+Σ+Μ+Ο+Σ=20+70+200+40+70+200= 600
ΑΚΟΙΜΙΣΤΟΣ (akoimistos waking) = Α+Κ+Ο+Ι+Μ+Ι+Σ+Τ+Ο+Σ = 1+20 +70+10+40+10+200+300+70 +200 = 600
ΣΥΜΠΑΝ (sympan universe) = Σ+Υ+Μ+Π+Α+Ν = 200+400+40+80+1+50 = 771
Η ΘΕΟΤΗΣ (hitheotesdivinity) = Η + Θ+Ε+Ο+Τ+Η+Σ=8 + 9+5+70+300+8+200 = 771
ΑΝΘΡΩΠΟΣ (anthropos man) = Α+Ν+Θ+Ρ+Ω+Π+Ο+Σ = 1+50+9+100+800+80+70+200 = 1310
ΦΥΣΙΣ (physis nature) = Φ+Υ+Σ+Ι+Σ = 500+400+200+10+200 = 1310
Η ΑΝΘΡΩΠΟΤΗΣ (hianthropotes humanity) = Η + Α+Ν+Θ+Ρ+Ω+Π+Ο+Τ+Η+Σ = 8 + 1+ 50+ 9+100+ 800+80+70+300+8+200 = 1626
ΤΕΤΡΑΚΤΥΣ (tetraktys tetras, Pythagorean name for the sum of the first four numbers = Τ+ Ε+ Τ+ Ρ+ Α+ Κ+ Τ+Υ+Σ = 300+5+300+100+1+20+300+400+200 = 1626
ΑΡΙΘΜΟΣ (arithmos number) = Α+Ρ+Ι+Θ+Μ+Ο+Σ = 1+100+10+9+40+70+200 = 430
ΜΑΘΗΜΑΤΙΚΑ (mathematica mathematics) = (Μ+Α+Θ+Η+Μ+Α+Τ+Ι+Κ+Α = 40+ 1+ 9+ 8+ 40+ 1+300+10+20+1 = 430
ΚΑΡΠΟΣ (carpos fruit cornseed) = Κ+Α+Ρ+Π+Ο+Σ = 20+1+100+80+70+200 = 471
ΚΑΡΚΙΝΟΣ (karkinos cancer) = Κ+Α+Ρ+Κ+Ι+Ν+Ο+Σ = 20+1+100+20+10+50+70+200 = 471
ΣΕΙΣΜΟΣ (seismos earthquake) ( Σ+Ε+Ι+Σ+Μ+Ο+Σ = 200+5+10+200+40+70+200 = 725
ΒΑΔΙΣΤΗΣ (vadistes goer runner) ( Β+Α+Δ+Ι+Σ+Τ+Η+Σ = 2+1+4+10+200+300+8+200 = 725
Aitema (postulate) 3d (lexarithmic and semasiological relation)
Every lexarithmic relation of two or more words in general which are considered us numbers or lexarithmoi (plural) is also semasiological between the being themselves reflected in those beings:
Geometry (earth measurement) as a science measures (γεωμετρεί geometrei) the earth via the triangles.
The names (or the words) ΤΡΙΓΩΝΑ (plural) and ΓΕΩΜΕΤΡΙΑ (singular) are equal in number namely the sum of the elements (letters) of each word is the same orthe lexarithmos of both names is 1264 (according to the alpharithmetic system).
⇒ This expresses the lexarithmic relation which here is the equality (isopsephe words).
According to 3d postulare every lexarithmic relation is semasiological as well namely there exists a semasiological relation between the two names. In fact the two notions are related in an unbroken way since the notion of geometry flows from the notion of the triangle:
It could be said that the triangle is the unit of geometry.
⇒ This expresses the semasiological relation which here is true.
So every lexarithmic relation – in order to be consequent – must be semasiological too.
The semasiological relation could be represented as follows:
Let us suppose now that an electronic recogniser (scanner) – by a special software – read seach word (name) [from α to ω] of a large lexicon of the Hellenic language computes its lexarithmos and after put sit on a line numbered from 1 to n (where n is the largest lexarithmos of the Hellenic language or the largest Helleni cword.
Then to every lexarithmos a group of isopsephe words (all the words give the same lexarithmos namely the one on the line) is corresponded. The line formed by this way is called line of the Hellenic language:
Note: In reality the line is not a straight one but a double logarithmic spira.
But every name possesses one or more meanings which are received from oral or written speech dictionaries texts slates popular proverbs etc. So we have the following axiom:
If we examine the meanings which a name takes in the course of time (diachronically) we will conclude that these meanings are described by the help of other names (their meanings).
Consequently we do not know the name’s real meaning (or meanings) but one or more icons (images). This real meaning behind or under each name is called cosmic meaning: it is a prototype one and all its icons belong to or are identified with it.
Axiom 1^{st} [diachronical meaning (DM) – cosmic meaning (CM)]
Every word (name) has or receives in the course of time one or more diachronical meanings (DM) which are not unrelated to each other but they are cut in a prototype meaning which is called cosmic meaning (CM) (i.e. this meaning belongs to every diachronical one):
Model (checking of the axiom 1^{st})
Name: Γλώσσα (glossa tongue language)
Cosmic meaning Þthe medium of intercourse between two things (beingsentities)
DM1: organ of the body (intercourse between man and food).
DM2: organ of the speech (intercourse between two men).
DM3: organ of the shoe (intercourse between foot and shoe).
DM4: smooth and narrow strip of land deeply into the sea (intercourse between land and sea).
DM5: the little tongue of the flute (intercourse between man and flute).
Definition of isopsephe group
As we saw above (line of the Hellenic language) to each lexarithmos corresponds a set of isopsephe words which is called isopsephe group and to every isopsephe word of course corresponds a set of diachronical meanings (see previously).
As the language follows its course more and more names are added on the language logarithmic spiral some isopsephe groups of which are increased in number. How now all these isopsephe meanings are related to each other and to corresponding lexarithmos?
We consider that the notion “arithmos” (number) and more strictly its ikon (image) namely the lexarithmos is inseparable component of every being: Naturally we refer to perceivable ones which man is able to know.
So the lexarithmos possesses in turn an archetype meaning which is preceded of its isopsephe meanings in their genesis.
From the above the follwing axiom is introduced:
Axiom 2^{d} [correlating of the cosmic meaning (CM) – Lexarche (Λ)]
In every isopsephe group the cosmic meanings CM also are not unrelated to each other but they are intersected at an archetype meaning which is called Λεξάρχη (Lexarche)Λ [(lexis+arche (origin) – word origin]:
Model (checking of the axiom 2^{d})
Lexarche ÞThe monodrom notion of the thing (being) [monos (single alone only) + dromos (way road)]
CM1: βάβαξ (lexarithmos 66) (vavax: a babblet chatterer / reveler / bawler).
CM2: διάβημα (lexarithmos 66) (diavema (metaphorically): a step across a step).
CM3: ζημία (lexarithmos 66) (zemia damia samia (Doric types) (metaphorically): loss damage / penalty in money fine / generally penalty / of what is bought too dearly bad bargain a dead loss.
CM4: ήθημα (lexarithmos 66) (ethema (metaphorically): that which is sifted or strained).
CM5: ίζημα (lexarithmos 66) (izema (metaphorically): subsidence sinking).
CM6: κήλη (lexarithmos 66) (kele kale: tumor rupture hermia hump on a buffalo’s back / hump in human beings as well).
Note: It is evident that the Lexarche belongs to every cosmic meaning of the isopsephe group. But because the cosmic meaning in turn belongs to every diachronical meaning it follows that finally the Lexarche is the intersection of all diachronical meanings of the isopsephe group or equivalently it belongs or it is contained in every one of them.
Analysis of the cosmic meanings
According to the 3^{d} aitema the Lexarche Λ (monodrom notion) must be contained in every cosmic meaning of the subgroup with lexarithmos 66 or alternatively every cosmic meaning of it belongs to or expresses some kind of monodrom motion:
⇒ After the point M the being cannot return to state A under the same psychological form
· The “vavax” (βάβαξ – babbler...) acts in a way that the object of his action cannot– by nature – return by his own forces in the prior (former previous) condition (state) under the same form f.e.: Þ the trapped bird is going either to be ate (the bird has got through a monodrom motion) or to be enclosed (shat in confined) in a cage and then its return to the prior state is impossible (again monodrom motion: the bird is not going to survive).
· The “diavema” (διάβημα – a step across...) belongs in an action according to which man either das not return to the life or he returns but completely changed altered ethicaly (morally) (a desperate step).
· The “zemia” (ζημία – damage...)  it is obvious  acts always in a monodrom way.
⇒ The man who has suffered a damage das not return to the former state the same at least.
⇒ The crops which have suffered a damage because of different causes do not return alike (same like similar) to the previous state.
· The “ethema” (ήθημα –sifted...) and the “izema” (ίζημα – subsidence...) compose the main parts of a solution which is divided into two monodrom motions:
During the filtration of a solution by the help of ethmoid material on the one hand the “ethema” goes through the pores (passages) of the ethmoid material but on the other hand the “izema” remains onto the bottom.
Both motions for the their formation are monodrom because once they will be formed they cannot be reunited (united again joined again connect again) for the recreation of the solution.
· The “kele” (κήλη – rupture...) finally  it is given – that it is not healed (cured remedied by the powers of the organism because it has traced (has depicted) a monodrom motion:
Kele can be get settled (be established) only by operation (surgery).
The operation defines the monodrom motion of the “kele”.
The isopsephe group 66 includes (counts in) also others words the detailed (elaborated) analysis of which is exposed in a book of mine^{x.}
Here a sample only is received (obtained taken part) towards faster understanding of the notions (meanings senses concepts conceptions) being implicated (entangled involved).
[see Markatos “ Η Λεξαριθμική Θεωρία  Η Γλώσσα των Γλωσσών” (The Lexarithmic Theory  The Glossa ton Glosson or The Language of the Languages).
This system is called Lexias (Λεξίας) being perceptible by Galaxy (Γαλαξίαςor Galaxias) where, the conceivable center of the Galaxy is the Lexarche, suns the Cosmic meanings and Planets the Diachronic meanings.
Aitema 2A (the equality to BeΕ, to BecomeΓ and the CosmosΚ: the UniverseΣ)
If two or more names (words) are lexarithmically equals or isopsephes then the beings represented by them belong into Be (E) the by virtue existing.
Examples
Note: The numbers 284, 350, 330 are thw lexarithmoi of the isopsephes words θεόςκαί ;aγιος / διογενήςκαίθεογενής / ;iσονκαi νόμιμον)
Definitions
· Gnomonic sequence is a sequence of natural numbers (positive integers) possessing the property every member from the third and further to be equal with the sum of the two previous
• Such a sequence is produced by the help of the mathematical algorithm (of the theory)
given a natural number 1 2 3 ... n (where n is the larger greek name [ see “ line of the Hellenic language”, p.4 )
•• The algorithm is based on the physical gnomonic sequence of the Hellenes (or Fibonacci sequence) 1 1 2 3 5 8 13 21 34 55 ...
Lexarithmic gnomonic sequence is a sequence of lexarithmoi (or words) being considered as numbers according of which every word equals with the sum of the previous two from the third and beyond.
The algorithm
Sentence: For every physical number L (positive integer greater than 0) consequently for every lexarithmos(name or word in general considered as a such number) there exist at most two sets of finished number of physical gnomonic sequence (or Fibonacci sequence), of which only one is accepted (see next).
F_{i}: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ... i=123...
such that the sum of the members for each set to be equal to L:
L = F_{i1}+F_{i2}+...+F_{ik+1} and L = F_{ι}_{1}+F_{ι}_{2}+...+F_{ik+1 }
Remark:
The two set are coincided (since F_{1}= F_{2 }=1) but in forming the asked gnomonic sequences from the given lexarithmos L they differ because on the one hand F_{1} is the first member of the sequence on the other hand F_{2} is the second [see further on].
Three are the main steps of the algorithm:
[a] If the number L is (equals) a number of the above sequence, then it belongs to F_{i} and then the algorithm is terminated.
[b] If the step [a] is not valued then we find the first term of the F_{i} which is maller than the given L (L=F_{i1}+N_{1}) and then the step [a] is repeated now for the difference F_{i1}N_{1} until the derived difference becomes a term of F_{i} and then the algorithm is terminated:
The successive differences may be written:
L = F_{i1}+N_{1} N_{1} not a F_{i }number
N_{1 }= F_{i2}+N_{2} N_{2} not a F_{i} number
N_{2 }= F_{i3}+N_{3} N_{3} not a F_{i} number
.................................................................
N_{k} = F_{ik+1}+N_{k+1} N_{k+1} a F_{i} number
Consequently we add these equations side by side:
L = F_{i1}+F_{i2}+F_{i3}+...+N_{k+1} Þ all F_{i} numbers
This is the analysis of the given lexarithmos in terms of F_{i} numbers.
Now we write the equation in a column
F_{i1}
F_{i2}
F_{i3}
.
.
.
N_{k+1}
L
and then we complete the F_{i} sequences to the left in order starting from the last difference and going to the first one. For the next terms to L we simply apply the law of the physical gnomonic sequence or Fibonacci one namely F_{i+1} = F_{0}+F_{i} (where F_{0}=F_{1}=1):
1 1 2 3 ... F_{k2} F_{k1 }F_{k} N_{k+1} F_{k+2 }F_{k+3} F_{k+4 }...
[c] We now distinguish two cases:
c_{1}. If N_{ik+1} is greater than the number 1 (namely 2 3 5 ...), then the given lexarithmos L produces (or belongs to) only one lexarithmic gnomonic sequence formed in connection with the successive terms of F_{i} as above.
c_{2} If N_{ik+1}=1 equals the number one, the given lexarithmos L produces (or belongs to) two lexarithmic gnomonic sequence
We write in column
F_{i11} F_{i1 }F_{i+11} ... F_{i1} F_{i}_{+1}_{1} F_{i}_{+2}_{1} ...
F_{i2 }F_{i2} F_{i+12} ... F_{i2} F_{i+12} F_{i+12} ...
F_{i3} F_{i3} F_{i+13} ... F_{i3} F_{i+13} F_{i+13} ...
. . . . . .
. . . . . .
. . . . . .
1 N_{ik+1}=1 2 3 5 ... N_{ik+1}=1 1 2 3 5 ...
First sequence (to both directions) Second sequence (to one direction)
Example1: L=361 (ΜΟΝΑΣ or MONAS, UNIT) [M=40 Ο=70 Ν=50 Α=1 Σ=200]
F_{i}: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 ...
[a] step: The given lexarithmos is not a F_{i} number.
[b] step: The first difference (namely 128) 361 = 233+128 is not a F_{i} number.
The second difference (namely 39) 128 = 89+39 is not a F_{i} number.
The third difference (namely 5) 39 = 34+5 is a F_{i }number.
We add the equations side by side:
L = 233+89+34+5Þ this is the analysis of 361 in term of F_{i}
[c] step: The last difference namely the number 5 (the fifth term of F_{i}) is greater than 1 so L produces only one sequence:
34

55

89

144

233

377

610 ...

13

21

34

55

89

144

233 ...

5

8

13

21

34

55

89 ...

1

1

2

3

5

8

13 ...

53 85 138 223 361 584 945 ...
Example2: L=773 (ΔΙΚΑΙΟΣΥΝΗ or DIKAIOSYNE JUSTICE ) [Δ=4 Ι=10 Κ=20 Α=1 Ο=70 Σ=200 Υ=400 Ν=50 Η=8]
F_{i}: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 ...
[a] step: The given lexarithmos is not a F_{i} number.
[b] step: The first difference (namely 163) 773 = 610+163 is not a F_{i} number.
The second difference (namely 19) 163 = 144+19 is not a F_{i} number.
The third difference (namely 6) 19 = 13+6 is not a F_{i }number.
The forth difference (namely 1) 6 = 5+1 is a F_{i }number.
We add the equations side by side:
L = 610+144+13+5+1 > this is the analysis of 773 in term of F_{i}
[c] step: The last difference namely the number 1 (first and second term of F_{i}) is equale to 1, so L produces two sequences
377

610

987

1597

2584 ...


610

987

1597

2584 ...

89

144

233

377

610 ...


144

233

377

610 ...

8

13

21

34

55 ...


13

21

34

55 ...

3

5

8

13

21 ...


5

8

13

21 ...

1

1

2

3

5 ...


1

1

2

3 ...

558 773 1251 2024 ... 773 1250 2023 3273 ...
First sequence (accepted) Second sequence (not accepted)
The article (the negative and positive form of being)
The alpharithm αΑ (the letter a as a number) is the least article:
The article (ο, ητοor ho hi to) is double (dual twofold) negative (steretic) and positive (collective accumulative thetic).
Consequently there is a semantic distance between the inarticulate word and the articulate one:
 The inarticulate name (word) or the inarticulate word generally represents the unformed
being, so as (καιωςεκτούτου) it belongs into Einai (to Be) in to by virtue of existence .
 The articulate name (word) or the articulate word generally represents the formed reality
(existance) and so it belongs into Gignesthai (to Become) the active existence.
The being formed entity by time receives its greatest development and then becomes formed being and so it belongs into Cosmos.
> As with the number, the article also articulates, connects two things so that they find themselves in a dynamic equilibrium.
Especially the article articulates in two ways the inarticulate name into the conceivable (perceptible) cosmos (to Einai) with the articulate name into real and unordered cosmos (To Gignesthai to Become) and into ordered cosmos as well (Cosmos).
> The articulate being is the elementary form of the inarticulate one.
> The elementary and articulate form (eidos or gender species) has the physical inclination to be summed up be accumulated towards creation synthesis and formation of the general and inarticulate form (genos genus or race).
Examples
The notion dynatos (δυνατός or powerful) represents the genus while on the one hand “the powerful” represents the species on the other hand the adynatos (ἀδύνατοςor weak, unpoweful) expresses the elementary powerful , the subspecies.
Other examples (every adjectival form)
> θρῆσκος (threskos religious) ὁθρῆσκος (ho threskos the religious) ἂθρησκος (athreskos unreligious).
> θεός (theos god) ὁθεός (ho theos the god) ἂθεος (atheos godless).
> νόμος (nomos law) ὁνόμος (the law ho nomos) ἂνομος (anomos unlawful).
> μονός (monos single) ὁμόνος (ho monos the single) ἂμονος (amonos unsingle).
> λόγος (logos reasonable) ὁλόγος (ho logos the reasonable) ἂλογος (alogos unreasonable).
> τομός (tomos being cut)) ὁ τομός (ho tomos the being cut) ἂτομος (atomos being uncut).
> Κόσμος (Kosmos Cosmos) ὁ Κόσμος (ho Kosmos the Cosmos) Ἂκοσμος (Akosmos unCosmos).
> δίκ(αι)ος (dikaios right) ὁ δίκαιος (ho dik(ai)os the right) ἂδικος (adikos unright), (etc).
The athroistic law (athristic or accumulative)
(relation between positive and negative form of the name)
The inarticulate and articulate being while they are not coincided are connected via the accumulative law of the theory which is formulated as follows:
Great sum under conditions too of negative (steretic) and positive forms (accumulative collective) of the being (species – eide είδη) is equivalent or is identified also in the limit with the being herself in his inarticulate form (race, genus, γένος):
In symbolic terms the law is expressed as follows:
α+Λ Σ(α+Λ) ≈ Λ
(where the alpharithm) α=1 is the least article giving the negative form of the being and Λ is the lexarithmos (name or word) in her articulate form.
Models
Note: The atom (of the matter) – according to the negative form of the being and the present law  is deprived of cutting (it is not cut, not divided) because it has by nature depicted a monodrom motion (not reversed).
Many atoms – when their numerous reach the critical number under condition as well (the notion of the athroistic or accumulative law) – are cut. This cutting (splitting) consists the splitting of the atom.
Consequently it would be more accurate to say “The splitting of the atoms.”
We speak about physical process in which man dos not intervene (interfere). But through man’s intervention  the intelligence of whom surpasses her (exceeds) in her parts – the critical number which will cause (generate induce) the splitting of the atoms remains unchanged (immutable). Besides “splitting of one only atom as powerless das not have so significance.
In a wide sense
In Hellenic language with the name (word) atom is meant the social and the genetic one [(βιομονάς: viomonas DNA, the double and complement helix) too.
The athroistic law continues to be valid and the extracted outcomes in any case from its application may be:
The social atom is not split off but a great number of social atoms under conditions causes their splitting.
Under the notion “social atom” the undivided personality is meant of every atom (person) in one social unity which is held as long as that sum (the number the numerous) is less than the critical number. When the sum or this numerous reach the critival number then occurs splitting and man is divided looses his personality and turns to a non man (inhuman).
The one genetic atom is not split of, namely, every atom is defined through its genetic code which is unique. But if in a test tube the sum of genetic cods of the same kind (uniformed) reach the critical number then the genetic code (the viomonas the DNA) is split into its component elements (genes).
In the present case the extensions and consequences of the spitting may be not controlled:
(1) It is probable a automatic recomposition (rearrangement) of the split human code into a new form of genetic code which will produce a new specie or new species as well.
(2) Possibly man may be master of the whole structure of the genetic cod and capable to recompose and create a new genetic cod “a new man”.
Aphorism
Every intervention in a physical process breeds dangers!
Aitema 2B (the similarity of beings in the lexarithmic gnomonic sequence)
The law of similarity
If three or more names (words) belong in a lexarithmic gnomonic sequence, then the depicted beings to them are similar as they belong in the same dialectic change.
The lexarithmic gnomic sequence follows the law of similarity of beings, according to which they are changed by the nature in similarity altering only the form.
A perceptibility of the law can by succeeded through a sequence of descendants from the same parent (father) e.g.
great grand father, grand father, father, son, great son, grandson, grandson ...
All the members are similar one to each other^{.} a member (e.g grandson) is similar to grand father but both could become “equal” if the grandson could be grow up in the same environment physical and dietetic as with his grand father*.
*geometry: Similar triangles are those with two equal angles.
Two different in size similar triangles can be coincide e.g. if the smaller will be magnified turned and transposed towards the bigger one.
Physical gnomon can be produced in the perfect pentagon: Two such as are formed from both sides of the golden triangle O_{1}AB which is revealed from the section of its three diagonal on one hand the two of them O_{1}O_{3} O_{1}O_{4} being driven from the same vertex say O_{1} on the other hand the third say O_{2}O_{5} being driven from the vertexes say O_{2 }O_{5 }on both sides of the vertexn O_{1}.
(* The letter Φ has been given by Kepler on the honor of Feidias (Φειδίας) the main sculpture of Parthenon).
> The golden number satisfies the equality Φ^{2} =Φ+1 which is unique also.
> The ratio of the two sides (greater to smaller) of the golden triangle is Φ.
> The law of the similarity of the beings is owed to the golden number Φ.
Starting with the golden triangle Ο_{1}ΑΒ (Ο_{1}BA) a double physical sequence of similar triangles is developed  one το the left ABO_{1}O_{5}O_{3} (anticlockwise) and the other το the right BAO_{1}O_{2}O_{4} (clockwise) correspondingly  to the infinity (externally and internally). It is proved that in this process all the vertexes lay on a logarithmic spira.
> All the similar triangles are golden changing only size and position.
> Every sequence of sides – if a unit will defined for the base of the original triangle (e.g. the number 1)  is gnomonic:
small side: 1 Φ 1+Φ 1+2Φ 2+3Φ 3+5Φ 5+8Φ 8+13Φ 13+21Φ 21+34Φ 34+55Φ ...
great side: Φ 1+Φ 1+2Φ 2+3Φ 3+5Φ 5+8Φ 8+13Φ 13+21Φ 21+34Φ 34+55Φ ...
From the exterior vertexes of the triangles for any sequence a spira is passed which as logarithmic is unique.
> Perceptibility of the logarithmic spira is met in the marine (nautilus) and land (terrestrial) nails (every section through a plane gives almost the mathematical logarithmic spira.
All the beings in the microkosmos (atom genetic cod) and in the macrocosmos (universe cosmos galaxies music nature water phyllotaxis) evidently or invisibly on energy or on power follow the golden section (the mesos and akros logos) and the logarithmic spira being subjected to the pancosmios gnomonic law of similarity.
So much in the sequence of the similar triangles or gnomons [expressing the incontinuity (non continuity) of the being] as too in the logarithmic spira (expressing the continuity of the being) beings remain always the same:
Παν Εν Είναι Αεί Ρεί (Pan En Einai Aei Reior Whole One Is Always Runs)*
The by commas separated names and words** are isopsephes:
Παν = Π+α+ν = 80+1+50 = 131
Εν Ειναι = Ε+ν Ε+ι+ν+α+ι = 5+50 5+10+50+1+10 = 131
Αει Ρει = Α+ε+ι Ρ+ε+ι = 1+5+10 100+5+10 = 131
* The first name (Παν) and the third word (Αειρει) are synthesized in a word (phrase) [in different syntax: Ta panta (plural of pan) Rei: every thing changes, are referred to Heracleitos.
** (word in general phrase).
> Every similar triangle (from the second and beyond) independently of size and position in the sequence can through diminution and enlargement transfer and turn to be identified with the original triangle (similarly proportionately and inversely if any triangle will be obtained (accidental triangle of the sequence) [see previously also].
While it seems that the being obtains indefinite forms in the continuous coming into being all these forms are brought to one to the original is one because of the law of similarity.
Τhe following diagram describes the step by step (successively) constructed sequence of the gnomons which depicts the physical gnomonic one or fibbonacci sequence
1 2 3 5 8 13 21 34 55 ...
put ν=1
The sequence of gnomons starts with the isosceles triangle (1,2,2) where 1 is the length of base and 2 are the lengths of the equal sides (base equal side equal side).
The second triangle is defined from the lengths (2,3,3) the third from the lengths (3,5,5) the fourth from the lengths 5,8,8) e.t. All the triangles are constructed so as to be isosceles moreover they are approximately similar. But as the sequence goes more and more (continually) to the greater numbers the triangles become more and more “similar” namely the approximation continually is increased. In the limit (mathematically) these triangles become similar: all the angles of the vertex of the triangles are made equal to 36 degrees.
Note: If the sequence of the gnomons is constructed with original triangle (1,Φ,Φ) or if as a sequence was obtained the following then all the triangles would be similar from the beginning:
1 Φ 1+Φ 1+2φ 2+3Φ 3+5φ 5+8Φ 8+13Φ 13+21Φ 21+34Φ 34+55Φ ...
> The sequence of gnomons is constructed for every lexarithmic one (given a name or a word in general since they are considered as numbers) e.g.
> (1) given lexarithmos: Κόσμος (Cosmos)=Κ+ο+σ+μ+ο+ς = 20+70+200+40+70+200 = 600 the mathematical algorithm produces one gnomonic sequence:
142 229 371 600 971 1571 ... K_{n} K_{n+1 } Limit (K_{n+1}/ K_{n}) = Φ ≈ 1.618
(142+ 229=371 229+371=600 371+600=971 ... 1571/971» 1.618.)
and the successive approximate similar triangles are:
(142 229 229) (229 371 371) (371 600 600) (600 971 971) (971 1571 1571) ...
(2) given the lexarithmos Ἓλλην (Hellen Greek) = Ε+λ+λ+η+ν = 5+30+30+8+50 = 123 the mathematical algorithm produces one gnomonic sequence:
1 3 4 7 11 18 29 47 76 123 199 322 521 ... E_{n} E_{n+1 } Limit (E_{n+1}/ E_{n}) = Φ ≈ 1.618
(1+3=4 3+4=7 7+11=18 ...521/322 »1618.)
and the successive approximate similar triangles are:
(1 3 3) (3 4 4) (4 7 7) (7,11,11) (11,18,18) (18,29,29) (29,47,47) (47,76.76) (76.123.123) ...
> From the vertexes of the triangles an approximate logarithmic spira passes which in the limit becomes accurate (mathematical): the logarithmic spira is an alternative naming for the law of similarity.
Note:
I. This gnomonic sequence in which belongs the lexarithmos Ελληνis called square gnomonic sequence because its members are the successive approximate powers of the golden number Φ: Φ^{1 }»1 Φ^{2 }»3 Φ^{3 }»4 Φ^{4 }»7 Φ^{5 }11 ... Φ^{10} »123 ... ( Φ=1.6180339)
II. Ontologically, philosophically and genetically on one hand the physical gnomonic sequence is referred to genus (γένος or genos) of the beings on the other hand the square one as a derivative one is referred to species (εἶδοςor eidos) of the beings.
> Consequently the “human” kind (homo) belongs to the first and physical sequence while the human species (ἂνθρωποςor anthropos man) belongs to the second and derivative one.
Examples of gnomonic sequences
The extractive deductions in every case are considered axiomatic since us mach the structure as the structuring of the theory are axiomatic [the theory is built based developed and formed on aitemas (postulates) and axioms].
The lexarithmic gnomonic sequence containing actively the golden mean Φ(mesos and akros logos Φ) – the ratio of every name (word). from the second and beyond to the previous one gives an approximation of Φ which proceeds increased.
But Φ is and considered the contributor of Dialectic of the being’s eternal (perpetual) changing which because of himthis change is similar a divine analogy (proportion):
The being is unbreakably connected with the inevitable necessity to change in similarity. She has no other exit except of the cases with abnormal change (violation of the law: e.g. birth of an abnormal galaxy creation of an abnormal planet system birth of abnormal children animals etc.).
In every isopsephe group of the lexarithmic gnomonic sequence one can distinguish gene (genders) and subgene species and subspecies.
The notion “genus” according to lexarithmic theory is extended to every direction and upon every matter of science [άλογος (alogos or unreasonable absurb) and reasonable organic and inorganic animate and inanimate beings etc.].
So in a lexarithmic gnomonic sequence:
Animals or plants classified in species or subspecies whose the names (considered as lexarithmoi  numbers) are members of it are expected to belong or be referred to the same genus or subgenus correspondingly.
Notions belonging to any gnostic object (ontology theology cosmology religion art literature mathematics sciences technique technology philosophy) and classified in species or subspecies whose the names (considered as lexarithmoi) are members of it are expected to belong or referred to the same genus or subgenus correspondingly.
Men or women of Hellene genus (or not) whose names (considered as lexarithmoi) are expected to belong or be referred to the same genus or to the same genealogical tree or to the same genarches (head of the race) primitive or intermediate.
Applied the theory on a lexarithmic gnomonic sequence it is able on occasion:
> to verify the mythology the history the archaeology or to complete them (e.g. localization of an ancient city or elucidation of a disputed).
> to verify the accumulative experience or to prolong it under the meaning of anticipation.
> to eternally verify the scientific knowledge to complete it and more to anticipate the next one.
