ΤΕΤΡΑΓΩΝΟΝ (tetragonon) — Lexarithmos 1679 · Etymology, Meanings

Definition

According to the Liddell-Scott-Jones Lexicon, τετράγωνον is defined as "a square figure, a square." The word, a compound of the numeral "τέσσαρες" (four) and the noun "γωνία" (angle), describes a geometric shape with four equal sides and four right angles. Its significance extends beyond mere description, as it embodies the ideas of symmetry, balance, and completeness, making it one of the most fundamental shapes in ancient Greek thought.

In arithmetic, τετράγωνον also refers to a "square number," i.e., the product of a number multiplied by itself (e.g., 3x3=9). The Pythagoreans attributed special importance to square numbers, considering them symbols of perfection and stability, as they could be geometrically represented as square arrangements of points. This interrelation of geometry and arithmetic was central to the Greek scientific tradition.

Beyond its strictly mathematical uses, τετράγωνον acquired metaphorical meanings. A "τετράγωνος ἀνήρ" (a square man), as described by Simonides and referenced by Aristotle, was a steadfast, upright, and unyielding individual, like a well-built square that is not easily shaken. This metaphorical usage underscores the cultural value ascribed to the quality of the square as a symbol of stability and integrity.

Etymology

tetra-gon- (compound root from τέσσαρες and γωνία)

The word "τετράγωνον" is a compound noun derived from the numeral "τέσσαρες" (four) and the noun "γωνία" (angle). This compound is transparent, directly describing a shape that possesses four angles. The root "τετρα-" stems from the Proto-Indo-European root *kwetwer- "four," while the root "γων-" (from PIE *g(e)neu- "knee, angle") denotes a bend or the meeting point of two lines.

Cognate words include the numeral "τέσσαρες" and the noun "γωνία," which serve as its constituent elements. From this compound, other derivatives arise, such as the adjective "τετράγωνος" (square), the verb "τετραγωνίζω" (to square, to calculate the square), and the noun "τετραγωνισμός" (the act of squaring).

Main Meanings

Geometric figure: the square — A plane figure with four equal sides and four right angles. The primary and most common meaning in ancient geometry.
Arithmetic: square number — The product of a number multiplied by itself (e.g., 9 is the square of 3). Significant in Pythagorean arithmology.
Metaphorical: firm, upright, complete — Describes a person of stable character, unyielding in their principles, like a well-constructed square. (e.g., "τετράγωνος ἀνήρ").
Architectural/Structural: squared stone — A stone hewn into a square shape, used in construction for stability and aesthetic appeal.
Astrology: square aspect — An aspect between two planets forming a 90-degree angle, often considered challenging or difficult.
Military: square formation — A formation of troops in a square shape for defense or attack, imparting stability and uniformity.

Word Family

tetra-gon- (compound root from τέσσαρες and γωνία)

The root tetra-gon- is a transparent compound that unites the concept of the number "four" with that of "angle," creating a family of words revolving around the geometric figure of the square and its properties. This compound highlights the Greek tendency for precise description and categorization, especially in mathematics. Each member of the family develops a specific aspect of the central idea, from quantity and shape to the action of squaring and the characteristics derived from it.

τέσσαρες numeral · lex. 1412

The basic numeral "four," which forms the first component of "τετράγωνον." It is fundamental to the definition of the shape, indicating the number of its sides and angles.

γωνία ἡ · noun · lex. 864

The "angle," the second component of the word, referring to the meeting point of two lines or planes. In the case of a square, the angles are four and right, defining its essential property.

τετράγωνος adjective · lex. 1829

The adjective meaning "having four angles" or "square." It is used to describe anything with the form of a square, such as "τετράγωνος λίθος" (square stone) or, metaphorically, "τετράγωνος ἀνήρ" (Simonides, Fragment 542).

τετραγωνίζω verb · lex. 2376

Means "to make square," "to calculate the square of a number," or "to square a figure." This verb is central to the concept of "squaring the circle," a famous unsolved problem in ancient geometry.

τετραγωνισμός ὁ · noun · lex. 2079

The act or process of squaring, i.e., transforming a figure into a square of equal area, or calculating the square of a number. It is frequently found in mathematical texts, such as in Plato's "Meno" (82B) concerning the squaring of a double square.

τετραγωνικός adjective · lex. 1859

Pertaining to a square or squaring. Used to describe properties or concepts related to squares, e.g., "square root" (though the term itself is more modern, the concept existed).

τετραγωνιστής ὁ · noun · lex. 2207

One who practices squaring, i.e., a geometer or mathematician who solves problems of squaring.

Philosophical Journey

The concept of the square, both as a geometric figure and an arithmetic property, has a long and rich history in Greek thought, shaping its scientific and philosophical development.

6th-5th C. BCE

Pythagoreans

The Pythagoreans extensively studied square numbers and their geometric representation, connecting them with the harmony and order of the cosmos.

5th C. BCE

Simonides of Ceos

The lyric poet Simonides used the phrase "τετράγωνος ἀνήρ" to describe a virtuous and steadfast man, a metaphor later adopted by Aristotle.

4th C. BCE

Plato

In the dialogue "Meno," Plato presents the problem of "squaring the square" as an example of the Socratic method, where an uneducated slave boy is led to discover geometric truths.

3rd C. BCE

Euclid

In his "Elements," Euclid provides the rigorous definition of a square (Definition 22, Book I) as a quadrilateral with equal sides and right angles, foundational to classical geometry.

1st C. BCE - 1st C. CE

Roman Period

The concept of the square continued to be used in architectural and engineering applications, as well as in astrological treatises for the "square aspects" of planets.

Byzantine Period

Byzantine Mathematicians

The study of geometry and square numbers persisted, with commentators on the works of Euclid and other ancient authors preserving and transmitting this knowledge.

In Ancient Texts

Three of the most significant passages highlighting the importance of the square in ancient Greek thought:

«οὐκοῦν τὸ μὲν τετράγωνον τοῦτο, ὦ Μένων, οὐκ ἔστι διπλάσιον τοῦ πρώτου;»

Then this square, Meno, is not double the first one?

Plato, Meno 82B

«τετράγωνος δ' ἀνὴρ ἄνευ ψόγου τετυγμένος, χερσὶ καὶ ποσὶ καὶ νόῳ τετράγωνος»

A square man, faultlessly fashioned, square in hands and feet and mind.

Simonides of Ceos, Fragment 542 (quoted by Aristotle, Rhetoric 1361a2)

«τετράγωνον δέ ἐστιν, ὃ ἰσόπλευρόν τέ ἐστι καὶ ὀρθογώνιον.»

A square is that which is equilateral and right-angled.

Euclid, Elements, Book I, Definition 22

Lexarithmic Analysis

The lexarithmos of the word ΤΕΤΡΑΓΩΝΟΝ is 1679, from the sum of its letter values:

Τ = 300

Tau

Ε = 5

Epsilon

Τ = 300

Tau

Ρ = 100

Rho

Α = 1

Alpha

Γ = 3

Gamma

Ω = 800

Omega

Ν = 50

Nu

Ο = 70

Omicron

Ν = 50

Nu

= 1679

Total

300 + 5 + 300 + 100 + 1 + 3 + 800 + 50 + 70 + 50 = 1679

1679 decomposes into 1600 (hundreds) + 70 (tens) + 9 (units).

The 18 Methods

Applying the 18 traditional lexarithmic methods to the word ΤΕΤΡΑΓΩΝΟΝ:

Method	Result	Meaning
Isopsephy	1679	Base lexarithmos
Decade Numerology	5	1+6+7+9 = 23 → 2+3 = 5 — Pentad, the number of completeness, harmony, and man as a microcosm.
Letter Count	10	10 letters — Decad, the number of perfection, the cosmos, and completion.
Cumulative	9/70/1600	Units 9 · Tens 70 · Hundreds 1600
Odd/Even	Odd	Masculine force
Left/Right Hand	Right	Divine (≥100)
Quotient	—	Comparative method
Notarikon	Τ-Ε-Τ-Ρ-Α-Γ-Ω-Ν-Ο-Ν	Telos Ergon Tes Rhizēs Archē Geometrias Ontōs Nomos Holoklērou Noēseōs (A perfect work of the root, beginning of geometry, truly a law of complete understanding).
Grammatical Groups	4V · 3S · 3M	4 vowels (E, A, Ω, O), 3 semivowels (R, N, N), 3 mutes (T, T, G). The balance of these groups suggests the stability and harmony of the shape.
Palindromes	No
Onomancy	—	Comparative
Sphere of Democritus	—	Divination with lunar day
Zodiacal Isopsephy	Saturn ♄ / Pisces ♓	1679 mod 7 = 6 · 1679 mod 12 = 11

Isopsephic Words (1679)

Words from the Liddell-Scott-Jones lexicon with the same lexarithmos (1679) as "τετράγωνον," but from different roots, highlighting numerical coincidence:

μαθηματοπωλικός

"pertaining to selling lessons" or "a seller of lessons." This word, though numerically identical, represents a commercial aspect of knowledge, contrasting with the pure, abstract geometry of the square.

φασματοφάνεια

"the appearance of a phantom or vision." This contrasts with the concrete, tangible, and logically defined nature of the square, introducing the concept of the vague and supernatural.

θεμερόφρων

"firm-minded, steadfast." This word has an interesting semantic connection to the metaphorical use of "τετράγωνος ἀνήρ," as both imply stability and an unshakeable character.

εὐστροφάλιγξ

"well-turning, easily revolving." This stands in opposition to the static and unchangeable nature of the square, suggesting movement and flexibility.

διαγωνιάω

"to be at an angle, to be diagonal." Although a geometric term, it refers to a different aspect of angles and relationships, not the wholeness and balance of the square.

ἐγκωμιαστικός

"eulogistic, laudatory." A word belonging to the realm of rhetoric and evaluation, far removed from the objective description of geometric shapes.

The LSJ lexicon contains a total of 41 words with lexarithmos 1679. For the full catalog and AI semantic filtering, see the interactive tool.

Sources & Bibliography

Liddell, H. G., Scott, R., Jones, H. S. — A Greek-English Lexicon. Oxford: Clarendon Press, 1940.
Plato — Meno.
Euclid — Elements.
Aristotle — Rhetoric.
Heath, T. L. — A History of Greek Mathematics, Vol. I: From Thales to Euclid. Oxford: Clarendon Press, 1921.
Burkert, W. — Lore and Science in Ancient Pythagoreanism. Cambridge, MA: Harvard University Press, 1972.
DK (Diels, H., Kranz, W.) — Die Fragmente der Vorsokratiker. Berlin: Weidmannsche Buchhandlung, 1951.