WHAT IT IS

After calculating the lexarithmos, you reduce the number to its Pythmene (1–9) by repeatedly summing its digits. The resulting number corresponds to one of the nine archetypal powers. The Decad (10) is not a result of reduction — it is the philosophical framework: the universe of numbers (1+2+3+4=10) that returns to the Monad.

1. Unified

The total of all words is reduced to a single digit — e.g. ΛΟΓΟΣ+ΕΣΤΙ=888 → 6 → Hexad. The phrase as a unified entity. This is frequently used in the Sibylline Oracles and by the Church Fathers.

2. Per word

Each word is reduced separately — e.g. ΛΟΓΟΣ=373 → 4 (Tetrad), ΕΣΤΙ=515 → 2 (Dyad). Reveals the "decadic relation" between words. According to Nicomachus, reduction applies to any number — so both methods are valid.

HISTORY & SOURCES

The Pythagorean decad (or Tetraktys) was not merely a counting system, but a map of cosmogony. For the Pythagoreans, the numbers 1 to 10 described the transition from the divine monad to material completion.

According to the tradition preserved by Nicomachus:
Monad (1): The point, the source of all things, the masculine element and mind.
Dyad (2): The line, differentiation, the feminine element and matter.
Triad (3): The plane (triangle), the first real number with beginning, middle and end.
Tetrad (4): The solid body, justice and the four elements of nature.
Decad (10): The "All-Complete" number containing all proportions and returning to unity (1+0=1).

Nicomachus of Gerasa (c. 60–120 AD) was one of the most important mathematicians and philosophers of antiquity, from Gerasa (modern Jordan). He is considered the founder of Neo-Pythagoreanism, a movement that revived Pythagoras' teachings with emphasis on the mystical and philosophical dimension of numbers.

His work Introduction to Arithmetic was the standard mathematics textbook for over 1,000 years. Through Boethius' Latin translations, it deeply influenced education in medieval Europe. In his Manual of Harmonics he connected numerical ratios to musical intervals. He believed numbers pre-existed in the mind of God as cosmic templates for the creation of the universe.

Hippolytus of Rome (c. 220–230 AD), in his anti-heretical work Refutatio Omnium Haeresium (Book IV), describes how the Gnostics (Marcosians, Ophites, followers of Basilides) used this method of reduction. Their aim was to find the "root" or "monad" — the divine One — from which all things proceeded. Hippolytus documents that the Gnostics borrowed from Pythagorean arithmosophy.

Fun fact: The satirist Lucian used the phrase "you calculate like Nicomachus" as the ultimate compliment for someone excellent at arithmetic.

THE TEN PRIMORDIAL POWERS

1	Monad → Divine Origin, the One
2	Dyad → Polarity, Matter
3	Triad → Perfection, Divine Triads
4	Tetrad → Four Elements, Cosmos
5	Pentad → Golden Ratio, Life
6	Hexad → Harmony \| Cosmos = 600
7	Heptad → Sacred Number, Planets
8	Ogdoad → Harmony, Justice (Christ. application: JESUS=888)
9	Ennead → Last Monad, Completion
10	Decad → Return to the Monad

HOW IT WORKS

Lexarithmos of ΛΟΓΟΣ = 373
3 + 7 + 3 = 13
1 + 3 = 4 (Tetrad)
The Tetrad: four elements, cosmos, stability — and structural component of the Decad, since 1+2+3+4=10 (Tetraktys)

CONCLUSION

The single-digit number shows which of the ten archetypal powers the word expresses — regardless of the size of the lexarithmos. Two words with the same pythmenical lexarithmos were considered "cognate" by Hippolytus — they carry the same hidden power at a deeper level.