The Enneagon and the Square of Nine

Nine is the Triad squared: 3 rows of 3, the first square after 4 and the last number before the Decad closes the count. Its polygon, like the heptagon's, lies beyond the compass; its 3 stars include 3 triangles turning on 1 centre; and its square of 9 cells, filled with the numbers 1 to 9 so that every line sums to 15, is the oldest magic square in the world. The Ennead of Heliopolis and the 9 Muses count the same completion.

I. The Figure

Τὸ Σχῆμα

The regular enneagon, or nonagon, has 9 equal sides and 9 angles of 140°. Its construction fails for the same reason the heptagon's did, with a sharper edge: to draw it one must trisect the 120° angle of the equilateral triangle, and the trisection of an arbitrary angle is the second of the 3 classical impossibilities that Wantzel settled in 1837.¹ Nine is 3 × 3, and the compass can bisect but never trisect. The enneagon can be reached by a neusis, as the heptagon was, and by the folding of paper; it can't be reached by Euclid's instruments.

What the enneagon lacks in construction it makes up in stars. Nine has 3 star polygons: {9/2}, a single line through every second point; {9/3}, which falls into 3 equilateral triangles, each turned 40° from the last; and {9/4}, a single line through every fourth point, the sharpest of the 3.² The triangles of {9/3} are the Triad's figure 3 times over, 3 Triads about 1 centre, which is 9 in the plainest geometric sense.

The Theology of Arithmetic calls 9 the greatest number before 10 and gives it the names Ocean and horizon, the boundary that encloses the field.³ The Temple's numerology receives the Ennead as finish, the image, the capstone and the threshold of return, and draws it as 3 rows of 3: the Triad extended into a complete and proportioned image.³ That figure of 9 points is the frame of the magic square below.

II. The Construction

Ἡ Κατασκευή

1. Enneagon. Draw a circle with centre O and mark 9 points at intervals of 40° with a protractor, or trisect the 120° arcs of an inscribed equilateral triangle by neusis. Join the 9 points in order.
2. Join every third point: 3 equilateral triangles, {9/3}. Join every second point for {9/2} and every fourth for {9/4}.
3. Square of 9. Draw a square and divide it into 3 rows of 3 cells. Set 5 in the centre.
4. Fill the cells with 1 to 9 so that every row, every column and both diagonals sum to 15. The arrangement 4 9 2 ⁄ 3 5 7 ⁄ 8 1 6 does it, and every other solution is this one turned or reflected: the 3 × 3 magic square is unique up to its 8 symmetries.⁴

The square's constant is forced. The 9 numbers sum to 45, the 3 rows share it equally, and 45 ⁄ 3 is 15. The centre must be 5 because 4 lines pass through it and 5 is the only number that can hold the middle of 4 pairs summing to 10: 1 and 9, 2 and 8, 3 and 7, 4 and 6. The Temple's Pentad page makes that very observation about 5 as the centre of 1 to 9, and the magic square is its proof laid out in cells.⁵

III. Number and Form

Ἀριθμὸς καὶ Σχῆμα

Nine is a square number, 3², and the Temple's Ennead page draws it as the Triad brought into an image: the number of rows answers to the number of points in each row, and the dimension and the whole bear 1 form.³ The same page explains the Ennead's return by the subtraction of nines: 10 leaves 1, 11 leaves 2, 100 leaves 1 after 11 nines are taken away. In the decimal count, casting out nines sends every number back to its digit sum, and 9 is the number at which the count turns over.

The magic square's total, 45, is the ninth triangular number, 1 + 2 + … + 9, the Tetractys of Layer 10 extended to 9 rows. Its constant, 15, is the fifth triangular number and the Pentad's own: 3 times 5. Nine, 5 and 3 share 1 figure. Agrippa made the square Saturn's, the first of his 7 planetary tables, "a square of three, containing the particular numbers of nine", each line making 15 and the whole 45.⁴ Saturn is Kronos, the father of Zeus and the outermost of the old planets, the Ennead's horizon in the sky.

IV. In the Ancient Witnesses

Παρὰ τοῖς Ἀρχαίοις

The Great Ennead of Heliopolis is the oldest 9 on record. At its head stands Atum, the creator and Sun God, who of himself brought forth Shu and Tefnut, air and moisture; they bore Geb and Nut, earth and sky; and from these came Osiris, Isis, Seth and Nephthys.⁶ One source and 8 descendants in 4 pairs: the Ennead of Egypt is the Monad of Layer 1 joined to the Octad of Layer 8, 1 + 8 = 9, the point and the cube together. The Pyramid Texts already invoke it, and the Temple's reading of the Ennead as the completed image of the source has its first instance in that family.

The Greek 9 are the Muses. Hesiod names them in the Theogony: Clio, Euterpe, Thalia, Melpomene, Terpsichore, Erato, Polyhymnia, Urania and Calliope, "who is the chiefest of them all".⁷ Nine daughters of Zeus and Memory, and 9 arts: history, music, comedy, tragedy, dance, lyric, hymn, astronomy and epic. The Muses are the Ennead of the mind, and the Temple's core meaning for 9, "the image", is what each of them makes.

The square of 9 is Chinese in its first record. The Lo Shu, the "writing of the river Lo", is the 3 × 3 square 4 9 2 ⁄ 3 5 7 ⁄ 8 1 6, known to the ancient Chinese and set by their tradition in the 3rd millennium BCE; in Europe the same square reaches Agrippa's Three Books of Occult Philosophy in 1533 as the table of Saturn.⁴ The Temple's own pages on the magickal squares give the square of Saturn, with its total of 45 and a working of 9 days, as the first of the planetary squares and set out how the squares are used, with exact repetition joined to directed affirmation and visualisation; this square is the smallest of them.⁸

V. The Zevist Reading

Ἡ Ζευϊστικὴ Ἀνάγνωσις

The Ennead is the capstone, and the 3 triangles of {9/3} are the Temple's trinity of the worlds drawn 3 times: the triad of Gods, the triad of the soul's channels and the triad of beginning, middle and end, each a Triad and the 3 together an Ennead.⁹ Where the Triad gave 1 action its full course, the Ennead gives a whole work its completed form, and the Temple's Words of Power meditation uses 9 affirmations after the vibration for exactly that reason: 9 is the count of endings.¹⁰

The magic square is Ma'at as arithmetic. Every line equal, every direction balanced, no cell able to change without breaking the whole: it's the Temple's teaching of justice as proportion, which the Tetrad introduced through the square of side 4, raised to the level of a completed image. The Temple's Ennead page asks for the final correction, the fulfilment of an obligation, the preservation of what should endure; a magic square is a figure in which every correction has already been made.³

Birth and death meet in the number because 9 is where the count returns. Add 1 to 9 and the digit falls to 0 and a new place opens. In the Temple's doctrine of Light and Darkness the death, life and rebirth of cells under Ma'at is health; the Ennead is that cycle in number, the end that becomes the condition of another beginning. Osiris stands in the Heliopolitan 9 as the God who dies and is restored, and the Temple's Djed pillar page reads his axis as the stability of resurrection.¹¹ The capstone is set, and the next course begins at 1.

VI. Measure

Τὸ Μέτρον

VII. Sources and Reading

Πηγαί

1. Eric W. Weisstein, "Regular Nonagon", MathWorld: mathworld.wolfram.com. MacTutor, Wantzel.
2. Eric W. Weisstein, "Star Polygon", MathWorld: mathworld.wolfram.com.
3. Temple of Zeus, 9 · The Ennead; The Theology of Arithmetic, On the Ennead, Greek text, Wikisource.
4. Eric W. Weisstein, "Magic Square", MathWorld: mathworld.wolfram.com. MacTutor, "Mathematical games and recreations": mathshistory.st-andrews.ac.uk. Heinrich Cornelius Agrippa, Three Books of Occult Philosophy II.22, edited by Joseph H. Peterson: esotericarchives.com.
5. Temple of Zeus, 5 · The Pentad.
6. Global Egyptian Museum, glossary, "Ennead": globalegyptianmuseum.org. Encyclopaedia Britannica, "Ennead": britannica.com.
7. Hesiod, Theogony 75 to 79. Greek, Perseus · English (Evelyn-White), theoi.com.
8. Temple of Zeus, Saturn 45, Magickal/Kabalistic Squares: How to Correctly Use These, Planetary Squares, and How to Use the Numbers in Magick & Meditation.
9. Temple of Zeus, 3 · The Triad.
10. Temple of Zeus, Words of Power, cited in How to Use the Numbers in Magick & Meditation.
11. Temple of Zeus, The Djed Pillar of Osiris.