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Planetary Kamea and Sigil Workbench

Seven tables of numbers, Saturn's three-by-three to the Moon's nine-by-nine, as Agrippa printed them in 1533, each set on an engraved plate. Every row, column and diagonal adds itself up in front of you; each table can be watched building itself from the plain count of its numbers; and Saturn's seal turns out to be nothing but its own count, one to nine, in three strokes. Spell a name in Hebrew and it is cut into the plate one stroke at a time; turn the plate to see the seal and the characters set apart.

The sums, the constructions and the gematria are arithmetic. The sigils depend on how large values are reduced to fit the square, and practice varied: the rule in use is named beside every result, and every traced figure is a reconstruction, not Agrippa's printed character.

The table of Saturn

שבתאי
492357816

Rows: four, nine, two; three, five, seven; eight, one, six.

The rim: the planet's sign, its name, and its four numbers, 3, 9, 15, 45.

Every line of the table of Saturn comes to 15.

Beneath the numbers, in red, the seal of Saturn: its count from one to nine, in three strokes.

The table · arithmetic checked live

Agrippa's four numbers

Agrippa gives each table four numbers, its side, its cells, its line and its whole, and over them sets “such names as fill up the numbers”.

Side · 3

  • אבAbAgrippa, 1651

    over the side, 3 · gematria 3 ✓

Cells · 9

  • הדHodAgrippa, 1651

    over the cells, 9 · gematria 9 ✓

Line · 15

  • יהIahAgrippa, 1651

    over the line, 15 · gematria 15 ✓

  • הודHodAgrippa, 1651

    over the line, 15 · gematria 15 ✓

Whole · 45

  • יוד הא ואו האJehovah extendedAgrippa, 1651the four letters of the Name, each spelled out

    over the whole, 45 · gematria 45 ✓

  • אגיאלAgiel

    Intelligence · received 45 · gematria 45 ✓

  • זאזלZazel

    Spirit · received 45 · gematria 45 ✓

Source: Received
Seven ladders

From the bottom thread up: side, cells, line, whole.

The four numbers of each table
TableSideCellsLineWhole
Saturn391545
Jupiter41634136
Mars52565325
Sun636111666
Venus7491751225
Mercury8642602080
Moon9813693321

  • "The squares are assigned to the seven planets in the order of their velocity": Saturn, the slowest, has the smallest, and the Moon, the swiftest, the largest.

    Source: Scholarly reading
    • K. A. Nowotny, "The Construction of Certain Seals and Characters in the Work of Agrippa of Nettesheim", Journal of the Warburg and Courtauld Institutes 12 (1949), p. 50

  • Saturn's 3 is the number of Binah, the third Sephirah, and the Moon's 9 the number of Yesod, the ninth: Agrippa's Scale of the number ten (II.13) and his tables coincide, and the Golden Dawn read the coincidence as design.

    Source: Scholarly reading

  • The order was not universal. Al-Mālaqī's reversal (c. 1135), with the three-by-three for the Moon, became by far the commoner in the Islamicate world, and Cardano (Practica arithmetice, 1539) used it again; Lehrich suspects he did so out of dislike for Agrippa.

    Source: Scholarly reading
    • B. Hallum, "New Light on Early Arabic Awfāq Literature", in Islamicate Occult Sciences in Theory and Practice (Leiden: Brill, 2021), on al-Mālaqī's "System 2"
    • K. A. Nowotny, "The Construction of Certain Seals and Characters in the Work of Agrippa of Nettesheim", Journal of the Warburg and Courtauld Institutes 12 (1949), p. 47, on Cardano's Practica arithmetice
    • C. I. Lehrich, The Language of Demons and Angels: Cornelius Agrippa's Occult Philosophy (Leiden: Brill, 2003), p. 103 n. 12

  • Saturn's square is the only one of its kind; Jupiter's is one of 880; Mars's one of 275,305,224; the Sun's one of some 1.8 × 10^19, counted only in 2024; beyond that nobody has counted.

    Source: Arithmetic
Each number
✓ 1–9 once each
Lines
✓ 8 of 8 make 15
Cells total
✓ 45

Hover, tap or focus a medallion on the rim to sum each row, column and diagonal. Every one comes to 15.

Prove each line sends light down every line in turn, clockwise from the top right, and each medallion counts up to its sum; Let it flow sends it down all of them at once. When the last line lands, the rule round the plate fills.

Every line comes to fifteen: 3 × (3² + 1) ÷ 2 = 3 × 5, the side times the centre.

The diagonals count: ↘︎ 4, 5, 6: counts by 1; ↗︎ read from the top right, 2, 5, 8: counts by 3.

Source: Arithmetic

Partners

Every number and the number opposite it through the centre make 10. Tap a cell to see its pair, or draw your own line: choose cells and each brings its mirror; any 3 cells drawn so come to 15, whatever their shape.

Why it works
  1. Every number and the number opposite it through the centre make 10.
  2. 3 cells drawn in mirror pairs are 1 pairs and the centre, and 1 × 10 + 5 = 15.
  3. So any figure closed under that mirror comes to the constant, whatever its shape: the rows, the columns and the diagonals are only the best known of them.

The counting thread

Joined in counting order, 1 to 9, the numbers make a single thread. On six of the tables the thread is its own half turn. Claude Bragdon called figures drawn through a magic square in counting order "magic lines" and used them as ornament.

Source: ArithmeticSource: Modern
  • Claude Bragdon, Projective Ornament (1915), on magic lines

What Saturn's table holds

Source: ReceivedSource: Scholarly readingSource: Arithmetic
Source: Scholarly readingSource: ReceivedSource: Arithmetic
Source: ReceivedSource: ArithmeticSource: Legend
Source: Received
All 8 lines, summed
  • Row 1: 4 + 9 + 2 = 15
  • Row 2: 3 + 5 + 7 = 15
  • Row 3: 8 + 1 + 6 = 15
  • Column 1: 4 + 3 + 8 = 15
  • Column 2: 9 + 5 + 1 = 15
  • Column 3: 2 + 7 + 6 = 15
  • Diagonal, top left to bottom right: 4 + 5 + 6 = 15
  • Diagonal, bottom left to top right: 8 + 5 + 2 = 15

The hour

Saturn in the day

Three Books of Occult Philosophy, II.22

Agrippa's words and his plate

The first of them is assigned to Saturn, and consists of a square of three, containing the particular numbers of nine, and in every line three every way, and through each Diameter making fifteen. Now the whole sum of numbers is fourty five.

“They say that this table being with a fortunate Saturn engraved on a plate of lead, doth help to bring forth, or birth, and to make a man safe, and powerfull …”

Beside it, always, his caution: “For materiall numbers, and figures can do nothing in the mysteries of hid things, but representatively by formall numbers, and figures, as they are governed, and informed by intelligencies, and divine numerations …”

Source: Received

The rim of this plate takes the colour of Agrippa's plate for this table: lead (II.22). The planet's metal in the common inheritance is lead.

“with a fortunate Saturn engraved on a plate of lead”

Source: Received

Agrippa prints the tables and, after each, its seals and characters; he gives no design for a plate. The two faces here follow that order. Nowotny describes a post-mediaeval medallion of the Moon in Vienna with the nine-by-nine square on one side and a character divided between both sides (1949, p. 56). The rim, the arrangement and the look of the plate are this site's.

The method he keeps back: “Now how the seals, and Characters of the Stars, and spirits are drawn from these tables, the wise searcher, and he which shall understand the verifying of these tables, shall easily find out.”

How this plate is drawn
  • A ring: the name turns on this cell. (this page's convention)
  • A ring broken by a line: the line passes over this cell without turning. (this page's convention)
  • A halo round a number: the line passes behind it. (this page's convention)
  • Strands bowed apart: the same stroke made again, once for each pass. (this page's convention)
  • A small circle: where the name begins. (Regardie's "little circle … on the first letter" (1972))
  • A bar: where it ends. (this page's convention)
  • Loops and ×n: a letter landing again on the same cell. (this page's convention)
  • Order numerals: which letters land on the cell, counted along the name. (this page's convention)

Line styles

  • Solid: the rule in force.
  • Long dash: drop zeros as needed.
  • Short dash: strike every zero.
  • Dotted: wrap (a modern comparison). Nothing else on the plate is dashed or dotted.

Inks

Red is used for a seal and nothing else: a seal is drawn from the square's construction. Goldis a character, drawn from a name. Hatching means “not reproduced here”.

Hallmarks

  • Agrippa's, as printed: the table's numbers. On the reverse, Saturn's seal, his figure as Nowotny and Lehrich identify it, redrawn here from the square.
  • Reconstructed, by a named rule or scholar.
  • This site's design.

On a phone the rim is too narrow for the hallmarks and the divine names, and the reverse too small for its hallmarks; they are given here, in the Table panel and under the plate.

The six tags

  • Source: ReceivedPrinted in Agrippa's tables (1533; quoted from the 1651 English) or another named primary source.
  • Source: ArithmeticComputed live from the table on this page.
  • Source: ReconstructionA recipe or figure that reproduces the table exactly; not a method the source states.
  • Source: Scholarly readingA named scholar's identification or interpretation.
  • Source: LegendA story told about the thing, not an attestation of it.
  • Source: ModernNo witness older than the modern author named.

Before and after Agrippa

No route is known from the Chinese square to the Arabic ones; there is only one three-by-three magic square.

  1. 1st c. CEThe nine numbers in the order of the three-by-three square, in the Bright Hall chapter of the Da Dai Liji.Source: Received
    • S. Cammann, "The Evolution of Magic Squares in China", Journal of the American Oriental Society 80 (1960), pp. 116–124
  2. c. 850Al-Ṭabarī's Firdaws al-ḥikma uses the square to ease childbirth: the earliest Islamic reference to a magic square.Source: Scholarly reading
    • B. Hallum, "New Light on Early Arabic Awfāq Literature", in Islamicate Occult Sciences in Theory and Practice (Leiden: Brill, 2021), on al-Ṭabarī
  3. 10th c.The Ikhwān al-Ṣafāʾ give squares of three to nine, and al-Anṭākī and al-Būzjānī write the earliest surviving treatises on making them.Source: Scholarly reading
    • J. Sesiano, Magic Squares in the Tenth Century: Two Arabic Treatises by Anṭākī and Būzjānī (Springer, 2017), introduction
  4. 11th c.Ibn al-Zarqālluh gives the seven squares, three to nine, to the seven planets, Saturn's the three.Source: Scholarly reading
    • B. Hallum, "New Light on Early Arabic Awfāq Literature", in Islamicate Occult Sciences in Theory and Practice (Leiden: Brill, 2021), on "System 1"
    • R. Comes, "The Transmission of Azarquiel's Magic Squares in Latin Europe", in Medieval Textual Cultures (de Gruyter, 2016), pp. 159–198
  5. c. 1135Al-Mālaqī reverses the order, giving the three-by-three to the Moon; his order became by far the commoner in the Islamicate world.Source: Scholarly reading
    • B. Hallum, "New Light on Early Arabic Awfāq Literature", in Islamicate Occult Sciences in Theory and Practice (Leiden: Brill, 2021), on "System 2"
    • C. I. Lehrich, The Language of Demons and Angels: Cornelius Agrippa's Occult Philosophy (Leiden: Brill, 2003), p. 103 n. 12
  6. 15th c.De septem quadraturis planetarum, a Latin treatise on the seven squares, is bound into the Cracow codex of the Latin Picatrix (Biblioteka Jagiellońska 793).Source: Scholarly reading
    • K. A. Nowotny, "The Construction of Certain Seals and Characters in the Work of Agrippa of Nettesheim", Journal of the Warburg and Courtauld Institutes 12 (1949), p. 46
    • C. I. Lehrich, The Language of Demons and Angels: Cornelius Agrippa's Occult Philosophy (Leiden: Brill, 2003), p. 103 n. 12
  7. 1514Dürer's Melencolia I sets a four-by-four square: Agrippa's later table of Jupiter turned upside down with its two middle columns exchanged, which puts 15 14 in the bottom row.Source: Received
  8. 1533Agrippa prints the seven tables with their seals, characters and names in De occulta philosophia, Book II, chapter 22; the 1531 printing had Book I only.Source: Received
    • Agrippa, De occulta philosophia libri tres (1533), II.22
    • C. I. Lehrich, The Language of Demons and Angels: Cornelius Agrippa's Occult Philosophy (Leiden: Brill, 2003), p. 104
  9. 1801Francis Barrett's The Magus reprints Agrippa's tables, seals and characters in engraved plates.Source: Received
    • Francis Barrett, The Magus (London, 1801), Book I, Part II, ch. XXVIII, pp. 142–147 and Plate 1
  10. 1910An Alpha et Omega board of the Moon's seals and names, executed (made) in September 1910, is marked "Hanging under Kamea on Banner of East".Source: Received
  11. 1949Nowotny shows that the seals come from the construction of the squares and the characters from the names.Source: Scholarly reading
    • K. A. Nowotny, "The Construction of Certain Seals and Characters in the Work of Agrippa of Nettesheim", Journal of the Warburg and Courtauld Institutes 12 (1949), pp. 46–57
  12. 1972Regardie's How to Make and Use Talismans traces names on the squares by the Nine Chambers, with a little circle on the first letter.Source: Received
    • Israel Regardie, How to Make and Use Talismans (Aquarian Press, 1972), ch. 1
  13. 2003Lehrich refines Nowotny's method and finds Agrippa's figures "in no way random or haphazard".Source: Scholarly reading
    • C. I. Lehrich, The Language of Demons and Angels: Cornelius Agrippa's Occult Philosophy (Leiden: Brill, 2003), pp. 100–111

The tradition

A planetary magic square sets the numbers 1 to n² in a grid so that every row, every column and both diagonals add to the same sum. Seven such squares, of orders three to nine, were given to the seven planets in their Chaldean order: Saturn's three-by-three, Jupiter's four-by-four, down to the Moon's nine-by-nine. Kamea, from a Hebrew word for an amulet, is the name the later English tradition uses for them; Agrippa calls them the tables of the planets.

Magic squares were long worked in the Islamic world, and the planetary set circulated in Latin manuscripts before Agrippa printed it in 1533. His tables give each square with numbers drawn from it, and the names of Intelligences and Spirits whose Hebrew letters mostly sum to those numbers; Venus is the clearest exception. The Hermetic Order of the Golden Dawn inherited the squares and names, and the practice of drawing a sigil by tracing a name letter by letter from cell to cell is associated with that inheritance.

Reading the instrument

  • Each table is set on an engraved plate with two faces. The front holds the table "in his compass", as Agrippa's caption has it; the reverse holds "The Seales or Characters" in the order of his captions, with the name last cut set in a roundel. Turn the plate with the button beside the modes. The rim takes a faint tint of the metal Agrippa names for each table: lead, silver, iron or gold.
  • Table reads the front. Every row, column and diagonal ends in a medallion on the rim holding its sum, computed live against the constant n(n² + 1) ÷ 2; hover, tap or focus one to light its line, or prove them all in turn and watch the rule round the plate fill. On a phone the larger tables are stepped through with the arrows in the dock. Partners joins each number to the one that makes n² + 1 with it, and Draw your own line shows that any figure mirrored through the centre comes to the constant. The four dies are Agrippa's four numbers, with the names that fill them; the notices are what each table holds, each tagged and cited.
  • Build plays one way the table can be made from the plain count of its numbers, step by step: the terrace or the step rule for the odd tables, keep and turn for Jupiter and Mercury, and the same turn mended by six exchanges for the Sun. Each recipe ends on Agrippa's table exactly, and none of them is his: he does not say how his tables were made.
  • Name cuts a name into the plate stroke by stroke. Pick a listed name, tap letters, tap a cell to see which letters land there, or type Hebrew, numbers or a name in Latin letters; a Latin name the tables print is matched to the Hebrew this page lists for it (the received spelling, corrected where the 1651 text as transcribed slips; see the variants), and anything else is spelled and marked as made here. Letters are counted by standard gematria, final forms at their ordinary value; the two Nine Chambers rules place final forms from 500 to 900 when reducing, as the Aiq Bekar table does, without changing the total. A value larger than the square is reduced by the rule in force, which is printed beside every result; compare the three rules, or load a sequence a source printed and see which rule reproduces it.
  • How the figure is drawn is given in the key under the plate. A ring is a turn; a ring broken by a line is a cell the line passes over; a halo is a number the line passes behind; bowed strands are the same stroke made again; the small circle (Regardie's) is the start and the bar the end; loops count a letter landing again on the same cell. Red is used only for Saturn's seal, which is drawn from the square's own count; gold is a character, drawn from a name. Hatching marks what is described or printed but not reproduced here.
  • The hour panel shows when the plate's planet rules the hour and the day at the place saved in your preferences, reckoned as the Planetary Hours instrument reckons them, with the caveat beside it that Agrippa asked for far more than the hour. The plate never switches to the hour's planet by itself.
  • The received numbers are not uniform. Most names equal their square's constant or total, but Hagiel's 49 is the count of Venus's cells, Venus's plural Intelligences, Bne Seraphim, are given 1252, which is none of the square's numbers and needs final Mem at 600, and the Moon's Intelligence is a compound title whose spelling varies.

Where it descends from

Three Books of Occult Philosophy
Agrippa's compendium, printed in full in 1533. Book II, chapter 22 gives the seven tables with their seals, the numbers taken from each, and the names of the Intelligences and Spirits with their Hebrew spellings. It does not explain how the sigils were drawn.
The Magus
Francis Barrett's 1801 compilation, which reprinted Agrippa's tables and seals for English readers.
The Golden Dawn
Israel Regardie's publication of the Order's papers, the route by which Agrippa's squares and names, and sigil-drawing on them, reached most modern practice.

What this does not claim

The squares, line sums and gematria are arithmetic and are checked live. The sigil is a reconstruction by a named rule. It is not a copy of the sigil Agrippa printed for that name: reconstructions match some printed sigils closely and others poorly, and the woodcuts vary between editions. Practice varied, and the different reduction rules put the same letter on different cells. Nothing here claims that a square or a sigil has any effect.

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