Magic Squares and Planetary Powers

If there is one field that is thought to be completely impervious to the inexplicable, it is mathematics. And yet, since the Renaissance, Agrippa’s magic squares have intrigued, fascinated, and inspired those interested in magic, esotericism, and the forces that shape the universe. Behind these mathematical arrangements lies a body of knowledge in which every number, every sum, and every shape follows a... sacred logic. For Agrippa, a great 16th-century Hermetic thinker, these squares represented far more than an intellectual game: they were a language in their own right, a way of contacting planetary forces and making them act in the world. An exploration.

The Origins of Magic Squares

The first known magic squares date back to ancient Asia. A Chinese legend from around 650 BCE tells of the mysterious appearance of a small three-by-three square on the back of a sacred turtle that emerged from the Luo River. This square, called Lo Shu (“writing of the Luo River”), contained the numbers 1 to 9 arranged so that their sum was always 15, symbolizing the balance between the forces of yin and yang. A true treasure of Chinese thought, the Lo Shu was seen as a symbol harmonizing Heaven and Earth. Other Asian civilizations also explored these figures: in India, for example, references to magic squares can be found in ancient treatises. One of them, the Garga Samhita, recommended using a 3×3 square to appease the influences of the nine celestial bodies (navagraha) in Hindu astrology. Thus, in Asia, the magic square was both an object of mathematical curiosity and a sacred sign, endowed with spiritual or astrological significance.

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Subsequently, knowledge of magic squares spread westward. Scholars of the Islamic Golden Age played a decisive role in this transmission. As early as the 8th century, scholars in the Arab world studied and compiled magic squares they had discovered in Indian and Chinese traditions. The Arab mathematician Thabit ibn Qurra discussed these squares in the 9th century, and the encyclopedia of the Brethren of Purity (around 990) describes examples of orders 3 through 9. In the Arab-Muslim world, these squares were called wafq (the harmonious arrangement of numbers) and were sometimes used as talismans, with letters inscribed in place of numbers. The great Sufi magician Ahmed al-Būnī taught the construction of magic squares and their association with the celestial spheres in his grimoire Shams al-Ma’arif in the 13th century. Through these exchanges, the concept of magic squares reached medieval Europe: as early as the 14th century, Latin manuscripts presented squares of orders 3 through 9, already associated with the seven traditional planets (Saturn, Jupiter, Mars, Sun, Venus, Mercury, Moon) in an astrological and talismanic context.

Mathematical Sciences and Occult Philosophy

During the Renaissance, magic squares captivated both mathematicians and followers of esotericism. It was an age of rediscovering ancient knowledge: European humanists gained access to Arabic and Eastern compilations and marveled at these strange mathematical objects. In 1514, the German artist Albrecht Dürer immortalized a 4×4 magic square (constant 34) in his famous engraving Melencolia I. Discreetly incorporated into the wall of the scene, this square is not there by chance: it evokes talismanic squares while demonstrating the intellectual mastery and melancholy of the genius (melancholy being associated with Saturn at the time). Dürer thus gave the magic square unexpected visibility in Western art, halfway between mathematical play and symbolic allusion.

Engraving Melancolia I, Albrecht Dürer. Source: Research Gate

At the same time, scholars such as Girolamo Cardano (in 1539) and Claude de Boissière (known as Bachet de Méziriac) took an interest in methods for constructing these “magic” squares. Cardano published the so-called “diamond” method for constructing odd-order squares, helping spread the concept among learned circles. However, beyond mathematical enjoyment, it was in Renaissance occultism that magic squares acquired a new dimension. They became central tools of the “occult philosophy,” which sought to unite science, mysticism, and astrology.

This is the context in which Heinrich Cornelius Agrippa von Nettesheim appears. In 1531, this German philosopher, physician, and mage published his treatise De Occulta Philosophia (Occult Philosophy), a monumental three-volume work that sought to synthesize the magical knowledge of his time. Agrippa devotes Chapter 22 of Book II to the seven planetary magic squares, inherited from the medieval tradition but presented in a structured and revised form for an educated audience. He calls them the “sacred tables of the planets,” asserting that they possess great celestial virtues and reflect the divine harmony of celestial numbers. Agrippa adopts the traditional correspondence between each planet and a square of a specific order (from 3 to 9 cells per side), associating each with an entire symbolic apparatus: divine names, angels (Intelligences) and demons (Spirits) governing each planet, seals, and mystical characters. By systematizing the esoteric use of magic squares in this way, Agrippa offers the Renaissance a bridge between mathematics and magic, in which numbers become keys for invoking astral influences.

Agrippa’s seven planetary squares

According to the traditional cosmology adopted by Agrippa, each planet of classical astrology (Saturn, Jupiter, Mars, Sun, Venus, Mercury, Moon¹) possesses an associated magic square, whose size and numbers reflect the attributes of the celestial body. These seven squares, ranging from order 3 to order 9, form a coherent set: the smallest square (3×3) corresponds to Saturn, the slowest and most distant planet, while the largest known square (9×9) is attributed to the Moon, the mobile celestial body of the first heaven. Each of these squares has a unique magic constant, as well as a series of sacred names whose numerical values correspond to the numbers in the square. Agrippa also explains how to draw a mystical sign or character for each planet, derived from the numerical pattern by connecting certain numbers in the square. This sign, along with the seals of the Intelligence (guardian angel) and the Genius (planetary spirit), serves as a graphic symbol summarizing the power of the square.

Note that several arrangements are possible for the same magic square. The tables presented here correspond to the form transmitted in the early editions of the work by Heinrich Cornelius Agrippa, but other versions may appear in some more modern books: they simply result from rotating or reflecting the square, which does not change its mathematical properties.

The square of Saturn (3×3)

The square of Saturn is the most famous and probably the oldest. It is a 3-by-3 square containing the numbers 1 through 9. Its magic constant is 15, since 1+5+9 = 15 in every column, as well as in every row and diagonal. The sum of all the numbers in the square is 45 (which also corresponds to 15×3). Agrippa presents Saturn as the “first table” and notes that regardless of which row or column is chosen, adding the three numbers invariably gives fifteen. This small perfect square (represented by the Chinese Lo Shu pattern) is associated with Saturn’s traditional qualities: time, austere wisdom, restriction, but also protection against evil. In Renaissance magic, Saturn is an ambivalent planet, considered malefic when poorly aspected, but potentially beneficial when mastered.

The seal of Saturn (an abstract geometric symbol representing the spirit of Saturn) and the character of Saturn, which can be engraved on a talisman, are derived from the Saturn square. Agrippa states that divine and angelic names can be deduced from this square, including an angel (the Intelligence Agiel) for good and a demon (the Spirit Zazel) for evil, each associated with the total sum 45.

The Saturn square was traditionally engraved on a lead plate (Saturnian metal) during a favorable hour of Saturn. When used correctly, it served as a talisman of protection and stability; it could ward off harmful influences, repel enemies or evil spirits, and confer the contemplative wisdom associated with Saturn. However, if activated under an unfavorable astrological configuration, it was said to produce the opposite effect (discord, obstacles, and curses).

The Square of Jupiter (4×4)

The Square of Jupiter is a fourth-order square, therefore containing 4 rows and 4 columns of numbers. It contains the numbers from 1 to 16 and has a magic constant of 34. In other words, the sum of each vertical or horizontal row is 34, and the sum of all the numbers in the square is 136. Agrippa describes it as “a square containing sixteen numbers [...] whose sum in each row is thirty-four, and whose total sum is one hundred and thirty-six.” In astrological symbolism, Jupiter is the planet of expansion, benevolence, and prosperity. Its magic square reflects these attributes through an order higher than Saturn’s square, signifying a more expansive power.

The Square of Jupiter also generates a character specific to the planet, along with seals associated with its Intelligence and Genius. The Hebrew names of Jupiter’s angel Yophiel and spirit Hismael each have a numerical value of 136, equal to the square’s total sum. Jupiter’s seal can be drawn by connecting the square’s cells according to a given sequence; this seal is then engraved on Jovian talismans.

Traditionally, Jupiter’s talisman is made by engraving its square on tin (the metal attributed to Jupiter), preferably on a Thursday during Jupiter’s hour, when the planet is astrologically “fortunate.” This talisman is reputed to attract luck, wealth, and divine favor. Renaissance magicians viewed Jupiter as a beneficent star: its square was supposed to bring success, honors, health, and inner peace to its wearer by channeling the “great fortune” governed by Jupiter.

The Square of Mars (5×5)

The Square of Mars is a fifth-order square comprising 25 cells filled with the numbers from 1 to 25. Its magic constant is 65: every row, column, and main diagonal adds up to 65, and the sum of all the numbers is 325. Agrippa states of it: “It is a square of twenty-five figures […] their total by column or row is sixty-five, their total sum three hundred and twenty-five.” As Mars is the planet of war, vigor, and iron, its magic square is associated with strength, courage, and protection in conflicts.

A character of the Hammer or Shield (symbolizing Martian energy), as well as the seal of the Intelligence Graphiel and that of the Genius Barzabel, spiritual entities associated with Mars, are extracted from the square of Mars. These symbols are engraved alongside the square on martial amulets.

Historical talisman engraved with the magic square of Mars (5×5) on the reverse. Source: Museum of Witchcraft and Magic

The talisman of Mars is engraved on iron or steel (Martian metal) on a Tuesday (the day of Mars) at an auspicious time. Its purpose is to symbolically arm its wearer: it was believed to grant courage in combat, victory over enemies, and protection against attacks. A striking example has come down to us: a 17th-century talisman found in England features the carefully inscribed square of Mars on one side, and the seal of Mars surrounded by various spiritual names on the other. This small metal object, likely worn by a soldier or traveler, concretely illustrates the faith placed in the square of Mars as a protective amulet.

The square of the Sun (6×6)

The square of the Sun is a sixth-order square containing the numbers 1 to 36. It is notable for its magic constant of 111: indeed, six numbers in each row add up to 111, and the sum of all the elements in the square is 666. Agrippa describes this solar square by noting that “these numbers, when added vertically or horizontally, give one hundred and eleven, and their total sum is six hundred and sixty-six.” The number 6 (the size of the square) is linked to the Sun because it represents balance and harmony at the center of the seven celestial bodies (the Sun being in the middle of the traditional planets). In addition, the number 666 evokes, in Christian symbolism, the perfection of Creation in 6 days, before later becoming associated with other connotations.

The square of the Sun provides the seal of the Sun (sometimes called Sorath’s seal, after the name of the solar spirit) and the solar character, represented by a circle containing a radiant cross. The names attributed to the Sun include the Intelligence Nachiel (value 111) and the Genius Sorath (value 666), among others.

Engraved on pure gold (the Sun’s metal) on a Sunday at sunrise, the solar square is a talisman of glory, vitality, and success. It strengthens physical and spiritual vitality, brings success in endeavors, and protects against depression or inner darkness.

The square of Venus (7×7)

The square of Venus is an order-7 square, comprising 49 cells containing the numbers 1 to 49. Its magic constant is 175: each series of 7 aligned numbers gives 175, and the sum of all the numbers in the square is 1225 (that is, 175×7). Agrippa presents it as “the fifth table, that of Venus: 49 numbers […] their addition by column or row gives one hundred and seventy-five; the total sum is one thousand two hundred and twenty-five.” Venus, the planet of love, beauty, and harmony, is assigned a more complex magic square, yet one whose balance remains perfect.

From it are derived the seal of Venus (which takes the form of a graceful interlacing) and the character of Venus, as well as sacred names such as the angel Hagiel and the spirit Kedemel, whose numerical values correspond to 49 and 175 respectively. 

The square of Venus, engraved on copper (a Venusian metal) on a Friday during the hour of Venus, serves as a talisman for love, harmony, and creativity. Practitioners of magic attribute to it the power to attract the affection and goodwill of others, stimulate beauty and artistic inspiration, and soothe quarrels. Carrying this square (or its seal) improves personal charm and fosters harmony in the home.

The square of Mercury (8×8)

The square of Mercury is an order-8 square comprising 64 cells occupied by the numbers 1 to 64. Its magic constant is 260, as each row of 8 numbers totals 260, while the sum of all the numbers is 2080 (that is, 260×8). Agrippa notes of it: “the sixth table is that of Mercury, dedicated to the number eight: it is a square […] the numbers appearing in it, when added by column or row, give two hundred and sixty; their total is two thousand and eighty.” Mercury, the planet of intelligence, communication, and movement, is associated with the largest square after the Moon, reflecting the mental agility and complexity of this messenger of the gods.

From Mercury’s square derive a specific symbol (the Seal of Mercury, sometimes depicted as a shape reminiscent of the caduceus or rapid writing) and Mercury’s character. Agrippa states that the names of its Intelligence, Tiriel (value 260), and its Spirit, Taphthartharath (value 2080), correspond exactly to the sums characterizing the square.

A Mercurial talisman was engraved on liquid quicksilver (Mercury) or on a consecrated alloy or paper, on Wednesday (Mercury’s day). It was attributed with the power to increase memory, eloquence, cunning, and speed. A student, merchant, or traveler might, for example, carry Mercury’s square to encourage success in examinations, commercial success, or protection while traveling, as Mercury was the patron of travelers. This square was also used to facilitate divination and prophetic dreams, because of Mercury’s psychopomp aspect, enabling him to travel between worlds.

The Square of the Moon (9×9)

The square of the Moon is the last in the series: an order& 9 square made up of 81 cells containing the numbers 1 to 81. Its magic constant is 369, meaning that each row or column totals 369, and the sum of all the numbers in the square reaches 3321 (369×9). Agrippa describes it as follows: « the seventh table is that of the Moon: it is a square based on the ennead (9), divided into eighty-one cells […] the numbers, added by column or row, give three hundred and sixty-nine; their sum is three thousand three hundred and twenty-one ». The Moon, heavenly body of change, imagination, and emotions, is associated with the largest square, reflecting its encompassing nature and its embrace of every current (as one might symbolically say, like the 81 possible aspects of personality).

From the lunar square, we derive the Seal of the Moon (a symbol shaped like a complex geometric figure or a gentle sigil) and the lunar character. Sacred names aligned with the Moon include, for example, the Intelligence Malka betharsith and the Spirit Chasmodai (or Hasmoday), whose numerical values correspond to the key numbers of the lunar square (9, 81, and 369 appear in certain combinations of the Hebrew letters).

The Moon talisman is engraved on pure silver (the lunar metal) on a Monday, ideally during a full-moon night. It is used for protection during nighttime travel, spiritual growth, dreams, and psychic receptivity. It is said to help develop intuition, promote fertility (the Moon being linked to female cycles and the germination of plants), and protect against dangerous lunar influences such as madness or enchantments. By wearing or placing the Moon square, the practitioner seeks to capture the changing benefits of the nocturnal star while protecting themselves from its instability.

Talismans and Practices with Magic Squares

Planetary magic squares, as systematized by Agrippa, were widely used in talismanic magic during the Renaissance and in subsequent centuries. Magicians conceived of these squares as condensers of astral influences: by inscribing them on suitable physical materials and consecrating them ritually, they hoped to capture the “virtue” of the corresponding planet and direct it toward a specific goal. In practice, making a planetary talisman followed an entire protocol: it was necessary to choose the right astrological moment (for example, when the planet was dominant in the sky and not afflicted by negative aspects), use the associated metal or material (lead for Saturn, tin for Jupiter, iron for Mars, gold for the Sun, copper for Venus, mercury² for Mercury, silver for the Moon), and sometimes also write sacred names or draw seals derived from the square. Once the square had been engraved or written, the talisman was consecrated with prayers, incense, and appropriate formulas in order to “fix” the planetary influence in the object.

Ritual uses could vary: some talismans were worn on the body as a personal amulet, while others were placed in the home or on the altar during astrological ceremonies. The Jupiter square could be worn around the neck to attract prosperity, or the Saturn square could be buried at the entrance to one’s property to seal its protection. Many grimoires and practical magic manuscripts from the 16th–17th centuries reproduce instructions for making such objects. The Petit Albert and the Heptameron, attributed to Pietro d’Abano (and inspired by earlier traditions), notably mention planetary talismans as means of obtaining favors from celestial intelligences.

Historically, a few physical examples of these practices have been found. In addition to the talisman of Mars mentioned above, magic squares engraved on jewelry or pendants have been discovered. In late medieval Europe, it was also customary to inscribe a magic square on parchment along with religious symbols and hang it in the home as household protection—a practice also found elsewhere, in pre-Islamic Arabia, where magic squares were hung in homes to ward off evil spirits. Far from being confined to books, Agrippa’s squares therefore had an active life in popular esoteric culture. They circulated among alchemists, astrologers, and even certain learned clergymen (who saw in them a form of number science given by God). Of course, their use was generally clandestine or discreet, since the boundary between a pious amulet and blameworthy superstition was a fine one. Nevertheless, these magic squares persisted in practice: they can be found even in certain Golden Dawn rituals in the 19th century and among 20th-century occultists, a sign that their reputation for power has endured through the ages.

Beyond Agrippa

Agrippa left a lasting mark on the Western tradition of planetary magic squares, to the point that these squares are sometimes called “kameas” in esotericism (from the Hebrew term for “seal”). However, the idea of associating numerical squares with higher influences exists in many cultures, independently of the system of the seven classical planets.

In China, the small Lo Shu square mentioned above gave rise to a rich numerological symbolism. It inspired the He Tu diagram and was connected to the eight trigrams of the Yi Jing (Classic of Changes). In Taoism and feng shui, this magic square is regarded as a model of universal harmony: it can be found in amulets, architectural layouts (the plans of certain imperial tombs or temples are designed according to a 3×3 magic square), and even in the ideal organization of the home to channel qi (vital energy). It is less a precise ritual tool than an omnipresent cosmological symbol, representing the perfect balance of yin and yang through numbers. Even today, in China and East Asia, the 3×3 square pattern with a sum of 15 is recognized as a bearer of good fortune and celestial order.

In India and in Hindu and Buddhist traditions, magic squares are also found in the form of yantras. One well-known example is the Kubera Kolam, a 3×3 square drawn at the entrance of homes in southern India, with a particular arrangement of nine numbers whose sum is 72 in every direction—this square being dedicated to the god of wealth, Kubera, to attract prosperity. More broadly, the Navagraha Yantras (talismans of the nine planets) sometimes incorporate magic squares: each planet in Vedic astrology (including Rahu and Ketu, the lunar nodes) can be represented by a geometric design containing a square of numbers used for incantatory purposes. Thus, the practice of using number squares as astrological remedies (to pacify a malefic planet or strengthen a beneficial influence) is documented in medieval Sanskrit texts and continues in popular astrology in India.

The Islamic and Arab-Persian world, for its part, developed an esoteric tradition of magic squares that partially diverged from the European tradition. Known as wafq or afaq, these squares could contain either numbers or letters of the Arabic alphabet, used for their numerical values according to the abjad system. For example, a very widespread 4×4 square featured the Arabic word B U D U H (corresponding to the numbers 2-6-4-5 in abjad) arranged so that the constant was 18, and was considered a powerful protective talisman in Islamic magic. Sufi masters, such as al-Būnī, mentioned earlier, as well as the authors of Ottoman grimoires, used these squares for various purposes: protection against the ʿayn (evil eye), healing illnesses, success in Quran recitation exams, ... These squares were sometimes incorporated into calligraphic talismans, interwoven with Quranic verses to increase their spiritual effectiveness tenfold. This parallel tradition shows that the magic square, as a concept, transcends religious boundaries: it can fit equally well within a monotheistic framework (as a medium for divinely inspired words expressed in numbers) or a pagan or esoteric one.

Passion Facade Sagrada Família. Source: Vacatis

Finally, in modern Western culture, magic squares continue to appear occasionally as esoteric symbols or winks. The renowned architect Antoni Gaudí included a 4×4 magic square on the façade of the Sagrada Família’s Passion façade in Barcelona (constant 33, symbolizing Christ’s age). Although this particular square is not directly related to Agrippa’s planets, its incorporation into a basilica shows how the idea of the magic square has permeated the collective imagination as a sign of order and mystery. Today, magic squares can be found in literary works, mathematical puzzles, or “new age” content, and contemporary occultists still take an interest in Agrippa’s planetary kameas, whether in ceremonial magic, Kabbalistic astrology, or eclectic witchcraft. The persistence of these squares across time and space testifies to their universal significance: born from a simple game of numbers, they have been transformed into a powerful symbol of humanity’s quest for harmony among the heavens, numbers, and earthly life.

Agrippa’s magic squares thus represent a fascinating crossroads between the science of numbers and Renaissance spirituality. Educational in their arithmetic simplicity, poetic in their symmetry, and mystical in the beliefs they inspired, they awakened beginners’ curiosity while challenging scholars’ analysis. Whether approached as a mathematical pastime or as an esoteric legacy, these squares remain a testament to the ancient conviction that number is the secret language of the Universe.

Notes:

¹ In Agrippa’s time, only seven “planetary” celestial bodies were known to influence the Earth: five planets in the astronomical sense (Saturn, Jupiter, Mars, Venus, Mercury), plus the Sun and the Moon, often called “planets” in ancient astrology.

² Since mercury (quicksilver) is liquid, some talismans of Mercury were engraved on a metal alloy containing mercury or on yellow-orange parchment supposedly representing this elusive metal.

Sources:

• De Occulta Philosophia Libri Tres, Heinrich Cornelius Agrippa, 1531 (Latin edition)

• Three Books of Occult Philosophy, translated by J.F., edited by Donald Tyson, Llewellyn Publications

• A History of Magic, Witchcraft and the Occult, DK Publishing, 2020

• The Magic Squares of the Planets in Occult Philosophy, Stephen Skinner, Golden Hoard Press

• The Occult Sciences in the Renaissance, Frances A. Yates, Éditions Gallimard

• Kabbalah and the Occult Sciences, Gershom Scholem, Payot

• Magic Squares: Their History and Use in Western Occultism, Journal of Esoteric Studies, Vol. 22, 2017