Graphiel

The planetary magic of Mars gathers the names, seals and number-square attached to that planet in the Western magical tradition. Its magic square, or kamea, is of order 5, a grid of 5 by 5 cells holding the numbers 1 to 25 so that every row, column and diagonal sums to 65.

Square, intelligence and spirit

From that square are derived, by gematria, the names of the planet's two governing powers: the Intelligence Graphiel, the ordering and benevolent aspect, and the Spirit Barzabel, the raw and unmediated force. This apparatus was set out by Heinrich Cornelius Agrippa in the Three Books of Occult Philosophy (1533), drawing on earlier Hebrew and Arabic number-magic. To the same planet is also joined an Olympic Spirit, here Phaleg, one of the seven planetary governors named in the Arbatel of Magic (1575).

These names, squares and hierarchies are described here as an element of the documented tradition of Renaissance ceremonial magic, studied historically; the spirits are not reified.

The Mars table is a five by five square of the numbers one to twenty five, adding to sixty five in every row, column and diagonal, and to three hundred and twenty five in total. Odd ordered squares like this one and Saturn's are the easy cases: a straightforward construction rule, known in several traditions independently, fills them without trial and error, which is part of why the odd planets' tables are the most stable across manuscripts.

The names attached to it follow the same derivation as the rest of the set. In Agrippa's scheme the intelligence of Mars is Graphiel and the spirit Barzabel, each built so that the numerical values of its Hebrew letters return a number from the square. A reader who wants to check the system rather than take it on trust can do so here with a pencil.

The wider point about the Mars table is what planetary magic thought it was doing with any of this. The squares were not decoration and not, in the intention of the writers, symbols. They were held to be the mathematical form of a planetary influence, so that engraving one on the correct metal at the correct hour produced a physical object in which that form was present. The atlas records the practice as documented rather than effective, and records the arithmetic because the arithmetic is real and checkable even where the claim built on it is not.

Agrippa published the tables in 1533. The manuscript tradition behind them is older and Arabic in large part.