Four elements and archangels, four directions.  GO.

Agrippa’s system (book II, chapter 7):

• East – Fire (Michael)
• West – Air (Raphael)
• North – Water (Gabriel)
• South – Earth (Auriel)

The logic for this can be understood by looking at a horoscope.  Let’s say that, at the start of the astrological year, the Sun rises at dawn on the spring equinox (0 Aries).  Aries, then, is ascending on the east; Libra descends on the west; Capricorn is at the midheaven; Cancer is at the nadir.  Aries is ruled by Fire, Libra Air, Capricorn Earth, and Cancer Water.  In this system, the hot elements (Fire and Air) are on the East-West axis, while the cold elements (Water and Earth) are on the North-South axis.  This makes sense to me, at least, though I haven’t seen this put forth as an official explanation of this. 

Something more official might go something like this: the order of the elements in the Zodiac is fire, earth, air, and water (Aries, Taurus, Gemini, Cancer…); the order of the elements, going around clockwise starting at East, is fire, earth, air, and water; the order of the four cherubim presented in Rev 4:7 is “lion, calf, man and eagle”, representing, respectively, fire, earth, air, and water.  Lots of stuff to chew on here, and definitely reverberates well.

Golden Dawn system:

• East – Air (Raphael)
• West – Water (Gabriel)
• North – Earth (Auriel)
• South – Fire (Michael)

These attributions seem to have originated with the Golden Dawn who were trying to expand on Dee’s work, and the four angels and angels from Dee were assigned their elements.  Later on, Gerald Gardner used Golden Dawn material in forming Wicca, which has propagated this system to many forms of neopagan cosmologies that exist today.

The Golden Dawn likely used a solar-based or geographic system to determine these elemental attributions.  At least from the northern hemisphere, when the Sun is at its highest point and hottest, it’s in the southern part of the sky (hot and dry, Fire).  At dawn and dusk, things are heating up and evaporating the dew (hot and moist, Air) or cooling down with sweat and rain (cold and moist, Water).  At nighttime, things are dark, cold, and hidden, all qualities associated with Earth (cold and dry).  Alternatively, from their primarily European frame of reference, they considered the southern lands to be hot and dry like the Sahara (Fire), and northern lands like Scotland or Scandinavia to be barren and frozen (Earth).  The Atlantic ocean was to the west (Water), while fertile lands and places of learning and civilization were out to the east (Air).  This system breaks down when applied to other parts of the world, especially the entire southern hemisphere, which might be better off switching Fire and Earth.  Indeed, this is one of the major issues with this system, and I’ve seen it applied, changed, or tweaked in various ways.

The angels ruling the elements are the same between the two systems, as are most of the other correspondences between the elements and other beings and attributions.  The Golden Dawn style of attribution is common, both in modern ceremonial and neopagan systems of magic, but I prefer the traditional Agrippan attributions and use that in my own work.  Since the attributions between elements and other things are largely the same regardless of the system, with just the directions and minor details switched around, I normally fit things into the Agrippa system whenever I adapt a ritual or tool into my own practice.  It’s a more classical or Renaissance system that I prefer, following in the steps of past and present traditional Hermetic magicians, and would seem to be independent of hemisphere, but YMMV.

For instance, my Circle of Art has four four-lettered names of God around the outer ring: Agla in the East, Eloah in the North, Eheieh in the West, and Adonai in the South.  I got these godnames from Fr. Osiris’ New Banishing Ritual of the Pentagram, which I find of much better than the standard LBRP; however, he follows the GD correspondences and uses Agla in the South, Eloah in the West, Eheieh in the East, and Adonai in the North.  Given the system I work within, I find that East=Fire/Michael/Agla works better than East=Air/Raphael/Eheieh.  Likewise, when I decide to do a personal Litany to the Holy Angels, I swap the order of the angels, calling on Michael, Auriel, Raphael, and Gabriel in turn instead of Raphael, Michael, Gabriel, and Auriel.

Correspondences differ between traditions.  That’s alright.  Different traditions operate on different views of the same cosmos, different interpretations of the same reality.  Within one system, complete and well-structured according to its own rules, things work well; swapping between systems without taking into account different correspondences may cause things to slighty out of tune or entirely out of order.