Here are the essential basics of how to do compatibility based on the mutual relationships of the signs of two people.
Start at Taurus, call it “1”, say the signs in order, numbering each one, till you get to “7” (from Taurus it will be scorpio) This sign is always the most COMPELLING attraction and compatibility.
Start over and do the same thing, this time note the signs that are numbered “5” and “9” (from Taurus it is Virgo and Capricorn). These signs are always RELIABLE compatibly.
Start over again and note the signs that are “3” and “11” (From taurus these are Cancer and Pisces) These signs are nice relationships, just rather mildly positive.
The signs just before and after the one you start at (2 and 12 – from Taurus these are Aries and Gemini) are incompatible… not terribly, mildly. *Here is the one you are asking about, Taurus/Aries. More on this after I finish up this number pattern.*
The signs just before and after the one “7” signs from where you start (6 and 8 – from Taurus these are Libra and Sagittarius) are more strongly incompatible, reliably, constantly not good.
There are only two signs left now, these are the ones “4” and “10” from wherever you start (from Taurus this would be Leo and Aquarius) these are the biggest compatibility clashes, the most incompatible.
Now, thats the THEORY. You also have to know how to apply this theory.
a) you have to do it for the signs of your ascendant, moon, sun AND venus – and weight the results all together (or separately to reveal separate aspects of compatibility)
b) you have to do it for the interactions between these four signs. (for example, Moon in Taurus and the other persons ascendant in Scorpio is very good for compatibility).
c) You have to consider the degrees, not just the signs. When two bodies are close together by degree, their positive or negative compatibility effect is much more pronounced than if those two bodies are far apart.
Hope this helps! =)
PS – This pattern system I described doesn’t at all contradict an element-based way of understanding compatibility and incompatibility. In fact the pattern arises because of the compatibility of the elements and modes of the signs.
I use this as the fundamental system of my compatibility readings (along with another, more traditional Indian system) and the results are very, very reliable and useful.