The prepositions seem to fall into two broad categories.

One, those that are definitively paired with either the accusative case (e.g. ich warte auf dich") or the dative case (e.g. ich suche nach dem Täter).

Two, those where you just have to learn what goes with what along the way (e.g. es besteht AUS zwei Protonen).

The preposition "an" frustratingly seems to bypass those possibilities. I'll quite often read an article where I was surprised at it's usage in that particular context, and then in addition, one can use "an" with both accusative and dative.

Here are a few examples from newspapers to show what I mean:

1) Zwei Streifen an seinem Ärmel weisen ihn als Corporal aus.

Question 1: could one also use "auf" here given the stripes lie "on" the sleeves?

2) Er hatte im Labor an einer Methode herumprobiert, die Hirnhäute einer Maus so freizulegen, dass sie im Ganzen sichtbar werden.

Question 2: could one also use "mit einer Methode" here?

3) All das lernen sie in einem dreimonatigen Einführungskurs an einer kleinen Hochschule.

Question 3: could one also use "in einer kleinen Hochschule" here or would that subtly change the meaning?

4) An dieses Gerät lasse ich keinen ran!

This is just to provide an example of the usage of "an" with accusative.

So my question pertains to the usage of "an" in the above examples, whether or not one COULD potentially substitute another preposition instead of using "an".

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    The accusative/dative usage is exactly the same as with the other spatial prepositions (note that auf is among them, it's not an accusative-only one). You see that in sentence 1 (position, fixed) vs. sentence 4 (movement, towards the machine). Otherwise, prepositions simply are hard. – chirlu Jun 11 '15 at 7:05
  • Other prepisitions which can go with both cases are among others: an, auf, neben, in, über, unter, zwischen, vor, hinter. – Bartłomiej Zalewski Jun 11 '15 at 7:10
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    Technically, yes you can use the others prepositions. This, however, may have a slightly different meaning. – Your question is a little broad, though. You've given three examples that we can elaborate on, but it's likely that it doesn't really help for future issues. If disappointing, but prepositions is one of the worst things about languages and it takes a long time to really use them appropriately. – Em1 Jun 11 '15 at 7:59
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    This question is similar to the question when to use the English preposition on. You can write a small book about this question. – rogermue Jun 11 '15 at 8:23
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    In the title you ask about when to use it. Then you start talking about which cases to use it with and ask about that and then you go back to examples for when to use it. I'm confused. What does the first paragraph of the question have to do with the problem ? – Emanuel Jun 11 '15 at 9:28

Prepositions in any and all languages are fixed expressions, derived from the dark depths of when the language evolved away from its close relatives.

There is no way to adequately guess which preposition will certainly be correct in a certain position. The only thing one can do is adequately guess, which prepositions might definitely be wrong in a given context:

[…] zwischen einer kleinen Hochschule.

To elaborate a very tiny bit on your questions:

Zwei Streifen an seinem Ärmel

Yes, you could use auf here, although an is preferred.

Er hatte im Labor an einer Methode herumprobiert

No, mit einer Methode would change the meaning. With an, he is modifying the method itself, with mit he is doing different things with the same method.

an einer kleinen Hochschule

Technically yes, you could use in here. It shifts the meaning from the course being offered by the college to the course being held inside the college’s building. With an, the course could also be held in the nearby church’s rooms and the sentence still be fine.

All of these answers are very specific to your cases, though.

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