Suppose you have an analogy like this:
CAT : TOMATO :: ______ : PLANT
This is what we might call an easy analogical inference; what you do is actual look at the ratio TOMATO : PLANT and see that PLANT is a more general description of TOMATO; then you do the same with CAT. Depending precisely how you understand the TOMATO : PLANT ratio, there are several things you could put in the blank. ANIMAL would be the more obvious one. You could try, of course, to work things out based on the CAT : TOMATO ratio, but that's a much harder route.
There are, however, much harder analogical inferences. For instance:
CAT : TOMATO :: ______ : POTATO
It's not so obvious what to put into the blank, is it? You have two ratios again, but neither of them is a softball like TOMATO : PLANT. Instead you have to figure out what about the ratio CAT : TOMATO or TOMATO : POTATO is relevant. Unlike TOMATO : PLANT neither of these involve a genus-species relationship; nor do they involve the other easy relation, part-whole, nor do they involve common associations, at least of any sort that would be useful. One could perhaps use similarity to find answer; since TOMATO and POTATO are both plants in some way, one could just pick an animal and fill the blank with that. DOG would be the most obvious, but there's really nothing obviously preventing you from writing in AARDVARK.
It's very noticeable that what we do when we try to solve analogical inferences is make use of topoi or commonplaces or loci communes. Topoi or commonplaces are traditionally used to find middle terms. In neither case above are we finding a middle term. But this is because the usefulness of topics as a field of logic derives from the fact that finding a middle term is merely a specific form of a more general activity, namely, finding a relevant term. And, of course, degenerate analogical inferences do begin to look a lot more like finding middle terms in a syllogism, e.g.,:
BIGGEST : BIGGER :: ______ : BIG
And the reason for this, again, is that they share the fact that they are forms of reasoning devoted to finding relevant terms.
Congratulations, by the way, if you figured that the optimal answer to the second analogy problem was ASH and that the inference was actually purely verbal -- as the scholastics might say, it requires reading the terms with material supposition. Tricky, wasn't it?