# Eureka! A Discovery of Proportions

I have been making a study of analogies and analogical reasoning, and recently saw a connection that I had not seen before. That connection is between what is called ordered-pair analogies, i.e. A is to B as C is to D (or more briefly A : B :: C : D) and mathematical fractions. I was fascinated by what I found. Let me explain.

#### Re-arranging analogy pairs

I first noticed that, in an ordered-pair analogy, corresponding parts had to be the same part of speech (noun, verb, adjective, etc). Either A & B and C & D had to be the same part of speech, or A & C and B & D had to be the same. For example, this is a good analogy:

drink : eat :: liquid : solid.

Here we have “verb is to verb as noun is to noun.” But an equally valid analogy is

drink : liquid :: eat : solid.

This is “verb is to noun as verb is to noun.” If the first analogy is A : B :: C : D, this second one is A : C :: B : D. Similarly, we can invert both pairs to get valid analogies, as in these examples:

eat : drink :: solid : liquid

liquid : drink :: solid : eat

These would be B : A :: D : C, and C : A :: D : B. We could also switch each pair around the double colon. All these work as good analogies.

#### The connection

Now, those of you reading closely who remember your basic fractions probably see the connection already. If this is a true equality,

A/B = C/D

then so are all these:

A/C = B/D

B/A = D/C

C/A = D/B.

These equalities follow the same patterns as the analogies above. You might see it clearer with specific numbers. If the first equality is true (and it is), then all the rest must be true:

16/24 = 6/9
16/6 = 24/9
24/16 = 9/6
6/16 = 9/24.

#### The question

Do you see it? Every re-arrangement that is valid for verbal analogies is equally valid for mathematical fractions, and vice versa. But why should this be so? What is the connection between these two very different kinds of proportions?

There may be some connection between reducing the numerical fraction and finding the fundamental relationship in the verbal analogy. Just as 16/24 = 6/9 because they both equal 2/3, so ‘eat : solid :: drink : liquid’ because they share the relationship of ‘mode of consuming : state of matter of what is consumed.’

I am confident that there is something deeper going on here. Can you find any other connections between verbal analogies and numerical fractions?