What is the origin of the phrase, "Put two and two together?"

I used the phrase, "She put two and two together..." the other day and, shortly after saying, wondered about its origin. My understanding is that it means to "connect the dots" or to figure the answer to a question, but I'm uncertain why "put two and two together" became a synonym.

My guess is that it originally had a longer form, like, "She put two and two together to get four," but that the "to get four" part has fallen out of usage.

A query by "put two and two together" on the COHA (1810s-2000s) shows that the phrase appeared in sentences without a longer form since 1848 at least (1848 is the year of the first result). There are occurrences of a longer form "put two and two together and make four". It's not clear whether the shorter form derived from the longer form, but I'd say that your hypothesis makes sense.

The earliest uses I can find are 1816 and 1820 and include the four.

Here's Charles Taylor in 1816, in Facts and evidences on the subject of Baptism in three additional letters:

How slightly do some very good people read their Bibles! — .The Scripture is plain enough, to proper attention. Any who can put two and two together, to make four, may, and indeed must understand it.

Here's William Cobbett in 1820, in Cobbett's Weekly Political Register: Volume 36:

I am no lawyer, and if it were possible, still less of a conjuror, but, being able to put two and two together, and to ascertain that they make four; being able to arrive at this conclusion, with mathematical certainty ...

And even more interesting, from the same volume:

My opinions have not often been contradicted by events ; and my decided opinion is, that, when you and your colleagues have carefully put two and two together ; and then put a one to the four, and by the total have multiplied twenty, the result will show you that it would much better not to proceed with the trial any more than with the Coronation.

Both from very similar times.

It's interesting to find some instances of putting two and two together in a literal sense, such as this 1820 example of how to pot and collar eels in The practice of cookery, pastry, confectionary, pickling, preserving, &c By Frazer (mrs.)

After taking off the skin, split them down from the shoulder to the tail, and bone them ; season them highly with salt, spices, and sweet herbs ; then put two and two together, with the shoulder of the one to the tail of the other;

The OED is where you want to go for most questions about first use that don't go back into Middle or Old English. It lists this under its entry for "two".

1695 J. Collier Misc. upon Moral Subj., p. 80:

The... Notion... is as clear as that Two and Two makes Four.

It's always possible the expression was calqued over from French or Latin, which is what I would assume they were using to teach arithmetic and logic at the time.