Hcfyk

Looking at a Latin dictionary, petitio usually means a requesting, asking, desiring, petition, solicitation. The word principii is the genitive declension of principium, which means beginning, commencement, origin. So petitio principium means a petition of the beginning, or an appeal to the thing you started with.

Here the thing you started with is the point you were trying to make, or the question. An appeal to it essentially means assuming it. So petitio principii means assuming the point you were trying to make, or "appealing to the question". However, begging is another translation for petitio. Using this word rather than appealing to, we get "begging the question". Since begging does not have the connotations that appealing to does in English, it is really an inferior (I would even say incorrect) translation.

How did begging the question become the accepted translation for petitio principii? Maybe it started out as a joke. I can't answer this.

ADDED: There is a Language Log blog post that discusses the phrase "begging the question", and goes into more detail.

The actual story about "beg the question" is far more complicated than virtually all authorities suppose. In its logician's use: it is always used absolutely, that is, "That just begs the question" (full stop: there's no further "question"). In that sense, it means "Assume the very thing you're trying to prove."

That is, of course, a bizarre construction, and it is only explainable on the basis of its historical origin, which descends from a phrase in Aristotle's logical works that can't really be understood without knowing the context in which it arose: a dialectical argument as portrayed in Book VIII of his Topics. As depicted there, a dialectical debate has two participants, a questioner and an answerer; the answerer begins the debate by undertaking to defend some proposition, and the questioner then tries to refute the answerer by asking questions that can be answered by yes or no of the answerer and, from these, deducing the contradictory of the answerer's initial proposition. In this context, Aristotle calls the initial proposition proposed by the answerer "the initial thing" (to en archêi).

One of the rules of this kind of debate is that the questioner cannot simply make the very thesis put forward by the answerer into a question and ask that: to do so is the error of "asking for the initial thing" (to to en archêi aiteisthai). It also counts as "asking for the initial thing" if the questioner asks something not identical to but more or less equivalent to the initial thing, though explaining how close it has to be is a problem for Aristotle.

Translated out of the context of dialectical debate, this same error can be described as "taking the conclusion you are trying to prove as one of the premises of your argument", and that is the original sense of "begging the question" when it enters English in the late 16th century ("question" having taken on the sense "the proposition being debated").

A complicating point in this history is that the Greek verb aitein, "ask", took on the sense "ask for as a premise" in logical and mathematical contexts and thus came to mean "assume" (as a premise). If we assume the same for "beg" in English, then "beg the question" can indeed be read as "assume the thing you're trying to prove", although that is hardly what an ordinary speaker of English would suppose it to mean.

(As a footnote: the translation "in the beginning to assume", given in some authorities as a translation of to en archêi, is grammatically impossible as a rendering of it in Aristotle, since it assumes the definite article to goes with the infinitive aiteisthai; Aristotle's usage, especially in those contexts in which the phrase begins with two definite articles, shows that it goes instead with "in the begining" (so, "to assume the in-the-beginning thing" would be much better).)

One possibility that I don't see considered is that the phrase is connected to 'beggar belief', which roughly means to defy belief (something can also beggar description, which means to defy attempts at description). This is more or less exactly the same meaning as 'beg the question' - namely defy the question, by presupposing an answer. I find it doubtful that this is a coincidence. All that would need to happen to get the word 'beggar' into the phrase 'beg the question' would be to shorten it - one of the most common transformations of a word over time.

If this is plausible (as it seems to me) the 'beg the question' would have the same origin as 'beggar belief' or 'beggar description'. A possibility there is that it comes from Shakespeare, in Antony & Cleopatra, 1616:

For her owne person It beggerd all discription.

The word 'beggerd' here is understood to have meant something like 'dispossessed' - to be beggered was to be dispossessed of ones belongings. In that case, the origin of 'beg the question' would be 'beggerd the question' which would mean to have been dispossessed of the question, by having presupposed the answer.

Begging the question is, as you stated, assuming an initial condition upon which the argument turns. Therefore, when people fall into this logical fallacy, they are literally begging the person with whom they are arguing to question the initial assumption.

To me this phrase makes sense if you interpret as "asking the question for the answer". In other words, obtaining your answer from the question directly, rather than any kind of reasoning process.

I'm thinking of something like "he came to beg me for money" as the template. Instead, "he went to the question begging for the answer".

In the context of begging the question, a question is not a question
one asks but

a point or topic to be investigated or discussed; a problem  (OED)

It is a proposition that one is trying to prove (a proposition in
question, the ''point at issue''). To beg the question is simply to ask
(or beg) one's (real or imaginary) interlocutor to grant one the
question, ie. to accept it as true prior to any demonstration. It is to
beg the question of him.

I don't see a problem here.

The Latin and Greek versions are a different question and more difficult
to understand, seeing as they have principium* and τὸ ἐν ἀρχῇ,
sc. beginning, instead of question.

Prof. Smith's answer is useful here, but I had some difficulty
understanding it. The only way I can make sense of it is if asking for
the beginning is in fact asking one's interlocutor directly to
contradict the statement with which he began, ie. to give up his
initial statement without fight to the man asking for it.

The Greek and Latin words of which begging is a translation mean ask
rather than beg. But, of course, in this situation of having to prove
something, asking to be absolved from that task, to receive what one had
to earn for free, is begging.

I had the sense from answers and comments that beg was taken very
literally (relatively to the common sense of the word) and therefore
seen as problematic. But begging does not have to involve actual
humility, kneeling, &c.:

(OED, beg v 3a:)

In beg pardon, beg excuse, beg leave, etc.: beside the strict sense as

in 2, the whole expression is often merely a courteous or apologetic

mode of asking what is expected, or even of taking as a matter of

course.

(The entry (cited in another answer) presenting begging in begging
the question as a special sense is not necessary for understanding the
expression. On the contrary, it lends itself to the misunderstanding
that this begging has nothing to do with begging in the other senses.
And it is a sense introduced only to account for expressions similar to
begging the question, as Peter Shor pointed out in a comment.)

• petitio principii, the begging of the question, corresponds to
  petere principium, to beg the question.