Why doesn't "its" have an apostrophe?

I know that its is the possessive and it's is the contraction, and know when to use them. But why doesn't the possessive have an apostrophe?

• "The bear's eating a fish." [contraction]
• "The bear's coat is brown." [possessive]
• "It's eating a fish." [contraction]
• "Its coat is brown." [possessive]
• "One's eating a fish." [contraction]
• "One's coat is brown." [possessive]

Wiktionary lists the etymology as "From it +‎ 's", and Online Etymology Dictionary says that this is actually the original form:

Originally written it's, and still deliberately spelled thus by some writers until early 1800s.

So what happened to the apostrophe? When did people stop using it, and why did they?

It seems that it's as the possessive is more natural, as most people do this until they're taught that it is wrong (or even after).

Update: Online Etymology Dictionary has been updated to include two potential explanations:

The apostrophe came to be omitted, perhaps because it's already was established as a contraction of it is, or by general habit of omitting apostrophes in personal pronouns (hers, yours, theirs, etc.).

Can anyone back up either of these arguments? The possessive one's still has the apostrophe, despite these.

Professor David Crystal explains it in his book The Fight for English: How language pundits ate, shot, and left (Crystal 2006), pp. 134-135:

Its is just as possessive as cat's, but it doesn't have an apostrophe. Why not? Because the printers and grammarians [of the nineteenth century - Alex B.] never thought the matter through [emphasis mine - Alex B.]. They applied their rule to nouns and forgot about pronouns, thus creating an exception (along with the food is hers, ours, yours, theirs) without realizing it. And even if they had noticed, they wouldn't have done anything about it, for it's was already taken, as it were, as the abbreviation of it is.

I highly recommend this book to anyone interested in languages. 

Charles Fries (Fries 1927) speculates that it could also be so because "their plural forms do not end in s" (cf. one - one's - ones or other - other's - others, ft. 7)

For an excellent summary of how the rules concerning apostrophe use developed, see

Sklar, E. (1976). The Possessive Apostrophe: The Development and Decline of a Crooked Mark. College English, 38(2), 175-183. doi:10.2307/376342

If you want to learn more about how the grammarians of the past arbitrarily imposed their confusing rules - and didn't stick to them - see pages 197-198 in Doctrine of correctness in English usage, 1700-1800 by S.A. Leonard (Leonard 1962); for instance, Joseph Priestly argued in The rudiments of English grammar, which was published in 1772, the following (pp. 86-87):

On the other hand, in the same book, on page 11, he lists all the possessive pronouns without an apostrophe and he treats its separately, as the genitive form:

Thirteen years later, J. Mennye in An English grammar ; being a compilation from the works of such grammarians as have acquired the approbation of the public [...] argued diametrically the opposite of the convention earlier proposed by Joseph Priestly.

But in 1823, T.O. Churchill says the following in A New Grammar of the English Language:

Here are some possessive pronouns:

• My neighbour
• Your friend
• His wife
• Her dog
• Its tail

None of these have apostrophes. See Martin Beckett's answer for the rest.

I was told that the possessive apostrophe was originally a contraction.

"The bear's coat" was originally "the bear his coat" and the apostrophe appeared as the his was gradually shortened — so it would make sense that it is had an apostrophe but its didn't.