When did throwing the ball around the horn after an out start and what exactly is its purpose?

There are several theories on this, I quote three of them:

The shape of the bases forms a curved horn (as in animal horns, not musical), and the action of the double play that goes 5-4-3 is considered to have gone 'around the horn'. A victory toss of the ball after a strike out that goes from 2-5-6-4-3-1 is also considered to be 'going around the horn'. http://www.funtrivia.com/askft/Question58466.html

According to Paul Dickson's "The New Baseball Dictionary", the term "around the horn", which refers to a double play started by the third baseman, or throwing the ball around the infield after a strikeout, has its origins in sailing. It refers to the long voyage from the Atlantic Ocean to the Pacific Ocean. Before the creation of the Panama Canal, ships had to travel south past the tip of South America, Cape Horn, to get from one ocean to the other. So the long journey from one ocean to the other, like the journey from third base to first, had to go "around the horn."

"Around the horn a 5-4-3 double play, in which the ball goes from the third baseman, to the second baseman, who tags second, to the first baseman, who tags first. Like many baseball terms, this originates from sailing. Until the Panama Canal was built, the quickest way from the North Atlantic Ocean to the Pacific Ocean was to sail around Cape Horn, the southernmost tip of South America. By extension, "going around the horn" refers to covering all or several angles of something in baseball. More commonly, this phrase refers to the tradition of tossing the ball around the infield after an out is recorded such that no runners are left on base. For example, after an out recorded at first base, if there are no runners on, the first baseman will toss to the second baseman, and the ball will subsequently be thrown to the shortstop and then the third baseman, and finally back to the pitcher. The order in which the ball is thrown "around the horn," can vary from team to team and from situation to situation, but many teams see this as the "way" to do it and it may even be considered bad luck to deviate from this standard pattern. Also, some teams use this only when a strikeout is recorded. This version starts with the pitch being thrown and the last strike recorded then is quickly thrown from the catcher to the first baseman who proceeds with the regular order. Now it is used more as a method to show off, than to ensure that no runner is left on."

http://www.sportspool.com/baseball/terms.php

Perhaps Bob Laz can shed some light?

I always felt better if I touched the game ball every inning. I think most infielders feel the same way. By passing it around after a strike-out, all the infielders got to "get a feel". It also gives the battery a few seconds to unwind and get ready for the next batter. To non-players, this may not seem important, but players know that each batter has presents a different set of challenges. (It also gives the catcher a chance to get out of that leg-numbing squat.)

Yes, that was always my understanding... it gives the pitcher a moment's respite and the catcher the opportunity to stretch his legs.

Can't add to what Sail mentioned as far as the origins..I know we do it at the youth level to give kids some pride/keep their minds in the game...