Shabbos, Chapter Eleven, Mishnah 1. This chapter deals with carrying by means of throwing, that is, propelling an object from one domain into another. The Mishnah opens: "Hazorek mireshus hayachid lireshus harabim" - one who throws an object from a private domain into a public domain, and likewise the reverse, from a public domain into a private domain, is liable.
Uprooting and setting down in a throw:
The novelty of the Mishnah is that liability is not limited to someone who carries the object out in his hand. As is known, in order to transgress the prohibition of carrying on Shabbos - whether by transferring from one domain to another or by carrying four cubits within a public domain - both an uprooting and a setting down are required. When one throws, the setting down seemingly happens on its own, but in fact it happens through the motion that the person himself set in place. And even though the thrower does not place the object down with his hands, but rather the setting down comes about through his throw, that is enough to be considered an uprooting and a setting down, and he is liable.
Throwing from one private domain to another private domain through a public domain:
If a person throws an object from one private domain into another private domain, with a public domain separating between them: the uprooting took place in a private domain and the setting down took place in a private domain, but on its way the object passed over a public domain. Is he liable or exempt? On this the Tanna'im disagree:
Rabbi Akiva - holds him liable.
The Sages - exempt him, since the object never came to rest in the public domain.
What is Rabbi Akiva's reasoning? Rabbi Akiva holds a unique position, maintaining that 'an object in the air is regarded as though it had been set down': when an object is within the airspace of a given place, we view it as though it had actually come to rest in that very place. Accordingly, the object thrown from the private domain, at the moment it passes through the public domain, is considered as having been set down in the public domain, and then it was uprooted again on its own as it continued along its path and was set down in the private domain. The result is that we have here a transfer from a private domain into a public domain, followed by a transfer from a public domain into a private domain, and for this reason Rabbi Akiva holds him liable.
The Sages exempt him, since they do not see the airspace as constituting any sort of stopping and setting down. In their view the object had an uprooting in a private domain and a setting down in a private domain, and its passage through the public domain is entirely irrelevant.
Above ten handbreadths:
All of the above refers to an object thrown through a public domain within the first ten handbreadths of its height, for a public domain extends only up to a height of ten handbreadths, and above ten handbreadths is an exempt area. Therefore, if the object passed over the public domain at a height above ten handbreadths, even Rabbi Akiva agrees that he is not liable, since it is not considered as having come to rest in the public domain.
In summary: In this Mishnah we learned that one who throws from a private domain into a public domain, or from a public domain into a private domain, is liable, since the throw itself combines an uprooting with a setting down. Where one throws from a private domain to a private domain with a public domain in between, Rabbi Akiva holds him liable based on the principle that 'an object in the air is regarded as though it had been set down', while the Sages exempt him because it never rested in the public domain; and where the object passes above ten handbreadths, which is an exempt area, even Rabbi Akiva agrees that he is exempt.