According to Restatement of Torts, Second (I've only scratched the surface), to be liable for battery, the actor must have an intent, though the intent might not have been to bring about the specific damaging outcome. Both the harmful and offensive contact varieties of battery require a bad intent: §13 "An actor is subject to liability to another for battery if (a) he acts intending to cause a harmful or offensive contact...", and per §16, "If an act is done with the intention of inflicting upon another an offensive but not a harmful bodily contact...the actor is liable to the other for a battery although the act was not done with the intention of bringing about the resulting bodily harm". §18 Battery: Offensive Contact says more or less the same thing with "offensive" inserted. Furthermore,
§18(2) An act which is not done with the intention stated in Subsection (1,a) does not make the actor liable to the other for a mere offensive contact with the other's person although the act involves an unresonable risk of inflicting it and, therefore, would be negligent or reckless if the risk threatened bodily harm.
So, if A throws a dart at a dartboard while B is right next to it, and hits B in the eye, it would seem that this is not battery, and poor B might be out of luck.
I think this would be generic negligence, where §281 "The actor is liable for an invasion of an interest of another, if: (a) the interest invaded is protected against unintentional invasion...". The problem is, the description of the interest in freedom from harmful bodily contact only protects against the interest w.r.t. intentional contact, so where does the protection of that interest come from in terms of unintentional contact. Specifically: is there a general rule that every protection against an intentional invasion of interest entails a protection against unintentional invasion of that interest? (I assume the "duty" is a generic and universal one).