I still ask, why not both??
Apparently, a- as a prefix comes from "at".
Webster's unabridged also tells us... (at http://dict.die.net/a/
A In process of; in the act of; into; to; -- used with
verbal substantives in -ing which begin with a consonant.
This is a shortened form of the preposition an (which was
used before the vowel sound); as in a hunting, a building,
a begging. "Jacob, when he was a dying" --Heb. xi. 21.
"We'll a birding together." " It was a doing." --Shak.
"He burst out a laughing." --Macaulay.
Note: The hyphen may be used to connect a with the verbal
substantive (as, a-hunting, a-building) or the words
may be written separately. This form of expression is
now for the most part obsolete, the a being omitted and
the verbal substantive treated as a participle.
My etymological wonderings were only wonderings, and probably nonsense, but why is "geritten" only a past participle if you can say "er kommt geritten"?