Rex, ne ipse rem tam tristem iudicaret ac secundum iudicium supplicii auctor esset, concilio populi advocato, “Duumviros” inquit, “qui Horatio perduellionem iudicent, secundum legem facio.”
The king so that he would not himself judge on such a tragic matter, then established an additional? / secondary? [secundum?] court of punishment [supplicium, -i (n)] and, having called a meeting of the people, said: I create with approving legislation the Duumvirii who will judge the treason of Horatius’
Secundum is being used as a preposition here in both cases (“according to”/“in accordance with”). So in the second (pardon the pun!) instance it’s “I appoint a committee of two in accordance with the law”.
For perduellio here, see L&S:
“for killing his sister, which was regarded as a usurpation of the prerogative of punishment, and hence as a crime against the state”
http://perseus.uchicago.edu/cgi-bin/philologic/getobject.pl?c.14:1549.lewisandshort
But this is L&S’ explanation rather than a translation, right?. I’m a bit confused about the subject-object action in the original sentence.
Duumviros" inquit, “qui Horatio perduellionem iudicent, secundum legem facio..”
He says: ‘I establish, according to the law, Duumviri who may judge Horatius [to have committed] treason…’
I’m taking it that Horatio is dative.
OK so here’s, I think, a better final translation of this paragraph.
Rex, ne ipse rem tam tristem iudicaret ac secundum iudicium [secundum iudicium = ex iudicio] supplicii auctor esset, concilio populi advocato, “Duumviros” inquit, “qui Horatio perduellionem iudicent, secundum legem facio.”
The king, lest he himself should judge in such a tragic case and, according to the law, be the author of the punishment, called together an assembly of the people; "I establish, according to the law, Duumviri, who may determine Horatius’ treason.