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Now it happened
This phrase is used here to mark a break in the main storyline. Here Luke starts to tell a new part of the story. AT: "Now when the baby was eight days old" or "Then, on the eighth day after the baby was born" (See: rc://en/ta/man/translate/writing-newevent)
on the eighth day
Here "eighth day" refers to the time after the birth of the baby. AT: "when the baby was eight days old" or "on the eighth day after the baby was born" (See: rc://en/ta/man/translate/translate-ordinal)
they came to circumcise the child
This was often a ceremony where one person circumcised the baby and friends were there to celebrate with the family. AT: "they came for the baby's circumcision ceremony" (See: rc://en/ta/man/translate/figs-explicit)
They would have called him
"They were going to name him" or "They wanted to give him the name"
after the name of his father
"his father's name"
by this name
"by that name" or "by the same name"