My understanding that the December 25th date was choosen because it’s symbolic, not because it was believed to be the actual date of birth. The symbolism was that the 25th is the 3rd day after the winter solcest(sp), the lowest point of the suns orbit in the southern hemisphere in a given year. Thus, just as Christ rose from the dead three days after his execution, the birth of Jesus is set on the 3rd day of the rising sun.