While Josephus does not mention the Herodian census, Luke carefully distinguishes the census at the time of Jesus' birth as "first," (i.e. first in a series of enrollments connected either with Quirinius or with the imperial policy inaugurated by the decree of Augustus).
The geographical work of [Herod] Agrippa, together with the interest of the emperor in the organization and finances of the empire and the attention which he gave to the provinces are indirectly corroborative of Luke's statement. Augustus himself conducted a census in Italy in and in Gaul in 727/27* [see
roman dating system, ‘auc'] and had a census taken in other provinces. For Egypt there is evidence of a regular periodic census every 14 years extending back to 773/20 and it is not improbable that this procedure was introduced by Augustus.
The time of the decree is stated only in general terms by Luke, and it may have been as early as 727/27 or later in 746-8, its execution in different provinces and subject kingdoms being carried out at different times. Luke dates the census in the kingdom of Herod specifically by connecting it with the administrative functions of Quirinius in Syria. But as P. Quintilius Varus was the legate of Syria just before and after the death of Herod from 748/6-750/4 and his predecessor was C. Sentius Saturninus from 745/9-748/6 there seems to be no place for Quirinius during the closing years of Herod's reign."