Quadratus was one of the first of the Christian apologists. He is said to have presented his apology to Hadrian while the emperor was in Athens attending the celebration of the Eleusinian mysteries. The period of the emperor Hadrian, during which Quadratus is said to have made his apology, was from 117 CE to 138 CE.
Here is the reference from Eusebius, Hist. Eccl. IV.3.
1 After Trajan had reigned for nineteen and a half years Aelius Adrian became his successor in the empire. To him Quadratus addressed a discourse containing an apology for our religion, because certain wicked men had attempted to trouble the Christians. The work is still in the hands of a great many of the brethren, as also in our own, and furnishes clear proofs of the man's understanding and of his apostolic orthodox. 2 He himself reveals the early date at which he lived in the following words: "But the works of our Saviour were always present, for they were genuine:-those that were healed, and those that were raised from the dead, who were seen not only when they were healed and when they were raised, but were also always present; and not merely while the Saviour was on earth, but also after his death, they were alive for quite a while, so that some of them lived even to our day." Such then was Quadratus.
3 Aristides also, a believer earnestly devoted to our religion, left, like Quadratus, an apology for the faith, addressed to Adrian. His work, too, has been preserved even to the present day by a great many persons.
Here is the reference from Jerome, Illustrious Men 19.
Quadratus, disciple of the apostles, after Publius bishop of Athens had been crowned with martyrdom on account of his faith in Christ, was substituted in his place, and by his faith and industry gathered the church scattered by reason of its great fear. And when Hadrian passed the winter at Athens to witness the Eleusinian mysteries and was initiated into almost all the sacred mysteries of Greece, those who hated the Christians took opportunity without instructions from the Emperor to harass the believers. At this time he presented to Hadrian a work composed in behalf of our religion, indispensable, full of sound argument and faith and worthy of the apostolic teaching. In which, illustrating the antiquity of his period, he says that he has seen many who, oppressed by various ills, were healed by the Lord in Judea as well as some who had been raised from the dead.
Robert M. Grant writes (The Anchor Bible Dictionary, v. 5):
The context of his argument [concerning the healed who remained alive in the time of Quadratus], regrettably not reported by Eusebius, could have lain in philosophical debates over men treated as gods beause of fictitious miracles, or in debates over Christ's miracles, or in both at once. About half a century later, Irenaeus may have relied on Quadratus for his own discussions of miracles (Haer. 2.31.2 and 2.32.4), later copied by Eusebius (Hist. Eccl. 5.7). Irenaeus claimed that in Christian churches there were those who "cure the sick by laying hands on them, and...the dead have been raised and remained with us for many years." It is not absolutely certain what time frames either Quadratus or Irenaeus had in view, for the latter spoke of the reign of Domitian, nearly a century earlier, as "not long ago but practically in our own generation" (Haer. 5.30.1; a passage known to Eusebius, Hist. Eccl. 5.8.6).
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