Eusebius of Caesarea, Ecclesiastical History

Eusebius (ca. 265 – ca. 340) lived at a pivotal time between past persecution and the recognition of Christianity as a lawful religion by Emperor Constantine. He became a priest at Caesarea in 300, before being bishop to 312. He is close to Constantine, and wrote the Vita Constantini (Life of Constantine). He is also considered the first historian of the Christian Church. However he is not primarily a historian but a in biblical and apologetic writer. His History is written in a religious perspective, and focus on Christian faith.
In this passage, Eusebius explains why the message of Jesus, conveyed by the apostles (Peter in this text), is written down. The main reason is the desire to preserve the message in the absence of the apostle. However, the extract also shows that writing down such a message is not obvious. The intervention of the Holy Spirit persuades Mark, the companion of Peter, to accept the task. Eusebius, like his contemporaries and Christians until recently, has no doubt that Mark was the author of the gospel. But if Mark is the author, it is inspired by the Holy Spirit. The extract quoted here shows that clerics consider that the authors of their sacred writings are human, inspired by God through the Holy Spirit. Hence the importance of kata, "according to" in the title of the gospels: there is one gospel (one message of Jesus), but "interpreted" by men.

Thus when the divine word had thus been established with them [the citizens of Rome], Simon’s power was extinguished and immediately destroyed, along with the man himself. So greatly did the lamp of piety enlighten the minds of Peter’s hearers that they were not satisfied with a single hearing of the unwritten teaching in the divine message, and with all kinds of appeals begged Mark, whose gospel is extant, as he was a follower of Peter, to provide them with a written record of the teaching they had received in oral form. Persisting in their request until they had persuaded him, they were thus responsible for the writing known as the Gospel according to Mark. 2. It is said that, when what had happened was revealed to him by the Spirit, the apostle was delighted by their enthusiasm, and authorized the work, for reading in the churches.

F. Watson, Gospel Writing: A Canonical Perspective, p. 442:

Gospel according to Matthew

The Gospel according to Matthew, the first writing of the New Testament canon was written after 70, probably by an author of Jewish origin; the final version probably dates from the 80's. It is written for a Syrian community, either in Antioch or in another city in Syria. This is the only passage of the Christian canon where the birth of Jesus is detailed. It may be interesting to compare with the sample passage from the Gospel of Pseudo-Matthew. This apocryphal gospel (whose original title was Nativity of Mary, the current title comes from its editor, Constantin von Tischendorf) is a reworking (in Latin) of several earlier texts. It was probably written in the late sixth century. It offers an amplified version of the birth of Jesus, with the addition of many details and informations and a more detailed timeline (two days, eight days, etc.). Like other apocryphal gospels, this text is a more detailed account of the life, and especially the youth of Jesus, and focus on its miraculous aspects (as soon as he was born, Jesus knows stand). Mages are three in number, according to an earlier oral tradition (see resource n°2b). The sarcophagus of the Nativity, now in the Musée de l’Arles antique (Museum of Ancient Arles) is marble sarcophagus (marble from St. Beatus in the Pyrenees) from the late fourth century. It is named after the central scene on it, especially in its upper register. Indeed, in the upper register show the Nativity: Mary is depicted standing under a tiled roof; before her, the cradle with the child wrapped in swaddling clothes, the ox and the ass leaned over the child, and a shepherd at the head of the cradle, making a gesture of cheer. The epiphany is shown in the lower register: the coming of the Magi, here three in number, wearing the Phrygian cap and each bearing a present; one seems to point the finger at the star that is located above the upper register of Mary.

After ¹ Jesus was born in Bethlehem in Judea, during the time of King Herod, Magi from the east came to Jerusalem& ² and asked, “Where is the one who has been born king of the Jews? We saw his star when it rose and have come to worship him.”
When King Herod heard this he was disturbed, and all Jerusalem with him […].
⁷ Then Herod called the Magi secretly and found out from them the exact time the star had appeared. ⁷ He sent them to Bethlehem and said, “Go and search carefully for the child. As soon as you find him, report to me, so that I too may go and worship him.”
⁹ After they had heard the king, they went on their way, and the star they had seen when it rose went ahead of them until it stopped over the place where the child was. ¹⁰ When they saw the star, they were overjoyed. ¹¹ On coming to the house, they saw the child with his mother Mary, and they bowed down and worshiped him. Then they opened their treasures and presented him with gifts of gold, frankincense and myrrh. ¹² And having been warned in a dream not to go back to Herod, they returned to their country by another route.

Gospel according to Matthew 2:1-12, New International Version (NIV):
https://www.biblegateway.com/passage/?search=Matthew%202:1-12 (08/12/2014)

Sarcophagus depicting the Nativity

Gospel of Pseudo-Matthew

The Gospel according to Matthew, the first writing of the New Testament canon was written after 70, probably by an author of Jewish origin; the final version probably dates from the 80's. It is written for a Syrian community, either in Antioch or in another city in Syria. This is the only passage of the Christian canon where the birth of Jesus is detailed. It may be interesting to compare with the sample passage from the Gospel of Pseudo-Matthew. This apocryphal gospel (whose original title was Nativity of Mary, the current title comes from its editor, Constantin von Tischendorf) is a reworking (in Latin) of several earlier texts. It was probably written in the late sixth century. It offers an amplified version of the birth of Jesus, with the addition of many details and informations and a more detailed timeline (two days, eight days, etc.). Like other apocryphal gospels, this text is a more detailed account of the life, and especially the youth of Jesus, and focus on its miraculous aspects (as soon as he was born, Jesus knows stand). Mages are three in number, according to an earlier oral tradition (see resource n°2b). The sarcophagus of the Nativity, now in the Musée de l’Arles antique (Museum of Ancient Arles) is marble sarcophagus (marble from St. Beatus in the Pyrenees) from the late fourth century. It is named after the central scene on it, especially in its upper register. Indeed, in the upper register show the Nativity: Mary is depicted standing under a tiled roof; before her, the cradle with the child wrapped in swaddling clothes, the ox and the ass leaned over the child, and a shepherd at the head of the cradle, making a gesture of cheer. The epiphany is shown in the lower register: the coming of the Magi, here three in number, wearing the Phrygian cap and each bearing a present; one seems to point the finger at the star that is located above the upper register of Mary.

13.2 -And when he had thus said, the angel ordered the beast to stand, for the time when she should bring forth was at hand; and he commanded the blessed Mary to come down off the animal, and go into a recess under a cavern, in which there never was light, but always darkness, because the light of day could not reach it. And when the blessed Mary had gone into it, it began to shine with as much brightness as if it were the sixth hour of the day. The light from God so shone in the cave, that neither by day nor night was light wanting as long as the blessed Mary was there. And there she brought forth a son, and the angels surrounded Him when He was being born. And as soon as He was born, He stood upon His feet, and the angels adored Him, saying: “Glory to God in the highest, and on earth peace to men of good pleasure.” […]
14. -And on the third day after the birth of our Lord Jesus Christ, the most blessed Mary went forth out of the cave, and entering a stable, placed the child in the stall, and the ox and the ass adored Him. Then was fulfilled that which was said by Isaiah the prophet, saying: “The ox knoweth his owner, and the ass his master's crib.” The very animals, therefore, the ox and the ass, having Him in their midst, incessantly adored Him. Then was fulfilled that which was said by Abacuc the prophet, saying: “Between two animals thou art made manifest.” In the same place Joseph remained with Mary three days.
[Joseph and his family go back to Jerusalem. Jesus is circumcised on the 8th day. The family seems to settle down in Bethlehem]
And when the second year was past, Magi came from the east to Jerusalem, bringing great gifts […]. And while the magi were going on their way, there appeared to them the star, which was, as it were, a guide to them, going before them until they came to where the child was. And when the magi saw the star, they rejoiced with great joy; and going into the house, they saw the child Jesus sitting in His mother's lap. Then they opened their treasures, and presented great gifts to the blessed Mary and Joseph. And to the child Himself they offered each of them a piece of gold. And likewise one gave gold, another frankincense, and the third myrrh. [3] And when they were going to return to King Herod, they were warned by an angel in their sleep not to go back to Herod; and they returned to their own country by another road.

[ENG] Gospel of Pseudo-Matthew. Trans. Alexander Roberts, Sir James Donaldson, Arthur Cleveland Coxe, Ante Nicene Fathers, vol. 8.