The Origin

In 1743, German Protestant, Paul Ernst Jablonski argued Christmas was placed on December 25 to correspond with the Roman solar holiday Dies Natalis Solis Invicti and was therefore a "paganization" that debased the true church.
["Christmas", Encarta, Roll, Susan K., Toward the Origins of Christmas, (Peeters Publishers, 1995), p.130.
Tighe, William J., "Calculating Christmas"]

In 1889, Louis Duchesne suggested that the date of Christmas was calculated as nine months after the Annunciation (March 25), the traditional date of the Incarnation.

Dies Natalis Solis Invicti means "the birthday of the unconquered Sun". The use of the title Sol Invictus allowed several solar deities to be worshipped collectively, including Elah-Gabal, a Syrian sun god; Sol, the god of Emperor Aurelian; and Mithras, a soldiers' god of Persian origin.
["Mithraism", The Catholic Encyclopedia, 1913.]

Emperor Elagabalus (218–222) introduced the festival, and it reached the height of its popularity under Aurelian, who promoted it as an empire-wide holiday. This day had held no significance in the Roman festive calendar until it was introduced in the third century.
[William J. Tighe, "Calculating Christmas"]

The festival was placed on the date of the solstice because this was on this day that the Sun reversed its southward retreat and proved itself to be "unconquered". Several early Christian writers connected the rebirth of the sun to the birth of Jesus.
["Christmas", The Catholic Encyclopedia, 1913.]

"O, how wonderfully acted Providence that on that day on which that Sun was born... Christ should be born", Cyprian wrote.
["Christmas", The Catholic Encyclopedia, 1913.]

John Chrysostom also commented on the connection: "They call it the 'Birthday of the Unconquered'. Who indeed is so unconquered as Our Lord...?"
["Christmas", The Catholic Encyclopedia, 1913.]