Far more interesting, however, is the correlation between Christmas and the birth of the Sun, we resume reading Macrobius’s Saturnalia Book 1.18.2, where the conversation attempts to define which God is exactly the God of the Sun, or the Sun God:
[2 ] You must bear in mind, replied Vettius, that the company of poets in their stories about the gods usually borrow the elements of these stories from the secret places of philosophy; certainly it is not empty superstition but divine reason that makes them relate almost all the gods—at any rate the celestial gods—to the sun.
Correlating all God’s to the Sun isn’t really that helpful, but as we will see he’s not kidding, this begins the same chapter section 7:
 I first maintained that Apollo is to be identified with the sun, and I afterward explained that Liber Pater is himself Apollo; and so there can be no doubt but that the sun and Liber Pater are to be regarded as manifestations of the same deity. Nevertheless the point shall be established distinctly by yet clearer proofs.  In the performance of sacred rites a mysterious rule of religion ordains that the sun shall be called Apollo when it is in the upper hemisphere, that is to say, by day, and be held to be Dionysus, or Liber Pater, when it is in the lower hemisphere, that is to say, at night.  Likewise, statues of Liber Pater represent him sometimes as a child and sometimes as a young man; again, as a man with a beard and also as an old man, as for example the statue of the god which the Greeks call Bassareus and Briseus, and that which in Campania the Neapolitans worship under the name Hebon.  These differences in age have reference to the sun, for at the winter solstice the sun would seem to be a little child, like that which the Egyptians bring forth from a shrine on an appointed day, since the day is then at its shortest and the god is accordingly shown as a tiny infant.” Afterward, however, as the days go on and lengthen, the sun at the spring equinox acquires strength in a way comparable to growth to adolescence, and so the god is given the appearance of a young man. Subsequently, he is represented in full maturity, with a beard, at the summer solstice, when the sun’s growth is completed. After that, the days shorten, as though with the approach of his old age—hence the fourth of the figures by which the god is portrayed.
So as it turns out, in Egypt in the 4/5th centuries the sun god, whoever that may be, was portrayed as an infant on the Solstice. Whether this predates December 25 the date as given by Hippolytus in 202-211AD remains unclear.
Macrobius continues to identify other gods with the sun as well as the 12 signs of the Zodiac in Saturnalia 1.21.12:
This is their sign for Osiris, and by it they indicate that this god is the sun, which with royal power looks down upon the world from on high. And indeed in ancient usage the sun is called the eye of Jupiter.
 Among the Egyptians Apollo (and he is the sun) is called Horus —whence the name “hours” (horae) has been given to 24 divisions which make up a day and a night and to the four seasons [ὥραι] which together complete the cycle of the year  It has also been a practice of the Egyptians, when they wish to dedicate a statue of the sun under its own name, to represent it with the head shaved except on the right side, where the hair is allowed to remain. The hair that is kept shows that the sun In never hidden from the world of nature, and the retention of the roots after the locks have been sure indicates that it is an essential property of the song even when it is invisible to us, to reappear like those locks.  This same attribute of a half-shorn head is also a symbol of the time when the light is reduced and when the sun, as though shorn of its growth and with a mere stubble, so to speak, remaining, comes to the shortest day (which the men of old called the winter solstice, using the word bruma for winter, from the shortness of the day, as though to say “short day.” But when the sun rises again from its narrow retreat, it reaches out to the summer hemisphere, growing in strength as though by a process of birth, and it is believed to have come then into its own realm. [ 16] That is why, among the signs of the zodiac, the Egyptians have dedicated an animal, the lion, in that part of the heavens where in its yearly course the sun’s powerful heat is hottest. And the Sign of the Lion there they call “The House of the Sun,” because a lion seems to derive its essential qualities from the natural properties of the sun.  For, in the first place, the lion by its energy and ardor surpasses other animals as the sun surpasses the rest of the stars. And then, just as a lion’s strength is in its breast and in the font part of its body, but its hinder limbs are weaker, so the might of the sun grows more powerful from the first part of the day up to noon or from the first part of the year, that is from the spring, ro summer; but afterward the sun grows weaker, as it declines to its setting (which would seem to be the hinder part of the day) or to the winter (the hinder part of the year). And the lion, too, always gazes with open fiery eyes, just as the sun regards the earth with the continuous and unwearied gaze of its open fiery eye.
 Again, not only the Lion, but every one of the signs of the zodiac as well, may properly be related to natural attributes of the sun. To begin with the Ram: the affinity here is well marked, for throughout the six winter months a ram…
So in the fourth and fifth centuries many gods were associated with the Sun and in Egypt the Sun was shown as an infant on the winter solstice, (December 25 was thought of as the winter solstice even though it may not have actually fallen on that date specifically). “Bruma” or “Brumalia” is another name for the winter solstice, Roger Pearse has an excellent blog post here and here about this and its relationship to Christmas.
But turning back to the idea that the sun was born on the winter solstice and its influence on Christmas; is there any evidence that the birth of the Sun was celebrated on the winter solstice before 202-211 AD (when Hippolytus first marked the birth of Jesus as December 25)?
Not that I can find, the closest is an inscription dated to 275 AD in Inscriptiones Latinae Selectae by Hermann Dessau 1969 (I have not checked this, I read it in “Toward the Origins of Christmas” by Susan Roll). The inscription apparently marks Emperor Aurelian’s attempts to subordinate all God’s under Sol (the Sun) and declares December 25 as the birth of this god.
I really would like to get a hold of a copy of this inscription (e-mail me or leave a comment if you have access), but for now it is obvious that this decree was issued more than 50 years after Hippolytus wrote, and that the reason for first establishing the birth of Jesus on December 25 was not because of this decree. Indeed, the real reason December 25 was chosen has to do with the date of the Passover, which you can read about here.