In addition to this, there are many other facets of New Year’s Eve celebrations which have mythical roots. One of these is the popular New Year’s Eve figure Father Time; traditionally depicted as an old man with a long white beard, an hour-glass or clock, and a scythe. Most mythographers believe that Father Time is based on the Greco-Roman character of Cronus (called Saturn by the Romans). Cronus was the ruthless leader of the Titans – a race of giants who ruled over the world before the advent of the Olympian gods – and who is most famously remembered for making a snack out of his own children. As Saturn the Romans worshipped Cronus as a harvest deity, which explains the scythe. The Romans also held a popular end of year celebration is honor of Saturn known as Saturnalia which began on December 17th and lasted a week, being something of a cross between Thanksgiving, Christmas, and New Year’s Eve.
Later on it appears that Cronus was confused or combined with another Greco-Roman god-like being known as Chronos; the personification of time and the source of such modern day words as chronology.
Another element to consider is that of New Year’s Day; January 1st. The month of January draws its name from the Roman god Janus who presided over gateways, beginnings, and endings. Janus is classically depicted in art as having two heads; one to look forward with and one to look backwards with, a gift from Saturn himself. Janus’ temple, which was located in the Roman Forum, was also unique in that it possessed two separate gateways; one to enter the temple and one to leave it. This was in opposition to the traditional temple model which only featured one gateway in and out.There are very few myths about Janus, the most famous being the tale of how Janus got a wife. According to Ovid’s Metamorphosis there was once a nymph by the name of Carna whose days were spent teasing men with her sexual advances only to run away as quick as a flash whenever said men attempted to make a move. One day Carna made the mistake of teasing Janus, not knowing the god literally had eyes in the back of his head. When Janus attempted to make a move on Carna the nymph once again attempted to run away, only this time Janus saw where she went and quickly gave case and furthermore caught the saucy nymph forcing her to become his bride. According to legend the King of Alba Long was their son.
Sources: The Penguin Dictionary of American Folklore (2001) by Alan Axelrod and Harry Oster, Don’t Know Much About Mythology (2005) by Kenneth C. Davis, The Ultimate Encyclopedia of Mythology (2001) by Arthur Cotterell & Rachel Storm, and the Metamorphosis by Ovid, translated by Charles Martin (2004).
 It could be speculated that the image of Cronus/Saturn devouring his own children so as to prevent them from overthrowing him is linked in some way to the modern day, sanitized image of Father Time and Baby New Year, though I haven’t done the research to prove this.
 As an additional perk Janus made Carna the goddess of doorhinges.