Here’s a Latin quiz:
Was the month of January seen as the first month of the year by the Romans?
The poet Ovid explains it in his first book of the “Fasti” (l. 39-44).
“Martis erat primus mensis, Venerisque secundus: (…)
tertius a senibus, iuvenum de nomine quartus,
quae sequitur, numero turba notata fuit.
at Numa nec Ianum nec avitas praeterit umbras,
mensibus antiquis praeposuit duos.”
In these lines Ovid says that the first month of the year used to be the month of the god Mars (= March), the second month was of the goddess Venus (= April); the third month took its name from old/ancient people (= May from the word “maiores”), the fourth from young people (= June from the word “iuvenes”); the rest of the months were named after numbers (= Quintilis, Sextilis ,etc).
So, according to Ovid, the year started originally in March and had 10 months, which explains why the nouns September (7th), October (8th), November (9th) and December (10th) don’t match our current months’ numbers, since for us the first month is January and not March.
At the end of this excerpt, Ovid adds that the (legendary) king Numa added 2 months before March, dedicated to the god Janus (=January) and the ancestral shades (=February, from the god Februus).
So, yes, the classic Romans did see January as the first month of the year. It makes sense since the god Janus was the god of gates, entrances and beginnings. February was named after the god Februus, and the month was dedicated to purification (“februum”) to appease the dead with sacrifices and offerings.
Now, here are few ways to say “happy new year” in Latin:
“Felix sit annus novus” (May the new year be happy)
“Annum novum faustum prosperumque tibi precor” (I wish you a new year auspicious and prosperous)
“Felix totus ut annus eat” (May the entire year run happy. Quote from Ovid, Fasti, I,26)
You can also just simply use the accusative of exclamation for short greetings:
Optimum annum ! ([Have a] great year!)
Bonum annum ! ([Have a] good year!)
Felicem annum ! ([Have a] happy year!)