The ancient Romans celebrated the beginning of a new year with very special holidays, as it couldn’t be otherwise: not for nothing is very important in the ancient classical world is a mistaken idea of cyclical time just constantly reborn. See http://www.antiquitatem.com/en/what-is-century

In the months of December and January they were held in Rome several feasts  (the annual calendar was full of them), the Saturnalia, New Year, … that have come together and made their mark on our celebrations of Christmas and New Year. See article  http://www.antiquitatem.com/en/christmas-birth-of-jesus-mithras

This survival  certainly helps us to better understand these ancient rites, despite the differences; Consider that in this way we celebrate the arrival of a new year over two thousand years.

As today, these festivals  had a public and official dimension (inauguration of the new consuls, expressing loyalty to the emperor by asking for it and making him gifts that are called "STRENA" (I will later comment on them); today's  they are the message of king or political leader, new budgets, public horseback riding, etc.);  and a private party (visits to the homes of relatives and friends, mutual congratulations, exchanging hugs, kisses, gifts and good wishes votes, all with some ritual value …).

In three days of rites, official and private acts and practices of purification and augural, the Romans lived closed the year and opened a new era under the protection of Iuppiter Optimus Maximus, bifrons Ianus and the Lares (household gods).

As I said, it is essential to start things right: the good start is a good premonition of further development and therefore the first name pronounced in a solemn celebration should be a good omen, the name of the first recruited soldier should express happiness, the name of the persons who led the victim to the slaughter should also mean happiness, etc. Therefore any action usually begins with the formula "quod bonum, faustum, felix, fortunatumque esset", "may it be good, fortunate, happy and prosperous."

The first day of the year, full of force, of a special "Virtus" also should be happy as a determinant premonition of a year full of goods and then  at that time gifts were made and wises of a happy new year with the formula "annum novum faustum felicem (tibi, vobis precor, adprecor, … etc.), sometimes with "annum novum faustum  felicem fortunatum”,  similar to the general formula, which are nothing but rites and magic acts aimed at ensuring a favorable future ..

We have certainty of this formula in at least two letters and on numerous objects, some of which are precisely the  gifts which were exchanged at this time, especially chandeliers or lamps (lucerna), about  which I will speak immediately.

In addition to an explicit reference in Pliny, the formula appears repeatedly in imperial coins and medals and in dedication of some monuments. We can conclude from all this that the formula "annum novum faustum felicem" was so prevalent at this time as our "happy new year".

The Vindolanda tablets are a wooden tablets the size of a postcard with texts written in ink, of the first and second centuries that appeared at the foot of the famous Hadrian's Wall that separates the warring tribes of northern Britain from dominated part by legions. They appeared in the fort  Vindolanda and military and  personal issues them of the  garrison encamped are there  reflected. They were discovered as late as 1973. There are 752 tablets (still they appear) that have been translated and published in 2010. They are the oldest documents written in Latin in the British Isles.

Note: For anyone interested on them,  I offer a link to the website where you can find complete information http://vindolanda.csad.ox.ac.uk/

Well, the tablet number 261 is testimony to the use of the formula in question.

"Hostilius Flavianus to his Cerealis, greetings. A fortunate and happy New Year …"

 1  Hostilius Flauianus Cereali / 2  suo salutem / 3 / 4  annum nouom faustum felicem / . . . . . . .

Probably this Hostilius Flavianus is a prefect who in his letter to Cerealis, whom is considered "brother", refers to a sacrifice on the occasion of the new year, as we infer from the tablet number 265, although nothing authorizes link the two tablets, which says:

"… to his Cerialis, greetings. Just as you wished, brother, I have consecrated the day of the Kalends by a sacrifice …"

1  ]..s Ceriali suo / 2  salutem / 3  ego, fráter, sacrificio diem / 4  Kalendarum sic- / 5  ut uoluerás dedi- / 6  traces / . . . . . . . .

Often the expression "diem Kalendarum"  without reference to the month, refer to the Kalendae par excellence, “Kalendas Ianuarias”, the New Year.

There is also a letter of Marcus Cornelius Fronto (approximately lived between 95 and 165), the greatest orator on his time, to Caesar Marcus Aurelius,  in which he congratulated the New Year:

Fronto: Ad M. Caesarem 5.45 [77 Hout; 1.228 Haines] (30)

To my Lord.

A happy New Year and a prosperous in all things that you rightly desire to you and our Lord your Father and your mother and your wife and daughter, and to all others who deservedly share your affection –that is my prayer! In my still feeble state of healt I was afraid to trust myself to the crowd and crush. I shall see you, please God, the day after to.morrow offering up your vows. Farewell, my most sweet Lord. Greet my Lady. (Translate by C.R. Haines, (1919) Loeb Classical Library)

Domino meo.
Annum novum faustum tibi et ad omnia, quae recte cupis, prosperum cum tibi tum domino nostro patri tuo et matri et uxori et filiae ceterisque omnibus quos merito diligis, precor. Metui ego invalido adhuc corpore turbae et inpressioni me committere. Si dei juvabunt, perendie vos vota nuncupantis videbo.
Vale, mi domine dulcissime. Dominam saluta.

Pliny attests us the custom of wishing a happy new year in Natural History, 28, 5 (22)

I would appeal, too, for confirmation on this subject, to the intimate experience of each individual. Why, in fact, upon the first day of the new year, do we accost one another with prayers for good fortune, and for luck’s sake, wish each other a happy new year?. (Translated by John Bostock M.D, y H.T. Riley

Libet hanc partem singulorum quoque conscientia coarguere. cur enim primum anni incipientis diem laetis precationibus invicem faustum ominamur?

Tibullus relates the rite of burning bay leaves and the observation  whether they  crackle on the fire, good signal, or do not it, with the omen  of a new year:

Tibullus, II, 5, 79 ss.

With these portents the former times were scard
But Phoebus kindly better Fates award:
These Prodegies avert and turn away,
Imers’d beneath the surges of the sea.
May cracking Laurel in the flame declare,
The omen of a sacred happy year.

(Translated by Mr. Dart. London. 1720)

sed tu iam mitis, Apollo,
prodigia indomitis merge sub aequoribus,
et succensa sacris crepitet bene laurea flammis,
omine quo felix et sacer annus erit.

The New Year's greeting is accompanied by special gifts: dates, dried figs and honey are given away, as still happens; all sweets products that I enjoyed especially when child, now extended with the inevitable nougat, as an expression of best wishes.

They are also given bronze coins with the image of the two-faced god Janus, god of the end and the beginning, god facing the past and the future,  and therefore closing the old year and opens again the new year.

Janus, Ianus, is precisely the god who gives name to the month of January, Ianuarius, with it is open the Roman year since 153 BC on the occasion of the Celtiberic wars (Spain); previously the year began in March.

This gift coin has its continuity in our Christmas bonus, “aguinaldo” on Spanish language, word whose origin  the dictionary of the Royal Spanish Academy says: "Perhaps from  Latin hoc in anno 'on this year'. "

These gifts were called STRENA, whence the Spanish “estrenar” and "estrena"  that the Dictionary of the Royal Academy defines as "" gift, jewelry or given  by sign and demonstration of taste, happiness or benefit received," and French "étrenne”.

The Latin word STRENA probably from Sabine origin, means "good omen" and especially "gift made as a sign of good omen" in the sense that says Festus, 410, 21:

Strenae: So it is called the gift  that was made in a day consecrated by religion as a sign of good omen, from the number that  meants  to come second and third in the same advantages as if it is said “trena”, prepending an "S" as the ancients used to do in the  words  “loco” (locus) and lite (litis).

Strenam vocamus, quae datur die religioso ominis boni gratia, a numero, quo significatur, alterum, tertiumque venturum similis commodi, veluti trenam, praeposita S littera, ut in loco, et lite solebant antiqui.

They are very special the gifts which at this time are made to the emperors in recognition of his  authority and government. Serve as an example Suetonius on Augustus.

Suetonius, Life of Augustus, 57:

How much he was beloved for his worthy conduct in all these respects, it is easy to imagine. I say nothing of the decrees of the senate in his honour, which may seem to have resulted from compulsion or deference. The Roman knights voluntarily, and with one accord, always celebrated his birth for two days together; and all ranks of the people yearly, in performance of a vow they had made, threw a piece of money into the Curtian lake,  as an offering for his welfare. They likewise, on the calends [first] of January, presented for his acceptance new-year's gifts in the capitol, though he was not present: with which donations he purchased some costly images of the Gods, which he erected in several streets of the city: as that of Apollo Sandaliarius, Jupiter Tragoedus,  and others. When his house on the Palatine hill was accidentally destroyed by fire, the veteran soldiers, the judges, the tribes, and even the people, individually, contributed, according to the ability of each, for rebuilding it; but he would accept only of some small portion out of the several sums collected, and refused to take from any one person more than a single denarius. Upon his return home from any of the provinces, they attended him not only with joyful acclamations, but with songs. It is also remarked, that as often as he entered the city, the infliction of punishment was suspended for the time. (Translation by  Alexander Thomson. Philadelphia. Gebbie & Co. 1889.)

Pro quibus meritis quanto opere dilectus sit, facile est aestimare. Omitto senatus consulta, quia possunt videri vel necessitate expressa vel verecundia. Equites R. natalem eius sponte atque consensu biduo semper celebrarunt. Omnes ordines in lacum Curti quotannis ex voto pro salute eius stipem iaciebant, item Kal. Ian. strenam in Capitolio etiam absenti, ex qua summa pretiosissima deorum simulacra mercatus vicatim dedicabat, ut Apollinem Sandaliarium et Iovem Tragoedum aliaque. In restitutionem Palatinae domus incendio absumptae veterani, decuriae, tribus atque etiam singillatim e cetero genere hominum libentes ac pro facultate quisque pecunias contulerunt, delibante tantum modo eo summarum acervos neque ex quoquam plus denario auferente. Revertentem ex provincia non solum faustis ominibus, sed et modulatis carminibus prosequebantur. Observatum etiam est, ne quotiens introiret urbem, supplicium de quoquam sumeretur.

Inscription on the pedestal of the sculpture dedicated to Vulcan by Augustus with the money paid as “strena” in new year (CIL 06 00457)

The emperor Caesar Augustus, son of the divine emperor Pontifex Maximus for the thirteenth time, consul for the eleventh, with tribunician power for the fifteenth, dedicated (this statue) to Vulcan with the money the Roman people gave him absent  in new year, at the consulate of Nero Claudius Drusus and  Titus Quintius Crispinus.

Imp(erator) Caesar divi f(ilius) Augustus / pontifex maximus imp(erator) XIII
co(n)s(ul) XI trib(unicia) potest(ate) XV / ex stipe quam populus
Romanus / anno novo apsenti contulit / Nerone Claudio Druso
co(n)s(ulibus) / T(ito) Quinctio Crispino / Volcano

The word STRENA probably is from Sabine origin. It is affirmed by Symmachus when he gives the custom of giving gifts to the Sabine king Tatius. The text appears in some older editions as the letter number 28 or number 35, according to the edition, of the tenth book of His letters. Modern the text appears as the Report 15 of his work "Relations". It also refers to the custom that remains three centuries after Augustus, to give the strena to emperor.

Symmachus. Epistulae, 10,  28 (al. 35), or Relationes, 15:

To our lords, Theodosius and Arcadius always august, Symmachus, very famous man, prefect of the City (384)

The practice of the strena  was developed almost from the birth of the City of Mars by the impulse of King Tatius, who was the first to accept some branches of a fruitful tree from sacred forest of Strenia, as good auspices for  the new year, my emperors lords. The name suggests that strenae  marry with the strong, “strenous”,  men due to their  value, and therefore a distinction of this kind is due to beings like you, whose divine spirit awaits  rather a testimony  to their surveillance than to  an omen. Receive therefore, advocates of public welfare, modest gifts made of gold, according to custom, not to you  rejoice with an offering of precious metal but to  testify  the opulence of a happy time with our gift. We consecrate  good acquired wealth as gifts to good princes. You who condemn the hidden payments, receive a manifest obsequiousness from the dignitaries. We offer rightly to you, because you are  the deities of our safety,  the traditional pateras with five solids (solidus=coin)  each one. With them they are not overwhelmed  your modesty or our property. Persist long time that practice of consideration for you and an endless succession of years  renew the homage of your mercy. The prefecture, which must be granted to strong (strenuos) men willingly will satisfy the traditional strenae (gifts).

DD. VALENTINIANO, THEODOSIO ET ARCADIO SEMPER AUG. SYMMACHUS V. C. PRAEF. URB.
Ab exortu pene urbis Martiae strenarum usus adolevit, auctoritate Tatii regis, qui verbenas felicis arboris ex luco Strenuae anni novi auspices primus accepit, DD. imperatores. Nomen indicio est viris strenuis haec convenire ob virtutem: atque ideo vobis hujusmodi insigne deberi; quorum divinus animus magis testimonium vigilantiae quam omen exspectat. Sumite igitur defensores publicae salutis solemniter auro ducta munuscula: non quia divitis metalli honore gaudetis, sed ut nostra devotio felicis saeculi testetur opulentiam. Bonis principibus bene parta libamus. Suscipite a judicibus aperta obsequia, qui pretia occulta damnatis. Merito vobis solemnes pateras, cum quinis solidis, ut numinibus integritatis offerimus: quibus nec vester pudor, nec noster census oneratur. Maneat aevum talis circa nos usus officii: et honorem clementiae vestrae interminus annorum recursus instauret. Libenter strenis solemnibus praefectura fungetur, strenuis deferendis
.

On the custom of giving gifts, especially to the guests of the banquets, read them articles, on Saturnalia  http://www.antiquitatem.com/en/christmas-birth-of-jesus-mithras

and on Christmas gifts http://www.antiquitatem.com/en/xenia-apophoreta-martial-petronius-satyr

We can see on these gifts, sometimes laden with irony, a precedent of "twisted roll of kings" that is widespread in Spain. In the roulade it is hidden a "bean", vegetable-laden deep and strange ritual value to the Romans, (the Pythagoreans  are forbidden to eat beans), and also it is hidden also another "positive" figure, which can be a king, probably reminding the figure of "king of the feast", who regulated  the intake of wine during the celebration.

I reproduce an epigram of Martial on which he refers to the custom of giving  gifts at New Year time.

Martial, VIII, 33

Paulus had sent M. a cup of such thin metal that it could hardly be called a cup. An epigram against paltry gifts.

From your praetor's crown, Paulus, you send me a leaf and require this to be called a bowl. With this film your platform  was lately coated, and the pale stream of red saffron  washed it away. Or rather was it a flake I think, belonging to the leg of your couch scraped off by the nail of a cunning slave ? It can from a distance feel the fluttering of a gnat, and be wafted by the wing of the very smallest butterfly ; it floats in air, kept up by the heat of a tiny lamp, and, splashed with wine even lightly sprinkled, it dissolves. With such a layer is coated on the Kalends of January the nut  which a shabby client brings as a gift together with small coin. Pliant Egyptian beans grow with a less slender filament, of thicker mould are lily leaves that fall beneath the overpowering sun ; nor does the spider dart about a web so slender, nor the pendulous silkworm ply a work so light. Denser stands the chalk on old Fabulla's face, denser swells the bubble in tumbled water, and stronger is the bladder-net that confines knotted locks, and the Batavian pomade l that transforms Latin tresses. With skin like this is clothed the chick in a swan's egg, such are the patches that rest on a crescent-plastered brow. What use have you for a bowl when you can send me a tablespoon, when you can send me even a snail-pick I am suggesting too great things when you can send me a snail-shell : in a word, when you, Paulus, can send me nothing? (Translated by Walter C.A. Ker, M.A.  The Loeb Classical Library)

De praetoricia folium mihi, Paule, corona
Mittis et hoc phialae nomen habere iubes.
Hac fuerat nuper nebula tibi pegma perunctum,
Pallida quam rubri diluit unda croci.
An magis astuti derasa est ungue ministri
Brattea, de fulcro quam reor esse tuo?
Illa potest culicem longe sentire volantem
Et minimi pinna papilionis agi;
Exiguae volitat suspensa vapore lucernae
Et leviter fuso rumpitur icta mero.
Hoc linitur sputo Iani caryota Kalendis,
Quam fert cum parco sordidus asse cliens.
Lenta minus gracili crescunt colocasia filo,
Plena magis nimio lilia sole cadunt;
Nec vaga tam tenui discurrit aranea tela,
Tam leve nec bombyx pendulus urguet opus.
Crassior in facie vetulae stat creta Fabullae,
Crassior offensae bulla tumescit aquae;
Fortior et tortos servat vesica capillos
Et mutat Latias spuma Batava comas.
Hac cute Ledaeo vestitur pullus in ovo,
Talia lunata splenia fronte sedent.
Quid tibi cum phiala, ligulam cum mittere possis,
Mittere cum possis vel cocleare mihi, –
Magna nimis loquimur – cocleam cum mittere possis,
Denique cum possis mittere, Paule, nihil?

Saint Agustine, criticizing the abundant pagan gods, refers to the goddess Strenia, cited in the text above, who makes strong the men because “strong” on Latin is said "strenuus". It is said in two chapters of the book IV of his work  De civitate Dei, 11 and 16.  I reproduce the chapter XVI:

Saint Agustíne, De civitate Dei (The City of God  , IV, 16

What Was the Reason Why the Romans, in Detailing Separate Gods for All Things and All Movements of the Mind, Chose to Have the Temple of Quiet Outside the Gates.
But I wonder very much, that while they assigned to separate gods single things, and (well near) all movements of the mind; that while they invoked the goddess Agenoria, who should excite to action; the goddess Stimula, who should stimulate to unusual action; the goddess Murcia, who should not move men beyond measure, but make them, as Pomponius says, murcid— that is, too slothful and inactive; the goddess Strenua, who should make them strenuous; and that while they offered to all these gods and goddesses solemn and public worship, they should yet have been unwilling to give public acknowledgment to her whom they name Quies because she makes men quiet, but built her temple outside the Colline gate. Whether was this a symptom of an unquiet mind, or rather was it thus intimated that he who should persevere in worshipping that crowd, not, to be sure, of gods, but of demons, could not dwell with quiet; to which the true Physician calls, saying, Learn of me, for I am meek and lowly in heart, and you shall find rest unto your souls?
(http://www.newadvent.org/fathers/1201.htm)

Miror autem plurimum, quod, cum deos singulos singulis rebus et paene singulis motibus adtribuerent, uocauerunt deam Agenoriam, quae ad agendum excitaret, deam Stimulam, quae ad agendum ultra modum stimularet, deam Murciam, quae praeter modum non moueret ac faceret hominem, ut ait Pomponius, murcidum, id est nimis desidiosum et inactuosum, deam Streniam, quae faceret strenuum, his omnibus diis et deabus publica sacra facere susceperunt, Quietem uero appellantes, quae faceret quietum, cum aedem haberet extra portam Collinam, publice illam suscipere noluerunt. Vtrum indicium fuit animi inquieti, an potius ita significatum est, qui illam turbam colere perseueraret non plane deorum, sed daemoniorum, eum quietem habere non posse? ad quam uocat uerus medicus dicens: Discite a me, quoniam mitis sum et humilis corde, et inuenietis requiem animabus uestris.

But the most special gift for the occasion is a Lucerne or oil lamp, usually ceramic, where the inscription in question appears: novum annum faustum felicem (tibi, …).

This gift is obviously in relation to the fact that  the ancient Romans celebrated the new year on December 25, the birthday of the "unconquered Sun", is the "dies natalis Solis invicti", "the birthday of the god unconquered Sun ". See http://www.antiquitatem.com/en/mithraisme-christianity-winter-solstice

The lamp represents the "new light", the "new Sun", the "Novum annum" that  begins at that day.

    

Two “lucernae”  very similar, one made of terracotta and the other of bronze. The terracotta  made is in the Museum of the City of Milan. The exceptional of bronze appears for sale on a website trade or auction of various objects, without context  and without special historical value because all archaeological information relating to the site is disappeared.

On them it appears a winged Victory with a pin or palm in his hand, breads, nuts and Janus coin, with the shield on which is inscribed the phrase :

A  happy new year  for you.  annum novum felicem tibi

Among the numerous inscriptions on lamps, in addition to the above, only two, I will reproduce this one from Spain. So this from Hispania Citerior (Madrid), of uncertain origin: CIL 02, 04,969.03; EDCS-20301000

Annum / nov(u)m Faus/tum fel/icem mih(i) / hu(n)c

Or this one of Bruttium et Lucania, Eboli / Eburum: CIE 10 08053,005ª; EDCS-22900527 in which the inscription is on the shield held  by the Victoria:

Annum / nov(u)m fau/stum fel/icem mi/hi

Other objects, like this glass engraved with the usual motifs are also given: dates, figs, bay leaves, coin with the image of Janus, a temple, etc

  

Or these terracotta medallions with the inscriptions we  try and one with the symbols above.

  

In the following text of Ovid these customs are explained, because  citizens of the time seem not to explain them. For “as” gift or bronze coin with bifrons Janus on the obverse and a ship on the back,  Ovid's contemporaries seem to show a complete lack of explanation. I reproduce the text, without the last detail of the ship on the reverse of the coin for not to be excessively long. In any case, the interested reader will find the text easily, or I can help if you show me your interest in it.

  

Ovid, Fasti, I, v. 145 y ss.

So far he spoke, and his benignant eye
Seemed to invite to further colloquy ;
I plucked up courage, made acknowledgment, 
And thus pursued, but with mine eyes down bent.
" O tell me wherefore is the new year born
In frost and cold, when all is chill and lorn ?
Surely its birthday should in Spring have been
When meads are blossoming and woods are green ; 
When the young buds are bursting from the bines,
And elms are gorgeous, garmented with vines,
When corn blades upward shoot and the sweet bird,
In purest ether, carolling is heard ;
When flocks are wanton, and the swallow guest 
Beneath the roof and rafter frames her nest ;
When sunshine blesses all, the steer and share,
Should not this be the birthday of the year ? "
So did I ask with fervency ; so he
Responded with contrasting brevity : — 
" The winter Solstice doth the years divide ;
Annus and Phoebus, both claim Brumaltide."
I pondered on and asked why New Year's-day
From legal pleas should not exempted be ?
" Then learn ; " he said ; " inestimable time 
Must not be lost, but seized on at the prime ;
A bad beginning would usurp the year :
Therefore I will, that each man in his sphere
Work ere he play — and show his willingness."
I asked him next — " Wherefore, when I address 
My prayers, Janus, to the gods on high,
Do I bring wine and frankincense to thee ? "
" Because I hold the threshold, keep the door ;
Access to them, through me, must you implore."
" Why, Janus, on thy Kalends do we greet 
With compliments and wishes those we meet ? "
Then he, incumbent on his staff, replied :
" Omens are from first indexes supplied ;
Your timid ears catch the first spoken word,
The Augur answers from the first seen bird, 
Then ears of gods are open, and their fanes,
And then the votary his wish obtains."
He paused — -my question trod on his reply —
" Say what the date and dry fig signify,
And candid honey in white earthen ware 
We offer then ? " " All that is sweet and fair ;
A sign prognostic that the year may run
Its destined course as blandly as begun."
" I see the cause of sweets, now tell the cause
For the small coin, and I shall know thy laws." 
He laughed and said : " thou of little wit,
The habits of all time account for it ;
Why e'en when Saturn reigned on earth, e'en then,
Nought sweeter was than money unto men.
The passion grew with time ; it grew, but now 
It culminates, and can no further grow.
Wealth now predominates ; not so of yore
When Rome was little and the Romans poor.
Quirinus, Mars-begotten, laid his head
In a thatched hut, beside the river's bed 
The reeds amidst.

(Translation  By JOHN BENSON EOSE.

Dixerat, et vultu, si plura requirere vellem, 145
  Se mihi difficilem non fore, fassus erat:
Sumpsi animum, gratesque deo non territus egi,
  Verbaque sum spectans pauca locutus humum:
Dic, age, frigoribus quare novus incipit annus,
  Qui melius per ver incipiendus erat? 150
Omnia tunc florent, tunc est nova temporis aetas,
  Et nova de gravido palmite gemma tumet,
Et modo formatis operitur frondibus arbos,
  Prodit et in summum seminis herba solum,
Et tepidum volucres concentibus aëra mulcent, 155
  Ludit et in pratis luxuriatque pecus.
Tum blandi soles, ignotaque prodit hirundo,
  Et luteum celsa sub trabe fingit opus.
Tum patitur cultus ager, et renovatur aratro.
  Haec anni novitas jure vocanda fuit. 160
Quaesieram multis: non multis ille moratus,
  Contulit in versus sic sua verba duos:
Bruma novi prima est, veterisque novissima solis:
  Principium capiunt Phoebus et annus idem.
Post ea mirabar, cur non sine litibus esset 165
  Prima dies. Causam percipe, Janus ait.
Tempora commisi nascentia rebus agendis,
  Totus ab auspicio ne foret annus iners.
Quisque suas artes ob idem delibat agendo,
  Nec plus quam solitum testificatur opus. 170
Mox ego: Cur, quamvis aliorum numina placem,
  Jane, tibi primo tura merumque fero?
Ut per me possis aditum, qui limina servo,
  Ad quoscumque voles, inquit, habere deos.
At cur laeta tuis dicuntur verba Kalendis, 175
  Et damus alternas accipimusque preces?
Tum deus incumbens baculo, quem dextra gerebat,
  Omina principiis, inquit, inesse solent.
Ad primam vocem timidas advertitis aures,
  Et primum visam consulit augur avem. 180
Templa patent auresque deûm, nec lingua caducas
  Concipit ulla preces, dictaque pondus habent.
Desierat Janus: nec longa silentia feci,
  Sed tetigi verbis ultima verba meis:
Quid vult palma sibi rugosaque carica, dixi, 185
  Et data sub niveo Candida mella cado?
Omen, ait, causa est, ut res sapor ille sequatur,
  Et peragat coeptum dulcis ut annus iter.
Dulcia cur dentur, video: stipis adjice causam,
  Pars mihi de festo ne labet ulla tuo. 190
Risit, et, O quam te fallunt tua saecula, dixit,
  Qui stipe mel sumpta dulcius esse putes!
Vix ego Saturno quemquam regnante videbam,
  Cujus non animo dulcia lucra forent.
Tempore crevit amor, qui nunc est summus, habendi; 195
  Vix ultra, quo jam progrediatur, habet.
Pluris opes nunc sunt, quam prisci temporis annis,
  Dum populus pauper, dura nova Roma fuit,
Dum casa Martigenam capiebat parva Quirinum,

And still Ovid is singing happy first age and explaining how there it is now an excessive desire for wealth and gold …

Seneca also refers to dried floured figs,  which remain today especially at New Year, on  Epistula 87.3:

Our repast was such, that nothing could be spared from it, nor did it take up much time in dressing. I am satisfied with a few dried figs and dates. When I have any bread, the figs serve me for a dainty dish; when I have no bread, they supply its place. They make me fancy it to be New-year’s day; which endeavour to render auspicious and happy, by harbouring good thoughts, and keeping up a greatness of foul; which is never greater, than when it hath withdrawn itself from all external things; and hath obtained  for itself peace, by fearing nothing, and wealth by coveting nothing. (Translation by Thomas Morrell,  D.D.

De prandio nihil detrahi potuit; paratum fuit non magis hora, nusquam sine caricis, numquam sine pugillaribus. Illae, si panem habeo, pro pulmentario sunt, si non habeo, pro pane. Cotidie mihi annum novum faciunt, quem ego faustum et felicem reddo bonis cogitationibus et animi magnitudine, qui numquam maior est, quam ubi aliena seposuit et fecit sibi pacem nihil timendo, fecit sibi divitias nihil concupiscendo.

The inscription discussed with initial A.N.F.F., which we transcribe as Annum Novum Faustum Felicem, also it appears on coins of several emperors, particularly of Hadrian, Antoninus Pius and Alexander Severus, usually with a laurel garland and sometimes accompanied with  OPTIMO PRINCIPI words (to our best Prince) . Remember the custom to congratulate the emperor and offer the "strenae"   as recognition. The inscription  can be:

S. P. Q. R. A. N. F. F. OPTIMO PRINCIPI (o HADRIANO AVG. P. P.),

Senatus Populusque Romanus, Annum Novum Faustum Felicem Optimo Principi [i.e. adprecatur.]

The Senate and the Roman People pray for a prosperous and happy New Year to the best prince.

 Photograph taken from the book  “Antonini Pii Augusti Nummus de Anni Novi Auspiciis explicatus. Romae, Typiis Iac. Dragondelli. 1676)

All these rites and formulas, extended even with others from the conquered provinces, as some Celtic as the cult of the god-deer, spread throughout the empire and have survived to this day in some way, despite the surprisingly shock with Christian celebration that inevitably was occurred when Christianity was imposed. Pagan experience of "natural time", which is renewed every year, clashes with the Christian conception of time that does not expect anything from concrete time in expectation of eternal life. The apologist Parents of the Church, as Ambrosius and Tertullian, argument strongly against paganism, and in part they were successful, although these rites and customs have largely survived to this day.

And St. Ambrosius (339-397) says in his "Commentarium in Epistolam Beati Pauli ad Galatas, IV, versiculum 10 (Migne, PLXVII):

You take care of  days and months and seasons and years (Galat.IV, v.10). He clears now the aforementioned elements. The sun makes the day, the courses of the moon make the months; the seasons  are spring, summer, autumn and winter; these four, once completed its own number, they make a year. Those who say, for example, we can not travel tomorrow, after tomorrow you should not start anything, and they so more and more often deceive themselves, take care of the days. However they worship the months who analyze  the movements of the moon and say, for example, in the seventh moon it should not be construed instruments; and again in the ninth moon, for example, it should not take home a purchased slave; and with these things they tend to easily take care of adversity. Instead they take care of the seasons stations when they say:  today is the beginning of spring, today is a holiday; Vulcanalia (feasts of Vulcan) are day after tomorrow . And they also say these things: the next day, you can not leave home. And they give worship to the years when they say, with the Kalends of January there is a new year, as if the years were not completed every day. But because they  keep the worship to the memory of that famous Janus, they are still serving this superstition, which should be away from the servants of God. For if you love God with all your heart, being he favorable, there should be no fear, no worry about these things. As it has happened just prosperously everything is done under the devotion of God.

“Dies observatis, et menses, et tempora, et annos” (Galat. IV) .  Quae elementa dixerit, nunc declarat.  Diem sol facit,menses cursus lunae; tempora vere sunt ver, aestas,autumnus, hiems: quatuor haec, completo numero proprio, faciunt annum. Dies ergo observant, qui dicunt, ut puta, crastino proficiscendum non est, post crastinum enim non debet aliquid inchoari, et sic solent magis decipi. Hi autem colunt menses, qui cursus perscrutantur lunae dicentes, ut puta, septima luna instrumenta confici non debent; nona iterum luna, ut puta emptum servum domum duci non oportet: et per haec facilius solent adversa provenire. (0263D) Tempora vero sic observant cum dicunt: Hodie veris initium est, festivitas est; post cras Vulcanalia sunt. Et talia iterum aiunt: Posterum est, domum egredi non licet. Annos sic colunt cum dicunt, Kalendis Ianuariis novus est annus quasi non quotidie anni impleantur. Sed ut Iani illius recolant memoriam  bifrontis, hac superstitione utuntur, quae longe debet esse a servis Dei. Si enim Deus ex toto corde diligitur, ipso propitio nulla debet esse formido, neque suspicio harum rerum. Prospere enim potest cedere, quidquid simpliciter sub Dei devotione fit.

And Du Cange, citing a text by Maximus Taurinensis, Homily on Kalendis Ianuariis, in his Glossarium mediae et infimæ latinitatis, on the term annus novus says:

New Year. Maximus of Turin. Homily for the Kalends of January:

In severe and absolute wickedness it has fallen the hearts of those who disregard God's warnings with sacrilegious speeches, love vanities and follow  falsities and besides all call a certain days New Year to complete his offense.  Although according to them conveniently, they call  New the Year, because they renew perversity and shamelessness of a false honesty with nefarious feasts. They call New the Year like the sky or the earth reveal something new. They call New Year  to Kalendas of January when they get dirty with always old error and horror.

Annus Novus. S. Maximus Taurinensis Hom. de Kalendis Januariis :
Gravi utique eorum cor est, atque omni impietate depressum, qui per sacrilegos logos divinis monitis illudentes, vana diligunt, et falsa sectantur, et post omnia ad offensionis plenitudinem dies ipsos Annum Novum vocant. Quamquam non inconvenienter secundum se, Novum appellant Annum : quoniam per nefandas ferias de honestate falsa turpitudo et perversitas innovatur. Novum vocant Annum, quasi novi aliquid aut cœlum tunc ostendat, aut terra. Novum Annum Januarias appellant Kalendas, cum vetusto semper errore et horrore sordescant.

Tertullian meanwhile refers for example to four feasts, probably the most popular in the city of Cartago where he lives: Roman Saturnalia, the Kalends of January, the winter solstice and Matronalia. Tertullian can not bear  that Christians participate in these celebrations, sometimes overflowing and with worship pagan gods. He says in his De Idolatria, (On Idolatry), 14,6:

Chapter XIV.-Of Blasphemy. One of St. Paul's Sayings.

But, however, the majority (of Christians) have by this time induced the belief in their mind that it is pardonable if at any time they do what the heathen do, for fear "the Name be blasphemed." Now the blasphemy which must quite be shunned by us in every way is, I take it, this: If any of us lead a heathen into blasphemy with good cause, either by fraud, or by injury, or by contumely, or any other matter of worthy complaint, in which "the Name" is deservedly impugned, so that the Lord, too, be deservedly angry. Else, if of all blasphemy it has been said, "By your means My Name is blasphemed," we all perish at once; since the whole circus, with no desert of ours, assails "the Name" with wicked suffrages. Let us cease (to be Christians) and it will not be blasphemed! On the contrary, while we are, let it be blasphemed: in the observance, not the overstepping, of discipline; while we are being approved, not while we are being reprobated. Oh blasphemy, bordering on martyrdom, which now attests me to be a Christian, while for that very account it detests me! The cursing of well-maintained Discipline is a blessing of the Name. "If," says he, "I wished to please men, I should not be Christ's servant." But the same apostle elsewhere bids us take care to please all: "As I," he says, "please all by all means." No doubt he used to please them by celebrating the Saturnalia and New-year's day! [Was it so] or was it by moderation and patience? by gravity, by kindness, by integrity? In like manner, when he is saying, "I have become all things to all, that I may gain all," does he mean "to idolaters an idolater? ""to heathens a heathen? ""to the worldly worldly? "But albeit he does not prohibit us from having our conversation with idolaters and adulterers, and the other criminals, saying, "Otherwise ye would go out from the world," of course he does not so slacken those reins of conversation that, since it is necessary for us both to live and to mingle with sinners, we may be able to sin with them too. Where there is the intercourse of life, which the apostle concedes, there is sinning, which no one permits. To live with heathens is lawful, to die with them is not. Let us live with all; let us be glad with them, out of community of nature, not of superstition. We are peers in soul, not in discipline; fellow-possessors of the world, not of error. But if we have no right of communion in matters of this kind with strangers, how far more wicked to celebrate them among brethren! Who can maintain or defend this? The Holy Spirit upbraids the Jews with their holy-days. "Your Sabbaths, and new moons, and ceremonies," says He, "My soul hateth." By us, to whom Sabbaths are strange, and the new moons and festivals formerly beloved by God, the Saturnalia and New-year's and Midwinter's festivals and Matronalia are frequented-presents come and go-New-year's gifts-games join their noise-banquets join their din! Oh better fidelity of the nations to their own sect, which claims no solemnity of the Christians for itself! Not the Lord's day, not Pentecost, even it they had known them, would they have shared with us; for they would fear lest they should seem to be Christians. We are not apprehensive lest we seem to be heathens! If any indulgence is to be granted to the flesh, you have it. I will not say your own days, but more too; for to the heathens each festive day occurs but once annually: you have a festive day every eighth day. Call out the individual solemnities of the nations, and set them out into a row, they will not be able to make up a Pentecost.  (Translated by the Rev. S. Thelwall.]

XIV. Sed enim plerique iam induxerunt animo ignoscendum esse, si quando, quae  ethnici, faciunt, ne nomen blasphemetur. Porro blasphemia, quae nobis omni modo deuitanda est, haec opinor est, si qui nostrum ad iustam blasphemiam ethnicum deducat aut fraude aut iniuria aut contumelia aliaue materia dignae querelae, in qua nomen merito percutitur, ut merito irascatur et dominus. Ceterum si de omni blasphemia dictum est, uestra causa nomen meum blasphematur, perimus uniuersi, cum totus circus scelestis suffragiis nullo merito nomen lacessit. Desinamus, et non blasphemabitur. Immo blasphemetur, dum sumus in obseruatione, non in exorbitatione disciplinae, dum probamur, non dum reprobamur.  O blasphemiam martyrii adfinem, quae tunc me testatur Christianum, cum propterea me detestatur ! Benedictio est nominis maledictio custoditae disciplinae. Si hominibus, inquit, uellem placere, seruus Christi non essem. Sed idem alibi iubet, omnibus placere curemus. Quemadmodum ego, inquit, omnibus per omnia placeo. Nimirum Saturnalia et Kalendas Ianuarias celebrans hominibus placebat ? An modestia et patientia ? An grauitate, an humanitate, an integritate ? Proinde cum dicit, omnibus omnia factus sum, ut omnes lucrifaciam: numquid idololatris idololatres ? Numquid ethnicis ethnicus ? Numquid saecularibus saecularis ? Sed etsi non prohibet nos conuersari cum idololatris et adulteris et ceteris criminosis dicens, ceterum de mundo exiretis, non utique eas habenas conuersationis inmittit, ut, quoniam necesse est et conuiuere nos et commisceri cum peccatoribus, idem et compeccare possimus. Vbi est commercium uitae, quod apostolus concedit, ibi peccare, quod nemo permittit. Licet conuiuere cum ethnicis, commori non licet. Conuiuamus cum omnibus; conlaetemur ex communione naturae, non superstitionis. Pares anima sumus, non disciplina, compossessores mundi, non erroris. Quod si nobis nullum ius est communionis in huiusmodi cum extraneis, quanto scelestius est haec inter fratres frequentare. Quis hoc sustinere aut defendere potest ? Iudaeis dies suos festos exprobrat spiritus sanctus. Sabbata, inquit, uestra et numenias et ceremonias odit anima mea. Nobis, quibus sabbata extranea sunt et numeniae et feriae a deo aliquando dilectae, Saturnalia et Ianuariae et Brumae et Matronales frequentantur, munera commeant et strenae, consonant lusus, conuiuia constrepunt.O melior fides nationum in suam sectam, quae nullam sollemnitatem Christianorum sibi uindicat ! Non dominicum diem, non pentecosten, etiamsi nossent, nobiscum communicassent ; timerent enim, ne Christiani uiderentur. Nos ne ethnici pronuntiemur, non ueremur. Si quid et carni indulgendum est, habes, non dicam tuos dies tantum, sed et plures. Nam ethnicis semel annuus dies quisque festus est, tibi octauo quoque die. Excerpe singulas sollemnitates nationum et in ordinem exsere : pentecosten implere non poterunt.

Note: some information and some images were obtained from http://hortushesperidum.blogspot.com/2013/12/annum-novum-faustum-felicem-tibi.html

Well,  Christianity triumphed somewhat in this controversy , since it is imposed as the official and only religion in the Roman Empire, ending the pagan religion, but he did it at the price of maintaining these customs and traditions which have not stopped held today and shows remarkable vitality, albeit under the guise of traditional Christian motifs.

Annum novum faustum felicem A good, happy, prosperous and fortunate New Year

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