This saving chalice too we bring,
receive it graciously, O King;
with fragrant odour may it rise
to your high throne above the skies.
— D. M. Coffey, "Receive, O Father, God of might", v. 2
Jungmann, in his magisterial work The Mass of the Roman Rite, has much to say of the prayer Offerimus tibi, which is used in the Traditional Latin Mass at the offering of the chalice; according to his sources, it first appears as a 9th or 10th century supplement to a manuscript Sacramentary. There, and in other early records, it did not yet include the stirring and dogmatically significant words pro nostra et totius mundi salute.
The Canon of the Mass, in the oblation after the Consecration, has the words offerimus præclaræ majestati tuæ... calicem salutis perpetuæ. In these words we find the proximate source of the prayer Offerimus tibi:
Offerimus tibi, Domine, calicem salutaris, tuam deprecantes clementiam: ut, in conspectu divinæ majestatis tuæ, pro nostra et totius mundi salute cum odore suavitatis ascendat. Amen.
(We offer unto Thee, Lord, the saving chalice, beseeching Thy clemency: that it may go up with an odour of sweetness in sight of Thy Divine Majesty, for our and the whole world's salvation. Amen.)
In Exodus, and repeatedly through Leviticus and Numbers, are found phrases about offering sacrifice as a sweetest savour to the Lord, just as once the Almighty smelt the odour of sweetness arising from Noë's sacrifice and was appeased (Gen. viii, 20-21). This finds its true meaning in the New Testament, where we read that "Christ loved us and handed over Himself for us, as an Oblation and Victim to God in the odour of sweetness", in odorem suavitatis (Eph. v, 2).
As Aquinas put it in one of his hymns for Corpus Christi, Christ "gave Himself with His own Hands" when He took the chalice and made the wine within become His Blood, the price of our salvation; and having made His Apostles Priests of the New Testament, well may all the ordained say Calicem salutaris accipiam, et nomen Domini invocabo, "I will take the saving chalice, and invoke the name of the Lord" (Ps 115:4). This chalice, wherein soon again Christ's Blood will be made present, is therefore seen as lifted up, and giving forth the sweet savour whereby the Deity is propitiated: and we know that "He is the propitiation for our sins, and not for ours alone, but for the whole world's" – pro totius mundi (I John ii, 2).
Originally, this prayer was said by the deacon, not the priest, when the former brought the chalice, already filled with wine, to the altar and said this prayer of offering himself. It was conceived that the priest offered the chalice through the ministry of the deacon, hence the usual opening word Offerimus, "We offer", not the expected Offero, "I offer" (which was used in some places); in process of time, it became the custom for both to say the prayer together, the deacon touching the chalice and supporting the priest's arm; thus was kept alive in the later Use of Rome the ancient dignity of the Deacon as minister of the chalice, long after Communion under both kinds died out.
One source, an 11th century missal from Hamburg, has, by prolepsis, sanguinem Filii tui ("the Blood of Thy Son") instead of calicem salutaris; another, a Westminster missal from the late 14th century, makes this a formula for offering up both elements with the words calicem et hostiam salutaris, and changing the verb to the plural ascendant. (It is easy to spot that the words "and host" are added, since normally one would say "host and chalice", not "chalice and host".)
Since my own encounters with priests celebrating the Dominican Rite – which, like most variants of the Roman Rite, does not contain the Offerimus tibi – I have rather neglected this prayer in favour of the Suscipe sancta Trinitas in its short Dominican form, which is one of my customary private prayers at Mass (adding per manus sacerdotis tui, "by the hands of Thy priest", to make it a fit prayer for a layman). But it is too beautiful to neglect wholly.
One suggestion is that, in some future revision of the Ordinary Form, the rather grand offertory prayers of the Extraordinary Form could be made alternatives to the untraditional formulas that I call Bugnini's bastard twins – "Blessed are you... through your goodness we have this bread / wine to offer..." – which seem intended to teach 'modern [urban] man' that human hands make bread and wine from the produce of plants; at least it is acknowledged that God, the Lord of Creation, ought be blessed for His goodness in providing that this be so; the prayers also presume to state that they will become "the bread of life" and "spiritual drink", in case anyone is unsure of their purpose; at least the basic verb "to offer" is retained.
How insufferably didactic! To take benedictions of God from Jewish table prayers, and then to clumsily adapt them to Christian use at Mass by alluding to the new purpose to which the elements will be put, is a wholly untraditional approach not seen before. I think Bugnini in his prolix account of the reform of the Roman liturgy rather ingenuously states that it was realized that some new offertory prayers just had to be made up, since otherwise (horribile dictu) the older ones would simply have continued to be said...
Offerimus tibi is only slightly longer than its replacement, and is certainly more poetic, as well as being more Scriptural: I would be delighted to hear it at ordinary Mass –
We offer you, Lord, the chalice of salvation, imploring your mercy, that it may arise with a sweet savour before your divine majesty, for our salvation and that of the whole world. Amen.