Objection 1. It seems that trine immersion is essential to Baptism. For Augustine says in a sermon on the Symbol, addressed to the Neophytes: "Rightly were you dipped three times, since you were baptized in the name of the Trinity. Rightly were you dipped three times, because you were baptized in the name of Jesus Christ, Who on the third day rose again from the dead. For that thrice repeated immersion reproduces the burial of the Lord by which you were buried with Christ in Baptism." Now both seem to be essential to Baptism, namely, that in Baptism the Trinity of Persons should be signified, and that we should be conformed to Christ's burial. Therefore it seems that trine immersion is essential to Baptism.
Objection 2. Further, the sacraments derive their efficacy from Christ's mandate. But trine immersion was commanded by Christ: for Pope Pelagius II wrote to Bishop Gaudentius: "The Gospel precept given by our Lord God Himself, our Saviour Jesus Christ, admonishes us to confer the sacrament of Baptism to each one in the name of the Trinity and also with trine immersion." Therefore, just as it is essential to Baptism to call on the name of the Trinity, so is it essential to baptize by trine immersion.
Objection 3. Further, if trine immersion be not essential to Baptism, it follows that the sacrament of Baptism is conferred at the first immersion; so that if a second or third immersion be added, it seems that Baptism is conferred a second or third time. which is absurd. Therefore one immersion does not suffice for the sacrament of Baptism, and trine immersion is essential thereto.
On the contrary, Gregory wrote to the Bishop Leander: "It cannot be in any way reprehensible to baptize an infant with either a trine or a single immersion: since the Trinity can be represented in the three immersions, and the unity of the Godhead in one immersion."
I answer that As stated above (7, ad 1), washing with water is of itself required for Baptism, being essential to the sacrament: whereas the mode of washing is accidental to the sacrament. Consequently, as Gregory in the words above quoted explains, both single and trine immersion are lawful considered in themselves; since one immersion signifies the oneness of Christ's death and of the Godhead; while trine immersion signifies the three days of Christ's burial, and also the Trinity of Persons.
But for various reasons, according as the Church has ordained, one mode has been in practice, at one time, the other at another time. For since from the very earliest days of the Church some have had false notions concerning the Trinity, holding that Christ is a mere man, and that He is not called the "Son of God" or "God" except by reason of His merit, which was chiefly in His death; for this reason they did not baptize in the name of the Trinity, but in memory of Christ's death, and with one immersion. And this was condemned in the early Church. Wherefore in the Apostolic Canons (xlix) we read: "If any priest or bishop confer baptism not with the trine immersion in the one administration, but with one immersion, which baptism is said to be conferred by some in the death of the Lord, let him be deposed": for our Lord did not say, "Baptize ye in My death," but "In the name of the Father and of the Son, and of the Holy Ghost."
Later on, however, there arose the error of certain schismatics and heretics who rebaptized: as Augustine (Super. Joan., cf. De Haeres. lxix) relates of the Donatists. Wherefore, in detestation of their error, only one immersion was ordered to be made, by the (fourth) council of Toledo, in the acts of which we read: "In order to avoid the scandal of schism or the practice of heretical teaching let us hold to the single baptismal immersion."
But now that this motive has ceased, trine immersion is universally observed in Baptism: and consequently anyone baptizing otherwise would sin gravely, through not following the ritual of the Church. It would, however, be valid Baptism.
Reply to Objection 1. The Trinity acts as principal agent in Baptism. Now the likeness of the agent enters into the effect, in regard to the form and not in regard to the matter. Wherefore the Trinity is signified in Baptism by the words of the form. Nor is it essential for the Trinity to be signified by the manner in which the matter is used; although this is done to make the signification clearer.
In like manner Christ's death is sufficiently represented in the one immersion. And the three days of His burial were not necessary for our salvation, because even if He had been buried or dead for one day, this would have been enough to consummate our redemption: yet those three days were ordained unto the manifestation of the reality of His death, as stated above (Question 53, Article 2). It is therefore clear that neither on the part of the Trinity, nor on the part of Christ's Passion, is the trine immersion essential to the sacrament.
Reply to Objection 2. Pope Pelagius understood the trine immersion to be ordained by Christ in its equivalent; in the sense that Christ commanded Baptism to be conferred "in the name of the Father, and of the Son, and of the Holy Ghost." Nor can we argue from the form to the use of the matter, as stated above (ad 1).
Reply to Objection 3. As stated above (Question 64, Article 8), the intention is essential to Baptism. Consequently, one Baptism results from the intention of the Church's minister, who intends to confer one Baptism by a trine immersion. Wherefore Jerome says on Ephesians 4:5-6: "Though the Baptism," i.e. the immersion, "be thrice repeated, on account of the mystery of the Trinity, yet it is reputed as one Baptism." If, however, the intention were to confer one Baptism at each immersion together with the repetition of the words of the form, it would be a sin, in itself, because it would be a repetition of Baptism.
Source: Summa Theologica (New Advent)