Objection 1. It seems that this is not a suitable form of Baptism: "I baptize thee in the name of the Father, and of the Son, and of the Holy Ghost." For action should be ascribed to the principal agent rather than to the minister. Now the minister of a sacrament acts as an instrument, as stated above (Question 64, Article 1); while the principal agent in Baptism is Christ, according to John 1:33, "He upon Whom thou shalt see the Spirit descending and remaining upon Him, He it is that baptizeth." It is therefore unbecoming for the minister to say, "I baptize thee": the more so that "Ego" [I] is understood in the word "baptizo" [I baptize, so that it seems redundant.
Objection 2. Further, there is no need for a man who does an action, to make mention of the action done; thus he who teaches, need not say, "I teach you." Now our Lord gave at the same time the precepts both of baptizing and of teaching, when He said (Matthew 28:19): "Going, teach ye all nations," etc. Therefore there is no need in the form of Baptism to mention the action of baptizing.
Objection 3. Further, the person baptized sometimes does not understand the words; for instance, if he be deaf, or a child. But it is useless to address such a one; according to Sirach 32:6: "Where there is no hearing, pour not out words." Therefore it is unfitting to address the person baptized with these words: "I baptize thee."
Objection 4. Further, it may happen that several are baptized by several at the same time; thus the apostles on one day baptized three thousand, and on another, five thousand (Acts 2, 4). Therefore the form of Baptism should not be limited to the singular number in the words, "I baptize thee": but one should be able to say, "We baptize you."
Objection 5. Further, Baptism derives its power from Christ's Passion. But Baptism is sanctified by the form. Therefore it seems that Christ's Passion should be mentioned in the form of Baptism.
Objection 6. Further, a name signifies a thing's property. But there are three Personal Properties of the Divine Persons, as stated in I, 32, 3. Therefore we should not say, "in the name," but "in the names of the Father, and of the Son, and of the Holy Ghost."
Objection 7. Further, the Person of the Father is designated not only by the name Father, but also by that of "Unbegotten and Begetter"; and the Son by those of "Word," "Image," and "Begotten"; and the Holy Ghost by those of "Gift," "Love," and the "Proceeding One." Therefore it seems that Baptism is valid if conferred in these names.
On the contrary, our Lord said (Matthew 28:19): "Going... teach ye all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost."
I answer that, Baptism receives its consecration from its form, according to Ephesians 5:26: "Cleansing it by the laver of water in the word of life." And Augustine says (De Unico Baptismo iv) that "Baptism is consecrated by the words of the Gospel." Consequently the cause of Baptism needs to be expressed in the baptismal form. Now this cause is twofold; the principal cause from which it derives its virtue, and this is the Blessed Trinity; and the instrumental cause, viz. the minister who confers the sacrament outwardly. Wherefore both causes should be expressed in the form of Baptism. Now the minister is designated by the words, "I baptize thee"; and the principal cause in the words, "in the name of the Father, and of the Son, and of the Holy Ghost." Therefore this is the suitable form of Baptism: "I baptize thee in the name of the Father, and of the Son, and of the Holy Ghost."
Reply to Objection 1. Action is attributed to an instrument as to the immediate agent; but to the principal agent inasmuch as the instrument acts in virtue thereof. Consequently it is fitting that in the baptismal form the minister should be mentioned as performing the act of baptizing, in the words, "I baptize thee"; indeed, our Lord attributed to the ministers the act of baptizing, when He said: "Baptizing them," etc. But the principal cause is indicated as conferring the sacrament by His own power, in the words, "in the name of the Father, and of the Son, and of the Holy Ghost": for Christ does not baptize without the Father and the Holy Ghost.
The Greeks, however, do not attribute the act of baptizing to the minister, in order to avoid the error of those who in the past ascribed the baptismal power to the baptizers, saying (1 Corinthians 1:12): "I am of Paul... and I of Cephas." Wherefore they use the form: "May the servant of Christ, N..., be baptized, in the name of the Father," etc. And since the action performed by the minister is expressed with the invocation of the Trinity, the sacrament is validly conferred. As to the addition of "Ego" in our form, it is not essential; but it is added in order to lay greater stress on the intention.
Reply to Objection 2. Since a man may be washed with water for several reasons, the purpose for which it is done must be expressed by the words of the form. And this is not done by saying: "In the name of the Father, and of the Son, and of the Holy Ghost"; because we are bound to do all things in that Name (Colossians 3:17). Wherefore unless the act of baptizing be expressed, either as we do, or as the Greeks do, the sacrament is not valid; according to the decretal of Alexander III: "If anyone dip a child thrice in the water in the name of the Father, and of the Son, and of the Holy Ghost, Amen, without saying, I baptize thee in the name of the Father, and of the Son, and of the Holy Ghost, Amen, the child is not baptized."
Reply to Objection 3. The words which are uttered in the sacramental forms, are said not merely for the purpose of signification, but also for the purpose of efficiency, inasmuch as they derive efficacy from that Word, by Whom "all things were made." Consequently they are becomingly addressed not only to men, but also to insensible creatures; for instance, when we say: "I exorcize thee, creature salt" (Roman Ritual).
Reply to Objection 4. Several cannot baptize one at the same time: because an action is multiplied according to the number of the agents, if it be done perfectly by each. So that if two were to combine, of whom one were mute, and unable to utter the words, and the other were without hands, and unable to perform the action, they could not both baptize at the same time, one saying the words and the other performing the action.
On the other hand, in a case of necessity, several could be baptized at the same time; for no single one of them would receive more than one baptism. But it would be necessary, in that case, to say: "I baptize ye." Nor would this be a change of form, because "ye" is the same as "thee and thee." Whereas "we" does not mean "I and I," but "I and thou"; so that this would be a change of form.
Likewise it would be a change of form to say, "I baptize myself": consequently no one can baptize himself. For this reason did Christ choose to be baptized by John (Extra, De Baptismo et ejus effectu, cap. Debitum).
Reply to Objection 5. Although Christ's Passion is the principal cause as compared to the minister, yet it is an instrumental cause as compared to the Blessed Trinity. For this reason the Trinity is mentioned rather than Christ's Passion.
Reply to Objection 6. Although there are three personal names of the three Persons, there is but one essential name. Now the Divine power which works in Baptism, pertains to the Essence; and therefore we say, "in the name," and not, "in the names."
Reply to Objection 7. Just as water is used in Baptism, because it is more commonly employed in washing, so for the purpose of designating the three Persons, in the form of Baptism, those names are chosen, which are generally used, in a particular language, to signify the Persons. Nor is the sacrament valid if conferred in any other names.
Source: Summa Theologica (New Advent)