The following article is taken from The Encyclopedia of Religion and Ethics, (1964) page 380 ff.

The main Scripture passages concerned are Mt 28:19, Mk 16:16, and John 3:5, of which Mt 28:19 is the central piece of evidence for the traditional view of the institution of baptism by Christ. It describes the risen Lord as saying to his Disciples, 'Go ye and make disciples of all the nations, baptizing them into the name of the Father, and of the Son, and of the Holy Spirit.' If it were undisputed, this would, of course, be decisive, but its trustworthiness is impugned on the grounds of textual criticism, literary criticism, and historical criticism.

"The facts are in summary that Eusebius quotes Mt 28:19 twenty-one times, either omitting everything between [please forgive me for taking the liberty to translate. For those of us who do not read Greek, it would be not only meaningless, but difficult for me to try and type it from the symbols, therefore I am going to go by an Interlinear to provide the interpretation] "nations" and "teaching" or in the form "Go ye and make disciples of all the nations in my name, teaching them...", the latter form being the more frequent. He also quotes it four times in the ordinary text; but it is significant that these four quotes are all in the latter writings of Eusebius [once in the Syriac Theophany, lv. 8 (Lee's tr. P.223) once in contra Marcellum, p. 30, once in the de Ecclesiastica Theologia, v. p. 174a, and once in the letter of Eusebius to the Church at Caesarea quoted by Socrates, H E i. 8.38; it should be noted that there is reason to think that the Syriac translator is giving, not the text of Eusebius, but the version to which he is accustomed (cf. Burkitt, Evangelion da Mepharreshe, ii. 171), and that the authorship of the contra Marcelluan and the de Ecclesiastica Theologia is doubtful (cf. Conybeare, NZTW, 1905, p. 250 ff., and a reply by Gerhard Loeschocke, ib. 1906, p. 69 ff.)] At first sight this evidence seems to prove that Eusebius, in his earlier writings at all events used MSS of the Gospels which omitted the command to baptize in Mt 28:19, but Riggenbach ('Die trinitarische Taaufbefehl, Beitrage zur Fordsrung christl. Theol. 1903) and Chase (JThSt, 1905, p.481 ff.) have argues that his method of quotation is due to the influence of the arcani disciplina. This suggestion does not seem to bear examination, for the quotations in Eusebius are not found in works intended for unbelievers or for catechumens. The most reasonable view seems to be that Conybeare has shown that the quotations in Eusebius point to a text which omitted the baptismal formula, though it is still open to question whether Eusebius knew also the traditional form. It is naturally important to ask whether there is any other evidence for the 'Eusebian' type of text. Conybeare thinks that he can see traces of it in Justin Martyr, Dial. Xxxix. 258, and liii. 272, and in Hermas, Simil. ix. 17.4; but none of these passages is convincing, and perhaps more striking than any of them is a passage in which Justin gives a description of the regeneration of Christian converts in connection with baptism (Apol. I. 61). Here he quotes a saying of Christ ('Except ye be born again ye shall not enter into the kingdom of heaven') as a proof of the necessity of regeneration, but falls back upon the use of Isaiah and Apostolic tradition to justify the practice of baptism and the use of the triune formula. This certainly suggests that Justin did not know the traditional text of Mt 28:19."

Whether the 'Eusebian' text, if its existence be granted, has any real claim to be regarded as a serious rival to the traditional form, is a wholly different question. The answer depends on the view taken of the general problem of textual criticism. If a high value be attached to the existing MSS of the NT, the traditional text, though no longer unassailed, must be accepted. But if it be thought (as many critics think) that no MSS represents more than a comparatively late recensions of the text, it is necessary to set against the mass of manuscript evidence the influence of baptismal practice. It seems easier to believe that the traditional text was brought about by the influence working on the 'Eusebian' text, than that the latter arose out of the former in spite of it.

"The accounts which we possess are Mt 28:18-20, Mk 16:15-18, Lk 24:44-49, and perhaps Jn 20:21-23. Of these Mk 16:15-18 is generally considered to be a patchwork composition based on Matthew and Luke. If this be so, it affords evidence that at the time when it was written baptism was connected with the preaching of the gospel. It does not support the triune formula, but rather the 'Eusebian' text (cf. "in my name" 16:17) and it is easy to think that the reference to baptism was derived from from contemporary usage as from Mt 28:19. Lk 24:47 is more closely allied with the Eusebian than with the traditional text, and both this passage and Jn 20:21-23 suggest that the earliest form of the tradition as to the Lord's parting words to the disciples said nothing about baptism. It may be argued that the idea of repentance and forgiveness of sin was for early Christians so closely connected with that of baptism that one implies the other. But this is not the point. It is probable that baptism and the preaching of the gospel went hand in hand from the beginning. The question is whether this was due to the direct association in the 'parting words' of the Lord, or to other causes. The evidence of Mt 28:19, if the traditional text is sound, point to the former alternative; but the Third and the Fourth Gospels suggest that the earliest tradition knew only of a command to preach the gospel of repentance for the forgiveness of sins. In the case of the Third gospel this argument is especially strong. Either Luke knew of the commission to baptize (whither in the triune name or not) and omitted it, or he did not know it. It seems impossible to find any reason why he should have omitted it. At first sight this argument hold equally good for the Fourth Gospel, but it is not nearly so strong, as the writer has not unreasonably been thought to show a tendency to omit the material side of the sacramental rites of early Christianity, because of a tendency to over-emphasis its importance. Hence he omits the institution of the Eucharist. So that his omission to connect baptism with the forgiveness of sins in Jn 20:23 is not so strong as the similar omission by Luke."

On the whole, then, the evidence of literary criticism is against the historical character of the traditional text of Mt 28:19.

"Christian baptism, when connected with the mention of a formula, is alluded to four times in the Acts (2:38, 8:16, 10:48, 19:5) and the formula is never that of Mt 28:19, but is twice as "in the name of Jesus Christ" (2:38, 10:48) and twice as "in the name of the Lord Jesus" (8:16, 19:5). That this was the usual formula of Christian baptism is supported by the evidence of the Pauline Epistles, which speak of being baptized only "to Christ" (Gal. 3:27) or "into Christ Jesus" (Rom. 6:3). Is it possible to reconcile these facts with the belief that Christ commanded the disciples to baptize into the triune name? The obvious explanation of the silence of the NT on the triune name, and the use of another formula in Acts and Paul, is that this other formula was earlier, and that the triune formula is a latter addition. It would require very strong argument to controvert this presumption, and none seems to exist ( a statement of curious attempts, ancient and modern, is given in 'Baptism' in Hastings' DB, vol. i, by Dr. A. Plummer).

The cumulative evidence of these three lines of criticism is thus distinctly against the view that Mt 28:19 represents the ipissima verba of Christ in instituting baptism. If this be so, it is plain that neither Mk 16:16 nor Jn 3:5 can prove the institution. Mk 16:16 has been incidentally dealt with; Jn 3:5 is more difficult. Doubts have been cast on the text of this verse, so far as the reference to water is concerned, but for the present it is enough to point out that, even if the reference to baptism be undisputed, it does not follow that it implies the institution of baptism by Christ; it rather suggest a practice which he found to be existing and accepted. It is also necessary to remember that the present position of the criticism of the Fourth Gospel no one can confidently build on historical statements which are found only in that document.

The case against the indirect evidence in support of the traditional view is less convincing. The position in defense of that view is that, even if the evidence of Acts be admitted to prove that baptism in the triune name was not instituted by Christ, it shows that from the beginning it was unquestioningly practiced by all Christians, and it is urged that this is unwarranted, and that some of the evidence of the Epistles, properly regarded, tells against rather than for the traditional view. The crucial passage is 1 Cor. 1:14-17;

It is urged with great force that Paul could not possibly have written this if Christ had given the definite command to baptize, related to Mt 28:19. It is possible to argue that Paul is speaking of himself, not of the other disciples; but this introduces a limitation into the commission to baptize which cannot be supported, and is also contrary to the constant claim of Paul that he has the Apostolic commission as fully as any of the other apostles.