Didache (1st Century) 7:1-4:
Now about baptism: this is how to baptize. Give public instruction on all these points, and then "baptize" in running water, "in the name of the Father and of the Son and of the Holy Spirit." 2If you do not have running water, baptize in some other. 3If you cannot in cold, then in warm. If you have neither, then pour water on the head three times "in the name of the Father, Son, and Holy Spirit." 4Before the baptism, moreover, the one who baptizes and the one being baptized must fast, and any others who can. And you must tell the one being baptized to fast for one or two days beforehand.
The Shepherd of Hermas (1st or 2nd Century), Similitude 9, Chapter 16:
“They were obliged,” he answered, “to ascend through water in order that they might be made alive; for, unless they laid aside the deadness of their life, they could not in any other way enter into the kingdom of God. Accordingly, those also who fell asleep received the seal of the Son of God. For,” he continued, “before a man bears the name of the Son of God he is dead; but when he receives the seal he lays aside his deadness, and obtains life. The seal, then, is the water: they descend into the water dead, and they arise alive. And to them, accordingly, was this seal preached, and they made use of it that they might enter into the kingdom of God.”
Justin Martyr’s Dialog with Trypho (Mid-2nd Century) Chapter 29:
What need have I of that other baptism, who have been baptized with the Holy Ghost?
Justin Martyr’s Dialog with Trypho (Mid-2nd Century) Chapter 33:
And we, who have approached God through Him, have received not carnal, but spiritual circumcision, which Enoch and those like him observed. And we have received it through baptism, since we were sinners, by God's mercy; and all men may equally obtain it.
Justin Martyr’s First Apology (Mid-2nd Century) Chapter 61:
I will also relate the manner in which we dedicated ourselves to God when we had been made new through Christ; lest, if we omit this, we seem to be unfair in the explanation we are making. As many as are persuaded and believe that what we teach and say is true, and undertake to be able to live accordingly, are instructed to pray and to entreat God with fasting, for the remission of their sins that are past, we praying and fasting with them. Then they are brought by us where there is water, and are regenerated in the same manner in which we were ourselves regenerated. For, in the name of God, the Father and Lord of the universe, and of our Saviour Jesus Christ, and of the Holy Spirit, they then receive the washing with water. For Christ also said, "Except ye be born again, ye shall not enter into the kingdom of heaven. Now, that it is impossible for those who have once been born to enter into their mothers' wombs, is manifest to all. And how those who have sinned and repent shall escape their sins, is declared by Esaias the prophet, as I wrote above; he thus speaks: "Wash you, make you clean; put away the evil of your doings from your souls; learn to do well; judge the fatherless, and plead for the widow: and come and let us reason together, saith the Lord. And though your sins be as scarlet, I will make them white like wool; and though they be as crimson, I will make them white as snow. But if ye refuse and rebel, the sword shall devour you: for the mouth of the Lord hath spoken it."
And for this [rite] we have learned from the apostles this reason. Since at our birth we were born without our own knowledge or choice, by our parents coming together, and were brought up in bad habits and wicked training; in order that we may not remain the children of necessity and of ignorance, but may become the children of choice and knowledge, and may obtain in the water the remission of sins formerly committed, there is pronounced over him who chooses to be born again, and has repented of his sins, the name of God the Father and Lord of the universe; he who leads to the layer the person that is to be washed calling him by this name alone. For no one can utter the name of the ineffable God; and if any one dare to say that there is a name, he raves with a hopeless madness. And this washing is called illumination, because they who learn these things are illuminated in their understandings. And in the name of Jesus Christ, who was crucified under Pontius Pilate, and in the name of the Holy Ghost, who through the prophets foretold all things about Jesus, he who is illuminated is washed.
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Find more of what the early Christians thought on my Christian History page!