Does the Didache reflect that the early church allowed the pouring of water over the head three times as a legitimate alternative when immersion wasn't possible? I'm certain that many people will try to use this as a argument that sprinkling is an acceptable form of baptism, but it seems pretty clear that it was accepted only in circumstances when immersion was not possible. I just want to confirm that I am reading it correctly. Thanks.
The Didache, or the Teachings of the Apostles, is believed to be second century writing. It is not about of the New Testament and has no authority. It can, however, be used to trace the history of digression. So, yes, it does show that baptism was being modified soon after the first century. This does not mean that pouring is an acceptable substitute for baptism.
Do keep in mind that while the document was mentioned by other early Christian writers, we don't have a complete old copy of the document.
Only one Greek text of the Didache has survived. It is the Jerusalem Codex discovered by Byrennios in 1873, and published by him in Constantinople ten years later. It was written by a scribe, Leo, in 1056. A photographic facsimile was published by J. Rendel Harris in 1887.
Two papyrus fragments of the Didache in Greek (chs. 1:3, 4 and 2:7 to 3:2) were edited by A. S. Hunt in Oxyrhynchus Papyri, 15, London, 1922, pp. 1215.
The Greek texts of the Epistle of Barnabas (chs. 18 to 20) and of the Apostolic Church Order (chs. 1 to 13) contain the "Two Ways" material in different forms. In the latter case there are many additions, and dependence on the "Two Ways" breaks off at the equivalent of Did. 4:8. The Greek text of the Apostolic Constitutions (ch. 7:132) contains almost the whole of the Didache with a number of changes and many insertions.
In Syriac there are citations from the Didache in the Didascalia, edited by R. H. Connolly, Oxford University Press, London, 1929.
In Latin there is a third century translation of the "Two Ways." A fragment was published by B. Pez in 1723. The complete text was edited from an eleventh century manuscript by J. Schlecht, Doctrina XII Apostolorum, Freiburg, 1900.
In Coptic there is a fifth century papyrus fragment of chs. 10:3b to 12:2a, edited by G. Horner in The Journal of Theological Studies, 25, 1924, pp. 225231. (It is notable for adding after the Eucharistic prayer a thanksgiving for myron, holy oil for confirmation.)
In Ethiopic the following parts of the Didache have been preserved in the Ecclesiastical Canons: chs. II:35, 711, 12; 12:15; 13:1, 37; 8:1, 2a, in that order. They are edited by G. Horner, Statutes of the Apostles, pp. 193, 194, London, 1904.
In Georgian there is a complete translation made in the fifth century by a scribe, Jeremias of Orhai. The variant readings were published by G. Peradze in Zeitschrift für die neutestamentliche Wissenschaft, pp. 111116, 1932, from a copy of an eleventh century manuscript in Constantinople.
Notice that while the section called "The Two Ways" (chapters 1-6) was popular, chapter 7 has less evidence. It is hard to prove that later beliefs where not introduced into the document. There are scholars who believe that what is now called the Didache was pieced together from several sources.