The woman caught in adultery
Absent from the earliest manuscripts and from nearly all early commentators (Didymus the Blind is the known exception); received in the committed text, it is a monument of mercy and a case study in canon and text.
John 7:53-8:11
53Everyone went to his own house,
1but Jesus went to the Mount of Olives.
2Now very early in the morning, he came again into the temple, and all the people came to him. He sat down and taught them.
3The scribes and the Pharisees brought a woman taken in adultery. Having set her in the middle,
4they told him, “Teacher, we found this woman in adultery, in the very act.
5Now in our law, Moses commanded us to stone such women. What then do you say about her?”
6They said this testing him, that they might have something to accuse him of. But Jesus stooped down and wrote on the ground with his finger.
7But when they continued asking him, he looked up and said to them, “He who is without sin among you, let him throw the first stone at her.”
8Again he stooped down and wrote on the ground with his finger.
9They, when they heard it, being convicted by their conscience, went out one by one, beginning from the oldest, even to the last. Jesus was left alone with the woman where she was, in the middle.
10Jesus, standing up, saw her and said, “Woman, where are your accusers? Did no one condemn you?”
11She said, “No one, Lord.” Jesus said, “Neither do I condemn you. Go your way. From now on, sin no more.”
What each tradition says
Rabbinic / Midrashic
The Jewish Encyclopedia suggests that Jesus' words regarding the adulteress, "He that is without sin among you, let him first cast a stone at her," may merely echo a rabbinical saying. That saying states that only when the husband is without sin will the ordeal of a wife suspected of adultery prove effective.
Executive Committee of the Editorial Board, in Isidore Singer (ed.), The Jewish Encyclopedia, s.v. 'New Testament' (New York: Funk & Wagnalls, 1901-1906) — New Testament, section on the sayings of Jesus (midrashic origins)
Patristic
Augustine says Jesus' words, "Let him who is without sin cast the first stone," are an answer of Wisdom that sends the accusers back to themselves. They stood outwardly to accuse and censure, but did not examine themselves inwardly: they saw the adulteress, not themselves.
Augustine of Hippo (354-430), Lectures or Tractates on the Gospel According to St. John (trans. John Gibb and James Innes), in Nicene and Post-Nicene Fathers, ser. 1, vol. 7 (Buffalo: Christian Literature Publishing Co., 1888) — Tractate XXXIII, §5 (on John 8:7)
Catholic
Reid recounts that Cajetan approvingly quoted Jerome’s unfavorable comment on Mark 16:9-20, and Erasmus rejected the section on the Adulterous Woman as unauthentic. At Trent, however, no doubt of authenticity was expressed; only the manner of reception was questioned. These portions were received like the deuterocanonical books, without the slightest distinction.
George J. Reid, The Catholic Encyclopedia, vol. 3, s.v. 'Canon of the New Testament' (New York: Robert Appleton Company, 1908) — Article, on the deuterocanonical parts received at Trent
Orthodox
No sourced position found. Checked: John of Damascus, Exposition of the Orthodox Faith (NPNF 2-09 — no comment); Chrysostom's Homilies on John (NPNF 1-14 — Homily LII on 7:45-46 passes straight to Homily LIII on 8:20, skipping the pericope, so he is evidence of its absence, not a reading of it); the 1672 Synod of Jerusalem (full text — silent); Apostolic Constitutions (ANF 7 — no citation); Didymus the Blind, the one early commentator who cites it, survives only in a modern critical edition (Tura papyrus, 1941), not public domain.
Reformed / Protestant
Calvin notes the passage was anciently unknown to the Greek Churches, and some conjecture it was brought from elsewhere and inserted here. Yet because the Latin Churches always received it, many old Greek manuscripts contain it, and it has nothing unworthy of an Apostolic Spirit, he sees no reason to refuse applying it for our advantage.
John Calvin (1509-1564), Commentary on the Gospel According to John, vol. 1 (trans. William Pringle, Calvin Translation Society, 1847) — Commentary on John 8:1-11, on verse 3
Evangelical
Barnes notes that many critics have thought this passage, along with the last verse of the preceding chapter, to be spurious. He reports that it is wanting in many ancient manuscripts and versions and was rejected by Erasmus, Calvin, Beza, Grotius, Wetstein, Tittman, Knapp, and many others.
Albert Barnes (1798-1870), Notes, Explanatory and Practical, on the New Testament: John (New York: Harper & Brothers), on John 8:11 — Notes on John 8:11 (closing remark on the passage)
Evangelical
Wesley explains that Jesus’ words “Neither do I condemn thee” mean he does not take upon himself to pass any such sentence. He adds that this deliverance should lead thee to repentance.
John Wesley (1703-1791), Explanatory Notes upon the New Testament (1755), on John 8 — Note on John 8:11