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Propitiation (from Latin propitiāre, "to appease;" from propitius, "gracious"), also called expiation, is the act of appeasing or making well disposed a deity, thus incurring divine favor to avoid divine retribution.
Propitiation is translated from the Greek hilasterion, meaning "that which expiates or propitiates" or "the gift which procures propitiation". "1 John 2:2 (KJV) And he is the propitiation for our sins: and not for ours only, but also for the sins of the whole world." The word is also used in the New Testament for the place of propitiation, the "mercy seat". Hebrews 9:5. There is frequent similar use of hilasterion in the Septuagint, Exodus 25:18 ff. The mercy seat was sprinkled with atoning blood on the Day of Atonement (Leviticus 16:14), representing that the righteous sentence of the Law had been executed, changing a judgment seat into a mercy seat (Hebrews 9:11–15; compare with "throne of grace" in Hebrews 4:14–16; place of communion, Exodus 25:21–22).
Another Greek word, hilasmos, is used for Christ as our propitiation. 1 John 2:2; 4:10, and for "atonement" in the Septuagint (Leviticus 25:9). The thought in the Old Testament sacrifices and in the New Testament fulfillment, is that Christ completely satisfied the just demands of our Holy Father for judgment on sin, by his death on the Cross of Calvary. (Hebrews 7:26-28)
God, in view of the cross, is declared righteous in having been able to justify sins in the Old Testament period, as well as in being able to forgive sinners under the New Covenant (Romans 3:25,26; cf. Exodus 29:33, note). 
The case for translating hilasterion as "expiation" instead of "propitiation" was put forward by C. H. Dodd in 1935 and at first gained wide support. As a result, hilasterion has been translated as "expiation" in the RSV and other modern versions. Dodd argued that in pagan Greek the translation of hilasterion was indeed to propitiate, but that in the Septuagint (the oldest Greek translation of the Hebrew Old Testament) that kapporeth (Hebrew for "atone") is often translated with words that mean "to cleanse or remove" (Dodd, "The Bible and the Greeks", p 93). This view was challenged by Leon Morris who argued that because of the focus in the book of Romans on God's wrath, that the concept of hilasterion needed to include the appeasement of God's wrath (Morris, Apostolic Preaching of the Cross, p 155). Dodd's study is also criticized by David Hill in his detailed semantic study of hilasterion, in the book Greek Words and Hebrew Meanings: Studies in the Semantics of Soteriological Terms. Hill claims that Dodd leaves out several Septuagint references to propitiation, and also cites apocryphal sources.
Theologians stress the idea of propitiation because it specifically addresses the aspect of the Atonement dealing with God's wrath. Critics of penal substitutionary atonement state that seeing the Atonement as appeasing God is a pagan idea that makes God seem tyrannical (See for example, Stricken by God?, ed. Brad Jersak, Eerdmans: 2007 or Be Ye Reconciled by Paul Peter Waldenstrom).
J.I. Packer in "Knowing God" designates a distinct difference between pagan and Christian propitiation: "In paganism, man propitiates his gods, and religion becomes a form of commercialism and, indeed, of bribery. In Christianity, however, God propitiates his wrath by his own action. He set forth Jesus Christ... to be the propitiation of our sins." 
John Stott writes that propitiation "does not make God gracious...God does not love us because Christ died for us, Christ died for us because God loves us" (The Cross of Christ, p 174). John Calvin, quoting Augustine from John's Gospel cx.6, writes, "Our being reconciled by the death of Christ must not be understood as if the Son reconciled us, in order that the Father, then hating, might begin to love us" (Institutes, II:16:4). Continuing the quote: "... but that we were reconciled to him already, loving, though at enmity with us because of sin. To the truth of both propositions we have the attestation of the Apostle, 'God commendeth his love toward us, in that while we were yet sinners, Christ died for us,' (Rom. 5: 8.) Therefore he had this love towards us even when, exercising enmity towards him, we were the workers of iniquity. Accordingly in a manner wondrous and divine, he loved even when he hated us."  See http://www.spurgeon.org/~phil/calvin/bk2ch16.html#four.htm
Packer also cites God's love as the impetus that provides Christ's sacrifice for the atonement of mankind and hence the removal of God's wrath. According to Packer, propitiation and the wrath of God that propitiation implies is necessary to properly define God's love; God could not be righteous and "His love would degenerate into sentimentality (without Christ's atonement containing aspects of propitiation).The wrath of God is as personal, and as potent, as his Love."
Thus the definition of Christian propitiation asserted by Calvin, Packer and Murray holds that within God there is a dichotomy of love and anger, but through propitiation love trumps anger, abolishing it. "'The doctrine of the propitiation is precisely this that God loved the objects of His wrath so much that He gave His own Son to the end that He by His blood should make provision for the removal of this wrath... (John Murray, The Atonement, p.15)'"