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it wrong for the Priest to offer incense during the Great Doxology and if
it is, then why?
Answer to Question 4.
Mattins and the Divine Liturgy are two totally independent services and during the Doxology the Priest and Deacon say silently the closing petitions, prayers and dismissal of Mattins. They also say in preparation of the Divine Liturgy "O heavenly King and Comforter..., "Glory to God in the highest..., O Lord, open thou my lips..., O Lord, Lord, open to us the gate of Thy mercy" and the small dialogue between the Priest and Deacon if a Deacon is also taking part. This then is the modern day practice in Greek parishes, but in older times the Liturgy did not immediately follow Mattins.
The ancient practice has been preserved in Monasteries which after Mattins follows the First Hour and then the morning break. On returning to Church the Third and Sixth Hours are read and then the Divine Liturgy. The Slavonic Churches do something similar. On the evening before they have a small vigil with the services of Vespers, Mattins and the First Hour. The next day they read the Third and Sixth Hours followed by the Liturgy. During the reading of the hours, the Priest prepares for the Divine Liturgy with the preparatory prayers called "Kairon", the Vesting and the Office of Oblation (Prothesis, Proskomidi).
With the ancient practice, the closing of the Proskomidi, coincided with the Start of the Liturgy which is actually one service. On censing the Holy Altar and all the Sanctuary as we do today in parishes, the Priest also censed the Iconostasis and all the Church and people present saying the verses: "In the tomb according to the flesh..., Thy life-giving tomb is revealed to us..., Hail, O hallowed and divine dwelling place of the Highest..., and the 50th Psalm." The present day practice of censing at the Doxology is therefore not to make the singing of the Doxology more grandeur, as some Priest think, but for the start of the Divine Liturgy.
is therefore correct to cense before the start of the Liturgy and the most
opportune time is at the Doxology of Mattins so that the Priest will have
enough time to say the closing prayers, the dismissal of Mattins and the
preparatory lines for the start of the Liturgy. If the Priest does cense
at the Doxology then he shouldn't cense the Holy Altar and Sanctuary at
the closing of the Proskomidi, which is in fact the same censing. If, as
is done occasionally in some places, only the Liturgy is served then it
goes without saying that the Priest must cense the whole Church before the
start of the service.