Here is the first quote
Going to Church.
Woman and man are to go to church decently attired, with natural step, embracing silence, possessing unfeigned love, pure in body, fit to pray to God. Let the woman observe this, further. Let her be entirely covered, unless she happen to be at home. For that style of dress is grave, and protects from being gazed at. And she will never fall, who puts before her eyes modesty, and her shawl; nor will she invite another to fall into sin by uncovering her face. For this is the wish of the Word, since it is becoming for her to pray veiled.
They say that the wife of AEneas, through excess of propriety, did not, even in her terror at the capture of Troy, uncover herself; but, though fleeing from the conflagration, remained veiled.
What is the context of this quote?
Why entirely covered?
1) it is appropriately "grave", (i.e., not frivolous), and
2) it is modest.
Modesty, in Clement's view, is linked to sexual sin. Covering a woman's face will protect her from falling into sin (the sin of pride? the sin of tempting others?), and it will protect others from sinning by looking at her face (presumably because of her beauty).
On what authority?
Clement then claims all this what Scripture says. He pulls out a positive affirmation based on 1 Corinthians 11:13:
First, nothing in the biblical passage talks about covering a woman's face.
Second, while Clement uses 1 Corinthians 11:10 to (wrongly) argue covering prevents lust (see below), he completely ignores the main reasons Paul presents in the biblical passage.
In fact, Clement's second appeal to an authority strongly suggests that his understanding of head covering owes a lot to pagan culture.
What do we do with this quote?
So, what use is this quote to advocates of the head covering tradition today?
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Here is the second quote
CEcumenius from Book III. On 1 Cor. xi. 10
https://www.newadvent.org/fathers/0211.htm
Sadly, based on the first quote, it seems to be his main take away from the whole passage.
What do we do with this quote?
Once again, we can use this quote to maintain that the head covering tradition was alive during Clement's time, but we can't say that it was particularly alive and well.