The key to understanding the first line of Donne's poem is to notice the word "as" that begins it. "As," in this case, is the beginning of a simile in which the speaker of the poem compares virtuous men dying to his departing from his beloved. The simile is not completed until the second stanza "So let us melt and make no noise" when the speaker refers to the two lovers who must depart from one another.
From the title we know that mourning will be forbidden. Donne begins the poem with a situation in which people often mourn--the death of a virtuous man. Ironically, however, Donne uses this situation to show that mourning is out of place in such a situation, because virtuous men pass "mildly away." They do not fear death because they are aware that their deaths are not an end but a beginning. They have faith that their souls will go to heaven; consequently their deaths are peaceful, even imperceptible to those around them.
This is the way the speaker wants to part from his beloved--so imperceptibly that no observers--"the laity"--are aware that anything is amiss. Tears and sighs show a lack of faith in their love for each other. As the speaker will continue to point out, their love does not depend on physical contact, but is rather a spiritual connection that is not weakened by physical separation. Because they have faith in each other and the strength of their love, (as the virtuous men have faith that their souls will go to heaven), they need not mourn when the speaker must depart. Religious men afraid of dying would be hypocrites. If they expressed their fear of dying on their death beds by crying and and making a scene they would be defiling their faith. Similarly, the two lovers would be "desecrating" their love--defiling it, cheapening it--if they mourned when they had to be apart from one another.
To answer your question then, the first line is the metaphorical part of the simile which has its literal reference in the second stanza. Good luck with the rest of the poem.
All it means by itself is that something happens while good men die. The word "As" can be troublesome because without some context we don't know if it connotes "while" or "when" or "because" or "just as." So it's not good to try to separate one small part of a sentence from its context and try to interpret it on its own. Here is the entire sentence:
As virtuous men pass mildly away,
And whisper to their souls to go,
Whilst some of their sad friends do say, The
breath goes now, and some say, No:
So let us melt, and make no noise,
No tear-floods, nor sigh-tempests move;
'Twere profanation of our joys
To tell the laity our love.
What Donne is saying is this: Just as people who have led good and virtuous lives have no fear of death and quietly pass away without fearful cries, then so should we part from each other; it would profane (or debase or tarnish) the joys of our love to let other people see how much we'll miss each other.
He didn't want his wife to cry and carry on when he left on his trip to France and Germany, so he wrote her this poem. It seems strange to us, but Donne considered the love between husband and wife to be similar to the love of God for humanity and vice versa. Uncontrolled human emotion somehow cheapens that love.
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This poem's meaning is in its title: A Valediction: Forbidding Mourning. Vale= Latin for "farewell" and Diction= "speaking". Thus, "A speaking farewell but forbidding mourning". Knowing this, the first line of the poem makes so much more sense. In this first stanza, Donne writes of a funeral. Virtuous men pass mildly away into the next life because they do not fear what is on the other side. They are virtuous, true, and God-fearing and well-behaved. In fact, they pass away so mildly that the people in the room argue about the exact moment of passing--"Whilst some of their sad friends do say, The breath goes now, and some say, No".
So, as they leave without fear of the future, so must the speaker of the poem and his wife leave one another without fear of their future.
In the remainder of the poem, he assures his wife that they are not the normal lovers--they are connected in more ways than just the physical. They are connected mentally, emotionally, spiritually and physically. Therefore, a brief absence will not cause their relationship to be damaged in any way, but only make it stronger.
Two metaphorical conceits he uses to compare their connection is that of a compass--she is the fixed foot and he the part with the pencil which travels out and returns to her after completing his circle back to the one who completes him.
The other is of gold--a malleable metal that only thins as it is beaten, never breaking