"Levigate" comes from Latin "levigatus," the past participle of the verb "levigare" ("to make smooth"). "Levigare" is derived in part from "levis," the Latin word for "smooth." "Alleviate" and "levity" can also be traced back to a Latin "levis," and the "levi-" root in both words might suggest a close relationship with "levigate." This is not the case, however. The Latin "levis" that gives us "alleviate" and "levity" does not mean "smooth," but "light" (in the sense of having little weight). One possible relative of "levigate" in English is "oblivion," which comes from the Latin "oblivisci" ("to forget"), a word which may be a combination of "ob-" ("in the way") and the "levis" that means "smooth."
The idea of polishing brought to mind the thought of a mirror. Mirrors are polished to make the reflection clearer and more accurate. We were created in the image of G-d to reflect His glory (Genesis 1:26-27). It seems as though He polishes our lives and removes the dust and smudges to make us better reflect who He is. The very first mention of mirrors in Scripture is found in Exodus 38:8 when Moshe is making the furniture for the tabernacle:
Moreover, he made the laver of bronze with its base of bronze, from the mirrors of the serving women who served at the doorway of the tent of meeting.
The midieval French commentator Rashi provides some explanation:
During the difficult times of the Egyptian bondage the Israelite men separated from their wives. They believed it was improper to bring children into a world of suffering. Their wives disagreed. They faithfully believed that G-d would soon bring an end to their suffering. It was best in their opinion to have children and preserve the continuity of the Israelite people. They used these mirrors to beautify themselves before their husbands in an attempt to persuade them to have children. Their arguments and actions prevailed and they succeeded in raising a new generation.
As instruments of faith these mirrors were incorporated into the order of worship that G-d ordained for His people. Whenever the priests would bend over to wash their hands and their feet during their service in the temple they would see the reflection of the sky and be reminded of the One Who is over all mankind. They were also reminded of the great faith of their ancestresses and be encouraged to walk in that great degree of faith (2 Cor 5:7).
This word brought to mind a passage from Luke:
And he [John] came into all the district around the Jordan, preaching a baptism of repentance for the forgiveness of sins; as it is written in the book of the words of Isaiah the prophet, "THE VOICE OF ONE CRYING IN THE WILDERNESS, 'MAKE READY THE WAY OF THE LORD, MAKE HIS PATHS STRAIGHT. 'EVERY RAVINE WILL BE FILLED, AND EVERY MOUNTAIN AND HILL WILL BE BROUGHT LOW; THE CROOKED WILL BECOME STRAIGHT, AND THE ROUGH ROADS SMOOTH; AND ALL FLESH WILL SEE THE SALVATION OF GOD.'"
If this passage were spoken in Hebrew or Aramaic then the concluding phrase would be "AND ALL FLESH WILL SEE THE YESHUA OF G-D". It is a play on words... a pun. "Yeshua" is the Hebrew name that G-d gave to Joseph in Matthew 1:21. Yeshua means "salvation" in Hebrew. John is quoting from the Septuagint and declaring that all humanity will see the Yeshua of G-d.
May that day spoken of in Isaiah 45:23 when "every knee shall bow, every tongue shall swear" come to pass speedily and in our days. Amen.