The first person I know of who is historically attributed to stating and explaining the fact that the Moon actually reflected sunlight (as opposed to it being self-illuminating like the Sun) was a dude named Anaxagoras - an ancient Greek philosopher. He lived between 510 and 428 BC, which was a few hundred years before Vitruvius was around.

This fact was determined by Anaxagoras by him personally observing and reading/hearing of the eyewitness accounts of others who had observed solar and lunar eclipses.

The darkening visual effects on the Moon's "disc" as it temporarily passed directly or partially in front of the Sun (a "solar" eclipse), or at times when the Earth's shadow blocked the sunlight from striking the Moon temporarily (a "lunar" eclipse), caused him to rightly reason that the Moon was not a self-illuminating ball in the sky like the Sun - "a red hot stone" as he described it - but that the Moon was in fact a reflective object in the Heavens, reflecting the light from the Sun itself.

Anaxagoras proposed that the moon shines by reflected light from the "red-hot stone" which was the sun, the first such recorded claim. Showing great genius he was also then able to take the next step and become the first to explain correctly the reason for eclipses of the sun and moon. His explanation of eclipses of the sun is completely correct but he did spoil his explanation of eclipses of the moon by proposing that in addition to being caused by the shadow of the earth, there were other dark bodies between the earth and the moon which also caused eclipses of the moon. It is a little unclear why he felt it necessary to postulate the existence of these bodies but it does not detract from this major breakthrough in mathematical astronomy. There is also other evidence to suggest that Anaxagoras had applied geometry to the study of astronomy.