Galileo, Sidereus nuncius (Venice, 1610), photograph of Moon engravings

The Aristotelians in Italian universities taught that the shading of the Moon was internal, its surface as smooth as a marble. Early telescopic observers remarked on the “strange spottedness” of the Moon, without doubting its spherical perfection. But Galileo observed the Moon with the eye of an artist, trained in the science of perspective. 

With a multitude of diagrams Galileo taught us how to observe the lunar surface. What appears as an isolated peak one night, may become a chain of mountains the next night, or converge in a circular structure after that. He was not mapping the moon, or implying that a crater of gigantic size is present in the bottom center of the lunar face, but proving how to detect topography on the Moon’s surface, contrary to the Aristotelians. In this respect, his diagrams were intended to train his readers to think in terms of the principles of perspective drawing.

“When the Moon displays herself with brilliant horns, the boundary dividing the bright from the dark part does not form a uniformly oval line, as would happen in a perfectly shaped spherical solid, but is marked by an uneven, rough, and very sinuous line.... what causes even greater wonder is that very many bright points appear within the dark part of the Moon... gradually these are increased in size and brightness [and...] joined with the rest of the bright part.... Now, on Earth, before sunrise, aren’t the peaks of the highest mountains illuminated by the Sun’s rays while shadows still cover the plain?” Galileo, Sidereus Nuncius, trans. Albert Van Helden (University of Chicago, 1989).

One misinterprets Galileo to accuse him of exaggerating the size of the small Aristarchus crater (bottom center) accurately depicted later by Hevelius. 

Galileo’s discovery entailed more than inferring the existence of mountains on the Moon: Galileo argued that the Moon and the Earth are similar kinds of bodies. They both have mountains, seas, atmospheres, and both shine by reflected light. All this suggests that the Earth, also, is a wandering planet.

Using simple geometry, Galileo estimated the height of lunar mountains (AD) at about 4 miles.