Eighteenth-century astronomer william herschel first began estimating the size of what would later be known as the milky way galaxy by

By counting stars.

Explanation:

During the late eighteenth century, there was an astronomer - William Herschel who tried to estimate the size as well as shape of the galaxy by on the sky. It was the time when astronomers didn't even know that there exist some galaxies outside the milky way. He counted all the stars and created a map of the galaxy.

However, in that attempt he didn't realize that stars he was counting were not clear from his view. With that attempt of incorrect nature, he concluded his work as, the galaxy is  disk-shaped, flat and contains sun at the center. Today, when the astronomers got a perfect idea of the vastness of the universe and galaxies, we are in a position to say that Herschel's idea of the galaxies was very tiny and underestimated.