The "Eureka" story about Archimedes and the bath tub was as well known in Galileo's day as it is in ours. Galileo, who was a great admirer of Archimedes and adopted many of his methods, probably read it in one of the editions of Vitruvius's The Ten Books on Architecture, which was very popular in Renaissance Europe. Supposedly, it was in the bath tub that Archimedes figured out the solution to the problem posed to him by the king of Syracuse: was a crown (or wreath) supposedly made of pure gold in fact entirely gold? He measured the amount of water displaced by the crown and by an equal weight of gold, and found that the crown displaced more water. Its specific gravity was thus less than that of gold, and therefore it had been adulterated with another metal.
Weighing precious metals in air and then in water was presumably a practice that was common among jewelers in Europe. Galileo had some ideas for refining the practice and, at the age of 22, he wrote a little tract about it, which he entitled La Bilancetta, or "The Little Balance." What Galileo described was an accurate balance for weighing things in air and water, in which the part of the arm on which the counter weight was hung was wrapped with metal wire. The amount by which the counterweight had to be moved when weighing in water could then be determined very accurately by counting the number of turns of the wire, and the proportion of, say, gold to silver in the object could be read off directly.
This little tract illustrates the mixture of the theoretical and practical that marks Galileo's science in contrast to that of most of his contemporaries.
Notes:  There are many editions of The Ten Books on Architecture. The story of Archimedes is related in the introduction to Book IX.
Sources: The tract is available in English translation in Laura Fermi and Gilberta Bernardini, Galileo and the Scientific Revolution (New York: Basic Books, 1961), pp. 133-143.
©1995 Al Van Helden