Final answer:
Prospectors predominantly found gold by panning in rivers and using sluice boxes to separate it from sediment, and they used mercury to collect fine gold particles.
Step-by-step explanation:
The two predominant ways prospectors tried to find gold during the gold rushes of the 19th century were by panning in rivers and streams and using sluice boxes. Panning was a method where gold seekers would fill a pan with river sediment and water, swirling it to allow the gold, being denser, to sink to the bottom. Sluice boxes worked on a similar principle of density separation, but on a larger scale, allowing for more material to be processed more efficiently. Large operations eventually gave way to hydraulic mining and other more invasive methods, but these two methods were the mainstays for individual prospectors.
Prospectors during this time also found techniques to recover finer particles of gold. They added mercury to their pans to form an amalgam with gold, which allowed for easier collection of even fine gold particulates. However, this practice led to environmental contamination as mercury was lost to the waterways.