Final answer:
Two schools of thought emerged regarding the age of the Earth and origin of mountains: catastrophism and uniformitarianism.
Step-by-step explanation:
The two schools of thought that emerged regarding the age of the Earth and origin of mountains by the start of the 1800s were the catastrophism and uniformitarianism theories.
The catastrophism theory suggested that the Earth's geological features were created by catastrophic events occurring during a relatively brief past. This view contrasted with uniformitarianism, which argued that contemporary geological processes, such as wind and rain, were responsible for the formation of mountains and other geological features over a long period of time.
These two schools of thought were popularized by geologists like James Hutton and Charles Lyell, and their ideas had a significant influence on the development of modern geology.