The Recovery of Archimedes' Method
In 1899, at the monastery of the Holy Sepulcher at Jerusalem, a 10th-century palimpset was discovered.
The parchment contained works of Archimedes that, sometime between the 12th and 14th centuries, had been partially erased and overwritten by religious text.
It included a text of the Method, a work of Archimedes previously thought lost.
Archimedes used his knowledge of levers and centers of gravity to envision ways of balancing various geometric figures against one another.
This allowed him to compare their areas or volumes.
Archimedes then used mathematics to rigorously prove the results of his Method investigations.
"... certain things first became clear to me by a mechanical method, although they had to be proved by geometry afterwards because their investigation by the said method did not furnish an actual proof. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge." (from the Method)