I think Archimedes had some great non-infinitesimal methods for discovering the area and volume of shapes. Some very visual methods involving his method of exhaustion for the volume of a sphere for example. These ideas were transcribed from some Palimpsest discovered with various drawings on them too but almost hidden. These methods are known now so why aren't they taught in high school (a way to introduce High School student to Calculus)?
I think I answered this question here:
Mathematics is very large. There are many beautiful things we could teach. But the time available is limited. Therefore we teach what is considered most important. Geometry (parts of Euclid) and algebra in high school. Calculus and linear algebra in the College. This is a very small part of mathematics.