History of geometry
Classic geometry was focused in compass and straightedge constructions. Geometry was revolutionized by Euclid, who introduced mathematical rigor and the axiomatic method still in use today. His book, "The Elements" is widely considered the most influential textbook of all time, and was known to all educated people in the West until the middle of the 20th century.
History of geography
Thales of Miletus is one of the first known philosophers known to have wondered about the shape of the world. He proposed that the world was based on water, and that all things grew out of it. He also laid down many of the astronomical and mathematical rules that would allow geography to be studied scientifically. His successor Anaximander is the first person known to have attempted to create a scale map of the known world and to have introduced the gnomon to Ancient Greece.
Hecataeus of Miletus initiated a different form of geography, avoiding the mathematical calculations of Thales and Anaximander he learnt about the world by gathering previous works and speaking to the sailors who came through the busy port of Miletus. From these accounts he wrote a detailed prose account of what was known of the world. A similar work, and one that mostly survives today, is Herodotus' "Histories". While primarily a work of history, the book contains a wealth of geographic descriptions covering much of the known world. Egypt, Scythia, Persia, and Asia Minor are all described, including a mention of India. The description of Africa as a whole are contentious, with Herodotus describing the land surrounded by a sea. Though, historically the Indian sea was thought of an inland sea which was that round of the southern part of Africa is surrounded by the eastern part of Asia by connecting land, which inference only after the circumnavigation of Africa by Vasco da Gama was abandoned by the western cartographers of the 15th century. Some, though, hold that the descriptions of areas such as India are mostly imaginary. Regardless, Herodotus made important observations about geography. He is the first to have noted the process by which large rivers, such as the Nile, build up deltas, and is also the first recorded as observing that winds tend to blow from colder regions to warmer ones.