Analytical Geometry (ca. 1629–1637): Geometry may be defined as the branch of mathematics that studies the two- and threedimensional positioning of shapes (circles, triangles, etc.). Derived from the Greek words for earth and measurement, geometry historically was used in the surveying of land. Architects throughout time have also used geometrical thinking in the design and construction of buildings and temples. While these uses of geometry predate the rise of the Greek civilization, it was the ancient Greeks who were the first to apply the processes of logical thinking to the study of geometry. One of the earliest, and most recognized, examples of Greek geometry was provided by Pythagoras (ca. 580 B.C.E.). Pythagoras formally defined the relationship between the hypotenuse and sides of a right triangle, although the Babylonians had understood this relationship centuries earlier. However, the major advances in Greek geometry may be attributed to the work of Euclid (ca. 330–ca. 270 B.C.E.). Euclid pioneered the use of an axiomatic system of geometry. In this system a series of true statements are established, called axioms, which are in turn then used to develop general geometric theorems. Euclid’s Elements (ca. 300 B.C.E.) was a compilation of Greek mathematics to date and listed hundreds of such theorems. Elements formed the basis of geometric thinking until the time of the Renaissance.
To the ancient Greeks geometry was a tool to examine algebraic relationships, a process sometimes called geometric algebra. However, in order to completely understand the properties of a geometric shape, it is first necessary to be able to define and manipulate the geometric figure in terms of an algebraic equation. This form of mathematics is called analytical geometry. The first step in identifying the algebraic