In his study of astronomy and astrophysics, Johannes Kepler found it necessary to develop new mathematics. Mathematical calculations sometimes lead to the discovery of new physical phenomena. Deviations in the motions of Neptune from the predictions of mathematical theory led to the conclusion that an unknown planet existed. Unlike arithmetic, which deals with specific numbers, algebra introduces 'variables' that greatly extend the generality and scope of arithmetic. Algebra may be described as a generalization and extension of arithmetic.
Geometry is concerned with the properties of space and of objects in space.
Geometry deals with sets of points in a plane or in space. In its most elementary form geometry is concerned with such metrical problems as determining the areas and diameters of two-dimensional figures and the surface areas and volumes of solids. The study of plane curves, angles, polygons, and lines is called plane geometry . The study of curves in three-dimensional space such as spheres, cones, cylinders, and polyhedra is called solid geometry . Analysis and Calculus Analysis is concerned with abstract objects such as sets of numbers, sets of geometric points, or sets of functions that map numbers into numbers or points into points and with the processes, called limit processes , that depend on a measure of closeness between numbers, points, or functions.