# A report onGeometry

Geometry is, with arithmetic, one of the oldest branches of mathematics.

- Geometry

## Mathematics

Mathematics is an area of knowledge that includes such topics as numbers (arithmetic, number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

## Euclidean geometry

Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.

## Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space.

## Archimedes

Approximation of pi; defining and investigating the spiral that now bears his name; and devising a system using exponentiation for expressing very large numbers.

Approximation of pi; defining and investigating the spiral that now bears his name; and devising a system using exponentiation for expressing very large numbers.

Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including: the area of a circle; the surface area and volume of a sphere; area of an ellipse; the area under a parabola; the volume of a segment of a paraboloid of revolution; the volume of a segment of a hyperboloid of revolution; and the area of a spiral.

## Differential geometry

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.

## Parallel postulate

In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry.

## Calculus

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

## Algebraic geometry

Branch of mathematics, classically studying zeros of multivariate polynomials.

Branch of mathematics, classically studying zeros of multivariate polynomials.

Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.