Geometry

Geometry comes from the Greek for "earth measurement" - from the words geo, "Earth," and metron, "measure,". Geometry is concerned with the properties of space and of objects in space, e.g. the shapes of objects, spatial relationships among objects, and the properties of surrounding space. It is one of the oldest branches of mathematics, having arisen to solve practical problems such as those in surveying. Geometry is a mathematical system that is usually concerned with points, lines, surfaces, and solids. All mathematical systems are based on undefined elements, assumed relations, unproved statements (postulates and assumptions), and proved statements (theorems). Different sets of assumptions give rise to different geometries.

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.

Other fields of geometry include analytic geometry, descriptive geometry, analysis situs or topology, the geometry of spaces having four or more dimensions, fractal geometry, and non-Euclidean geometry. Modern abstract geometry deals with very general questions of space, shape, size, and other properties of figures.

Analytic geometry is the study of geometry using algebraic methods. Analytic geometry combines the generality of algebra with the precision of geometry. It is sometimes called Cartesian geometry, after Descartes, who was the first to exploit the methods of algebra in geometry. Analytic geometry addresses geometric problems from an algebraic point of view by associating any curve with variables by means of a coordinate system.

Differential geometry applies techniques of calculus and studies such local properties as tangents and curvature.

Topology, developed in the 20th century, is a study of generalized geometric elements and such properties as connectedness and compactness.

Early written records show that geometry arose in ancient Egypt and Mesopotamia as an art associated with such practical problems as surveying. But it was the Greeks of the 1st millennium BC who investigated it in a more systematic and general way; they began the transformation of the subject from an empirical art to a systematic science.

Euclidean Geometry

The most common geometry is Euclidean geometry, which appears to explain the universe in which we live. Applications of it are found in nearly all aspects of daily activity as well as in the development of all industrial and scientific products.

Each of the 13 books of Euclid's Elements opens with a statement of the definitions required in that book. In the first book the definitions are followed by the assumptions (postulates and common notions) that are to be used. There follows a set of propositions (theorems) with proofs. The logical structure of the exposition of the proofs has influenced all scientific thinking since Euclid's time. This logical structure is essentially as follows:

• A statement of the proposition.
• A statement of the given data (usually with a diagram).
• An indication of the use that is to be made of the data.
• A construction of any additional lines or figures.
• A synthetic proof.
• A conclusion stating what has been done.

In about 300 � Euclid established a set of axioms for geometry. One of them assumes the existence of one and only one line that is both parallel to a given line and contains a given point that is not a point of the line; or, given a line and a point not on the line, one and only one coplanar line can be drawn through the point parallel to the given line."

Non-Euclidean geometres have been developed by denying the validity of the famous fifth postulate (parallel postulate), and are based on alternatives to it.

The discovery of non-Euclidean geometries inspired a new approach to the subject by presenting theorems in terms of axioms applied to properties assigned to undefined elements called points and lines, not necessarily limited to the classical study of flat surfaces (plane geometry) and rigid three-dimensional objects (solid geometry). This led to many new different types of geometries such as, for example, hyperbolic geometry, projective geometry, Riemannian geometry, elliptical, and parabolic geometries.

Trigonometry

Trigonometry is the study of triangles, angles, and specific functions of angles and their application to calculations in geometry. Trigonometry is the branch of mathematics used in computing, rather than directly measuring, distances. The trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant) can be defined as the ratios of lengths of sides of right triangles, or in terms of coordinates of terminal points of arcs on the unit circle (circular functions). For example, if a right-angled triangle contains an angle, symbolized here by the Greek letter alpha, α, the ratio of the side of the triangle opposite to α to the side opposite the right angle (the hypotenuse) is called the sine of α. The ratio of the side adjacent to α to the hypotenuse is the cosine of α. These functions are properties of the angle α, and calculated values have been tabulated for many angles. These are useful in determining unknown angles and distances from known or measured angles in geometric figures. The subject developed from a need to compute angles and distances in such fields as astronomy, map making, surveying, and artillery range finding.