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Enoch Root to Clarke: "Tell your brother to show the boy Euclid and let him find his own way"

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Euclid

From Wikipedia, the free encyclopedia.

Euclid of Alexandria (Greek: Eukleides) (circa 365 -275 BCE) was a Greek mathematician who lived in the 3rd century BCE in Alexandria. His most famous work is the Elements, a book in which he deduces the properties of geometrical objects and integers from a set of axioms, thereby anticipating the axiomatic method of modern mathematics. Although many of the results in the Elements originated with earlier mathematicians, one of Euclid's major accomplishments was to present them in a single logically coherent framework. The geometry of Euclid was known for ages as "the" geometry, but is nowadays referred to as Euclidean geometry. Euclid.jpg
EUCLID

Euclidean geometry

In mathematics, Euclidean geometry is the familiar kind of geometry of at most three dimensions ; it is the kind usually taught in high school. It is named after the Greek mathematician Euclid. Euclid's text Elements is an early systematic treatment of this kind of geometry, based on axioms (or postulates). In differential geometry, and in constrast to the main types of non-Euclidean geometry it is also called " flat " geometry, or " parabolic " geometry because it is between elliptic geometry which is positively curved, and hyperbolic geometry which is negatively curved.

The traditional presentation of Euclidean geometry is as an axiomatic system, which hoped to prove all the "true statements" as theorems in geometry from a set of finite number of axioms.

The five postulates/axioms of the Euclidean system are: 1. Any two points can be joined by a straight line. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

Euclidean geometry is distinguished from other geometries by the parallel postulate, which is more easily phrased as follows: Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.

The parallel postulate was a subject of deep contention among modern mathematicians in the middle ages. Many mathematicians, argued that the fifth postulate was redundant, and could be deduced independently from the first four postulates. They tried to prove this fact by "not assuming" the postulate to hold true, and try and arrive at this postulate as a theorem by means of inference rules alone.

Other mathematicians proposed that the statement was indeed a postulate, and tried to prove their suggestions, by "negating the postulate" (call the negation NOT P) as an axiom, hoping to arrive at a contradiction. If a contradiction is achieved with Y, a known theorem, (previously deduced from the other postulates) ie : if it leads to a situation - Y AND NOT Y, or if a contradiction is achieved directly with the assumed NOT P —— ie : if we arrive at P AND NOT P, this would mean that the assumption of negation of P was wrong (Proof by contradiction), and hence the parallel postulate needs to be "assumed to be true".

However, both factions of mathematicians were stumped in their efforts to achieve a definite answer to the question of "whether the parallel postulate is an axiom of geometry". The disbelievers, could not successfully prove that it is not. Neither could the believers arrive at a contradiction, by negating it. Curiously, however, by negating the fifth postulate in various ways, they extended geometry to represent non-planar universes.

In particular, this postulate separates Euclidean geometry from hyperbolic geometry, where many parallel lines could be drawn through the point, and from elliptic and projective geometry, where no parallel lines exist. (Euclidean geometry does, however, share the parallel postulate with some other geometries, such as certain finite geometries and affine geometry. )

Since Euclid's time, other mathematicians have laid out more thorough axiomatic systems for Euclidean geometry, such as David Hilbert and George Birkhoff.

Modern Concept of Euclidean Geometry

Today Euclidean geometry is usually constructed rather than axiomatized, by means of analytic geometry. A rectangular coordinate system maps each point in Euclidean space with a unique list of n real numbers ((x1,...,xn), so we can define it to be the set of all such lists (Rn). We also define a metric (distance function) d by

d(x,y)^2=(x_1-y_1)^2+\ldots+(x_n-y_n)^2

which you might recognise as an application of the Pythagorean Theorem (see also Euclidean distance). This turns Rn into a metric space. Maps that preserve the distance between all pairs of points are called isometries, and include reflections, rotations, translations, and compositions thereof. In matrix notation any of these have the form

x'=\mathbf{A}x+b

where A is an orthogonal matrix and bis a column vector. Isometries are taken as the congruences of Euclidean geometry - that is, we only consider properties preserved by them. That way we do not have to worry about the precise origin or axes, but still consider distances, angles, and so forth.

The Parallel Postulate

The fifth postulate of the Euclidean geometry, called the parallel postulate, states that if a straight line (note: in Euclid's terminology a line may be finite) intersects two other straight lines, and the sum of the interior angles on one side of the line is less than 180 degrees (literally "two right angles"), then the two lines, if they are lengthened indefintely, will intersect on the same side on the line as the interior angles. Since this axiom is less obvious than the others, many mathematicians tried to derive it from the others. Then, in the 19th century, Janos Bolyai (and probably Carl Friedrich Gauss before him) realized that its negation leads to consistent non-euclidean geometries, which were later developed by Lobachevsky, Riemann and Poincaré. See Non-Euclidean geometry for further explication)

In addition to a treatment of plane geometry, including proofs of the Pythagorean theorem and a version of the more general law of cosines, Euclid's book also contains the beginnings of elementary number theory, such as the notion of divisibility, the greatest common divisor and the Euclidean algorithm to determine it, and the infinity of prime numbers. Later chapters deal with three-dimensional geometry and the platonic solids. The book also contains proofs that the area of a circle is proportional to the square of its radius, and that the volume of a sphere is proportional to the cube of its radius.

While the Elements was still used in the 20th century as a geometry text book and has been considered a fine example of the formally precise axiomatic method, Euclid's treatment does not hold up to modern standards and some logically necessary axioms are missing. The first correct axiomatic treatment of geometry was provided by Hilbert in 1899.

¡¡Disambiguation!!

Euclid of Alexandria has occasionally been confused with the philosopher Eukleides (Euclid) of Megara who lived about a century earlier. Such confusion may have begun with Valerius Maximus (circa 20 BCE - circa 50 CE), a contemporary of Tiberius, who claimed (search for "Eucliden") that Plato turned to Euclid for help with a geometrical problem.

Euclid is also a programming language developed at the University of Toronto by Holt et al, originally for the Motorola 6809 microprocessor. The Fujitsu FM-9 used two 6809 CPUs and was a successful Japanese Apple II clone. Also see HCF which was implemented in the hardware.

In a number of American cities — because of triskaidekaphobia, where there are numbered streets — Euclid is often substituted for 13TH street. The rule used to have avenues named for Blaise Pascal because of the arithmatical triangle bearing his name, though this may have fallen into disuse.