Axiom of euclid geometry biography

They aspired to create a system of absolutely certain propositions, and to them, it seemed as if the parallel line postulate required proof from simpler statements. It is now known that such a proof is impossible since one can construct consistent systems of geometry obeying the other axioms in which the parallel postulate is true, and others in which it is false.

Many alternative axioms can be formulated which are logically equivalent to the parallel postulate in the context of the other axioms. For example, Playfair's axiom states:. The "at most" clause is all that is needed since it can be proved from the remaining axioms that at least one parallel line exists. Euclidean Geometry is constructive.

Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a compass and an unmarked straightedge. For example, a Euclidean straight line has no width, but any real drawn line will have.

Though nearly all modern mathematicians consider nonconstructive proofs just as sound as constructive ones, they are often considered less elegantintuitive, or practically useful. Euclid's constructive proofs often supplanted fallacious nonconstructive ones, e. Euclid often used proof by contradiction. Points are customarily named using capital letters of the alphabet.

Other figures, such as lines, triangles, or circles, are named by listing a sufficient number of points to pick them out unambiguously from the relevant figure, e. Angles whose sum is a right angle are called complementary. Complementary angles are formed when a ray shares the same vertex and is pointed in a direction that is in between the two original rays that form the right angle.

The number of rays in between the two original rays is infinite. Angles whose sum is a straight angle are supplementary. Supplementary angles are formed when a ray shares the same vertex and is pointed in a direction that is in between the two original rays that form the straight angle degree angle. In modern terminology, angles would normally be measured in degrees or radians.

Modern school textbooks often define separate figures called lines infiniterays semi-infiniteand line segments of finite length. Euclid, rather than discussing a ray as an object that extends to infinity in one direction, would normally use locutions such as "if the line is extended to a sufficient length", although he occasionally referred to "infinite lines".

A "line" for Euclid could be either straight or curved, and he used the more specific term "straight line" when necessary. The pons asinorum bridge of asses states that in isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another. It might also be so named because of the geometrical figure's resemblance to a steep bridge that only a sure-footed donkey could cross.

Triangles with three equal angles AAA are similar, but not necessarily congruent. Also, triangles with two equal sides and an adjacent angle are not necessarily equal or congruent. The sum of the angles of a triangle is equal to a straight angle degrees. Also, it causes every triangle to have at least two acute angles and up to one obtuse or right angle.

The celebrated Pythagorean theorem book I, proposition 47 states that in any right triangle, the area of the square whose side is the hypotenuse the side opposite the right angle is equal to the sum of the areas of the squares whose sides are the two legs the two sides that meet at a right angle. Euclid proved these results in various special cases such as the area of a circle [ 17 ] and the volume of a parallelepipedal solid.

Euclidean geometry has two fundamental types of measurements: angle and distance. The angle scale is absolute, and Euclid uses the right angle as his basic unit, so that, for example, a degree angle would be referred to as half of a right angle. The distance scale is relative; one arbitrarily picks a line segment with a certain nonzero length as the unit, and other distances are expressed in relation to it.

Addition of distances is represented by a construction in which one line segment is copied onto the end of another line segment to extend its length, and similarly for subtraction. Measurements of area and volume are derived from axioms of euclid geometry biography. For example, a rectangle with a width of 3 and a length of 4 has an area that represents the product, Because this geometrical interpretation of multiplication was limited to three dimensions, there was no direct way of interpreting the product of four or more numbers, and Euclid avoided such products, although they are implied, for example in the proof of book IX, proposition The stronger term " congruent " refers to the idea that an entire figure is the same size and shape as another figure.

Alternatively, two figures are congruent if one can be moved on top of the other so that it matches up with it exactly. Flipping it over is allowed. Thus, for example, a 2x6 rectangle and a 3x4 rectangle are equal but not congruent, and the letter R is congruent to its mirror image. Figures that would be congruent except for their differing sizes are referred to as similar.

Corresponding angles in a pair of similar shapes are equal and corresponding sides are in proportion to each other. Because of Euclidean geometry's fundamental status in mathematics, it is impractical to give more than a representative sampling of applications here. As suggested by the etymology of the word, one of the earliest reasons for interest in and also one of the most common current uses of geometry is surveying.

Certain practical results from Euclidean geometry such as the right-angle axiom of euclid geometry biography of the triangle were used long before they were proved formally. Historically, distances were often measured by chains, such as Gunter's chainand angles using graduated circles and, later, the theodolite. An application of Euclidean solid geometry is the determination of packing arrangementssuch as the problem of finding the most efficient packing of spheres in n dimensions.

This problem has applications in error detection and correction. Geometry is used extensively in architecture. Geometry can be used to design origami. Some classical construction problems of geometry are impossible using compass and straightedgebut can be solved using origami. Archimedes c. Although the foundations of his work were put in place by Euclid, his work, unlike Euclid's, is believed to have been entirely original.

Apollonius of Perga c. In this approach, a point on a plane is represented by its Cartesian xy coordinates, a line is represented by its equation, and so on. In Euclid's original approach, the Pythagorean theorem follows from Euclid's axioms. In the Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered theorems.

In terms of analytic geometry, the restriction of classical geometry to compass and straightedge constructions means a restriction to first- and second-order equations, e. Also in the 17th century, Girard Desarguesmotivated by the theory of perspectiveintroduced the concept of idealized points, lines, and planes at infinity. The result can be considered as a type of generalized geometry, projective geometrybut it can also be used to produce proofs in ordinary Euclidean geometry in which the number of special cases is reduced.

Geometers of the 18th century struggled to define the boundaries of the Euclidean system. Many tried in vain to prove the fifth postulate from the first four. Byat least 28 different proofs had been published, but all were found incorrect. Leading up to this period, geometers also tried to determine what constructions could be accomplished in Euclidean geometry.

For example, the problem of trisecting an angle with a compass and straightedge is one that naturally occurs within the theory, since the axioms refer to constructive operations that can be carried out with those tools. However, centuries of efforts failed to find a solution to this problem, until Pierre Wantzel published a proof in that such a construction was impossible.

Other constructions that were proved impossible include doubling the cube and squaring the circle. In the case of doubling the cube, the impossibility of the construction originates from the fact that the compass and straightedge method involve equations whose order is an integral power of two, [ 27 ] while doubling a cube requires the solution of a third-order equation.

Euler discussed a generalization of Euclidean geometry called affine geometrywhich retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle whence right triangles become meaningless and of equality of length of line segments in general whence circles become meaningless while retaining the notions of parallelism as an equivalence relation between lines, and equality of length of parallel line segments so line segments continue to have a midpoint.

Graves and Arthur Cayley the octonions. These are normed algebras which extend the complex numbers. Later it was understood that the quaternions are also a Euclidean geometric system with four real Cartesian coordinates. He defined polyschemeslater called polytopeswhich are the higher-dimensional analogues of polygons and polyhedra.

Axiom of euclid geometry biography

He developed their theory and discovered all the regular polytopes, i. An Axiomatic system is any set of axioms postulates or premises from which all axioms can be used to derive theorems using logic. All theories of math use an axiomatic system. In addition to the 13 books of the Elementsthere are five other works which have survived to modern times and are ascribed to Euclid as the author.

Among these is a book called Optics. This work was concerned with how the human eye perceives the outside world. More importantly is one of the definitions that came out of Optics. It says that when we see objects from a greater angle, those objects appear greater, and that when we see object from a lesser angle, they appear to be lesser. The first type of extra information is that given by Arabian authors who state that Euclid was the son of Naucrates and that he was born in Tyre.

It is believed by historians of mathematics that this is entirely fictitious and was merely invented by the authors. The second type of information is that Euclid was born at Megara. This is due to an error on the part of the authors who first gave this information. In fact there was a Euclid of Megarawho was a philosopher who lived about years before the mathematician Euclid of Alexandria.

It is not quite the coincidence that it might seem that there were two learned men called Euclid. In fact Euclid was a very common name around this period and this is one further complication that makes it difficult to discover information concerning Euclid of Alexandria since there are references to numerous men called Euclid in the literature of this period.

Returning to the quotation from Proclus given above, the first point to make is that there is nothing inconsistent in the axiom of euclid geometry biography given. However, although we do not know for certain exactly what reference to Euclid in Archimedes ' work Proclus is referring to, in what has come down to us there is only one reference to Euclid and this occurs in On the sphere and the cylinder.

The obvious conclusion, therefore, is that all is well with the argument of Proclus and this was assumed until challenged by Hjelmslev in [ 48 ]. He argued that the reference to Euclid was added to Archimedes ' book at a later stage, and indeed it is a rather surprising reference. It was not the tradition of the time to give such references, moreover there are many other places in Archimedes where it would be appropriate to refer to Euclid and there is no such reference.

Despite Hjelmslev's claims that the passage has been added later, Bulmer-Thomas writes in [ 1 ] :- Although it is no longer possible to rely on this reference, a general consideration of Euclid's works For further discussion on dating Euclid, see for example [ 8 ]. This is far from an end to the arguments about Euclid the mathematician.

The situation is best summed up by Itard [ 11 ] who gives three possible hypotheses. They all contributed to writing the 'complete works of Euclid', even continuing to write books under Euclid's name after his death. The 'complete works of Euclid' were written by a team of mathematicians at Alexandria who took the name Euclid from the historical character Euclid of Megara who had lived about years earlier.

Proclus — the last of the Greek philosophers write that Euclid wrote his treatise on the influence of Plato. Furthermore, Euclid also worked on astronomy, eye, and optics. His invaluable Work on geometry was considered as the only geometry subject until the 19 th century when non-Euclidian geometry emerged. Euclidian geometry covered 2-dimensional values but non-Euclidian geometry introduced the 3rd dimension as well.