Elementary Euclidean Geometry
The branch of maths that has to do with proofs Incorrect. The measure of the earth This was the old defintion, but it is accepted. The branch of maths that has to do with spatial relationships Correct. Process of reaching a conclusion based on previous observations Correct. Process of reaching a conclusion by hearing it from somewhere else that you trust Incorrect. Process of reaching a conclusion by combining known truths to create a new truth Incorrect. Process of reaching a conclusion in which the argument supports the conclusion based upon a rule Incorrect.
Process of reaching a conclusion based on previous observations Incorrect.
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Process of reaching a conclusion by combining known truths to create a new truth Correct. Process of reaching a conclusion in which the assumption of an argument supports the conclusion, but does not ensure it Incorrect. Points, circles, and planes Incorrect. Points, lines, and planes Correct.
Points, line segments, and planes Incorrect. Points, lines, and circles Incorrect. A dot Incorrect. A point in space that has a location but no dimensions Incorrect. An infinitely small point that exists on a line Incorrect. This is a trick question Correct. A statement that proves itself Incorrect.
A statement which is taken to be self-evident, and cannot be proved Correct. Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid , which he described in his textbook on geometry : the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms , and deducing many other propositions theorems from these.
Although many of Euclid's results had been stated by earlier mathematicians,  Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory , explained in geometrical language.
For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious with the possible exception of the parallel postulate that any theorem proved from them was deemed true in an absolute, often metaphysical, sense.
Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein 's theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only over short distances relative to the strength of the gravitational field.
Euclidean geometry is an example of synthetic geometry , in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects, all without the use of coordinates to specify those objects. This is in contrast to analytic geometry , which uses coordinates to translate geometric propositions into algebraic formulas. The Elements is mainly a systematization of earlier knowledge of geometry.
Its improvement over earlier treatments was rapidly recognized, with the result that there was little interest in preserving the earlier ones, and they are now nearly all lost. Many results about plane figures are proved, for example "In any triangle two angles taken together in any manner are less than two right angles. Books V and VII—X deal with number theory , with numbers treated geometrically as lengths of line segments or areas of regions. Notions such as prime numbers and rational and irrational numbers are introduced. It is proved that there are infinitely many prime numbers.
A typical result is the ratio between the volume of a cone and a cylinder with the same height and base. The platonic solids are constructed. Euclidean geometry is an axiomatic system , in which all theorems "true statements" are derived from a small number of simple axioms. Until the advent of non-Euclidean geometry , these axioms were considered to be obviously true in the physical world, so that all the theorems would be equally true. However, Euclid's reasoning from assumptions to conclusions remains valid independent of their physical reality.
Near the beginning of the first book of the Elements , Euclid gives five postulates axioms for plane geometry, stated in terms of constructions as translated by Thomas Heath : . Although Euclid only explicitly asserts the existence of the constructed objects, in his reasoning they are implicitly assumed to be unique. Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation.
To the ancients, the parallel postulate seemed less obvious than the others. 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. Though nearly all modern mathematicians consider nonconstructive methods just as sound as constructive ones, Euclid's constructive proofs often supplanted fallacious nonconstructive ones—e.
Euclid often used proof by contradiction. Euclidean geometry also allows the method of superposition, in which a figure is transferred to another point in space. For example, proposition I. Some modern treatments add a sixth postulate, the rigidity of the triangle, which can be used as an alternative to superposition. 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 distances. 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 congruent and corresponding sides are in proportion to each other.
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 infinite , rays semi-infinite , and line segments of finite length.
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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" in Euclid could be either straight or curved, and he used the more specific term "straight line" when necessary. The Pythagorean theorem states that the sum of the areas of the two squares on the legs a and b of a right triangle equals the area of the square on the hypotenuse c. Thales' theorem states that if AC is a diameter, then the angle at B is a right angle.
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.
Elementary Euclidean Geometry: An Introduction - C. G. Gibson - Google книги
Euclid proved these results in various special cases such as the area of a circle  and the volume of a parallelepipedal solid. Because of Euclidean geometry's fundamental status in mathematics, it is impractical to give more than a representative sampling of applications here. Sphere packing applies to a stack of oranges. As suggested by the etymology of the word, one of the earliest reasons for interest in geometry was surveying ,  and certain practical results from Euclidean geometry, such as the right-angle property of the triangle, were used long before they were proved formally.
Historically, distances were often measured by chains, such as Gunter's chain , and angles using graduated circles and, later, the theodolite. An application of Euclidean solid geometry is the determination of packing arrangements , such as the problem of finding the most efficient packing of spheres in n dimensions. This problem has applications in error detection and correction. Geometric optics uses Euclidean geometry to analyze the focusing of light by lenses and mirrors.
The water tower consists of a cone, a cylinder, and a hemisphere. Its volume can be calculated using solid geometry. Geometry is used extensively in architecture. Apr 22 '11 at This time I beat you by some seconds : The last sentence, while certainly correct sounds a bit funny modern Maybe I'll win by some seconds the next time around. Well, this happens sometimes: I had this same experience with Matt Emerton some time ago, when over half an hour we answered the same three questions and posted within a few seconds of each other. Good luck! But strangely it contains very few proofs or examples to guide the reader into the subject.
So beginners must rely on other books to fill in the gap while learning the subject. Cheng Apr 17 '14 at You will need to give all proceeds to charity, of course. Have you seen fedja at mathoverflow. From the Preface: The present book takes a clear position: The Elements are read, interpreted, and commented upon from the point of view of modern mathematics.
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