First, let`s turn to a person commonly referred to as the „father of geometry“: the ancient Greek mathematician Euclid. Euclid`s work is the first example of a systematic approach to geometry. If you make a general statement in geometry, such as the Pythagorean theorem, you should prove this statement by deriving it from statements that you are convinced are self-evident using the rules of logic. For 2000 years, Euclid`s systematic approach seemed to prove truths about geometric objects and thus gain certainty. Apollonius of Perga (c. 262 BC – c. 190 BC) is best known for his study of conical sections. The modern formulation of the proof by induction was not developed until the 17th century, but some later commentators consider it implicit in some of Euclid`s proofs, for example.dem proof of the infinity of prime numbers. [45] Geometry can be used to design origami.
Some classic geometry design problems are impossible with the compass and right edge, but can be solved with origami. [22] Euclid often used contradictory proof. Euclidean geometry also allows the method of superposition, in which a figure is transferred to another point in space. For example, theorem I.4, lateral congruence at lateral angles of triangles, is proved by moving one of the two triangles so that one of its sides coincides with the same side of the other triangle, and then proving that the other sides also coincide. Some modern treatments add a sixth postulate, triangle rigidity, which can be used as an alternative to layering. [11] Angles whose sum is a right angle are said to be complementary. Complementary angles occur when a beam splits the same vertex and points in a direction between the two original radii that form the correct angle. The number of rays between the two original rays is infinite. Euclid refers to a pair of lines or a pair of flat or fixed figures as „equal“ (ἴσος) if their lengths, faces or volumes are equal, and similar for angles.
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 over the other to fit it exactly. (Flipping is allowed.) For example, a 2×6 rectangle and a 3×4 rectangle are identical but not congruent, and the letter R is congruent to its mirror image. Figures that would be congruent except for their different sizes are said to be similar. The corresponding angles in a pair of similar shapes are congruent and the corresponding sides are related to each other. Periods are usually named with capital letters of the alphabet. Other figures, such as lines, triangles or circles, are named by listing a sufficient number of points to filter them clearly out of the relevant map, for example the triangle ABC would usually be a triangle with vertices at points A, B and C. Euclid`s geometry is the generally accepted model of perfect intellectual authority. Or is it the case? In the second article, we will blow all this out of the water.
18th century surveyors struggled to define the boundaries of the Euclidean system. Many have tried in vain to prove the fifth postulate of the first four. By 1763, at least 28 different proofs had been published, but all turned out to be false. [26] It turns out that Euclid`s proof dates from the 5th century. Postulate required. So this theorem about the sum of angles requires that the space be Euclidean. Kant does not say this, but he says that there is only one space. For Kant, no alternative to Euclid seems conceivable. Triangles are congruent if they have equal the three sides (SSS), two sides and the equal angle between them (SAS), or two angles and one equal side (ASA) (Book I, sentences 4, 8 and 26). Triangles with three equal angles (AAA) are similar, but not necessarily congruent.
In addition, triangles with two equal sides and an adjacent angle are not necessarily identical or congruent. The art and architecture of the early modern period also reflect Euclidean conception of space. Here is the first important perspective painting of the Renaissance: the Trinity of Masaccio. Euclidean geometry has two basic types of measurements: angle and distance. The angle scale is absolute, and Euclid uses the right angle as the base unit, so, for example, a 45-degree angle is called half the right angle. The distance scale is relative; One arbitrarily chooses a line segment with a certain non-zero length as a unit, and other distances are expressed from it. Distance addition is represented by a construct in which a line segment is copied to the end of another line segment to extend its length, and likewise for subtraction. The elements also contain the following five „common terms“: One application of Euclidean solid geometry is the determination of packaging arrangements, such as the problem of finding the most efficient settlement of n-dimensional spheres. This problem has applications in error detection and correction. Books I to IV and VI deal with plane geometry.
Many results on flat figures are proven, for example: „In any triangle, two angles taken together in one way or another are less than two right angles.“ (Book I Thesis 17) and the Pythagorean theorem „In right triangles, the square of the side that undermines the right angle is equal to the squares of the sides containing the right angle.“ (Book I, Proposition 47) The ambiguous nature of axioms, as originally formulated by Euclid, allows various commentators to discuss some of their other implications for the structure of space, such as whether it is infinite or not[38] (see below) and what its topology is.
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