By the late fourth century appeared a book entitled The Elements to immortalize a mathematician of ancient Greece named Euclid, which he summarized in thirteen books (which are now known just as the thirteen chapters of the book) the developments that had made at the time the first major branch of mathematics developed in a logical and formal: the flat geometry (also known as mapping ). The work was divided into the following sections:
Books I-IV: The Geometry of Triangles and Circles
Books V-VI: Theory of Geometric Ratios
Books VII-IX: Theory of Numbers
Book X: Theory of Irrational Numbers
Books XI-XIII:
Solid Geometry Euclid's book established the methodology rigorous still used today from the most basic math high school through advanced mathematics with which conducted the latest research and developments in the universities: it begins with a set of definitions and axioms (truths so obvious that nobody will put into question) with which to begin to carry out demonstrations of other propositions whose truth is not so evident propositions calls theorems. For a theorem can be accepted as valid, it is required that can be deducted from the basic axioms (or postulates) used as a starting point. Entanto such thing can not be achieved, the truth of the theorem will into question, and rather than be called theorem will be called simply a guess .
The fundamental building blocks with which Euclid built his work, his point of departure, its ten principles are as follows:
AXIOMS \u200b\u200bof Euclidean Geometry
Postulate 1: Between two points can only draw a straight line.
Postulate 2: A line can be extended indefinitely in both directions.
Postulate 3: You can draw a circle using any point as a center, which can have any radius.
Postulate 4: All right angles are equal.
Postulate 5: If two straight lines, intersecting with third, unilateral internal angles whose sum is less than two right angles (less than 180 degrees), it turns out these two lines, to extend them indefinitely, will meet by one side that this sum is less than two right angles.
Postulate 6: Two things are equal to one third equal.
Postulate 7: If we add same same, the resulting sums will also be equal.
Postulate 8: If we subtract them equal equal, the remnant will be equal.
Postulate 9: Figures that can be matched are equal.
Postulate 10: The whole is greater than the sum of its parts.
All, absolutely all the theorems of geometry "classic" can be obtained from the ten principles given by Euclid.
However, if we read the ten principles, stands out immediately that the fifth postulate seems to be too elaborate, too complicated. This immediately raises a question: Could it be possible to postulate that a combination of several ways the other nine principles? Below are shown more clearly what we meant with his fifth postulate Euclid:
In the diagram, if two straight lines (in this case the line AB and line CD), the which obviously are not parallel lines , to intersect with a third (in this case the line EF), unilateral internal angles whose sum is less than two right angles (note that the sum of internal angles and m n is less than 180 degrees, which can be verified by measuring the angles with a protractor and adding), the extend them indefinitely, it will cross one side that this sum is less than two right angles, or in this case to be extended will be found on the right side.
not hard to see that if the straight lines AB and CD have been drawn in such a way that would be what we call straight parallel, then the third line intersecting the line EF have formed internal angles and m n whose sum would have been equal to two right angles, or equal to 180 degrees, defined by construction and alternate interior angles , in which case the straight lines will never be found or the right side or left side to extend them indefinitely, which is a slightly elaborated to say that two parallel lines never cross . And as this happens only when the sum of the two internal angles and m n is exactly 180 degrees, this means that you can only draw a line parallel to another given line. (This is a good opportunity to point out that the equality symbol both used in mathematical expressions like A = B actually comes from the representation of two small parallel lines, thereby emphasizing that since the time of the Middle Ages believed that there was nothing like each other than two parallel lines.)
As enunciated by Euclid the "fifth postulate", may not be familiar to many until we replace it by another statement that is completely equivalent, the postulate parallel is taught in schools, designed by John Playfair (1748-1819), which reads: through a point P located outside a straight line can only draw a line parallel to the given line . In fact, they involve different ways of talking about the same thing.
can demonstrate, in Euclidean geometry and using simple arguments, that all the following statements equivalent to the fifth postulate, leads each of them individually to any of the other. 're Actually the same thing but with different clothes:
(1) view a line L and P that point is not situated in that line, there just a line on plane that found the line l and point P not intersect with l .
(2) The sum of the interior angles in any triangle is 180 degrees (two right angles).
(3) The ratio of the circumference to the diameter of a circle (pi) is the same for any circle, regardless size.
(4) Given any triangle, there are more and arbitrarily arbitrary triangles whose sides are smaller in proportion relative to each other (similarity ratio).
(5) The Pythagorean Theorem: In any triangle, the square of the hypotenuse equals the sum of the squares of the other sides .
As he said, made the fifth postulate suspect that instead of being a true "fundamental" is something that can possibly be deduced from the most basic things, in this case an intelligent combination of nine axioms, in which case the assumption would be an assumption and would become a theorem. As was stated originally, the fifth postulate does not seem to be a truth so obvious, there seems to be an axiom, rather seems to be a theorem can be shown to other elementary principles. And the attempt to derive the fifth postulate from the other nine went on to become an obsession that consumed the lives of many mateméticos of antiquity even to the dawn of the Renaissance. The English mathematician John Wallis (1616-1703) dedicated his book, "From the fifth postulate" to gather the many attempts that had been carried out to try to prove the fifth postulate. From 1607 to 1880, more than a thousand books, essays, and reports had been devoted to the fifth postulate. However, all attempts to prove it from other basic axioms were found to be defective, none of them could withstand serious examination. As an example, in 1834 in volume 11 of the Journal für Mathematik "was published that tried to be a serious demonstration of good faith, the fifth postulate, based on the following figure:
This "demonstration" proceeded as follows. The surveyors drew the line for B BY parallel to CD, and built the angles ABN, NBO, OBP and PBQ, each equal to the angle YBA. Correctly reasoned that whatever the magnitude of the angle YBA, could, building enough angles ABN, NBO, etc., And finally to one side of which (in the figure, BQ) falls below the BZ. Suppose there is a number n of these angles (in the figure four). He noted then n segments, CE, EG, GJ, etc., Each equal to BC, and the points of division drew the lines EF, GH, JK, etc., Parallel to CD. So far all the steps are correct. But the problem is that it started to compare infinite areas. Maintained, for example, that the area bounded by straight lines and BA BY indefinite equals the infinite area bounded by the lines BA and BN, and the infinite area limited on three sides by the line segment and lines BC and CD BY indefinite equals the infinite area bounded on three sides by the line segment CE and open-ended straight CD and EF. Or as he said that the area YBA, ABN is equal to the area and the area equals the area YBCD DIF.
Assume for a moment that such reasoning and reasoning about infinite fields are valid. The rest of the show is then: YBLM area is equal to n times the area YBCD and YBQ area is equal to n times the area YBA. But the area YBLM is only one part of the YBZ, while the area YBZ is itself only part of the YBQ. Hence, n times the area is less than YBCD YBZ area, which in turn is less than n times the area YBA. Ie n * (YBCD) is less than n * (YBA), or is less than YBCD YBA. But if that is the case, has to cut AB CD. For if AB does not cut a CD, YBCD would equal the sum of YBA and ABCD, and therefore would be greater than YBA.
The problem with this "show", as stated, is that the areas considered are endless. Two finite areas can be said that the former is lower, equal or greater than the second. But it is impossible to compare two infinite area, of whom one can only say they are endless.
The wrong kind of show more widespread was his replacement by another equivalent proposition such as "the perpendicular and oblique to a straight cut," there is a triangle similar to the triangle given but not equal to it, " "the locus of points equidistant from a given line, if you are on the same side of it, is a straight line," through any three points can be drawn or a line or a circle. " But if the parallel axiom of Euclid does not occur, then all these propositions are wrong. Therefore, assuming any of these propositions as an axiom, then the fifth postulate is fair, ie, we assume the correctness of what we wanted to prove. To date, nobody has been able to derive the fifth postulate of Euclid's other postulates, and there is now a general consensus that this is not possible. And even as modified by John Playfair a seemingly simple expression ( through a point outside a line can only draw a line parallel to the given line ), the problem is that the statement is supposed to always be valid when straight lines are extended to infinity , and it is in the mismanagement of the infinite mathematical where renowned scientists have made their worst mistakes incur its greatest failures. We do not know what lies beyond what our most powerful telescopes like the Hubble satellite from view, we know which properties can have the space to distances as great. Even if space and everything in it is going to shrink or expand to large distances, that is something that is impossible to derive the fifth postulate. The fifth postulate in the version that has been given by Playfair assumed that the properties of space in which we live will remain the same to infinity, and that is an extremely reckless generalization that in fact it is impossible to prove. So instead it is the space that defines the validity of the fifth postulate Euclid's fifth postulate of Euclid in Playfair's version that defines the properties that must have space for large distances, called to replace the experimental evidence for purely mental conceptions. In fact, the original version given by Euclid, but more elaborate, is superior to the version given by Playfair, by the fact that the version of Euclid does not claim that two parallel lines will not be at infinity, instead of specifying conditions under which any two lines drawn on a plan if you come to find.
Euclid may not have been very happy with the inclusion of his fifth postulate in his Elements for the reasons cited. However, it is not possible to resort to a simplistic and work to eliminate only the nine remaining postulates, because there are many theorems in Euclidean geometry can not be shown if you delete the fifth postulate. The fifth postulate, despite the suspicions aroused, was (and remains) a necessary evil.
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