Next to the Bible - probably the book most played and studied the history of the Western world. It was the most influential text of all time, so striking that the successors of Euclid called "elements."
This work is considered one of the biggest bestsellers of all time. Work admired by mathematicians and philosophers of all countries and all times by the geometric purity of style and conciseness luminous form, logical model for all sciences and the accuracy of statements made by the way are the foundations of geometry.
There are few books that have been so edited, translated and annotated as Euclid's Elements. In ancient Greece, this work was commented on by Proclus (410-485) Heron (c. 10 - 75) and Simplicius (490-560) in the Middle Ages was translated into Latin and Arabic, after the discovery of printing, there have been numerous editions of it in all European languages. The first of these issues was to Campano (1220 - 1296), in Latin, published in 1482 edition used by Pedro Nunes (1502-1578), who quoted numerous times in his works.
In Portugal, Angelo Brunelli published in 1768 a Portuguese translation of the first six books, the eleventh and twelfth. For this translation has used the Latin version of Frederick Comandini and did it follow some notes that Roberto Sinson (1687 - 1768) had illustrated this release. This book, was once widely used in Portuguese schools which is why we have new editions of the translation of Brunelli in 1790, 1792, 1824, 1835, 1839, 1852, 1855 and 1862.
Euclid's Elements have an exceptional importance in the history of mathematics. Indeed, it has no geometry as a mere cluster of unconnected data, but rather as a logical system. The definitions, axioms or postulates (concepts and propositions are accepted without proof that the fundamentals specifically geometric and secure the existence of fundamental entities: point, line and plane) and the theorems do not appear grouped at random, but rather exposed in perfect order. Each theorem is clear from the definitions, axioms and theorems earlier, according to a rigorous proof.
Euclid was the first to use this method, called axiomatic. Thus, its elements are the first and most noble example of a logical system, an ideal which mimicked many other sciences and continue to emulate. However, we must not forget that Euclid endeavored axiomatize geometry with the resources it had at the time. It therefore easy to understand that the system chosen has some shortcomings. Unintentionally, some of its financial results admitted, often intuitive, without proof.
The thirteen books
* Books I-IV deal with elementary plane geometry. Using the most basic properties of lines and angles lead to the congruence of triangles, equal areas, the Pythagorean Theorem (Book I, Proposition 47) and its reciprocal (Book I, Proposition 48), the construction of a square area equal to a given rectangle, the golden section, the circle and regular polygons. The Pythagorean theorem and the golden section are introduced as properties of areas.
Like most of the thirteen books, the Book I begins with a list of settings (23 in total) without comment, for example, point, line, circle, triangle, angle, parallelism and perpendicularity of lines such as
"is a point which has no part,"
"a line is a length without breadth"
"is a surface which has only length and width."
Following the definitions, there are the postulates and common notions or axioms, in that order. Postulates are specific geometric propositions. "Participate" means "ask to accept." Thus, Euclid asks the reader to accept the five propositions formulated in geometric postulates:
1. Given two points, there is a line segment that joins them;
2. A line segment may be extended indefinitely to build a line;
3. Given a point any distance and any one can construct a circle at center point and radius equal to the distance given;
4. All right angles are equal;
5. If a straight line cutting two other lines so that the sum of two interior angles on the same side is less than two brackets, so these two lines, if sufficiently prolonged, intersect on the same side on which are these two angles
(This is the famous 5th postulate of Euclid)
Thus, three core concepts - the point , of the line and the circle - and postulates relating to them, are the basis for all of Euclidean geometry.
V * The book presents the theory of proportions of Eudoxus (408 BC - 355 BC) in its purely geometric and
* Book VI applies to the similarity of plane figures. Here we return to the Pythagorean theorem and the golden section (Book VI, proposition 31 and 30) but now as theorems concerning the reasons for greatness. Of particular interest is the theorem (Book VI, proposition 27) that contains the maximum of the first problem that came to us with proof that the square is, of all rectangles of a given perimeter, which has maximum area.
* Books VII-IX are devoted to number theory such as the divisibility of integers, the addition of geometric series, some properties of prime numbers and proof of the irrationality of the number. Here we find both the 'Euclidean algorithm' for finding the greatest common divisor of two numbers, such as "Euclid" according to which there is an infinity of primes (Book IX, proposition 20).
X * The book, the longest of all and often considered the most difficult, containing the geometric classification of irrational quadratic and quadratic roots.
* Books XI-XIII are concerned with the solid geometry, leading through the solid angles, the volume of the parallelepiped, prism and pyramid, the ball and what seems to have been considered the climax - the discussion of five regular polyhedra ('Platonic') and proof that these exist only five regular polyhedra.
Final Thoughts
When writing the Elements, Euclid was to collect a text three major findings of their recent past: the theory of proportions of Eudoxus, the theory of irrational Theaetetus (417 BC - 369 BC) and the theory of the five regular solids, which occupied an important place in Plato's cosmology.
in the Elements Euclid compiled all the known geometry of his day. But not content to gather all the knowledge of geometry, and ordered him framed him as a science. That is, from a few axioms developed and demonstrated the theorems and geometric propositions and give further statements when the former does not adapted themselves to the new order that was given to the proposals. Also, thoroughly studied by him the properties of geometric figures, areas and volumes and established the concept of locus.
While some elements have logical weaknesses, by current standards, these shortcomings have been overlooked for over two millennia. The critical movement began perhaps in the late seventeenth century, John Wallis (1616-1703), still a little fuzzy during the next century, the Jesuit Abbe Saccheri (1667-1733) and mathematicians Lambert (1728-1777) and Gauss (1777-1855). It is already well into the nineteenth century that the critical is assumed to Euclid to the extreme, culminating in the proposed or alternative geometries by Bolyai (1802 - 1860), Lobachewski (1792 - 1856) and Riemann (1826 - 1866), either in a full review of the fundamentals of Euclidean geometry by Pasch ( 1843 - 1930) and Hilbert (1862 - 1943), or in the emergence of new concepts about the classification of geometries by Felix Klein (1849-1925).
None of this value removes the monumental work of Euclid. As they say Borsuk (1905 - 1982) and Szmielew (Foundations of Geometry, 1960):
"If the value of a scientific work can be measured by time during which it retains its importance, then Euclid's Elements are the most valid scientific work of all time. "
source:" http://www.educ.fc.ul.pt/docentes/opombo/seminario/ "
The thirteen books
* Books I-IV deal with elementary plane geometry. Using the most basic properties of lines and angles lead to the congruence of triangles, equal areas, the Pythagorean Theorem (Book I, Proposition 47) and its reciprocal (Book I, Proposition 48), the construction of a square area equal to a given rectangle, the golden section, the circle and regular polygons. The Pythagorean theorem and the golden section are introduced as properties of areas.
Like most of the thirteen books, the Book I begins with a list of settings (23 in total) without comment, for example, point, line, circle, triangle, angle, parallelism and perpendicularity of lines such as
"is a point which has no part,"
"a line is a length without breadth"
"is a surface which has only length and width."
Following the definitions, there are the postulates and common notions or axioms, in that order. Postulates are specific geometric propositions. "Participate" means "ask to accept." Thus, Euclid asks the reader to accept the five propositions formulated in geometric postulates:
1. Given two points, there is a line segment that joins them;
2. A line segment may be extended indefinitely to build a line;
3. Given a point any distance and any one can construct a circle at center point and radius equal to the distance given;
4. All right angles are equal;
5. If a straight line cutting two other lines so that the sum of two interior angles on the same side is less than two brackets, so these two lines, if sufficiently prolonged, intersect on the same side on which are these two angles
(This is the famous 5th postulate of Euclid)
Thus, three core concepts - the point , of the line and the circle - and postulates relating to them, are the basis for all of Euclidean geometry.
V * The book presents the theory of proportions of Eudoxus (408 BC - 355 BC) in its purely geometric and
* Book VI applies to the similarity of plane figures. Here we return to the Pythagorean theorem and the golden section (Book VI, proposition 31 and 30) but now as theorems concerning the reasons for greatness. Of particular interest is the theorem (Book VI, proposition 27) that contains the maximum of the first problem that came to us with proof that the square is, of all rectangles of a given perimeter, which has maximum area.
* Books VII-IX are devoted to number theory such as the divisibility of integers, the addition of geometric series, some properties of prime numbers and proof of the irrationality of the number. Here we find both the 'Euclidean algorithm' for finding the greatest common divisor of two numbers, such as "Euclid" according to which there is an infinity of primes (Book IX, proposition 20).
X * The book, the longest of all and often considered the most difficult, containing the geometric classification of irrational quadratic and quadratic roots.
* Books XI-XIII are concerned with the solid geometry, leading through the solid angles, the volume of the parallelepiped, prism and pyramid, the ball and what seems to have been considered the climax - the discussion of five regular polyhedra ('Platonic') and proof that these exist only five regular polyhedra.
Final Thoughts
When writing the Elements, Euclid was to collect a text three major findings of their recent past: the theory of proportions of Eudoxus, the theory of irrational Theaetetus (417 BC - 369 BC) and the theory of the five regular solids, which occupied an important place in Plato's cosmology.
in the Elements Euclid compiled all the known geometry of his day. But not content to gather all the knowledge of geometry, and ordered him framed him as a science. That is, from a few axioms developed and demonstrated the theorems and geometric propositions and give further statements when the former does not adapted themselves to the new order that was given to the proposals. Also, thoroughly studied by him the properties of geometric figures, areas and volumes and established the concept of locus.
While some elements have logical weaknesses, by current standards, these shortcomings have been overlooked for over two millennia. The critical movement began perhaps in the late seventeenth century, John Wallis (1616-1703), still a little fuzzy during the next century, the Jesuit Abbe Saccheri (1667-1733) and mathematicians Lambert (1728-1777) and Gauss (1777-1855). It is already well into the nineteenth century that the critical is assumed to Euclid to the extreme, culminating in the proposed or alternative geometries by Bolyai (1802 - 1860), Lobachewski (1792 - 1856) and Riemann (1826 - 1866), either in a full review of the fundamentals of Euclidean geometry by Pasch ( 1843 - 1930) and Hilbert (1862 - 1943), or in the emergence of new concepts about the classification of geometries by Felix Klein (1849-1925).
None of this value removes the monumental work of Euclid. As they say Borsuk (1905 - 1982) and Szmielew (Foundations of Geometry, 1960):
"If the value of a scientific work can be measured by time during which it retains its importance, then Euclid's Elements are the most valid scientific work of all time. "
source:" http://www.educ.fc.ul.pt/docentes/opombo/seminario/ "
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