The geometry , With antecedents from the time of the Egyptian pharaohs, is the branch of mathematics that studies the properties and the figures in a plane or space.
There are texts belonging to Herodotus and Strabo and one of the most important treatises of geometry, The elements Of Euclid, was written in the third century BC. By the Greek mathematician. This treaty gave way to a form of study of geometry that lasted for several centuries, being known as Euclidean geometry.
For more than a millennium Euclidean geometry was used to study the astronomy And cartography. It practically did not undergo any modification until Rene Descartes arrived in century XVII.
Descartes' studies linking geometry with algebra implied a shift in the prevailing paradigm of geometry.
Later, the advances discovered by Euler allowed greater precision in geometric calculus, where algebra and geometry begin to be inseparable. The mathematical and geometric developments begin to be linked until the arrival to our days.
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Early history of geometry
Geometry in Egypt
The ancient Greeks said that it was the Egyptians who had taught them the basic principles of geometry.
The basic knowledge of geometry they had basically served to measure plots of land, hence the name of geometry, which in ancient Greek means measurement of the earth.
Greek Geometry
The Greeks were the first to use geometry as a formal science and began to use geometric forms to define habitual forms of things.
Thales of Miletus Was one of the first Greeks to contribute to the advancement of geometry. He spent a lot of time in Egypt and from these he learned the basic knowledge. He was the first to establish formulas for measuring geometry.
Thales of Miletus
He managed to measure the height of Egypt's pyramids by measuring his shadow at the exact moment when his height was equal to the extent of his shadow.
Then came Pythagoras and his disciples, the Pythagoreans, who made important advances in geometry that are still used today. They still made no distinction between geometry and mathematics.
Later Euclid appeared, being the first to establish a clear vision of geometry. It was based on several postulates that were considered truthful to be intuitive and deduced from them the other results.
After Euclid was Archimedes, who made studies of curves and introduced the figure of the spiral. In addition to calculating the sphere based on calculations performed with cones and cylinders.
Anaxagoras tried unsuccessfully to square a circle. This meant finding a square whose area measured the same as a given circle, leaving that problem for later geometricians.
Geometry in the Middle Ages
The Arabs and Hindus were responsible for developing logic and algebra in later centuries, but there is no great contribution to the field of geometry.
Geometry was studied in universities and schools, but there was no mention of geometry during the period of the Middle Ages
Geometry in the Renaissance
It is in this period is when you start to use the geometry in a projective way. One tries to look for the geometric properties of the objects to create new forms, mainly in the art.
Leonardo da Vinci studies where geometry knowledge is applied to use perspectives and sections in their designs.
It is known as projective geometry, because it tried to copy the geometric properties to create new objects.
Vitruvian man by Da Vinci
Geometry in the Modern Age
Geometry as we know it undergoes a breakthrough in the Modern Age with the emergence of analytic geometry.
Descartes is in charge of promoting a new method for solving geometric problems. Algebraic equations are used to solve geometry problems. These equations are easily represented on an axis of Cartesian coordinates.
This model of geometry also allowed to represent objects in the form of algebraic functions, where the lines can be represented as algebraic functions of first degree and the circumferences and rest of curves like equations of second degree.
Descartes' theory was later supplemented, since at that time, negative numbers were not yet used.
New methods in geometry
With the breakthrough in Descartes' analytical geometry, a new paradigm of geometry begins. The new paradigm establishes an algebraic resolution of problems, instead of using axioms and definitions and from them to obtain theorems, which is known as synthetic method.
The synthetic method ceases to be used gradually, disappearing as a research formula of geometry towards the twentieth century, remaining in the background and as a discipline already closed, which still use formulas for geometric calculations.
Advances in algebra that develop from the fifteenth century, help geometry to solve third and fourth degree equations.
This allows you to analyze new forms of curves that until now were impossible to obtain mathematically and could not be drawn with ruler and compass.
Rene Descartes
With the algebraic advances it is begun to use a third axis in the axis of coordinates that helps to develop the idea of tangents with respect to curves.
Advances in geometry also helped to develop infinitesimal calculus. Euler began postulating the difference between curve and function of two variables. In addition to developing the study of surfaces.
Until the appearance of Gauss geometry was used for the mechanics and branches of physics through differential equations, which were used for the measurement of orthogonal curves.
After all these advances, Huygens and Clairaut arrived to discover the calculation of the curvature of a flat curve, and to develop the Implicit Function Theorem.
References
- BOI, Luciano; FLAMENT, Dominique; SALANSKIS, Jean-Michel (ed.), 1830-1930: the century of geometry: epistemology, history and mathematics. Springer, 1992.
- KATZ, Victor J. History of mathematics. Pearson, 2014.
- LACHTERMAN, David Rapport. The ethics of geometry: the genealogy of modernity.
- BOYER, Carl B.History of analytic geometry. Courier Corporation, 2012.
- MARIOTTI, Maria A., et al. Approaching Geometry theorems in contexts: from history and epistemology to cognition.
- STILLWELL, John. Mathematics and its History. The Australian Mathem. Soc, 2002, p. 168.
- HENDERSON, David Wilson; TAIMINA, Daina.Experiencing geometry: Euclidean and non-Euclidean with history. Prentice Hall, 2005.