WHAT IS THE BACKGROUND OF GEOMETRY? - SCIENCE - 2024

The geometry, with a history since the time of the Egyptian pharaohs, is the branch of mathematics that studies the properties and figures in a plane or space.

There are texts belonging to Herodotus and Strabo and one of the most important treatises on geometry, The elements of Euclid, was written in the 3rd century BC by the Greek mathematician. This treatise 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 astronomy and cartography. It practically did not undergo any modification until René Descartes arrived in the seventeenth century.

Descartes's studies linking geometry with algebra brought about 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. Mathematical and geometric developments begin to be linked until the arrival of our days.

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Early Geometry Backgrounds

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 was basically used to measure parcels of land, that is where the name of geometry comes from, which in ancient Greek means measurement of the land.

Greek geometry

The Greeks were the first to use geometry as a formal science, and they began to use geometric shapes to define forms of common things.

Thales of Miletus was one of the first Greeks to contribute to the advancement of geometry. He spent a long 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 the pyramids of Egypt, measuring their shadow at the exact moment when their height was equal to the measure of their shadow.

Then came Pythagoras and his disciples, the Pythagoreans, who made important advances in geometry that are still used today. They still did not distinguish 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 true for being intuitive and deduced the other results from them.

After Euclid was Archimedes, who made studies of curves and introduced the figure of the spiral. In addition to the calculation of the sphere based on calculations that are made with cones and cylinders.

Anaxagoras tried unsuccessfully to square a circle. This involved finding a square whose area measured the same as a given circle, leaving that problem for later geometers.

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 no notable geometrist appeared during the Middle Ages.

Geometry in the Renaissance

It is in this period that geometry begins to be used projectively. An attempt is made to find the geometric properties of objects to create new forms, especially in art.

Leonardo da Vinci's studies stand out where knowledge of geometry is applied to use perspectives and sections in his designs.

It is known as projective geometry, because it tried to copy geometric properties to create new objects.

The Vitruvian Man by Da Vinci

Geometry in the Modern Age

Geometry as we know it underwent a breakthrough in the Modern Age with the appearance of analytical geometry.

Descartes is in charge of promoting a new method to solve geometric problems. Algebraic equations begin to be used to solve geometry problems. These equations are easily representable on a Cartesian coordinate axis.

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 the first degree and the circles and other curves as equations of the second degree.

Descartes' theory was later supplemented, since negative numbers were not yet used in his time.

New methods in geometry

With Descartes's advance in analytical geometry, a new paradigm of geometry begins. The new paradigm establishes an algebraic resolution of the problems, instead of using axioms and definitions and from them obtaining the theorems, which is known as the synthetic method.

The synthetic method gradually ceased to be used, disappearing as a geometry research formula towards the 20th century, remaining in the background and as a closed discipline, of which formulas are still used for geometric calculations.

Advances in algebra that have developed since the 15th century help geometry to solve equations of the third and fourth degree.

This allows new shapes of curves to be analyzed that until now were impossible to obtain mathematically and that could not be drawn with a ruler and compass.

Rene Descartes

With the algebraic advances, a third axis is used in the coordinate axis that helps to develop the idea of ​​tangents with respect to curves.

Advances in geometry also helped develop the infinitesimal calculus. Euler began to postulate the difference between a curve and a function of two variables. In addition to developing the study of surfaces.

Until the appearance of Gauss, geometry was used for 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 plane curve, and to develop the Implicit Function Theorem.

References

1. BOI, Luciano; FLAMENT, Dominique; SALANSKIS, Jean-Michel (ed.). 1830-1930: a century of geometry: epistemology, history and mathematics. Springer, 1992.
2. KATZ, Victor J. History of mathematics. Pearson, 2014.
3. LACHTERMAN, David Rapport. The ethics of geometry: a genealogy of modernity.
4. BOYER, Carl B. History of analytic geometry. Courier Corporation, 2012.
5. MARIOTTI, Maria A., et al. Approaching Geometry theorems in contexts: from history and epistemology to cognition.
6. STILLWELL, John. Mathematics and its History.The Australian Mathem. Soc, 2002, p. 168.
7. HENDERSON, David Wilson; TAIMINA, Daina.Experiencing geometry: Euclidean and non-Euclidean with history. Prentice Hall, 2005.