Geometry is a crucial component of everyday life. With many applications, geometry is a cornerstone to a myriad of professions, such as construction, architecture, interior designers, and those in computer design. Students studying geometry are not introduced or pre-exposed to the early development of this subject. Teachers often go straight to the concepts and problems when presenting data about it. The subject of geometry spins back to 3,000 BC era in ancient Egypt (Berlinghoff and Gouvea 2). Geometry has evolved for a considerable period since the ancient Egyptian era to the modern non-Euclidean time. Theorists, such as Euclid, Rene Descartes, and Gauss, have had a substantial contribution to the development of geometry. Thus, it is interesting to analyze how geometry has evolved through the ages.

Ancient Egyptian Application

Geometry can trace its origin to the ancient Egyptian mathematicians who predominantly used crude geometrical concepts in construction works. Egyptian geometry was developed and used in from 3,000BC to 300BC (Berlinghoff and Gouvea 66). Ancient Egyptians were crucial in the development of geometrical concepts that are predominantly used today, including area and volume. As argued by Van der Waerden , ancient Egyptian scribes wrote their problems using multiple parts to solve the area of representation (25). The mathematicians would provide titles to their problems and the data needed to solve them before answering and verifying that the solution was correct. However, the scribes did not use any variables, as questions were represented in prose form.

The Egyptians would calculate the area of triangle, rectangles, and circles. For example, the ancient Egyptians would calculate the area of a triangle using the formula area equals half-base multiplied by width, and the triangle has base multiplied by width. The ancient Egyptian scribes were very practical with their mathematics, and most problems were solved on construction sites. It is well documented how Egyptians built monuments in the form of huge geometric objects that were spanning for meters in length and height (Van der Waerden 72). The Great Pyramid of Giza demonstrates a perfect use of geometry to design and construct an object. The base is a near perfect square, the four sides are perfectly aligned to the four key compass points, and the ratio of the perimeter of the square to the height of the monument is near 2-pie. Thus, the Egyptians significantly contributed to the modern geometry through the practical application of concepts in prose form to make geometrically-sound objects.

Euclid’s Elements

The next substantial contribution to geometry was witnessed during the period of Euclid. Euclid’s elements are one of the most influential scientific works in the history of humanity. The beauty emanates from the element's logical development of geometry. Euclid wrote a text titled Elements in 300 BC. In it, he postulated ideal axiomatic forms where small sets of statements accepted as true would be used to prove propositions (Berlinghoff and Gouvea 142). Euclid was able to create planar geometry by using the five postulates in is elements. The elements show that a straight-line piece can be drawn joining any two points. With a straight-line segment, a person can draw a circle with the segment as its radius and one end-point as the center of the circle. Euclid postulated that all right-angled triangles are congruent, and if two lines are drawn intersecting at the third line in a manner that the sum of their inner angles is less than two right angles on their one side, then the two lines so drawn should crisscross one another inevitably on the side where they are extrapolated.

The elements in 13 books written by Euclid remain the key blueprint in the application of logic to mathematics, specifically geometry. From a historical context, Euclid’s elements have influenced great scientists and mathematicians throughout the history of science. Renowned personalities, such as Copernicus, Galilei, Newton, and Kepler, were great followers of Euclid’s elements and even applied some of his principles in their works (Berlinghoff and Gouvea 166). It is, therefore, safe to outline that Euclid laid a strong foundation to mathematics, especially the field of geometry. The principle’s constructive propositions demonstrate the existence of a figure by detailing the procedures one can use to construct an object.

Rene Descartes’ Coordinate Geometry

The contribution of Descartes to the subject of mathematics in general and the field of geometry cannot be wished away when discussing the development of geometry. Rene Descartes developed the coordinate system, commonly known as the Cartesian system, in 17th century. According to Berlinghoff and Gouvea (170), a person can easily locate a certain point on a plane or map by using the Cartesian coordinates system through identification of its relative distance from perpendicular intersecting planes. All lines, points, and figures are drawn in a coordinate plane in Cartesian geometry. It allows any point, line, or figure to be easily and precisely located using two coordinates. In Rene Descartes’ coordinate geometry system, the first coordinate value is denoted by x (called x-coordinate), and it is found on the horizontal axis of the Cartesian plane. The second coordinate is known as the y-coordinate and is found on the vertical axis of the plane. With clearly defined coordinates, one can easily duplicate geometric figures with a great level of precision. A triangle, for example, can be replicated by using its coordinates to find the three vertices. Such vertices can then be joined with straight lines to form the duplicate triangle.

Descartes’ Cartesian coordinates have a myriad of applications in the real world, as they have been adopted in other fields of science, including physics and mathematics. Rene Descartes’ invention of the Cartesian coordinates revolutionized the way how the mathematics is understood, and solutions to mathematical problems are solved (Berlinghoff and Gouvea 173). The Cartesian geometry provided the first systematic link between algebra and Euclidean geometry. It is possible to describe a geometric phenomenon, including the nature of shapes and figures, using the Cartesian coordinates model. Mathematicians can define definite shapes, such as curves, if they have specific coordinates for identifying the shapes on the Cartesian plane. It would involve the formation and interpretation of algebraic equations that have coordinates of the points lying on the plane. For example, an equation x2 + y2 = 4 describes a circle of radius, two units centered at the origin of the Cartesian plane. Cartesian coordinates have revolutionaries the field of geometry. The principles are the foundation of analytic geometry, as they provide enlightening interpretation of other areas of mathematics that can be utilized in geometric projections and computations (Dunham 158). Cartesian geometry is used in linear algebra, multivariate calculus, differential geometry, and group theory among others. Many concepts in physics now require the knowledge of coordinate geometry, which is explained by the Cartesian coordinate system. The study of electromagnetism presupposes Cartesian knowledge since an electron projected with its velocity parallel to an electric field undergoes parabolic trajectory. Examining this trajectory needs coordinate geometry knowledge.

Development of non-Euclidian Geometry

The modern geometry is attributed to the contributions of Friedrich Gauss, who revolutionized the analysis of surfaces and laid the foundations for non-Euclidean geometry. Gauss, as a young man, has already devised a way to construct a 17-sided rectangular polygon using the compass and straightedge tools only. As a surveyor, Gauss mapped out a large part of Europe (Dunham 236). Gauss triangulated different locations into mapped regions, being divided by great circle arcs. Gauss was not alone in the pursuit of non-Euclidean geometry but was flanked by other mathematicians, such as Janos Bolyai and Nicolai Ivanovic from Hungary and Russia respectively. The three non-Euclidean geometrics realized that it was possible to achieve a two-dimensional geometry using the first four axioms as postulated by Euclid but impossible with the fifth axiom. According to the parallel postulate, for any point not on a given line, there is one line through the point that would not meet the given line. The non-Euclidean geometry pioneers found systems based on the alternatives to the Euclidean fifth element.

The non-Euclidean founders developed an alternative axiom stating that there could be more than one line passing through a given point but not meeting a given line, leading to the founding of the hyperbolic geometry. Even though the theorem postulated by Bolyai and Nicolai seemed strange, they were similar in consistency (Dunham 263). The diagrams these theorists used to illustrate the alternative axiom to Euclidean theory were not identical to Euclidean text. Scientist Herman von Helmholtz was one of the first expositors of the non-Euclidean geometry. Helmholtz used the dimensional analogy to illustrate the concept of non-Euclidean two-dimensional geometry. Accordingly, the expositor requested his readers to consider a two—dimensional creature that made strictly slide along a piece of marble, taking measurements of its curves an angle size. The expositor suggested the use of a pseudosphere surface, which has a sharp edge but illustrated most of the critical characteristics to hyperbolic geometry (Daus 13). The alternative axiom postulated by the non-Euclidean geometers led to the development of elliptic geometry. The elliptic geometry was reminiscent of the geometry of the sphere, where two great circles converge. Nevertheless, the spherical geometry has imitations, as the straight lines met twice. As a solution to this, the sphere would be cut into two, threw away half the points of the sphere, and retain the southern hemisphere.

Non-Euclidean geometry has profound historical and practical significance. The development of this modern geometry theory caused a huge revolution in mathematics science as well as philosophy. Non-Euclidean geometry is significant in philosophy since it explains the link between various aspects of philosophy, such as the relationship among science, mathematics, and observations. Initially, many scientists and mathematicians have assumed that the Euclidean geometry correctly explained the concept of physical space (Dunham 214). Several theorists have attempted to find the way in which Euclidean geometry is used to describe variables in physical space. For example, Kant attempted to explain this phenomenon based on the basic elements postulated by Euclid. Even Gauss himself comprehended that the only way to ascertain the truth of Euclidean postulate was through its subjection to experiment. Gauss went ahead to measure the triangles angles made by three mountain peaks to determine whether they all added to 180. Various inconsistencies marred the results, thus being inconclusive. The non-Euclidean geometry also has scientific significance, as it paved the way for the development of Riemannian geometry. The Riemannian geometry gave way for the development of Albert Einstein's General Theory of Relativity.

Non-Euclidean geometry is any kind or type of geometry that is fundamentally different from Euclidean geometry. Non-Euclidean geometers have done studies in the field and come up with different modern geometrical concepts, including spherical and hyperbolic geometries. However, Daus mentions that the fundamental difference between the two geometries stems from the nature of the parallel lines (14). The Euclidean geometry has been defined by the exact words, as the originator postulated the axiom that with a straight-line segment, a person can draw a circle with the segment as its radius and one end-point as the center of the circle. Spherical geometry negates that there exists such a line. Hyperbolic geometry, on the other hand, states that more than two lines passing through a point are parallel with one another.

Conclusion

The paper has discussed the development of geometry since its crude periods in the ancient Egyptian era to the modern geometry learned in institutions and practiced in daily applications. It has been established that geometry has come a long way. The ancient Egyptians used geometry to make monumental structures that have outlived history. Even thought their geometry was not conveyed and used in the forms of variables but was rather a prose, the precision with which structures were constructed points to the accuracy of their applied geometrical knowledge. Euclid then revolutionized the way geometry is understood and used in a mathematical and scientific application. Euclidean geometry has shaped many scientific and mathematical theories since its postulation. Rene Descartes used the Euclidean geometry as the basis for his coordinate geometry system, popularly known as the Cartesian plane. The Cartesian plane is now a concept strongly enshrined in mathematics and science. Later on, there came post-Euclidean geometers who found subtle faults in theory and suggested new axioms that could be used to cure the limitations. The non-Euclidean geometers managed to bring about the development of contemporary geometries, including spherical geometry and hyperbolic geometry. Therefore, as many problems continue to emerge, the field of geometry still faces new challenges and attracts the need for more modern axioms to solve these emerging issues. Even though geometry has come a long and exciting way, there is still much to be expected in the future.

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