In the ancient days, people were willing to know the volume of solid shapes to store goods and also for construction purposes. The theory of conic sections, which was a part of Greek geometry found its application in astronomy and optics. Another important and notable discovery was that of Pythagoras. Pythagoras was an ancient Greek mathematician, who discovered that there is a relationship between the sides of a right-angled triangle and gave it proof.
The Pythagoras theorem finds its application in the field of trigonometry. It uses the concepts of both algebra and geometry. Euclidean geometry deals with the study of length, area, and volume of solid shapes based on certain axioms and theorems. It was developed by a Greek mathematician called Euclid. This branch of geometry deals with terms like points, lines, surfaces, dimensions of the solids, etc.
Analytical geometry also referred to as coordinate geometry or cartesian geometry deals with the coordinate system to represent lines and points. In analytical geometry a point is represented by two or three numbers to denote its position on a plane, this is called a coordinate point. It is written in the form of 3,4 , where 3 is the x-coordinate and 4 is the y-coordinate.
We can also have a point 2, 3, 4 where 2 is the x-coordinate, 3 is the y-coordinate and 4 is the z-coordinate. This branch of geometry uses algebraic equations and methods to solve problems. It also deals with midpoint, parallel and perpendicular lines, line equations, distances between two linear paths. The figure below shows a point 3,4 on a coordinate plane. A branch of geometry that deals with geometric images when they are projected into another surface.
It is more inclined towards the point of view of an object. Also, projective geometry does not involve any angle measures. It involves only construction using straight lines and points. A branch of geometry that deals with curved surfaces and investigating geometrical structures, calculating variations in manifolds, and many more. It uses the concepts of differential calculus.
It is mainly used in physics and chemistry for various calculations. Topology is a branch of geometry, which deals with the study of properties of objects that are stretched, resized, and deformed.
Topology deals with curves, surfaces, and objects in a three-dimensional surface or a plane. Check out these interesting articles to know more about the origin of geometry and its related topics. Example 1: Find the area of a circle with radius of 7 units. The fundamental relationships were based on ideas of 'projection and section' which means that any rigid Euclidean shape can be transformed into another 'similar' shape by a perspective transformation.
A square can be transformed into a parallelogram think of shadow play and while the number and order of the sides remain the same, their length varies. In the late 18th century Desargues' work was rediscovered, and developed both theoretically and practically into a coherent system, with central concepts of invariance and duality. In Projective geometry lengths, and ratios of lengths, angles and the shapes of figures, can all change under projection.
Parallel lines do not exist because any pair of distinct lines intersect in a point. Another important concept in projective geometry is duality. In the plane, the terms 'point' and 'line' are dual and can be interchanged in any valid statement to yield another valid statement.
In spite of the practical inventions of Spherical Trigonometry by Arab Astronomers, of Perspective Geometry by Renaissance Painters, and Projective Geometry by Desargues and later 18th century mathematicians, Euclidean Geometry was still held to be the true geometry of the real world. Nevertheless, mathematicians still worried about the validity of the parallel postulate. In the English mathematician John Wallis had translated the work of al-Tusi and followed his line of reasoning.
To prove the fifth postulate he assumed that for every figure there is a similar one of arbitrary size. However, Wallis realized that his proof was based on an assumption equivalent to the parallel postulate. Saccheri's title page. Girolamo Saccheri entered the Jesuit Order in He went to Milan, studied philosophy and theology and mathematics.
He became a priest and taught at various Jesuit Colleges, finally teaching philosophy and theology at Pavia, and holding the chair of mathematics there until his death. Saccheri knew about the work of the Arab mathematicians and followed the reasoning of al-Tusi in his investigation of the parallel postulate, and in he published his famous book, Euclid Freed from Every Flaw. Saccheri assumes that i a straight line divides the plane into two separate regions and ii that straight line can be infinite in extent.
These assumptions are incompatible with the obtuse angle case, and so this is rejected. However, they are compatible with the acute angle case, and we can see from his diagram fig. The irony is that in the next twenty or so pages, in order to show that the acute angle case is impossible, he demonstrates a number of elegant theorems of non-Euclidean geometry!
It was clear that Saccheri could not cope with a perfectly logical conclusion that appeared to him to be against common sense. Saccheri's work was virtually unknown until when it was discovered and republished by the Italian mathematician, Eugenio Beltrami As far as we know it had no influence on Lambert, Legendre or Gauss. Johan Heinrich Lambert Johan Heinrich Lambert followed a similar plan to Saccheri.
He investigated the hypothesis of the acute angle without obtaining a contradiction. Lambert noticed the curious fact that, in this new geometry, the angle sum of a triangle increased as the area of the triangle decreased.
Many of the consequences of the Parallel Postulate, taken with the other four axioms for plane geometry, can be shown logically to imply the Parallel Postulate. For example, these statements can also be regarded as equivalent to the Parallel Postulate.
Carl Friedrich Gauss Gauss was the first person to truly understand the problem of parallels. He began work on the fifth postulate by attempting to prove it from the other four. But by he was convinced that the fifth postulate was independent of the other four, and then began to work on a geometry where more than one line can be drawn through a given point parallel to a given line.
He told one or two close friends about his work, though he never published it and in a private letter of he wrote:. The final breakthrough was made quite independently by two men, and it is clear that both Bolyai and Lobachevski were completely unaware of each other's work.
Nikolai Ivanovich Lobachevski Nikolai Ivanovich Lobachevski did not try to prove the fifth postulate but worked on a geometry where the fifth postulate does not necessarily hold. Lobachevski thought of Euclidean geometry as a special case of this more general geometry, and so was more open to strange and unusual possibilities.
In he published the first account of his investigations in Russian in a journal of the university of Kazan but it was not noticed. His original work, Geometriya had already been completed in , but not published until Lobachevski explained how his geometry works, "All straight lines which in a plane go out from a point can, with reference to a given straight line in the same plane, be divided into two classes - into cutting and non-cutting.
The boundary lines of the one and the other class of those lines will be called parallel to the given line.
Farkas had worked on the problem of the fifth postulate, but had not been able to make any headway. He continued to work in mathematics, presenting some original ideas, but his enthusiasm and health deteriorated and he never published again.
Lobachevski and Bolyai had discovered what we now call Hyperbolic Geometry. This is the geometry of the acute angle hypothesis where a 'line' is no longer a straight line and there are many possible lines through a given point which do not intersect another line. This is very difficult to visualize, and for people brought up to believe Euclidean geometry was 'true' this was counter-intuitive and unacceptable. Eugenio Beltrami It was not until Beltrami produced the first model for hyperbolic geometry on the surface of a pseudo-sphere in that many mathematicians began to accept this strange new geometry.
Imagine a circular polar grid like a dart board pulled up from the origin. The theory states that the sum of the squares of the legs of a right triangle equals the square of the hypotenuse. Euclid is best known for his book treatise The Elements. The Elements is one of the most important works in history and had a profound impact on the development of Western civilization.
Euclid began The Elements with just a few basics, 23 definitions, 5 postulates, and 5 common notions or general axioms.
An axiom is a statement that is accepted as true. From these basics, he proved his first proposition. Once proof was established for his first proposition, it could then be used as part of the proof of a second proposition, then a third, and on it went.
This process is known as the axiomatic approach. Archimedes of Syracuse — BC is regarded as the greatest of the Greek mathematicians and was also the inventor of many mechanical devices including the screw, the pulley, and the lever. The Archimedean screw — a device for raising water from a low level to a higher one — is an invention that is still in use today.
Archimedes works include his treatise Measurement of a Circle , which was an analysis of circular area, and his masterpiece On the Sphere and the Cylinder in which he determined the volumes and surface areas of spheres and cylinders.
There were no major developments in geometry until the appearance of Rene Descartes — In his famous treatise Discourse on the Method of Rightly Conducting the Reason in the Search for Truth in the Sciences , Descartes combined algebra and geometry to create analytic geometry.
0コメント