This idea of proof still dominates pure mathematics in the modern world. Archimedes was a great mathematician and was a master at visualising and manipulating space.
He perfected the methods of integration and devised formulae to calculate the areas of many shapes and the volumes of many solids. He often used the method of exhaustion to uncover formulae. For example, he found a way to mathematically calculate the area underneath a parabolic curve; calculated a value for Pi more accurately than any previous mathematician; and proved that the area of a circle is equal to Pi multiplied by the square of its radius.
He also showed that the volume of a sphere is two thirds the volume of a cylinder with the same height and radius. This last discovery was engraved into his tombstone. Apollonius was a mathematician and astronomer, and he wrote a treatise called 'Conic Sections.
He also wrote extensively on the ideas of tangents to curves, and his work on conics and parabolas would influence the later Islamic scholars and their work on optics. Greek geometry eventually passed into the hands of the great Islamic scholars, who translated it and added to it. In this study of Greek geometry, there were many more Greek mathematicians and geometers who contributed to the history of geometry, but these names are the true giants, the ones that developed geometry as we know it today.
Martyn Shuttleworth Jan 8, Greek Geometry. Retrieved Nov 11, from Explorable. The text in this article is licensed under the Creative Commons-License Attribution 4. That is it. You can use it freely with some kind of link , and we're also okay with people reprinting in publications like books, blogs, newsletters, course-material, papers, wikipedia and presentations with clear attribution. Menu Search. Menu Search Login Sign Up. You must have JavaScript enabled to use this form. Sign up Forgot password.
Leave this field blank :. Search over articles on psychology, science, and experiments. Search form Search :. Reasoning Philosophy Ethics History. Psychology Biology Physics Medicine Anthropology. Martyn Shuttleworth Don't miss these related articles:. Augustine 8. Euclid, illustrating geometry in "The School of Athens", by Raffaello Sanzio Public Domain Certainly, for measuring boundaries and for erecting buildings, humans need to have some inbuilt mechanism and instinct for judging distances, angles, and height.
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No problem, save it as a course and come back to it later. Add to my courses. In his treatise he introduced many new fundamental concepts. The term 'point at infinity' the vanishing point appears for the first time.
He also uses the ideas of a 'cone of vision' and talks about 'pencils of lines', like the lines emanating from the vanishing point, and if you can have a point at infinity, why not more, to make lines at infinity? Legendre proved that the fifth postulate is equivalent to the statement that the sum of the angles of a triangle is equal to two right angles. Legendre also obtained a number of consistent but counter-intuitive results in his investigations, but was unable to bring these ideas together into a consistent system.
Lobachevski tried to get his work Geometrical investigations on the theory of parallels recognized, and an account in French in brought his work on non-Euclidean geometry to a wide audience but the mathematical community was not yet ready to accept these revolutionary ideas.
On this basis he set up two systems of geometry, and searched for theorems that could be valid in both. Clarendon Press, Oxford. Oxford University Press. Main menu Search. Geometry: A History from Practice to Abstraction. Ancient and Classical Geometries As an essential part of their daily lives, ancient cultures knew a considerable amount of geometry as practical measurement and as rules for dividing and combining shapes of different kinds for building temples, palaces and for civil engineering.
For their everyday practical purposes, people lived on a 'flat' Earth. A 'straight line' was a tightly stretched rope, and a circle could be drawn by tracing round a fixed point. Much of the knowledge of these peoples was well-known around the Mediterranean, and when the Greek civilisation began to assert itself in the 4th century BCE, philosophers like Aristotle BCE , developed a particular way of thinking, and promoted a mode of discussion which required the participants to state as clearly as possible the basis of their argument.
In this atmosphere, Greek Logic was born. During this period, Alexandria became one of the important centres of Greek learning and this is where Euclid's Elements of Mathematics was written in about BCE. Following Aristotle's principles, Euclid based his mathematics on a series of definitions of basic objects like points, straight lines, surfaces, angles, circles and triangles, and axioms or postulates.
These were the agreed starting points for his development of mathematics. Almost as soon as Euclid put his pen down, mathematicians and philosophers were having difficulty with the fifth postulate. In contrast to the short statements of the first four, the fifth looked as though it ought to be a theorem, not an axiom, meaning that it ought to be deducible from the other axioms.
We know this from various logical analyses written by other mathematicians. Playfair's Axiom. John Playfair Today, this is known as Playfair's axiom, after the English mathematician John Playfair who wrote an important work on Euclid in , even though this axiom had been known for over years!
Arab mathematicians studied the Greek works, logically analysed the relatively complex statement of the fifth postulate, and produced their own versions. Abul Wafa developed some important ideas in trigonometry and is said to have devised a wall quadrant [See Note 1 below] for the accurate measurement of the declination of stars.
All this was done as part of an investigation into the Moon's orbit in his Theories of the Moon. The Abul Wafa crater is named after him. As a result of his trigonometric investigations, he developed ways of solving some problems of spherical triangles.
Greek astronomers had long since introduced a geometrical model of the universe. Abul Wafa was the first Arab astronomer to use the idea of a spherical triangle to develop ways of measuring the distance between stars on the inside of a sphere. In the accompanying diagram, the blue triangle with sides a, b, and c represents the distances between stars on the inside of a sphere. The apex where the three angles are marked is the position of the observer.
Spherical Triangle. Famous for his poetry, Omar Khayyam was also an outstanding astronomer and mathematician who wrote Commentaries on the difficult postulates of Euclid's book. He tried to prove the fifth postulate and found that he had discovered some non-Euclidean properties of figures.
Omar Khayyam Omar Khayyam Quadrilateral. Omar Khayyam constructed the quadrilateral shown in the figure in an effort to prove that Euclid's fifth postulate could be deduced from the other four. He recognized that if, by connecting C and D, he could prove that the internal angles at the top of the quadrilateral are right angles, then he would have shown that DC is parallel to AB.
Although he showed that the internal angles at the top are equal try it yourself he could not prove that they were right angles. Al-Tusi wrote commentaries on many Greek texts and his work on Euclid's fifth postulate was translated into Latin and can be found in John Wallis' work of Al-Tusi's argument looked at the second part of the statement.
On each side of these perpendiculars, one angle is acute towards A , and the other obtuse towards B. Clearly the perpendicular PQ is longer than each of the others and finally longer than XY. The opposite is also true; perpendicular XY is shorter than all those up to and including EF. So, if any pair of these perpendiculars is chosen to make a rectangle, the rectangle will contain an acute angle on the A side and an obtuse angle on the B side.
So how can we ensure that the perpendiculars are the same length, or show that both angles are right angles? One of al-Tusi's most important mathematical contributions was to show that the whole system of plane and spherical trigonometry was an independent branch of mathematics. In setting up the system, he discussed the comparison of curved lines and straight lines.
The 'sine formula' for plane triangles had been known for some time, and Al-Tusi established an analogous formula for spherical triangles:. Great Circles Triangle. The important idea here is that Abul Wafa and al-Tusi were dealing with the real problems of astronomy and between them they produced the first real-world non-Euclidean geometry which required calculation for its justification as well as logical argument.
It was the ' Geometry of the Inside of a Sphere '. In the Middle Ages the function of Christian Art was largely hierarchical. Important people were made larger than others in the picture, and sometimes to give the impression of depth, groups of saints or angels were lined up in rows one behind the other like on a football terrace.
Euclid's Optics provided a theoretical geometry of vision, but when the optical work of Al-Haytham became known, artists began to develop new techniques. Pictures in correct perspective appear in the fourteenth century, and methods of constructing the 'pavement' were no doubt handed down from master to apprentice.
Leone Battista Alberti published the first description of the method in , and dedicated his book to Fillipo Brunelleschi who is the person who gave the first correct method for constructing linear perspective and was clearly using this method by Leone Battista Alberti Alberti Perspective Construction. Alberti's method here is called distance point construction. In the centre of the picture plane, mark a line H the horizon and on it mark V the vanishing point. Draw a series of equally spaced lines from V to the bottom of the picture.
Then mark any point Z on the horizon line and draw a line from Z to the corner of the frame underneath H. This line will intersect all the lines from V. The points of intersection give the correct spaces for drawing the horizontal lines of the 'pavement' on which the painting will be based. Piero della Francesca was a highly competent mathematician who wrote treatises on arithmetic and algebra and a classic work on perspective in which he demonstrates the important converse of proposition 21 in Euclid Book VI:b.
Piero's converse showed that if a pair of unequal parallel segments are divided into equal parts, the lines joining corresponding points converge to the vanishing point. Piero della Francesca Piero Euclid VI, 21 diagram. This implies that all the converging lines meet at A, the vanishing point at infinity. Durer "Reclining woman" perspective picture. Albrecht Durer This was a completely new kind of geometry.
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.
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