History of Geometry

HISTORY OF GEOMETRY

ANCIENT GEOMETRY (30000BCE – 500BCE)

Many ancient civilizations like the Babylonians, Egyptians, Hindus and Chinese, laid the foundation for geometry as practiced today. Before recorded history, geometry existed as simply, the science of measuring land and storage containers. The first concepts of geometry “had their origin in simple observations stemming from human ability to recognize physical form and to compare shapes and sizes” (Historical Topics, 165). These types of situations brought into being the geometric concepts of vertical, parallel and perpendicular.

BABYLONIAN (4000 – 500BCE) AND EGYPTIAN (5000 – 500BCE) The geometry of the ancient days was a collection of procedures followed after experimenting, observing, guessing and sometimes even intuition was done. In Egyptian times, geometry was used to conduct the re-surveying of the banks of the River Nile. Due to constant flooding of the river bank, the Egyptians were forced to solve the problem of changes to the size and boundaries of land along the river. This resulted in the discovery of a system of making boundaries, measuring lengths and areas. A chief source of information on Egyptian mathematics can be found in a document called a papyrus. One of the surviving documents of this time is the Moscow Papyrus. About 1900BC, the Moscow papyrus was well known for some of its geometry problems. Problems 10 and 14 computed a surface area and volume of the fructum, a portion of a solid (normally a cone or pyramid) that lies between two parallel planes cutting it, respectively. Another such document, the Rhind papyrus, was written by the Egyptian scribe, Ahmes, who claims to have merely scribed the document. He says that the material comes from work done before 2400BC. The Rhind papyrus, also called the ‘Ahmes Papyrus’, came to a British museum in 1863. It contained rules for division, and has eighty-seven (87) problems including solution to equation, progressions, volumes of granaries, etc.

INDIA (1500 – 200BCE) The science of geometry originated in India with the construction of alters meant for Vedic sacrifices. The Sulbas or the Sulba Sutras are the manuals for the construction of these altars for worship and describe the geometric history of the Indians. These alters had different shapes but occupied the same area. The Salbas contained the earliest verbal expression for the Pythagoras theorem although it was already known by the Babylonians. The postulates of the Sulba are connected with the division of figures such as straight lines, rectangles, circles and triangles.

GREEK (600BC – 400AD) The Greeks worked on geometry in a different way from their predecessors. They transformed geometry from a scientific approach to a more systematic one. The Greeks believed that geometric truth should be derived from studying and proof rather than experimentation. The history of Greek geometry is based on manuscripts that were written many years after the history had begun to develop. According to these manuscripts, one of the first to indulge in finding geometric results using logic and reasoning was Thales of Miletus. He used computations from the Egyptian and Babylonian mathematicians and determined logically which deductions were correct. Another great Greek geometer is Pythagoras of Samos. He founded a school which was committed to the study of philosophy, mathematics and natural science. Under his guidance, student of the school proved that the sum of the angles in a triangle is equivalent to two right angles. They also studied similar figures. One of the students of the Pythagorean School, Hippocrates of Chois, has been credited with writing the first “Elements of Geometry”. The book included early methods of finding solutions to quadratic equations and integral problems. He also studied squaring a circle and squaring a lune. Hippocrates of Chois also was the first to show that the ratio of the areas of two circles was equal to the ratio of the squares of their radii. Plato, although not making any mathematical contribution, insisted on accurate definitions and clear hypotheses, setting a high standard for geometers like Euclid. Euclid collected work from Pythagoras, Hippocrates, and many others and compiled a book called “The Elements”. This book became quite popular and is one of the most read books in history. However, with time several geometer can to content the fifth postulate in the book, the Parallel Postualate. This postulate states that if two straight lines in a plane are crossed by another straight line (called the transversal), and the interior angles between the two lines and the transversal lying on one side of the transversal add up to less than two right angles, then on that side of the transversal, the two lines extended will intersect. Euclid’s approach to geometry, the axiomatic method, is the basis for all “pure mathematics”. In this instance, “pure” means that no physical experiments are necessary. All statements can be deduced through reasoning of demonstrations. Another great mathematician, Archemedes of Syracuse, had many mathematical achievements as well as being the inventor of the screw, the pulley, the lever, and other mechanical devices. He perfected integration using the method of exhaustion. He was also able to find the areas and volumes of many shapes. The Greek geometer, Apollonius, gave us terms like “eclipse”, “parabola” and “hyperbola”. He is credited for having constructed a circle with a tangent to three objects. He also made an approximation of π that was closer than Archimedes of Syracuse. The first woman that made a substantial contribution to mathematics was Hypatia of Alexandria. She worked with her father to produce a new version of Euclid’s Elements and wrote commentaries on the works of other great geometers. She was the first woman in history to be recognized as a professional geometer and mathematician. MEDIEVAL GEOMETRY (600AD – 1500AD) ARABIC/ISLAMIC (600AD – 1500AD) Islamic geometry during this period was more algebraic than geometric. Muslim mathematicians were famed for their work in algebra, number theory and number systems but they made contributions to geometry and mathematical astronomy. Muhammed ibn Mūsā al-Ḵwārizmī was a Persian scholar, mathematician and astronomer who invented algorithm in mathematics the key for computer science. His successors used this systematic approach and applied algebra t the mathematical concepts of that time resulting in the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numeric solutions to equations, the new elementary theory of numbers and the geometric construction of equations. Al-Karaji completely separated algebra from geometry and replaced them with the arithmetical type operations we use today. In the early 11th century the first attempt at proving Euclid’s parallel postulate was made by Idn al-Haytham (Alhazen) using proof by contradiction. The concept of motion and transformation in geometry was introduced by him. Omar Khayyam was known for inventing a general method of solving cubic equations by intersecting a parabola with a circle. He also discovered the binomial expansion and openly critised Euclid’s theories of parallels which later lead to the development of non-Euclidean geometry. Omar also provided methods of solving algebraic equations by geometric means. Muslim craftsmen used their knowledge of geometry to create geometrical designs in the production of tiles, mosaic, plaster and wooden patterns that were used on the buildings.

CHINESE (100BCE – 1400AD) The first definitive work and the oldest in existence on the geometry of China was the ‘Mo Jing’. It was a compilation of work done by philosopher Mozi, produced after his death around 330BC. Due to the infamous ‘Burning of Books’ in the Chinese history, much written literature of that time was burnt. The concepts presented I the ‘Mo Jing’ are advanced and suggests that a previous mathematical based existed. The Han Dynasty (202BCE – 220AD) saw many developments in mathematics. Suan shu shu, the oldest Chinese texts, presents geometric progressions. The mathematician, Zhang Heng used geometrical formulas to create a more accurate estimation for pi (π).

MODERN GEOMETRY (1600 – 2000AD) THE 17TH CENTURY There were two major developments in the early seventeenth century. Firstly, there was the creation of analytical geometry, geometry with coordinates and equations, by Rene Descartes and Pierre de Fermat. Secondly, Girard Desargues began studying a type of geometry called projective geometry, the study of how points align with each other without measurement. In the late seventeenth century Isaac Newton and Gottfried Wilhelm von Leibniz independently and almost simultaneously developed calculus into what is now called analysis. It is not considered a branch of geometry but is applicable to geometry.

THE 18TH AND 19TH CENTURY During this period, many geometers attempted to prove Euclid’s Fifth Postulate, the “Parallel Postulate”. However, all were deemed faulty. They continued the work started by Omar Khayyam who had crititised Euclid’s theories of parallels and proved properties of figures using non-Euclidean geometries. These proofs would lead to the development of non-Euclidean geometry. In the 19th century, geometers such as Carl Friedrich Gauss, Johann Bolyai and Lobatchewsky, each working independently, took a different approach to the “parallel” problem. They began to think that the Parallel Postulate may be impossible to prove and therefore set out to prove that it is false. They were successful and we therefore created the first non- Euclidean geometry. By 1854, Bernhard Riemann, a student of Gauss, applied methods of calculus to develop a geometry based on all smooth surfaces. This became a different non-Euclidean geometry and formed the basis for Einstein’s theory of relativity.

THE 20TH CENTURY In this period, there were developments in algebraic geometry. The study of curves and other surfaces over finite fields were demonstrated in the works of André Weil, Alexander Grothendieck and Jean-Pierre Serre among others.

REFERENCES http://www.softsurfer.com/history.htm http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Ahmes.html http://hubpages.com/hub/History-of-Geometry http://www.google.tt/#q=history+of+geometry+timeline&hl=en&sa=X&biw=1276&bih= 563&tbs=tl:1,tl_num:20&prmd=ivns&ei=wpo5Te2hMMP88AbH482mCg&ved=0CFYQ ywEoAg&fp=a021805f138d5a4b http://www.enotes.com/topic/History_of_geometry