Geometry and Trigonometry

Geometry and Trigonometry — page 5 — EUCLIDEAN GEOMETRY he ancient branch of mathematics known as geometry deals with points lines sur faces and solids and their relationships In particular geometry may be thought of as offering W precise definitions of many dif ferent figures 2 construction methods for drawing figures 3 a wealth of facts about the figures and most important 4 ways to prove the facts Geometry was thoroughly organized in about 300 BCE when the Greek mathemati cian Euclid gathered what was known at the time added original work of his own and arranged 465 propositions into 13 books col lectively called Elements The books covered not only plane and solid geometry but also much of what is now known as algebra trigo nometry and advanced arithmetic 10

Geometry and Trigonometry — page 6 — EUCLIDEAN GEOMETRY Down through the ages the propositions have been rearranged and many of the proofs are different but the basic idea presented in the Elements has not changed In the work facts are not just cataloged but are developed in an orderly way starting with statements defini tions common notions and postulates that seem perfectly self evident with each succes sive theorem proved by using only previously shown facts This mode of reasoning known as the axiomatic method has profoundly influ enced epistemology the study of the nature origin and limits of human knowledge and education Even in 300 BCE geometry was recognized to be not just for mathematicians Anyone can benefit from the basic teachings of geometry which are how to follow lines of reasoning how to say precisely what is intended and especially how to prove basic concepts by fol lowing these lines of reasoning Taking a course in geometry is beneficial for all students who will find that learning to reason and prove convincingly is necessary for every profession It is true that not everyone must prove things but everyone is exposed to proof Politicians advertisers and many other people try to offer convincing arguments Anyone who cannot tell a good proof from a bad one may easily be persuaded in the wrong direction Geometry 11

Geometry and Trigonometry — page 7 — Euclid works with a compass a tool used in drawing circles in a 14th century marble panel on the Campanile bell tower of Santa Maria del Fiore in Florence Italy A Dagli Orti De Agostini Picture Library Getty Images 12

Geometry and Trigonometry — page 8 — EUCLIDEAN GEOMETRY EUCLID It has been said that next to the Bible the Elements of Euclid is the most translated published and studied tome in the Western world Of the author himself almost nothing is known It is recorded that he founded and taught at a school of mathematics in Alexandria Egypt during the reign of Ptolemy I Soter who ruled from 323 to about 283 BCE It is assumed from his books that he was not a first class mathematician but he was a first rate teacher of geometry and arithmetic The Elements remained unchallenged for more than 2 000 years Not until the mid 19th century was a non Euclidean geometry devised To compile his Elements Euclid relied on the work of several predecessors so the treatise is of uneven quality Once it was published the Elements superseded all pre vious mathematical treatises and became the standard text During the Middle Ages three Arabic translations were made of the work and it was through these that it became known in Europe The English traveler and philosopher Adelard of Bath went to Spain disguised as a Muslim student and obtained an Arabic copy from which he made a Latin translation in 1120 The first Latin translation of the Greek without an Arabic intermedi ary was made by Bartolomeo Zamberti and published in Venice in 1505 There have been several more recent translations into other languages Among Euclid's other works on geometry were Data and On Divisions He also wrote Optics and The Elements of Music Some of his writings have been lost while other works are wrongly credited to him 13

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