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
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relied on the work of several predecessors so the treatise is of uneven quality Once it was published the Elements superseded all pre vious mathematical treatises and became the standard text During the Middle Ages three Arabic translations were made of the work and it was through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
Geometry and Trigonometry provides a simplified universe where points and lines obey believable rules and where conclusions are easily verified By first study ing how to reason in this simplified universe people can eventually through practice and experience learn how to reason in a compli cated world Geometry in ancient times was recognized as part of everyone s education Early Greek philos ophers asked that no one come to their schools who had not learned the Elements of Euclid There were and still are many who resisted this kind of education It is said that Ptolemy I asked Euclid for an easier way to learn the material Euclid told him there was no royal road to geometry The same message applies to readers today They will not learn what geometry is all about What they will learn is the basic shapes of some of the fig ures dealt with in geometry and a few facts about them It takes a geometry course with textbook and teacher to show the complete and orderly arrangement of the facts and how each is proved POINTS LINES AND ANGLES Geometric figures are based on points and lines Euclid began his Elements with a series of definitions starting with these two A point is that which has no part A line is breadthless length 14
EUCLIDEAN GEOMETRY Because he had not explained the words part breadth or length he was not defining these figures in terms of simpler things The fact is that points and lines are so simple that they cannot be defined perfectly but other things can be defined in terms of them Definitions in this volume will not be stated in full detail as they would be in a geometry text book but they will be described briefly to show how they are linked lead ing from points and lines to very special figures A line is usually drawn with arrowheads to show that it extends without end in both directions A ray is half of a line with just one Some different geometric figures are from top endpoint Two rays to bottom a line a ray an angle a straight with the same endpoint angle and a right angle The latter two are also shown as an open book Encyclopaedia form an angle If the Britannica Inc rays are the two halves 15
