Euclid

Euclid (c. 325 BCE c. 265 CE) was of the best selling mathematics textbook of all time. The Elements has introduced millions to the owers of logic and language. !s a struggling law"er at age fort"# !braham $incoln reflected% &' said# ($incoln# "ou can ne)er make a law"er if "ou do not understand what demonstrate means*+ and ' left m" situation in , ringfield# went home to m" father*s house# and sta"ed there till ' could gi)e an" ro osition in the six books of Euclid at sight. ' then found out what (demonstrate*# means# and went back to m" law studies.-

Euclid did not call his work *geometr"+* the word didn*t a

ear an" lace in his thirteen books#

ossibl" because at the time geometr" referred to land measure. Euclid*s Elements# written some time around 3.. BCE# was in use for some twent"/three centuries. 0is name is better known than that of an" other mathematician who e)er li)ed. 1oda"*s high school geometr" students do not use the historic Elements# and it*s robabl" a good thing. 'n his book Dots and Lines# 2ichard 3. 1rudeau notes what isn*t and what is found in The Elements% &4eo le who read The Elements for the first time often get a feeling that things are missing% it has no reface or introduction# no statement of ob5ecti)es# and it offers no moti)ation or commentar". 6ost strikingl"# there is no mention of the scientific and technological uses to which man" of the theorems can be ut# nor an" warning that large sections of the work will ha)e no ractical use at all7. 1he theorems are included for their own sake# because the" are interesting in themsel)es. 1his attitude of self/sufficienc" is the hallmark of ure mathematics.-

Euclid was the first ma5or scholar at the $ibrar" of !lexandria in Eg" t# founded b" 8ing 4tolem"# the illegitimate son and successor of !lexander the 9reat. 1he $ibrar" / sometimes called the 6useum# coined because the institution was dedicated to the se)en muses / was a center of learning as well as a re ositor" of knowledge. Estimates of the number of its scrolls when !lexandria was the cultural center of the 0ellenistic range from 2..#... to :..#.... 't re resented the world*s knowledge at that time. 4tolem"*s successors went to great lengths to obtain scrolls that could be co ied and become art of the collection. ;esiring a 9reek translation of the 0ebrew Bible# 4tolem" '' im risoned se)ent" 3ewish scholars in cells on the island of 4haros until the" com leted the work. 2e ortedl"# 4tolem" ''' wrote to all the world*s rulers# asking to borrow their books# and the ones that were sent to him# he ke t.

<f the man# little is known. ,ome scholars ha)e ex lored the idea that Euclid was a leader of a band of mathematicians who working together roduced the Elements# making him a sort of earl" da" =icolas Bourbaki. >hile there isn*t an" e)idence to refute this there isn*t an" credible reason to belie)e it either. 1here is e)en less reason to belie)e another suggestion that Euclid was not a historical character and that the grou of mathematicians who re ared the Elements named its author in honor of the hiloso her Euclid of 6egara# who li)ed ?.. "ears earlier. 1he im ortant thing is what The Elements has meant to mathematics. ! arentl" its author atientl" collected all the geometrical facts known in

his da"# arranged the theorems in a logical order# im ro)ed their roofs where necessar"# and added theorems from his own in)estigations. 'n his roofs# he 5ustified the ste s b" references to re)iousl" ro)ed results. 0is organi@ation and logical demonstrations remained a model for mathematical reasoning for some 2... "ears. !s !lfred 0oo er wrote in Makers of Mathematics, &0e was the master mind that was able to collect all the muddled# confused ieces of a )ast mathematical 5igsaw u@@le and ut them together in such a wa" that a clear and beautiful icture suddenl" emerged from what had been a welter of odds and ends of mathematical knowledge. &

Euclid stated 23 definitions# fi)e geometric ostulates# fi)e additional axioms# which he called common notions# and ro)ed A65 theorems# all the geometrical knowledge known in his da". Each of the thirteen books of the Elements was written on a se arate roll of archment. Bor twent" centuries the first six books were the student*s usual introduction to lane geometr". Books se)en to nine dealt with number theor"# book ten treated the theor" of irrational numbers# and books ele)en to thirteen featured solid geometr". Euclid did not claim that all of the theorems and roofs found in The Elements were original with him# although some surel" were. 0owe)er the book*s format was all due to him. Euclid wasn*t the first to ha)e made a com ilation of geometr". 0i ocrates of Chios is said to ha)e written

Elements of Geometry in the mid fifth centur" BCE. 1heaetetus of !thens# who li)ed from about A?: BCE to 36C BCE made man" im ortant contributions to mathematics# which are described in Book D and Book D''' of Euclid*s Elements.

1he rinci al translator of Euclid into $atin was !nicius 6anlius ,e)erinus Boethius in the 6th centur" CE. 0e ga)e onl" the definitions and theorems of Book ' with no roofs. 0e did include a large number of ractical a lications of the selected ro ositions. 0is Geometry and Arithmetic were the basis for

most of the mathematical teaching of the earl" medie)al times in Euro e. $ate in the Eth centur"# an !rab cali h came into ossession of a co " of the Elements in the original 9reek. ,e)eral !rabic translations were made from it# including one at the time 0arun al 2ashid# the cali h known to osterit" from ,chehera@ade*s Arabian Nights Tales. 'n ??2. !delard of Bath made a $atin translation from an !rabic )ersion. ,e)eral other $atin translations were made# including the first im ortant mathematical book to be rinted F?AE2). 't contains beautiful figures engra)ed on its wide margins. ,ince this first rinted )ersion# o)er a thousand editions of the work ha)e been ublished. 1he first English translation of the Elements a eared in ?5:..

'n addition to the Elements, Euclid is known to ha)e written an elementar" treatise on conic sections# a

book on surface loci# a collection of geometrical fallacies# and a treatise on orisms Fthat is# ro ositions deri)ed from others). !ll of these are lost. !mong his existing works is a book on o tics# treated geometricall"# which consists of 6? ro ositions based on ?2 assum tions. Euclid also wrote the Phaenomena# a treatise consisting of ?E ro ositions# concerning the basic geometr" of the celestial s here. 0is Data contains CA ro ositions that demonstrate that when certain as ects of a figure are gi)en# then other as ects of the figure are also gi)en. 1hese ro ositions ma" be considered elementar" exercises in anal"sis# su lementing the theorems and roblems found in The Elements On Divisions of

ig!res consists of 36 ro ositions concerning the di)ision of )arious figures into two or more eGual arts or arts in gi)en ratios.

1here was a time# a )er" long time# when Euclid*s Elements was considered to be a sort of secular )ersion of the Bible in that it contained absolute truths. Bor instance# in Prolegomena to any !t!re Meta"hysics, 'mmanuel 8ant wrote# &1here is no single book about meta h"sics like we ha)e in mathematics. 'f "ou want to know what mathematics is# 5ust look at Euclid*s Elements.- >hen non/ Euclidean geometries were created# The Elements was downgraded to the stature of relati)e roof. !t least Euclid had Guite a run. >hether he belie)ed he was dealing with absolute truth# relati)e truth or an" kind of truth# we*ll ne)er know. 0e certainl" didn*t com ile the thirteen books with an" antici ation of them ha)ing an" ractical use. !s !lfred 0oo er suggested# he seemed to be one who had the man" ieces of a 5igsaw u@@le s read out around him# determined to fit them together to re)eal a beautiful whole. 0e wasn*t com letel" successful in this endea)or# for his work was not logicall" erfect and not all of his roofs can be acce ted as such an" more# but it seems ick" to critici@e Euclid for not knowing twent"/three hundred "ears ago what is known now. 'n the 2.th centur"# ;a)id 0ilbert com leted Euclid*s geometr" b" increasing the number of ostulates to 2. and ro)ed the resulting s"stem was com lete and consistent.

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