,transcrib]'s notes3 7#a7 ,numb]s foll[+ lett]s 7)\t space7 l ,c#b 7 orig9ally pr9t$ 9 subscript4 ,lett] subscripts are prec$$ by an und]score1 l ,cn4 7#b7 ,*>act]s foll[+ a c>at 77 7 orig9ally pr9t$ 9 sup]script4 7#c7 root /&s = ! root symbol2 alpha1 beta1 etc4 = greek lett]s4 ,illu/ration3 ,ari/ippus ,philosophus ,socraticus1 nausragio cum ejectus ad ,rhodi5sium litus animadv]tisses ,geometrica ,s*emata descripta1 exclamavisse ad comites ita dicitur1 ,b5e ,sp]emus1 ,hom9um enim ve/igia video4 ,vitruv4 ,ar*itect lib4 #f4 ,prief4 del9 ,m,burghers sculpt,univ4 ,oxon4 ,,pione]s ,,( ,,progress ,,m5 ,,( ,,sci5ce ,,edit$ ,,by ,s4 ,,*apman1 ,m4,a41 ,d4,,sc41 ,f4,r4,s4 ,,ar*im$es ,,by ,,sir ,,?omas ,,hea? ,k4,c4,b41 ,k4,c4,v4,o41 ,f4,r4,s42 ,,sc4,d41 ,,camb4 ,,hon4 ,d4,,sc41 ,,ox=d ,greek3 ,dos moi p\ /o1 kai k9o t5 g5 ,,london3 ,,society ,,= ,,promot+ ,,*ri/ian ,,k ,,new ,,york3 ,,! ,,macmillan ,,co4 #aibj ,,cont5ts4 ,,*ap4 ,,page ,i4 ,,ar*im$es #a ,,ii4 ,,greek ,,geometry ,,to ,,ar*im$es #g ,,iii4 ,,! ,,works ,,( ,,ar*im$es #bd ,,iv4 ,,geometry ,,9 ,,ar*im$es #bi ,v4 ,,! ,,s&reckon] #de ,,vi4 ,,me*anics #ej ,,vii4 ,,hydro/atics #ec ,,bibliography #eg ,,*ronology #eh ,,*apt] ,i4 ,,ar*im$es4 ,if ! ord9>y p]son 7 ask$ to say (f-h& :at he knew ( ,ar*im$es1 he w\ld probably1 at ! mo/1 be able to quote "o or o!r ( ! well-kn[n /ories ab\t him3 h[1 aft] discov]+ ! solution ( "s problem 9 ! ba?1 he 0 s ov]joy$ t he ran nak$ to 8 h\se1 %\t+ ,greek3 eureka1 eureka 7or1 z we might say1 ,i've got x1 ,i've got x72 or h[ he said ,give me a place to /& on & ,i w move ! e>?2 or aga9 h[ he 0 kill$1 at ! capture ( ,syracuse 9 ! ,second ,punic ,w>1 by a ,roman soldi] :o res5t$ be+ told to get away f a diagram drawn on ! gr\nd :i* he 0 /udy+4 ,& x is to be fe>$ t few :o are n exp]ts 9 ! hi/ory ( ma!matics h any acqua9tance ) ! details ( ! orig9al discov]ies 9 ma!matics ( ! greate/ ma!matician ( antiquity1 p]haps ! greate/ ma!matical g5ius t ! _w has ev] se54 ,hi/ory & tradition "k ,ar*im$es almo/ exclusively z ! inv5tor ( a numb] ( +5i\s me*anical appliances1 ?+s :i* naturally appeal m to ! popul> imag9ation ?an ! subtleties ( pure ma!matics4 ,almo/ all t is told ( ,ar*im$es rea*es u ?r\gh ! acc\nts by ,polybius & ,plut>* ( ! siege ( ,syracuse by ,m>cellus4 ,he p]i%$ 9 ! sack ( t city 9 #bab ,b4,c41 &1 z he 0 !n an old man 7p]haps #ge ye>s old71 he mu/ h be5 born ab\t #bhg ,b4,c4 ,he 0 ! son ( ,phidias1 an a/ronom]1 & 0 a fri5d & k9sman ( ,k+ ,hi]on ( ,syracuse & 8 son ,gelon4 ,he sp5t "s "t at ,alex&ria /udy+ ) ! successors ( ,euclid 7,euclid :o fl\ri%$ ab\t #cjj ,b4,c4 0 !n no long] liv+74 ,x 0 d\btless at ,alex&ria t he made ! acqua9tance ( ,conon ( ,samos1 :om he admir$ z a ma!matician & *]i%$ z a fri5d1 z well z ( ,erato/h5es2 to ! =m]1 & to ! latt] dur+ 8 e>ly p]iod he 0 9 ! habit ( communicat+ 8 discov]ies be=e _! publication4 ,x 0 also probably 9 ,egypt t he inv5t$ ! wat]-screw kn[n by 8 "n1 ! imm$iate purpose be+ ! draw+ ( wat] = irrigat+ fields4 ,aft] 8 return to ,syracuse he liv$ a life entirely devot$ to ma!matical rese>*4 ,incid5tally he became fam\s ?r\gh 8 clev] me*anical inv5tions4 ,^! ?+s 71 h[ev]1 9 8 case ! div]sions ( geometry at play1 & he atta*$ no importance to !m4 ,9 ! words ( ,plut>*1 he possess$ s l(ty a _s1 s pr(\nd a s\l1 & su* a weal? ( sci5tific k t1 al?\gh ^! inv5tions _h won = him ! r5[n ( m ?an human sagacity1 yet he w\ld n cons5t to leave beh9d him any writt5 "w on su* subjects1 b1 reg>d+ z ignoble & sordid ! bus9ess ( me*anics & e sort ( art :i* is direct$ to practical utility1 he plac$ 8 :ole ambition 9 ^? speculations 9 ! beauty & subtlety ( :i* !re is no admixture ( ! common ne$s ( life4 ,dur+ ! siege ( ,syracuse ,ar*im$es contriv$ all sorts ( eng9es aga9/ ! ,roman besieg]s4 ,!re 7 catapults s +5i\sly con/ruct$ z to be equally s]viceable at long or %ort range1 & ma*9es = dis*>g+ %[]s ( missiles ?r\gh holes made 9 ! walls4 ,o!r ma*9es consi/$ ( long movable poles project+ beyond ! walls2 "s ( ^! dropp$ heavy weights ^u ! enemy's %ips & on ! con/ructions :i* !y call$ sambuca1 f _! resemblance to a musical in/rum5t ( t "n1 & :i* consi/$ ( a protect$ ladd] ) "o end re/+ on two qu9qu]emes la%$ toge!r side by side z base1 & capable ( be+ rais$ by a w9dlass2 o!rs 7 fitt$ ) an iron h& or a beak l t ( a crane1 :i* grappl$ ! pr[s ( %ips1 !n lift$ !m into ! air & let !m fall aga94 ,m>cellus is said to h d]id$ 8 own eng9e]s & artific]s ) ! words1 ,%all we n make an end ( fight+ ) ?is geometrical ,bri>eus :o uses our %ips l cups to ladle wat] f ! sea1 drives our sambuca (f ignom9i\sly ) cudgel-bl[s1 &1 by ! multitude ( missiles t he hurls at u all at once1 outdoes ! hundr$-h&$ giants ( my?ology8 ,b ! exhortation _h no effect1 ! ,romans be+ 9 su* abject t]ror t1 if !y did b see a piece ( rope or wood project+ above ! wall !y w\ld cry '!re x is1' decl>+ t ,ar*im$es 0 sett+ "s eng9e 9 motion aga9/ !m1 & w\ld turn _! backs & run away1 insomu* t ,m>cellus desi/$ f all fight+ & assault1 putt+ all 8 hope 9 a long siege4 ,ar*im$es di$1 z he _h liv$1 absorb$ 9 ma!matical contemplation4 ,! acc\nts ( ! circum/ances ( 8 dea? diff] 9 "s details4 ,plut>* gives m ?an "o v]sion 9 ! foll[+ passage3 ,m>cellus 0 mo/ ( all afflict$ at ! dea? ( ,ar*im$es1 =1 z fate w\ld h x1 he 0 int5t on work+ out "s problem ) a diagram1 &1 8 m9d & 8 eyes be+ alike fix$ on 8 inve/igation1 he nev] notic$ ! incursion ( ! ,romans nor ! capture ( ! city4 ,& :5 a soldi] came up to him sudd5ly & bade him foll[ to ,m>cellus1 he refus$ to d s until he _h work$ out 8 problem to a demon/ration2 :]eat ! soldi] 0 s enrag$ t he drew 8 sword & slew him4 ,o!rs say t ! ,roman ran up to him ) a drawn sword1 ?reat5+ to kill him2 &1 :5 ,ar*im$es saw him1 he begg$ him e>ne/ly to wait a little :ile 9 ord] t he might n leave 8 problem incomplete & unsolv$1 b ! o!r took no notice & kill$ him4 ,aga91 !re is a ?ird acc\nt to ! effect t1 z he 0 c>ry+ to ,m>cellus "s ( 8 ma!matical in/rum5ts1 sundials1 sph]es1 & angles adju/$ to ! app>5t size ( ! sun to ! sight1 "s soldi]s met him &1 be+ "u ! impression t he c>ri$ gold 9 ! vessel1 kill$ him4 ,! mo/ picturesque v]sion ( ! /ory is t :i* repres5ts him z say+ to a ,roman soldi] :o came too close1 ,/& away1 fell[1 f my diagram1 :]eat ! man 0 s enrag$ t he kill$ him4 ,ar*im$es is said to h reque/$ 8 fri5ds & relatives to place ^u 8 tomb a repres5tation ( a cyl9d] circumscrib+ a sph]e )9 x1 toge!r ) an inscription giv+ ! ratio 7#c_/b7 :i* ! cyl9d] be>s to ! sph]e2 f :i* we may inf] t he himself reg>d$ ! discov]y ( ?is ratio z 8 greate/ a*ievem5t4 ,cic]o1 :5 quae/or 9 ,sicily1 f\nd ! tomb 9 a neglect$ /ate & re/or$ x4 ,9 mod]n times n ! slighte/ trace ( x has be5 f\nd4 ,beyond ! above p>ticul>s ( ! life ( ,ar*im$es1 we h no?+ b a numb] ( /ories :i*1 if p]haps n lit]ally accurate1 yet help u to a conception ( ! p]sonality ( ! man :i* we w\ld n will+ly h alt]$4 ,?us1 9 illu/ration ( 8 entire preoccupation by 8 ab/ract /udies1 we are told t he w\ld =get all ab\t 8 food & su* necessities ( life1 & w\ld be draw+ geometrical figures 9 ! a%es ( ! fire1 or1 :5 ano9t+ himself1 9 ! oil on 8 body4 ,( ! same k9d is ! /ory m5tion$ above1 t1 hav+ discov]$ :ile 9 a ba? ! solution ( ! "q ref]r$ to him by ,hi]on z to :e!r a c]ta9 cr[n suppos$ to h be5 made ( gold did n 9 fact conta9 a c]ta9 proportion ( silv]1 he ran nak$ ?r\gh ! /reet to 8 home %\t+ ,greek3 eureka1 eureka4 ,x 0 9 connexion ) 8 discov]y ( ! solution ( ! problem ,to move a giv5 weight by a giv5 =ce t ,ar*im$es utt]$ ! fam\s say+1 ,give me a place to /& on1 & ,i c move ! e>? 7,greek3 dos moi p\ /o kai k9o t5 g51 or 9 8 broad ,doric1 z "o v]sion has x1 ,greek3 pa bo kai k9o tan gan74 ,plut>* repres5ts him z decl>+ to ,hi]on t any giv5 weight c\ld be mov$ by a giv5 =ce1 & boa/+1 9 reliance on ! cog5cy ( 8 demon/ration1 t1 if he 7 giv5 ano!r e>?1 he w\ld cross ov] to x & move ?is "o4 ,& :5 ,hi]on 0 /ruck ) amazem5t & ask$ him to r$uce ! problem to practice & to %[ him "s great weight mov$ by a small =ce1 he fix$ on a %ip ( burd5 ) ?ree ma/s f ! k+'s ars5al :i* _h only be5 drawn up by ! great lab\r ( _m m52 & load+ h] ) _m pass5g]s & a full freight1 sitt+ himself ! :ile af> (f1 ) no great ef=t b quietly sett+ 9 motion ) 8 h& a comp\nd pulley1 he drew ! %ip t[>ds him smoo?ly & safely z if %e 7 mov+ ?r\gh ! sea4 ,hi]on1 we are told else:]e1 0 s mu* a/oni%$ t he decl>$ t1 f t day =?1 ,ar*im$es's ^w 0 to be accept$ on e subject6 ,ano!r v]sion ( ! /ory describes ! ma*9e us$ z a helix2 ?is t]m mu/ be suppos$ to ref] to a screw 9 ! %ape ( a cyl9drical helix turn$ by a h&le & act+ on a cog-:eel ) oblique tee? fitt+ on ! screw4 ,ano!r inv5tion 0 t ( a sph]e con/ruct$ s z to imitate ! motions ( ! sun1 ! moon1 & ! five planets 9 ! heav5s4 ,cic]o actually saw ?is contrivance1 & he gives a description ( x1 /at+ t x repres5t$ ! p]iods ( ! moon & ! app>5t motion ( ! sun ) su* accuracy t x w\ld ev5 7ov] a %ort p]iod7 %[ ! eclipses ( ! sun & moon4 ,x may h be5 mov$ by wat]1 = ,pappus speaks 9 "o place ( ^? :o und]/& ! mak+ ( sph]es & produce a model ( ! heav5s by means ( ! regul> circul> motion ( wat]4 ,9 any case x is c]ta9 t ,ar*im$es 0 mu* occupi$ ) a/ronomy4 ,livy calls him unicus spectator caeli sid]umque4 ,hipp>*us says1 ,f ^! obs]vations x is cle> t ! diff]5ces 9 ! ye>s are altoge!r small1 b1 z to ! sol/ices1 ,i almo/ ?9k t bo? ,i & ,ar*im$es h err$ to ! ext5t ( a qu>t] ( a day bo? 9 obs]vation & 9 ! d$uction !refrom4 ,x appe>s1 !re=e1 t ,ar*im$es _h consid]$ ! "q ( ! l5g? ( ! ye>4 ,macrobius says t he discov]$ ! di/ances ( ! planets4 ,ar*im$es himself describes 9 ! ,s&reckon] ! app>atus by :i* he measur$ ! app>5t diamet] ( ! sun1 i4e4 ! angle subt5d$ by x at ! eye4 ,! /ory t he set ! ,roman %ips on fire by an arrangem5t ( burn+-glasses or concave mirrors is n f\nd 9 any au?ority e>li] ?an ,lucian 7second c5tury ,a4,d472 b !re is no improbability 9 ! idea t he discov]$ "s =m ( burn+-mirror1 e4g4 a p>aboloid ( revolution1 :i* w\ld reflect to "o po9t all rays fall+ on its concave surface 9 a direction p>allel to its axis4 ,,*apt] ,,ii4 ,,greek ,,geometry ,,to ,,ar*im$es4 ,9 ord] to enable ! read] to arrive at a correct und]/&+ ( ! place ( ,ar*im$es & ( ! significance ( 8 "w x is necess>y to pass 9 review ! c\rse ( developm5t ( ,greek geometry f its fir/ beg9n+s d[n to ! "t ( ,euclid & ,ar*im$es4 ,greek au?ors f ,h]odotus d[nw>ds agree 9 say+ t geometry 0 inv5t$ by ! ,egyptians & t x came into ,greece f ,egypt4 ,"o acc\nt says3-- ,geometry is said by _m to h be5 inv5t$ among ! ,egyptians1 its orig9 be+ due to ! measurem5t ( plots ( l&4 ,?is 0 necess>y !re because ( ! ris+ ( ! ,nile1 :i* oblit]at$ ! b\nd>ies app]ta9+ to sep>ate own]s4 ,nor is x m>vell\s t ! discov]y ( ?is & ! o!r sci5ces %\ld h aris5 f su* an occasion1 s9ce ev]y?+ :i* moves 9 ! s5se ( developm5t w advance f ! imp]fect to ! p]fect4 ,f s5se-p]ception to reason+1 & f reason+ to und]/&+1 is a natural transition4 ,j z among ! ,pho5icians1 ?r\gh comm]ce & ex*ange1 an accurate k ( numb]s 0 orig9at$1 s also among ! ,egyptians geometry 0 inv5t$ = ! reason above /at$4 ,?ales fir/ w5t to ,egypt & !nce introduc$ ?is /udy into ,greece4 ,b x is cle> t ! geometry ( ! ,egyptians 0 almo/ entirely practical & did n g beyond ! requirem5ts ( ! l&-surveyor1 f>m] or m]*ant4 ,!y did inde$ "k1 z f> back z #bjjj ,b4,c41 t 9 a triangle :i* has its sides proportional to #c1 #d1 #e ! angle conta9$ by ! two small] sides is a "r angle1 & !y us$ su* a triangle z a practical means ( draw+ "r angles4 ,!y _h =mulae1 m or less inaccurate1 = c]ta9 measurem5ts1 e4g4 = ! areas ( c]ta9 triangles1 p>allel-trapezia1 & circles4 ,!y _h1 fur!r1 9 _! con/ruction ( pyramids1 to use ! notion ( simil> "r-angl$ triangles2 !y ev5 _h a "n1 se-qet1 = ! ratio ( ! half ( ! side ( ! base to ! height1 t is1 = :at we %\ld call ! co-tang5t ( ! angle ( slope4 ,b n a s+le g5]al !orem 9 geometry c be trac$ to ! ,egyptians4 ,_! k t ! triangle 7#c1 #d1 #e7 is "r angl$ is f> f imply+ any k ( ! g5]al proposition 7,eucl4 ,i41 #dg7 kn[n by ! "n ( ,py?agoras4 ,! sci5ce ( geometry1 9 fact1 rema9$ to be discov]$2 & ?is requir$ ! g5ius = pure speculation :i* ! ,greeks possess$ 9 ! l>ge/ measure among all ! nations ( ! _w4 ,?ales1 :o _h travell$ 9 ,egypt & !re le>nt :at ! prie/s c\ld tea* him on ! subject1 introduc$ geometry into ,greece4 ,almo/ ! :ole ( ,greek sci5ce & philosophy beg9s ) ,?ales4 ,8 date 0 ab\t #fbd-edg ,b4,c4 ,fir/ ( ! ,ionian philosoph]s1 & decl>$ "o ( ! ,sev5 ,wise ,m5 9 #ehb-eha1 he %one 9 all fields1 z a/ronom]1 ma!matician1 eng9e]1 /atesman & man ( bus9ess4 ,9 a/ronomy he pr$ict$ ! sol> eclipse ( #bh ,may1 #ehe1 discov]$ ! inequality ( ! f\r a/ronomical seasons1 & c\nsell$ ! use ( ! ,little ,be> in/ead ( ! ,great ,be> z a means ( f9d+ ! pole4 ,9 geometry ! foll[+ !orems are attribut$ to him--& _! "* %[s h[ ! ,greeks _h to beg9 at ! v beg9n+ ( ! !ory--7#a7 t a circle is bisect$ by any diamet] 7,eucl4 ,i41 ,def4 #ag71 7#b7 t ! angles at ! base ( an isosceles triangle are equal 7,eucl4 ,i41 #e71 7#c7 t1 if two /raight l9es cut "o ano!r1 ! v]tically opposite angles are equal 7,eucl4 ,i41 #ae71 7#d7 t1 if two triangles h two angles & "o side respectively equal1 ! triangles are equal 9 all respects 7,eucl4 ,i41 #bf74 ,he is said 7#e7 to h be5 ! fir/ to inscribe a "r-angl$ triangle 9 a circle3 :i* mu/ mean t he 0 ! fir/ to discov] t ! angle 9 a semicircle is a "r angle4 ,he also solv$ two problems 9 practical geometry3 7#a7 he %[$ h[ to measure ! di/ance f ! l& ( a %ip at sea 7= ?is he is said to h us$ ! proposition numb]$ 7#d7 above71 & 7#b7 he measur$ ! heights ( pyramids by means ( ! %ad[ ?r[n on ! gr\nd 7?is implies ! use ( simil> triangles 9 ! way t ! ,egyptians _h us$ !m 9 ! con/ruction ( pyramids74 ,aft] ,?ales come ! ,py?agoreans4 ,we are told t ! ,py?agoreans 7 ! fir/ to use ! t]m ,greek3 ma!mata 7lit]ally subjects ( in/ruction7 9 ! specialis$ s5se ( ma!matics2 !y1 too1 fir/ advanc$ ma!matics z a /udy pursu$ = its own sake & made x a "p ( a lib]al education4 ,py?agoras1 son ( ,mnes>*us1 0 born 9 ,samos ab\t #egb ,b4,c41 & di$ at a great age 7#ge or #hj7 at ,metapontum4 ,8 int]e/s 7 z v>i\s z ^? ( ,?ales2 8 travels1 all und]tak5 9 pursuit ( k1 7 probably ev5 m ext5d$4 ,l ,?ales1 & p]haps at 8 sugge/ion1 he visit$ ,egypt & /udi$ !re = a long p]iod 7#bb ye>s1 "s say74 ,x is difficult to dis5tangle f ! body ( ,py?agorean doctr9es ! portions :i* are due to ,py?agoras himself because ( ! habit :i* ! memb]s ( ! s*ool _h ( attribut+ ev]y?+ to ! ,ma/] 7,greek3 autos epha1 ipse dixit74 ,9 a/ronomy two ?+s at lea/ may safely be attribut$ to him2 he held t ! e>? is sph]ical 9 %ape1 & he recognis$ t ! sun1 moon & planets h an indep5d5t motion ( _! own 9 a direction contr>y to t ( ! daily rotation2 he seems1 h[ev]1 to h adh]$ to ! geoc5tric view ( ! univ]se1 & x 0 8 successors :o evolv$ ! !ory t ! e>? does n rema9 at ! c5tre b revolves1 l ! o!r planets & ! sun & moon1 ab\t ! c5tral fire4 ,p]haps 8 mo/ rem>kable discov]y 0 ! dep5d5ce ( ! musical int]vals on ! l5g?s ( vibrat+ /r+s1 ! proportion = ! octave be+ #b 3 #a1 = ! fif? #c 3 #b & = ! f\r? #d 3 #c4 ,9 ari?metic he 0 ! fir/ to exp\nd ! !ory ( means & ( proportion z appli$ to comm5surable quantities4 ,he laid ! f\ndation ( ! !ory ( numb]s by consid]+ ! prop]ties ( numb]s z su*1 namely1 prime numb]s1 odd & ev5 numb]s1 etc4 ,by means ( figur$ numb]s1 squ>e1 oblong1 triangul>1 etc4 7repres5t$ by dots arrang$ 9 ! =m ( ! v>i\s figures7 he %[$ ! connexion betwe5 numb]s & geometry4 ,9 view ( all ^! prop]ties ( numb]s1 we c easily und]/& h[ ! ,py?agoreans came to lik5 all ?+s to numb]s & to f9d 9 ! pr9ciples ( numb]s ! pr9ciples ( all ?+s 7all ?+s are numb]s74 ,we come n[ to ,py?agoras's a*ievem5ts 9 geometry4 ,!re is a /ory t1 :5 he came home f ,egypt & tri$ to f\nd a s*ool at ,samos1 he f\nd ! ,samians indiff]5t1 s t he _h to take special measures to ensure t 8 geometry might n p]i% ) him4 ,go+ to ! gymnasium1 he s\ght out a well-fav\r$ y\? :o seem$ likely to suit 8 purpose1 & 0 )al poor1 & brib$ him to le>n geometry by promis+ him sixp5ce = e proposition t he ma/]$4 ,v soon ! y\? got fasc9at$ by ! subject = its own sake1 & ,py?agoras rightly judg$ t he w\ld gladly g on )\t ! sixp5ce4 ,he h9t$1 !re=e1 t he himself 0 poor & mu/ try to e>n 8 liv+ in/ead ( do+ ma!matics2 :]eupon ! y\?1 r ?an give up ! /udy1 volunte]$ to pay sixp5ce to ,py?agoras = ea* proposition4 ,9 geometry ,py?agoras set himself to lay ! f\ndations ( ! subject1 beg9n+ ) c]ta9 important def9itions & inve/igat+ ! fundam5tal pr9ciples4 ,( propositions attribut$ to him ! mo/ fam\s is1 ( c\rse1 ! !orem t 9 a "r-angl$ triangle ! squ>e on ! hypot5use is equal to ! sum ( ! squ>es on ! sides ab\t ! "r angle 7,eucl4 ,i41 #dg72 &1 see+ t ,greek tradition univ]sally cr$its him ) ! pro( ( ?is !orem1 we pref] to believe t tradition is "r4 ,?is is to "s ext5t confirm$ by ano!r tradition t ,py?agoras discov]$ a g5]al =mula = f9d+ two numb]s su* t ! sum ( _! squ>es is a squ>e numb]4 ,?is dep5ds on ! !ory ( ! gnomon1 :i* at fir/ _h an ari?metical signification correspond+ to ! geometrical use ( x 9 ,euclid1 ,book ,,ii4 ,a figure 9 ! %ape ( a gnomon put r\nd two sides ( a squ>e makes x into a l>g] squ>e4 ,n[ consid] ! numb] #a repres5t$ by a dot4 ,r\nd ?is place ?ree o!r dots s t ! f\r dots =m a squ>e 7#a #c #bb74 ,r\nd ! f\r dots 7on two adjac5t sides ( ! squ>e7 place five dots at regul> & equal di/ances1 & we h ano!r squ>e 7#a #c #e #cb72 & s on4 ,! successive odd numb]s #a1 #c1 #e 444 7 call$ gnomons1 & ! g5]al =mula is #a #c #e 444 7#bn - #a7 n#b4 ,add ! next odd numb]1 i4e4 #bn #a1 & we h n#b 7#bn #a7 7n #a7b4 ,9 ord]1 !n1 to get two squ>e numb]s su* t _! sum is a squ>e we h only to see t #bn #a is a squ>e4 ,suppose t #bn #a m#b2 !n n #a_/b7m#b - #a71 & we h #a_/b7m#b - #a7b m#b #a_/b7m#b #a7b1 :]e m is any odd numb]2 & ?is is ! g5]al =mula attribut$ to ,py?agoras4 ,proclus also attributes to ,py?agoras ! !ory ( proportionals & ! con/ruction ( ! five cosmic figures1 ! five regul> solids4 ,"o ( ! said solids1 ! dodecah$ron1 has twelve p5tagonal faces1 & ! con/ruction ( a regul> p5tagon involves ! cutt+ ( a /raight l9e 9 extreme & mean ratio 7,eucl4 ,,ii41 #aa1 & ,,vi41 #cj71 :i* is a p>ticul> case ( ! me?od kn[n z ! application ( areas4 ,h[ mu* ( ?is 0 due to ,py?agoras himself we d n "k2 b ! :ole me?od 0 at all ev5ts fully work$ out by ! ,py?agoreans & prov$ "o ( ! mo/ p[]ful ( geometrical me?ods4 ,! mo/ elem5t>y case appe>s 9 ,euclid1 ,i41 #dd1 #de1 :]e x is %[n h[ to apply to a giv5 /raight l9e z base a p>allelogram hav+ a giv5 angle 7say a rectangle7 & equal 9 area to any rectil9eal figure2 ?is con/ruction is ! geometrical equival5t ( ari?metical division4 ,! g5]al case is t 9 :i* ! p>allelogram1 ?\gh appli$ to ! /raight l9e1 ov]laps x or falls %ort ( x 9 su* a way t ! "p ( ! p>allelogram :i* ext5ds beyond1 or falls %ort (1 ! p>allelogram ( ! same angle & bread? on ! giv5 /raight l9e itself 7exactly7 z base is simil> to ano!r giv5 p>allelogram 7,eucl4 ,,vi41 #bh1 #bi74 ,?is is ! geometrical equival5t ( ! mo/ g5]al =m ( quadratic equation ax - mx#b ,c1 s f> z x has real roots2 :ile ! condition t ! roots may be real 0 also work$ out 7 ,eucl4 ,,vi41 #bg74 ,x is important to note t ?is me?od ( application ( areas 0 directly us$ by ,apollonius ( ,p]ga 9 =mulat+ ! fundam5tal prop]ties ( ! ?ree conic sections1 :i* prop]ties correspond to ! equations ( ! conics 9 ,c>tesian co-ord9ates2 & ! names giv5 by ,apollonius 7= ! fir/ "t7 to ! respective conics are tak5 f ! !ory1 p>abola 7,greek3 p>abole7 mean+ application 7i4e4 9 ?is case ! p>allelogram is appli$ to ! /raight l9e exactly71 hyp]bola 7,greek3 hyp]bole71 exce$+ 7i4e4 9 ?is case ! p>allelogram exce$s or ov]laps ! /raight l9e71 ellipse 7,greek3 elleipsis71 fall+ %ort 7i4e4 ! p>allelogram falls %ort ( ! /raight l9e74 ,ano!r problem solv$ by ! ,py?agoreans is t ( draw+ a rectil9eal figure equal 9 area to "o giv5 rectil9eal figure & simil> to ano!r4 ,plut>* m5tions a d\bt z to :e!r x 0 ?is problem or ! proposition ( ,euclid ,i41 #dg1 on ! /r5g? ( :i* ,py?agoras 0 said to h sacrific$ an ox4 ,! ma9 p>ticul> applications ( ! !orem ( ! squ>e on ! hypot5use 7e4g4 ^? 9 ,euclid1 ,book ,,ii47 7 also ,py?agorean2 ! con/ruction ( a squ>e equal to a giv5 rectangle 7,eucl4 ,,ii41 #ad7 is "o ( !m & corresponds to ! solution ( ! pure quadratic equation x#b ab4 ,! ,py?agoreans prov$ ! !orem t ! sum ( ! angles ( any triangle is equal to two "r angles 7,eucl4 ,i41 #cb74 ,speak+ g5]ally1 we may say t ! ,py?agorean geometry cov]$ ! bulk ( ! subject-matt] ( ,books ,i41 ,,ii41 ,,iv41 & ,,vi4 ( ,euclid 7) ! qualification1 z reg>ds ,book ,,vi41 t ! ,py?agorean !ory ( proportion appli$ only to comm5surable magnitudes74 ,our in=mation ab\t ! orig9 ( ! propositions ( ,euclid1 ,book ,,iii41 is n s complete2 b x is c]ta9 t ! mo/ important ( !m 7 well kn[n to ,hippocrates ( ,*ios 7:o fl\ri%$ 9 ! second half ( ! fif? c5tury1 & liv$ p]haps f ab\t #dgj to #djj ,b4,c471 :5ce we conclude t ! ma9 propositions ( ,book ,,iii4 7 also includ$ 9 ! ,py?agorean geometry4 ,la/ly1 ! ,py?agoreans discov]$ ! exi/5ce ( incomm5surable l9es1 or ( irrationals4 ,?is 01 d\btless1 fir/ discov]$ ) ref]5ce to ! diagonal ( a squ>e :i* is incomm5surable ) ! side1 be+ 9 ! ratio to x ( root#b to #a4 ,! ,py?agorean pro( ( ?is p>ticul> case survives 9 ,ari/otle & 9 a proposition int]polat$ 9 ,euclid's ,book ,x42 x is by a r$uctio ad absurdum prov+ t1 if ! diagonal is comm5surable ) ! side1 ! same numb] mu/ be bo? odd & ev54 ,?is discov]y ( ! incomm5surable 0 b\nd to cause geomet]s a great %ock1 because x %[$ t ! !ory ( proportion inv5t$ by ,py?agoras 0 n ( univ]sal application1 & !re=e t propositions prov$ by means ( x 7 n really e/abli%$4 ,h5ce ! /ories t ! discov]y ( ! irrational 0 = a "t kept secret1 & t ! fir/ p]son :o divulg$ x p]i%$ by %ipwreck4 ,! fatal flaw ?us reveal$ 9 ! body ( geometry 0 n remov$ till ,eudoxus 7#djh-cee ,b4,c47 discov]$ ! great !ory ( proportion 7exp\nd$ 9 ,euclid's ,book ,v471 :i* is applicable to incomm5surable z well z to comm5surable magnitudes4 ,by ! "t ( ,hippocrates ( ,*ios ! scope ( ,greek geometry 0 no long] ev5 limit$ to ! ,elem5ts2 c]ta9 special problems 7 also attack$ :i* 7 beyond ! p[] ( ! geometry ( ! /raight l9e & circle1 & :i* 7 de/9$ to play a great "p 9 det]m9+ ! direction tak5 by ,greek geometry 9 its highe/ flights4 ,! ma9 problems 9 "q 7 ?ree3 7#a7 ! d\bl+ ( ! cube1 7#b7 ! trisection ( any angle1 7#c7 ! squ>+ ( ! circle2 & f ! "t ( ,hippocrates onw>ds ! inve/igation ( ^! problems proce$$ p>i passu ) ! completion ( ! body ( ! ,elem5ts4 ,hippocrates himself is an example ( ! concurr5t /udy ( ! two dep>tm5ts4 ,on ! "o h&1 he 0 ! fir/ ( ! ,greeks :o is kn[n to h compil$ a book ( ,elem5ts4 ,?is book1 we may be sure1 conta9$ 9 p>ticul> ! mo/ important propositions ab\t ! circle includ$ 9 ,euclid1 ,book ,,iii4 ,b a mu* m important proposition is attribut$ to ,hippocrates2 he is said to h be5 ! fir/ to prove t circles are to "o ano!r z ! squ>es on _! diamet]s 7 ,eucl4 ,,xii41 #b71 ) ! d$uction t simil> segm5ts ( circles are to "o ano!r z ! squ>es on _! bases4 ,^! propositions 7 us$ by him 9 8 tract on ! squ>+ ( lunes1 :i* 0 int5d$ to lead up to ! squ>+ ( ! circle4 ,! latt] problem is "o :i* mu/ h ex]cis$ practical geomet]s f "t immemorial4 ,anaxagoras = in/ance 7ab\t #ejj-dbh ,b4,c47 is said to h work$ at ! problem :ile 9 prison4 ,! ess5tial portions ( ,hippocrates's tract are pres]v$ 9 a passage ( ,simplicius 7on ,ari/otle's ,physics71 :i* conta9s sub/antial fragm5ts f ,eudemus's ,hi/ory ( ,geometry4 ,hippocrates %[$ h[ to squ>e ?ree p>ticul> lunes ( diff]5t =ms1 & !n1 la/ly1 he squ>$ ! sum ( a c]ta9 circle & a c]ta9 lune4 ,un=tunately1 h[ev]1 ! la/-m5tion$ lune 0 n "o ( ^? :i* c be squ>$1 & s ! attempt to squ>e ! circle 9 ?is way fail$ aft] all4 ,hippocrates also attack$ ! problem ( d\bl+ ! cube4 ,!re are two v]sions ( ! orig9 ( ?is fam\s problem4 ,accord+ to "o ( !m1 an old tragic poet repres5t$ ,m9os z hav+ be5 dissatisfi$ ) ! size ( a tomb erect$ = 8 son ,glaucus1 & hav+ told ! ar*itect to make x d\ble ! size1 reta9+1 h[ev]1 ! cubical =m4 ,accord+ to ! o!r1 ! ,delians1 suff]+ f a pe/il5ce1 7 told by ! oracle to d\ble a c]ta9 cubical alt> z a means ( /ay+ ! plague4 ,hippocrates did n1 inde$1 solve ! problem1 b he succe$$ 9 r$uc+ x to ano!r1 namely1 ! problem ( f9d+ two mean proportionals 9 cont9u$ proportion betwe5 two giv5 /raight l9es1 i4e4 f9d+ x1 y su* t a 3 x x 3 y y 3 b1 :]e a1 b are ! two giv5 /raight l9es4 ,x is easy to see t1 if a 3 x x 3 y y 3 b1 !n b_/a 7x_/a7#c1 &1 z a p>ticul> case1 if b #b;a1 x#c #b;a#c1 s t ! side ( ! cube :i* is d\ble ( ! cube ( side a is f\nd4 ,! problem ( d\bl+ ! cube 0 h5ce=? tri$ exclusively 9 ! =m ( ! problem ( ! two mean proportionals4 ,two significant e>ly solutions are on record4 7#a7 ,ar*ytas ( ,t>5tum 7:o fl\ri%$ 9 fir/ half ( f\r? c5tury ,b4,c47 f\nd ! two mean proportionals by a v /rik+ con/ruction 9 ?ree dim5sions1 :i* %[s t solid geometry1 9 ! h&s ( ,ar*ytas at lea/1 0 already well advanc$4 ,! con/ruction 0 usually call$ me*anical1 :i* x no d\bt 0 9 =m1 ?\gh 9 reality x 0 9 ! highe/ degree !oretical4 ,x consi/$ 9 det]m9+ a po9t 9 space z ! int]section ( ?ree surfaces3 7a7 a cyl9d]1 7b7 a cone1 7c7 an an*or-r+ ) int]nal radius #j4 7#b7 ,m5ae*mus1 a pupil ( ,eudoxus1 & a contempor>y ( ,plato1 f\nd ! two mean proportionals by means ( conic sections1 9 two ways1 7alpha7 by ! int]section ( two p>abolas1 ! equations ( :i* 9 ,c>tesian co-ord9ates w\ld be x#b ay1 y#b bx1 & 7beta7 by ! int]section ( a p>abola & a rectangul> hyp]bola1 ! correspond+ equations be+ x#b ay1 & xy ab respectively4 ,x w\ld appe> t x 0 9 ! ef=t to solve ?is problem t ,m5ae*mus discov]$ ! conic sections1 :i* are call$1 9 an epigram by ,erato/h5es1 ! triads ( ,m5ae*mus4 ,! trisection ( an angle 0 effect$ by means ( a curve discov]$ by ,hippias ( ,elis1 ! sophi/1 a contempor>y ( ,hippocrates z well z ( ,democritus & ,socrates 7#dgj-cii ,b4,c474 ,! curve 0 call$ ! quadratrix because x also s]v$ 79 ! h&s1 z we are told1 ( ,d9o/ratus1 bro!r ( ,m5ae*mus1 & ( ,nicom$es7 = squ>+ ! circle4 ,x 0 !oretically con/ruct$ z ! locus ( ! po9t ( int]section ( two /raight l9es mov+ at uni=m spe$s & 9 ! same "t1 "o motion be+ angul> & ! o!r rectil9e>4 ,suppose ,,oa1 ,,ob are two radii ( a circle at "r angles to "o ano!r4 ,tang5ts to ! circle at ,a & ,b1 meet+ at ,c1 =m ) ! two radii ! squ>e ,,oacb4 ,! radius ,,oa is made to move uni=mly ab\t ,o1 ! c5tre1 s z to describe ! angle ,,aob 9 a c]ta9 "t4 ,simultane\sly ,,ac moves p>allel to itself at uni=m spe$ su* t ,a j describes ! l9e ,,ao 9 ! same l5g? ( "t4 ,! int]section ( ! mov+ radius & ,,ac 9 _! v>i\s positions traces out ! quadratrix4 ,! re/ ( ! geometry :i* conc]ns u 0 mo/ly ! "w ( a few m51 ,democritus ( ,abd]a1 ,!odorus ( ,cyr5e 7! ma!matical tea*] ( ,plato71 ,!aetetus1 ,eudoxus1 & ,euclid4 ,! actual writ]s ( ,elem5ts ( :om we he> 7 ! foll[+4 ,leon1 a little y\ng] ?an ,eudoxus 7#djh-cee ,b4,c471 0 ! au?or ( a collection ( propositions m num]\s & m s]viceable ?an ^? collect$ by ,hippocrates4 ,!udius ( ,magnesia1 a contempor>y ( ,m5ae*mus & ,d9o/ratus1 put toge!r ! elem5ts admirably1 mak+ _m p>tial or limit$ propositions m g5]al4 ,!udius's book 0 no d\bt ! geometrical text-book ( ! ,academy & t us$ by ,ari/otle4 ,!odorus ( ,cyr5e & ,!aetetus g5]alis$ ! !ory ( irrationals1 & we may safely conclude t a great "p ( ! sub/ance ( ,euclid's ,book ,x4 7on irrationals7 0 due to ,!aetetus4 ,!aetetus also wrote on ! five regul> solids 7! tetrah$ron1 cube1 octah$ron1 dodecah$ron1 & icosah$ron71 & ,euclid 0 !re=e no d\bt equally indebt$ to ,!aetetus = ! cont5ts ( 8 ,book ,,xiii4 ,9 ! matt] ( ,book ,,xii4 ,eudoxus 0 ! pione]4 ,^! facts are confirm$ by ! rem>k ( ,proclus t ,euclid1 9 compil+ 8 ,elem5ts1 collect$ _m ( ! !orems ( ,eudoxus1 p]fect$ _m o!rs by ,!aetetus1 & br\ght to irrefragable demon/ration ! propositions :i* _h only be5 some:at loosely prov$ by 8 pr$ecessors4 ,eudoxus 7ab\t #djh-cee ,b4,c47 0 p]haps ! greate/ ( all ,ar*im$es's pr$ecessors1 & x is 8 a*ievem5ts1 especially ! discov]y ( ! me?od ( exhau/ion1 :i* int]e/ u 9 connexion ) ,ar*im$es4 ,9 a/ronomy ,eudoxus is fam\s = ! beautiful !ory ( conc5tric sph]es :i* he inv5t$ to expla9 ! app>5t motions ( ! planets1 &1 p>ticul>ly1 _! app>5t /ation>y po9ts & retrogradations4 ,! !ory appli$ also to ! sun & moon1 = :i* ,eudoxus requir$ only ?ree sph]es 9 ea* case4 ,he repres5t$ ! motion ( ea* planet z comp\nd$ ( ! rotations ( f\r int]connect$ sph]es ab\t diamet]s1 all ( :i* pass ?r\gh ! c5tre ( ! e>?4 ,! out]mo/ sph]e repres5ts ! daily rotation1 ! second a motion along ! zodiac circle or ecliptic2 ! poles ( ! ?ird sph]e1 ab\t :i* t sph]e revolves1 are fix$ at two opposite po9ts on ! zodiac circle1 & are c>ri$ r\nd 9 ! motion ( ! second sph]e2 & on ! surface ( ! ?ird sph]e ! poles ( ! f\r? sph]e are fix$2 ! f\r? sph]e1 revolv+ ab\t ! diamet] jo9+ its two poles1 c>ries ! planet :i* is fix$ at a po9t on its equator4 ,! poles & ! spe$s & directions ( rotation are s *os5 t ! planet actually describes a hippop$e1 or horse-fett]1 z x 0 call$ 7i4e4 a figure ( eight71 :i* lies along & is longitud9ally bisect$ by ! zodiac circle1 & is c>ri$ r\nd t circle4 ,z a t\r de =ce ( geometrical imag9ation x w\ld be difficult to p>allel ?is hypo!sis4 ,9 geometry ,eudoxus discov]$ ! great !ory ( proportion1 applicable to incomm5surable z well z comm5surable magnitudes1 :i* is exp\nd$ 9 ,euclid1 ,book ,v41 & :i* /ill holds its own & w d s = all "t4 ,he also solv$ ! problem ( ! two mean proportionals by means ( c]ta9 curves1 ! nature ( :i*1 9 ! abs5ce ( any description ( !m 9 our s\rces1 c only be conjectur$4 ,la/ ( all1 & mo/ important = our purpose1 is 8 use ( ! fam\s me?od ( exhau/ion = ! measurem5t ( ! areas ( curves & ! volumes ( solids4 ,! example ( ?is me?od :i* w be mo/ famili> to ! read] is ! pro( 9 ,euclid ,,xii41 #b1 ( ! !orem t ! areas ( circles are to "o ano!r z ! squ>es on _! diamet]s4 ,! pro( 9 ?is & 9 all cases dep5ds on a lemma :i* =ms ,prop4 #a ( ,euclid's ,book ,x4 to ! effect t1 if !re are two unequal magnitudes ( ! same k9d & f ! great] y subtract n less ?an its half1 !n f ! rema9d] n less ?an its half1 & s on cont9ually1 y w at l5g? h rema9+ a magnitude less ?an ! less] ( ! two magnitudes set out1 h[ev] small x is4 ,ar*im$es says t ! !orem ( ,euclid ,,xii41 #b1 0 prov$ by means ( a c]ta9 lemma to ! effect t1 if we h two unequal magnitudes 7i4e4 l9es1 surfaces1 or solids respectively71 ! great] exce$s ! less] by su* a magnitude z is capable1 if add$ cont9ually to itself1 ( exce$+ any magnitude ( ! same k9d z ! orig9al magnitudes4 ,?is assumption is kn[n z ! ,axiom or ,po/ulate ( ,ar*im$es1 ?\gh1 z he /ates1 x 0 assum$ be=e 8 "t by ^? :o us$ ! me?od ( exhau/ion4 ,x is 9 reality us$ 9 ,euclid's lemma 7,eucl4 ,x41 #a7 on :i* ,euclid ,,xii41 #b1 dep5ds1 & only diff]s 9 /atem5t f ,def4 #d ( ,euclid1 ,book ,v41 :i* is no d\bt due to ,eudoxus4 ,! me?od ( exhau/ion 0 n discov]$ all at once2 we f9d traces ( grop+s aft] su* a me?od be=e x 0 actually evolv$4 ,x 0 p]haps ,antiphon1 ! sophi/1 ( ,a!ns1 a contempor>y ( ,socrates 7#dgj-cii ,b4,c471 :o took ! fir/ /ep4 ,he inscrib$ a squ>e 7or1 accord+ to ano!r acc\nt1 an equilat]al triangle7 9 a circle1 !n bisect$ ! arcs subt5d$ by ! sides1 & s inscrib$ a polygon ( d\ble ! numb] ( sides2 he !n repeat$ ! process1 & ma9ta9$ t1 by cont9u+ x1 we %\ld at la/ arrive at a polygon ) sides s small z to make ! polygon co9cid5t ) ! circle4 ,?\gh ?is 0 =mally incorrect1 x nev]!less conta9$ ! g]m ( ! me?od ( exhau/ion4 ,hippocrates1 z we h se51 is said to h prov$ ! !orem t circles are to "o ano!r z ! squ>es on _! diamet]s1 & x is difficult to see h[ he c\ld h done ?is except by "s =m1 or anticipation1 ( ! me?od4 ,!re is1 h[ev]1 no d\bt ab\t ! "p tak5 by ,eudoxus2 he n only bas$ ! me?od on rigor\s demon/ration by means ( ! lemma or lemmas a=esaid1 b he actually appli$ ! me?od to f9d ! volumes 7#a7 ( any pyramid1 7#b7 ( ! cone1 prov+ 7#a7 t any pyramid is "o ?ird "p ( ! prism :i* has ! same base & equal height1 & 7#b7 t any cone is "o ?ird "p ( ! cyl9d] :i* has ! same base & equal height4 ,ar*im$es1 h[ev]1 tells u ! rem>kable fact t ^! two !orems 7 fir/ discov]$ by ,democritus 7:o fl\ri%$ t[>ds ! end ( ! fif? c5tury ,b4,c471 ?\gh he 0 n able to prove !m 7:i* no d\bt means1 n t he gave no sort ( pro(1 b t he 0 n able to e/abli% ! propositions by ! rigor\s me?od ( ,eudoxus74 ,ar*im$es adds t we mu/ give no small %>e ( ! cr$it = ^! !orems to ,democritus2 & ?is is ano!r te/imony to ! m>vell\s p[]s1 9 ma!matics z well z 9 o!r subjects1 ( ! great man :o1 9 ! words ( ,ari/otle1 seems to h ?\ght ( ev]y?+4 ,we "k f o!r s\rces t ,democritus wrote on irrationals2 he is also said to h discuss$ ! "q ( two p>allel sections ( a cone 7:i* 7 evid5tly suppos$ to be indef9itely close toge!r71 ask+ :e!r we are to reg>d !m z unequal or equal3 = if !y are unequal !y w make ! cone irregul> z hav+ _m ind5tations1 l /eps1 & unev5nesses1 b1 if !y are equal1 ! cone w appe> to h ! prop]ty ( ! cyl9d] & to be made up ( equal1 n unequal1 circles1 :i* is v absurd4 ,?is explanation %[s t ,democritus 0 already close on ! track ( inf9itesimals4 ,ar*im$es says fur!r t ! !orem t sph]es are 9 ! triplicate ratio ( _! diamet]s 0 prov$ by means ( ! same lemma4 ,! pro(s ( ! propositions ab\t ! volumes ( pyramids1 cones & sph]es are1 ( c\rse1 conta9$ 9 ,euclid1 ,book ,,xii4 7,props4 #c-g ,cor41 #aj1 #af-ah respectively74 ,x is no d\bt desirable to illu/rate ,eudoxus's me?od by "o example4 ,we w take "o ( ! simple/1 ! proposition 7,eucl4 ,,xii41 #aj7 ab\t ! cone4 ,giv5 ,,abcd1 ! circul> base ( ! cyl9d] :i* has ! same base z ! cone & equal height1 we inscribe ! squ>e ,,abcd2 we !n bisect ! arcs subt5d$ by ! sides1 & draw ! regul> inscrib$ polygon ( eight sides1 !n simil>ly we draw ! regul> inscrib$ polygon ( sixte5 sides1 & s on4 ,we erect on ea* regul> polygon ! prism :i* has ! polygon = base1 !reby obta9+ successive prisms inscrib$ 9 ! cyl9d]1 & ( ! same height ) x4 ,ea* "t we d\ble ! numb] ( sides 9 ! base ( ! prism we take away m ?an half ( ! volume by :i* ! cyl9d] exce$s ! prism 7s9ce we take away m ?an half ( ! excess ( ! area ( ! circul> base ov] t ( ! inscrib$ polygon1 z 9 ,euclid ,,xii41 #b74 ,suppose n[ t ,v is ! volume ( ! cone1 ,c t ( ! cyl9d]4 ,we h to prove t ,c #c,v4 ,if ,c is n equal to #c,v1 x is ei!r great] or less ?an #c,v4 ,suppose 7#a7 t ,c #c,v1 & t ,c #c,v ,e4 ,cont9ue ! con/ruction ( prisms inscrib$ 9 ! cyl9d] until ! p>ts ( ! cyl9d] left ov] outside ! f9al prism 7( volume ,p7 are toge!r less ?an ,e4 ,!n ,c - ,p ,e4 ,b ,c - #c,v ,e2 ,!re=e ,p #c,v4 ,b x has be5 prov$ 9 e>li] propositions t ,p is equal to ?ree times ! pyramid ) ! same base z ! prism & equal height4 ,!re=e t pyramid is great] ?an ,v1 ! volume ( ! cone3 :i* is impossible1 s9ce ! cone encloses ! pyramid4 ,!re=e ,c is n great] ?an #c,v4 ,next 7#b7 suppose t ,c #c,v1 s t1 inv]sely1 ,v #a_/c ,c4 ,?is "t we inscribe successive pyramids 9 ! cone until we arrive at a pyramid su* t ! portions ( ! cone left ov] outside x are toge!r less ?an ! excess ( ,v ov] #a_/c ,c4 ,x foll[s t ! pyramid is great] ?an #a_/c ,c4 ,h5ce ! prism on ! same base z ! pyramid & inscrib$ 9 ! cyl9d] 7:i* prism is ?ree times ! pyramid7 is great] ?an ,c3 :i* is impossible1 s9ce ! prism is enclos$ by ! cyl9d]1 & is !re=e less ?an x4 ,!re=e ,v is n great] ?an #a_/c ,c1 or ,c is n less ?an #c,v4 ,accord+ly ,c1 be+ nei!r great] nor less ?an #c,v1 mu/ be equal to x2 t is1 ,v #a_/c ,c4 ,x only rema9s to add t ,ar*im$es is fully acqua9t$ ) ! ma9 prop]ties ( ! conic sections4 ,^! _h already be5 prov$ 9 e>li] treatises1 :i* ,ar*im$es ref]s to z ! ,elem5ts ( ,conics4 ,we "k ( two su* treatises1 7#a7 ,euclid's f\r ,books on ,conics1 7#b7 a "w by "o ,ari/aeus call$ ,solid ,loci1 probably a treatise on conics reg>d$ z loci4 ,bo? ^! treatises are lo/2 ! =m] 01 ( c\rse1 sup]s$$ by ,apollonius's great "w on ,conics 9 eight ,books4 ,,*apt] ,,iii4 ,,! ,,works ,,( ,,ar*im$es4 ,! range ( ,ar*im$es's writ+s w be ga!r$ f ! li/ ( 8 v>i\s treatises4 ,an extraord9>ily l>ge proportion ( _! cont5ts repres5ts entirely new discov]ies ( 8 own4 ,he 0 no compil] or writ] ( text-books1 & 9 ?is respect he diff]s f ,euclid & ,apollonius1 ^: "w l>gely consi/$ 9 sy/ematis+ & g5]alis+ ! me?ods us$ & ! results obta9$ by e>li] geomet]s4 ,!re is 9 ,ar*im$es no m]e work+-up ( exi/+ mat]ial2 8 objective is always some?+ new1 "s def9ite addition to ! sum ( k4 ,confirmation ( ?is is f\nd 9 ! introductory lett]s prefix$ to mo/ ( 8 treatises4 ,9 !m we see ! directness1 simplicity & humanity ( ! man4 ,!re is full & g5]\s recognition ( ! "w ( pr$ecessors & contempor>ies2 8 e/imate ( ! relation ( 8 own discov]ies to !irs is obvi\sly j & free f any %ade ( egoism4 ,8 mann] is to /ate :at p>ticul> discov]ies made by 8 pr$ecessors _h sugge/$ to him ! possibility ( ext5d+ !m 9 new directions2 ?us he says t1 9 connexion ) ! ef=ts ( e>li] geomet]s to squ>e ! circle1 x occurr$ to him t no "o _h tri$ to squ>e a p>abolic segm5t2 he accord+ly attempt$ ! problem & f9ally solv$ x4 ,simil>ly he describes 8 discov]ies ab\t ! volumes & surfaces ( sph]es & cyl9d]s z supplem5t+ ! !orems ( ,eudoxus ab\t ! pyramid1 ! cone & ! cyl9d]4 ,he does n hesitate to say t c]ta9 problems baffl$ him = a long "t2 9 "o place he positively insi/s1 = ! purpose ( po9t+ a moral1 on specify+ two propositions :i* he _h enunciat$ b :i* on fur!r inve/igation prov$ to be wrong4 ,! ord9>y ,,mss4 ( ! ,greek text ( ,ar*im$es give 8 works 9 ! foll[+ ord]3-- #a4 ,on ! ,sph]e & ,cyl9d] 7two books74 #b4 ,measurem5t ( a ,circle4 #c4 ,on ,conoids & ,sph]oids4 #d4 ,on ,spirals4 #e4 ,on ,plane ,equilibriums 7two books74 #f4 ,! ,s&reckon]4 #g4 ,quadrature ( a ,p>abola4 ,a mo/ important addition to ?is li/ has be5 made 9 rec5t ye>s ?r\gh an extraord9>y piece ( good =tune4 ,9 #aijf ,j4 ,l4 ,heib]g1 ! mo/ rec5t editor ( ! text ( ,ar*im$es1 discov]$ a palimpse/ ( ma!matical cont5t 9 ! ,j]usalemic ,libr>y ( "o ,papadop\los ,k]ameus at ,con/ant9ople4 ,?is prov$ to conta9 writ+s ( ,ar*im$es copi$ 9 a good h& ( ! t5? c5tury4 ,an attempt _h be5 made 7=tunately ) only p>tial success7 to wa% out ! old writ+1 & !n ! p>*m5t 0 us$ aga9 to write a ,eu*ologion ^u4 ,h[ev]1 on mo/ ( ! leaves ! e>li] writ+ rema9s m or less legible4 ,! important fact ab\t ! ,,ms4 is t x conta9s1 besides sub/antial portions ( ! treatises previ\sly kn[n1 7#a7 a consid]able portion ( ! "w1 9 two books1 ,on ,float+ ,bodies1 :i* 0 =m]ly suppos$ to h be5 lo/ 9 ,greek & only to h surviv$ 9 ! translation by ,wilhelm ( ,morbeke1 & 7#b7 mo/ preci\s ( all1 ! great] "p ( ! book call$ ,! ,me?od1 treat+ ( ,me*anical ,problems & address$ to ,erato/h5es4 ,! important treatise s happily recov]$ is n[ includ$ 9 ,heib]g's new 7second7 edition ( ! ,greek text ( ,ar*im$es 7,teubn]1 #aiaj-ae71 & "s acc\nt ( x w be giv5 9 ! next *apt]4 ,! ord] 9 :i* ! treatises appe> 9 ! ,,mss4 0 n ! ord] ( composition2 b f ! v>i\s prefaces & f int]nal evid5ce g5]ally we are able to e/abli% ! foll[+ z be+ approximately ! *ronological sequ5ce3-- #a4 ,on ,plane ,equilibriums1 ,i4 #b4 ,quadrature ( a ,p>abola4 #c4 ,on ,plane ,equilibriums1 ,,ii4 #d4 ,! ,me?od4 #e4 ,on ! ,sph]e & ,cyl9d]1 ,i1 ,,ii4 #f4 ,on ,spirals4 #g4 ,on ,conoids & ,sph]oids4 #h4 ,on ,float+ ,bodies1 ,i1 ,,ii4 #i4 ,measurem5t ( a ,circle4 #aj4 ,! ,s&reckon]4 ,9 addition to ! above we h a collection ( geometrical propositions :i* has rea*$ u ?r\gh ! ,arabic ) ! title ,lib] assumptorum ,ar*im$is4 ,!y 7 n writt5 by ,ar*im$es 9 _! pres5t =m1 b 7 probably collect$ by "s lat] ,greek writ] = ! purpose ( illu/rat+ "s anci5t "w4 ,x is1 h[ev]1 q likely t "s ( ! propositions1 :i* are rem>kably elegant1 7 ( ,ar*im$ean orig91 notably ^? conc]n+ ! geometrical figures made ) ?ree & f\r semicircles respectively & call$ 7f _! %ape7 7#a7 ! %oemak]'s knife & 7#b7 ! ,sal9on or salt-cell>1 & ano!r !orem :i* be>s on ! trisection ( an angle4 ,an int]e/+ fact :i* we n[ "k f ,arabian s\rces is t ! =mula = ! area ( any triangle 9 t]ms ( its sides :i* we write 9 ! =m ,delta roots7s - a77s - b77s - c71 & :i* 0 suppos$ to be ,h]on's because ,h]on gives ! geometrical pro( ( x1 0 really due to ,ar*im$es4 ,ar*im$es is fur!r cr$it$ ) ! au?or%ip ( ! fam\s ,cattle-,problem enunciat$ 9 a ,greek epigram edit$ by ,less+ 9 #aggc4 ,accord+ to its head+ ! problem 0 communicat$ by ,ar*im$es to ! ma!maticians at ,alex&ria 9 a lett] to ,erato/h5es2 & a s*olium to ,plato's ,*>mides speaks ( ! problem call$ by ,ar*im$es ! ,cattle-,problem4 ,x is an extraord9>ily difficult problem 9 indet]m9ate analysis1 ! solution ( :i* involves enorm\s figures4 ,( lo/ works ( ,ar*im$es ! foll[+ c be id5tifi$3-- #a4 ,inve/igations relat+ to polyh$ra are ref]r$ to by ,pappus1 :o1 aft] speak+ ( ! five regul> solids1 gives a description ( ?irte5 o!r polyh$ra discov]$ by ,ar*im$es :i* are semi-regul>1 be+ conta9$ by polygons equilat]al & equiangul> b n simil>4 ,"o at lea/ ( ^! semi-regul> solids 01 h[ev]1 already kn[n to ,plato4 #b4 ,a book ( ari?metical cont5t entitl$ ,pr9ciples dealt1 z we le>n f ,ar*im$es himself1 ) ! nam+ ( numb]s1 & exp\nd$ a sy/em ( express+ l>ge numb]s :i* c\ld n be writt5 9 ! ord9>y ,greek notation4 ,9 sett+ out ! same sy/em 9 ! ,s&reckon] 7see ,*apt] ,v4 bel[71 ,ar*im$es expla9s t he does s = ! b5efit ( ^? :o _h n se5 ! e>li] "w4 #c4 ,on ,balances 7or p]haps lev]s74 ,pappus says t 9 ?is "w ,ar*im$es prov$ t great] circles ov]p[] less] circles :5 !y rotate ab\t ! same c5tre4 #d4 ,a book ,on ,c5tres ( ,gravity is allud$ to by ,simplicius4 ,x is n1 h[ev]1 c]ta9 t ?is & ! la/-m5tion$ "w 7 sep>ate treatises1 ,possibly ,book ,i4 ,on ,plane ,equilibriums may h be5 "p ( a l>g] "w 7call$ p]haps ,elem5ts ( ,me*anics71 & ,on ,balances may h be5 an alt]native title4 ,! title ,on ,c5tres ( ,gravity may be a loose way ( ref]r+ to ! same treatise4 #e4 ,catoptrica1 an optical "w f :i* ,!on ( ,alex&ria quotes a rem>k ab\t refraction4 #f4 ,on ,sph]e-mak+1 a me*anical "w on ! con/ruction ( a sph]e to repres5t ! motions ( ! heav5ly bodies 7cf4 pp4 #e-f above74 ,arabian writ]s attribute yet fur!r works to ,ar*im$es1 7#a7 ,on ! circle1 7#b7 ,on a heptagon 9 a circle1 7#c7 ,on circles t\*+ "o ano!r1 7#d7 ,on p>allel l9es1 7#e7 ,on triangles1 7#f7 ,on ! prop]ties ( "r-angl$ triangles1 7#g7 a book ( ,data2 b we h no confirmation ( ^! /atem5ts4 ,,*apt] ,,iv4 ,,geometry ,,9 ,,ar*im$es4 ,! fam\s ,fr5* geomet]1 ,*asles1 drew an in/ructive di/9ction betwe5 ! pr$om9ant features ( ! geometry ( ! two great successors ( ,euclid1 namely1 ,ar*im$es & ,apollonius ( ,p]ga 7! great geomet]1 & au?or ( ! classical treatise on ,conics74 ,! works ( ^! two m5 may1 says ,*asles1 be reg>d$ z ! orig9 & basis ( two great inquiries :i* seem to %>e betwe5 !m ! doma9 ( geometry4 ,apollonius is conc]n$ ) ! ,geometry ( ,=ms & ,situations1 :ile 9 ,ar*im$es we f9d ! ,geometry ( ,measurem5ts1 deal+ ) ! quadrature ( curvil9e> plane figures & ) ! quadrature & cubature ( curv$ surfaces1 inve/igations :i* gave bir? to ! calculus ( ! inf9ite conceiv$ & br\ght to p]fection by ,kepl]1 ,cavali]i1 ,f]mat1 ,leibniz & ,newton4 ,9 geometry ,ar*im$es /&s1 z x 71 on ! %\ld]s ( ,eudoxus 9 t he appli$ ! me?od ( exhau/ion to new & m difficult cases ( quadrature & cubature4 ,fur!r1 9 8 use ( ! me?od he introduc$ an int]e/+ v>iation ( ! proc$ure z we "k x f ,euclid4 ,euclid 7& presumably ,eudoxus also7 only us$ inscrib$ figures1 exhau/+ ! figure to be measur$1 & _h to inv]t ! second half ( ! r$uctio ad absurdum to enable approximation f bel[ 7s to speak7 to be appli$ 9 t case also4 ,ar*im$es1 on ! o!r h&1 approximates f above z well z f bel[2 he approa*es ! area or volume to be measur$ by tak+ clos] & clos] circumscrib$ figures1 z well z inscrib$1 & !reby compress+1 z x 71 ! inscrib$ & circumscrib$ figure into "o1 s t !y ultimately co9cide ) "o ano!r & ) ! figure to be measur$4 ,b he foll[s ! cauti\s me?od to :i* ! ,greeks always adh]$2 he nev] says t a giv5 curve or surface is ! limit+ =m ( ! inscrib$ or circumscrib$ figure2 all t he ass]ts is t we c approa* ! curve or surface z ne>ly z we please4 ,! d$uctive =m ( pro( by ! me?od ( exhau/ion is apt to obscure n only ! way 9 :i* ! results 7 arriv$ at b also ! real "* ( ! proc$ure foll[$4 ,:at ,ar*im$es actually does 9 c]ta9 cases is to p]=m :at are se51 :5 ! analytical equival5ts are set d[n1 to be real integrations2 ?is rem>k applies to 8 inve/igation ( ! areas ( a p>abolic segm5t & a spiral respectively1 ! surface & volume respectively ( a sph]e & a segm5t ( a sph]e1 & ! volume ( any segm5ts ( ! solids ( revolution ( ! second degree4 ,! result is1 z a rule1 only obta9$ aft] a long s]ies ( prelim9>y propositions1 all ( :i* are l9ks 9 a *a9 ( argum5t elaborately =g$ = ! "o purpose4 ,! me?od sugge/s ! tactics ( "s ma/] ( /rategy :o =esees ev]y?+1 elim9ates ev]y?+ n imm$iately conducive to ! execution ( 8 plan1 ma/]s e position 9 its ord]1 & !n sudd5ly 7:5 ! v elaboration ( ! s*eme has almo/ obscur$1 9 ! m9d ( ! onlook]1 its ultimate object7 /rikes ! f9al bl[4 ,?us we read 9 ,ar*im$es proposition aft] proposition ! be>+ ( :i* is n imm$iately obvi\s b :i* we f9d infallibly us$ lat] on2 & we are l$ on by su* easy /ages t ! difficulty ( ! orig9al problem1 z pres5t$ at ! outset1 is sc>cely appreciat$4 ,z ,plut>* says1 ,x is n possible to f9d 9 geometry m difficult & tr\blesome que/ions1 or m simple & lucid explanations4 ,b x is decid$ly a rhetorical exagg]ation :5 ,plut>* goes on to say t we are deceiv$ by ! eas9ess ( ! successive /eps into ! belief t any "o c\ld h discov]$ !m = himself4 ,on ! contr>y1 ! /udi$ simplicity & ! p]fect f9i% ( ! treatises involve at ! same "t an elem5t ( my/]y4 ,al?\gh ea* /ep dep5ds ^u ! prec$+ ones1 we are left 9 ! d>k z to h[ !y 7 sugge/$ to ,ar*im$es4 ,!re is1 9 fact1 mu* tru? 9 a rem>k ( ,wallis to ! effect t he seems z x 7 ( set purpose to h cov]$ up ! traces ( 8 inve/igation z if he _h grudg$ po/]ity ! secret ( 8 me?od ( inquiry :ile he wi%$ to extort f !m ass5t to 8 results4 ,a p>tial exception is n[ furni%$ by ! ,me?od2 = "h we h 7z x 77 a lift+ ( ! veil & a glimpse ( ! int]ior ( ,ar*im$es's work%op4 ,he tells u h[ he discov]$ c]ta9 !orems 9 quadrature & cubature1 & he is at ! same "t c>eful to insi/ on ! diff]5ce betwe5 7#a7 ! means :i* may s]ve to sugge/ ! tru? ( !orems1 al?\gh n furni%+ sci5tific pro(s ( !m1 & 7#b7 ! rigor\s demon/rations ( !m by approv$ geometrical me?ods :i* mu/ foll[ be=e !y c be f9ally accept$ z e/abli%$4 ,writ+ to ,erato/h5es he says3 ,see+ 9 y1 z ,i say1 an e>ne/ /ud5t1 a man ( consid]able em95ce 9 philosophy & an admir] ( ma!matical inquiry :5 x comes y\r way1 ,i h ?\ght fit to write out = y & expla9 9 detail 9 ! same book ! peculi>ity ( a c]ta9 me?od1 :i*1 :5 y see x1 w put y 9 possession ( a means :]eby y c inve/igate "s ( ! problems ( ma!matics by me*anics4 ,?is proc$ure is1 ,i am p]suad$1 no less useful = ! pro(s ( ! actual !orems z well4 ,= c]ta9 ?+s :i* fir/ became cle> to me by a me*anical me?od _h aft]w>ds to be demon/rat$ by geometry1 because _! inve/igation by ! said me?od did n furni% an actual demon/ration4 ,b x is ( c\rse easi]1 :5 we h previ\sly acquir$ by ! me?od "s k ( ! que/ions1 to supply ! pro( ?an x is to f9d ! pro( )\t any previ\s k4 ,?is is a reason :y1 9 ! case ( ! !orems ! pro( ( :i* ,eudoxus 0 ! fir/ to discov]1 namely1 t ! cone is a ?ird "p ( ! cyl9d]1 & ! pyramid a ?ird "p ( ! prism1 hav+ ! same base & equal height1 we %\ld give no small %>e ( ! cr$it to ,democritus1 :o 0 ! fir/ to ass]t ?is tru? ) reg>d to ! said figures1 ?\gh he did n prove x4 ,i am myself 9 ! position ( hav+ made ! discov]y ( ! !orem n[ to be publi%$ 9 ! same way z ,i made my e>li] discov]ies2 & ,i ?\ght x desirable n[ to write out & publi% ! me?od1 p>tly because ,i h already spok5 ( x & ,i d n want to be ?\ght to h utt]$ va9 words1 b p>tly also because ,i am p]suad$ t x w be ( no little s]vice to ma!matics2 = ,i appreh5d t "s1 ei!r ( my contempor>ies or ( my successors1 w1 by means ( ! me?od :5 once e/abli%$1 be able to discov] o!r !orems 9 addition1 :i* h n occurr$ to me4 ,fir/ !n ,i w set out ! v fir/ !orem :i* became kn[n to me by means ( me*anics1 namely1 t ,any segm5t ( a section ( a "r-angl$ cone i4e4 a p>abola is f\r-?irds ( ! triangle :i* has ! same base & equal height2 & aft] ?is ,i w give ea* ( ! o!r !orems inve/igat$ by ! same me?od4 ,!n1 at ! end ( ! book1 ,i w give ! geometrical pro(s ( ! propositions4 ,! foll[+ description w1 ,i hope1 give an idea ( ! g5]al features ( ! me*anical me?od employ$ by ,ar*im$es4 ,suppose t ,x is ! plane or solid figure ! area or cont5t ( :i* is to be f\nd4 ,! me?od 9 ! simple/ case is to weigh inf9itesimal elem5ts ( ,x aga9/ ! correspond+ elem5ts ( ano!r figure1 ,b say1 be+ su* a figure t its area or cont5t & ! position ( its c5tre ( gravity are already kn[n4 ,! diamet] or axis ( ! figure ,x be+ drawn1 ! inf9itesimal elem5ts tak5 are p>allel sections ( ,x 9 g5]al1 b n always1 at "r angles to ! axis or diamet]1 s t ! c5tres ( gravity ( all ! sections lie at "o po9t or o!r ( ! axis or diamet] & _! weights c !re=e be tak5 z act+ at ! sev]al po9ts ( ! diamet] or axis4 ,9 ! case ( a plane figure ! inf9itesimal sections are spok5 ( z p>allel /raight l9es & 9 ! case ( a solid figure z p>allel planes1 & ! aggregate ( ! inf9ite numb] ( sections is said to make up ! :ole figure ,x4 7,al?\gh ! sections are s spok5 ( z /raight l9es or planes1 !y are really indef9itely n>r[ plane /rips or indef9itely ?9 lam9ae respectively47 ,! diamet] or axis is produc$ 9 ! direction away f ! figure to be measur$1 & ! diamet] or axis z produc$ is imag9$ to be ! b> or lev] ( a balance4 ,! object is n[ to apply all ! sep>ate elem5ts ( ,x at "o po9t on ! lev]1 :ile ! correspond+ elem5ts ( ! kn[n figure ,b op]ate at diff]5t po9ts1 namely1 :]e !y actually are 9 ! fir/ in/ance4 ,ar*im$es contrives1 !re=e1 to move ! elem5ts ( ,x away f _! orig9al position & to conc5trate !m at "o po9t on ! lev]1 su* t ea* ( ! elem5ts balances1 ab\t ! po9t ( susp5sion ( ! lev]1 ! correspond+ elem5t ( ,b act+ at its c5tre ( gravity4 ,! elem5ts ( ,x & ,b respectively balance ab\t ! po9t ( susp5sion 9 accordance ) ! prop]ty ( ! lev] t ! weights are inv]sely proportional to ! di/ances f ! fulcrum or po9t ( susp5sion4 ,n[ ! c5tre ( gravity ( ,b z a :ole is kn[n1 & x may !n be suppos$ to act z "o mass at its c5tre ( gravity4 7,ar*im$es assumes z kn[n t ! sum ( ! mom5ts1 z we call !m1 ( all ! elem5ts ( ! figure ,b1 act+ sev]ally at ! po9ts :]e !y actually are1 is equal to ! mom5t ( ! :ole figure appli$ z "o mass at "o po9t1 its c5tre ( gravity47 ,moreov] all ! elem5ts ( ,x are conc5trat$ at ! "o fix$ po9t on ! b> or lev]4 ,if ?is fix$ po9t is ,h1 & ,g is ! c5tre ( gravity ( ! figure ,b1 :ile ,c is ! po9t ( susp5sion1 ,x 3 ,b ,,cg 3 ,,*4 ,?us ! area or cont5t ( ,x is f\nd4 ,conv]sely1 ! me?od c be us$ to f9d ! c5tre ( gravity ( ,x :5 its area or volume is kn[n be=eh&4 ,9 ?is case ! elem5ts ( ,x1 & ,x itself1 h to be appli$ :]e !y are1 & ! elem5ts ( ! kn[n figure or figures h to be appli$ at ! "o fix$ po9t ,h on ! o!r side ( ,c1 & s9ce ,x1 ,b & ,,* are kn[n1 ! proportion ,b 3 ,x ,,cg 3 ,,* det]m9es ,,cg1 :]e ,g is ! c5tre ( gravity ( ,x4 ,! me*anical me?od is us$ = f9d+ 7#a7 ! area ( any p>abolic segm5t1 7#b7 ! volume ( a sph]e & a sph]oid1 7#c7 ! volume ( a segm5t ( a sph]e & ! volume ( a "r segm5t ( ea* ( ! ?ree conicoids ( revolution1 7#d7 ! c5tre ( gravity 7a7 ( a hemisph]e1 7b7 ( any segm5t ( a sph]e1 7c7 ( any "r segm5t ( a sph]oid & a p>aboloid ( revolution1 & 7d7 ( a half-cyl9d]1 or1 9 o!r words1 ( a semicircle4 ,ar*im$es !n proce$s to f9d ! volumes ( two solid figures1 :i* are ! special subject ( ! treatise4 ,! solids arise z foll[s3-- 7#a7 ,giv5 a cyl9d] inscrib$ 9 a rectangul> p>allelepip$ on a squ>e base 9 su* a way t ! two bases ( ! cyl9d] are circles inscrib$ 9 ! opposite squ>e faces1 suppose a plane drawn ?r\gh "o side ( ! squ>e conta9+ "o base ( ! cyl9d] & ?r\gh ! p>allel diamet] ( ! opposite base ( ! cyl9d]4 ,! plane cuts (f a solid ) a surface resembl+ t ( a horse's ho(4 ,ar*im$es proves t ! volume ( ! solid s cut (f is "o six? "p ( ! volume ( ! p>allelepip$4 7#b7 ,a cyl9d] is inscrib$ 9 a cube 9 su* a way t ! bases ( ! cyl9d] are circles inscrib$ 9 two opposite squ>e faces4 ,ano!r cyl9d] is inscrib$ :i* is simil>ly relat$ to ano!r pair ( opposite faces4 ,! two cyl9d]s include betwe5 !m a solid ) all its angles r\nd$ (f2 & ,ar*im$es proves t ! volume ( ?is solid is two-?irds ( t ( ! cube4 ,hav+ prov$ ^! facts by ! me*anical me?od1 ,ar*im$es conclud$ ! treatise ) a rigor\s geometrical pro( ( bo? propositions by ! me?od ( exhau/ion4 ,! ,,ms4 is un=tunately some:at mutilat$ at ! end1 s t a c]ta9 am\nt ( re/oration is necess>y4 ,i %all n[ attempt to give a %ort acc\nt ( ! o!r treatises ( ,ar*im$es 9 ! ord] 9 :i* !y appe> 9 ! editions4 ,! fir/ is-- ,on ! ,sph]e & ,cyl9d]4 ,book ,i4 beg9s ) a preface address$ to ,dosi!us 7a pupil ( ,conon71 :i* rem9ds him t on a =m] occasion he _h communicat$ to him ! treatise prov+ t any segm5t ( a section ( a "r-angl$ cone 7i4e4 a p>abola7 is f\r-?irds ( ! triangle ) ! same base & height1 & adds t he is n[ s5d+ ! pro(s ( c]ta9 !orems :i* he has s9ce discov]$1 & :i* seem to him to be wor?y ( comp>ison ) ,eudoxus's propositions ab\t ! volumes ( a pyramid & a cone4 ,! !orems are 7#a7 t ! surface ( a sph]e is equal to f\r times its greate/ circle 7i4e4 :at we call a great circle ( ! sph]e72 7#b7 t ! surface ( any segm5t ( a sph]e is equal to a circle ) radius equal to ! /raight l9e drawn f ! v]tex ( ! segm5t to a po9t on ! circle :i* is ! base ( ! segm5t2 7#c7 t1 if we h a cyl9d] circumscrib$ to a sph]e & ) height equal to ! diamet]1 !n 7a7 ! volume ( ! cyl9d] is #a-a_/b times t ( ! sph]e & 7b7 ! surface ( ! cyl9d]1 includ+ its bases1 is #a-a_/b times ! surface ( ! sph]e4 ,next come a few def9itions1 foll[$ by c]ta9 ,assumptions1 two ( :i* are well kn[n1 namely3-- #a4 ,( all l9es :i* h ! same extremities ! /raight l9e is ! lea/ 7?is has be5 made ! basis ( an alt]native def9ition ( a /raight l9e74 #b4 ,( unequal l9es1 unequal surfaces & unequal solids ! great] exce$s ! less by su* a magnitude z1 :5 7cont9ually7 add$ to itself1 c be made to exce$ any assign$ magnitude among ^? :i* are comp>able ) x & ) "o ano!r 7i4e4 are ( ! same k9d74 ,?is is ! ,po/ulate ( ,ar*im$es4 ,he also assumes t1 ( pairs ( l9es 7includ+ brok5 l9es7 & pairs ( surfaces1 concave 9 ! same direction & b\nd$ by ! same extremities1 ! out] is great] ?an ! inn]4 ,^! assumptions are fundam5tal to 8 inve/igation1 :i* proce$s ?r\gh\t by means ( figures inscrib$ & circumscrib$ to ! curv$ l9es or surfaces t h to be measur$4 ,aft] "s prelim9>y propositions ,ar*im$es f9ds 7,props4 #ac1 #ad7 ! area ( ! surfaces 7#a7 ( a "r cyl9d]1 7#b7 ( a "r cone4 ,!n1 aft] quot+ c]ta9 ,euclidean propositions ab\t cones & cyl9d]s1 he passes to ! ma9 bus9ess ( ! book1 ! measurem5t ( ! volume & surface ( a sph]e & a segm5t ( a sph]e4 ,by circumscrib+ & inscrib+ to a great circle a regul> polygon ( an ev5 numb] ( sides & mak+ x revolve ab\t a diamet] connect+ two opposite angul> po9ts he obta9s solids ( revolution great] & less respectively ?an ! sph]e4 ,9 a s]ies ( propositions he f9ds expressions = 7a7 ! surfaces1 7b7 ! volumes1 ( ! figures s inscrib$ & circumscrib$ to ! sph]e4 ,next he proves 7,prop4 #cb7 t1 if ! inscrib$ & circumscrib$ polygons :i*1 by _! revolution1 g5]ate ! figures are simil>1 ! surfaces ( ! figures are 9 ! duplicate ratio1 & _! volumes 9 ! triplicate ratio1 ( _! sides4 ,!n he proves t ! surfaces & volumes ( ! inscrib$ & circumscrib$ figures respectively are less & great] ?an ! surface & volume respectively to :i* ! ma9 propositions decl>e ! surface & volume ( ! sph]e to be equal 7,props4 #be1 #bg1 #cj1 #ca ,cor474 ,he has n[ all ! mat]ial = apply+ ! me?od ( exhau/ion & s proves ! ma9 propositions ab\t ! surface & volume ( ! sph]e4 ,! re/ ( ! book applies ! same proc$ure to a segm5t ( ! sph]e4 ,surfaces ( revolution are inscrib$ & circumscrib$ to a segm5t less ?an a hemisph]e1 & ! !orem ab\t ! surface ( ! segm5t is f9ally prov$ 9 ,prop4 #db4 ,prop4 #dc d$uces ! surface ( a segm5t great] ?an a hemisph]e4 ,prop4 #dd gives ! volume ( ! sector ( ! sph]e :i* includes any segm5t4 ,book ,,ii beg9s ) ! problem ( f9d+ a sph]e equal 9 volume to a giv5 cone or cyl9d]2 ?is requires ! solution ( ! problem ( ! two mean proportionals1 :i* is accord+ly assum$4 ,prop4 #b d$uces1 by means ( #a41 #dd1 an expression = ! volume ( a segm5t ( a sph]e1 & ,props4 #c1 #d solve ! important problems ( cutt+ a giv5 sph]e by a plane s t 7a7 ! surfaces1 7b7 ! volumes1 ( ! segm5ts may h to "o ano!r a giv5 ratio4 ,! solution ( ! second problem 7,prop4 #d7 is difficult4 ,ar*im$es r$uces x to ! problem ( divid+ a /raight l9e ,,ab into two p>ts at a po9t ,m su* t ,,mb 3 7a giv5 l5g?7 7a giv5 area7 3 ,,am#b4 ,! solution ( ?is problem ) a det]m9ation ( ! limits ( possibility are giv5 9 a fragm5t by ,ar*im$es1 discov]$ & pres]v$ = u by ,eutocius 9 8 comm5t>y on ! book2 !y are effect$ by means ( ! po9ts ( int]section ( two conics1 a p>abola & a rectangul> hyp]bola4 ,?ree problems ( con/ruction foll[1 ! fir/ two ( :i* are to con/ruct a segm5t ( a sph]e simil> to "o giv5 segm5t1 & hav+ 7a7 its volume1 7b7 its surface1 equal to t ( ano!r giv5 segm5t ( a sph]e4 ,! la/ two propositions are int]e/+4 ,prop4 #h proves t1 if ,v1 ,v' be ! volumes1 & ,s1 ,s' ! surfaces1 ( two segm5ts into :i* a sph]e is divid$ by a plane1 ,v & ,s belong+ to ! great] segm5t1 !n ,s#b 3 ,s'#b ,v 3 ,v' ,s7#c_/b7 3 ,s'7#c_/b74 ,prop4 #i proves t1 ( all segm5ts ( sph]es :i* h equal surfaces1 ! hemisph]e is ! greate/ 9 volume4 ,! ,measurem5t ( a ,circle4 ,?is treatise1 9 ! =m 9 :i* x has come d[n to u1 conta9s only ?ree propositions2 ! second1 be+ an easy d$uction f ,props4 #a & #c1 is out ( place 9 s f> z x uses ! result ( ,prop4 #c4 ,9 ,prop4 #a ,ar*im$es inscribes & circumscribes to a circle a s]ies ( successive regul> polygons1 beg9n+ ) a squ>e1 & cont9ually d\bl+ ! numb] ( sides2 he !n proves 9 ! or?odox mann] by ! me?od ( exhau/ion t ! area ( ! circle is equal to t ( a "r-angl$ triangle1 9 :i* ! p]p5dicul> is equal to ! radius1 & ! base equal to ! circumf]5ce1 ( ! circle4 ,prop4 #c is ! fam\s proposition 9 :i* ,ar*im$es f9ds by %e] calculation upp] & l[] ari?metical limits to ! ratio ( ! circumf]5ce ( a circle to its diamet]1 or :at we call pi2 ! result obta9$ is #c-a_/g pi #c-aj_/ga4 ,ar*im$es inscribes & circumscribes successive regul> polygons1 beg9n+ ) hexagons1 & d\bl+ ! numb] ( sides cont9ually1 until he arrives at inscrib$ & circumscrib$ regul> polygons ) #if sides2 see+ !n t ! l5g? ( ! circumf]5ce ( ! circle is int]m$iate betwe5 ! p]imet]s ( ! two polygons1 he calculates ! two p]imet]s 9 t]ms ( ! diamet] ( ! circle4 ,8 calculation is bas$ on two close approximations 7an upp] & a l[]7 to ! value ( root#c1 t be+ ! cotang5t ( ! angle ( #cj deg41 f :i* he beg9s to "w4 ,he assumes z kn[n t #bfe_/aec root#c #acea_/ghj4 ,9 ! text1 z we h x1 only ! results ( ! /eps 9 ! calculation are giv51 b !y involve ! f9d+ ( approximations to ! squ>e roots ( sev]al l>ge numb]s3 ?us #aagb-a_/h is giv5 z ! approximate value ( root7#acgcidc-cc_/fd71 #cjac-c_/d z t ( root7#ijhbcba7 & #ahch-i_/aa z t ( root7#cchjibi74 ,9 ?is way ,ar*im$es arrives at #adfhh_/7dfgc-a_/b7 z ! ratio ( ! p]imet] ( ! circumscrib$ polygon ( #if sides to ! diamet] ( ! circle2 ?is is ! figure :i* he r\nds up into #c-a_/g4 ,! correspond+ figure = ! inscrib$ polygon is #fccf_/7bjag-a_/d71 :i*1 he says1 is #c-aj_/ga4 ,?is example %[s h[ little ! ,greeks 7 emb>rass$ 9 ari?metical calculations by _! alphabetical sy/em ( num]als4 ,on ,conoids & ,sph]oids4 ,! preface address$ to ,dosi!us %[s1 z we may also inf] f int]nal evid5ce1 t ! :ole ( ?is book also 0 orig9al4 ,ar*im$es fir/ expla9s :at 8 conoids & sph]oids are1 & !n1 aft] ea* description1 /ates ! ma9 results :i* x is ! aim ( ! treatise to prove4 ,! conoids are two4 ,! fir/ is ! "r-angl$ conoid1 a "n adapt$ f ! old "n 7section ( a "r-angl$ cone7 = a p>abola2 ?is conoid is !re=e a p>aboloid ( revolution4 ,! second is ! obtuse-angl$ conoid1 :i* is a hyp]boloid ( revolution describ$ by ! revolution ( a hyp]bola 7a section ( an obtuse-angl$ cone7 ab\t its transv]se axis4 ,! sph]oids are two1 be+ ! solids ( revolution describ$ by ! revolution ( an ellipse 7a section ( an acute-angl$ cone7 ab\t 7#a7 its major axis & 7#b7 its m9or axis2 ! fir/ is call$ ! oblong 7or oblate7 sph]oid1 ! second ! flat 7or prolate7 sph]oid4 ,z ! volumes ( oblique segm5ts ( conoids & sph]oids are aft]w>ds f\nd 9 t]ms ( ! volume ( ! conical figure ) ! base ( ! segm5t z base & ! v]tex ( ! segm5t z v]tex1 & z ! said base is ?us an elliptic section ( an oblique circul> cone1 ,ar*im$es calls ! conical figure ) an elliptic base a segm5t ( a cone z di/9ct f a cone4 ,z usual1 a s]ies ( prelim9>y propositions is requir$4 ,ar*im$es fir/ sums1 9 geometrical =m1 c]ta9 s]ies1 includ+ ! ari?metical progression1 a1 #b;a1 #c;a1 444 na1 & ! s]ies =m$ by ! squ>es ( ^! t]ms 79 o!r words ! s]ies #ab1 #bb1 #cb1 444 n#b72 ^! summations are requir$ = ! f9al addition ( an indef9ite numb] ( elem5ts ( ea* figure1 :i* am\nts to an integration4 ,next come two prop]ties ( conics 7,prop4 #c71 !n ! det]m9ation by ! me?od ( exhau/ion ( ! area ( an ellipse 7,prop4 #d74 ,?ree propositions foll[1 ! fir/ two ( :i* 7,props4 #g1 #h7 %[ t ! conical figure above ref]r$ to is really a segm5t ( an oblique circul> cone2 ?is is done by actually f9d+ ! circul> sections4 ,prop4 #i gives a simil> pro( t ea* elliptic section ( a conoid or sph]oid is a section ( a c]ta9 oblique circul> cyl9d] 7) axis p>allel to ! axis ( ! segm5t ( ! conoid or sph]oid cut (f by ! said elliptic section74 ,props4 #aa-ah %[ ! nature ( ! v>i\s sections :i* cut (f segm5ts ( ea* conoid & sph]oid & :i* are circles or ellipses accord+ z ! section is p]p5dicul> or obliquely incl9$ to ! axis ( ! solid2 !y include also c]ta9 prop]ties ( tang5t planes1 etc4 ,! real bus9ess ( ! treatise beg9s ) ,props4 #ai1 #bj2 "h x is %[n h[1 by draw+ _m plane sections equidi/ant f "o ano!r & all p>allel to ! base ( ! segm5t ( ! solid1 & describ+ cyl9d]s 79 g5]al oblique7 ?r\gh ea* plane section ) g5]ators p>allel to ! axis ( ! segm5t & t]m9at$ by ! contigu\s sections on ei!r side1 we c make figures circumscrib$ & inscrib$ to ! segm5t1 made up ( segm5ts ( cyl9d]s ) p>allel faces & pres5t+ ! appe>ance ( ! /eps ( a /aircase4 ,add+ ! elem5ts ( ! inscrib$ & circumscrib$ figures respectively & us+ ! me?od ( exhau/ion1 ,ar*im$es f9ds ! volumes ( ! respective segm5ts ( ! solids 9 ! approv$ mann] 7,props4 #ba1 #bb = ! p>aboloid1 ,props4 #be1 #bf = ! hyp]boloid1 & ,props4 #bg-cj = ! sph]oids74 ,! results are /at$ 9 ?is =m3 7#a7 ,any segm5t ( a p>aboloid ( revolution is half z l>ge aga9 z ! cone or segm5t ( a cone :i* has ! same base & axis2 7#b7 ,any segm5t ( a hyp]boloid ( revolution or ( a sph]oid is to ! cone or segm5t ( a cone ) ! same base & axis 9 ! ratio ( ,,ad #c;,,ca to ,,ad #b;,,ca 9 ! case ( ! hyp]boloid1 & ( #c;,,ca - ,,ad to #b;,,ca - ,,ad 9 ! case ( ! sph]oid1 :]e ,c is ! c5tre1 ,a ! v]tex ( ! segm5t1 & ,,ad ! axis ( ! segm5t 7suppos$ 9 ! case ( ! sph]oid to be n great] ?an half ! sph]oid74 ,on ,spirals4 ,! preface address$ to ,dosi!us is ( "s l5g? & conta9s1 fir/1 a tribute to ! memory ( ,conon1 & next a summ>y ( ! !orems ab\t ! sph]e & ! conoids & sph]oids includ$ 9 ! above two treatises4 ,ar*im$es !n passes to ! spiral :i*1 he says1 pres5ts ano!r sort ( problem1 hav+ no?+ 9 common ) ! =ego+4 ,aft] a def9ition ( ! spiral he enunciates ! ma9 propositions ab\t x :i* are to be prov$ 9 ! treatise4 ,! spiral 7n[ kn[n z ! ,spiral ( ,ar*im$es7 is def9$ z ! locus ( a po9t />t+ f a giv5 po9t 7call$ ! orig97 on a giv5 /raight l9e & mov+ along ! /raight l9e at uni=m spe$1 :ile ! l9e itself revolves at uni=m spe$ ab\t ! orig9 z a fix$ po9t4 ,props4 #a-aa are prelim9>y1 ! la/ two am\nt+ to ! summation ( c]ta9 s]ies requir$ = ! f9al addition ( an indef9ite numb] ( elem5t-areas1 :i* aga9 am\nts to integration1 9 ord] to f9d ! area ( ! figure cut (f betwe5 any portion ( ! curve & ! two radii vectores drawn to its extremities4 ,props4 #ac-bj are int]e/+ & difficult propositions e/abli%+ ! prop]ties ( tang5ts to ! spiral4 ,props4 #ba-bc %[ h[ to inscribe & circumscribe to any portion ( ! spiral figures consi/+ ( a multitude ( elem5ts :i* are n>r[ sectors ( circles ) ! orig9 z c5tre2 ! area ( ! spiral is int]m$iate betwe5 ! areas ( ! inscrib$ & circumscrib$ figures1 & by ! usual me?od ( exhau/ion ,ar*im$es f9ds ! areas requir$4 ,prop4 #bd gives ! area ( ! fir/ complete turn ( ! spiral 7 #a_/cpi7#bpia7#b1 :]e ! spiral is r a!ta71 & ( any portion ( x up to ,,op :]e ,p is any po9t on ! fir/ turn4 ,props4 #be1 #bf deal simil>ly ) ! second turn ( ! spiral & ) ! area subt5d$ by any arc 7n be+ great] ?an a complete turn7 on any turn4 ,prop4 #bg proves ! int]e/+ prop]ty t1 if ,r#a be ! area ( ! fir/ turn ( ! spiral b\nd$ by ! initial l9e1 ,r#b ! area ( ! r+ add$ by ! second complete turn1 ,r#c ! area ( ! r+ add$ by ! ?ird turn1 & s on1 !n ,r#c #b,r#b1 ,r#d #c,r#b1 ,r#e #d,r#b1 & s on to ,rn 7n - #a7,r#b1 :ile ,r#b1 #f,r#a4 ,quadrature ( ! ,p>abola4 ,! title ( ?is "w seems orig9ally to h be5 ,on ! ,section ( a ,"r-angl$ ,cone & to h be5 *ang$ aft] ! "t ( ,apollonius1 :o 0 ! fir/ to call a p>abola by t "n4 ,! preface address$ to ,dosi!us 0 evid5tly ! fir/ communication f ,ar*im$es to him aft] ! dea? ( ,conon4 ,x beg9s ) a feel+ allusion to 8 lo/ fri5d1 to :om ! treatise 0 orig9ally to h be5 s5t4 ,x is 9 ?is preface t ,ar*im$es alludes to ! lemma us$ by e>li] geomet]s z ! basis ( ! me?od ( exhau/ion 7! ,po/ulate ( ,ar*im$es1 or ! !orem ( ,euclid ,x41 #a74 ,he m5tions z hav+ be5 prov$ by means ( x 7#a7 ! !orems t ! areas ( circles are to "o ano!r 9 ! duplicate ratio ( _! diamet]s1 & t ! volumes ( sph]es are 9 ! triplicate ratio ( _! diamet]s1 & 7#b7 ! propositions prov$ by ,eudoxus ab\t ! volumes ( a cone & a pyramid4 ,no "o1 he says1 s f> z he is aw>e1 has yet tri$ to squ>e ! segm5t b\nd$ by a /raight l9e & a section ( a "r-angl$ cone 7a p>abola72 b he has succe$$ 9 prov+1 by means ( ! same lemma1 t ! p>abolic segm5t is equal to f\r-?irds ( ! triangle on ! same base & ( equal height1 & he s5ds ! pro(s1 fir/ z inve/igat$ by means ( me*anics & secondly z demon/rat$ by geometry4 ,! phraseology %[s t "h1 z 9 ! ,me?od1 ,ar*im$es reg>d$ ! me*anical inve/igation z furni%+ evid5ce r ?an pro( ( ! tru? ( ! proposition1 pure geometry alone furni%+ ! absolute pro( requir$4 ,! me*anical pro( ) ! necess>y prelim9>y propositions ab\t ! p>abola 7"s ( :i* are m]ely quot$1 :ile two1 evid5tly orig9al1 are prov$1 ,props4 #d1 #e7 ext5ds d[n to ,prop4 #ag2 ! geometrical pro( ) o!r auxili>y propositions completes ! book 7,props4 #ah-bd74 ,! me*anical pro( recalls t ( ! ,me?od 9 "s respects1 b is m elaborate 9 t ! elem5ts ( ! area ( ! p>abola to be measur$ are n /raight l9es b n>r[ /rips4 ,! figures inscrib$ & circumscrib$ to ! segm5t are made up ( su* n>r[ /rips & h a saw-l edge2 all ! elem5ts are trapezia except two1 :i* are triangles1 "o 9 ea* figure4 ,ea* trapezium 7or triangle7 is weigh$ :]e x is aga9/ ano!r area hung at a fix$ po9t ( an assum$ lev]2 ?us ! :ole ( ! inscrib$ & circumscrib$ figures respectively are weigh$ aga9/ ! sum ( an indef9ite numb] ( areas all susp5d$ f "o po9t on ! lev]4 ,! result is obta9$ by a real integration1 confirm$ z usual by a pro( by ! me?od ( exhau/ion4 ,! geometrical pro( proce$s ?us4 ,draw+ 9 ! segm5t ! inscrib$ triangle ) ! same base & height z ! segm5t1 ,ar*im$es next inscribes triangles 9 precisely ! same way 9 ea* ( ! segm5ts left ov]1 & proves t ! sum ( ! two new triangles is #a_/d ( ! orig9al inscrib$ triangle4 ,aga91 draw+ triangles inscrib$ 9 ! same way 9 ! f\r segm5ts left ov]1 he proves t _! sum is #a_/d ( ! sum ( ! prec$+ pair ( triangles & !re=e 7#a_/d7b ( ! orig9al inscrib$ triangle4 ,proce$+ ?us1 we h a s]ies ( areas exhau/+ ! p>abolic segm5t4 ,_! sum1 if we d5ote ! fir/ inscrib$ triangle by ,delta1 is ,delta#a #a_/d 7#a_/d7b 7#a_/d7c 4 4 4 4 ,ar*im$es proves geometrically 9 ,prop4 #bc t ! sum ( ?is inf9ite s]ies is #d_/c;,delta1 & !n confirms by r$uctio ad absurdum ! equality ( ! area ( ! p>abolic segm5t to ?is area4 ,,*apt] ,v4 ,,! ,,s&reckon]4 ,! ,s&reckon] des]ves a place by itself4 ,x is n ma!matically v important2 b x is an ari?metical curiosity :i* illu/rates ! v]satility & g5ius ( ,ar*im$es1 & x conta9s "s preci\s details ( ! hi/ory ( ,greek a/ronomy :i*1 com+ f su* a s\rce & at fir/ h&1 possess unique au?ority4 ,we w beg9 ) ! a/ronomical data4 ,!y are conta9$ 9 ! preface address$ to ,k+ ,gelon ( ,syracuse1 :i* beg9s z foll[s3-- ,!re are "s1 ,k+ ,gelon1 :o ?9k t ! numb] ( ! s& is inf9ite 9 multitude2 & ,i mean by ! s& n only t :i* exi/s ab\t ,syracuse & ! re/ ( ,sicily b also t :i* is f\nd 9 e region :e!r inhabit$ or un9habit$4 ,aga91 !re are "s :o1 )\t reg>d+ x z inf9ite1 yet ?9k t no numb] has be5 nam$ :i* is great en\gh to exce$ its multitude4 ,& x is cle> t !y :o hold ?is view1 if !y imag9$ a mass made up ( s& 9 o!r respects z l>ge z ! mass ( ! e>?1 includ+ 9 x all ! seas & ! holl[s ( ! e>? fill$ up to a height equal to t ( ! highe/ ( ! m\nta9s1 w\ld be _m times fur!r /ill f recognis+ t any numb] c\ld be express$ :i* exce$$ ! multitude ( ! s& s tak54 ,b ,i w try to %[ y1 by means ( geometrical pro(s :i* y w be able to foll[1 t1 ( ! numb]s nam$ by me & giv5 9 ! "w :i* ,i s5t to ,zeuxippus1 "s exce$ n only ! numb] ( ! mass ( s& equal 9 size to ! e>? fill$ up 9 ! way describ$1 b also t ( a mass equal 9 size to ! univ]se4 ,n[ y are aw>e t 'univ]se' is ! "n giv5 by mo/ a/ronom]s to ! sph]e ! c5tre ( :i* is ! c5tre ( ! e>?1 :ile ! radius is equal to ! /raight l9e betwe5 ! c5tre ( ! sun & ! c5tre ( ! e>?4 ,?is is ! common acc\nt1 z y h he>d f a/ronom]s4 ,b ,ari/>*us ( ,samos br\ght out a book consi/+ ( "s hypo!ses1 9 :i* ! premises lead to ! conclusion t ! univ]se is _m times great] ?an t n[ s call$4 ,8 hypo!ses are t ! fix$ />s & ! sun rema9 unmov$1 t ! e>? revolves ab\t ! sun 9 ! circumf]5ce ( a circle1 ! sun ly+ 9 ! c5tre ( ! orbit1 & t ! sph]e ( ! fix$ />s1 situat$ ab\t ! same c5tre z ! sun1 is s great t ! circle 9 :i* he supposes ! e>? to revolve be>s su* a ratio to ! di/ance ( ! fix$ />s z ! c5tre ( ! sph]e be>s to its surface4 ,"h !n is absolute & practically contempor>y evid5ce t ! ,greeks1 9 ! p]son ( ,ari/>*us ( ,samos 7ab\t #caj-bcj ,b4,c471 _h anticipat$ ,cop]nicus4 ,by ! la/ words quot$ ,ari/>*us only meant to say t ! size ( ! e>? is negligible 9 comp>ison ) ! imm5sity ( ! univ]se4 ,?is1 h[ev]1 does n suit ,ar*im$es's purpose1 because he has to assume a def9ite size1 h[ev] l>ge1 = ! univ]se4 ,consequ5tly he takes a lib]ty ) ,ari/>*us4 ,he says t ! c5tre 7a ma!matical po9t7 c h no ratio :atev] to ! surface ( ! sph]e1 & t we mu/ !re=e take ,ari/>*us to mean t ! size ( ! e>? is to t ( ! s-call$ univ]se z ! size ( ! s-call$ univ]se is to t ( ! real univ]se 9 ! new s5se4 ,next1 he has to assume c]ta9 dim5sions = ! e>?1 ! moon & ! sun1 & to e/imate ! angle subt5d$ at ! c5tre ( ! e>? by ! sun's diamet]2 & 9 ea* case he has to exagg]ate ! probable figures s z to be on ! safe side4 ,:ile !re=e 7he says7 "s h tri$ to prove t ! p]imet] ( ! e>? is #cjj1jjj /adia 7,erato/h5es1 8 contempor>y1 made x #beb1jjj /adia1 say #bd1ffb miles1 giv+ a diamet] ( ab\t #g1hej miles71 he w assume x to be t5 times z great or #c1jjj1jjj /adia4 ,! diamet] ( ! e>?1 he cont9ues1 is great] ?an t ( ! moon & t ( ! sun is great] ?an t ( ! e>?4 ,( ! diamet] ( ! sun he obs]ves t ,eudoxus _h decl>$ x to be n9e times t ( ! moon1 & 8 own fa!r1 ,phidias1 _h made x twelve times1 :ile ,ari/>*us _h tri$ to prove t ! diamet] ( ! sun is great] ?an eighte5 times b less ?an tw5ty times ! diamet] ( ! moon 7?is 0 9 ! treatise ( ,ari/>*us ,on ! ,sizes & ,di/ances ( ! ,sun & ,moon1 :i* is /ill extant1 & is an admirable piece ( geometry1 prov+ rigor\sly1 on ! basis ( c]ta9 assumptions1 ! result /at$74 ,ar*im$es aga9 int5ds to be on ! safe side1 s he takes ! diamet] ( ! sun to be ?irty times t ( ! moon & n great]4 ,la/ly1 he says t ,ari/>*us discov]$ t ! diamet] ( ! sun appe>$ to be ab\t #a_/gbj? "p ( ! zodiac circle1 i4e4 to subt5d an angle ( ab\t half a degree2 & he describes a simple in/rum5t by :i* he himself f\nd t ! angle subt5d$ by ! diamet] ( ! sun at ! "t :5 x _h j ris5 0 less ?an #a_/afd? "p & great] ?an #a_/bjj? "p ( a "r angle4 ,tak+ ?is z ! size ( ! angle subt5d$ at ! eye ( ! obs]v] on ! surface ( ! e>?1 he works out1 by an int]e/+ geometrical proposition1 ! size ( ! angle subt5d$ at ! c5tre ( ! e>?1 :i* he f9ds to be #a_/bjcrd "p ( a "r angle4 ,consequ5tly ! diamet] ( ! sun is great] ?an ! side ( a regul> polygon ( #hab sides inscrib$ 9 a great circle ( ! s-call$ univ]se1 & a =tiori great] ?an ! side ( a regul> *iliagon 7polygon ( #ajjj sides7 inscrib$ 9 t circle4 ,on ^! assumptions1 & see+ t ! p]imet] ( a regul> *iliagon 7z ( any o!r regul> polygon ( m ?an six sides7 inscrib$ 9 a circle is m ?an #c times ! l5g? ( ! diamet] ( ! circle1 x easily foll[s t1 :ile ! diamet] ( ! e>? is less ?an #a1jjj1jjj /adia1 ! diamet] ( ! s-call$ univ]se is less ?an #aj1jjj times ! diamet] ( ! e>?1 & !re=e less ?an #aj1jjj1jjj1jjj /adia4 ,la/ly1 ,ar*im$es assumes t a quantity ( s& n great] ?an a poppy-se$ conta9s n m ?an #aj1jjj gra9s1 & t ! diamet] ( a poppy-se$ is n less ?an #a_/dj? ( a dactylus 7:ile a /adium is less ?an #aj1jjj dactyli74 ,ar*im$es is n[ ready to "w out 8 calculation1 b = ! inadequacy ( ! alphabetic sy/em ( num]als to express su* l>ge numb]s z are requir$4 ,he1 !re=e1 develops 8 rem>kable t]m9ology = express+ l>ge numb]s4 ,! ,greek has names = all numb]s up to a myriad 7#aj1jjj72 !re 01 !re=e1 no difficulty 9 express+ ) ! ord9>y num]als all numb]s up to a myriad myriads 7#ajj1jjj1jjj74 ,let u1 says ,ar*im$es1 call all ^! numb]s numb]s ( ! fir/ ord]4 ,let ! second ord] ( numb]s beg9 ) #ajj1jjj1jjj1 & end ) #ajj1jjj1jjjb4 ,let #ajj1jjj1jjjb be ! fir/ numb] ( ! ?ird ord]1 & let ?is ext5d to #ajj1jjj1jjjc2 & s on1 to ! myriad-myriad? ord]1 beg9n+ ) #ajj1jjj1jjj7ii1iii1iii7 & end+ ) #ajj1jjj1jjj7ajj1jjj1jjj71 :i* = brevity we w call ,p4 ,let all ! numb]s ( all ! ord]s up to ,p =m ! fir/ p]iod1 & let ! fir/ ord] ( ! second p]iod beg9 ) ,p & end ) #ajj1jjj1jjj ,p2 let ! second ord] beg9 ) ?is1 ! ?ird ord] ) #ajj1jjj1jjjb ,p1 & s on up to ! #ajj1jjj1jjj? ord] ( ! second p]iod1 end+ ) #a1jjj1jjj1jjj7ajj1jjj1jjj7 ,p or ,p#b4 ,! fir/ ord] ( ! ?ird p]iod beg9s ) ,p#b1 & ! ord]s proce$ z be=e4 ,cont9u+ ! s]ies ( p]iods & ord]s ( ea* p]iod1 we f9ally arrive at ! #ajj1jjj1jjj? p]iod end+ ) ,p7#ajj1jjj1jjj74 ,! prodigi\s ext5t ( ?is s*eme is se5 :5 x is consid]$ t ! la/ numb] ( ! fir/ p]iod w\ld n[ be repres5t$ by #a foll[$ by #hjj1jjj1jjj ciph]s1 :ile ! la/ numb] ( ! #ajj1jjj1jjj? p]iod w\ld require #ajj1jjj1jjj times z _m ciph]s1 i4e4 #hj1jjj million million ciph]s4 ,z a matt] ( fact1 ,ar*im$es does n ne$1 9 ord] to express ! numb] ( ! s&1 to g beyond ! eigh? ord] ( ! fir/ p]iod4 ,! ord]s ( ! fir/ p]iod beg9 respectively ) #a1 #ajh1 #ajaf1 #ajbd1 444 7#ajh77ii1iii1iii72 & we c express all ! numb]s requir$ 9 p[]s ( #aj4 ,s9ce ! diamet] ( a poppy-se$ is n less ?an #a_/dj? ( a dactylus1 & sph]es are to "o ano!r 9 ! triplicate ratio ( _! diamet]s1 a sph]e ( diamet] #a dactylus is n great] ?an #fd1jjj poppy-se$s1 &1 !re=e1 conta9s n m ?an #fd1jjj ,x #aj1jjj gra9s ( s&1 & a =tiori n m ?an #a1jjj1jjj1jjj1 or #aji gra9s ( s&4 ,ar*im$es multiplies ! diamet] ( ! sph]e cont9ually by #ajj1 & /ates ! correspond+ numb] ( gra9s ( s&4 ,a sph]e ( diamet] #aj1jjj dactyli & a =tiori ( "o /adium conta9s less ?an #ajba gra9s2 & proce$+ 9 ?is way to sph]es ( diamet] #ajj /adia1 #aj1jjj /adia & s on1 he arrives at ! numb] ( gra9s ( s& 9 a sph]e ( diamet] #aj1jjj1jjj1jjj /adia1 :i* is ! size ( ! s-call$ univ]se2 ! correspond+ numb] ( gra9s ( s& is #ajea4 ,! diamet] ( ! real univ]se be+ #aj1jjj times t ( ! s-call$ univ]se1 ! f9al numb] ( gra9s ( s& 9 ! real univ]se is f\nd to be #ajfc1 :i* 9 ,ar*im$es's t]m9ology is a myriad-myriad units ( ! eigh? ord] ( numb]s4 ,,*apt] ,,vi4 ,,me*anics4 ,x is said t ,ar*ytas 0 ! fir/ to treat me*anics 9 a sy/ematic way by ! aid ( ma!matical pr9ciples2 b no trace survives ( any su* "w by him4 ,9 practical me*anics he is said to h con/ruct$ a me*anical dove :i* w\ld fly1 & also a rattle to amuse *ildr5 & keep !m f break+ ?+s ab\t ! h\se 7s says ,ari/otle1 add+ = x is impossible = *ildr5 to keep /ill74 ,9 ! ,ari/otelian ,me*anica we f9d a rem>k on ! m>vel ( a great weight be+ mov$ by a small =ce1 & ! problems discuss$ br+ 9 ! lev] 9 v>i\s =ms z a means ( do+ ?is4 ,we are told also t practically all movem5ts 9 me*anics r$uce to ! lev] & ! pr9ciple ( ! lev] 7t ! weight & ! =ce are 9 inv]se proportion to ! di/ances f ! po9t ( susp5sion or fulcrum ( ! po9ts at :i* !y act1 x be+ assum$ t !y act 9 directions p]p5dicul> to ! lev]74 ,b ! lev] is m]ely ref]r$ to ! circle2 ! =ce :i* acts at ! great] di/ance f ! fulcrum is said to move a weight m easily because x describes a great] circle4 ,!re is1 !re=e1 no pro( "h4 ,x 0 res]v$ = ,ar*im$es to prove ! prop]ty ( ! lev] or balance ma!matically1 on ! basis ( c]ta9 po/ulates precisely =mulat$ & mak+ no l>ge dem& on ! fai? ( ! le>n]4 ,! treatise ,on ,plane ,equilibriums 9 two books is1 z ! title implies1 a "w on /atics only2 &1 aft] ! pr9ciple ( ! lev] or balance has be5 e/abli%$ 9 ,props4 #f1 #g ( ,book ,i41 ! re/ ( ! treatise is devot$ to f9d+ ! c5tre ( gravity ( c]ta9 figures4 ,!re is no dynamics 9 ! "w & !re=e no room = ! p>allelogram ( velocities1 :i* is giv5 ) a fairly adequate pro( 9 ! ,ari/otelian ,me*anica4 ,ar*im$es's po/ulates include assumptions to ! foll[+ effect3 7#a7 ,equal weights at equal di/ances are 9 equilibrium1 & equal weights at unequal di/ances are n 9 equilibrium1 b ! sy/em 9 t case incl9es t[>ds ! weight :i* is at ! great] di/ance1 9 o!r words1 ! action ( ! weight :i* is at ! great] di/ance produces motion 9 ! direction 9 :i* x acts2 7#b7 & 7#c7 ,if :5 weights are 9 equilibrium some?+ is add$ to or subtract$ f "o ( ! weights1 ! sy/em w incl9e t[>ds ! weight :i* is add$ to or ! weight f :i* no?+ is tak5 respectively2 7#d7 & 7#e7 ,if equal & simil> figures be appli$ to "o ano!r s z to co9cide ?r\gh\t1 _! c5tres ( gravity also co9cide2 if figures be unequal b simil>1 _! c5tres ( gravity are simil>ly situat$ ) reg>d to ! figures4 ,! ma9 proposition1 t two magnitudes balance at di/ances reciprocally proportional to ! magnitudes1 is prov$ fir/ = comm5surable & !n = incomm5surable magnitudes4 ,prelim9>y propositions h dealt ) equal magnitudes dispos$ at equal di/ances on a /raight l9e & odd or ev5 9 numb]1 & h %[n :]e ! c5tre ( gravity ( ! :ole sy/em lies4 ,take fir/ ! case ( comm5surable magnitudes4 ,if ,a1 ,b be ! weights act+ at ,e1 ,d on ! /raight l9e ,,ed respectively1 & ,,ed be divid$ at ,c s t ,a 3 ,b ,,dc 3 ,,ce1 ,ar*im$es has to prove t ! sy/em is 9 equilibrium ab\t ,c4 ,he produces ,,ed to ,k1 s t ,,dk ,,ec1 & ,,de to ,l s t ,,el ,,cd2 ,,lk is !n a /raight l9e bisect$ at ,c4 ,aga91 let ,h be tak5 on ,,lk su* t ,,lh #b,,le or #b;,,cd1 & x foll[s t ! rema9d] ,,hk #b;,,dk or #b;,,ec4 ,s9ce ,a1 ,b are comm5surable1 s are ,,ec1 ,,cd4 ,let x be a common measure ( ,,ec1 ,,cd4 ,take a weight w su* t w is ! same "p ( ,a t x is ( ,,lh4 ,x foll[s t w is ! same "p ( ,b t x is ( ,,hk4 ,ar*im$es n[ divides ,,lh1 ,,hk into p>ts equal to x1 & ,a ,b into p>ts equal to w1 & places ! w's at ! middle po9ts ( ! x's respectively4 ,all ! w's are !n 9 equilibrium ab\t ,c4 ,b all ! w's act+ at ! sev]al po9ts along ,,lh are equival5t to ,a act+ z a :ole at ! po9t ,e4 ,simil>ly ! w's act+ at ! sev]al po9ts on ,,hk are equival5t to ,b act+ at ,d4 ,!re=e ,a1 ,b plac$ at ,e1 ,d respectively balance ab\t ,c4 ,prop4 #g d$uces by r$uctio ad absurdum ! same result 9 ! case :]e ,a1 ,b are incomm5surable4 ,prop4 #h %[s h[ to f9d ! c5tre ( gravity ( ! rema9d] ( a magnitude :5 ! c5tre ( gravity ( ! :ole & ( a "p respectively are kn[n4 ,props4 #i-ae f9d ! c5tres ( gravity ( a p>allelogram1 a triangle & a p>allel-trapezium respectively4 ,book ,,ii41 9 t5 propositions1 is entirely devot$ to f9d+ ! c5tre ( gravity ( a p>abolic segm5t1 an elegant b difficult piece ( geometrical "w :i* is z usual confirm$ by ! me?od ( exhau/ion4 ,,*apt] ,,vii4 ,,hydro/atics4 ,! sci5ce ( hydro/atics is1 ev5 m ?an t ( /atics1 ! orig9al creation ( ,ar*im$es4 ,9 hydro/atics he seems to h _h no pr$ecessors4 ,only "o ( ! facts prov$ 9 8 "w ,on ,float+ ,bodies1 9 two books1 is giv5 ) a sort ( pro( 9 ,ari/otle4 ,?is is ! proposition t ! surface ( a fluid at re/ is t ( a sph]e ) its c5tre at ! c5tre ( ! e>?4 ,ar*im$es f\nds 8 :ole !ory on two po/ulates1 "o ( :i* comes at ! beg9n+ & ! o!r aft] ,prop4 #g ( ,book ,i4 ,po/ulate #a is z foll[s3-- ,let u assume t a fluid has ! prop]ty t1 if its p>ts lie ev5ly & are cont9u\s1 ! "p :i* is less compress$ is expell$ by t :i* is m compress$1 & ea* ( its p>ts is compress$ by ! fluid above x p]p5dicul>ly1 unless ! fluid is %ut up 9 some?+ & compress$ by some?+ else4 ,po/ulate #b is3 ,let u assume t any body :i* is borne upw>ds 9 wat] is c>ri$ along ! p]p5dicul> to ! surface :i* passes ?r\gh ! c5tre ( gravity ( ! body4 ,9 ,prop4 #b ,ar*im$es proves t ! surface ( any fluid at re/ is ! surface ( a sph]e ! c5tre ( :i* is ! c5tre ( ! e>?4 ,props4 #c-g deal ) ! behavi\r1 :5 plac$ 9 fluids1 ( solids 7#a7 j z heavy z ! fluid1 7#b7 light] ?an ! fluid1 7#c7 heavi] ?an ! fluid4 ,x is prov$ 7,props4 #e1 #f7 t1 if ! solid is light] ?an ! fluid1 x w n be completely imm]s$ b only s f> t ! weight ( ! solid w be equal to t ( ! fluid displac$1 &1 if x be =cibly imm]s$1 ! solid w be driv5 upw>ds by a =ce equal to ! diff]5ce betwe5 ! weight ( ! solid & t ( ! fluid displac$4 ,if ! solid is heavi] ?an ! fluid1 x w1 if plac$ 9 ! fluid1 desc5d to ! bottom &1 if weigh$ 9 ! fluid1 ! solid w be light] ?an its true weight by ! weight ( ! fluid displac$ 7,prop4 #g74 ,! la/-m5tion$ !orem naturally connects itself ) ! /ory ( ! cr[n made = ,hi]on4 ,x 0 suspect$ t ?is 0 n :olly ( gold b conta9$ an admixture ( silv]1 & ,hi]on put to ,ar*im$es ! problem ( det]m9+ ! proportions 9 :i* ! metals 7 mix$4 ,x 0 ! discov]y ( ! solution ( ?is problem :5 9 ! ba? t made ,ar*im$es run home nak$1 %\t+ ,greek3 eureka1 eureka4 ,"o acc\nt ( ! solution makes ,ar*im$es use ! proposition la/ quot$2 b on ! :ole x seems m likely t ! actual discov]y 0 made by a m elem5t>y me?od describ$ by ,vitruvius4 ,obs]v+1 z he is said to h done1 t1 if he /epp$ into ! ba? :5 x 0 full1 a volume ( wat] 0 spilt equal to ! volume ( 8 body1 he ?\ght ( apply+ ! same idea to ! case ( ! cr[n & measur+ ! volumes ( wat] displac$ respectively 7#a7 by ! cr[n itself1 7#b7 by ! same weight ( pure gold1 & 7#c7 by ! same weight ( pure silv]4 ,?is gives an easy means ( solution4 ,suppose t ! weight ( ! cr[n is ,w1 & t x conta9s weights w#a & w#b1 ( gold & silv] respectively4 ,n[ exp]im5t %[s 7#a7 t ! cr[n itself displaces a c]ta9 volume ( wat]1 ,v say1 7#b7 t a weight ,w ( gold displaces a c]ta9 o!r volume ( wat]1 ,v#a say1 & 7#c7 t a weight ,w ( silv] displaces a volume ,v#b4 ,f 7#b7 x foll[s1 by proportion1 t a weight w#a ( gold w displace w#a_/,w 4 ,v#a ( ! fluid1 & f 7#c7 x foll[s t a weight w#b ( silv] displaces w#b_/,w 4 ,v#b ( ! fluid4 ,h5ce ,v w#a_/,w 4 ,v#a w#b_/,w 4 ,v#b2 !re=e ,,wv w#a,v#a w#b,v#b1 t is1 7w#a w#b7,v w#a,v#a w#b,v#b1 s t w#a_/w#b 7,v#b - ,v7_/7,v - ,v#a71 :i* gives ! requir$ ratio ( ! weights ( gold & silv] conta9$ 9 ! cr[n4 ,! la/ two propositions ( ,book ,i4 inve/igate ! case ( a segm5t ( a sph]e float+ 9 a fluid :5 ! base ( ! segm5t is 7#a7 entirely above & 7#b7 entirely bel[ ! surface ( ! fluid2 & x is %[n t ! segm5t w 9 ei!r case be 9 equilibrium 9 ! position 9 :i* ! axis is v]tical1 ! equilibrium be+ 9 ! fir/ case /able4 ,book ,,ii4 is a geometrical t\r de =ce4 ,"h1 by ! me?ods ( pure geometry1 ,ar*im$es inve/igates ! positions ( re/ & /ability ( a "r segm5t ( a p>aboloid ( revolution float+ ) its base upw>ds or d[nw>ds 7b completely above or completely bel[ ! surface7 = a numb] ( cases diff]+ 7#a7 accord+ to ! relation betwe5 ! l5g? ( ! axis ( ! p>aboloid & ! pr9cipal p>amet] ( ! g5]at+ p>abola1 & 7#b7 accord+ to ! specific gravity ( ! solid 9 relation to ! fluid2 :]e ! position ( re/ & /ability is su* t ! axis ( ! solid is n v]tical1 ! angle at :i* x is incl9$ to ! v]tical is fully det]m9$4 ,! idea ( specific gravity appe>s all ?r\gh1 ?\gh ?is actual t]m is n us$4 ,ar*im$es speaks ( ! solid be+ light] or heavi] ?an ! fluid or equally heavy ) x1 or :5 a ratio has to be express$1 he speaks ( a solid ! weight ( :i* 7= an equal volume7 has a c]ta9 ratio to t ( ! fluid4 ,,bibliography4 ,! editio pr9ceps ( ! works ( ,ar*im$es ) ! comm5t>ies ( ,eutocius 0 br\ght out by ,h]vagius 7,h]wag57 at ,basel 9 #aedd4 ,d4 ,rivault 7,p>is1 #afae7 gave ! enunciations 9 ,greek & ! pro(s 9 ,lat9 some:at ret\*$4 ,! ,ar5>ius 7,s&reckon]7 & ! ,dim5sio circuli ) ,eutocius's comm5t>y 7 edit$ ) ,lat9 translation & notes by ,wallis 9 #afgh 7,ox=d74 ,torelli's monum5tal edition 7,ox=d1 #agib7 ( ! ,greek text ( ! complete works & ( ! comm5t>ies ( ,eutocius1 ) a new ,lat9 translation1 rema9$ ! /&>d text until rec5t ye>s2 x is n[ sup]s$$ by ! def9itive text ) ,lat9 translation ( ! complete works1 ,eutocius's comm5t>ies1 ! fragm5ts1 s*olia1 etc41 edit$ by ,heib]g 9 ?ree volumes 7,teubn]1 ,leipzig1 fir/ edition1 #ahhj-a2 second edition1 includ+ ! newly discov]$ ,me?od1 etc41 #aiaj-ae74 ,( translations ! foll[+ may be m5tion$4 ,! ,ald9e edition ( #aeeh1 #dto1 conta9s ! ,lat9 translation by ,comm&9us ( ! ,measurem5t ( a ,circle1 ,on ,spirals1 ,quadrature ( ! ,p>abola1 ,on ,conoids & ,sph]oids1 ,! ,s&reckon]4 ,isaac ,b>r['s v]sion 0 conta9$ 9 ,op]a ,ar*im$is1 ,apollonii ,p]goei conicorum libri1 ,!odosii ,spho]ica1 me?odo novo illu/rata et demon/rata 7,london1 #afge74 ,! fir/ ,fr5* v]sion ( ! works 0 by ,peyr>d 9 two volumes 7second edition1 #ahjh74 ,a valuable ,g]man translation1 ) notes1 by ,e4 ,nizze1 0 publi%$ at ,/ralsund 9 #ahbd4 ,!re is a complete edition 9 mod]n notation by ,t4 ,l4 ,hea? 7,! ,works ( ,ar*im$es1 ,cambridge1 #ahig1 supplem5t$ by ,! ,me?od ( ,ar*im$es1 ,cambridge1 #aiab74 ,,*ronology4 7,,approximate ,,9 ,,"s ,,cases47 ,b4,c4 #fbd-edg ,?ales #egb-dig ,py?agoras #ejj-dbh ,anaxagoras #dgj-djj _/ ,hippocrates ( ,*ios ,hippias ( ,elis #dgj-chj ,democritus #dfj-che ,!odorus ( ,cyr5e #dcj-cfj ,ar*ytas ( ,t>as 7,t>5tum7 #dbg-cdg ,plato #dae-cfi ,!aetetus #djh-cee ,eudoxus ( ,cnidos _/ ,leon fl4 ab\t #cej ,m5ae*mus ,d9o/ratus ,!udius fl4 #cjj ,euclid #caj-bcj ,ari/>*us ( ,samos #bhg-bab ,ar*im$es #bhd-bjc ,erato/h5es #bfe-aij ,apollonius ( ,p]ga