,,pione]s ,,( ,,progress    ,,m5 ,,(    
,,sci5ce   ,,edit$ ,,by ,s4 ,,*apman1   
,m4,a41 ,d4,sc41 ,f4,r4,s4     ,,!      
,,cop]nicus ,,( ,,antiquity             
7,,ari/>*us ,,( ,,samos7     ,,by       
,,sir ,,?omas ,,hea?   ,k4,c4,b41       
,k4,c4,v4,o41 ,f4,r4,s42 ,,sc4,d41      
,,camb42 ,,hon4 ,d4,,sc41 ,,ox=d        
,,london3   ,,society ,,= ,,promot+     
,,*ri/ian ,,k   ,,new ,,york3 ,,!       
,,macmillan ,,company   #aibj           
,,cont5ts     ,,"p ,i   ,,greek         
,,a/ronomy ,,to ,,ari/>*us              
,,page   ,,?ales                        
#f    ,,anaxim&]                        
#aj    ,,anaxim5es                      
#ac    ,,py?agoras                      
#ad    ,,p>m5ides                       
#af    ,,anaxagoras                     
#ah    ,,emp$ocles                      
#ba    ,,! ,,py?agoreans                
#bb    ,,nopides ,,( ,,*ios             
#bd    ,,plato                          
#be    ,,eudoxus1 ,,callippus1          
,,ari/otle                              

#bh    ,,h]aclides ,,( ,,pontus         
#cc     ,,"p ,,ii   ,,ari/>*us ,,(      
,,samos    ,,! ,,helioc5tric ,,hypo!sis 
#ci    ,,on ,,! ,,app>5t ,,diamet] ,,(  
,,! ,,sun                       #db     
,,on ,,! ,,sizes ,,& ,,di/ances ,,( ,,! 
,,sun ,,& ,,moon            #dc    ,,on 
,,! ,,ye> ,,& 8,,great ,,ye>0           
#ec    ,,lat] ,,improvem5ts ,,on        
,,ari/>*us',s ,,figures                 
#ed     ,,bibliography                  
#eg    ,,*ronology                      
#ei   ,illu/ration3 ,,cop]nicus',s      
,,diagram ,,( ,,8 ,,sy/em 7,,anticipat$ 
,,by ,,ari/>*us74     ,,"p ,i4  ,,greek 
,,a/ronomy ,,to ,,ari/>*us4   ,!        
title-page ( ?is book necess>ily be>s ! 
"n ( "o man2 b ! read] w f9d 9 its pages
 ! /ory1 or "p ( ! /ory1 ( _m o!r       
,pione]s ( ,progress4 ,! cr[n+ a*ievem5t
 ( anticipat+ ! hypo!sis ( ,cop]nicus   
belongs to ,ari/>*us ( ,samos alone2 b  
to see x 9 its prop] sett+ x is necess>y
 to h foll[$ 9 ! foot/eps ( ! e>li]     
pione]s :o1 by "o bold speculation aft] 

ano!r1 br\ght ! solution ( ! problem    
ne>]1 ?\gh no "o be=e ,ari/>*us actually
 hit ^u ! tru?4 ,?is is :y ! writ] has  
?\ght x useful to prefix to 8 acc\nt (  
,ari/>*us a %ort sket* ( ! hi/ory ( !   
developm5t ( a/ronomy 9 ,greece d[n to  
,ari/>*us's "t1 :i* is inde$ ! mo/      
fasc9at+ portion ( ! /ory ( ,greek      
a/ronomy4  ,! extraord9>y advance 9     
a/ronomy made by ! ,greeks 9 a p]iod (  
little m ?an ?ree c5turies is a wor?y   
p>allel to ! rapid developm5t1 9 _! h&s1
 ( pure geometry1 :i*1 creat$ by !m z a 
!oretical sci5ce ab\t ! same "t1 _h by !
 "t ( ,ari/>*us cov]$ ! gr\nd ( !       
,elem5ts 7includ+ solid geometry & !    
geometry ( ! sph]e71 _h e/abli%$ ! ma9  
prop]ties ( ! ?ree conic sections1 _h   
solv$ problems :i* 7 beyond ! geometry (
 ! /raight l9e & circle1 & f9ally1 be=e 
! end ( ! ?ird c5tury ,b4,c41 _h be5    
c>ri$ to its highe/ p]fection by ! g5ius
 ( ,ar*im$es1 :o measur$ ! areas (      
curves & ! surfaces & volumes ( curv$   
surfaces by geometrical me?ods          

practically anticipat+ ! integral       
calculus4  ,to und]/& h[ all ?is 0      
possible we h to rememb] t ! ,greeks1   
pre-em95tly among all ! nations ( ! _w1 
possess$ j ^? gifts :i* are ess5tial to 
! initiation & developm5t ( philosophy &
 sci5ce4 ,!y _h 9 ! fir/ place a        
rem>kable p[] ( accurate obs]vation2 &  
to ?is 7 add$ cle>ness ( intellect to   
see ?+s z !y are1 a passionate love ( k 
= its own sake1 & a g5ius = speculation 
:i* /&s unrivall$ to ?is day4 ,no?+ t is
 p]ceptible to ! s5ses seems to h escap$
 !m2 & :5 ! app>5t facts _h be5         
accurately asc]ta9$1 !y want$ to "k ! :y
 & ! :]e=e1 nev] re/+ satisfi$ until !y 
_h giv5 a rational explanation1 or :at  
seem$ to !m to be su*1 ( ! ph5om5a      
obs]v$4 ,obs]vation or exp]im5t & !ory  
w5t h& 9 h&4 ,s x 0 t !y develop$ su*   
subjects z m$ic9e & a/ronomy4 ,9        
a/ronomy _! guid+ pr9ciple 01 9 _! own  
expressive words1 to 8save ! ph5om5a04  
,?is meant t1 z m & m facts became kn[n1
 _! !ories 7 cont9ually revis$ to fit   

!m4  ,x w\ld be easy to multiply        
in/ances2 x mu/ suffice 9 ?is place to  
m5tion "o1 :i* illu/rates n only !      
c]ta9ty ) :i* ! ,greeks detect$ !       
occurr5ce ( ev5 ! r>e/ ph5om5a1 b also !
 p]si/5ce ) :i* !y s\ght = ! true       
explanation4  ,cleom$es 7second c5tury  
,a4,d47 m5tions t !re 7 /ories (        
extraord9>y eclipses :i* 8! m anci5t ( !
 ma!maticians0 _h va9ly tri$ to expla92 
! suppos$ 8p>adoxical0 case 0 t 9 :i*1  
:ile ! sun seems to be /ill above !     
we/]n horizon1 ! eclips$ moon is se5 to 
rise 9 ! ea/4 ,! ph5om5on appe>$ to be  
inconsi/5t ) ! explanation ( lun>       
eclipses by ! entry ( ! moon into !     
e>?'s %ad[2 h[ c\ld ?is be if bo? bodies
 7 above ! horizon at ! same "t8 ,! 8m  
anci5t0 ma!maticians essay$ a           
geometrical explanation2 !y tri$ to     
argue t x 0 possible t a spectator /&+  
on an em95ce ( ! sph]ical e>? might see 
along ! g5]ators ( a cone i4e4 a little 
d[nw>ds on all sides in/ead ( m]ely 9 ! 
plane ( ! horizon1 & s might see bo? !  

sun & ! moon ev5 :5 ! latt] 0 9 ! e>?'s 
%ad[4 ,cleom$es d5ies ?is & pref]s to   
reg>d ! :ole /ory ( su* cases z a       
fiction design$ m]ely = ! purpose (     
plagu+ a/ronom]s & philosoph]s2 no      
,*aldan1 he says1 no ,egyptian1 & no    
ma!matician or philosoph] has record$   
su* a case4 ,b ! ph5om5on is possible1 &
 x is c]ta9 t x _h be5 obs]v$ 9 ,greece 
& t ! ,greek a/ronom]s did n re/ until  
!y _h f\nd out ! solution ( ! puzzle2 = 
,cleom$es himself gives ! explanation1  
namely t ! ph5om5on is due to atmosph]ic
 refraction4 ,obs]v+ t su* cases (      
atmosph]ic refraction 7 especially      
noticeable 9 ! neighb\rhood ( ! ,black  
,sea1 ,cleom$es goes on to say t x is   
possible t ! visual rays go+ out f our  
eyes are refract$ ?r\gh fall+ on wet &  
damp air1 & s rea* ! sun al?\gh x is    
already bel[ ! horizon2 & he comp>es !  
well-kn[n exp]im5t ( ! r+ at ! bottom ( 
a jug1 :]e ! r+1 j out ( sight :5 ! jug 
is empty1 is br\ght into view :5 wat] is
 p\r$ 94  ,! g5ius ( ! race be+ :at x 01

 ! ,greeks mu/ f ! e>lie/ times h be5 9 
! habit ( scann+ ! heav5s1 &1 z might be
 expect$1 we f9d ! beg9n+s ( a/ronomical
 k 9 ! e>lie/ ,greek lit]ature4  ,9 !   
,hom]ic poems & 9 ,hesiod ! e>? is a    
flat circul> disc2 r\nd ?is disc runs ! 
riv] ,oceanus1 encircl+ ! e>? & fl[+    
back into itself4 ,! flat e>? has above 
x ! vault ( heav51 l a sort (           
hemisph]ical dome exactly cov]+ x2 ?is  
vault rema9s = ev] 9 "o position2 ! sun1
 moon & />s move r\nd "u x1 ris+ f      
,oceanus 9 ! ea/ & plung+ into x aga9 9 
! we/4  ,hom] m5tions1 9 addition to !  
sun & moon1 ! ,morn+ ,/>1 ! ,ev5+ ,/>1 !
 ,pleiades1 ! ,hyades1 ,orion1 ! ,great 
,be> 78:i* is also call$ by ! "n ( !    
,wa9071 ,sirius1 ! late-sett+ ,botes 7! 
pl\ghman driv+ ! ,wa971 i4e4 ,arcturus1 
z x 0 fir/ call$ by ,hesiod4 ,( ! ,great
 ,be> ,hom] says t x turns r\nd on !    
same spot & wat*es ,orion2 x alone is   
)\t lot 9 ,oceanus's ba? 7i4e4 x nev]   
sets74 ,) reg>d to ! la/ /atem5t x is to
 be not$ t "s ( ! pr9cipal />s ( !      

,great ,be> d n[ set 9 ! ,m$it]ranean1  
e4g4 9 places fur!r s\? ?an ,rhodes     
7lat4 #cf71 1 ! h9d foot1 & 1 ! tip ( ! 
tail1 & at ,alex&ria all ! sev5 />s     
except 1 ! head4 ,x might be suppos$ t  
"h 0 a case ( ,hom] 8nodd+04 ,b no2 !   
old poet 0 p]fectly "r2 ! diff]5ce      
betwe5 ! facts z obs]v$ by him & z se5  
by u respectively is due to !           
,precession ( ! ,equ9oxes1 ! gradual    
movem5t ( ! fix$ />s !mselves ab\t !    
pole ( ! ecliptic1 :i* 0 discov]$ by    
,hipp>*us 7second c5tury ,b4,c474 ,we "k
 f ! orig9al writ+s ( ! ,greek a/ronom]s
 t 9 ,eudoxus's "t 7say #chj ,b4,c47 !  
:ole ( ! ,great ,be> rema9$ always well 
above ! horizon1 :ile 9 ! "t ( ,proclus 
7say ,a4,d4 #dfj7 ! ,great ,be> 8graz$0 
! horizon4  ,9 ,hom] a/ronomical ph5om5a
 are only vaguely us$ = su* purposes z  
fix+ localities or m>k+ times ( day or  
night4 ,sometimes con/ellations are us$ 
9 giv+ sail+ directions1 z :5 ,calypso  
directs ,odysseus to sail 9 su* a way z 
always to keep ! ,great ,be> on 8 left4 

,hesiod m5tions practically ! same />s z
 ,hom]1 b makes m use ( cele/ial ph5om5a
 = det]m9+ times & seasons4 ,= example1 
he m>k$ ! "t = s[+ at ! beg9n+ ( w9t] by
 ! sett+ ( ! ,pleiades 9 ! e>ly         
twilight1 or aga9 by ! e>ly sett+ ( !   
,hyades or ,orion1 :i* means ! #crd1    
#g?1 or #ae? ,novemb] 9 ! ,julian cal5d>
 accord+ to ! p>ticul> />s tak52 ! "t = 
h>ve/ he fix$ by ! e>ly ris+ ( !        
,pleiades 7#ai? ,may71 ?re%+ "t by !    
e>ly ris+ ( ,orion 7#i? ,july71 v9tage  
"t by ! e>ly ris+ ( ,arcturus 7#ah?     
,septemb]71 & s on4 ,hesiod makes spr+  
beg9 sixty days aft] ! w9t] sol/ice1 & !
 e>ly summ] fifty days aft] ! summ]     
sol/ice4 ,?us he knew ab\t ! sol/ices1  
?\gh he says no?+ ( ! equ9oxes4 ,he _h  
an approximate notion ( ! moon's p]iod1 
:i* he put at ?irty days4  ,b ?is use ( 
a/ronomical facts = ! purpose ( det]m9+ 
times & seasons or d$uc+ wea!r          
indications is a v diff]5t ?+ f ! sci5ce
 ( a/ronomy1 :i* seeks to expla9 !      
heav5ly ph5om5a & _! causes4 ,! hi/ory (

 ?is sci5ce1 z ( ,greek philosophy 9    
g5]al1 beg9s ) ,?ales4  ,! ,ionian      
,greeks 7 9 ! mo/ fav\rable position =  
initiat+ philosophy4 ,=emo/ among !     
,greeks 9 ! love ( adv5ture & ! in/9ct (
 new discov]y 7z is %[n by _! leav+ _!  
homes to f\nd settlem5ts 9 di/ant l&s71 
& fir$1 l all ,greeks1 ) a passion = k1 
!y ne$$ little impulse to set !m on !   
road ( indep5d5t ?\ght & speculation4   
,?is impulse 0 furni%$ by _! contact )  
two anci5t civilisations1 ! ,egyptian & 
! ,babylonian4 ,acquir+ f !m c]ta9      
elem5t>y facts & rules 9 ma!matics &    
a/ronomy :i* _h be5 h&$ d[n ?r\gh !     
prie/hood f remote antiquity1 !y built  
^u !m ! f\ndation ( ! sci5ce1 z di/9ct f
 ! m]e r\t9e1 ( ! subjects 9 "q4        
,,?ales4  ,?ales ( ,miletus 7ab\t       
#fbd-edg ,b4,c47 0 a man ( extraord9>y  
v]satility2 philosoph]1 ma!matician1    
a/ronom]1 /atesman1 eng9e]1 & man (     
bus9ess1 he 0 decl>$ "o ( ! ,sev5 ,wise 
,m5 9 #ehb-eha ,b4,c4 ,8 prop5sity to   
/>-gaz+ is atte/$ by ! /ory ( 8 hav+    

fall5 into a well :ile wat*+ ! />s1     
insomu* t 7z ,plato has x7 he 0 ralli$  
by a clev] & pretty maids]vant f ,?race 
= be+ s 8eag] to "k :at goes on 9 !     
heav5s :5 he c\ld n see :at 0 9 front ( 
him1 nay at 8 v feet04  ,?ales's claim  
to a place 9 ! hi/ory ( sci5tific       
a/ronomy re/s on "o a*ievem5t attribut$ 
to him1 t ( pr$ict+ an eclipse ( ! sun4 
,! evid5ce = ?is is fairly conclusive1  
?\gh ! acc\nts ( x diff] slightly4      
,eudemus1 ! pupil ( ,ari/otle1 :o wrote 
hi/ories ( ,greek geometry & a/ronomy1  
is quot$ by ?ree diff]5t ,greek writ]s z
 ! au?ority = ! /ory4 ,b !re is te/imony
 mu* e>li] ?an ?is4 ,h]odotus1 speak+ ( 
a w> betwe5 ! ,lydians & ! ,m$es1 says  
t1 8:5 9 ! six? ye> !y enc\nt]$ "o      
ano!r1 x fell out t1 aft] !y _h jo9$    
battle1 ! day sudd5ly turn$ into night4 
,n[ t ?is *ange ( day into night w\ld   
occur 0 =etold to ! ,ionians by ,?ales (
 ,miletus1 :o fix$ z ! limit ( "t ?is v 
ye> 9 :i* ! *ange took place40 ,moreov] 
,x5ophanes1 :o 0 born "s tw5ty-?ree ye>s

 be=e ,?ales's dea?1 is said to h laud$ 
,?ales's a*ievem5t2 ?is w\ld am\nt to   
almo/ contempor>y evid5ce4  ,c\ld ,?ales
 h kn[n ! cause ( sol> eclipses8 ,atius 
7,a4,d4 #ajj71 ! au?or ( an epitome ( an
 old] collection ( ! op9ions (          
philosoph]s1 says t ,?ales 0 ! fir/ to  
decl>e t ! sun is eclips$ :5 ! moon     
comes 9 a direct l9e bel[ x1 ! image ( !
 moon !n appe>+ on ! sun's disc z on a  
mirror2 he also associates ,?ales )     
,anaxagoras1 ,plato1 ,ari/otle1 & !     
,/oics z hold+ t ! moon is eclips$ by   
reason ( its fall+ into ! %ad[ made by !
 e>? :5 ! e>? is betwe5 ! sun & ! moon4 
,b1 z reg>ds ! eclipse ( ! moon1 ,?ales 
c\ld n h giv5 ?is explanation1 because  
he held t ! e>? 7:i* he presumably      
reg>d$ z a flat disc7 float$ on ! wat] l
 a log4 ,& if he _h giv5 ! true         
explanation ( a sol> eclipse1 x is      
impossible t all ! succe$+ ,ionian      
philosoph]s %\ld h exhau/$ _!           
imag9ations 9 o!r fanciful explanations 
su* z we f9d record$4  ,! key to !      

puzzle may be af=d$ by ! passage (      
,h]odotus accord+ to :i* ! pr$iction 0 a
 r\gh "o1 only specify+ t ! eclipse w\ld
 occur )9 a c]ta9 ye>4 ,! pr$iction 0   
probably "o ( ! same k9d z _h long be5  
made by ! ,*aldans4 ,! ,*aldans1 no d\bt
 z ! result ( obs]vations cont9u$ ?r\gh 
_m c5turies1 _h discov]$ ! p]iod ( #bbc 
lunations aft] :i* lun> eclipses recur4 
7,?is me?od w\ld v (t5 fail = sol>      
eclipses because no acc\nt 0 tak5 (     
p>allax2 & ,assyrian cunei=m            
inscriptions record failures z well z   
successful pr$ictions47 ,?ales1 !n1     
probably le>nt ab\t ! p]iod ( #bbc      
lunations ei!r 9 ,egypt or m directly   
?r\gh ,lydia1 :i* 0 an outpo/ (         
,assyrio-,babylonian culture4 ,if !re   
happ5$ to be a numb] ( possible sol>    
eclipses 9 ! ye> :i* 7accord+ to        
,h]odotus7 ,?ales fix$ = ! eclipse1 he  
01 9 us+ ! ,*aldan rule1 n tak+ an undue
 risk2 b x 0 great luck t ! eclipse %\ld
 h be5 total4 ,x seems practically c]ta9
 t ! eclipse 9 "q 0 t ( ! 7,julian7 #bh?

 ,may1 #ehe4  ,?ales1 z we h se51 made !
 e>? a circul> or cyl9drical disc float+
 on ! wat] l a log or a cork &1 s f> z  
we c judge ( 8 g5]al conception ( !     
univ]se1 he w\ld appe> to h reg>d$ x z a
 mass ( wat] 7t on :i* ! e>? floats7 ) !
 heav5s encircl+ x 9 ! =m ( a hemisph]e 
& also b\nd$ by ! primeval wat]4 ,?is   
view ( ! _w has be5 comp>$ ) t f\nd 9   
anci5t ,egyptian papyri4 ,9 ! beg9n+    
exi/$ ! ,n1 a primordial liquid mass 9 !
 limitless dep?s ( :i* float$ ! g]ms (  
?+s4 ,:5 ! sun began to %9e1 ! e>? 0    
flatt5$ out & ! wat] sep>at$ into two   
masses4 ,! "o gave rise to ! riv]s & !  
ocean1 ! o!r1 susp5d$ above1 =m$ ! vault
 ( heav51 ! wat]s above1 on :i* ! />s & 
! gods1 borne by an et]nal curr5t1 began
 to float4 ,! sun1 /&+ upright 9 8 sacr$
 b>que :i* _h endur$ = millions ( ye>s1 
glides sl[ly1 conduct$ by an army (     
second>y gods1 ! planets & ! fix$ />s4  
,! assumption ( an upp] & l[] ocean is  
also old ,babylonian 7cf4 ! division 9  
,g5esis #a4 #g ( ! wat]s :i* 7 "u !     

firmam5t f ! wat]s :i* 7 above !        
firmam5t74  ,x w\ld foll[ f ,?ales's    
g5]al view ( ! univ]se t ! sun1 moon &  
/>s did n1 betwe5 _! sett+ & ris+ aga91 
cont9ue _! circul> pa? bel[ ! e>? b 7z )
 ,anaxim5es lat]7 mov$ lat]ally r\nd !  
e>?4  ,?ales's fur!r contributions to   
obs]vational a/ronomy may be %ortly     
/at$4 ,he wrote two works ,on ! sol/ice 
& ,on ! equ9ox1 & he is said by ,eudemus
 to h discov]$ t 8! p]iod ( ! sun )     
respect to ! sol/ices is n always !     
same10 :i* probably means t he discov]$ 
! inequality ( ! f\r a/ronomical        
seasons4 ,8 division ( ! ye> into #cfe  
days he probably le>nt f ! ,egyptians4  
,he said ( ! ,hyades t !re are two1 "o  
nor? & ! o!r s\?4 ,he obs]v$ ! ,little  
,be> & us$ x z a means ( f9d+ ! pole2 he
 advis$ ! ,greeks to foll[ ! ,phnician  
plan ( sail+ by ! ,little ,be> 9        
pref]5ce to _! own habit ( /e]+ by !    
,great ,be>4  ,limit$ z ! c]ta9         
contributions ( ,?ales to a/ronomy are1 
x became ! habit ( ! ,greek ,doxographi1

 or retail]s ( ! op9ions ( philosoph]s1 
to attribute to ,?ales1 9 common ) o!r  
a/ronom]s 9 ea* case1 a numb] (         
discov]ies :i* 7 n made till lat]4 ,!   
foll[+ is a li/1 ) 79 brackets7 ! names 
( ! a/ronom]s to :om ! respective       
discov]ies may ) mo/ c]ta9ty be assign$3
 7#a7 ! fact t ! moon takes its light f 
! sun 7,anaxagoras71 7#b7 ! sph]icity ( 
! e>? 7,py?agoras71 7#c7 ! division ( ! 
heav5ly sph]e into five zones           
7,py?agoras & ,p>m5ides71 7#d7 !        
obliquity ( ! ecliptic 7nopides (       
,*ios71 & 7#e7 ! e/imate ( ! sun's      
app>5t diamet] z #a_/gbj? ( ! sun's     
circle 7,ari/>*us ( ,samos74            
,,anaxim&]4  ,anaxim&] 7ab\t #faa-edg   
,b4,c471 a contempor>y & fell[-citiz5 ( 
,?ales1 0 a rem>kably orig9al ?9k]4 ,he 
0 ! fir/ ,greek philosoph] :o v5tur$ to 
put =w>d 8 views 9 a =mal writt5        
treatise4 ,?is 0 a "w ,ab\t ,nature & 0 
n giv5 to ! _w till he 0 ab\t sixty-f\r 
ye>s old4 ,8 orig9ality is illu/rat$ by 
8 !ory ( evolution4 ,accord+ to him     

animals fir/ arose f slime evaporat$ by 
! sun2 !y liv$ 9 ! sea & _h prickly     
cov]+s2 m5 at fir/ resembl$ fi%es4  ,b 8
 a/ronomical views 7 n less rem>kable4  
,anaxim&] boldly ma9ta9$ t ! e>? is 9 ! 
c5tre ( ! univ]se1 susp5d$ freely & )\t 
support1 :]eas ,?ales reg>d$ x z re/+ on
 ! wat] & ,anaxim5es z support$ by !    
air4 ,x rema9s 9 its position1 said     
,anaxim&]1 because x is at an equal     
di/ance f all ! re/ ( ! heav5ly bodies4 
,! e>? 01 accord+ to him1 cyl9d]-%ap$1  
r\nd 8l a /one pill>02 "o ( its two     
plane faces is t on :i* we /&2 its dep? 
is "o-?ird ( its bread?4  ,anaxim&]     
po/ulat$ z 8 fir/ pr9ciple1 n wat] 7l   
,?ales7 or any ( ! elem5ts1 b !         
,inf9ite2 ?is 0 a sub/ance1 n fur!r     
def9$1 f :i* all ! heav5s & ! worlds 9  
!m 7 produc$2 accord+ to him ! worlds   
!mselves 7 inf9ite 9 numb]1 & !re 7     
always "s worlds com+ into be+ & o!rs   
pass+ away ad inf9itum4 ,! orig9 ( !    
/>s1 & _! nature1 he expla9$ z foll[s4  
8,t :i* is capable ( begett+ ! hot & !  

cold out ( ! et]nal 0 sep>at$ (f dur+ ! 
com+ into be+ ( our _w1 & f ! flame ?us 
produc$ a sort ( sph]e 0 made :i* grew  
r\nd ! air ab\t ! e>? z ! b>k r\nd !    
tree2 !n ?is sph]e 0 torn (f & became   
enclos$ 9 c]ta9 circles or r+s1 & ?us 7 
=m$ ! sun1 ! moon & ! />s40 8,! />s are 
produc$ z a circle ( fire1 sep>at$ (f f 
! fire 9 ! univ]se & enclos$ by air4 ,!y
 h z v5ts c]ta9 pipe-%ap$ passages at   
:i* ! />s are se540 8,! />s are         
compress$ portions ( air1 9 ! %ape (    
:eels fill$ ) fire1 & !y emit flames at 
"s po9t f small op5+s40 8,! />s are     
borne r\nd by ! circles 9 :i* !y are    
enclos$40 8,! sun is a circle           
tw5ty-eight times 7v4 l4 #bg times7 !   
size ( ! e>?2 x is l a :eel ( a *>iot ! 
rim ( :i* is holl[ & full ( fire & lets 
! fire %9e out at a c]ta9 po9t 9 x ?r\gh
 an op5+ l ! tube ( a bl[-pipe2 su* is !
 sun40 8,! sun is equal to ! e>?40 8,!  
eclipses ( ! sun occur ?r\gh ! op5+ by  
:i* ! fire f9ds v5t be+ %ut up40 8,!    
moon is a circle n9ete5 times ! size ( !

 e>?2 x is simil> to a *>iot-:eel ! rim 
( :i* is holl[ & full ( fire l ! circle 
( ! sun1 & x is plac$ obliquely l ! o!r2
 x has "o v5t l ! tube ( a bl[-pipe2 !  
eclipses ( ! moon dep5d on ! turn+s ( ! 
:eel40 8,! moon is eclips$ :5 ! op5+ 9 !
 rim ( ! :eel is /opp$ up40 8,! moon    
appe>s sometimes z wax+1 sometimes z    
wan+1 to an ext5t correspond+ to ! clos+
 or op5+ ( ! passages40 8,! sun is plac$
 highe/ ( all1 aft] x ! moon1 & "u !m ! 
fix$ />s & ! planets40  ,x has be5 po9t$
 out t ! idea ( ! =mation ( tubes (     
compress$ air )9 :i* ! fire ( ea* /> is 
%ut up except = ! "o op5+ ?r\gh :i* !   
flame %[s 7l a gas-jet1 z x 77 is n     
unlike ,laplace's hypo!sis ) ref]5ce to 
! orig9 ( ,saturn's r+s4 ,9 any case x  
is a suffici5tly orig9al conception4    
,:5 ,anaxim&] says t ! hoops c>ry+ ! sun
 & moon 8lie obliquely10 ?is is no d\bt 
an attempt to expla91 9 addition to !   
daily rotation1 ! annual movem5t ( ! sun
 & ! mon?ly movem5t ( ! moon4  ,we h "h 
too ! fir/ speculation ab\t ! sizes &   

di/ances ( ! heav5ly bodies4 ,! sun is z
 l>ge z ! e>?4 ,! ambiguity betwe5 !    
e/imates ( ! size ( ! sun's circle z    
tw5ty-sev5 or tw5ty-eight times ! size (
 ! e>? sugge/s t x is a "q betwe5 tak+ !
 inn] & out] circumf]5ces ( ! sun's r+  
respectively1 & a simil> ambiguity may  
acc\nt = ! circle ( ! moon be+ /at$ to  
be n9ete5 times1 n eighte5 times1 ! size
 ( ! e>?4 ,no e/imate is giv5 ( !       
di/ance ( ! planets f ! e>?1 b z1       
accord+ to ,anaxim&]1 !y are ne>] to !  
e>? ?an ! sun & moon are1 x is possible 
t1 if a figure _h be5 /at$1 x w\ld h be5
 n9e times ! size ( ! e>?1 9 :i* case we
 %\ld h _h ! numb]s #i1 #ah1 #bg1 ?ree  
t]ms 9 ari?metical progression & all (  
!m multiples ( #i1 ! squ>e ( #c4 ,x     
seems probable t ^! figures 7 n arriv$  
at by any calculation bas$ on           
geometrical consid]ations1 b t we h "h  
m]ely an illu/ration ( ! anci5t cult ( !
 sacr$ numb]s #c & #i4 ,?ree is ! sacr$ 
numb] 9 ,hom]1 #i 9 ,!ognis4 ,! cult (  
#c & its multiples #i & #bg is f\nd     

among ! ,aryans1 !n among ! ,f9ns &     
,t>t>s & !n aga9 among ! ,etruscans4    
,!re=e ,anaxim&]'s figures probably say 
little m ?an :at ! ,indians tell u1     
namely1 t ?ree ,vi%nu-/eps rea* f e>? to
 heav54  ,anaxim&] is said to h be5 !   
fir/ to discov] ! gnomon 7or sun-dial ) 
a v]tical ne$le74 ,?is is1 h[ev]1       
incorrect1 = ,h]odotus says t ! ,greeks 
le>nt ! use ( ! gnomon & ! polos f !    
,babylonians4 ,anaxim&] may h be5 ! fir/
 to introduce ! gnomon into ,greece4 ,he
 is said to h set x up 9 ,sp>ta & to h  
%[n on x 8! sol/ices1 ! times1 !        
seasons1 & ! equ9ox04  ,b ,anaxim&] has 
ano!r title to fame4 ,he 0 ! fir/ :o    
v5tur$ to draw a map ( ! inhabit$ e>?4  
,! ,egyptians inde$ _h drawn maps be=e1 
b only ( special di/ricts4 ,anaxim&]    
boldly plann$ out ! :ole _w ) 8!        
circumf]5ce ( ! e>? & ( ! sea04         
,hecataeus1 a mu*-travell$ man1 is said 
to h correct$ ,anaxim&]'s map s t x     
became ! object ( g5]al admiration4     
,,anaxim5es4  ,) ,anaxim5es ( ,miletus  

7ab\t #ehe-ebh_/d ,b4,c47 ! e>? is /ill 
flat l a table1 b1 in/ead ( be+ susp5d$ 
freely )\t support z ) ,anaxim&]1 x is  
support$ by ! air1 rid+ on x z x 74 ,!  
sun1 moon & />s are all made ( fire & 7l
 ! e>?7 !y ride on ! air because ( _!   
bread?4 ,! sun is flat l a leaf4        
,anaxim5es also held t ! />s are fa/5$  
on a cry/al sph]e l nails or /uds4 ,x   
seems cle> !re=e t by ! />s :i* 8ride on
 ! air because ( _! bread?0 he meant !  
planets only4 ,a l app>5t inconsi/5cy   
applies to ! motion ( ! />s4 ,if ! />s  
are fix$ 9 ! cry/al sph]e l nails1 !y   
mu/ be c>ri$ r\nd complete circles by ! 
revolution ( ! sph]e ab\t a diamet]4    
,yet ,anaxim5es also said t ! />s d n   
move or revolve "u ! e>? z "s suppose1 b
 r\nd ! e>?1 j z a cap c be turn$ r\nd  
on ! head4 ,! sun is hidd5 f sight1 n   
because x is "u ! e>?1 b because x is   
cov]$ by ! high] p>ts ( ! e>? & because 
its di/ance f u is great]4 ,ari/otle    
adds ! detail t ! sun is c>ri$ r\nd !   
nor!rn portion ( ! e>? & produces night 

because ! e>? is l(ty t[>ds ! nor?4 ,we 
mu/ aga9 conclude t ! />s :i*1 l ! sun &
 moon1 move lat]ally r\nd ! e>? betwe5  
_! sett+ & ris+ aga9 are ! planets1 z   
di/9ct f ! fix$ />s4 ,x w\ld !re=e seem 
t ,anaxim5es 0 ! fir/ to di/+ui% !      
planets f ! fix$ />s 9 respect ( _!     
irregul> movem5ts4 ,he improv$ on       
,anaxim&] 9 t he relegat$ ! fix$ />s to 
! region mo/ di/ant f ! e>?4  ,anaxim5es
 0 also orig9al 9 hold+ t1 9 ! region   
occupi$ by ! />s1 bodies ( an e>?y      
nature are c>ri$ r\nd along ) !m4 ,!    
object ( ^! invisible bodies ( an e>?y  
nature c>ri$ r\nd along ) ! />s is      
cle>ly to expla9 ! eclipses & phases ( !
 moon4 ,x 0 d\btless ?is conception :i*1
 9 ! h&s ( ,anaxagoras & o!rs1          
ultimately l$ to ! true explanation (   
eclipses4  ,! "o feature ( ,anaxim5es's 
sy/em :i* 0 de/9$ to an endur+ triumph 0
 ! conception ( ! />s be+ fix$ on a     
cry/al sph]e z 9 a rigid frame4 ,?is    
really rema9$ ! fundam5tal pr9ciple 9   
all a/ronomy d[n to ,cop]nicus4         

,,py?agoras4  ,) ,py?agoras & !         
,py?agoreans we come to a diff]5t ord] (
 ?+s4 ,py?agoras1 born at ,samos ab\t   
#egb ,b4,c41 is und\bt$ly "o ( ! greate/
 names 9 ! hi/ory ( sci5ce4 ,he 0 a     
ma!matician ( brilliant a*ievem5ts2 he 0
 also ! inv5tor ( ! sci5ce ( ac\/ics1 an
 a/ronom] ( great orig9ality1 a !ologian
 & moral re=m]1 & ! f\nd] ( a bro!rhood 
:i* admits comp>ison ) ! ord]s ( m$ival 
*ivalry4 ,p]haps 8 mo/ epo*-mak+        
discov]y 0 t ( ! dep5d5ce ( musical     
tones on num]ical proportions1 ! octave 
repres5t+ ! proportion ( #b 3 #a 9 l5g? 
( /r+ at ! same t5sion1 ! fif? #c 3 #b1 
& ! f\r? #d 3 #c4 ,ma!maticians "k him z
 ! reput$ discov]] ( ! fam\s !orem ab\t 
! squ>e on ! hypot5use ( a "r-angl$     
triangle 7 ,euclid ,i4 #dg72 b he 0 also
 ! fir/ to make geometry a "p ( a lib]al
 education & to explore its fir/        
pr9ciples 7def9itions1 etc474           
,py?agoras is said to h be5 ! fir/ to   
ma9ta9 t ! e>? is sph]ical 9 %ape2 on   
:at gr\nd1 is unc]ta94 ,"o sugge/ion is 

t he may h argu$ f ! r\ndness ( ! %ad[  
ca/ by ! e>? 9 ! eclipses ( ! moon2 b   
,anaxagoras 0 ! fir/ to give ! true     
explanation ( su* eclipses4 ,probably   
,py?agoras attribut$ sph]ical %ape to ! 
e>? = ! ma!matical or ma!matico-/hetical
 reason t ! sph]e is ! mo/ beautiful (  
all solid figures4 ,x is probable too1 &
 = ! same reason1 t ,py?agoras gave !   
same sph]ical %ape to ! sun & moon1 &   
ev5 to ! />s1 9 :i* case ! way lay op5 =
 ! discov]y ( ! true cause ( eclipses & 
( ! phases ( ! moon4 ,py?agoras is also 
said to h di/+ui%$ five zones 9 ! e>?4  
,x is true t ! fir/ decl>ation t ! e>?  
is sph]ical & t x has five zones is     
alt]natively attribut$ to ,p>m5ides     
7born p]haps ab\t #eaf or #ead ,b4,c471 
on ! good au?ority ( ,!ophra/us4 ,x is  
possible t1 al?\gh ,py?agoras 0 ! real  
au?or ( ^! views1 ,p>m5ides 0 ! fir/ to 
/ate !m 9 public4  ,py?agoras reg>d$ !  
univ]se z liv+1 intellig5t1 sph]ical1   
enclos+ ! e>? at ! c5tre1 & rotat+ ab\t 
an axis pass+ ?r\gh ! c5tre ( ! e>?1 !  

e>? rema9+ at re/4  ,he is said to h be5
 ! fir/ to obs]ve t ! planets h an      
indep5d5t motion ( _! own 9 a direction 
opposite to t ( ! fix$ />s1 i4e4 ! daily
 rotation4 ,alt]natively ) ,p>m5ides he 
is said to h be5 ! fir/ to recognise t !
 ,morn+ & ! ,ev5+ ,/>s are "o & ! same4 
,py?agoras is h>dly likely to h kn[n ?is
 z ! result ( obs]vations ( 8 own2 he   
may h le>nt x f ,egypt or ,*alda along )
 o!r facts ab\t ! planets4   ,,p>m5ides4
  ,we h se5 t c]ta9 views are           
alt]natively ascrib$ to ,py?agoras &    
,p>m5ides4 ,! sy/em ( ,p>m5ides 0 9 fact
 a k9d ( bl5d ( ! !ories ( ,py?agoras & 
,anaxim&]4 ,9 giv+ ! e>? sph]ical =m )  
five zones he agre$ ) ,py?agoras4       
,py?agoras1 h[ev]1 made ! sph]ical      
univ]se rotate ab\t an axis ?r\gh !     
c5tre ( ! e>?2 ?is impli$ t ! univ]se is
 itself limit$1 b t some?+ exi/s r\nd x1
 & 9 fact t beyond ! f9ite rotat+ sph]e 
!re is limitless void or empty space4   
,p>m5ides1 on ! o!r h&1 d5i$ ! exi/5ce (
 ! inf9ite void & 0 !re=e oblig$ to make

 8 f9ite sph]e motionless & to hold t   
its app>5t rotation is only an illusion4
  ,9 o!r portions ( 8 sy/em ,p>m5ides   
foll[$ ! lead ( ,anaxim&]4 ,l ,anaxim&] 
7& ,democritus lat]7 he argu$ t ! e>?   
rema9s 9 ! c5tre because1 be+ equidi/ant
 f all po9ts on ! sph]e ( ! univ]se1 x  
is 9 equilibrium & !re is no m reason :y
 x %\ld t5d to move 9 "o direction ?an 9
 ano!r4 ,p>m5ides also _h a sy/em (     
wrea?s or b&s r\nd ! sph]e ( ! univ]se  
:i* conta9$ ! sun1 ! moon & ! />s2 !    
wrea?s rem9d u ( ! hoops ( ,anaxim&]1 b 
_! nature is diff]5t4 ,! wrea?s1 accord+
 to ! mo/ probable int]pretation ( !    
texts1 are1 />t+ f ! outside1 7#a7 a    
solid envelope l a wall2 7#b7 a b& (    
fire 7! !r-fire72 7#c7 mix$ b&s1 made up
 ( light & d>kness 9 comb9ation1 :i*    
exhibit ! ph5om5on ( 8fire %9+ out "h & 
!re10 ^! mix$ b&s includ+ ! ,milky ,way 
z well z ! sun1 moon & planets2 7#d7 a  
b& ( fire1 ! inn] side ( :i* is our     
atmosph]e1 t\*+ ! e>?4 ,except t        
,p>m5ides plac$ ! ,morn+ ,/> fir/ 9 ! !r

 & !re=e above ! sun1 he did n app>5tly 
diff] f ,anaxim&]'s view ( ! relative   
di/ances ( ! heav5ly bodies1 accord+ to 
:i* bo? ! planets & ! o!r />s are all   
plac$ bel[ ! sun & moon4  ,two l9es f   
,p>m5ides's poem h be5 quot$ to %[ t he 
decl>$ t ! moon is illum9at$ by ! sun4  
,! fir/ l9e speaks ( ! moon z 8a        
night-%9+ =eign light w&]+ r\nd ! e>?02 
b1 ev5 if ! l9e is g5u9e1 8=eign0 ne$ n 
mean 8borr[$04 ,! o!r l9e speaks ( !    
moon z 8always fix+ its gaze on ! sun02 
b1 ?\gh ?is /ates an obs]v$ fact1 x is  
f> f expla9+ ! cause4 ,we h1 moreov]1   
positive evid5ce aga9/ ! attribution ( !
 discov]y ( ! opacity ( ! moon to       
,p>m5ides4 ,x is "p ( ! connect$ prose  
description ( 8 sy/em t ! moon is a     
mixture ( air & fire1 & 9 o!r passages  
we are told t he held ! moon to be (    
fire4 ,la/ly1 ,plato speaks ( 8! fact   
:i* ,anaxagoras lately ass]t$1 t ! moon 
has its light f ! sun04 ,x seems        
impossible t ,plato w\ld speak 9 su*    
t]ms if ! fact 9 "q _h be5 /at$ = ! fir/

 "t ei!r by ,p>m5ides or by !           
,py?agoreans4   ,,anaxagoras4           
,anaxagoras1 a man ( sci5ce if ev] !re 0
 "o1 0 born at ,clazom5ae 9 !           
neighb\rhood ( ,smyrna ab\t #ejj ,b4,c4 
,he neglect$ 8 possessions1 :i* 7       
consid]able1 9 ord] to devote himself to
 sci5ce4 ,someone once ask$ him :at 0 ! 
object ( be+ born1 & he repli$1 8,!     
inve/igation ( sun1 moon & heav504 ,he  
took up 8 abode at ,a!ns1 :]e he enjoy$ 
! fri5d%ip ( ,p]icles4 ,:5 ,p]icles     
became unpopul> %ortly be=e ! outbreak (
 ! ,peloponnesian w>1 he 0 attack$ ?r\gh
 8 fri5ds1 & ,anaxagoras 0 accus$ (     
impiety = decl>+ t ! sun 0 a r$-hot /one
 & ! moon made ( e>?4 ,"o acc\nt says t 
he 0 f9$ & bani%$2 ano!r t he 0         
imprison$1 & t x 0 int5d$ to put him to 
dea?1 b t ,p]icles obta9$ 8 release2 he 
retir$ to ,lampsacus1 :]e he di$ at !   
age ( sev5ty-two4  ,"o epo*-mak+        
discov]y belongs to him1 namely1 t !    
moon does n %9e by its own light b      
receives its light f ! sun3 ,plato1 z we

 h se51 is "o au?ority = ?is /atem5t4   
,plut>* also 9 8 ,de facie 9 orbe lunae 
says1 8,n[ :5 our comrade 9 8 disc\rse  
_h exp\nd$ t proposition ( ,anaxagoras t
 #8! sun places ! brightness 9 ! moon1' 
he 0 greatly applaud$04  ,?is discov]y  
enabl$ ,anaxagoras to say t 8!          
obscurations ( ! moon mon? by mon? 7 due
 to its foll[+ ! c\rse ( ! sun by :i* x 
is illum9at$1 & ! eclipses ( ! moon 7   
caus$ by its fall+ )9 ! %ad[ ( ! e>? :i*
 !n comes betwe5 ! sun & ! moon1 :ile ! 
eclipses ( ! sun 7 due to ! int]position
 ( ! moon04 ,anaxagoras 0 !re=e ! fir/  
to give ! true explanation ( eclipses4  
,z reg>ds ! phases ( ! moon1 8          
explanation c\ld only h be5 complete if 
he _h kn[n t ! moon is sph]ical2 9 fact1
 h[ev]1 he consid]$ ! e>? 7& d\btless ! 
o!r heav5ly bodies also7 to be flat4 ,to
 8 true !ory ( eclipses ,anaxagoras add$
 ! unnecess>y assumption t ! moon 0     
sometimes eclips$ by o!r e>?y bodies    
bel[ ! moon b invisible to u4 ,9 ?is    
latt] assumption he foll[$ ! lead (     

,anaxim5es4 ,! o!r bodies 9 "q 7        
probably inv5t$ to expla9 :y ! eclipses 
( ! moon are se5 (t5] ?an ^? ( ! sun4   
,anaxagoras's cosmogony conta9$ "s      
fruitful ideas4 ,accord+ to him1 !      
=mation ( ! _w began ) a vortex set up1 
9 a portion ( ! mix$ mass 9 :i* 8all ?+s
 7 toge!r10 by ,m9d4 ,?is rotatory      
movem5t began at "o po9t & !n gradually 
spread1 tak+ 9 wid] & wid] circles4 ,!  
fir/ effect 0 to sep>ate two great      
masses1 "o consi/+ ( ! r>e1 hot1 light1 
dry1 call$ ! !r1 & ! o!r ( ! opposite   
categories & call$ air4 ,! !r took !    
out] place1 ! air ! inn]4 ,out ( ! air 7
 sep>at$ successively cl\ds1 wat]1 e>?1 
& /ones4 ,! d5se1 ! moi/1 ! d>k & cold1 
& all ! heavie/ ?+s1 collect 9 ! c5tre z
 ! result ( ! circul> motion1 & x is f  
^! elem5ts :5 consolidat$ t ! e>? is    
=m$4 ,b aft] ?is1 89 consequ5ce ( !     
viol5ce ( ! :irl+ motion1 ! surr\nd+    
fi]y !r tore /ones away f ! e>? & k9dl$ 
!m into />s04 ,anaxagoras conceiv$ !re=e
 ! idea ( a c5trifugal =ce1 z di/9ct f t

 ( conc5tration br\ght ab\t by ! motion 
( ! vortex1 & he assum$ a s]ies (       
projections or 8hurl+s-(f0 ( precisely !
 same k9d z ! !ory ( ,kant & ,laplace   
assum$ = ! =mation ( ! sol> sy/em4  ,9  
o!r matt]s ?an ! above ,anaxagoras did n
 make mu* advance on ! crude ,ionian    
!ories4 8,! sun is a r$-hot mass or a   
/one on fire40 8,x is l>g] 7or #8_m     
times l>g]'7 ?an ! ,peloponnese40 ,he   
consid]$ t 8! />s 7 orig9ally c>ri$ r\nd
 7lat]ally7 l a dome1 ! pole :i* is     
always visible be+ ?us v]tically above !
 e>?1 & x 0 only aft]w>ds t _! c\rse    
became incl9$04  ,b he put =w>d a       
rem>kable & orig9al hypo!sis to expla9 !
 ,milky ,way4 ,he ?\ght ! sun to be     
small] ?an ! e>?4 ,consequ5tly1 :5 ! sun
 9 its revolution passes bel[ ! e>?1 !  
%ad[ ca/ by ! e>? ext5ds )\t limit4 ,!  
trace ( ?is %ad[ on ! heav5s is ! ,milky
 ,way4 ,! />s )9 ?is %ad[ are n int]f]$ 
) by ! light ( ! sun1 & we !re=e see !m 
%9+2 ^? />s1 on ! o!r h&1 :i* are       
outside ! %ad[ are ov]p[]$ by ! light ( 

! sun :i* %9es on !m ev5 dur+ ! night1 s
 t we _c see !m4 ,ari/otle easily       
disposes ( ?is !ory by obs]v+ t1 ! sun  
be+ mu* l>g] ?an ! e>?1 & ! di/ance ( ! 
/>s f ! e>? be+ _m times great] ?an !   
di/ance ( ! sun1 ! sun's %ad[ w\ld =m a 
cone ) its v]tex n v f> f ! e>?1 s t !  
%ad[ ( ! e>?1 :i* we call night1 w\ld n 
rea* ! />s at all4   ,,emp$ocles4       
,emp$ocles ( ,agrig5tum 7ab\t #did-dcd  
,b4,c47 w\ld h>dly des]ve m5tion = 8    
a/ronomy alone1 s crude 7 8 views :]e !y
 diff]$ f ^? ( 8 pr$ecessors4 ,! e>?1   
accord+ to ,emp$ocles1 is kept 9 its    
place by ! swiftness ( ! revolution ( ! 
heav51 j z we may sw+ a cup ) wat] 9 x  
r\nd & r\nd s t 9 "s positions ! top ( !
 cup may ev5 be turn$ d[nw>ds )\t ! wat]
 escap+4 ,day & night he expla9$ z      
foll[s4 ,)9 ! cry/al sph]e to :i* ! fix$
 />s are atta*$ 7z ,anaxim5es held71 &  
fill+ x1 is a sph]e consi/+ ( two       
hemisph]es1 "o ( :i* is :olly ( fire &  
!re=e light1 :ile ! o!r is a mixture (  
air ) a little fire1 :i* mixture is     

d>kness or night4 ,! revolution ( ^! two
 hemisph]es r\nd ! e>? produces at ea*  
po9t on its surface ! succession ( day &
 night4 ,emp$ocles held ! sun to be1 n  
fire1 b a reflection ( fire simil> to t 
:i* takes place f ! surface ( wat]1 !   
fire ( a :ole hemisph]e ( ! _w be+ b5t  
back f ! e>?1 :i* is circul>1 &         
conc5trat$ into ! cry/all9e sun :i* is  
c>ri$ r\nd by ! motion ( ! fi]y         
hemisph]e4  ,emp$ocles's "o important   
sci5tific a*ievem5t 0 8 !ory t light    
travels & takes "t to pass f "o po9t to 
ano!r4 ,! !ory is allud$ to by          
,ari/otle1 :o says t1 accord+ to        
,emp$ocles1 ! light f ! sun rea*es !    
int]v5+ space be=e x rea*es ! eye or !  
e>?2 !re 0 !re=e a "t :5 ! ray 0 n yet  
se51 b 0 be+ transmitt$ ?r\gh ! m$ium4  
,,! ,,py?agoreans4  ,we h se5 t         
,py?agoras 0 ! fir/ to give sph]ical =m 
to ! e>? & probably to ! heav5ly bodies 
g5]ally1 & to assign to ! planets a     
revolution ( _! own 9 a s5se opposite to
 t ( ! daily rotation ( ! fix$ />s ab\t 

! e>? z c5tre4  ,b a mu* m rem>kable    
developm5t 0 to foll[ 9 ! ,py?agorean   
s*ool4 ,?is 0 no?+ less ?an ! ab&onm5t (
 ! geoc5tric hypo!sis & ! r$uction ( !  
e>? to ! /atus ( a planet l ! o!rs4 ,!  
result+ sy/em1 kn[n z ! ,py?agorean1 is 
attribut$ 7on ! au?ority probably (     
,!ophra/us7 to ,philolaus2 b ,diog5es   
,la]tius & ,atius ref] to "o ,hicetas ( 
,syracuse 9 ?is connection2 ,ari/otle   
attributes ! sy/em to 8! ,py?agoreans04 
,x is a p>tial anticipation ( ! !ory (  
,cop]nicus b diff]s f x 9 t ! e>? & !   
planets d n revolve r\nd ! sun b ab\t an
 assum$ 8c5tral fire10 & ! sun itself z 
well z ! moon does ! same4 ,!re 7 ?us   
eight heav5ly bodies1 9 addition to !   
sph]e ( ! fix$ />s1 all revolv+ ab\t !  
c5tral fire4 ,! numb] ( revolutions be+ 
?us increas$ to n9e1 ! ,py?agoreans     
po/ulat$ yet ano!r1 mak+ t54 ,! t5? body
 !y call$ ! c\nt]-e>?1 & its "* & object
 w appe> f ! foll[+ g5]al description ( 
! sy/em4  ,! univ]se is sph]ical 9 %ape 
& f9ite 9 size4 ,outside x is inf9ite   

void1 :i* enables ! univ]se to brea!1 z 
x 74 ,at ! c5tre is ! c5tral fire1 !    
,he>? ( ! ,univ]se1 call$ by v>i\s names
 su* z ! ,t[] or ,wat*-t[] ( ,zeus1 !   
,?rone ( ,zeus1 ! ,"m ( ! ,gods4 ,9 ?is 
c5tral fire is locat$ ! gov]n+ pr9ciple1
 ! =ce :i* directs ! movem5t & activity 
( ! univ]se4 ,! outside b\nd>y ( ! sph]e
 is an envelope ( fire2 ?is is call$    
,olympus1 & 9 ?is region ! elem5ts are  
f\nd 9 all _! purity2 bel[ ?is is !     
univ]se4 ,9 ! univ]se !re revolve 9     
circles r\nd ! c5tral fire ! foll[+     
bodies3 ne>e/ to ! c5tral fire !        
c\nt]-e>? :i* always accompanies ! e>?1 
!n ! e>?1 !n ! moon1 !n ! sun1 next to !
 sun ! five planets1 & la/ ( all1       
outside ! orbits ( ! planets1 ! sph]e ( 
! fix$ />s4 ,! c\nt]-e>?1 :i*           
accompanies ! e>? b revolves 9 a small] 
orbit1 is n se5 by u because ! hemisph]e
 on :i* we live is turn$ away f !       
c\nt]-e>?4 ,x foll[s t our hemisph]e is 
always turn$ away f ! c5tral fire1 t is1
 x faces outw>ds f ! orbit t[>ds        

,olympus 7! analogy ( ! moon :i* always 
turns "o side t[>ds u may h sugge/$     
?is72 ?is involves a rotation ( ! e>?   
ab\t its axis complet$ 9 ! same "t z x  
takes ! e>? to complete a revolution    
ab\t ! c5tral fire4  ,al?\gh !re 0 a    
!ory t ! c\nt]-e>? 0 introduc$ 9 ord] to
 br+ ! numb] ( ! mov+ bodies up to t51 !
 p]fect numb] accord+ to ! ,py?agoreans1
 x is cle> f a passage ( ,ari/otle t ?is
 0 n ! real reason4 ,ari/otle says1     
namely1 t ! eclipses ( ! moon 7 consid]$
 to be due sometimes to ! int]position (
 ! e>?1 sometimes to ! int]position ( ! 
c\nt]-e>?4 ,evid5tly !re=e ! purpose ( !
 c\nt]-e>? 0 to expla9 ! frequ5cy ) :i* 
eclipses ( ! moon occur4  ,!            
,py?agoreans held t ! e>?1 revolv+1 l "o
 ( ! />s1 ab\t ! c5tral fire1 makes     
night & day accord+ to its position     
relatively to ! sun2 x is !re=e day 9 t 
region :i* is lit up by ! sun & night 9 
! cone =m$ by ! e>?'s %ad[4 ,z ! same   
hemisph]e is always turn$ outw>ds1 x    
foll[s t ! e>? completes "o revolution  

ab\t ! c5tral fire 9 a day & a night or 
9 ab\t tw5ty-f\r h\rs4 ,?is w\ld acc\nt 
= ! app>5t diurnal rotation ( ! heav5s f
 ea/ to we/2 b = p>allax 7( :i*1 if we  
may believe ,ari/otle1 ! ,py?agoreans   
made light71 x w\ld be equival5t to !   
rotation ( ! e>? on its own axis once 9 
tw5ty-f\r h\rs4 ,?is w\ld make !        
revolution ( ! sph]e ( ! fix$ />s       
unnecess>y4 ,yet ! ,py?agoreans c]ta9ly 
gave "s motion to ! latt] sph]e4 ,:at x 
0 rema9s a puzzle4 ,x _c h be5 !        
precession ( ! equ9oxes1 = t 0 fir/     
discov]$ by ,hipp>*us 7second c5tury    
,b4,c474 ,p]haps !re 0 a real           
incompatibility betwe5 ! two revolutions
 :i* 0 unnotic$ by ! au?ors ( ! sy/em4  
,,nopides ,,( ,,*ios4  nopides ( ,*ios  
7a little y\ng] ?an ,anaxagoras7 is     
cr$it$ ) two discov]ies4 ,! fir/1 :i* 0 
important1 0 t ( ! obliquity ( ! zodiac 
circle or ! ecliptic2 ! second 0 t ( a  
,great ,ye>1 :i* nopides put at         
fifty-n9e ye>s4 ,he also 7s we are told7
 f\nd ! l5g? ( ! ye> to be #cfe-bb_/ei  

days4 ,he seems to h obta9$ ?is figure  
by a sort ( circul> argum5t4 ,/>t+ fir/ 
) #cfe days z ! l5g? ( a ye> & #bi days 
z ! l5g? ( ! lun> mon?1 approximate     
figures kn[n be=e 8 "t1 he _h to f9d !  
lea/ integral numb] ( complete ye>s     
conta9+ an exact numb] ( lun> mon?s2 ?is
 is cle>ly fifty-n9e ye>s1 :i* conta9   
twice #cfe or #gcj lun> mon?s4 nopides  
seems by 8 k ( ! cal5d> to h det]m9$ !  
numb] ( days 9 #gcj lun> mon?s to be    
#ba1eeg1 & ?is numb] divid$ by          
fifty-n9e1 ! numb] ( ye>s1 gives        
#cfe-bb_/ei z ! numb] ( days 9 ! ye>4   
,,plato4  ,we come n[ to ,plato         
7#dbg-cdg ,b4,c474 ,9 ! a/ronomy (      
,plato1 z we f9d x 9 ! ,dialogues1 !re  
is s l>ge an admixture ( my? & poetry t 
x is impossible to be sure :at 8 real   
views 7 on c]ta9 po9ts ( detail4 ,9 !   
,phd we h c]ta9 /atem5ts ab\t ! e>? to !
 effect t x is ( v l>ge dim5sions1 !    
app>5t holl[ 7accord+ to ,plato7 9 :i*  
we live be+ a v small portion ( ! :ole1 
& t x is 9 ! middle ( ! heav51 9        

equilibrium1 )\t any support1 by virtue 
( ! uni=mity 9 ! sub/ance ( ! heav54 ,9 
! ,republic we h a glimpse ( a m        
complete a/ronomical sy/em4 ,! out]mo/  
revolution is t ( ! sph]e ( ! fix$ />s1 
:i* c>ries r\nd ) x ! :ole univ]se      
includ+ ! sun1 moon & planets2 ! latt]  
sev5 bodies1 :ile !y are s c>ri$ r\nd by
 ! g5]al rotation1 h sl[] revolutions ( 
_! own 9 addition1 "o inside ! o!r1 ^!  
revolutions be+ at diff]5t spe$s b all 9
 ! opposite s5se to ! g5]al rotation ( !
 univ]se4 ,! quicke/ rotation is t ( !  
fix$ />s & ! univ]se1 :i* takes place   
once 9 ab\t tw5ty-f\r h\rs4 ,! sl[]     
spe$s ( ! sun1 moon & planets are n     
absolute b relative to ! sph]e ( ! fix$ 
/>s reg>d$ z /ation>y4 ,! e>? 9 ! c5tre 
is unmov$2 ! successive revolutions ab\t
 x & )9 ! sph]e ( ! fix$ />s are        
7reckon+ f ! e>? outw>ds7 ^? ( ! moon1 !
 sun1 ,v5us1 ,m]cury1 ,m>s1 ,jupit]1    
,saturn2 ! spe$ ( ! moon is ! quicke/1 t
 ( ! sun ! next quicke/1 :ile ,v5us &   
,m]cury travel ) ! sun & h ! same spe$1 

tak+ ab\t a ye> to describe _! orbits2  
aft] ^! 9 spe$ comes ,m>s1 !n ,jupit] &1
 la/ & sl[e/ ( all1 ,saturn4 ,!re is    
no?+ said 9 ! ,republic ab\t ! sev5     
bodies revolv+ 9 a circle diff]5t f &   
incl9$ to ! equator ( ! sph]e ( ! fix$  
/>s2 t is1 ! obliquity ( ! ecliptic does
 n appe>2 h5ce ! /&po9t ( ! :ole sy/em  
is t ( ,py?agoras z di/9ct f t ( !      
,py?agoreans4  ,plato's a/ronomical     
sy/em is1 h[ev]1 mo/ fully develop$ 9 ! 
,timu4 ,:ile o!r details rema9          
sub/antially ! same1 ! zodiac circle 9  
:i* ! sun1 moon & planets revolve is    
di/+ui%$ f ! equator ( ! sph]e ( ! fix$ 
/>s4 ,! latt] is call$ ! circle ( !     
,same1 ! =m] t ( ! ,o!r1 & we are told  
7q correctly7 t1 s9ce ! revolution ( !  
univ]se 9 ! circle ( ! ,same c>ries all 
! o!r revolutions ) x1 ! effect on ea* (
 ! sev5 bodies is to turn _! actual     
motions 9 space into spirals4 ,!re is a 
difficulty 9 int]pret+ a phrase 9       
,plato's description :i* says t ,v5us & 
,m]cury1 ?\gh mov+ 9 a circle hav+ equal

 spe$ ) ! sun1 8h ! contr>y t5d5cy to   
x04 ,lit]ally ?is w\ld seem to mean t   
,v5us & ,m]cury describe _! circles !   
opposite way to ! sun1 b ?is is s       
contradict$ by obs]vation t ,plato c\ld 
h>dly h ma9ta9$ x2 h5ce ! words h be5   
?\ght to convey a vague ref]5ce to !    
app>5t irregul>ities 9 ! motion ( ,v5us 
& ,m]cury1 _! /&+s-/ill &               
retrogradations4  ,b ! mo/ disput$ po9t 
9 ! sy/em is ! "p assign$ 9 x to ! e>?4 
,an expression is us$ ) reg>d to its    
relation to ! axis ( ! heav5ly sph]e :i*
 might mean ei!r 7#a7 t x is wrapp$ or  
glob$ ab\t t axis b )\t motion1 or 7#b7 
t x revolves r\nd ! axis4 ,if ! ^w means
 revolv+ ab\t ! axis ( ! sph]e1 !       
revolution w\ld be ei!r 7a7 rotation    
ab\t its own axis suppos$ to be id5tical
 ) t ( ! sph]e1 or 7b7 revolution ab\t !
 axis ( ! heav5ly sph]e 9 ! same way t !
 sun1 moon & planets revolve ab\t an    
axis obliquely incl9$ to t axis4 ,b 7a7 
if ! e>? rotat$ ab\t its own axis1 ?is  
w\ld make unnecess>y ! rotation ( !     

sph]e ( ! fix$ />s once 9 tw5ty-f\r     
h\rs1 :i*1 h[ev]1 is expressly includ$ z
 "p ( ! sy/em4 ,! hypo!sis 7b7 w\ld make
 ! sy/em simil> to ! ,py?agorean except 
t ! e>? w\ld revolve ab\t ! axis ( !    
heav5ly sph]e in/ead ( r\nd ! c5tral    
fire4 ,! support]s ( ?is hypo!sis cite  
two passages ( ,plut>* to ! effect t    
,plato 0 said 9 8 old age to h rep5t$ ( 
hav+ giv5 ! e>? ! middle place 9 !      
univ]se in/ead ( plac+ x else:]e & giv+ 
! middle & *iefe/ place to "s wor?i]    
occupant4 ,x is a suffici5t answ] to ?is
 argum5t t1 if ,plato really meant 9 !  
passage ( ! ,timu to say t ! e>?        
revolves ab\t ! axis ( ! heav5ly sph]e1 
he _h no?+ to rep5t (4 ,we mu/ !re=e1 = 
our "p1 conclude t 9 8 writt5 ,dialogues
 ,plato reg>d$ ! e>? z at re/ 9 ! c5tre 
( ! univ]se4  ,we h x on good au?ority t
 ,plato set x z a problem to all e>ne/  
/ud5ts 8to f9d :at are ! uni=m & ord]$  
movem5ts by ! assumption ( :i* ! app>5t 
movem5ts ( ! planets c be acc\nt$ =04 ,!
 same au?ority adds t ,eudoxus 0 ! fir/ 

to =mulate a !ory ) ?is object2 &       
,h]aclides ( ,pontus foll[$ ) an        
entirely new hypo!sis4 ,bo? 7 pupils (  
,plato &1 9 s f> z ! /atem5t ( 8 problem
 0 a /imulus to ^! speculations1 he     
r5d]$ an important s]vice to ! sci5ce ( 
a/ronomy4   ,,eudoxus1 ,,callippus1     
,,ari/otle4  ,eudoxus ( ,cnidos 7ab\t   
#djh-cee ,b4,c47 0 "o ( ! v greate/ ( ! 
,greek ma!maticians4 ,he 0 ! discov]] & 
elaborator ( ! great !ory ( proportion  
applicable to all magnitudes :e!r       
comm5surable or incomm5surable :i* is   
giv5 9 ,euclid's ,book ,v4 ,he 0 also ! 
orig9ator ( ! p[]ful me?od ( exhau/ion  
us$ by all lat] ,greek geomet]s = !     
purpose ( f9d+ ! areas ( curves & !     
volumes ( pyramids1 cones1 sph]es & o!r 
curv$ surfaces4 ,x is n !re=e surpris+ t
 he %\ld h inv5t$ a rem>kable           
geometrical hypo!sis = expla9+ !        
irregul> movem5ts ( ! planets4 ,!       
problem 0 to f9d ! necess>y numb] (     
circul> motions :i* by _! comb9ation    
w\ld produce ! motions ( ! planets z    

actually obs]v$1 & 9 p>ticul> !         
v>iations 9 _! app>5t spe$s1 _! /ations 
& retrogradations & _! movem5ts 9       
latitude4 ,?is ,eudoxus endeav\r$ to d  
by comb9+ ! motions ( sev]al conc5tric  
sph]es1 "o inside ! o!r1 & revolv+ ab\t 
diff]5t axes1 ea* sph]e revolv+ on its  
own acc\nt b also be+ c>ri$ r\nd bodily 
by ! revolution ( ! next sph]e encircl+ 
x4 ,we are dep5d5t on passages f        
,ari/otle & ,simplicius = our k (       
,eudoxus's sy/em1 :i* he _h set out 9 a 
"w ,on ,spe$s1 n[ lo/4 ,eudoxus assum$  
?ree revolv+ sph]es = produc+ ! app>5t  
motions ( ! sun & moon respectively1 &  
f\r = t ( ea* ( ! planets4 ,9 8 hypo!sis
 = ! sun he seems delib]ately to h      
ignor$ ! discov]y made by ,meton &      
,euctemon "s sixty or sev5ty ye>s be=e t
 ! sun does n take ! same "t to describe
 ! f\r quadrants ( its orbit betwe5 !   
equ9octial & sol/itial po9ts4  ,x %\ld  
be obs]v$ t ! :ole hypo!sis ( !         
conc5tric sph]es is pure geometry1 & !re
 is no me*anics 9 x4 ,we w %ortly       

describe ! arrangem5t ( ! f\r sph]es :i*
 by _! revolution produc$ ! motion ( a  
planet4 ,! fir/ & out]mo/ sph]e produc$ 
! daily rotation 9 tw5ty-f\r h\rs2 !    
second sph]e revolv$ ab\t an axis       
p]p5dicul> to ! plane ( ! zodiac or     
ecliptic1 !reby produc+ ! motion along !
 zodiac 89 ! respective p]iods 9 :i* !  
planets appe> to describe ! zodiac      
circle10 i4e4 9 ! case ( ! sup]ior      
planets1 ! sid]eal p]iods ( revolution1 
& 9 ! case ( ,m]cury & ,v5us 7on a      
geoc5tric sy/em7 "o ye>4 ,! ?ird sph]e  
_h its poles at two opposite po9ts on ! 
zodiac circle1 ! poles be+ c>ri$ r\nd 9 
! motion ( ! second sph]e2 ! revolution 
( ! ?ird sph]e ab\t ! axis connect+ !   
two poles 0 aga9 uni=m & took place 9 a 
p]iod equal to ! synodic p]iod ( !      
planet1 or ! "t elaps+ betwe5 two       
successive oppositions or conjunctions )
 ! sun4  ,! poles ( ! ?ird sph]e 7      
diff]5t = all ! planets1 except t =     
,m]cury & ,v5us !y 7 ! same4 ,on !      
surface ( ! ?ird sph]e ! poles ( ! f\r? 

sph]e 7 fix$1 & its axis ( revolution 0 
incl9$ to t ( ! =m] at an angle con/ant 
= ea* planet b diff]5t = ! diff]5t      
planets4 ,! planet 0 fix$ at a po9t on !
 equator ( ! f\r? sph]e4 ,! ?ird & f\r? 
sph]es toge!r cause ! planet's movem5t 9
 latitude4 ,simplicius expla9s cle>ly ! 
effect ( ^! two rotations4 ,if1 he says1
 ! planet _h be5 on ! ?ird sph]e 7by    
itself71 x w\ld actually h arriv$ at !  
poles ( ! zodiac circle2 b1 z ?+s are1 !
 f\r? sph]e1 :i* turns ab\t ! poles ( ! 
incl9$ circle c>ry+ ! planet & rotates 9
 ! opposite s5se to ! ?ird1 i4e4 f ea/  
to we/1 b 9 ! same p]iod1 w prev5t any  
consid]able div]g5ce on ! "p ( ! planet 
f ! zodiac circle1 & w cause ! planet to
 describe ab\t ?is same zodiac circle ! 
curve call$ by ,eudoxus ! hippop$e      
7horse-fett]71 s t ! bread? ( ?is curve 
w be ! maximum am\nt ( ! app>5t         
deviation ( ! planet 9 latitude4 ,!     
curve 9 "q is an elongat$ figure-(-eight
 ly+ along & bisect$ by ! zodiac circle4
 ,! motion !n r\nd ?is figure-(-eight   

comb9$ ) ! motion 9 ! zodiac circle     
produces ! accel]ation & ret>dation ( ! 
motion ( ! planet1 caus+ ! /ations &    
retrogradations4 ,ma!maticians w        
appreciate ! wond]ful +5uity & beauty ( 
! con/ruction4  ,eudoxus sp5t sixte5    
mon?s 9 ,egypt ab\t #cha-chj ,b4,c41 &1 
:ile !re1 he assimilat$ ! a/ronomical k 
( ! prie/s ( ,heliopolis & himself made 
obs]vations4 ,! ,obs]vatory betwe5      
,heliopolis & ,c]cesura us$ by him 0    
/ill po9t$ out 9 ,augu/us's "t2 he also 
_h "o built at ,cnidos4 ,he wrote two   
books entitl$ respectively ! ,mirror & !
 ,phnom5a2 ! poem ( ,aratus 01 s f> z   
v]ses #ai-gcb are conc]n$1 drawn f !    
,phnom5a ( ,eudoxus4 ,he is also cr$it$ 
) ! inv5tion ( ! ara*ne 7spid]'s web7   
:i*1 h[ev]1 is alt]natively attribut$ to
 ,apollonius1 & :i* seems to h be5 a    
sun-clock ( "s k9d4  ,eudoxus's sy/em ( 
conc5tric sph]es 0 improv$ ^u by        
,callippus 7ab\t #cgj-cjj ,b4,c471 :o   
add$ two m sph]es = ! sun & ! moon1 & "o
 m 9 ! case ( ea* ( ! ?ree ne>] planets1

 ,m]cury1 ,v5us & ,m>s4 ,! two          
additional sph]es 9 ! case ( ! sun 7    
introduc$ 9 ord] to acc\nt = ! unequal  
motion ( ! sun 9 longitude2 & ! purpose 
9 ! case ( ! moon 0 presumably simil>4  
,callippus made ! l5g? ( ! seasons1     
beg9n+ ) ! v]nal equ9ox1 n9ety-f\r1     
n9ety-two1 eighty-n9e & n9ety days      
respectively1 figures mu* m accurate ?an
 ^? giv5 by ,euctemon a hundr$ ye>s     
e>li]1 :i* 7 n9ety-?ree1 n9ety1 n9ety & 
n9ety-two days respectively4  ,)        
,callippus z well z ,eudoxus ! sy/em (  
conc5tric sph]es 0 purely geometrical4  
,ari/otle 7#chd-cbb ,b4,c47 ?\ght x     
necess>y to alt] x 9 a me*anical s5se2  
he made ! sph]es into sph]ical %ells    
actually 9 contact ) "o ano!r1 & ?is    
made x almo/ necess>y1 in/ead ( hav+    
indep5d5t sets ( sph]es1 "o set = ea*   
planet1 to make all ! sets "p ( "o      
cont9u\s sy/em ( sph]es4 ,= ?is purpose 
he assum$ sets ( react+ sph]es betwe5   
successive sets ( ! orig9al sph]es4     
,e4g4 ,saturn be+ c>ri$ by a set ( f\r  

sph]es1 he _h ?ree react+ sph]es to     
neutralise ! la/ ?ree1 9 ord] to re/ore 
! out]mo/ sph]e to act z ! fir/ ( ! f\r 
sph]es produc+ ! motion ( ! next l[]    
planet1 ,jupit]1 & s on4 ,! *ange 0     
h>dly an improvem5t4  ,ari/otle's o!r   
ideas 9 a/ronomy d n require detail$    
notice1 except 8 views ab\t ! e>?4      
,al?\gh he held firmly to ! old belief t
 ! e>? is 9 ! c5tre & rema9s motionless1
 he 0 cle> t its %ape 7l t ( ! />s & !  
univ]se7 is sph]ical1 & he _h arriv$ at 
views ab\t its size s\nd] ?an ^? (      
,plato4 ,9 support ( ! sph]ical %ape ( !
 e>? he us$ "s good argum5ts bas$ on    
obs]vation4 7#a7 ,9 p>tial eclipses ( ! 
moon ! l9e sep>at+ ! d>k & bright       
portions is always circul>--unlike ! l9e
 ( dem>cation 9 ! phases ( ! moon :i*   
may be /raight4 7#b7 ,c]ta9 />s se5     
above ! horizon 9 ,egypt & 9 ,cyprus are
 n visible fur!r nor?1 &1 on ! o!r h&1  
c]ta9 />s set !re :i* 9 m nor!rn        
latitudes rema9 always above ! horizon4 
,z !re is s p]ceptible a *ange ( horizon

 betwe5 places s ne> to ea* o!r1 x      
foll[s n only t ! e>? is sph]ical b also
 t x is n a v l>ge sph]e4 ,ari/otle adds
 t p are n improbably "r :5 !y say t !  
region ab\t ! ,pill>s ( ,h]acles is jo9$
 on to ,india1 "o sea connect+ !m4 ,he  
quotes a result arriv$ at by !          
ma!maticians ( 8 "t1 t ! circumf]5ce ( !
 e>? is #djj1jjj /ades4 ,he is cle> t ! 
e>? is mu* small] ?an "s ( ! />s1 b t ! 
moon is small] ?an ! e>?4  ,! sy/ems (  
conc5tric sph]es 7 n de/9$ to hold _!   
gr\nd = long4 ,9 ^! sy/ems ! sun1 moon &
 planets 7 ( necessity always at ! same 
di/ances f ! e>? respectively4 ,b x 0   
soon recognis$ t !y did n 8save !       
ph5om5a10 s9ce x 0 se5 t ! planets      
appe>$ to be at "o "t ne>] & at ano!r "t
 fur!r (f4 ,autolycus ( ,pitane 7:o     
fl\ri%$ ab\t #caj ,b4,c47 knew ?is & is 
said to h tri$ to expla9 x2 inde$ x c   
h>dly h be5 unkn[n ev5 to ! au?ors ( !  
conc5tric !ory !mselves1 = ,polem>*us ( 
,cyzicus1 almo/ contempor>y ) ,eudoxus1 
is said to h be5 aw>e ( x b to h m9imis$

 ! difficulty because he pref]r$ !      
hypo!sis ( ! conc5tric sph]es to o!rs4  
,developm5t along ! l9es ( ,eudoxus's   
!ory be+ ?us block$1 ! alt]native 0 op5 
( see+ :e!r any modification ( !        
,py?agorean sy/em w\ld give bett]       
results4 ,we actually h evid5ce (       
revisions ( ! ,py?agorean !ory4 ,! fir/ 
/ep 0 to get rid ( ! c\nt]-e>?1 & "s    
,py?agoreans did ?is by id5tify+ !      
c\nt]-e>? ) ! moon4 ,we he> too ( a     
,py?agorean sy/em 9 :i* ! c5tral fire 0 
n outside ! e>? b 9 ! c5tre ( ! e>?     
itself4 ,! descriptions ( ?is sy/em r   
indicate t 9 x ! e>? 0 suppos$ to be at 
re/1 )\t any rotation1 9 ! c5tre ( !    
univ]se4 ,?is 0 practically a return to 
! /&po9t ( ,py?agoras himself4 ,b x is  
cle> t1 if ! sy/em ( ,philolaus 7or     
,hicetas7 be tak5 & ! c5tral fire be    
transf]r$ to ! c5tre ( ! e>? 7!         
c\nt]-e>? be+ also elim9at$71 & if !    
movem5ts ( ! e>?1 sun1 moon & planets   
r\nd ! c5tre be reta9$ )\t any          
modification save t :i* is ma!matically 

involv$ by ! transf] ( ! c5tral fire to 
! c5tre ( ! e>?1 ! daily revolution ( ! 
e>? ab\t ! c5tral fire is necess>ily    
trans=m$ into a rotation ( ! e>? ab\t   
its own axis 9 ab\t tw5ty-f\r h\rs4     
,,h]aclides ,,( ,,pontus4  ,all         
au?orities agree t ! !ory ( ! daily     
rotation ( ! e>? ab\t its own axis 0 put
 =w>d by ,h]aclides ( ,pontus 7ab\t     
#chh-cae ,b4,c471 a pupil ( ,plato2 )   
him 9 "s acc\nts is associat$ ! "n ( "o 
,ecphantus1 a ,py?agorean4 ,we are told 
t ,ecphantus ass]t$ 8t ! e>?1 be+ 9 !   
c5tre ( ! univ]se1 moves ab\t its own   
c5tre 9 an ea/w>d direction10 & t       
8,h]aclides ( ,pontus & ,ecphantus !    
,py?agorean make ! e>? move1 n 9 ! s5se 
( translation1 b by way ( turn+ z on an 
axle1 l a :eel1 f we/ to ea/1 ab\t its  
own c5tre04  ,h]aclides 0 born at       
,h]aclea 9 ,pontus4 ,he w5t to ,a!ns n  
lat] ?an #cfd ,b4,c41 & !re met         
,speusippus1 :o introduc$ him into !    
s*ool ( ,plato4 ,on ! dea? ( ,speusippus
 7!n at ! head ( ! s*ool7 9 #cch1       

,x5ocrates 0 elect$ to succe$ him2 at   
?is election ,h]aclides 0 also a c&idate
 & 0 only defeat$ by a few votes4 ,he 0 
! au?or ( dialogues1 brilliant &        
orig9al1 on all sorts ( subjects1 :i* 7 
mu* read & imitat$ at ,rome1 e4g4 by    
,v>ro & ,cic]o4 ,two ( !m 8,on ,nature0 
& 8,on ! ,heav5s0 may h dealt )         
a/ronomy4  ,9 8 view t ! e>? rotates    
ab\t its own axis ,h]aclides is         
associat$ ) ,ari/>*us ( ,samos2 ?us     
,simplicius says3 8,!re h be5 "s1 l     
,h]aclides ( ,pontus & ,ari/>*us1 :o    
suppos$ t ! ph5om5a c be sav$ if ! heav5
 & ! />s are at re/ :ile ! e>? moves    
ab\t ! poles ( ! equ9octial circle f !  
we/ to ! ea/1 complet+ "o revolution ea*
 day1 approximately2 ! #8approximately' 
is add$ because ( ! daily motion ( ! sun
 to ! ext5t ( "o degree04  ,h]aclides   
made ano!r important advance t[>ds !    
,cop]nican hypo!sis4 ,he discov]$ ! fact
 t ,v5us & ,m]cury revolve ab\t ! sun z 
c5tre4 ,s mu* is c]ta92 b a fur!r "q    
naturally arises4 ,hav+ made ,v5us &    

,m]cury revolve r\nd ! sun l satellites1
 did ,h]aclides proce$ to draw ! same   
inf]5ce ) reg>d to ! o!r1 ! sup]ior1    
planets8 ,! "q is int]e/+ because1 _h x 
be5 laid d[n t all ! five planets alike 
revolve r\nd ! sun1 ! comb9ation ( ?is  
hypo!sis ) ,h]aclides's assumption t !  
e>? rotates ab\t its own axis 9         
tw5ty-f\r h\rs w\ld h am\nt$ to an      
anticipation ( ! sy/em ( ,ty*o ,brahe1 b
 ) ! improvem5t ( ! sub/itution ( !     
daily rotation ( ! e>? = ! daily        
revolution ( ! :ole sy/em ab\t ! e>?    
suppos$ at re/4 ,s*iap>elli dealt ) ! "q
 9 two pap]s entitl$ ,i precursori di   
,cop]nico nell' anti*it 7#ahgc71 &      
,orig9e del si/ema planet>io elioc5trico
 presso i ,greci 7#ahih74 ,s*iap>elli   
tri$ to %[ t ,h]aclides did arrive at ! 
conclusion t ! sup]ior planets z well z 
,m]cury & ,v5us revolve r\nd ! sun2 b   
mo/ p]sons w probably agree t 8 argum5t 
is n conv9c+4 ,! difficulties seem too  
great4 ,! circles describ$ by ,m]cury & 
,v5us ab\t ! sun are relatively small   

circles & are entirely on "o side ( !   
e>?4 ,b :5 ! possibility (1 say1 ,m>s   
revolv+ ab\t ! sun came to be consid]$1 
x w\ld be at once obvi\s t ! precise    
hypo!sis adopt$ = ,m]cury & ,v5us w\ld n
 apply4 ,x w\ld be se5 t ,m>s is        
brighte/ :5 x occupies a position 9 !   
zodiac opposite to ! sun2 x mu/ !re=e be
 ne>e/ to ! e>? at t "t4 ,consequ5tly ! 
circle describ$ by ,m>s1 in/ead ( be+ on
 "o side ( ! e>?1 mu/ compreh5d ! e>?   
:i* is inside x4 ,:]eas !re=e ! circles 
describ$ by ,m]cury & ,v5us 7 :at !     
,greeks call$ epicycles ab\t a mat]ial  
c5tre1 ! sun 7itself mov+ 9 a circle    
r\nd ! e>?71 :at 0 want$ 9 ! case ( ,m>s
 7if ! circle describ$ by ,m>s 0 to h ! 
sun = c5tre7 0 :at ! ,greeks call$ an   
ecc5tric circle1 ) a c5tre :i* itself   
moves 9 a circle ab\t ! e>?1 & ) a      
radius great] ?an t ( ! sun's orbit4    
,?\gh ! same motion c\ld h be5 produc$  
by an epicycle1 ! epicycle w\ld h _h to 
h a ma!matical po9t 7n ! mat]ial sun7 z 
c5tre4 ,b ! idea ( us+ non-mat]ial po9ts

 z c5tres = epicycles 0 probably fir/   
?\ght (1 at a lat] /age1 by "s ( ! great
 ma!maticians su* z ,apollonius ( ,p]ga 
7ab\t #bfe-aij ,b4,c474  ,n only does   
,s*iap>elli ma9ta9 t ! complete 7b      
improv$7 ,ty*onic hypo!sis 0 put =w>d by
 ,h]aclides or at lea/ 9 ,h]aclides's   
"t2 he goes fur!r & makes a /ill great] 
claim on behalf ( ,h]aclides1 namely1 t 
x 0 he1 & n ,ari/>*us ( ,samos1 :o fir/ 
/at$ z a possibility ! ,cop]nican       
hypo!sis4 ,n[ x 0 mu* to discov]1 z     
,h]aclides did1 t ! e>? rotates ab\t its
 own axis & t ,m]cury & ,v5us revolve   
r\nd ! sun l satellites2 & x seems a    
priori incr$ible t "o man %\ld n only h 
rea*$1 & improv$ ^u1 ! hypo!sis ( ,ty*o 
,brahe b %\ld also h sugge/$ !          
,cop]nican hypo!sis4 ,x is !re=e        
necess>y to exam9e briefly ! evid5ce on 
:i* ,s*iap>elli reli$4 ,8 argum5t re/s  
entirely ^u "o passage1 a s5t5ce =m+ "p 
( a quotation f a summ>y by ,gem9us ( ! 
,meteorologica ( ,posidonius1 :i*       
,simplicius copi$ f ,alex&]             

,aphrodisi5sis & embodi$ 9 8 comm5t>y on
 ! ,physics ( ,ari/otle4 ,! s5t5ce 9 "q1
 accord+ to ! read+ ( ! ,,mss41 is z    
foll[s3 8,h5ce we actually f9d a c]ta9  
p]son1 ,h]aclides ( ,pontus1 com+ =w>d &
 say+ t1 ev5 on ! assumption t ! e>?    
moves 9 a c]ta9 way1 :ile ! sun is 9 a  
c]ta9 way at re/1 ! app>5t irregul>ity )
 ref]5ce to ! sun c be sav$04 7,! prec$+
 s5t5ce is ab\t possible answ]s to ! "q1
 :y d ! sun1 ! moon & ! planets appe> to
 move irregul>ly8 & says1 8we may answ] 
t1 if we assume t _! orbits are ecc5tric
 circles or t ! />s describe an         
epicycle1 _! app>5t irregul>ity w be    
sav$1 & x w be necess>y to g fur!r &    
exam9e 9 h[ _m diff]5t ways x is        
possible = ^! ph5om5a to be br\ght      
ab\t047  ,n[ x is impossible t ,gem9us  
himself c h spok5 ( an a/ronom] ( !     
di/9ction ( ,h]aclides z 8a c]ta9       
,h]aclides ( ,pontus04 ,consequ5tly !re 
h be5 diff]5t attempts made to em5d !   
read+ ( ! ,,mss4 ,all ! em5dations      
propos$ are op5 to s]i\s objections1 &  

we are ?r[n back on ! read+ ( ! ,,mss4  
,n[ x 8leaps to ! eyes0 t1 if ! "n (    
,h]aclides ( ,pontus is left out1 ev]y?+
 is 9 ord]4 8,?is is :y "o a/ronom] has 
actually sugge/$ t1 by assum+ ! e>? to  
move 9 a c]ta9 way1 & ! sun to be 9 a   
c]ta9 way at re/1 ! app>5t irregul>ity )
 ref]5ce to ! sun w be sav$40 ,?is seems
 to be ! solution ( ! puzzle sugge/$ by 
! ord9>y pr9ciples ( textual criticism1 
& is s simple & natural t x w surely    
c>ry conviction to ! m9ds ( unbiass$    
p]sons4 ,gem9us1 9 fact1 m5tion$ no "n b
 meant ,ari/>*us ( ,samos1 & "s s*olia/1
 rememb]+ t ,h]aclides _h giv5 a c]ta9  
motion to ! e>? 7namely1 rotation ab\t  
its axis71 imm$iately ?\ght ( ,h]aclides
 & ins]t$ 8 "n 9 ! m>g91 f :i* x        
aft]w>ds crept into ! text4  ,x is only 
necess>y to add t ,ar*im$es is n likely 
to h be5 wrong :5 he attribut$ ! fir/   
sugge/ion ( ! ,cop]nican hypo!sis to    
,ari/>*us ( ,samos 9 express t]ms2 & ?is
 is confirm$ by ano!r positive /atem5t  
by ,atius1 already quot$1 t 8,h]aclides 

( ,pontus & ,ecphantus ! ,py?agorean    
made ! e>? move1 n 9 ! s5se (           
translation1 b ) a movem5t ( rotation04 
,,"p ,,ii4  ,,ari/>*us ,,( ,,samos4     
,we are told t ,ari/>*us ( ,samos 0 a   
pupil ( ,/rato ( ,lampsacus1 a natural  
philosoph] ( orig9ality1 :o succe$$     
,!ophra/us z head ( ! ,p]ipatetic s*ool 
9 #bhh or #bhg ,b4,c41 & held t position
 = eighte5 ye>s4 ,two o!r facts enable u
 to fix ,ari/>*us's date approximately4 
,9 #bha-bhj he made an obs]vation ( !   
summ] sol/ice2 & ! book 9 :i* he =mulat$
 8 helioc5tric hypo!sis 0 publi%$ be=e !
 date ( ,ar*im$es's ,psammites or       
,s&reckon]1 a "w writt5 be=e #baf ,b4,c4
 ,ari/>*us !re=e probably liv$ circa    
#caj-bcj ,b4,c41 t is1 he came ab\t     
sev5ty-five ye>s lat] ?an ,h]aclides & 0
 old] ?an ,ar*im$es by ab\t tw5ty-five  
ye>s4  ,ari/>*us 0 call$ 8!             
ma!matician10 no d\bt 9 ord] to di/+ui% 
him f ! _m o!r p]sons ( ! same "n2      
,vitruvius includes him among ! few     
great m5 :o possess$ an equally pr(\nd k

 ( all bran*es ( sci5ce1 geometry1      
a/ronomy1 music1 etc4 8,m5 ( ?is type   
are r>e1 m5 su* z 7 9 times pa/         
,ari/>*us ( ,samos1 ,philolaus &        
,ar*ytas ( ,t>5tum1 ,apollonius ( ,p]ga1
 ,erato/h5es ( ,cyr5e1 ,ar*im$es &      
,scop9as ( ,syracuse1 :o left to po/]ity
 _m me*anical & gnomonic appliances :i* 
!y inv5t$ & expla9$ on ma!matical &     
natural pr9ciples40 ,t ,ari/>*us 0 a v  
capable geomet] is prov$ by 8 extant    
book1 ,on ! sizes & di/ances ( ! sun &  
moon1 pres5tly to be describ$4 ,9 !     
me*anical l9e he is cr$it$ ) ! inv5tion 
( an improv$ sun-dial1 ! s-call$ scaphe1
 :i* _h n a plane b a concave           
hemisph]ical surface1 ) a po9t] erect$  
v]tically 9 ! middle1 ?r[+ %ad[s & s    
enabl+ ! direction & height ( ! sun to  
be read (f by means ( l9es m>k$ on !    
surface ( ! hemisph]e4 ,he also wrote on
 vision1 light1 & col\rs4 ,8 views on ! 
latt] subjects 7 no d\bt l>gely influ5c$
 by ! tea*+ ( ,/rato4 ,/rato held t     
col\rs 7 emanations f bodies1 mat]ial   

molecules z x 71 :i* imp>t$ to ! int]v5+
 air ! same col\r z t possess$ by !     
body4 ,ari/>*us said t col\rs are 8%apes
 or =ms /amp+ ! air ) impressions l     
!mselves z x 710 t 8col\rs 9 d>kness h  
no col\r+10 & t 8light is ! col\r imp++ 
on a sub/ratum04   ,,! ,,helioc5tric    
,,hypo!sis4  ,!re is no d\bt :atev] t   
,ari/>*us put =w>d ! helioc5tric        
hypo!sis4 ,anci5t te/imony is unanim\s  
on ! po9t1 & ! fir/ witness is ,ar*im$es
 :o 0 a y\ng] contempor>y ( ,ari/>*us1 s
 t !re is no possibility ( a mi/ake4    
,cop]nicus himself admitt$ t ! !ory 0   
attribut$ to ,ari/>*us1 ?\gh ?is does n 
seem to be g5]ally kn[n4 ,cop]nicus     
ref]s 9 two passages ( 8 "w1 ,de        
revolutionibus caele/ibus1 to ! op9ions 
( ! anci5ts ab\t ! motion ( ! e>?4 ,9 ! 
d$icatory lett] to ,pope ,paul ,,iii he 
m5tions t he fir/ le>nt f ,cic]o t "o   
,nicetas 7i4e4 ,hicetas7 _h attribut$   
motion to ! e>?1 & t he aft]w>ds read 9 
,plut>* t c]ta9 o!rs held t op9ion2 he  
!n quotes ! ,placita philosophorum      

accord+ to :i* 8,philolaus ! ,py?agorean
 ass]t$ t ! e>? mov$ r\nd ! fire 9 an   
oblique circle 9 ! same way z ! sun &   
moon04 ,9 ,book ,i4 c4 #e ( 8 "w        
,cop]nicus alludes to ! views (         
,h]aclides1 ,ecphantus1 & ,hicetas1 :o  
made ! e>? rotate ab\t its own axis1 &  
!n goes on to say t x w\ld n be v       
surpris+ if any "o %\ld attribute to !  
e>? ano!r motion besides rotation1      
namely1 revolution 9 an orbit 9 space3  
8atque etiam 7t]ram7 pluribus motibus   
vagantem et unam ex a/ris ,philolaus    
,py?agoricus s5sisse f]tur1 ,ma!maticus 
non vulg>is04 ,"h1 h[ev]1 !re is no "q (
 ! e>? revolv+ r\nd ! sun1 & !re is no  
m5tion ( ,ari/>*us4 ,b ,cop]nicus did   
m5tion ! !ory ( ,ari/>*us 9 a passage   
:i* he aft]w>ds suppress$3 8,cr$ibile e/
 hisce similibusque causis ,philolaum   
mobilitatem t]rae s5sisse1 quod etiam   
nonnulli ,ari/>*um ,samium f]unt 9 eadem
 fuisse s5t5tia04  ,x is desirable to   
quote ! :ole passage ( ,ar*im$es 9 :i* !
 allusion to ,ari/>*us's helioc5tric    

hypo!sis occurs1 9 ord] to %[ ! :ole    
context4  8,y are aw>e #8y' be+ ,k+     
,gelon t #8univ]se' is ! "n giv5 by mo/ 
a/ronom]s to ! sph]e ! c5tre ( :i* is ! 
c5tre ( ! e>?1 :ile its radius is equal 
to ! /raight l9e betwe5 ! c5tre ( ! sun 
& ! c5tre ( ! e>?4 ,?is is ! common     
acc\nt z y h he>d f a/ronom]s4 ,b       
,ari/>*us br\ght out a book consi/+ (   
c]ta9 hypo!ses1 :]e9 x appe>s1 z a      
consequ5ce ( ! assumptions made1 t !    
univ]se is _m times great] ?an !        
#8univ]se' j m5tion$4 ,8 hypo!ses are t 
! fix$ />s & ! sun rema9 unmov$1 t ! e>?
 revolves ab\t ! sun 9 ! circumf]5ce ( a
 circle1 ! sun ly+ 9 ! middle ( ! orbit1
 & t ! sph]e ( ! fix$ />s1 situat$ ab\t 
! same c5tre z ! sun1 is s great t !    
circle 9 :i* he supposes ! e>? to       
revolve be>s su* a proportion to !      
di/ance ( ! fix$ />s z ! c5tre ( ! sph]e
 be>s to its surface40  ,! helioc5tric  
hypo!sis is "h /at$ 9 language :i*      
leaves no room = d\bt ab\t its mean+4 ,!
 sun1 l ! fix$ />s1 rema9s unmov$ & =ms 

! c5tre ( a circul> orbit 9 :i* ! e>?   
moves r\nd x2 ! sph]e ( ! fix$ />s has  
its c5tre at ! c5tre ( ! sun4  ,we h    
fur!r evid5ce 9 a passage ( ,plut>*'s   
tract1 ,on ! face 9 ! moon's orb3 8,only
 d n1 my de> fell[1 ent] an action =    
impiety aga9/ me 9 ! /yle ( ,clean!s1 :o
 ?\ght x 0 ! duty ( ,greeks to indict   
,ari/>*us on ! *>ge ( impiety = putt+ 9 
motion ! ,he>? ( ! ,univ]se1 ?is be+ !  
effect ( 8 attempt to save ! ph5om5a by 
suppos+ ! heav5 to rema9 at re/ & ! e>? 
to revolve 9 an oblique circle1 :ile x  
rotates1 at ! same "t1 ab\t its own     
axis04  ,"h we h ! additional detail t  
,ari/>*us foll[$ ,h]aclides 9 attribut+ 
to ! e>? ! daily rotation ab\t its axis2
 ,ar*im$es does n /ate ?is 9 s _m words1
 b x is cle>ly involv$ by 8 rem>k t     
,ari/>*us suppos$ ! fix$ />s z well z ! 
sun to rema9 unmov$ 9 space4 ,a tract   
8,aga9/ ,ari/>*us0 is m5tion$ by        
,diog5es ,la]tius among ,clean!s's      
works2 & x 0 evid5tly publi%$ dur+      
,ari/>*us's lifetime 7,clean!s di$ ab\t 

#bcb ,b4,c474  ,we le>n f ano!r passage 
( ,plut>* t ! hypo!sis ( ,ari/>*us 0    
adopt$1 ab\t a c5tury lat]1 by          
,seleucus1 ( ,seleucia on ! ,tigris1 a  
,*aldan or ,babylonian1 :o also wrote on
 ! subject ( ! tides ab\t #aej ,b4,c4 ,!
 passage is int]e/+ because x also      
alludes to ! d\bt ab\t ,plato's f9al    
views4 8,did ,plato put ! e>? 9 motion z
 he did ! sun1 ! moon & ! five planets  
:i* he call$ ! #8in/rum5ts ( "t' on     
acc\nt ( _! turn+s1 & 0 x necess>y to   
conceive t ! e>? #8:i* is glob$ ab\t !  
axis /ret*$ f pole to pole ?r\gh ! :ole 
univ]se' 0 n repres5t$ z be+ 7m]ely7    
held toge!r & at re/ b z turn+ &        
revolv+1 z ,ari/>*us & ,seleucus        
aft]w>ds ma9ta9$ t x did1 ! =m] ( :om   
/at$ ?is z only a hypo!sis1 ! latt] z a 
def9ite op9ion80  ,no "o aft] ,seleucus 
is m5tion$ by "n z hav+ accept$ !       
doctr9e ( ,ari/>*us &1 if o!r ,greek    
a/ronom]s ref] to x1 !y d s only to     
d5\nce x4 ,hipp>*us1 himself a          
contempor>y ( ,seleucus1 def9itely      

rev]t$ to ! geoc5tric sy/em1 & x 0      
d\btless 8 au?ority :i* seal$ ! fate ( !
 helioc5tric hypo!sis = s _m c5turies4  
,! reasons :i* weigh$ ) ,hipp>*us 7     
presumably ! facts t ! sy/em 9 :i* ! e>?
 revolv$ 9 a circle ( :i* ! sun 0 !     
exact c5tre fail$ to 8save ! ph5om5a10 &
 9 p>ticul> to acc\nt = ! v>iations (   
di/ance & ! irregul>ities ( ! motions1  
:i* became m & m pat5t z me?ods (       
obs]vation improv$2 t1 on ! o!r h&1 !   
!ory ( epicycles did suffice to repres5t
 ! ph5om5a ) consid]able accuracy2 & t !
 latt] !ory c\ld be reconcil$ ) !       
immobility ( ! e>?4   ,,on ,,! ,,app>5t 
,,diamet] ,,( ,,! ,,sun4  ,ar*im$es     
tells u 9 ! same treatise t 8,ari/>*us  
discov]$ t ! sun's app>5t size is ab\t  
#a_/gbj? "p ( ! zodiac circle02 t is to 
say1 he obs]v$ t ! angle subt5d$ at !   
e>? by ! diamet] ( ! sun is ab\t half a 
degree4   ,,on ,,! ,,sizes ,,&          
,,di/ances ,,( ,,! ,,sun ,,& ,,moon4    
,ar*im$es also says t1 :]eas ! ratio ( !
 diamet] ( ! sun to t ( ! moon _h be5   

e/imat$ by ,eudoxus at #i 3 #a & by 8   
own fa!r ,phidias at #ab 3 #a1 ,ari/>*us
 made ! ratio great] ?an #ah 3 #a b less
 ?an #bj 3 #a4 ,=tunately we possess 9  
,greek ! %ort treatise 9 :i* ,ari/>*us  
prov$ ^! conclusions2 on ! o!r matt] ( !
 app>5t diamet] ( ! sun ,ar*im$es's     
/atem5t is our only evid5ce4  ,x is     
notewor?y t 9 ,ari/>*us's extant        
treatise ,on ! sizes & di/ances ( ! sun 
& moon !re is no h9t ( ! helioc5tric    
hypo!sis1 :ile ! app>5t diamet] ( ! sun 
is !re assum$ to be1 n 1 b ! v          
inaccurate figure ( #b4 ,bo?            
circum/ances are expla9$ if we assume t 
! treatise 0 an e>ly "w writt5 be=e !   
hypo!ses describ$ by ,ar*im$es 7 put    
=w>d4 ,9 ! treatise ,ari/>*us f9ds !    
ratio ( ! diamet] ( ! sun to ! diamet] (
 ! e>? to lie betwe5 #ai 3 #c & #dc 3   
#f2 ?is w\ld make ! volume ( ! sun ab\t 
#cjj times t ( ! e>?1 & x may be t !    
great size ( ! sun 9 comp>ison ) ! e>?1 
z ?us br\ght out1 0 "o ( ! consid]ations
 :i* l$ ,ari/>*us to place ! sun r ?an !

 e>? 9 ! c5tre ( ! univ]se1 s9ce x might
 ev5 at t day seem absurd to make ! body
 :i* 0 s mu* l>g] revolve ab\t ! small]4
  ,!re is no reason to d\bt t 9 8       
helioc5tric sy/em ,ari/>*us reta9$ !    
moon z a satellite ( ! e>? revolv+ r\nd 
x z c5tre2 h5ce ev5 9 8 sy/em !re 0 "o  
epicycle4  ,! treatise ,on sizes &      
di/ances be+ ! only "w ( ,ari/>*us :i*  
has surviv$1 x w be fitt+ to give "h a  
description ( its cont5ts & special     
features4  ,! /yle ( ,ari/>*us is       
?or\ghly classical z befits an able     
geomet] int]m$iate 9 date betwe5 ,euclid
 & ,ar*im$es1 & 8 demon/rations are     
work$ out ) ! same rig\r z ^? ( 8       
pr$ecessor & successor4 ,! propositions 
( ,euclid's ,elem5ts are1 ( c\rse1 tak5 
= grant$1 b o!r ?+s are tacitly assum$  
:i* g beyond :at we f9d 9 ,euclid4 ,?us 
! trans=mations ( ratios def9$ 9        
,euclid1 ,book ,v1 & d5ot$ by ! t]ms    
inv]sely1 alt]nately1 compon5do1        
conv]t5do1 etc41 are regul>ly us$ 9     
deal+ ) unequal ratios1 :]eas 9 ,euclid 

!y are only us$ 9 proportions1 i4e4     
cases ( equality ( ratios4 ,b !         
propositions ( ,ari/>*us are also (     
p>ticul> ma!matical int]e/ because !    
ratios ( ! sizes & di/ances :i* h to be 
calculat$ are really trigonometrical    
ratios1 s9es1 cos9es1 etc41 al?\gh at ! 
"t ( ,ari/>*us trigonometry _h n be5    
inv5t$1 & no reasonably close           
approximation to ! value ( 1 ! ratio ( !
 circumf]5ce ( any circle to its        
diamet]1 _h be5 made 7x 0 ,ar*im$es :o  
fir/ obta9$ ! approximation #bb_/g74    
,exact calculation ( ! trigonometrical  
ratios be+ !re=e impossible = ,ari/>*us1
 he set himself to f9d upp] & l[] limits
 = !m1 & he succe$$ 9 locat+ ^? :i*     
em]ge 9 8 propositions )9 tol]ably n>r[ 
limits1 ?\gh n always ! n>r[e/ )9 :i* x 
w\ld h be5 possible1 ev5 = him1 to      
conf9e !m4 ,9 ?is species (             
approximation to trigonometry he tacitly
 assumes propositions comp>+ ! ratio    
betwe5 a great] & a less] angle 9 a     
figure ) ! ratio betwe5 two /raight     

l9es1 propositions :i* are =mally prov$ 
by ,ptolemy at ! beg9n+ ( 8 ,syntaxis4  
,"h aga9 we h pro( t textbooks conta9+  
su* propositions exi/$ be=e ,ari/>*us's 
"t1 & probably mu* e>li]1 al?\gh !y h n 
surviv$4  ,ari/>*us necess>ily beg9s by 
lay+ d[n1 z ! basis = 8 treatise1 c]ta9 
assumptions4 ,!y are six 9 numb]1 & he  
ref]s to !m z hypo!ses4 ,we _c d bett]  
?an quote !m 9 full1 along ) ! s5t5ces  
imm$iately foll[+1 9 :i* he /ates ! ma9 
results to be e/abli%$ 9 ! treatise3--  
,,hypo!ses4  #a4 ,t ! moon receives its 
light f ! sun4  #b4 ,t ! e>? is 9 !     
relation ( a po9t & c5tre to ! sph]e 9  
:i* ! moon moves4  #c4 ,t1 :5 ! moon    
appe>s to u halv$1 ! great circle :i*   
divides ! d>k & ! bright portions ( !   
moon is 9 ! direction ( our eye4  #d4   
,t1 :5 ! moon appe>s to u halv$1 its    
di/ance f ! sun is !n less ?an a        
quadrant by "o-?irtie? ( a quadrant4    
#e4 ,t ! bread? ( ! 7e>?'s7 %ad[ is 7t7 
( two moons4  #f4 ,t ! moon subt5ds     
"o-fifte5? "p ( a sign ( ! zodiac4  ,we 

are n[ 9 a position to prove ! foll[+   
propositions3--  #a4 ,! di/ance ( ! sun 
f ! e>? is great] ?an eighte5 times1 b  
less ?an tw5ty times1 ! di/ance ( ! moon
 7f ! e>?72 ?is foll[s f ! hypo!sis ab\t
 ! halv$ moon4  #b4 ,! diamet] ( ! sun  
has ! same ratio 7z a=esaid7 to !       
diamet] ( ! moon4  #c4 ,! diamet] ( !   
sun has to ! diamet] ( ! e>? a ratio    
great] ?an t :i* #ai has to #c1 b less  
?an t :i* #dc has to #f2 ?is foll[s f ! 
ratio ?us discov]$ betwe5 ! di/ances1 ! 
hypo!sis ab\t ! %ad[1 & ! hypo!sis t !  
moon subt5ds "o-fifte5? "p ( a sign ( ! 
zodiac4  ,! fir/ assumption is          
,anaxagoras's discov]y4 ,! second       
assumption is no d\bt an exagg]ation2 b 
x is made 9 ord] to avoid hav+ to all[ =
 ! fact t ! ph5om5a z se5 by an obs]v]  
on ! surface ( ! e>? are slightly       
diff]5t f :at w\ld be se5 if ! obs]v]'s 
eye 7 at ! c5tre ( ! e>?4 ,ari/>*us1 t  
is1 takes ! e>? to be l a po9t 9 ord] to
 avoid ! complication ( p>allax4  ,!    
mean+ ( ! ?ird hypo!sis is t ! plane ( !

 great circle 9 "q passes ?r\gh ! po9t  
:]e ! eye ( ! obs]v] is situat$2 t is to
 say1 we see ! circle end on1 z x 71 & x
 looks l a /raight l9e4  ,hypo!sis #d4  
,if ,s be ! sun1 ,m ! moon & ,e ! e>?1 !
 triangle ,,sme is1 at ! mom5t :5 ! moon
 appe>s to u halv$1 "r-angl$ at ,m2 & ! 
hypo!sis /ates t ! angle at ,e 9 ?is    
triangle is #hg1 or1 9 o!r words1 !     
angle ,,mse1 t is1 ! angle subt5d$ at ! 
sun by ! l9e jo9+ ,m to ,e1 is #c4 ,^!  
e/imates are decid$ly inaccurate1 = !   
true value ( ! angle ,,mes is #hi #ej1 &
 t ( ! angle ,,mse is !re=e #aj4 ,!re is
 no?+ to %[ h[ ,ari/>*us came to e/imate
 ! angle ,,mse at #c1 & none ( 8        
successors seem to h made any direct    
e/imate ( ! size ( ! angle4  ,!         
assumption 9 ,hypo!sis #e 0 improv$ ^u  
lat]4 ,hipp>*us made ! ratio ( ! diamet]
 ( ! circle ( ! e>?'s %ad[ to ! diamet] 
( ! moon to be1 n #b1 b #b at ! moon's  
mean di/ance at ! conjunctions2 ,ptolemy
 made x1 at ! moon's greate/ di/ance1 to
 be inappreciably less ?an #b4  ,! six? 

hypo!sis /ates t ! diamet] ( ! moon     
subt5ds at our eye an angle :i* is      
#a_/ae? ( #cj1 i4e4 #b1 :]eas ,ar*im$es1
 z we h se51 tells u t ,ari/>*us f\nd ! 
angle subt5d$ by ! diamet] ( ! sun to be
  7,ar*im$es 9 ! same tract describes a 
r\gh in/rum5t by means ( :i* he himself 
f\nd t ! diamet] ( ! sun subt5d$ an     
angle less ?an #a_/afd?1 b great] ?an   
#a_/bjj? ( a "r angle74 ,ev5 !          
,babylonians _h1 _m c5turies be=e1      
arriv$ at #a z ! app>5t angul> diamet] (
 ! sun4 ,x is n cle> :y ,ari/>*us took a
 value s inaccurate z #b4 ,x has be5    
sugge/$ t he m]ely int5d$ to give a     
specim5 ( ! calculations :i* w\ld h to  
be made on ! basis ( m exact exp]im5tal 
obs]vations1 & to %[ t1 = ! solution ( !
 problem1 "o ( ! data c\ld be *os5 almo/
 arbitr>ily1 by :i* proce$+ he secur$   
himself aga9/ c]ta9 objections :i* might
 h be5 rais$4 ,p]haps ?is is too +5i\s1 
& x may be t1 9 view ( ! difficulty (   
work+ out ! geometry if ! two angles 9  
"q are v small1 he took #c & #b z be+ ! 

smalle/ ) :i* he c\ld conv5i5tly deal4  
,c]ta9 x is t ! me?od ( ,ari/>*us is    
p]fectly correct &1 if he c\ld h        
sub/itut$ ! true values 7z we "k !m     
to-day7 = ! inaccurate values :i* he    
assumes1 & c\ld h c>ri$ f> en\gh 8      
geometrical sub/itute = trigonometry1 he
 w\ld h obta9$ close limits = ! true    
sizes & di/ances4  ,! book conta9s      
eighte5 propositions4 ,prop4 #a proves t
 we c draw "o cyl9d] to t\* two equal   
sph]es1 & "o cone to t\* two unequal    
sph]es1 ! planes ( ! circles ( contact  
be+ at "r angles to ! axis ( ! cyl9d] or
 cone4 ,next 7,prop4 #b7 x is %[n t1 if 
a less] sph]e be illum9at$ by a great]1 
! illum9at$ portion ( ! =m] w be great] 
?an a hemisph]e4 ,prop4 #c proves t !   
circle 9 ! moon :i* divides ! d>k & !   
bright portions 7we w 9 future1 = %ort1 
call ?is 8! divid+ circle07 is lea/ :5 !
 cone :i* t\*es ! sun & ! moon has its  
v]tex at our eye4 ,9 ,prop4 #d x is %[n 
t ! divid+ circle is n p]ceptibly       
diff]5t f a great circle 9 ! moon4 ,if  

,,cd is a diamet] ( ! divid+ circle1    
,,ef ! p>allel diamet] ( ! p>allel great
 circle 9 ! moon1 ,o ! c5tre ( ! moon1  
,a ! obs]v]'s eye1 ,,fdg ! great circle 
9 ! moon ! plane ( :i* passes ?r\gh ,a1 
& ,g ! po9t :]e ,,oa meets ! latt] great
 circle1 ,ari/>*us takes an arc ( !     
great circle ,,gh on "o side ( ,g1 &    
ano!r ,,gk on ! o!r side ( ,g1 su* t    
,,gh  ,,gk   7! arc ,,fd71 & proves t ! 
angle subt5d$ at ,a by ! arc ,,hk is    
less ?an #a_/dd2 consequ5tly1 he says1 !
 arc w\ld be imp]ceptible at ,a ev5 9 t 
position1 & a =tiori ! arc ,,fd 7:i* is 
ne>ly 9 a /raight l9e ) ! tang5t ,,ad7  
is q imp]ceptible to ! obs]v] at ,a4    
,h5ce 7,prop4 #e71 :5 ! moon appe>s to u
 halv$1 we c take ! plane ( ! great     
circle 9 ! moon :i* is p>allel to !     
divid+ circle z pass+ ?r\gh our eye4 7,x
 is tacitly assum$ 9 ,props4 #c1 #d1 &  
?r\gh\t1 t ! diamet]s ( ! sun & moon    
respectively subt5d ! same angle at our 
eye47 ,! pro( ( ,prop4 #d assumes z kn[n
 ! equival5t ( ! proposition 9          

trigonometry t1 if ea* ( ! angles 1  is 
n great] ?an a "r angle1 &   1 !n    tan
  _/ tan   _/  s9  _/ s9 4  ,prop4 #f   
proves t ! moon's orbit is 8l[]0 7i4e4  
small]7 ?an t ( ! sun1 & t1 :5 ! moon   
appe>s to u halv$1 x is di/ant less ?an 
a quadrant f ! sun4 ,prop4 #g is ! ma9  
proposition 9 ! treatise4 ,x proves t1  
on ! assumptions made1 ! di/ance ( ! sun
 f ! e>? is great] ?an eighte5 times1 b 
less ?an tw5ty times1 ! di/ance ( ! moon
 f ! e>?4 ,! pro( is simple & elegant & 
%\ld delight any ma!matician2 its two   
p>ts dep5d respectively on ! geometrical
 equival5ts ( ! two inequalities /at$ 9 
! =mula quot$ above1 namely1    tan  _/ 
tan   _/  s9  _/ s9 1  :]e 1  are angles
 n great] ?an a "r angle &   4 ,ari/>*us
 also1 9 ?is proposition1 cites #g_/e z 
an approximation by defect to ! value ( 
#b1 an approximation f\nd by !          
,py?agoreans & quot$ by ,plato4 ,!      
trigonometrical equival5t ( ! result    
obta9$ 9 ,prop4 #g is    #a_/ah  s9 #c  
#a_/bj4  ,prop4 #h /ates t1 :5 ! sun is 

totally eclips$1 ! sun & moon are       
compreh5d$ by "o & ! same cone :i* has  
its v]tex at our eye4 ,ari/>*us supports
 ?is by ! argum5ts 7#a7 t1 if ! sun     
ov]lapp$ ! moon1 x w\ld n be totally    
eclips$1 & 7#b7 t1 if ! sun fell %ort   
7i4e4 0 m ?an cov]$71 x w\ld rema9      
totally eclips$ = "s "t1 :i* x does n   
7?is1 he says1 is manife/ f obs]vation74
 ,x is cle> f ?is reason+ t ,ari/>*us _h
 n obs]v$ ! ph5om5on ( an annul> eclipse
 ( ! sun2 & x is curi\s t ! fir/ m5tion 
( an annul> eclipse seems to be t quot$ 
by ,simplicius f ,sosig5es 7second      
c5tury1 ,a4,d471 ! tea*] ( ,alex&]      
,aphrodisi5sis4  ,x foll[s 7,prop4 #i7 f
 ,prop4 #h t ! diamet]s ( ! sun & moon  
are 9 ! same ratio z _! di/ances f ! e>?
 respectively1 t is to say 7,prop4 #g7 9
 a ratio great] ?an #ah 3 #a b less ?an 
#bj 3 #a4 ,h5ce 7,prop4 #aj7 ! volume ( 
! sun is m ?an #ehcb times & less ?an   
#hjjj times t ( ! moon4  ,by ! usual    
geometrical sub/itute = trigonometry    
,ari/>*us proves 9 ,prop4 #aa t !       

diamet] ( ! moon has to ! di/ance betwe5
 ! c5tre ( ! moon & our eye a ratio :i* 
is less ?an #b_/de?s b great] ?an       
#a_/cj?4 ,s9ce ! angle subt5d$ by !     
moon's diamet] at ! obs]v]'s eye is     
assum$ to be #b1 ?is proposition is     
equival5t to ! trigonometrical =mula    
#a_/de  s9 #a  #a_/fj4  ,hav+ prov$ 9   
,prop4 #d t1 s f> z our p]ception goes1 
! divid+ circle 9 ! moon is             
indi/+ui%able f a great circle1         
,ari/>*us goes beh9d p]ception & proves 
9 ,prop4 #ab t ! diamet] ( ! divid+     
circle is less ?an ! diamet] ( ! moon b 
great] ?an #hi_/ij?s ( x4 ,?is is aga9  
because half ! angle subt5d$ by ! moon  
at ! eye is assum$ to be #a or #a_/ij? (
 a "r angle4 ,! proposition is equival5t
 to ! trigonometrical =mula    #a  cos  
#a  #hi_/ij4  ,we come n[ to            
propositions :i* dep5d on ,hypo!sis #e t
 8! bread? ( ! e>?'s %ad[ is t ( two    
moons04 ,prop4 #ac is ab\t ! diamet] ( !
 circul> section ( ! cone =m$ by ! e>?'s
 %ad[ at ! place :]e ! moon passes ?r\gh

 x 9 an eclipse1 & x is wor? :ile to    
notice ! extreme accuracy ) :i*         
,ari/>*us describes ! diamet] 9 "q4 ,x  
is ) him 8! /raight l9e subt5d+ !       
portion int]cept$ )9 ! e>?'s %ad[ ( !   
circumf]5ce ( ! circle 9 :i* !          
extremities ( ! diamet] ( ! circle      
divid+ ! d>k & ! bright portions 9 !    
moon move40 ,ari/>*us proves t ! l5g? ( 
! /raight l9e 9 "q has to ! diamet] ( ! 
moon a ratio less ?an #b b great] ?an   
#hh 3 #de1 & has to ! diamet] ( ! sun a 
ratio less ?an #a 3 #i b great] ?an #bb 
3 #bbe4 ,! ratio ( ! /raight l9e to !   
diamet] ( ! moon is1 9 po9t ( fact1 #b  
cos #a or #b s9 #hi1 & ,ari/>*us !re=e  
proves ! equival5t (    #b  #b cos #a   
7#hi_/de7 or #giba_/djej4  ,he !n       
obs]ves 7)\t explanation7 t #giba_/djej 
#hh_/de 7an approximation easily obta9$ 
by develop+ #giba_/djej z a cont9u$     
fraction 7 #a  7#a    #a    #a7_/7a  #ba
  #b7772 8 result is !re=e equival5t to 
#a  cos #a  #dd_/de4  ,! next           
propositions are ! equival5ts ( m       

complicat$ trigonometrical =mul4 ,prop4 
#ad is an auxili>y proposition to ,prop4
 #ae4 ,! diamet] ( ! %ad[ dealt ) 9     
,prop4 #ac divides into two p>ts !      
/raight l9e jo9+ ! c5tre ( ! e>? to !   
c5tre ( ! moon1 & ,prop4 #ad %[s t !    
:ole l5g? ( ?is l9e is m ?an #fge times 
! "p ( x t]m9at+ 9 ! c5tre ( ! moon4 ,) 
! aid ( ,props4 #g1 #ac1 & #ad ,ari/>*us
 is n[ able1 9 ,prop4 #ae1 to prove     
ano!r ( 8 ma9 results1 namely1 t !      
diamet] ( ! sun has to ! diamet] ( ! e>?
 a ratio great] ?an #ai 3 #c b less ?an 
#dc 3 #f4 ,9 ! second half ( ! pro( he  
has to h&le q l>ge numb]s4 ,if ,a be !  
c5tre ( ! sun1 ,b ! c5tre ( ! e>?1 & ,m 
! v]tex ( ! cone =m$ by ! e>?'s %ad[1 he
 proves t ,,ma 3 ,,ab is great] ?an     
7#ajabe  #gjhg7 3 7#iadf  #fgej7 or     
#gageehge 3 #fagceejj1 & !n adds1 )\t   
any ^w ( explanation1 t ! latt] ratio is
 great] ?an #dc 3 #cg4 ,"h aga9 x is    
difficult n to see 9 #dc 3 #cg ! cont9u$
 fraction #a  #aa_/7ff72 & al?\gh we _c 
suppose t ! ,greeks c\ld actually       

develop #gageehge_/fagceejj or          
#babfa_/ahbib z a cont9u$ fraction 79   
=m71 8we h "h an important pro( ( !     
employm5t by ! anci5ts ( a me?od (      
calculation1 ! !ory ( :i* unque/ionably 
belongs to ! mod]ns1 b ! fir/           
applications ( :i* are too simple n to h
 orig9at$ 9 v remote times0 7,paul      
,tann]y74  ,! rema9+ propositions conta9
 no m ?an ari?metical inf]5ces f !      
=ego+4 ,prop4 #af is to ! effect t !    
volume ( ! sun has to ! volume ( ! e>? a
 ratio great] ?an #fhei 3 #bg b less ?an
 #giejg 3 #baf 7! numb]s are ! cubes (  
^? 9 ,prop4 #ae72 ,prop4 #ag proves t ! 
diamet] ( ! e>? is to t ( ! moon 9 a    
ratio great] ?an #ajh 3 #dc b less ?an  
#fj 3 #ai 7ratios comp\nd$ ( ^? 9       
,props4 #i & #ae71 & ,prop4 #ah proves t
 ! volume ( ! e>? is to t ( ! moon 9 a  
ratio great] ?an #abeigab 3 #giejg b    
less ?an #bafjjj 3 #fhei4   ,,ari/>*us  
,,on ,,! ,,ye> ,,& 8,,great ,,ye>04     
,ari/>*us is said to h increas$ by      
#a_/afbcrd ( a day ,callippus's figure (

 #cfe days z ! l5g? ( ! sol> ye>1 & to h
 giv5 #bdhd ye>s z ! l5g? ( ! ,great    
,ye> or ! p]iod aft] :i* ! sun1 ! moon &
 ! five planets return to ! same        
position 9 ! heav5s4 ,tann]y has %[n    
reason = ?9k+ t #bdhd is a wrong read+ =
 #bdcd ye>s1 & he gives an explanation  
:i* seems conv9c+ ( ! way 9 :i*         
,ari/>*us arriv$ at #bdcd ye>s z ! l5g? 
( ! ,great ,ye>4 ,! ,*aldan p]iod ( #bbc
 lunations 0 well kn[n 9 ,greece4 ,its  
l5g? 0 calculat$ to be #fehe days1 & 9  
?is p]iod ! sun 0 e/imat$ to describe   
#aj ( its circle 9 addition to #ah      
sid]eal revolutions4 ,! ,greeks us$ !   
p]iod call$ by !m exeligmus :i* 0 ?ree  
times ! p]iod ( #bbc lunations & conta9$
 a :ole numb] ( days1 namely1 #aigef1   
dur+ :i* ! sun describ$ #cb 9 addition  
to #ed sid]eal revolutions4 ,x foll[$ t 
! numb] ( days 9 ! sid]eal ye> 0--      
#aigef_/7ed  #cb_/cfj7  #aigef_/7ed     
#d_/de7  7#de  #aigef7_/bdcd            
#hhijbj_/bdcd  #cfe  #c_/dhfh4  ,n[     
#dhfh_/c  #afbc - 1 & ,ari/>*us seems to

 h m]ely replac$ #c_/dhfh by ! close    
approximation #a_/afbc4 ,! calculation  
01 h[ev]1 ( no value because ! e/imate (
 #aj ov] #ah sid]eal revolutions seems  
to h be5 an approximation bas$ m]ely on 
! diff]5ce betwe5 #fehe days & #ah ye>s 
( #cfe days1 i4e4 #fegd days2 ?us ! #aj 
itself probably dep5d$ on a sol> ye> (  
#cfe days1 & ,ari/>*us's evaluation ( x 
z #cfe #a_/afbc 0 really a sort (       
circul> argum5t l ! simil> calculation (
 ! l5g? ( ! ye> made by nopides ( ,*ios4
   ,,lat] ,,improvem5ts ,,on            
,,ari/>*us',s ,,figures4  ,x may int]e/ 
! read] to "k h[ f> ,ari/>*us's e/imates
 ( sizes & di/ances 7 improv$ ^u by lat]
 ,greek a/ronom]s4 ,we are n in=m$ h[   
l>ge he conceiv$ ! e>? to be2 b         
,ar*im$es tells u t 8"s h tri$ to prove 
t ! circumf]5ce ( ! e>? is ab\t #cjj1jjj
 /ades & n great]10 & x may be presum$ t
 ,ari/>*us w\ld1 l ,ar*im$es1 be cont5t 
) ?is e/imate4 ,x is probable t x 0     
,dicae>*us :o 7ab\t #cjj ,b4,c47 arriv$ 
at ?is value1 & t x 0 obta9$ by tak+ #bd

 7#a_/ae? ( ! :ole m]idian circle7 z !  
diff]5ce ( latitude betwe5 ,sy5e &      
,lysima*ia 7on ! same m]idian7 & #bj1jjj
 /ades z ! actual di/ance betwe5 ! two  
places4 ,erato/h5es1 born a few ye>s    
aft] ,ar*im$es1 say #bhd ,b4,c41 is     
fam\s = a bett] measurem5t ( ! e>? :i* 0
 bas$ on sci5tific pr9ciples4 ,he f\nd t
 at noon at ! summ] sol/ice ! sun ?rew  
no %ad[ at ,sy5e1 :]eas at ! same h\r at
 ,alex&ria 7:i* he took to be on ! same 
m]idian7 a v]tical /ick ca/ a %ad[      
correspond+ to #a_/ej? ( ! m]idian      
circle4 ,assum+ !n t ! sun's rays at !  
two places are p>allel 9 direction1 &   
kn[+ ! di/ance betwe5 !m to be #ejjj    
/ades1 he _h only to take #ej times     
#ejjj /ades to get ! circumf]5ce ( !    
e>?4 ,he seems1 = "s reason1 to h alt]$ 
#bej1jjj into #beb1jjj /ades1 & ?is1    
accord+ to ,pl9y's acc\nt ( ! k9d ( /ade
 us$1 works out to ab\t #bd1ffb miles1  
giv+ = ! diamet] ( ! e>? a l5g? ( #ghej 
miles1 a surpris+ly close approximation1
 h[ev] mu* x owes to happy accid5ts 9 ! 

calculation4  ,erato/h5es's e/imates ( !
 sizes & di/ances ( ! sun & moon _c be  
re/or$ ) c]ta9ty 9 view ( ! defective   
/ate ( ! texts ( our au?orities4 ,we are
 bett] in=m$ ( ,hipp>*us's results4 ,9 !
 fir/ book ( a treatise on sizes &      
di/ances ,hipp>*us bas$ himself on an   
obs]vation ( an eclipse ( ! sun1        
probably t ( #bj? ,novemb] 9 ! ye> #abi 
,b4,c41 :i* 0 exactly total 9 ! region  
ab\t ! ,hellespont1 :]eas at ,alex&ria  
ab\t ?s only ( ! diamet] 0 obscur$4 ,f  
^! facts ,hipp>*us d$uc$ t1 if ! radius 
( ! e>? be ! unit1 ! lea/ di/ance ( !   
moon conta9s #ga1 & ! greate/ #hc ( ^!  
units1 ! mean ?us conta9+ #gg4 ,b he    
rev]t$ to ! "q 9 ! second book & prov$  
8f _m consid]ations0 t ! mean di/ance ( 
! moon is #fg times ! radius ( ! e>?1 & 
also t ! di/ance ( ! sun is #bdij times 
! radius ( ! e>?4 ,hipp>*us also made ! 
size 7mean+ !reby ! solid cont5t7 ( !   
sun to be #ahhj times t ( ! e>?1 & !    
size ( ! e>? to be #bg times t ( ! moon4
 ,! cube root ( #ahhj be+ ab\t #ab1 !   

diamet]s ( ! sun1 e>? & moon w\ld be 9 !
 ratio ( ! numb]s #ab1 #a1 4 ,hipp>*us  
seems to h accept$ ,erato/h5es's e/imate
 ( #beb1jjj /ades = ! circumf]5ce ( !   
e>?4  ,x is curi\s t ,posidonius        
7#ace-ea ,b4,c471 :o 0 mu* less ( an    
a/ronom]1 made a mu* bett] guess at !   
di/ance ( ! sun f ! e>?4 ,he made x     
#ejj1jjj1jjj /ades4 ,z he also e/imat$ !
 circumf]5ce ( ! e>? at #bdj1jjj /ades1 
we may take ! diamet] ( ! e>? to be1    
accord+ to ,posidonius1 ab\t #gf1djj    
/ades2 consequ5tly1 if ,d be t diamet]1 
,posidonius made ! di/ance ( ! sun to be
 equal to #fede;,d z comp>$ )           
,hipp>*us's #abde;,d4  ,ptolemy does n  
m5tion ,hipp>*us's figures4 ,8 own      
e/imate ( ! sun's di/ance 0 #fje;,d1 s t
 ,hipp>*us 0 f> ne>] ! tru?4 ,b         
,hipp>*us's e/imate rema9$ unkn[n &     
,ptolemy's held ! field = _m c5turies2  
ev5 ,cop]nicus only made ! di/ance ( !  
sun #gej times ! e>?'s diamet]1 & x 0 n 
till #afga-c t a sub/antial improvem5t 0
 made2 obs]vations ( ,m>s c>ri$ out 9 ^?

 ye>s by ,ri*] enabl$ ,cass9i to        
conclude t ! sun's p>allax 0 ab\t #ie   
correspond+ to a di/ance betwe5 ! sun & 
! e>? ( #hg1jjj1jjj miles4  ,ptolemy    
made ! di/ance ( ! moon f ! e>? to be   
#bi times ! e>?'s diamet]1 & ! diamet] (
 ! e>? to be #c times t ( ! moon4 ,he   
e/imat$ ! diamet] ( ! sun at #ah times t
 ( ! moon & !re=e ab\t #e times t ( !   
e>?1 a figure aga9 mu* inf]ior to t giv5
 by ,hipp>*us4     ,,bibliography4   ,on
 ! a/ronomy ( ! e>ly ,greek philosoph]s 
mu* in=mation is giv5 by ,ari/otle      
7especially 9 ! ,de ,caelo72 = ,ari/otle
 0 =tunately 9 ! habit ( /at+ ! views ( 
e>li] ?9k]s z a prelim9>y to enunciat+ 8
 own4 ,ap>t f :at we le>n f ,ari/otle1  
we are ma9ly dep5d5t on ! fragm5t>y     
acc\nts ( ! op9ions ( philosoph]s :i* 7 
collect$ 9 ! ,doxographi ,graeci (      
,h]mann ,diels 7,b]l91 #ahgi71 to :i*   
mu/ be add$1 = ! p]iod be=e ,socrates1  
,die ,fragm5te d] ,vorsokratik] by !    
same editor 7#bnd edition1 ) index1     
#aijf-aj1 #crd edition1 #aiab74 ,!      

doxographic data & ! fragm5ts = ! p]iod 
f ,?ales to ,emp$ocles 7 translat$ &    
expla9$ by ,paul ,tann]y 9 ,p\r         
l'hi/oire de la ,sci5ce ,hellne 7,p>is1 
#ahhg74 ,a ,hi/ory ( ,a/ronomy 0 writt5 
by ,eudemus ( ,rhodes1 a pupil (        
,ari/otle2 ?is is lo/1 b a fair numb] ( 
fragm5ts are pres]v$ by lat] writ]s4 ,( 
! !ory ( conc5tric sph]es we h a %ort   
acc\nt 9 ,ari/otle's ,metaphysics1 b a m
 elaborate & detail$ description is     
conta9$ 9 ,simplicius's comm5t>y on !   
,de ,caelo2 ,simplicius quotes l>gely f 
,sosig5es ! ,p]ipatetic 7second c5tury  
,a4,d47 :o drew ^u ,eudemus4  ,!re are a
 numb] ( valuable hi/ories ( ,greek     
a/ronomy4 ,9 ,g]man we h ,s*auba*1      
,ges*i*te d] grie*is*5 ,a/ronomie bis   
auf ,erato/h5es1 #ahji2 ,r4 ,wolf1      
,ges*i*te d] ,a/ronomie1 #ahgg2 & two   
admirable epitomes 7#a7 by ,siegmund    
,gn!r 9 ,w9delb&'s ,ges*i*te d] alt5    
,philosophie 7,iwan von ,mll]'s ,h&bu*  
d] klassis*5 ,alt]tumswiss5s*aft1 ,vol4 
,v1 ,pt4 #a71 #bnd edition1 #ahid1 &    

7#b7 by ,fri$ri* ,hults* 9              
,pauly-,wiss[a's ,real-,encyclopdie d]  
classis*5 ,alt]tumswiss5s*aft 7,art4    
,a/ronomie 9 ,vol4 ,,ii1 #b1 #ahif74  ,9
 ,fr5*1 besides ! great "w ( ,delambre1 
,hi/oire de l'a/ronomie anci5ne1 #ahag1 
we h ! valuable /udies ( ,paul ,tann]y 9
 ,re*]*es sur l'hi/oire de l'a/ronomie  
anci5ne1 #ahic1 & ,pi]re ,duhem1 ,le    
,sy/me du ,monde1 ,vol4 ,i1 #aiac4  ,9  
,engli%1 ref]5ce may be made to ,sir ,g4
 ,cornewall ,lewis1 ,an ,hi/orical      
,survey ( ! ,a/ronomy ( ! ,anci5ts1     
#ahfc2 ,j4 ,l4 ,e4 ,drey]1 ,hi/ory ( !  
,planet>y ,sy/ems f ,?ales to ,kepl]1   
,cambridge1 #aijf2 & ! hi/orical portion
 ( ,sir ,?omas ,hea?'s ,ari/>*us (      
,samos1 ! anci5t ,cop]nicus1 ,ox=d1     
#aiac4  ,ari/>*us's treatise ,on !      
,sizes & ,di/ances ( ! ,sun & ,moon fir/
 appe>$ 9 a ,lat9 translation by ,george
 ,valla 9 #adhh & #adih1 & next 9 a     
,lat9 translation by ,comm&9us 7#aegb74 
,! editio pr9ceps ( ! ,greek text 0     
br\ght out by ,john ,wallis1 ,ox=d1     

#afhh1 & 0 repr9t$ 9 ,johannis ,wallis  
,op]a ,ma!matica1 #afic-afii1 ,vol4     
,,iii1 9 bo? cases along ) ,comm&9us's  
translation4 ,9 #ahaj !re appe>$ an     
edition by ! ,comte de ,=tia d',urban1  
,hi/oire d',ari/>que de ,samos 444      
includ+ ! ,greek text & ,comm&9us's     
translation b )\t figures2 a ,fr5*      
translation by ,=tia d',urban foll[$ 9  
#ahbc4 ,! treatise 0 translat$ into     
,g]man by ,a4 ,nokk 9 #ahed4 ,f9ally1   
,sir ,?omas ,hea?'s "w above cit$       
conta9s a new ,greek text ) ,engli%     
translation & notes4     ,,*ronology4   
7,approximate :]e precise dates are n   
kn[n47               ,b4,c4             
#fbd-edg       ,?ales4                  
#faj-edf       ,anaxim&]4               
#ehe-ebf       ,anaxim5es4              
#egb-dig       ,py?agoras4   born #eaf  
or #ead       ,p>m5ides4   7possibly    
#edj74           #ejj-dbh               
,anaxagoras4           #did-dcd         
,emp$ocles4       #e? c5tury            
nopides ( ,*ios4                        

,philolaus4           #dbg-cdg          
,plato4           #djh-cee              
,eudoxus4           #chh-cae            
,h]aclides ( ,pontus4           #chd-cbb
       ,ari/otle4           #cgj-cjj    
,callippus4           #caj-bcj          
,ari/>*us ( ,samos4           #bhg-bab  
,ar*im$es4           #bhd-bjc           
,erato/h5es4           #bfe-aij         
,apollonius ( ,p]ga4       #crd c5tury  
,aratus4           fl4 #aej             
,hipp>*us4            #ace-ea           
,posidonius4             ,a4,d4         
#ej-abe       ,plut>*4                  
#ajj-agh       ,ptolemy4   ,,ab]de53 ,,!
 ,,univ]sity ,,press     ,transcrib]'s  
,notes   ,punctuation & spell+ 7 made   
consi/5t :5 a pr$om9ant pref]5ce 0 f\nd 
9 ?is book2 o!rwise !y 7 n *ang$4       
,simple typographical errors 7 correct$4
 ,"o unbalanc$ p>5!sis 0 rem$i$4        
,italics are indicat$ by und]scores4