,,pioneers ,,of ,,progress    ,,men ,,of
 ,,science   ,,edited ,,by ,s4          
,,chapman1 ,m4,a41 ,d4,sc41 ,f4,r4,s4   
,,the   ,,copernicus ,,of ,,antiquity   
7,,aristarchus ,,of ,,samos7     ,,by   
,,sir ,,thomas ,,heath   ,k4,c4,b41     
,k4,c4,v4,o41 ,f4,r4,s42 ,,sc4,d41      
,,camb42 ,,hon4 ,d4,,sc41 ,,oxford      
,,london3   ,,society ,,for ,,promoting 
,,christian ,,knowledge   ,,new ,,york3 
,,the ,,macmillan ,,company   #aibj     
,,contents     ,,part ,i   ,,greek      
,,astronomy ,,to ,,aristarchus          
,,page   ,,thales                       
#f    ,,anaximander                     
#aj    ,,anaximenes                     
#ac    ,,pythagoras                     
#ad    ,,parmenides                     
#af    ,,anaxagoras                     
#ah    ,,empedocles                     
#ba    ,,the ,,pythagoreans             
#bb    ,,nopides ,,of ,,chios           
#bd    ,,plato                          
#be    ,,eudoxus1 ,,callippus1          
,,aristotle                             

#bh    ,,heraclides ,,of ,,pontus       
#cc     ,,part ,,ii   ,,aristarchus ,,of
 ,,samos    ,,the ,,heliocentric        
,,hypothesis                            
#ci    ,,on ,,the ,,apparent ,,diameter 
,,of ,,the ,,sun                        
#db    ,,on ,,the ,,sizes ,,and         
,,distances ,,of ,,the ,,sun ,,and      
,,moon            #dc    ,,on ,,the     
,,year ,,and 8,,great ,,year0           
#ec    ,,later ,,improvements ,,on      
,,aristarchus',s ,,figures              
#ed     ,,bibliography                  
#eg    ,,chronology                     
#ei   ,illustration3 ,,copernicus',s    
,,diagram ,,of ,,his ,,system           
7,,anticipated ,,by ,,aristarchus74     
,,part ,i4  ,,greek ,,astronomy ,,to    
,,aristarchus4   ,the title-page of this
 book necessarily bears the name of one 
man2 but the reader will find in its    
pages the story1 or part of the story1  
of many other ,pioneers of ,progress4   
,the crowning achievement of            
anticipating the hypothesis of          

,copernicus belongs to ,aristarchus of  
,samos alone2 but to see it in its      
proper setting it is necessary to have  
followed in the footsteps of the earlier
 pioneers who1 by one bold speculation  
after another1 brought the solution of  
the problem nearer1 though no one before
 ,aristarchus actually hit upon the     
truth4 ,this is why the writer has      
thought it useful to prefix to his      
account of ,aristarchus a short sketch  
of the history of the development of    
astronomy in ,greece down to            
,aristarchus's time1 which is indeed the
 most fascinating portion of the story  
of ,greek astronomy4  ,the extraordinary
 advance in astronomy made by the       
,greeks in a period of little more than 
three centuries is a worthy parallel to 
the rapid development1 in their hands1  
of pure geometry1 which1 created by them
 as a theoretical science about the same
 time1 had by the time of ,aristarchus  
covered the ground of the ,elements     
7including solid geometry and the       

geometry of the sphere71 had established
 the main properties of the three conic 
sections1 had solved problems which were
 beyond the geometry of the straight    
line and circle1 and finally1 before the
 end of the third century ,b4,c41 had   
been carried to its highest perfection  
by the genius of ,archimedes1 who       
measured the areas of curves and the    
surfaces and volumes of curved surfaces 
by geometrical methods practically      
anticipating the integral calculus4  ,to
 understand how all this was possible we
 have to remember that the ,greeks1     
pre-eminently among all the nations of  
the world1 possessed just those gifts   
which are essential to the initiation   
and development of philosophy and       
science4 ,they had in the first place a 
remarkable power of accurate            
observation2 and to this were added     
clearness of intellect to see things as 
they are1 a passionate love of knowledge
 for its own sake1 and a genius for     
speculation which stands unrivalled to  

this day4 ,nothing that is perceptible  
to the senses seems to have escaped     
them2 and when the apparent facts had   
been accurately ascertained1 they wanted
 to know the why and the wherefore1     
never resting satisfied until they had  
given a rational explanation1 or what   
seemed to them to be such1 of the       
phenomena observed4 ,observation or     
experiment and theory went hand in hand4
 ,so it was that they developed such    
subjects as medicine and astronomy4 ,in 
astronomy their guiding principle was1  
in their own expressive words1 to 8save 
the phenomena04 ,this meant that1 as    
more and more facts became known1 their 
theories were continually revised to fit
 them4  ,it would be easy to multiply   
instances2 it must suffice in this place
 to mention one1 which illustrates not  
only the certainty with which the       
,greeks detected the occurrence of even 
the rarest phenomena1 but also the      
persistence with which they sought for  
the true explanation4  ,cleomedes       

7second century ,a4,d47 mentions that   
there were stories of extraordinary     
eclipses which 8the more ancient of the 
mathematicians0 had vainly tried to     
explain2 the supposed 8paradoxical0 case
 was that in which1 while the sun seems 
to be still above the western horizon1  
the eclipsed moon is seen to rise in the
 east4 ,the phenomenon appeared to be   
inconsistent with the explanation of    
lunar eclipses by the entry of the moon 
into the earth's shadow2 how could this 
be if both bodies were above the horizon
 at the same time8 ,the 8more ancient0  
mathematicians essayed a geometrical    
explanation2 they tried to argue that it
 was possible that a spectator standing 
on an eminence of the spherical earth   
might see along the generators of a cone
 i4e4 a little downwards on all sides   
instead of merely in the plane of the   
horizon1 and so might see both the sun  
and the moon even when the latter was in
 the earth's shadow4 ,cleomedes denies  
this and prefers to regard the whole    

story of such cases as a fiction        
designed merely for the purpose of      
plaguing astronomers and philosophers2  
no ,chaldan1 he says1 no ,egyptian1 and 
no mathematician or philosopher has     
recorded such a case4 ,but the          
phenomenon is possible1 and it is       
certain that it had been observed in    
,greece and that the ,greek astronomers 
did not rest until they had found out   
the solution of the puzzle2 for         
,cleomedes himself gives the            
explanation1 namely that the phenomenon 
is due to atmospheric refraction4       
,observing that such cases of           
atmospheric refraction were especially  
noticeable in the neighbourhood of the  
,black ,sea1 ,cleomedes goes on to say  
that it is possible that the visual rays
 going out from our eyes are refracted  
through falling on wet and damp air1 and
 so reach the sun although it is already
 below the horizon2 and he compares the 
well-known experiment of the ring at the
 bottom of a jug1 where the ring1 just  

out of sight when the jug is empty1 is  
brought into view when water is poured  
in4  ,the genius of the race being what 
it was1 the ,greeks must from the       
earliest times have been in the habit of
 scanning the heavens1 and1 as might be 
expected1 we find the beginnings of     
astronomical knowledge in the earliest  
,greek literature4  ,in the ,homeric    
poems and in ,hesiod the earth is a flat
 circular disc2 round this disc runs the
 river ,oceanus1 encircling the earth   
and flowing back into itself4 ,the flat 
earth has above it the vault of heaven1 
like a sort of hemispherical dome       
exactly covering it2 this vault remains 
for ever in one position2 the sun1 moon 
and stars move round under it1 rising   
from ,oceanus in the east and plunging  
into it again in the west4  ,homer      
mentions1 in addition to the sun and    
moon1 the ,morning ,star1 the ,evening  
,star1 the ,pleiades1 the ,hyades1      
,orion1 the ,great ,bear 78which is also
 called by the name of the ,wain071     

,sirius1 the late-setting ,botes 7the   
ploughman driving the ,wain71 i4e4      
,arcturus1 as it was first called by    
,hesiod4 ,of the ,great ,bear ,homer    
says that it turns round on the same    
spot and watches ,orion2 it alone is    
without lot in ,oceanus's bath 7i4e4 it 
never sets74 ,with regard to the last   
statement it is to be noted that some of
 the principal stars of the ,great ,bear
 do now set in the ,mediterranean1 e4g4 
in places further south than ,rhodes    
7lat4 #cf71 1 the hind foot1 and 1 the  
tip of the tail1 and at ,alexandria all 
the seven stars except 1 the head4 ,it  
might be supposed that here was a case  
of ,homer 8nodding04 ,but no2 the old   
poet was perfectly right2 the difference
 between the facts as observed by him   
and as seen by us respectively is due to
 the ,precession of the ,equinoxes1 the 
gradual movement of the fixed stars     
themselves about the pole of the        
ecliptic1 which was discovered by       
,hipparchus 7second century ,b4,c474 ,we

 know from the original writings of the 
,greek astronomers that in ,eudoxus's   
time 7say #chj ,b4,c47 the whole of the 
,great ,bear remained always well above 
the horizon1 while in the time of       
,proclus 7say ,a4,d4 #dfj7 the ,great   
,bear 8grazed0 the horizon4  ,in ,homer 
astronomical phenomena are only vaguely 
used for such purposes as fixing        
localities or marking times of day or   
night4 ,sometimes constellations are    
used in giving sailing directions1 as   
when ,calypso directs ,odysseus to sail 
in such a way as always to keep the     
,great ,bear on his left4  ,hesiod      
mentions practically the same stars as  
,homer1 but makes more use of celestial 
phenomena for determining times and     
seasons4 ,for example1 he marked the    
time for sowing at the beginning of     
winter by the setting of the ,pleiades  
in the early twilight1 or again by the  
early setting of the ,hyades or ,orion1 
which means the #crd1 #gth1 or #aeth    
,november in the ,julian calendar       

according to the particular stars taken2
 the time for harvest he fixed by the   
early rising of the ,pleiades 7#aith    
,may71 threshing time by the early      
rising of ,orion 7#ith ,july71 vintage  
time by the early rising of ,arcturus   
7#ahth ,september71 and so on4 ,hesiod  
makes spring begin sixty days after the 
winter solstice1 and the early summer   
fifty days after the summer solstice4   
,thus he knew about the solstices1      
though he says nothing of the equinoxes4
 ,he had an approximate notion of the   
moon's period1 which he put at thirty   
days4  ,but this use of astronomical    
facts for the purpose of determining    
times and seasons or deducing weather   
indications is a very different thing   
from the science of astronomy1 which    
seeks to explain the heavenly phenomena 
and their causes4 ,the history of this  
science1 as of ,greek philosophy in     
general1 begins with ,thales4  ,the     
,ionian ,greeks were in the most        
favourable position for initiating      

philosophy4 ,foremost among the ,greeks 
in the love of adventure and the        
instinct of new discovery 7as is shown  
by their leaving their homes to found   
settlements in distant lands71 and      
fired1 like all ,greeks1 with a passion 
for knowledge1 they needed little       
impulse to set them on the road of      
independent thought and speculation4    
,this impulse was furnished by their    
contact with two ancient civilisations1 
the ,egyptian and the ,babylonian4      
,acquiring from them certain elementary 
facts and rules in mathematics and      
astronomy which had been handed down    
through the priesthood from remote      
antiquity1 they built upon them the     
foundation of the science1 as distinct  
from the mere routine1 of the subjects  
in question4   ,,thales4  ,thales of    
,miletus 7about #fbd-edg ,b4,c47 was a  
man of extraordinary versatility2       
philosopher1 mathematician1 astronomer1 
statesman1 engineer1 and man of         
business1 he was declared one of the    

,seven ,wise ,men in #ehb-eha ,b4,c4    
,his propensity to star-gazing is       
attested by the story of his having     
fallen into a well while watching the   
stars1 insomuch that 7as ,plato has it7 
he was rallied by a clever and pretty   
maidservant from ,thrace for being so   
8eager to know what goes on in the      
heavens when he could not see what was  
in front of him1 nay at his very feet04 
,thales's claim to a place in the       
history of scientific astronomy rests on
 one achievement attributed to him1 that
 of predicting an eclipse of the sun4   
,the evidence for this is fairly        
conclusive1 though the accounts of it   
differ slightly4 ,eudemus1 the pupil of 
,aristotle1 who wrote histories of      
,greek geometry and astronomy1 is quoted
 by three different ,greek writers as   
the authority for the story4 ,but there 
is testimony much earlier than this4    
,herodotus1 speaking of a war between   
the ,lydians and the ,medes1 says that1 
8when in the sixth year they encountered

 one another1 it fell out that1 after   
they had joined battle1 the day suddenly
 turned into night4 ,now that this      
change of day into night would occur was
 foretold to the ,ionians by ,thales of 
,miletus1 who fixed as the limit of time
 this very year in which the change took
 place40 ,moreover ,xenophanes1 who was 
born some twenty-three years before     
,thales's death1 is said to have lauded 
,thales's achievement2 this would amount
 to almost contemporary evidence4       
,could ,thales have known the cause of  
solar eclipses8 ,atius 7,a4,d4 #ajj71   
the author of an epitome of an older    
collection of the opinions of           
philosophers1 says that ,thales was the 
first to declare that the sun is        
eclipsed when the moon comes in a direct
 line below it1 the image of the moon   
then appearing on the sun's disc as on a
 mirror2 he also associates ,thales with
 ,anaxagoras1 ,plato1 ,aristotle1 and   
the ,stoics as holding that the moon is 
eclipsed by reason of its falling into  

the shadow made by the earth when the   
earth is between the sun and the moon4  
,but1 as regards the eclipse of the     
moon1 ,thales could not have given this 
explanation1 because he held that the   
earth 7which he presumably regarded as a
 flat disc7 floated on the water like a 
log4 ,and if he had given the true      
explanation of a solar eclipse1 it is   
impossible that all the succeeding      
,ionian philosophers should have        
exhausted their imaginations in other   
fanciful explanations such as we find   
recorded4  ,the key to the puzzle may be
 afforded by the passage of ,herodotus  
according to which the prediction was a 
rough one1 only specifying that the     
eclipse would occur within a certain    
year4 ,the prediction was probably one  
of the same kind as had long been made  
by the ,chaldans4 ,the ,chaldans1 no    
doubt as the result of observations     
continued through many centuries1 had   
discovered the period of #bbc lunations 
after which lunar eclipses recur4 7,this

 method would very often fail for solar 
eclipses because no account was taken of
 parallax2 and ,assyrian cuneiform      
inscriptions record failures as well as 
successful predictions47 ,thales1 then1 
probably learnt about the period of #bbc
 lunations either in ,egypt or more     
directly through ,lydia1 which was an   
outpost of ,assyrio-,babylonian culture4
 ,if there happened to be a number of   
possible solar eclipses in the year     
which 7according to ,herodotus7 ,thales 
fixed for the eclipse1 he was1 in using 
the ,chaldan rule1 not taking an undue  
risk2 but it was great luck that the    
eclipse should have been total4 ,it     
seems practically certain that the      
eclipse in question was that of the     
7,julian7 #bhth ,may1 #ehe4  ,thales1 as
 we have seen1 made the earth a circular
 or cylindrical disc floating on the    
water like a log or a cork and1 so far  
as we can judge of his general          
conception of the universe1 he would    
appear to have regarded it as a mass of 

water 7that on which the earth floats7  
with the heavens encircling it in the   
form of a hemisphere and also bounded by
 the primeval water4 ,this view of the  
world has been compared with that found 
in ancient ,egyptian papyri4 ,in the    
beginning existed the ,n1 a primordial  
liquid mass in the limitless depths of  
which floated the germs of things4 ,when
 the sun began to shine1 the earth was  
flattened out and the water separated   
into two masses4 ,the one gave rise to  
the rivers and the ocean1 the other1    
suspended above1 formed the vault of    
heaven1 the waters above1 on which the  
stars and the gods1 borne by an eternal 
current1 began to float4 ,the sun1      
standing upright in his sacred barque   
which had endured for millions of years1
 glides slowly1 conducted by an army of 
secondary gods1 the planets and the     
fixed stars4 ,the assumption of an upper
 and lower ocean is also old ,babylonian
 7cf4 the division in ,genesis #a4 #g of
 the waters which were under the        

firmament from the waters which were    
above the firmament74  ,it would follow 
from ,thales's general view of the      
universe that the sun1 moon and stars   
did not1 between their setting and      
rising again1 continue their circular   
path below the earth but 7as with       
,anaximenes later7 moved laterally round
 the earth4  ,thales's further          
contributions to observational astronomy
 may be shortly stated4 ,he wrote two   
works ,on the solstice and ,on the      
equinox1 and he is said by ,eudemus to  
have discovered that 8the period of the 
sun with respect to the solstices is not
 always the same10 which probably means 
that he discovered the inequality of the
 four astronomical seasons4 ,his        
division of the year into #cfe days he  
probably learnt from the ,egyptians4 ,he
 said of the ,hyades that there are two1
 one north and the other south4 ,he     
observed the ,little ,bear and used it  
as a means of finding the pole2 he      
advised the ,greeks to follow the       

,phnician plan of sailing by the ,little
 ,bear in preference to their own habit 
of steering by the ,great ,bear4        
,limited as the certain contributions of
 ,thales to astronomy are1 it became the
 habit of the ,greek ,doxographi1 or    
retailers of the opinions of            
philosophers1 to attribute to ,thales1  
in common with other astronomers in each
 case1 a number of discoveries which    
were not made till later4 ,the following
 is a list1 with 7in brackets7 the names
 of the astronomers to whom the         
respective discoveries may with most    
certainty be assigned3 7#a7 the fact    
that the moon takes its light from the  
sun 7,anaxagoras71 7#b7 the sphericity  
of the earth 7,pythagoras71 7#c7 the    
division of the heavenly sphere into    
five zones 7,pythagoras and             
,parmenides71 7#d7 the obliquity of the 
ecliptic 7nopides of ,chios71 and 7#e7  
the estimate of the sun's apparent      
diameter as #a_/gbjth of the sun's      
circle 7,aristarchus of ,samos74        

,,anaximander4  ,anaximander 7about     
#faa-edg ,b4,c471 a contemporary and    
fellow-citizen of ,thales1 was a        
remarkably original thinker4 ,he was the
 first ,greek philosopher who ventured  
to put forward his views in a formal    
written treatise4 ,this was a work      
,about ,nature and was not given to the 
world till he was about sixty-four years
 old4 ,his originality is illustrated by
 his theory of evolution4 ,according to 
him animals first arose from slime      
evaporated by the sun2 they lived in the
 sea and had prickly coverings2 men at  
first resembled fishes4  ,but his       
astronomical views were not less        
remarkable4 ,anaximander boldly         
maintained that the earth is in the     
centre of the universe1 suspended freely
 and without support1 whereas ,thales   
regarded it as resting on the water and 
,anaximenes as supported by the air4 ,it
 remains in its position1 said          
,anaximander1 because it is at an equal 
distance from all the rest of the       

heavenly bodies4 ,the earth was1        
according to him1 cylinder-shaped1 round
 8like a stone pillar02 one of its two  
plane faces is that on which we stand2  
its depth is one-third of its breadth4  
,anaximander postulated as his first    
principle1 not water 7like ,thales7 or  
any of the elements1 but the ,infinite2 
this was a substance1 not further       
defined1 from which all the heavens and 
the worlds in them were produced2       
according to him the worlds themselves  
were infinite in number1 and there were 
always some worlds coming into being and
 others passing away ad infinitum4 ,the 
origin of the stars1 and their nature1  
he explained as follows4 8,that which is
 capable of begetting the hot and the   
cold out of the eternal was separated   
off during the coming into being of our 
world1 and from the flame thus produced 
a sort of sphere was made which grew    
round the air about the earth as the    
bark round the tree2 then this sphere   
was torn off and became enclosed in     

certain circles or rings1 and thus were 
formed the sun1 the moon and the stars40
 8,the stars are produced as a circle of
 fire1 separated off from the fire in   
the universe and enclosed by air4 ,they 
have as vents certain pipe-shaped       
passages at which the stars are seen40  
8,the stars are compressed portions of  
air1 in the shape of wheels filled with 
fire1 and they emit flames at some point
 from small openings40 8,the stars are  
borne round by the circles in which they
 are enclosed40 8,the sun is a circle   
twenty-eight times 7v4 l4 #bg times7 the
 size of the earth2 it is like a wheel  
of a chariot the rim of which is hollow 
and full of fire and lets the fire shine
 out at a certain point in it through an
 opening like the tube of a blow-pipe2  
such is the sun40 8,the sun is equal to 
the earth40 8,the eclipses of the sun   
occur through the opening by which the  
fire finds vent being shut up40 8,the   
moon is a circle nineteen times the size
 of the earth2 it is similar to a       

chariot-wheel the rim of which is hollow
 and full of fire like the circle of the
 sun1 and it is placed obliquely like   
the other2 it has one vent like the tube
 of a blow-pipe2 the eclipses of the    
moon depend on the turnings of the      
wheel40 8,the moon is eclipsed when the 
opening in the rim of the wheel is      
stopped up40 8,the moon appears         
sometimes as waxing1 sometimes as       
waning1 to an extent corresponding to   
the closing or opening of the passages40
 8,the sun is placed highest of all1    
after it the moon1 and under them the   
fixed stars and the planets40  ,it has  
been pointed out that the idea of the   
formation of tubes of compressed air    
within which the fire of each star is   
shut up except for the one opening      
through which the flame shows 7like a   
gas-jet1 as it were7 is not unlike      
,laplace's hypothesis with reference to 
the origin of ,saturn's rings4 ,in any  
case it is a sufficiently original      
conception4  ,when ,anaximander says    

that the hoops carrying the sun and moon
 8lie obliquely10 this is no doubt an   
attempt to explain1 in addition to the  
daily rotation1 the annual movement of  
the sun and the monthly movement of the 
moon4  ,we have here too the first      
speculation about the sizes and         
distances of the heavenly bodies4 ,the  
sun is as large as the earth4 ,the      
ambiguity between the estimates of the  
size of the sun's circle as twenty-seven
 or twenty-eight times the size of the  
earth suggests that it is a question    
between taking the inner and outer      
circumferences of the sun's ring        
respectively1 and a similar ambiguity   
may account for the circle of the moon  
being stated to be nineteen times1 not  
eighteen times1 the size of the earth4  
,no estimate is given of the distance of
 the planets from the earth1 but as1    
according to ,anaximander1 they are     
nearer to the earth than the sun and    
moon are1 it is possible that1 if a     
figure had been stated1 it would have   

been nine times the size of the earth1  
in which case we should have had the    
numbers #i1 #ah1 #bg1 three terms in    
arithmetical progression and all of them
 multiples of #i1 the square of #c4 ,it 
seems probable that these figures were  
not arrived at by any calculation based 
on geometrical considerations1 but that 
we have here merely an illustration of  
the ancient cult of the sacred numbers  
#c and #i4 ,three is the sacred number  
in ,homer1 #i in ,theognis4 ,the cult of
 #c and its multiples #i and #bg is     
found among the ,aryans1 then among the 
,finns and ,tartars and then again among
 the ,etruscans4 ,therefore             
,anaximander's figures probably say     
little more than what the ,indians tell 
us1 namely1 that three ,vishnu-steps    
reach from earth to heaven4             
,anaximander is said to have been the   
first to discover the gnomon 7or        
sun-dial with a vertical needle74 ,this 
is1 however1 incorrect1 for ,herodotus  
says that the ,greeks learnt the use of 

the gnomon and the polos from the       
,babylonians4 ,anaximander may have been
 the first to introduce the gnomon into 
,greece4 ,he is said to have set it up  
in ,sparta and to have shown on it 8the 
solstices1 the times1 the seasons1 and  
the equinox04  ,but ,anaximander has    
another title to fame4 ,he was the first
 who ventured to draw a map of the      
inhabited earth4 ,the ,egyptians indeed 
had drawn maps before1 but only of      
special districts4 ,anaximander boldly  
planned out the whole world with 8the   
circumference of the earth and of the   
sea04 ,hecataeus1 a much-travelled man1 
is said to have corrected ,anaximander's
 map so that it became the object of    
general admiration4   ,,anaximenes4     
,with ,anaximenes of ,miletus 7about    
#ehe-ebh_/d ,b4,c47 the earth is still  
flat like a table1 but1 instead of being
 suspended freely without support as    
with ,anaximander1 it is supported by   
the air1 riding on it as it were4 ,the  
sun1 moon and stars are all made of fire

 and 7like the earth7 they ride on the  
air because of their breadth4 ,the sun  
is flat like a leaf4 ,anaximenes also   
held that the stars are fastened on a   
crystal sphere like nails or studs4 ,it 
seems clear therefore that by the stars 
which 8ride on the air because of their 
breadth0 he meant the planets only4 ,a  
like apparent inconsistency applies to  
the motion of the stars4 ,if the stars  
are fixed in the crystal sphere like    
nails1 they must be carried round       
complete circles by the revolution of   
the sphere about a diameter4 ,yet       
,anaximenes also said that the stars do 
not move or revolve under the earth as  
some suppose1 but round the earth1 just 
as a cap can be turned round on the     
head4 ,the sun is hidden from sight1 not
 because it is under the earth1 but     
because it is covered by the higher     
parts of the earth and because its      
distance from us is greater4 ,aristotle 
adds the detail that the sun is carried 
round the northern portion of the earth 

and produces night because the earth is 
lofty towards the north4 ,we must again 
conclude that the stars which1 like the 
sun and moon1 move laterally round the  
earth between their setting and rising  
again are the planets1 as distinct from 
the fixed stars4 ,it would therefore    
seem that ,anaximenes was the first to  
distinguish the planets from the fixed  
stars in respect of their irregular     
movements4 ,he improved on ,anaximander 
in that he relegated the fixed stars to 
the region most distant from the earth4 
,anaximenes was also original in holding
 that1 in the region occupied by the    
stars1 bodies of an earthy nature are   
carried round along with them4 ,the     
object of these invisible bodies of an  
earthy nature carried round along with  
the stars is clearly to explain the     
eclipses and phases of the moon4 ,it was
 doubtless this conception which1 in the
 hands of ,anaxagoras and others1       
ultimately led to the true explanation  
of eclipses4  ,the one feature of       

,anaximenes's system which was destined 
to an enduring triumph was the          
conception of the stars being fixed on a
 crystal sphere as in a rigid frame4    
,this really remained the fundamental   
principle in all astronomy down to      
,copernicus4   ,,pythagoras4  ,with     
,pythagoras and the ,pythagoreans we    
come to a different order of things4    
,pythagoras1 born at ,samos about #egb  
,b4,c41 is undoubtedly one of the       
greatest names in the history of        
science4 ,he was a mathematician of     
brilliant achievements2 he was also the 
inventor of the science of acoustics1 an
 astronomer of great originality1 a     
theologian and moral reformer1 and the  
founder of a brotherhood which admits   
comparison with the orders of medival   
chivalry4 ,perhaps his most epoch-making
 discovery was that of the dependence of
 musical tones on numerical proportions1
 the octave representing the proportion 
of #b 3 #a in length of string at the   
same tension1 the fifth #c 3 #b1 and the

 fourth #d 3 #c4 ,mathematicians know   
him as the reputed discoverer of the    
famous theorem about the square on the  
hypotenuse of a right-angled triangle 7 
,euclid ,i4 #dg72 but he was also the   
first to make geometry a part of a      
liberal education and to explore its    
first principles 7definitions1 etc474   
,pythagoras is said to have been the    
first to maintain that the earth is     
spherical in shape2 on what ground1 is  
uncertain4 ,one suggestion is that he   
may have argued from the roundness of   
the shadow cast by the earth in the     
eclipses of the moon2 but ,anaxagoras   
was the first to give the true          
explanation of such eclipses4 ,probably 
,pythagoras attributed spherical shape  
to the earth for the mathematical or    
mathematico-sthetical reason that the   
sphere is the most beautiful of all     
solid figures4 ,it is probable too1 and 
for the same reason1 that ,pythagoras   
gave the same spherical shape to the sun
 and moon1 and even to the stars1 in    

which case the way lay open for the     
discovery of the true cause of eclipses 
and of the phases of the moon4          
,pythagoras is also said to have        
distinguished five zones in the earth4  
,it is true that the first declaration  
that the earth is spherical and that it 
has five zones is alternatively         
attributed to ,parmenides 7born perhaps 
about #eaf or #ead ,b4,c471 on the good 
authority of ,theophrastus4 ,it is      
possible that1 although ,pythagoras was 
the real author of these views1         
,parmenides was the first to state them 
in public4  ,pythagoras regarded the    
universe as living1 intelligent1        
spherical1 enclosing the earth at the   
centre1 and rotating about an axis      
passing through the centre of the earth1
 the earth remaining at rest4  ,he is   
said to have been the first to observe  
that the planets have an independent    
motion of their own in a direction      
opposite to that of the fixed stars1    
i4e4 the daily rotation4 ,alternatively 

with ,parmenides he is said to have been
 the first to recognise that the        
,morning and the ,evening ,stars are one
 and the same4 ,pythagoras is hardly    
likely to have known this as the result 
of observations of his own2 he may have 
learnt it from ,egypt or ,chalda along  
with other facts about the planets4     
,,parmenides4  ,we have seen that       
certain views are alternatively ascribed
 to ,pythagoras and ,parmenides4 ,the   
system of ,parmenides was in fact a kind
 of blend of the theories of ,pythagoras
 and ,anaximander4 ,in giving the earth 
spherical form with five zones he agreed
 with ,pythagoras4 ,pythagoras1 however1
 made the spherical universe rotate     
about an axis through the centre of the 
earth2 this implied that the universe is
 itself limited1 but that something     
exists round it1 and in fact that beyond
 the finite rotating sphere there is    
limitless void or empty space4          
,parmenides1 on the other hand1 denied  
the existence of the infinite void and  

was therefore obliged to make his finite
 sphere motionless and to hold that its 
apparent rotation is only an illusion4  
,in other portions of his system        
,parmenides followed the lead of        
,anaximander4 ,like ,anaximander 7and   
,democritus later7 he argued that the   
earth remains in the centre because1    
being equidistant from all points on the
 sphere of the universe1 it is in       
equilibrium and there is no more reason 
why it should tend to move in one       
direction than in another4 ,parmenides  
also had a system of wreaths or bands   
round the sphere of the universe which  
contained the sun1 the moon and the     
stars2 the wreaths remind us of the     
hoops of ,anaximander1 but their nature 
is different4 ,the wreaths1 according to
 the most probable interpretation of the
 texts1 are1 starting from the outside1 
7#a7 a solid envelope like a wall2 7#b7 
a band of fire 7the ther-fire72 7#c7    
mixed bands1 made up of light and       
darkness in combination1 which exhibit  

the phenomenon of 8fire shining out here
 and there10 these mixed bands including
 the ,milky ,way as well as the sun1    
moon and planets2 7#d7 a band of fire1  
the inner side of which is our          
atmosphere1 touching the earth4 ,except 
that ,parmenides placed the ,morning    
,star first in the ther and therefore   
above the sun1 he did not apparently    
differ from ,anaximander's view of the  
relative distances of the heavenly      
bodies1 according to which both the     
planets and the other stars are all     
placed below the sun and moon4  ,two    
lines from ,parmenides's poem have been 
quoted to show that he declared that the
 moon is illuminated by the sun4 ,the   
first line speaks of the moon as 8a     
night-shining foreign light wandering   
round the earth02 but1 even if the line 
is genuine1 8foreign0 need not mean     
8borrowed04 ,the other line speaks of   
the moon as 8always fixing its gaze on  
the sun02 but1 though this states an    
observed fact1 it is far from explaining

 the cause4 ,we have1 moreover1 positive
 evidence against the attribution of the
 discovery of the opacity of the moon to
 ,parmenides4 ,it is part of the        
connected prose description of his      
system that the moon is a mixture of air
 and fire1 and in other passages we are 
told that he held the moon to be of     
fire4 ,lastly1 ,plato speaks of 8the    
fact which ,anaxagoras lately asserted1 
that the moon has its light from the    
sun04 ,it seems impossible that ,plato  
would speak in such terms if the fact in
 question had been stated for the first 
time either by ,parmenides or by the    
,pythagoreans4   ,,anaxagoras4          
,anaxagoras1 a man of science if ever   
there was one1 was born at ,clazomenae  
in the neighbourhood of ,smyrna about   
#ejj ,b4,c4 ,he neglected his           
possessions1 which were considerable1 in
 order to devote himself to science4    
,someone once asked him what was the    
object of being born1 and he replied1   
8,the investigation of sun1 moon and    

heaven04 ,he took up his abode at       
,athens1 where he enjoyed the friendship
 of ,pericles4 ,when ,pericles became   
unpopular shortly before the outbreak of
 the ,peloponnesian war1 he was attacked
 through his friends1 and ,anaxagoras   
was accused of impiety for declaring    
that the sun was a red-hot stone and the
 moon made of earth4 ,one account says  
that he was fined and banished2 another 
that he was imprisoned1 and that it was 
intended to put him to death1 but that  
,pericles obtained his release2 he      
retired to ,lampsacus1 where he died at 
the age of seventy-two4  ,one           
epoch-making discovery belongs to him1  
namely1 that the moon does not shine by 
its own light but receives its light    
from the sun3 ,plato1 as we have seen1  
is one authority for this statement4    
,plutarch also in his ,de facie in orbe 
lunae says1 8,now when our comrade in   
his discourse had expounded that        
proposition of ,anaxagoras that #8the   
sun places the brightness in the moon1' 

he was greatly applauded04  ,this       
discovery enabled ,anaxagoras to say    
that 8the obscurations of the moon month
 by month were due to its following the 
course of the sun by which it is        
illuminated1 and the eclipses of the    
moon were caused by its falling within  
the shadow of the earth which then comes
 between the sun and the moon1 while the
 eclipses of the sun were due to the    
interposition of the moon04 ,anaxagoras 
was therefore the first to give the true
 explanation of eclipses4 ,as regards   
the phases of the moon1 his explanation 
could only have been complete if he had 
known that the moon is spherical2 in    
fact1 however1 he considered the earth  
7and doubtless the other heavenly bodies
 also7 to be flat4 ,to his true theory  
of eclipses ,anaxagoras added the       
unnecessary assumption that the moon was
 sometimes eclipsed by other earthy     
bodies below the moon but invisible to  
us4 ,in this latter assumption he       
followed the lead of ,anaximenes4 ,the  

other bodies in question were probably  
invented to explain why the eclipses of 
the moon are seen oftener than those of 
the sun4  ,anaxagoras's cosmogony       
contained some fruitful ideas4          
,according to him1 the formation of the 
world began with a vortex set up1 in a  
portion of the mixed mass in which 8all 
things were together10 by ,mind4 ,this  
rotatory movement began at one point and
 then gradually spread1 taking in wider 
and wider circles4 ,the first effect was
 to separate two great masses1 one      
consisting of the rare1 hot1 light1 dry1
 called the ther1 and the other of the  
opposite categories and called air4 ,the
 ther took the outer place1 the air the 
inner4 ,out of the air were separated   
successively clouds1 water1 earth1 and  
stones4 ,the dense1 the moist1 the dark 
and cold1 and all the heaviest things1  
collect in the centre as the result of  
the circular motion1 and it is from     
these elements when consolidated that   
the earth is formed4 ,but after this1   

8in consequence of the violence of the  
whirling motion1 the surrounding fiery  
ther tore stones away from the earth and
 kindled them into stars04 ,anaxagoras  
conceived therefore the idea of a       
centrifugal force1 as distinct from that
 of concentration brought about by the  
motion of the vortex1 and he assumed a  
series of projections or 8hurlings-off0 
of precisely the same kind as the theory
 of ,kant and ,laplace assumed for the  
formation of the solar system4  ,in     
other matters than the above ,anaxagoras
 did not make much advance on the crude 
,ionian theories4 8,the sun is a red-hot
 mass or a stone on fire40 8,it is      
larger 7or #8many times larger'7 than   
the ,peloponnese40 ,he considered that  
8the stars were originally carried round
 7laterally7 like a dome1 the pole which
 is always visible being thus vertically
 above the earth1 and it was only       
afterwards that their course became     
inclined04  ,but he put forward a       
remarkable and original hypothesis to   

explain the ,milky ,way4 ,he thought the
 sun to be smaller than the earth4      
,consequently1 when the sun in its      
revolution passes below the earth1 the  
shadow cast by the earth extends without
 limit4 ,the trace of this shadow on the
 heavens is the ,milky ,way4 ,the stars 
within this shadow are not interfered   
with by the light of the sun1 and we    
therefore see them shining2 those stars1
 on the other hand1 which are outside   
the shadow are overpowered by the light 
of the sun which shines on them even    
during the night1 so that we cannot see 
them4 ,aristotle easily disposes of this
 theory by observing that1 the sun being
 much larger than the earth1 and the    
distance of the stars from the earth    
being many times greater than the       
distance of the sun1 the sun's shadow   
would form a cone with its vertex not   
very far from the earth1 so that the    
shadow of the earth1 which we call      
night1 would not reach the stars at all4
   ,,empedocles4  ,empedocles of        

,agrigentum 7about #did-dcd ,b4,c47     
would hardly deserve mention for his    
astronomy alone1 so crude were his views
 where they differed from those of his  
predecessors4 ,the earth1 according to  
,empedocles1 is kept in its place by the
 swiftness of the revolution of the     
heaven1 just as we may swing a cup with 
water in it round and round so that in  
some positions the top of the cup may   
even be turned downwards without the    
water escaping4 ,day and night he       
explained as follows4 ,within the       
crystal sphere to which the fixed stars 
are attached 7as ,anaximenes held71 and 
filling it1 is a sphere consisting of   
two hemispheres1 one of which is wholly 
of fire and therefore light1 while the  
other is a mixture of air with a little 
fire1 which mixture is darkness or      
night4 ,the revolution of these two     
hemispheres round the earth produces at 
each point on its surface the succession
 of day and night4 ,empedocles held the 
sun to be1 not fire1 but a reflection of

 fire similar to that which takes place 
from the surface of water1 the fire of a
 whole hemisphere of the world being    
bent back from the earth1 which is      
circular1 and concentrated into the     
crystalline sun which is carried round  
by the motion of the fiery hemisphere4  
,empedocles's one important scientific  
achievement was his theory that light   
travels and takes time to pass from one 
point to another4 ,the theory is alluded
 to by ,aristotle1 who says that1       
according to ,empedocles1 the light from
 the sun reaches the intervening space  
before it reaches the eye or the earth2 
there was therefore a time when the ray 
was not yet seen1 but was being         
transmitted through the medium4   ,,the 
,,pythagoreans4  ,we have seen that     
,pythagoras was the first to give       
spherical form to the earth and probably
 to the heavenly bodies generally1 and  
to assign to the planets a revolution of
 their own in a sense opposite to that  
of the daily rotation of the fixed stars

 about the earth as centre4  ,but a much
 more remarkable development was to     
follow in the ,pythagorean school4 ,this
 was nothing less than the abandonment  
of the geocentric hypothesis and the    
reduction of the earth to the status of 
a planet like the others4 ,the resulting
 system1 known as the ,pythagorean1 is  
attributed 7on the authority probably of
 ,theophrastus7 to ,philolaus2 but      
,diogenes ,laertius and ,atius refer to 
one ,hicetas of ,syracuse in this       
connection2 ,aristotle attributes the   
system to 8the ,pythagoreans04 ,it is a 
partial anticipation of the theory of   
,copernicus but differs from it in that 
the earth and the planets do not revolve
 round the sun but about an assumed     
8central fire10 and the sun itself as   
well as the moon does the same4 ,there  
were thus eight heavenly bodies1 in     
addition to the sphere of the fixed     
stars1 all revolving about the central  
fire4 ,the number of revolutions being  
thus increased to nine1 the             

,pythagoreans postulated yet another1   
making ten4 ,the tenth body they called 
the counter-earth1 and its character and
 object will appear from the following  
general description of the system4  ,the
 universe is spherical in shape and     
finite in size4 ,outside it is infinite 
void1 which enables the universe to     
breathe1 as it were4 ,at the centre is  
the central fire1 the ,hearth of the    
,universe1 called by various names such 
as the ,tower or ,watch-tower of ,zeus1 
the ,throne of ,zeus1 the ,mother of the
 ,gods4 ,in this central fire is located
 the governing principle1 the force     
which directs the movement and activity 
of the universe4 ,the outside boundary  
of the sphere is an envelope of fire2   
this is called ,olympus1 and in this    
region the elements are found in all    
their purity2 below this is the         
universe4 ,in the universe there revolve
 in circles round the central fire the  
following bodies3 nearest to the central
 fire the counter-earth which always    

accompanies the earth1 then the earth1  
then the moon1 then the sun1 next to the
 sun the five planets1 and last of all1 
outside the orbits of the planets1 the  
sphere of the fixed stars4 ,the         
counter-earth1 which accompanies the    
earth but revolves in a smaller orbit1  
is not seen by us because the hemisphere
 on which we live is turned away from   
the counter-earth4 ,it follows that our 
hemisphere is always turned away from   
the central fire1 that is1 it faces     
outwards from the orbit towards ,olympus
 7the analogy of the moon which always  
turns one side towards us may have      
suggested this72 this involves a        
rotation of the earth about its axis    
completed in the same time as it takes  
the earth to complete a revolution about
 the central fire4  ,although there was 
a theory that the counter-earth was     
introduced in order to bring the number 
of the moving bodies up to ten1 the     
perfect number according to the         
,pythagoreans1 it is clear from a       

passage of ,aristotle that this was not 
the real reason4 ,aristotle says1       
namely1 that the eclipses of the moon   
were considered to be due sometimes to  
the interposition of the earth1         
sometimes to the interposition of the   
counter-earth4 ,evidently therefore the 
purpose of the counter-earth was to     
explain the frequency with which        
eclipses of the moon occur4  ,the       
,pythagoreans held that the earth1      
revolving1 like one of the stars1 about 
the central fire1 makes night and day   
according to its position relatively to 
the sun2 it is therefore day in that    
region which is lit up by the sun and   
night in the cone formed by the earth's 
shadow4 ,as the same hemisphere is      
always turned outwards1 it follows that 
the earth completes one revolution about
 the central fire in a day and a night  
or in about twenty-four hours4 ,this    
would account for the apparent diurnal  
rotation of the heavens from east to    
west2 but for parallax 7of which1 if we 

may believe ,aristotle1 the             
,pythagoreans made light71 it would be  
equivalent to the rotation of the earth 
on its own axis once in twenty-four     
hours4 ,this would make the revolution  
of the sphere of the fixed stars        
unnecessary4 ,yet the ,pythagoreans     
certainly gave some motion to the latter
 sphere4 ,what it was remains a puzzle4 
,it cannot have been the precession of  
the equinoxes1 for that was first       
discovered by ,hipparchus 7second       
century ,b4,c474 ,perhaps there was a   
real incompatibility between the two    
revolutions which was unnoticed by the  
authors of the system4   ,,nopides ,,of 
,,chios4  nopides of ,chios 7a little   
younger than ,anaxagoras7 is credited   
with two discoveries4 ,the first1 which 
was important1 was that of the obliquity
 of the zodiac circle or the ecliptic2  
the second was that of a ,great ,year1  
which nopides put at fifty-nine years4  
,he also 7so we are told7 found the     
length of the year to be #cfe-bb_/ei    

days4 ,he seems to have obtained this   
figure by a sort of circular argument4  
,starting first with #cfe days as the   
length of a year and #bi days as the    
length of the lunar month1 approximate  
figures known before his time1 he had to
 find the least integral number of      
complete years containing an exact      
number of lunar months2 this is clearly 
fifty-nine years1 which contain twice   
#cfe or #gcj lunar months4 nopides seems
 by his knowledge of the calendar to    
have determined the number of days in   
#gcj lunar months to be #ba1eeg1 and    
this number divided by fifty-nine1 the  
number of years1 gives #cfe-bb_/ei as   
the number of days in the year4         
,,plato4  ,we come now to ,plato        
7#dbg-cdg ,b4,c474 ,in the astronomy of 
,plato1 as we find it in the ,dialogues1
 there is so large an admixture of myth 
and poetry that it is impossible to be  
sure what his real views were on certain
 points of detail4 ,in the ,phdo we have
 certain statements about the earth to  

the effect that it is of very large     
dimensions1 the apparent hollow         
7according to ,plato7 in which we live  
being a very small portion of the whole1
 and that it is in the middle of the    
heaven1 in equilibrium1 without any     
support1 by virtue of the uniformity in 
the substance of the heaven4 ,in the    
,republic we have a glimpse of a more   
complete astronomical system4 ,the      
outermost revolution is that of the     
sphere of the fixed stars1 which carries
 round with it the whole universe       
including the sun1 moon and planets2 the
 latter seven bodies1 while they are so 
carried round by the general rotation1  
have slower revolutions of their own in 
addition1 one inside the other1 these   
revolutions being at different speeds   
but all in the opposite sense to the    
general rotation of the universe4 ,the  
quickest rotation is that of the fixed  
stars and the universe1 which takes     
place once in about twenty-four hours4  
,the slower speeds of the sun1 moon and 

planets are not absolute but relative to
 the sphere of the fixed stars regarded 
as stationary4 ,the earth in the centre 
is unmoved2 the successive revolutions  
about it and within the sphere of the   
fixed stars are 7reckoning from the     
earth outwards7 those of the moon1 the  
sun1 ,venus1 ,mercury1 ,mars1 ,jupiter1 
,saturn2 the speed of the moon is the   
quickest1 that of the sun the next      
quickest1 while ,venus and ,mercury     
travel with the sun and have the same   
speed1 taking about a year to describe  
their orbits2 after these in speed comes
 ,mars1 then ,jupiter and1 last and     
slowest of all1 ,saturn4 ,there is      
nothing said in the ,republic about the 
seven bodies revolving in a circle      
different from and inclined to the      
equator of the sphere of the fixed      
stars2 that is1 the obliquity of the    
ecliptic does not appear2 hence the     
standpoint of the whole system is that  
of ,pythagoras as distinct from that of 
the ,pythagoreans4  ,plato's            

astronomical system is1 however1 most   
fully developed in the ,timus4 ,while   
other details remain substantially the  
same1 the zodiac circle in which the    
sun1 moon and planets revolve is        
distinguished from the equator of the   
sphere of the fixed stars4 ,the latter  
is called the circle of the ,same1 the  
former that of the ,other1 and we are   
told 7quite correctly7 that1 since the  
revolution of the universe in the circle
 of the ,same carries all the other     
revolutions with it1 the effect on each 
of the seven bodies is to turn their    
actual motions in space into spirals4   
,there is a difficulty in interpreting a
 phrase in ,plato's description which   
says that ,venus and ,mercury1 though   
moving in a circle having equal speed   
with the sun1 8have the contrary        
tendency to it04 ,literally this would  
seem to mean that ,venus and ,mercury   
describe their circles the opposite way 
to the sun1 but this is so contradicted 
by observation that ,plato could hardly 

have maintained it2 hence the words have
 been thought to convey a vague         
reference to the apparent irregularities
 in the motion of ,venus and ,mercury1  
their standings-still and               
retrogradations4  ,but the most disputed
 point in the system is the part        
assigned in it to the earth4 ,an        
expression is used with regard to its   
relation to the axis of the heavenly    
sphere which might mean either 7#a7 that
 it is wrapped or globed about that axis
 but without motion1 or 7#b7 that it    
revolves round the axis4 ,if the word   
means revolving about the axis of the   
sphere1 the revolution would be either  
7a7 rotation about its own axis supposed
 to be identical with that of the       
sphere1 or 7b7 revolution about the axis
 of the heavenly sphere in the same way 
that the sun1 moon and planets revolve  
about an axis obliquely inclined to that
 axis4 ,but 7a7 if the earth rotated    
about its own axis1 this would make     
unnecessary the rotation of the sphere  

of the fixed stars once in twenty-four  
hours1 which1 however1 is expressly     
included as part of the system4 ,the    
hypothesis 7b7 would make the system    
similar to the ,pythagorean except that 
the earth would revolve about the axis  
of the heavenly sphere instead of round 
the central fire4 ,the supporters of    
this hypothesis cite two passages of    
,plutarch to the effect that ,plato was 
said in his old age to have repented of 
having given the earth the middle place 
in the universe instead of placing it   
elsewhere and giving the middle and     
chiefest place to some worthier         
occupant4 ,it is a sufficient answer to 
this argument that1 if ,plato really    
meant in the passage of the ,timus to   
say that the earth revolves about the   
axis of the heavenly sphere1 he had     
nothing to repent of4 ,we must          
therefore1 for our part1 conclude that  
in his written ,dialogues ,plato        
regarded the earth as at rest in the    
centre of the universe4  ,we have it on 

good authority that ,plato set it as a  
problem to all earnest students 8to find
 what are the uniform and ordered       
movements by the assumption of which the
 apparent movements of the planets can  
be accounted for04 ,the same authority  
adds that ,eudoxus was the first to     
formulate a theory with this object2 and
 ,heraclides of ,pontus followed with an
 entirely new hypothesis4 ,both were    
pupils of ,plato and1 in so far as the  
statement of his problem was a stimulus 
to these speculations1 he rendered an   
important service to the science of     
astronomy4   ,,eudoxus1 ,,callippus1    
,,aristotle4  ,eudoxus of ,cnidos 7about
 #djh-cee ,b4,c47 was one of the very   
greatest of the ,greek mathematicians4  
,he was the discoverer and elaborator of
 the great theory of proportion         
applicable to all magnitudes whether    
commensurable or incommensurable which  
is given in ,euclid's ,book ,v4 ,he was 
also the originator of the powerful     
method of exhaustion used by all later  

,greek geometers for the purpose of     
finding the areas of curves and the     
volumes of pyramids1 cones1 spheres and 
other curved surfaces4 ,it is not       
therefore surprising that he should have
 invented a remarkable geometrical      
hypothesis for explaining the irregular 
movements of the planets4 ,the problem  
was to find the necessary number of     
circular motions which by their         
combination would produce the motions of
 the planets as actually observed1 and  
in particular the variations in their   
apparent speeds1 their stations and     
retrogradations and their movements in  
latitude4 ,this ,eudoxus endeavoured to 
do by combining the motions of several  
concentric spheres1 one inside the      
other1 and revolving about different    
axes1 each sphere revolving on its own  
account but also being carried round    
bodily by the revolution of the next    
sphere encircling it4 ,we are dependent 
on passages from ,aristotle and         
,simplicius for our knowledge of        

,eudoxus's system1 which he had set out 
in a work ,on ,speeds1 now lost4        
,eudoxus assumed three revolving spheres
 for producing the apparent motions of  
the sun and moon respectively1 and four 
for that of each of the planets4 ,in his
 hypothesis for the sun he seems        
deliberately to have ignored the        
discovery made by ,meton and ,euctemon  
some sixty or seventy years before that 
the sun does not take the same time to  
describe the four quadrants of its orbit
 between the equinoctial and solstitial 
points4  ,it should be observed that the
 whole hypothesis of the concentric     
spheres is pure geometry1 and there is  
no mechanics in it4 ,we will shortly    
describe the arrangement of the four    
spheres which by their revolution       
produced the motion of a planet4 ,the   
first and outermost sphere produced the 
daily rotation in twenty-four hours2 the
 second sphere revolved about an axis   
perpendicular to the plane of the zodiac
 or ecliptic1 thereby producing the     

motion along the zodiac 8in the         
respective periods in which the planets 
appear to describe the zodiac circle10  
i4e4 in the case of the superior        
planets1 the sidereal periods of        
revolution1 and in the case of ,mercury 
and ,venus 7on a geocentric system7 one 
year4 ,the third sphere had its poles at
 two opposite points on the zodiac      
circle1 the poles being carried round in
 the motion of the second sphere2 the   
revolution of the third sphere about the
 axis connecting the two poles was again
 uniform and took place in a period     
equal to the synodic period of the      
planet1 or the time elapsing between two
 successive oppositions or conjunctions 
with the sun4  ,the poles of the third  
sphere were different for all the       
planets1 except that for ,mercury and   
,venus they were the same4 ,on the      
surface of the third sphere the poles of
 the fourth sphere were fixed1 and its  
axis of revolution was inclined to that 
of the former at an angle constant for  

each planet but different for the       
different planets4 ,the planet was fixed
 at a point on the equator of the fourth
 sphere4 ,the third and fourth spheres  
together cause the planet's movement in 
latitude4 ,simplicius explains clearly  
the effect of these two rotations4 ,if1 
he says1 the planet had been on the     
third sphere 7by itself71 it would      
actually have arrived at the poles of   
the zodiac circle2 but1 as things are1  
the fourth sphere1 which turns about the
 poles of the inclined circle carrying  
the planet and rotates in the opposite  
sense to the third1 i4e4 from east to   
west1 but in the same period1 will      
prevent any considerable divergence on  
the part of the planet from the zodiac  
circle1 and will cause the planet to    
describe about this same zodiac circle  
the curve called by ,eudoxus the        
hippopede 7horse-fetter71 so that the   
breadth of this curve will be the       
maximum amount of the apparent deviation
 of the planet in latitude4 ,the curve  

in question is an elongated             
figure-of-eight lying along and bisected
 by the zodiac circle4 ,the motion then 
round this figure-of-eight combined with
 the motion in the zodiac circle        
produces the acceleration and           
retardation of the motion of the planet1
 causing the stations and               
retrogradations4 ,mathematicians will   
appreciate the wonderful ingenuity and  
beauty of the construction4  ,eudoxus   
spent sixteen months in ,egypt about    
#cha-chj ,b4,c41 and1 while there1 he   
assimilated the astronomical knowledge  
of the priests of ,heliopolis and       
himself made observations4 ,the         
,observatory between ,heliopolis and    
,cercesura used by him was still pointed
 out in ,augustus's time2 he also had   
one built at ,cnidos4 ,he wrote two     
books entitled respectively the ,mirror 
and the ,phnomena2 the poem of ,aratus  
was1 so far as verses #ai-gcb are       
concerned1 drawn from the ,phnomena of  
,eudoxus4 ,he is also credited with the 

invention of the arachne 7spider's web7 
which1 however1 is alternatively        
attributed to ,apollonius1 and which    
seems to have been a sun-clock of some  
kind4  ,eudoxus's system of concentric  
spheres was improved upon by ,callippus 
7about #cgj-cjj ,b4,c471 who added two  
more spheres for the sun and the moon1  
and one more in the case of each of the 
three nearer planets1 ,mercury1 ,venus  
and ,mars4 ,the two additional spheres  
in the case of the sun were introduced  
in order to account for the unequal     
motion of the sun in longitude2 and the 
purpose in the case of the moon was     
presumably similar4 ,callippus made the 
length of the seasons1 beginning with   
the vernal equinox1 ninety-four1        
ninety-two1 eighty-nine and ninety days 
respectively1 figures much more accurate
 than those given by ,euctemon a hundred
 years earlier1 which were ninety-three1
 ninety1 ninety and ninety-two days     
respectively4  ,with ,callippus as well 
as ,eudoxus the system of concentric    

spheres was purely geometrical4         
,aristotle 7#chd-cbb ,b4,c47 thought it 
necessary to alter it in a mechanical   
sense2 he made the spheres into         
spherical shells actually in contact    
with one another1 and this made it      
almost necessary1 instead of having     
independent sets of spheres1 one set for
 each planet1 to make all the sets part 
of one continuous system of spheres4    
,for this purpose he assumed sets of    
reacting spheres between successive sets
 of the original spheres4 ,e4g4 ,saturn 
being carried by a set of four spheres1 
he had three reacting spheres to        
neutralise the last three1 in order to  
restore the outermost sphere to act as  
the first of the four spheres producing 
the motion of the next lower planet1    
,jupiter1 and so on4 ,the change was    
hardly an improvement4  ,aristotle's    
other ideas in astronomy do not require 
detailed notice1 except his views about 
the earth4 ,although he held firmly to  
the old belief that the earth is in the 

centre and remains motionless1 he was   
clear that its shape 7like that of the  
stars and the universe7 is spherical1   
and he had arrived at views about its   
size sounder than those of ,plato4 ,in  
support of the spherical shape of the   
earth he used some good arguments based 
on observation4 7#a7 ,in partial        
eclipses of the moon the line separating
 the dark and bright portions is always 
circular--unlike the line of demarcation
 in the phases of the moon which may be 
straight4 7#b7 ,certain stars seen above
 the horizon in ,egypt and in ,cyprus   
are not visible further north1 and1 on  
the other hand1 certain stars set there 
which in more northern latitudes remain 
always above the horizon4 ,as there is  
so perceptible a change of horizon      
between places so near to each other1 it
 follows not only that the earth is     
spherical but also that it is not a very
 large sphere4 ,aristotle adds that     
people are not improbably right when    
they say that the region about the      

,pillars of ,heracles is joined on to   
,india1 one sea connecting them4 ,he    
quotes a result arrived at by the       
mathematicians of his time1 that the    
circumference of the earth is #djj1jjj  
stades4 ,he is clear that the earth is  
much smaller than some of the stars1 but
 that the moon is smaller than the      
earth4  ,the systems of concentric      
spheres were not destined to hold their 
ground for long4 ,in these systems the  
sun1 moon and planets were of necessity 
always at the same distances from the   
earth respectively4 ,but it was soon    
recognised that they did not 8save the  
phenomena10 since it was seen that the  
planets appeared to be at one time      
nearer and at another time further off4 
,autolycus of ,pitane 7who flourished   
about #caj ,b4,c47 knew this and is said
 to have tried to explain it2 indeed it 
can hardly have been unknown even to the
 authors of the concentric theory       
themselves1 for ,polemarchus of         
,cyzicus1 almost contemporary with      

,eudoxus1 is said to have been aware of 
it but to have minimised the difficulty 
because he preferred the hypothesis of  
the concentric spheres to others4       
,development along the lines of         
,eudoxus's theory being thus blocked1   
the alternative was open of seeing      
whether any modification of the         
,pythagorean system would give better   
results4 ,we actually have evidence of  
revisions of the ,pythagorean theory4   
,the first step was to get rid of the   
counter-earth1 and some ,pythagoreans   
did this by identifying the             
counter-earth with the moon4 ,we hear   
too of a ,pythagorean system in which   
the central fire was not outside the    
earth but in the centre of the earth    
itself4 ,the descriptions of this system
 rather indicate that in it the earth   
was supposed to be at rest1 without any 
rotation1 in the centre of the universe4
 ,this was practically a return to the  
standpoint of ,pythagoras himself4 ,but 
it is clear that1 if the system of      

,philolaus 7or ,hicetas7 be taken and   
the central fire be transferred to the  
centre of the earth 7the counter-earth  
being also eliminated71 and if the      
movements of the earth1 sun1 moon and   
planets round the centre be retained    
without any modification save that which
 is mathematically involved by the      
transfer of the central fire to the     
centre of the earth1 the daily          
revolution of the earth about the       
central fire is necessarily transformed 
into a rotation of the earth about its  
own axis in about twenty-four hours4    
,,heraclides ,,of ,,pontus4  ,all       
authorities agree that the theory of the
 daily rotation of the earth about its  
own axis was put forward by ,heraclides 
of ,pontus 7about #chh-cae ,b4,c471 a   
pupil of ,plato2 with him in some       
accounts is associated the name of one  
,ecphantus1 a ,pythagorean4 ,we are told
 that ,ecphantus asserted 8that the     
earth1 being in the centre of the       
universe1 moves about its own centre in 

an eastward direction10 and that        
8,heraclides of ,pontus and ,ecphantus  
the ,pythagorean make the earth move1   
not in the sense of translation1 but by 
way of turning as on an axle1 like a    
wheel1 from west to east1 about its own 
centre04  ,heraclides was born at       
,heraclea in ,pontus4 ,he went to       
,athens not later than #cfd ,b4,c41 and 
there met ,speusippus1 who introduced   
him into the school of ,plato4 ,on the  
death of ,speusippus 7then at the head  
of the school7 in #cch1 ,xenocrates was 
elected to succeed him2 at this election
 ,heraclides was also a candidate and   
was only defeated by a few votes4 ,he   
was the author of dialogues1 brilliant  
and original1 on all sorts of subjects1 
which were much read and imitated at    
,rome1 e4g4 by ,varro and ,cicero4 ,two 
of them 8,on ,nature0 and 8,on the      
,heavens0 may have dealt with astronomy4
  ,in his view that the earth rotates   
about its own axis ,heraclides is       
associated with ,aristarchus of ,samos2 

thus ,simplicius says3 8,there have been
 some1 like ,heraclides of ,pontus and  
,aristarchus1 who supposed that the     
phenomena can be saved if the heaven and
 the stars are at rest while the earth  
moves about the poles of the equinoctial
 circle from the west to the east1      
completing one revolution each day1     
approximately2 the #8approximately' is  
added because of the daily motion of the
 sun to the extent of one degree04      
,heraclides made another important      
advance towards the ,copernican         
hypothesis4 ,he discovered the fact that
 ,venus and ,mercury revolve about the  
sun as centre4 ,so much is certain2 but 
a further question naturally arises4    
,having made ,venus and ,mercury revolve
 round the sun like satellites1 did     
,heraclides proceed to draw the same    
inference with regard to the other1 the 
superior1 planets8 ,the question is     
interesting because1 had it been laid   
down that all the five planets alike    
revolve round the sun1 the combination  

of this hypothesis with ,heraclides's   
assumption that the earth rotates about 
its own axis in twenty-four hours would 
have amounted to an anticipation of the 
system of ,tycho ,brahe1 but with the   
improvement of the substitution of the  
daily rotation of the earth for the     
daily revolution of the whole system    
about the earth supposed at rest4       
,schiaparelli dealt with the question in
 two papers entitled ,i precursori di   
,copernico nell' antichit 7#ahgc71 and  
,origine del sistema planetario         
eliocentrico presso i ,greci 7#ahih74   
,schiaparelli tried to show that        
,heraclides did arrive at the conclusion
 that the superior planets as well as   
,mercury and ,venus revolve round the   
sun2 but most persons will probably     
agree that his argument is not          
convincing4 ,the difficulties seem too  
great4 ,the circles described by        
,mercury and ,venus about the sun are   
relatively small circles and are        
entirely on one side of the earth4 ,but 

when the possibility of1 say1 ,mars     
revolving about the sun came to be      
considered1 it would be at once obvious 
that the precise hypothesis adopted for 
,mercury and ,venus would not apply4 ,it
 would be seen that ,mars is brightest  
when it occupies a position in the      
zodiac opposite to the sun2 it must     
therefore be nearest to the earth at    
that time4 ,consequently the circle     
described by ,mars1 instead of being on 
one side of the earth1 must comprehend  
the earth which is inside it4 ,whereas  
therefore the circles described by      
,mercury and ,venus were what the       
,greeks called epicycles about a        
material centre1 the sun 7itself moving 
in a circle round the earth71 what was  
wanted in the case of ,mars 7if the     
circle described by ,mars was to have   
the sun for centre7 was what the ,greeks
 called an eccentric circle1 with a     
centre which itself moves in a circle   
about the earth1 and with a radius      
greater than that of the sun's orbit4   

,though the same motion could have been 
produced by an epicycle1 the epicycle   
would have had to have a mathematical   
point 7not the material sun7 as centre4 
,but the idea of using non-material     
points as centres for epicycles was     
probably first thought of1 at a later   
stage1 by some of the great             
mathematicians such as ,apollonius of   
,perga 7about #bfe-aij ,b4,c474  ,not   
only does ,schiaparelli maintain that   
the complete 7but improved7 ,tychonic   
hypothesis was put forward by           
,heraclides or at least in ,heraclides's
 time2 he goes further and makes a still
 greater claim on behalf of ,heraclides1
 namely1 that it was he1 and not        
,aristarchus of ,samos1 who first stated
 as a possibility the ,copernican       
hypothesis4 ,now it was much to         
discover1 as ,heraclides did1 that the  
earth rotates about its own axis and    
that ,mercury and ,venus revolve round  
the sun like satellites2 and it seems a 
priori incredible that one man should   

not only have reached1 and improved     
upon1 the hypothesis of ,tycho ,brahe   
but should also have suggested the      
,copernican hypothesis4 ,it is therefore
 necessary to examine briefly the       
evidence on which ,schiaparelli relied4 
,his argument rests entirely upon one   
passage1 a sentence forming part of a   
quotation from a summary by ,geminus of 
the ,meteorologica of ,posidonius1 which
 ,simplicius copied from ,alexander     
,aphrodisiensis and embodied in his     
commentary on the ,physics of           
,aristotle4 ,the sentence in question1  
according to the reading of the ,,mss41 
is as follows3 8,hence we actually find 
a certain person1 ,heraclides of        
,pontus1 coming forward and saying that1
 even on the assumption that the earth  
moves in a certain way1 while the sun is
 in a certain way at rest1 the apparent 
irregularity with reference to the sun  
can be saved04 7,the preceding sentence 
is about possible answers to the        
question1 why do the sun1 the moon and  

the planets appear to move irregularly8 
and says1 8we may answer that1 if we    
assume that their orbits are eccentric  
circles or that the stars describe an   
epicycle1 their apparent irregularity   
will be saved1 and it will be necessary 
to go further and examine in how many   
different ways it is possible for these 
phenomena to be brought about047  ,now  
it is impossible that ,geminus himself  
can have spoken of an astronomer of the 
distinction of ,heraclides as 8a certain
 ,heraclides of ,pontus04 ,consequently 
there have been different attempts made 
to emend the reading of the ,,mss4 ,all 
the emendations proposed are open to    
serious objections1 and we are thrown   
back on the reading of the ,,mss4 ,now  
it 8leaps to the eyes0 that1 if the name
 of ,heraclides of ,pontus is left out1 
everything is in order4 8,this is why   
one astronomer has actually suggested   
that1 by assuming the earth to move in a
 certain way1 and the sun to be in a    
certain way at rest1 the apparent       

irregularity with reference to the sun  
will be saved40 ,this seems to be the   
solution of the puzzle suggested by the 
ordinary principles of textual          
criticism1 and is so simple and natural 
that it will surely carry conviction to 
the minds of unbiassed persons4         
,geminus1 in fact1 mentioned no name but
 meant ,aristarchus of ,samos1 and some 
scholiast1 remembering that ,heraclides 
had given a certain motion to the earth 
7namely1 rotation about its axis71      
immediately thought of ,heraclides and  
inserted his name in the margin1 from   
which it afterwards crept into the text4
  ,it is only necessary to add that     
,archimedes is not likely to have been  
wrong when he attributed the first      
suggestion of the ,copernican hypothesis
 to ,aristarchus of ,samos in express   
terms2 and this is confirmed by another 
positive statement by ,atius1 already   
quoted1 that 8,heraclides of ,pontus and
 ,ecphantus the ,pythagorean made the   
earth move1 not in the sense of         

translation1 but with a movement of     
rotation04     ,,part ,,ii4             
,,aristarchus ,,of ,,samos4   ,we are   
told that ,aristarchus of ,samos was a  
pupil of ,strato of ,lampsacus1 a       
natural philosopher of originality1 who 
succeeded ,theophrastus as head of the  
,peripatetic school in #bhh or #bhg     
,b4,c41 and held that position for      
eighteen years4 ,two other facts enable 
us to fix ,aristarchus's date           
approximately4 ,in #bha-bhj he made an  
observation of the summer solstice2 and 
the book in which he formulated his     
heliocentric hypothesis was published   
before the date of ,archimedes's        
,psammites or ,sandreckoner1 a work     
written before #baf ,b4,c4 ,aristarchus 
therefore probably lived circa #caj-bcj 
,b4,c41 that is1 he came about          
seventy-five years later than           
,heraclides and was older than          
,archimedes by about twenty-five years4 
,aristarchus was called 8the            
mathematician10 no doubt in order to    

distinguish him from the many other     
persons of the same name2 ,vitruvius    
includes him among the few great men who
 possessed an equally profound knowledge
 of all branches of science1 geometry1  
astronomy1 music1 etc4 8,men of this    
type are rare1 men such as were in times
 past ,aristarchus of ,samos1 ,philolaus
 and ,archytas of ,tarentum1 ,apollonius
 of ,perga1 ,eratosthenes of ,cyrene1   
,archimedes and ,scopinas of ,syracuse1 
who left to posterity many mechanical   
and gnomonic appliances which they      
invented and explained on mathematical  
and natural principles40 ,that          
,aristarchus was a very capable geometer
 is proved by his extant book1 ,on the  
sizes and distances of the sun and moon1
 presently to be described4 ,in the     
mechanical line he is credited with the 
invention of an improved sun-dial1 the  
so-called scaphe1 which had not a plane 
but a concave hemispherical surface1    
with a pointer erected vertically in the
 middle1 throwing shadows and so        

enabling the direction and height of the
 sun to be read off by means of lines   
marked on the surface of the hemisphere4
 ,he also wrote on vision1 light1 and   
colours4 ,his views on the latter       
subjects were no doubt largely          
influenced by the teaching of ,strato4  
,strato held that colours were          
emanations from bodies1 material        
molecules as it were1 which imparted to 
the intervening air the same colour as  
that possessed by the body4 ,aristarchus
 said that colours are 8shapes or forms 
stamping the air with impressions like  
themselves as it were10 that 8colours in
 darkness have no colouring10 and that  
8light is the colour impinging on a     
substratum04   ,,the ,,heliocentric     
,,hypothesis4  ,there is no doubt       
whatever that ,aristarchus put forward  
the heliocentric hypothesis4 ,ancient   
testimony is unanimous on the point1 and
 the first witness is ,archimedes who   
was a younger contemporary of           
,aristarchus1 so that there is no       

possibility of a mistake4 ,copernicus   
himself admitted that the theory was    
attributed to ,aristarchus1 though this 
does not seem to be generally known4    
,copernicus refers in two passages of   
his work1 ,de revolutionibus            
caelestibus1 to the opinions of the     
ancients about the motion of the earth4 
,in the dedicatory letter to ,pope ,paul
 ,,iii he mentions that he first learnt 
from ,cicero that one ,nicetas 7i4e4    
,hicetas7 had attributed motion to the  
earth1 and that he afterwards read in   
,plutarch that certain others held that 
opinion2 he then quotes the ,placita    
philosophorum according to which        
8,philolaus the ,pythagorean asserted   
that the earth moved round the fire in  
an oblique circle in the same way as the
 sun and moon04 ,in ,book ,i4 c4 #e of  
his work ,copernicus alludes to the     
views of ,heraclides1 ,ecphantus1 and   
,hicetas1 who made the earth rotate     
about its own axis1 and then goes on to 
say that it would not be very surprising

 if any one should attribute to the     
earth another motion besides rotation1  
namely1 revolution in an orbit in space3
 8atque etiam 7terram7 pluribus motibus 
vagantem et unam ex astris ,philolaus   
,pythagoricus sensisse fertur1          
,mathematicus non vulgaris04 ,here1     
however1 there is no question of the    
earth revolving round the sun1 and there
 is no mention of ,aristarchus4 ,but    
,copernicus did mention the theory of   
,aristarchus in a passage which he      
afterwards suppressed3 8,credibile est  
hisce similibusque causis ,philolaum    
mobilitatem terrae sensisse1 quod etiam 
nonnulli ,aristarchum ,samium ferunt in 
eadem fuisse sententia04  ,it is        
desirable to quote the whole passage of 
,archimedes in which the allusion to    
,aristarchus's heliocentric hypothesis  
occurs1 in order to show the whole      
context4  8,you are aware #8you' being  
,king ,gelon that #8universe' is the    
name given by most astronomers to the   
sphere the centre of which is the centre

 of the earth1 while its radius is equal
 to the straight line between the centre
 of the sun and the centre of the earth4
 ,this is the common account as you have
 heard from astronomers4 ,but           
,aristarchus brought out a book         
consisting of certain hypotheses1       
wherein it appears1 as a consequence of 
the assumptions made1 that the universe 
is many times greater than the          
#8universe' just mentioned4 ,his        
hypotheses are that the fixed stars and 
the sun remain unmoved1 that the earth  
revolves about the sun in the           
circumference of a circle1 the sun lying
 in the middle of the orbit1 and that   
the sphere of the fixed stars1 situated 
about the same centre as the sun1 is so 
great that the circle in which he       
supposes the earth to revolve bears such
 a proportion to the distance of the    
fixed stars as the centre of the sphere 
bears to its surface40  ,the            
heliocentric hypothesis is here stated  
in language which leaves no room for    

doubt about its meaning4 ,the sun1 like 
the fixed stars1 remains unmoved and    
forms the centre of a circular orbit in 
which the earth moves round it2 the     
sphere of the fixed stars has its centre
 at the centre of the sun4  ,we have    
further evidence in a passage of        
,plutarch's tract1 ,on the face in the  
moon's orb3 8,only do not1 my dear      
fellow1 enter an action for impiety     
against me in the style of ,cleanthes1  
who thought it was the duty of ,greeks  
to indict ,aristarchus on the charge of 
impiety for putting in motion the       
,hearth of the ,universe1 this being the
 effect of his attempt to save the      
phenomena by supposing the heaven to    
remain at rest and the earth to revolve 
in an oblique circle1 while it rotates1 
at the same time1 about its own axis04  
,here we have the additional detail that
 ,aristarchus followed ,heraclides in   
attributing to the earth the daily      
rotation about its axis2 ,archimedes    
does not state this in so many words1   

but it is clearly involved by his remark
 that ,aristarchus supposed the fixed   
stars as well as the sun to remain      
unmoved in space4 ,a tract 8,against    
,aristarchus0 is mentioned by ,diogenes 
,laertius among ,cleanthes's works2 and 
it was evidently published during       
,aristarchus's lifetime 7,cleanthes died
 about #bcb ,b4,c474  ,we learn from    
another passage of ,plutarch that the   
hypothesis of ,aristarchus was adopted1 
about a century later1 by ,seleucus1 of 
,seleucia on the ,tigris1 a ,chaldan or 
,babylonian1 who also wrote on the      
subject of the tides about #aej ,b4,c4  
,the passage is interesting because it  
also alludes to the doubt about ,plato's
 final views4 8,did ,plato put the earth
 in motion as he did the sun1 the moon  
and the five planets which he called the
 #8instruments of time' on account of   
their turnings1 and was it necessary to 
conceive that the earth #8which is      
globed about the axis stretched from    
pole to pole through the whole universe'

 was not represented as being 7merely7  
held together and at rest but as turning
 and revolving1 as ,aristarchus and     
,seleucus afterwards maintained that it 
did1 the former of whom stated this as  
only a hypothesis1 the latter as a      
definite opinion80  ,no one after       
,seleucus is mentioned by name as having
 accepted the doctrine of ,aristarchus  
and1 if other ,greek astronomers refer  
to it1 they do so only to denounce it4  
,hipparchus1 himself a contemporary of  
,seleucus1 definitely reverted to the   
geocentric system1 and it was doubtless 
his authority which sealed the fate of  
the heliocentric hypothesis for so many 
centuries4  ,the reasons which weighed  
with ,hipparchus were presumably the    
facts that the system in which the earth
 revolved in a circle of which the sun  
was the exact centre failed to 8save the
 phenomena10 and in particular to       
account for the variations of distance  
and the irregularities of the motions1  
which became more and more patent as    

methods of observation improved2 that1  
on the other hand1 the theory of        
epicycles did suffice to represent the  
phenomena with considerable accuracy2   
and that the latter theory could be     
reconciled with the immobility of the   
earth4   ,,on ,,the ,,apparent          
,,diameter ,,of ,,the ,,sun4            
,archimedes tells us in the same        
treatise that 8,aristarchus discovered  
that the sun's apparent size is about   
#a_/gbjth part of the zodiac circle02   
that is to say1 he observed that the    
angle subtended at the earth by the     
diameter of the sun is about half a     
degree4   ,,on ,,the ,,sizes ,,and      
,,distances ,,of ,,the ,,sun ,,and      
,,moon4  ,archimedes also says that1    
whereas the ratio of the diameter of the
 sun to that of the moon had been       
estimated by ,eudoxus at #i 3 #a and by 
his own father ,phidias at #ab 3 #a1    
,aristarchus made the ratio greater than
 #ah 3 #a but less than #bj 3 #a4       
,fortunately we possess in ,greek the   

short treatise in which ,aristarchus    
proved these conclusions2 on the other  
matter of the apparent diameter of the  
sun ,archimedes's statement is our only 
evidence4  ,it is noteworthy that in    
,aristarchus's extant treatise ,on the  
sizes and distances of the sun and moon 
there is no hint of the heliocentric    
hypothesis1 while the apparent diameter 
of the sun is there assumed to be1 not 1
 but the very inaccurate figure of #b4  
,both circumstances are explained if we 
assume that the treatise was an early   
work written before the hypotheses      
described by ,archimedes were put       
forward4 ,in the treatise ,aristarchus  
finds the ratio of the diameter of the  
sun to the diameter of the earth to lie 
between #ai 3 #c and #dc 3 #f2 this     
would make the volume of the sun about  
#cjj times that of the earth1 and it may
 be that the great size of the sun in   
comparison with the earth1 as thus      
brought out1 was one of the             
considerations which led ,aristarchus to

 place the sun rather than the earth in 
the centre of the universe1 since it    
might even at that day seem absurd to   
make the body which was so much larger  
revolve about the smaller4  ,there is no
 reason to doubt that in his            
heliocentric system ,aristarchus        
retained the moon as a satellite of the 
earth revolving round it as centre2     
hence even in his system there was one  
epicycle4  ,the treatise ,on sizes and  
distances being the only work of        
,aristarchus which has survived1 it will
 be fitting to give here a description  
of its contents and special features4   
,the style of ,aristarchus is thoroughly
 classical as befits an able geometer   
intermediate in date between ,euclid and
 ,archimedes1 and his demonstrations are
 worked out with the same rigour as     
those of his predecessor and successor4 
,the propositions of ,euclid's ,elements
 are1 of course1 taken for granted1 but 
other things are tacitly assumed which  
go beyond what we find in ,euclid4 ,thus

 the transformations of ratios defined  
in ,euclid1 ,book ,v1 and denoted by the
 terms inversely1 alternately1          
componendo1 convertendo1 etc41 are      
regularly used in dealing with unequal  
ratios1 whereas in ,euclid they are only
 used in proportions1 i4e4 cases of     
equality of ratios4 ,but the            
propositions of ,aristarchus are also of
 particular mathematical interest       
because the ratios of the sizes and     
distances which have to be calculated   
are really trigonometrical ratios1      
sines1 cosines1 etc41 although at the   
time of ,aristarchus trigonometry had   
not been invented1 and no reasonably    
close approximation to the value of 1   
the ratio of the circumference of any   
circle to its diameter1 had been made   
7it was ,archimedes who first obtained  
the approximation #bb_/g74 ,exact       
calculation of the trigonometrical      
ratios being therefore impossible for   
,aristarchus1 he set himself to find    
upper and lower limits for them1 and he 

succeeded in locating those which emerge
 in his propositions within tolerably   
narrow limits1 though not always the    
narrowest within which it would have    
been possible1 even for him1 to confine 
them4 ,in this species of approximation 
to trigonometry he tacitly assumes      
propositions comparing the ratio between
 a greater and a lesser angle in a      
figure with the ratio between two       
straight lines1 propositions which are  
formally proved by ,ptolemy at the      
beginning of his ,syntaxis4 ,here again 
we have proof that textbooks containing 
such propositions existed before        
,aristarchus's time1 and probably much  
earlier1 although they have not         
survived4  ,aristarchus necessarily     
begins by laying down1 as the basis for 
his treatise1 certain assumptions4 ,they
 are six in number1 and he refers to    
them as hypotheses4 ,we cannot do better
 than quote them in full1 along with the
 sentences immediately following1 in    
which he states the main results to be  

established in the treatise3--          
,,hypotheses4  #a4 ,that the moon       
receives its light from the sun4  #b4   
,that the earth is in the relation of a 
point and centre to the sphere in which 
the moon moves4  #c4 ,that1 when the    
moon appears to us halved1 the great    
circle which divides the dark and the   
bright portions of the moon is in the   
direction of our eye4  #d4 ,that1 when  
the moon appears to us halved1 its      
distance from the sun is then less than 
a quadrant by one-thirtieth of a        
quadrant4  #e4 ,that the breadth of the 
7earth's7 shadow is 7that7 of two moons4
  #f4 ,that the moon subtends           
one-fifteenth part of a sign of the     
zodiac4  ,we are now in a position to   
prove the following propositions3--  #a4
 ,the distance of the sun from the earth
 is greater than eighteen times1 but    
less than twenty times1 the distance of 
the moon 7from the earth72 this follows 
from the hypothesis about the halved    
moon4  #b4 ,the diameter of the sun has 

the same ratio 7as aforesaid7 to the    
diameter of the moon4  #c4 ,the diameter
 of the sun has to the diameter of the  
earth a ratio greater than that which   
#ai has to #c1 but less than that which 
#dc has to #f2 this follows from the    
ratio thus discovered between the       
distances1 the hypothesis about the     
shadow1 and the hypothesis that the moon
 subtends one-fifteenth part of a sign  
of the zodiac4  ,the first assumption is
 ,anaxagoras's discovery4 ,the second   
assumption is no doubt an exaggeration2 
but it is made in order to avoid having 
to allow for the fact that the phenomena
 as seen by an observer on the surface  
of the earth are slightly different from
 what would be seen if the observer's   
eye were at the centre of the earth4    
,aristarchus1 that is1 takes the earth  
to be like a point in order to avoid the
 complication of parallax4  ,the meaning
 of the third hypothesis is that the    
plane of the great circle in question   
passes through the point where the eye  

of the observer is situated2 that is to 
say1 we see the circle end on1 as it    
were1 and it looks like a straight line4
  ,hypothesis #d4 ,if ,s be the sun1 ,m 
the moon and ,e the earth1 the triangle 
,,sme is1 at the moment when the moon   
appears to us halved1 right-angled at   
,m2 and the hypothesis states that the  
angle at ,e in this triangle is #hg1 or1
 in other words1 the angle ,,mse1 that  
is1 the angle subtended at the sun by   
the line joining ,m to ,e1 is #c4 ,these
 estimates are decidedly inaccurate1 for
 the true value of the angle ,,mes is   
#hi #ej1 and that of the angle ,,mse is 
therefore #aj4 ,there is nothing to show
 how ,aristarchus came to estimate the  
angle ,,mse at #c1 and none of his      
successors seem to have made any direct 
estimate of the size of the angle4  ,the
 assumption in ,hypothesis #e was       
improved upon later4 ,hipparchus made   
the ratio of the diameter of the circle 
of the earth's shadow to the diameter of
 the moon to be1 not #b1 but #b at the  

moon's mean distance at the             
conjunctions2 ,ptolemy made it1 at the  
moon's greatest distance1 to be         
inappreciably less than #b4  ,the sixth 
hypothesis states that the diameter of  
the moon subtends at our eye an angle   
which is #a_/aeth of #cj1 i4e4 #b1      
whereas ,archimedes1 as we have seen1   
tells us that ,aristarchus found the    
angle subtended by the diameter of the  
sun to be  7,archimedes in the same     
tract describes a rough instrument by   
means of which he himself found that the
 diameter of the sun subtended an angle 
less than #a_/afdth1 but greater than   
#a_/bjjth of a right angle74 ,even the  
,babylonians had1 many centuries before1
 arrived at #a as the apparent angular  
diameter of the sun4 ,it is not clear   
why ,aristarchus took a value so        
inaccurate as #b4 ,it has been suggested
 that he merely intended to give a      
specimen of the calculations which would
 have to be made on the basis of more   
exact experimental observations1 and to 

show that1 for the solution of the      
problem1 one of the data could be chosen
 almost arbitrarily1 by which proceeding
 he secured himself against certain     
objections which might have been raised4
 ,perhaps this is too ingenious1 and it 
may be that1 in view of the difficulty  
of working out the geometry if the two  
angles in question are very small1 he   
took #c and #b as being the smallest    
with which he could conveniently deal4  
,certain it is that the method of       
,aristarchus is perfectly correct and1  
if he could have substituted the true   
values 7as we know them to-day7 for the 
inaccurate values which he assumes1 and 
could have carried far enough his       
geometrical substitute for trigonometry1
 he would have obtained close limits for
 the true sizes and distances4  ,the    
book contains eighteen propositions4    
,prop4 #a proves that we can draw one   
cylinder to touch two equal spheres1 and
 one cone to touch two unequal spheres1 
the planes of the circles of contact    

being at right angles to the axis of the
 cylinder or cone4 ,next 7,prop4 #b7 it 
is shown that1 if a lesser sphere be    
illuminated by a greater1 the           
illuminated portion of the former will  
be greater than a hemisphere4 ,prop4 #c 
proves that the circle in the moon which
 divides the dark and the bright        
portions 7we will in future1 for short1 
call this 8the dividing circle07 is     
least when the cone which touches the   
sun and the moon has its vertex at our  
eye4 ,in ,prop4 #d it is shown that the 
dividing circle is not perceptibly      
different from a great circle in the    
moon4 ,if ,,cd is a diameter of the     
dividing circle1 ,,ef the parallel      
diameter of the parallel great circle in
 the moon1 ,o the centre of the moon1 ,a
 the observer's eye1 ,,fdg the great    
circle in the moon the plane of which   
passes through ,a1 and ,g the point     
where ,,oa meets the latter great       
circle1 ,aristarchus takes an arc of the
 great circle ,,gh on one side of ,g1   

and another ,,gk on the other side of   
,g1 such that ,,gh  ,,gk   7the arc     
,,fd71 and proves that the angle        
subtended at ,a by the arc ,,hk is less 
than #a_/dd2 consequently1 he says1 the 
arc would be imperceptible at ,a even in
 that position1 and a fortiori the arc  
,,fd 7which is nearly in a straight line
 with the tangent ,,ad7 is quite        
imperceptible to the observer at ,a4    
,hence 7,prop4 #e71 when the moon       
appears to us halved1 we can take the   
plane of the great circle in the moon   
which is parallel to the dividing circle
 as passing through our eye4 7,it is    
tacitly assumed in ,props4 #c1 #d1 and  
throughout1 that the diameters of the   
sun and moon respectively subtend the   
same angle at our eye47 ,the proof of   
,prop4 #d assumes as known the          
equivalent of the proposition in        
trigonometry that1 if each of the angles
 1  is not greater than a right angle1  
and   1 then    tan  _/ tan   _/  sin   
_/ sin 4  ,prop4 #f proves that the     

moon's orbit is 8lower0 7i4e4 smaller7  
than that of the sun1 and that1 when the
 moon appears to us halved1 it is       
distant less than a quadrant from the   
sun4 ,prop4 #g is the main proposition  
in the treatise4 ,it proves that1 on the
 assumptions made1 the distance of the  
sun from the earth is greater than      
eighteen times1 but less than twenty    
times1 the distance of the moon from the
 earth4 ,the proof is simple and elegant
 and should delight any mathematician2  
its two parts depend respectively on the
 geometrical equivalents of the two     
inequalities stated in the formula      
quoted above1 namely1    tan  _/ tan    
_/  sin  _/ sin 1  where 1  are angles  
not greater than a right angle and   4  
,aristarchus also1 in this proposition1 
cites #g_/e as an approximation by      
defect to the value of #b1 an           
approximation found by the ,pythagoreans
 and quoted by ,plato4 ,the             
trigonometrical equivalent of the result
 obtained in ,prop4 #g is    #a_/ah  sin

 #c  #a_/bj4  ,prop4 #h states that1    
when the sun is totally eclipsed1 the   
sun and moon are comprehended by one and
 the same cone which has its vertex at  
our eye4 ,aristarchus supports this by  
the arguments 7#a7 that1 if the sun     
overlapped the moon1 it would not be    
totally eclipsed1 and 7#b7 that1 if the 
sun fell short 7i4e4 was more than      
covered71 it would remain totally       
eclipsed for some time1 which it does   
not 7this1 he says1 is manifest from    
observation74 ,it is clear from this    
reasoning that ,aristarchus had not     
observed the phenomenon of an annular   
eclipse of the sun2 and it is curious   
that the first mention of an annular    
eclipse seems to be that quoted by      
,simplicius from ,sosigenes 7second     
century1 ,a4,d471 the teacher of        
,alexander ,aphrodisiensis4  ,it follows
 7,prop4 #i7 from ,prop4 #h that the    
diameters of the sun and moon are in the
 same ratio as their distances from the 
earth respectively1 that is to say      

7,prop4 #g7 in a ratio greater than #ah 
3 #a but less than #bj 3 #a4 ,hence     
7,prop4 #aj7 the volume of the sun is   
more than #ehcb times and less than     
#hjjj times that of the moon4  ,by the  
usual geometrical substitute for        
trigonometry ,aristarchus proves in     
,prop4 #aa that the diameter of the moon
 has to the distance between the centre 
of the moon and our eye a ratio which is
 less than #b_/deths but greater than   
#a_/cjth4 ,since the angle subtended by 
the moon's diameter at the observer's   
eye is assumed to be #b1 this           
proposition is equivalent to the        
trigonometrical formula    #a_/de  sin  
#a  #a_/fj4  ,having proved in ,prop4 #d
 that1 so far as our perception goes1   
the dividing circle in the moon is      
indistinguishable from a great circle1  
,aristarchus goes behind perception and 
proves in ,prop4 #ab that the diameter  
of the dividing circle is less than the 
diameter of the moon but greater than   
#hi_/ijths of it4 ,this is again because

 half the angle subtended by the moon at
 the eye is assumed to be #a or #a_/ijth
 of a right angle4 ,the proposition is  
equivalent to the trigonometrical       
formula    #a  cos #a  #hi_/ij4  ,we    
come now to propositions which depend on
 ,hypothesis #e that 8the breadth of the
 earth's shadow is that of two moons04  
,prop4 #ac is about the diameter of the 
circular section of the cone formed by  
the earth's shadow at the place where   
the moon passes through it in an        
eclipse1 and it is worth while to notice
 the extreme accuracy with which        
,aristarchus describes the diameter in  
question4 ,it is with him 8the straight 
line subtending the portion intercepted 
within the earth's shadow of the        
circumference of the circle in which the
 extremities of the diameter of the     
circle dividing the dark and the bright 
portions in the moon move40 ,aristarchus
 proves that the length of the straight 
line in question has to the diameter of 
the moon a ratio less than #b but       

greater than #hh 3 #de1 and has to the  
diameter of the sun a ratio less than #a
 3 #i but greater than #bb 3 #bbe4 ,the 
ratio of the straight line to the       
diameter of the moon is1 in point of    
fact1 #b cos #a or #b sin #hi1 and      
,aristarchus therefore proves the       
equivalent of    #b  #b cos #a          
7#hi_/de7 or #giba_/djej4  ,he then     
observes 7without explanation7 that     
#giba_/djej  #hh_/de 7an approximation  
easily obtained by developing           
#giba_/djej as a continued fraction 7 #a
  7#a    #a    #a7_/7a  #ba  #b7772 his 
result is therefore equivalent to    #a 
cos #a  #dd_/de4  ,the next propositions
 are the equivalents of more complicated
 trigonometrical formul4 ,prop4 #ad is  
an auxiliary proposition to ,prop4 #ae4 
,the diameter of the shadow dealt with  
in ,prop4 #ac divides into two parts the
 straight line joining the centre of the
 earth to the centre of the moon1 and   
,prop4 #ad shows that the whole length  
of this line is more than #fge times the

 part of it terminating in the centre of
 the moon4 ,with the aid of ,props4 #g1 
#ac1 and #ad ,aristarchus is now able1  
in ,prop4 #ae1 to prove another of his  
main results1 namely1 that the diameter 
of the sun has to the diameter of the   
earth a ratio greater than #ai 3 #c but 
less than #dc 3 #f4 ,in the second half 
of the proof he has to handle quite     
large numbers4 ,if ,a be the centre of  
the sun1 ,b the centre of the earth1 and
 ,m the vertex of the cone formed by the
 earth's shadow1 he proves that ,,ma 3  
,,ab is greater than 7#ajabe  #gjhg7 3  
7#iadf  #fgej7 or #gageehge 3 #fagceejj1
 and then adds1 without any word of     
explanation1 that the latter ratio is   
greater than #dc 3 #cg4 ,here again it  
is difficult not to see in #dc 3 #cg the
 continued fraction #a  #aa_/7ff72 and  
although we cannot suppose that the     
,greeks could actually develop          
#gageehge_/fagceejj or #babfa_/ahbib as 
a continued fraction 7in form71 8we have
 here an important proof of the         

employment by the ancients of a method  
of calculation1 the theory of which     
unquestionably belongs to the moderns1  
but the first applications of which are 
too simple not to have originated in    
very remote times0 7,paul ,tannery74    
,the remaining propositions contain no  
more than arithmetical inferences from  
the foregoing4 ,prop4 #af is to the     
effect that the volume of the sun has to
 the volume of the earth a ratio greater
 than #fhei 3 #bg but less than #giejg 3
 #baf 7the numbers are the cubes of     
those in ,prop4 #ae72 ,prop4 #ag proves 
that the diameter of the earth is to    
that of the moon in a ratio greater than
 #ajh 3 #dc but less than #fj 3 #ai     
7ratios compounded of those in ,props4  
#i and #ae71 and ,prop4 #ah proves that 
the volume of the earth is to that of   
the moon in a ratio greater than        
#abeigab 3 #giejg but less than #bafjjj 
3 #fhei4   ,,aristarchus ,,on ,,the     
,,year ,,and 8,,great ,,year04          
,aristarchus is said to have increased  

by #a_/afbcrd of a day ,callippus's     
figure of #cfe days as the length of the
 solar year1 and to have given #bdhd    
years as the length of the ,great ,year 
or the period after which the sun1 the  
moon and the five planets return to the 
same position in the heavens4 ,tannery  
has shown reason for thinking that #bdhd
 is a wrong reading for #bdcd years1 and
 he gives an explanation which seems    
convincing of the way in which          
,aristarchus arrived at #bdcd years as  
the length of the ,great ,year4 ,the    
,chaldan period of #bbc lunations was   
well known in ,greece4 ,its length was  
calculated to be #fehe days1 and in this
 period the sun was estimated to        
describe #aj of its circle in addition  
to #ah sidereal revolutions4 ,the       
,greeks used the period called by them  
exeligmus which was three times the     
period of #bbc lunations and contained a
 whole number of days1 namely1 #aigef1  
during which the sun described #cb in   
addition to #ed sidereal revolutions4   

,it followed that the number of days in 
the sidereal year was--    #aigef_/7ed  
#cb_/cfj7  #aigef_/7ed  #d_/de7  7#de   
#aigef7_/bdcd         #hhijbj_/bdcd     
#cfe  #c_/dhfh4  ,now #dhfh_/c  #afbc - 
1 and ,aristarchus seems to have merely 
replaced #c_/dhfh by the close          
approximation #a_/afbc4 ,the calculation
 was1 however1 of no value because the  
estimate of #aj over #ah sidereal       
revolutions seems to have been an       
approximation based merely on the       
difference between #fehe days and #ah   
years of #cfe days1 i4e4 #fegd days2    
thus the #aj itself probably depended on
 a solar year of #cfe days1 and         
,aristarchus's evaluation of it as #cfe 
#a_/afbc was really a sort of circular  
argument like the similar calculation of
 the length of the year made by nopides 
of ,chios4   ,,later ,,improvements ,,on
 ,,aristarchus',s ,,figures4  ,it may   
interest the reader to know how far     
,aristarchus's estimates of sizes and   
distances were improved upon by later   

,greek astronomers4 ,we are not informed
 how large he conceived the earth to be2
 but ,archimedes tells us that 8some    
have tried to prove that the            
circumference of the earth is about     
#cjj1jjj stades and not greater10 and it
 may be presumed that ,aristarchus      
would1 like ,archimedes1 be content with
 this estimate4 ,it is probable that it 
was ,dicaearchus who 7about #cjj ,b4,c47
 arrived at this value1 and that it was 
obtained by taking #bd 7#a_/aeth of the 
whole meridian circle7 as the difference
 of latitude between ,syene and         
,lysimachia 7on the same meridian7 and  
#bj1jjj stades as the actual distance   
between the two places4 ,eratosthenes1  
born a few years after ,archimedes1 say 
#bhd ,b4,c41 is famous for a better     
measurement of the earth which was based
 on scientific principles4 ,he found    
that at noon at the summer solstice the 
sun threw no shadow at ,syene1 whereas  
at the same hour at ,alexandria 7which  
he took to be on the same meridian7 a   

vertical stick cast a shadow            
corresponding to #a_/ejth of the        
meridian circle4 ,assuming then that the
 sun's rays at the two places are       
parallel in direction1 and knowing the  
distance between them to be #ejjj       
stades1 he had only to take #ej times   
#ejjj stades to get the circumference of
 the earth4 ,he seems1 for some reason1 
to have altered #bej1jjj into #beb1jjj  
stades1 and this1 according to ,pliny's 
account of the kind of stade used1 works
 out to about #bd1ffb miles1 giving for 
the diameter of the earth a length of   
#ghej miles1 a surprisingly close       
approximation1 however much it owes to  
happy accidents in the calculation4     
,eratosthenes's estimates of the sizes  
and distances of the sun and moon cannot
 be restored with certainty in view of  
the defective state of the texts of our 
authorities4 ,we are better informed of 
,hipparchus's results4 ,in the first    
book of a treatise on sizes and         
distances ,hipparchus based himself on  

an observation of an eclipse of the sun1
 probably that of #bjth ,november in the
 year #abi ,b4,c41 which was exactly    
total in the region about the           
,hellespont1 whereas at ,alexandria     
about ths only of the diameter was      
obscured4 ,from these facts ,hipparchus 
deduced that1 if the radius of the earth
 be the unit1 the least distance of the 
moon contains #ga1 and the greatest #hc 
of these units1 the mean thus containing
 #gg4 ,but he reverted to the question  
in the second book and proved 8from many
 considerations0 that the mean distance 
of the moon is #fg times the radius of  
the earth1 and also that the distance of
 the sun is #bdij times the radius of   
the earth4 ,hipparchus also made the    
size 7meaning thereby the solid content7
 of the sun to be #ahhj times that of   
the earth1 and the size of the earth to 
be #bg times that of the moon4 ,the cube
 root of #ahhj being about #ab1 the     
diameters of the sun1 earth and moon    
would be in the ratio of the numbers    

#ab1 #a1 4 ,hipparchus seems to have    
accepted ,eratosthenes's estimate of    
#beb1jjj stades for the circumference of
 the earth4  ,it is curious that        
,posidonius 7#ace-ea ,b4,c471 who was   
much less of an astronomer1 made a much 
better guess at the distance of the sun 
from the earth4 ,he made it #ejj1jjj1jjj
 stades4 ,as he also estimated the      
circumference of the earth at #bdj1jjj  
stades1 we may take the diameter of the 
earth to be1 according to ,posidonius1  
about #gf1djj stades2 consequently1 if  
,d be that diameter1 ,posidonius made   
the distance of the sun to be equal to  
#fede;,d as compared with ,hipparchus's 
#abde;,d4  ,ptolemy does not mention    
,hipparchus's figures4 ,his own estimate
 of the sun's distance was #fje;,d1 so  
that ,hipparchus was far nearer the     
truth4 ,but ,hipparchus's estimate      
remained unknown and ,ptolemy's held the
 field for many centuries2 even         
,copernicus only made the distance of   
the sun #gej times the earth's diameter1

 and it was not till #afga-c that a     
substantial improvement was made2       
observations of ,mars carried out in    
those years by ,richer enabled ,cassini 
to conclude that the sun's parallax was 
about #ie corresponding to a distance   
between the sun and the earth of        
#hg1jjj1jjj miles4  ,ptolemy made the   
distance of the moon from the earth to  
be #bi times the earth's diameter1 and  
the diameter of the earth to be #c times
 that of the moon4 ,he estimated the    
diameter of the sun at #ah times that of
 the moon and therefore about #e times  
that of the earth1 a figure again much  
inferior to that given by ,hipparchus4  
,,bibliography4   ,on the astronomy of  
the early ,greek philosophers much      
information is given by ,aristotle      
7especially in the ,de ,caelo72 for     
,aristotle was fortunately in the habit 
of stating the views of earlier thinkers
 as a preliminary to enunciating his    
own4 ,apart from what we learn from     
,aristotle1 we are mainly dependent on  

the fragmentary accounts of the opinions
 of philosophers which were collected in
 the ,doxographi ,graeci of ,hermann    
,diels 7,berlin1 #ahgi71 to which must  
be added1 for the period before         
,socrates1 ,die ,fragmente der          
,vorsokratiker by the same editor 7#bnd 
edition1 with index1 #aijf-aj1 #crd     
edition1 #aiab74 ,the doxographic data  
and the fragments for the period from   
,thales to ,empedocles were translated  
and explained by ,paul ,tannery in ,pour
 l'histoire de la ,science ,hellne      
7,paris1 #ahhg74 ,a ,history of         
,astronomy was written by ,eudemus of   
,rhodes1 a pupil of ,aristotle2 this is 
lost1 but a fair number of fragments are
 preserved by later writers4 ,of the    
theory of concentric spheres we have a  
short account in ,aristotle's           
,metaphysics1 but a more elaborate and  
detailed description is contained in    
,simplicius's commentary on the ,de     
,caelo2 ,simplicius quotes largely from 
,sosigenes the ,peripatetic 7second     

century ,a4,d47 who drew upon ,eudemus4 
,there are a number of valuable         
histories of ,greek astronomy4 ,in      
,german we have ,schaubach1 ,geschichte 
der griechischen ,astronomie bis auf    
,eratosthenes1 #ahji2 ,r4 ,wolf1        
,geschichte der ,astronomie1 #ahgg2 and 
two admirable epitomes 7#a7 by ,siegmund
 ,gnther in ,windelband's ,geschichte   
der alten ,philosophie 7,iwan von       
,mller's ,handbuch der klassischen      
,altertumswissenschaft1 ,vol4 ,v1 ,pt4  
#a71 #bnd edition1 #ahid1 and 7#b7 by   
,friedrich ,hultsch in ,pauly-,wissowa's
 ,real-,encyclopdie der classischen     
,altertumswissenschaft 7,art4           
,astronomie in ,vol4 ,,ii1 #b1 #ahif74  
,in ,french1 besides the great work of  
,delambre1 ,histoire de l'astronomie    
ancienne1 #ahag1 we have the valuable   
studies of ,paul ,tannery in ,recherches
 sur l'histoire de l'astronomie         
ancienne1 #ahic1 and ,pierre ,duhem1 ,le
 ,systme du ,monde1 ,vol4 ,i1 #aiac4    
,in ,english1 reference may be made to  

,sir ,g4 ,cornewall ,lewis1 ,an         
,historical ,survey of the ,astronomy of
 the ,ancients1 #ahfc2 ,j4 ,l4 ,e4      
,dreyer1 ,history of the ,planetary     
,systems from ,thales to ,kepler1       
,cambridge1 #aijf2 and the historical   
portion of ,sir ,thomas ,heath's        
,aristarchus of ,samos1 the ancient     
,copernicus1 ,oxford1 #aiac4            
,aristarchus's treatise ,on the ,sizes  
and ,distances of the ,sun and ,moon    
first appeared in a ,latin translation  
by ,george ,valla in #adhh and #adih1   
and next in a ,latin translation by     
,commandinus 7#aegb74 ,the editio       
princeps of the ,greek text was brought 
out by ,john ,wallis1 ,oxford1 #afhh1   
and was reprinted in ,johannis ,wallis  
,opera ,mathematica1 #afic-afii1 ,vol4  
,,iii1 in both cases along with         
,commandinus's translation4 ,in #ahaj   
there appeared an edition by the ,comte 
de ,fortia d',urban1 ,histoire          
d',aristarque de ,samos 444 including   
the ,greek text and ,commandinus's      

translation but without figures2 a      
,french translation by ,fortia d',urban 
followed in #ahbc4 ,the treatise was    
translated into ,german by ,a4 ,nokk in 
#ahed4 ,finally1 ,sir ,thomas ,heath's  
work above cited contains a new ,greek  
text with ,english translation and      
notes4     ,,chronology4  7,approximate 
where precise dates are not known47     
,b4,c4           #fbd-edg       ,thales4
           #faj-edf       ,anaximander4 
#ehe-ebf       ,anaximenes4             
#egb-dig       ,pythagoras4   born #eaf 
or #ead       ,parmenides4   7possibly  
#edj74           #ejj-dbh               
,anaxagoras4           #did-dcd         
,empedocles4       #eth century         
nopides of ,chios4                      
,philolaus4           #dbg-cdg          
,plato4           #djh-cee              
,eudoxus4           #chh-cae            
,heraclides of ,pontus4                 
#chd-cbb       ,aristotle4              
#cgj-cjj       ,callippus4              
#caj-bcj       ,aristarchus of ,samos4  

#bhg-bab       ,archimedes4             
#bhd-bjc       ,eratosthenes4           
#bfe-aij       ,apollonius of ,perga4   
#crd century       ,aratus4             
fl4 #aej       ,hipparchus4             
#ace-ea       ,posidonius4              
,a4,d4            #ej-abe               
,plutarch4           #ajj-agh           
,ptolemy4   ,,aberdeen3 ,,the           
,,university ,,press     ,transcriber's 
,notes   ,punctuation and spelling were 
made consistent when a predominant      
preference was found in this book2      
otherwise they were not changed4        
,simple typographical errors were       
corrected4 ,one unbalanced parenthesis  
was remedied4  ,italics are indicated by
 underscores4