,transcriber's notes3  7#a7 ,numbers    
following letters 7without space7 like  
,c#b were originally       printed in   
subscript4 ,letter subscripts are       
preceded by an       underscore1 like   
,cn4  7#b7 ,characters following a carat
 77 were originally printed in          
superscript4  7#c7 root stands for the  
root symbol2 alpha1 beta1 etc4 for greek
       letters4     ,illustration3      
,aristippus ,philosophus ,socraticus1   
nausragio cum ejectus ad ,rhodiensium   
litus animadvertisses ,geometrica       
,schemata descripta1 exclamavisse ad    
comites ita dicitur1 ,bene ,speremus1   
,hominum enim vestigia video4  ,vitruv4 
,architect lib4 #f4 ,prief4  delin      
,m,burghers sculpt,univ4 ,oxon4         
,,pioneers ,,of ,,progress    ,,men ,,of
 ,,science    ,,edited ,,by ,s4         
,,chapman1 ,m4,a41 ,d4,,sc41 ,f4,r4,s4  
,,archimedes     ,,by    ,,sir ,,thomas 
,,heath   ,k4,c4,b41 ,k4,c4,v4,o41      
,f4,r4,s42 ,,sc4,d41 ,,camb4   ,,hon4   
,d4,,sc41 ,,oxford    ,greek3 ,dos moi  

pou sto1 kai kino ten gen     ,,london3 
,,society ,,for ,,promoting             
,,christian ,,knowledge   ,,new ,,york3 
,,the ,,macmillan ,,co4   #aibj         
,,contents4    ,,chap4                  
,,page     ,i4 ,,archimedes             
#a     ,,ii4 ,,greek ,,geometry ,,to    
,,archimedes    #g    ,,iii4 ,,the      
,,works ,,of ,,archimedes        #bd    
,,iv4 ,,geometry ,,in ,,archimedes      
#bi      ,v4 ,,the ,,sandreckoner       
#de     ,,vi4 ,,mechanics               
#ej    ,,vii4 ,,hydrostatics            
#ec    ,,bibliography                   
#eg    ,,chronology                     
#eh     ,,chapter ,i4  ,,archimedes4    
,if the ordinary person were asked to   
say off-hand what he knew of            
,archimedes1 he would probably1 at the  
most1 be able to quote one or other of  
the well-known stories about him3 how1  
after discovering the solution of some  
problem in the bath1 he was so overjoyed
 that he ran naked to his house1        
shouting ,greek3 eureka1 eureka 7or1 as 

we might say1 ,i've got it1 ,i've got   
it72 or how he said ,give me a place to 
stand on and ,i will move the earth2 or 
again how he was killed1 at the capture 
of ,syracuse in the ,second ,punic ,war1
 by a ,roman soldier who resented being 
told to get away from a diagram drawn on
 the ground which he was studying4  ,and
 it is to be feared that few who are not
 experts in the history of mathematics  
have any acquaintance with the details  
of the original discoveries in          
mathematics of the greatest             
mathematician of antiquity1 perhaps the 
greatest mathematical genius that the   
world has ever seen4  ,history and      
tradition know ,archimedes almost       
exclusively as the inventor of a number 
of ingenious mechanical appliances1     
things which naturally appeal more to   
the popular imagination than the        
subtleties of pure mathematics4  ,almost
 all that is told of ,archimedes reaches
 us through the accounts by ,polybius   
and ,plutarch of the siege of ,syracuse 

by ,marcellus4 ,he perished in the sack 
of that city in #bab ,b4,c41 and1 as he 
was then an old man 7perhaps #ge years  
old71 he must have been born about #bhg 
,b4,c4 ,he was the son of ,phidias1 an  
astronomer1 and was a friend and kinsman
 of ,king ,hieron of ,syracuse and his  
son ,gelon4 ,he spent some time at      
,alexandria studying with the successors
 of ,euclid 7,euclid who flourished     
about #cjj ,b4,c4 was then no longer    
living74 ,it was doubtless at           
,alexandria that he made the            
acquaintance of ,conon of ,samos1 whom  
he admired as a mathematician and       
cherished as a friend1 as well as of    
,eratosthenes2 to the former1 and to the
 latter during his early period he was  
in the habit of communicating his       
discoveries before their publication4   
,it was also probably in ,egypt that he 
invented the water-screw known by his   
name1 the immediate purpose being the   
drawing of water for irrigating fields4 
,after his return to ,syracuse he lived 

a life entirely devoted to mathematical 
research4 ,incidentally he became famous
 through his clever mechanical          
inventions4 ,these things were1 however1
 in his case the diversions of geometry 
at play1 and he attached no importance  
to them4 ,in the words of ,plutarch1 he 
possessed so lofty a spirit1 so profound
 a soul1 and such a wealth of scientific
 knowledge that1 although these         
inventions had won for him the renown of
 more than human sagacity1 yet he would 
not consent to leave behind him any     
written work on such subjects1 but1     
regarding as ignoble and sordid the     
business of mechanics and every sort of 
art which is directed to practical      
utility1 he placed his whole ambition in
 those speculations in the beauty and   
subtlety of which there is no admixture 
of the common needs of life4  ,during   
the siege of ,syracuse ,archimedes      
contrived all sorts of engines against  
the ,roman besiegers4 ,there were       
catapults so ingeniously constructed as 

to be equally serviceable at long or    
short range1 and machines for           
discharging showers of missiles through 
holes made in the walls4 ,other machines
 consisted of long movable poles        
projecting beyond the walls2 some of    
these dropped heavy weights upon the    
enemy's ships and on the constructions  
which they called sambuca1 from their   
resemblance to a musical instrument of  
that name1 and which consisted of a     
protected ladder with one end resting on
 two quinqueremes lashed together side  
by side as base1 and capable of being   
raised by a windlass2 others were fitted
 with an iron hand or a beak like that  
of a crane1 which grappled the prows of 
ships1 then lifted them into the air and
 let them fall again4 ,marcellus is said
 to have derided his own engineers and  
artificers with the words1 ,shall we not
 make an end of fighting with this      
geometrical ,briareus who uses our ships
 like cups to ladle water from the sea1 
drives our sambuca off ignominiously    

with cudgel-blows1 and1 by the multitude
 of missiles that he hurls at us all at 
once1 outdoes the hundred-handed giants 
of mythology8 ,but the exhortation had  
no effect1 the ,romans being in such    
abject terror that1 if they did but see 
a piece of rope or wood projecting above
 the wall they would cry 'there it is1' 
declaring that ,archimedes was setting  
some engine in motion against them1 and 
would turn their backs and run away1    
insomuch that ,marcellus desisted from  
all fighting and assault1 putting all   
his hope in a long siege4  ,archimedes  
died1 as he had lived1 absorbed in      
mathematical contemplation4 ,the        
accounts of the circumstances of his    
death differ in some details4 ,plutarch 
gives more than one version in the      
following passage3 ,marcellus was most  
of all afflicted at the death of        
,archimedes1 for1 as fate would have it1
 he was intent on working out some      
problem with a diagram1 and1 his mind   
and his eyes being alike fixed on his   

investigation1 he never noticed the     
incursion of the ,romans nor the capture
 of the city4 ,and when a soldier came  
up to him suddenly and bade him follow  
to ,marcellus1 he refused to do so until
 he had worked out his problem to a     
demonstration2 whereat the soldier was  
so enraged that he drew his sword and   
slew him4 ,others say that the ,roman   
ran up to him with a drawn sword1       
threatening to kill him2 and1 when      
,archimedes saw him1 he begged him      
earnestly to wait a little while in     
order that he might not leave his       
problem incomplete and unsolved1 but the
 other took no notice and killed him4   
,again1 there is a third account to the 
effect that1 as he was carrying to      
,marcellus some of his mathematical     
instruments1 sundials1 spheres1 and     
angles adjusted to the apparent size of 
the sun to the sight1 some soldiers met 
him and1 being under the impression that
 he carried gold in the vessel1 killed  
him4 ,the most picturesque version of   

the story is that which represents him  
as saying to a ,roman soldier who came  
too close1 ,stand away1 fellow1 from my 
diagram1 whereat the man was so enraged 
that he killed him4  ,archimedes is said
 to have requested his friends and      
relatives to place upon his tomb a      
representation of a cylinder            
circumscribing a sphere within it1      
together with an inscription giving the 
ratio 7#c_/b7 which the cylinder bears  
to the sphere2 from which we may infer  
that he himself regarded the discovery  
of this ratio as his greatest           
achievement4 ,cicero1 when quaestor in  
,sicily1 found the tomb in a neglected  
state and restored it4 ,in modern times 
not the slightest trace of it has been  
found4  ,beyond the above particulars of
 the life of ,archimedes1 we have       
nothing but a number of stories which1  
if perhaps not literally accurate1 yet  
help us to a conception of the          
personality of the man which we would   
not willingly have altered4 ,thus1 in   

illustration of his entire preoccupation
 by his abstract studies1 we are told   
that he would forget all about his food 
and such necessities of life1 and would 
be drawing geometrical figures in the   
ashes of the fire1 or1 when anointing   
himself1 in the oil on his body4 ,of the
 same kind is the story mentioned above1
 that1 having discovered while in a bath
 the solution of the question referred  
to him by ,hieron as to whether a       
certain crown supposed to have been made
 of gold did not in fact contain a      
certain proportion of silver1 he ran    
naked through the street to his home    
shouting ,greek3 eureka1 eureka4  ,it   
was in connexion with his discovery of  
the solution of the problem ,to move a  
given weight by a given force that      
,archimedes uttered the famous saying1  
,give me a place to stand on1 and ,i can
 move the earth 7,greek3 dos moi pou sto
 kai kino ten gen1 or in his broad      
,doric1 as one version has it1 ,greek3  
pa bo kai kino tan gan74 ,plutarch      

represents him as declaring to ,hieron  
that any given weight could be moved by 
a given force1 and boasting1 in reliance
 on the cogency of his demonstration1   
that1 if he were given another earth1 he
 would cross over to it and move this   
one4 ,and when ,hieron was struck with  
amazement and asked him to reduce the   
problem to practice and to show him some
 great weight moved by a small force1 he
 fixed on a ship of burden with three   
masts from the king's arsenal which had 
only been drawn up by the great labour  
of many men2 and loading her with many  
passengers and a full freight1 sitting  
himself the while afar off1 with no     
great effort but quietly setting in     
motion with his hand a compound pulley1 
he drew the ship towards him smoothly   
and safely as if she were moving through
 the sea4 ,hieron1 we are told          
elsewhere1 was so much astonished that  
he declared that1 from that day forth1  
,archimedes's word was to be accepted on
 every subject6 ,another version of the 

story describes the machine used as a   
helix2 this term must be supposed to    
refer to a screw in the shape of a      
cylindrical helix turned by a handle and
 acting on a cog-wheel with oblique     
teeth fitting on the screw4  ,another   
invention was that of a sphere          
constructed so as to imitate the motions
 of the sun1 the moon1 and the five     
planets in the heavens4 ,cicero actually
 saw this contrivance1 and he gives a   
description of it1 stating that it      
represented the periods of the moon and 
the apparent motion of the sun with such
 accuracy that it would even 7over a    
short period7 show the eclipses of the  
sun and moon4 ,it may have been moved by
 water1 for ,pappus speaks in one place 
of those who understand the making of   
spheres and produce a model of the      
heavens by means of the regular circular
 motion of water4 ,in any case it is    
certain that ,archimedes was much       
occupied with astronomy4 ,livy calls him
 unicus spectator caeli siderumque4     

,hipparchus says1 ,from these           
observations it is clear that the       
differences in the years are altogether 
small1 but1 as to the solstices1 ,i     
almost think that both ,i and           
,archimedes have erred to the extent of 
a quarter of a day both in observation  
and in the deduction therefrom4 ,it     
appears1 therefore1 that ,archimedes had
 considered the question of the length  
of the year4 ,macrobius says that he    
discovered the distances of the planets4
 ,archimedes himself describes in the   
,sandreckoner the apparatus by which he 
measured the apparent diameter of the   
sun1 i4e4 the angle subtended by it at  
the eye4  ,the story that he set the    
,roman ships on fire by an arrangement  
of burning-glasses or concave mirrors is
 not found in any authority earlier than
 ,lucian 7second century ,a4,d472 but   
there is no improbability in the idea   
that he discovered some form of         
burning-mirror1 e4g4 a paraboloid of    
revolution1 which would reflect to one  

point all rays falling on its concave   
surface in a direction parallel to its  
axis4     ,,chapter ,,ii4  ,,greek      
,,geometry ,,to ,,archimedes4   ,in     
order to enable the reader to arrive at 
a correct understanding of the place of 
,archimedes and of the significance of  
his work it is necessary to pass in     
review the course of development of     
,greek geometry from its first          
beginnings down to the time of ,euclid  
and ,archimedes4  ,greek authors from   
,herodotus downwards agree in saying    
that geometry was invented by the       
,egyptians and that it came into ,greece
 from ,egypt4 ,one account says3--      
,geometry is said by many to have been  
invented among the ,egyptians1 its      
origin being due to the measurement of  
plots of land4 ,this was necessary there
 because of the rising of the ,nile1    
which obliterated the boundaries        
appertaining to separate owners4 ,nor is
 it marvellous that the discovery of    
this and the other sciences should have 

arisen from such an occasion1 since     
everything which moves in the sense of  
development will advance from the       
imperfect to the perfect4 ,from         
sense-perception to reasoning1 and from 
reasoning to understanding1 is a natural
 transition4 ,just as among the         
,phoenicians1 through commerce and      
exchange1 an accurate knowledge of      
numbers was originated1 so also among   
the ,egyptians geometry was invented for
 the reason above stated4  ,thales first
 went to ,egypt and thence introduced   
this study into ,greece4  ,but it is    
clear that the geometry of the          
,egyptians was almost entirely practical
 and did not go beyond the requirements 
of the land-surveyor1 farmer or         
merchant4 ,they did indeed know1 as far 
back as #bjjj ,b4,c41 that in a triangle
 which has its sides proportional to #c1
 #d1 #e the angle contained by the two  
smaller sides is a right angle1 and they
 used such a triangle as a practical    
means of drawing right angles4 ,they had

 formulae1 more or less inaccurate1 for 
certain measurements1 e4g4 for the areas
 of certain triangles1                  
parallel-trapezia1 and circles4 ,they   
had1 further1 in their construction of  
pyramids1 to use the notion of similar  
right-angled triangles2 they even had a 
name1 se-qet1 for the ratio of the half 
of the side of the base to the height1  
that is1 for what we should call the    
co-tangent of the angle of slope4 ,but  
not a single general theorem in geometry
 can be traced to the ,egyptians4 ,their
 knowledge that the triangle 7#c1 #d1   
#e7 is right angled is far from implying
 any knowledge of the general           
proposition 7,eucl4 ,i41 #dg7 known by  
the name of ,pythagoras4 ,the science of
 geometry1 in fact1 remained to be      
discovered2 and this required the genius
 for pure speculation which the ,greeks 
possessed in the largest measure among  
all the nations of the world4  ,thales1 
who had travelled in ,egypt and there   
learnt what the priests could teach him 

on the subject1 introduced geometry into
 ,greece4 ,almost the whole of ,greek   
science and philosophy begins with      
,thales4 ,his date was about #fbd-edg   
,b4,c4 ,first of the ,ionian            
philosophers1 and declared one of the   
,seven ,wise ,men in #ehb-eha1 he shone 
in all fields1 as astronomer1           
mathematician1 engineer1 statesman and  
man of business4 ,in astronomy he       
predicted the solar eclipse of #bh ,may1
 #ehe1 discovered the inequality of the 
four astronomical seasons1 and          
counselled the use of the ,little ,bear 
instead of the ,great ,bear as a means  
of finding the pole4 ,in geometry the   
following theorems are attributed to    
him--and their character shows how the  
,greeks had to begin at the very        
beginning of the theory--7#a7 that a    
circle is bisected by any diameter      
7,eucl4 ,i41 ,def4 #ag71 7#b7 that the  
angles at the base of an isosceles      
triangle are equal 7,eucl4 ,i41 #e71    
7#c7 that1 if two straight lines cut one

 another1 the vertically opposite angles
 are equal 7,eucl4 ,i41 #ae71 7#d7 that1
 if two triangles have two angles and   
one side respectively equal1 the        
triangles are equal in all respects     
7,eucl4 ,i41 #bf74 ,he is said 7#e7 to  
have been the first to inscribe a       
right-angled triangle in a circle3 which
 must mean that he was the first to     
discover that the angle in a semicircle 
is a right angle4 ,he also solved two   
problems in practical geometry3 7#a7 he 
showed how to measure the distance from 
the land of a ship at sea 7for this he  
is said to have used the proposition    
numbered 7#d7 above71 and 7#b7 he       
measured the heights of pyramids by     
means of the shadow thrown on the ground
 7this implies the use of similar       
triangles in the way that the ,egyptians
 had used them in the construction of   
pyramids74  ,after ,thales come the     
,pythagoreans4 ,we are told that the    
,pythagoreans were the first to use the 
term ,greek3 mathemata 7literally       

subjects of instruction7 in the         
specialised sense of mathematics2 they1 
too1 first advanced mathematics as a    
study pursued for its own sake and made 
it a part of a liberal education4       
,pythagoras1 son of ,mnesarchus1 was    
born in ,samos about #egb ,b4,c41 and   
died at a great age 7#ge or #hj7 at     
,metapontum4 ,his interests were as     
various as those of ,thales2 his        
travels1 all undertaken in pursuit of   
knowledge1 were probably even more      
extended4 ,like ,thales1 and perhaps at 
his suggestion1 he visited ,egypt and   
studied there for a long period 7#bb    
years1 some say74  ,it is difficult to  
disentangle from the body of            
,pythagorean doctrines the portions     
which are due to ,pythagoras himself    
because of the habit which the members  
of the school had of attributing        
everything to the ,master 7,greek3 autos
 epha1 ipse dixit74 ,in astronomy two   
things at least may safely be attributed
 to him2 he held that the earth is      

spherical in shape1 and he recognised   
that the sun1 moon and planets have an  
independent motion of their own in a    
direction contrary to that of the daily 
rotation2 he seems1 however1 to have    
adhered to the geocentric view of the   
universe1 and it was his successors who 
evolved the theory that the earth does  
not remain at the centre but revolves1  
like the other planets and the sun and  
moon1 about the central fire4 ,perhaps  
his most remarkable discovery was the   
dependence of the musical intervals on  
the lengths of vibrating strings1 the   
proportion for the octave being #b 3 #a1
 for the fifth #c 3 #b and for the      
fourth #d 3 #c4 ,in arithmetic he was   
the first to expound the theory of means
 and of proportion as applied to        
commensurable quantities4 ,he laid the  
foundation of the theory of numbers by  
considering the properties of numbers as
 such1 namely1 prime numbers1 odd and   
even numbers1 etc4 ,by means of figured 
numbers1 square1 oblong1 triangular1    

etc4 7represented by dots arranged in   
the form of the various figures7 he     
showed the connexion between numbers and
 geometry4 ,in view of all these        
properties of numbers1 we can easily    
understand how the ,pythagoreans came to
 liken all things to numbers and to find
 in the principles of numbers the       
principles of all things 7all things are
 numbers74  ,we come now to             
,pythagoras's achievements in geometry4 
,there is a story that1 when he came    
home from ,egypt and tried to found a   
school at ,samos1 he found the ,samians 
indifferent1 so that he had to take     
special measures to ensure that his     
geometry might not perish with him4     
,going to the gymnasium1 he sought out a
 well-favoured youth who seemed likely  
to suit his purpose1 and was withal     
poor1 and bribed him to learn geometry  
by promising him sixpence for every     
proposition that he mastered4 ,very soon
 the youth got fascinated by the subject
 for its own sake1 and ,pythagoras      

rightly judged that he would gladly go  
on without the sixpence4 ,he hinted1    
therefore1 that he himself was poor and 
must try to earn his living instead of  
doing mathematics2 whereupon the youth1 
rather than give up the study1          
volunteered to pay sixpence to          
,pythagoras for each proposition4  ,in  
geometry ,pythagoras set himself to lay 
the foundations of the subject1         
beginning with certain important        
definitions and investigating the       
fundamental principles4 ,of propositions
 attributed to him the most famous is1  
of course1 the theorem that in a        
right-angled triangle the square on the 
hypotenuse is equal to the sum of the   
squares on the sides about the right    
angle 7,eucl4 ,i41 #dg72 and1 seeing    
that ,greek tradition universally       
credits him with the proof of this      
theorem1 we prefer to believe that      
tradition is right4 ,this is to some    
extent confirmed by another tradition   
that ,pythagoras discovered a general   

formula for finding two numbers such    
that the sum of their squares is a      
square number4 ,this depends on the     
theory of the gnomon1 which at first had
 an arithmetical signification          
corresponding to the geometrical use of 
it in ,euclid1 ,book ,,ii4 ,a figure in 
the shape of a gnomon put round two     
sides of a square makes it into a larger
 square4 ,now consider the number #a    
represented by a dot4 ,round this place 
three other dots so that the four dots  
form a square 7#a  #c  #bb74 ,round the 
four dots 7on two adjacent sides of the 
square7 place five dots at regular and  
equal distances1 and we have another    
square 7#a  #c  #e  #cb72 and so on4    
,the successive odd numbers #a1 #c1 #e  
444 were called gnomons1 and the general
 formula is    #a  #c  #e  444  7#bn -  
#a7  n#b4  ,add the next odd number1    
i4e4 #bn  #a1 and we have n#b  7#bn  #a7
  7n  #a7b4 ,in order1 then1 to get two 
square numbers such that their sum is a 
square we have only to see that #bn  #a 

is a square4 ,suppose that #bn  #a  m#b2
 then n  #a_/b7m#b - #a71 and we have   
#a_/b7m#b - #a7b  m#b  #a_/b7m#b  #a7b1 
where m is any odd number2 and this is  
the general formula attributed to       
,pythagoras4  ,proclus also attributes  
to ,pythagoras the theory of            
proportionals and the construction of   
the five cosmic figures1 the five       
regular solids4  ,one of the said       
solids1 the dodecahedron1 has twelve    
pentagonal faces1 and the construction  
of a regular pentagon involves the      
cutting of a straight line in extreme   
and mean ratio 7,eucl4 ,,ii41 #aa1 and  
,,vi41 #cj71 which is a particular case 
of the method known as the application  
of areas4 ,how much of this was due to  
,pythagoras himself we do not know2 but 
the whole method was at all events fully
 worked out by the ,pythagoreans and    
proved one of the most powerful of      
geometrical methods4 ,the most          
elementary case appears in ,euclid1 ,i41
 #dd1 #de1 where it is shown how to     

apply to a given straight line as base a
 parallelogram having a given angle 7say
 a rectangle7 and equal in area to any  
rectilineal figure2 this construction is
 the geometrical equivalent of          
arithmetical division4 ,the general case
 is that in which the parallelogram1    
though applied to the straight line1    
overlaps it or falls short of it in such
 a way that the part of the             
parallelogram which extends beyond1 or  
falls short of1 the parallelogram of the
 same angle and breadth on the given    
straight line itself 7exactly7 as base  
is similar to another given             
parallelogram 7,eucl4 ,,vi41 #bh1 #bi74 
,this is the geometrical equivalent of  
the most general form of quadratic      
equation ax - mx#b  ,c1 so far as it has
 real roots2 while the condition that   
the roots may be real was also worked   
out 7 ,eucl4 ,,vi41 #bg74 ,it is        
important to note that this method of   
application of areas was directly used  
by ,apollonius of ,perga in formulating 

the fundamental properties of the three 
conic sections1 which properties        
correspond to the equations of the      
conics in ,cartesian co-ordinates2 and  
the names given by ,apollonius 7for the 
first time7 to the respective conics are
 taken from the theory1 parabola        
7,greek3 parabole7 meaning application  
7i4e4 in this case the parallelogram is 
applied to the straight line exactly71  
hyperbola 7,greek3 hyperbole71 exceeding
 7i4e4 in this case the parallelogram   
exceeds or overlaps the straight line71 
ellipse 7,greek3 elleipsis71 falling    
short 7i4e4 the parallelogram falls     
short of the straight line74  ,another  
problem solved by the ,pythagoreans is  
that of drawing a rectilineal figure    
equal in area to one given rectilineal  
figure and similar to another4 ,plutarch
 mentions a doubt as to whether it was  
this problem or the proposition of      
,euclid ,i41 #dg1 on the strength of    
which ,pythagoras was said to have      
sacrificed an ox4  ,the main particular 

applications of the theorem of the      
square on the hypotenuse 7e4g4 those in 
,euclid1 ,book ,,ii47 were also         
,pythagorean2 the construction of a     
square equal to a given rectangle       
7,eucl4 ,,ii41 #ad7 is one of them and  
corresponds to the solution of the pure 
quadratic equation x#b  ab4  ,the       
,pythagoreans proved the theorem that   
the sum of the angles of any triangle is
 equal to two right angles 7,eucl4 ,i41 
#cb74  ,speaking generally1 we may say  
that the ,pythagorean geometry covered  
the bulk of the subject-matter of ,books
 ,i41 ,,ii41 ,,iv41 and ,,vi4 of ,euclid
 7with the qualification1 as regards    
,book ,,vi41 that the ,pythagorean      
theory of proportion applied only to    
commensurable magnitudes74 ,our         
information about the origin of the     
propositions of ,euclid1 ,book ,,iii41  
is not so complete2 but it is certain   
that the most important of them were    
well known to ,hippocrates of ,chios    
7who flourished in the second half of   

the fifth century1 and lived perhaps    
from about #dgj to #djj ,b4,c471 whence 
we conclude that the main propositions  
of ,book ,,iii4 were also included in   
the ,pythagorean geometry4  ,lastly1 the
 ,pythagoreans discovered the existence 
of incommensurable lines1 or of         
irrationals4 ,this was1 doubtless1 first
 discovered with reference to the       
diagonal of a square which is           
incommensurable with the side1 being in 
the ratio to it of root#b to #a4 ,the   
,pythagorean proof of this particular   
case survives in ,aristotle and in a    
proposition interpolated in ,euclid's   
,book ,x42 it is by a reductio ad       
absurdum proving that1 if the diagonal  
is commensurable with the side1 the same
 number must be both odd and even4 ,this
 discovery of the incommensurable was   
bound to cause geometers a great shock1 
because it showed that the theory of    
proportion invented by ,pythagoras was  
not of universal application1 and       
therefore that propositions proved by   

means of it were not really established4
 ,hence the stories that the discovery  
of the irrational was for a time kept   
secret1 and that the first person who   
divulged it perished by shipwreck4 ,the 
fatal flaw thus revealed in the body of 
geometry was not removed till ,eudoxus  
7#djh-cee ,b4,c47 discovered the great  
theory of proportion 7expounded in      
,euclid's ,book ,v471 which is          
applicable to incommensurable as well as
 to commensurable magnitudes4  ,by the  
time of ,hippocrates of ,chios the scope
 of ,greek geometry was no longer even  
limited to the ,elements2 certain       
special problems were also attacked     
which were beyond the power of the      
geometry of the straight line and       
circle1 and which were destined to play 
a great part in determining the         
direction taken by ,greek geometry in   
its highest flights4 ,the main problems 
in question were three3 7#a7 the        
doubling of the cube1 7#b7 the          
trisection of any angle1 7#c7 the       

squaring of the circle2 and from the    
time of ,hippocrates onwards the        
investigation of these problems         
proceeded pari passu with the completion
 of the body of the ,elements4          
,hippocrates himself is an example of   
the concurrent study of the two         
departments4 ,on the one hand1 he was   
the first of the ,greeks who is known to
 have compiled a book of ,elements4     
,this book1 we may be sure1 contained in
 particular the most important          
propositions about the circle included  
in ,euclid1 ,book ,,iii4 ,but a much    
more important proposition is attributed
 to ,hippocrates2 he is said to have    
been the first to prove that circles are
 to one another as the squares on their 
diameters 7 ,eucl4 ,,xii41 #b71 with the
 deduction that similar segments of     
circles are to one another as the       
squares on their bases4 ,these          
propositions were used by him in his    
tract on the squaring of lunes1 which   
was intended to lead up to the squaring 

of the circle4 ,the latter problem is   
one which must have exercised practical 
geometers from time immemorial4         
,anaxagoras for instance 7about #ejj-dbh
 ,b4,c47 is said to have worked at the  
problem while in prison4 ,the essential 
portions of ,hippocrates's tract are    
preserved in a passage of ,simplicius   
7on ,aristotle's ,physics71 which       
contains substantial fragments from     
,eudemus's ,history of ,geometry4       
,hippocrates showed how to square three 
particular lunes of different forms1 and
 then1 lastly1 he squared the sum of a  
certain circle and a certain lune4      
,unfortunately1 however1 the            
last-mentioned lune was not one of those
 which can be squared1 and so the       
attempt to square the circle in this way
 failed after all4  ,hippocrates also   
attacked the problem of doubling the    
cube4 ,there are two versions of the    
origin of this famous problem4          
,according to one of them1 an old tragic
 poet represented ,minos as having been 

dissatisfied with the size of a tomb    
erected for his son ,glaucus1 and having
 told the architect to make it double   
the size1 retaining1 however1 the       
cubical form4 ,according to the other1  
the ,delians1 suffering from a          
pestilence1 were told by the oracle to  
double a certain cubical altar as a     
means of staying the plague4            
,hippocrates did not1 indeed1 solve the 
problem1 but he succeeded in reducing it
 to another1 namely1 the problem of     
finding two mean proportionals in       
continued proportion between two given  
straight lines1 i4e4 finding x1 y such  
that a 3 x  x 3 y  y 3 b1 where a1 b are
 the two given straight lines4 ,it is   
easy to see that1 if a 3 x  x 3 y  y 3  
b1 then b_/a  7x_/a7#c1 and1 as a       
particular case1 if b  #b;a1 x#c        
#b;a#c1 so that the side of the cube    
which is double of the cube of side a is
 found4  ,the problem of doubling the   
cube was henceforth tried exclusively in
 the form of the problem of the two mean

 proportionals4 ,two significant early  
solutions are on record4  7#a7 ,archytas
 of ,tarentum 7who flourished in first  
half of fourth century ,b4,c47 found the
 two mean proportionals by a very       
striking construction in three          
dimensions1 which shows that solid      
geometry1 in the hands of ,archytas at  
least1 was already well advanced4 ,the  
construction was usually called         
mechanical1 which it no doubt was in    
form1 though in reality it was in the   
highest degree theoretical4 ,it         
consisted in determining a point in     
space as the intersection of three      
surfaces3 7a7 a cylinder1 7b7 a cone1   
7c7 an anchor-ring with internal radius 
#j4 7#b7 ,menaechmus1 a pupil of        
,eudoxus1 and a contemporary of ,plato1 
found the two mean proportionals by     
means of conic sections1 in two ways1   
7alpha7 by the intersection of two      
parabolas1 the equations of which in    
,cartesian co-ordinates would be x#b    
ay1 y#b  bx1 and 7beta7 by the          

intersection of a parabola and a        
rectangular hyperbola1 the corresponding
 equations being x#b  ay1 and xy  ab    
respectively4 ,it would appear that it  
was in the effort to solve this problem 
that ,menaechmus discovered the conic   
sections1 which are called1 in an       
epigram by ,eratosthenes1 the triads of 
,menaechmus4  ,the trisection of an     
angle was effected by means of a curve  
discovered by ,hippias of ,elis1 the    
sophist1 a contemporary of ,hippocrates 
as well as of ,democritus and ,socrates 
7#dgj-cii ,b4,c474 ,the curve was called
 the quadratrix because it also served  
7in the hands1 as we are told1 of       
,dinostratus1 brother of ,menaechmus1   
and of ,nicomedes7 for squaring the     
circle4 ,it was theoretically           
constructed as the locus of the point of
 intersection of two straight lines     
moving at uniform speeds and in the same
 time1 one motion being angular and the 
other rectilinear4 ,suppose ,,oa1 ,,ob  
are two radii of a circle at right      

angles to one another4 ,tangents to the 
circle at ,a and ,b1 meeting at ,c1 form
 with the two radii the square ,,oacb4  
,the radius ,,oa is made to move        
uniformly about ,o1 the centre1 so as to
 describe the angle ,,aob in a certain  
time4 ,simultaneously ,,ac moves        
parallel to itself at uniform speed such
 that ,a just describes the line ,,ao in
 the same length of time4 ,the          
intersection of the moving radius and   
,,ac in their various positions traces  
out the quadratrix4  ,the rest of the   
geometry which concerns us was mostly   
the work of a few men1 ,democritus of   
,abdera1 ,theodorus of ,cyrene 7the     
mathematical teacher of ,plato71        
,theaetetus1 ,eudoxus1 and ,euclid4 ,the
 actual writers of ,elements of whom we 
hear were the following4 ,leon1 a little
 younger than ,eudoxus 7#djh-cee        
,b4,c471 was the author of a collection 
of propositions more numerous and more  
serviceable than those collected by     
,hippocrates4 ,theudius of ,magnesia1 a 

contemporary of ,menaechmus and         
,dinostratus1 put together the elements 
admirably1 making many partial or       
limited propositions more general4      
,theudius's book was no doubt the       
geometrical text-book of the ,academy   
and that used by ,aristotle4  ,theodorus
 of ,cyrene and ,theaetetus generalised 
the theory of irrationals1 and we may   
safely conclude that a great part of the
 substance of ,euclid's ,book ,x4 7on   
irrationals7 was due to ,theaetetus4    
,theaetetus also wrote on the five      
regular solids 7the tetrahedron1 cube1  
octahedron1 dodecahedron1 and           
icosahedron71 and ,euclid was therefore 
no doubt equally indebted to ,theaetetus
 for the contents of his ,book ,,xiii4  
,in the matter of ,book ,,xii4 ,eudoxus 
was the pioneer4 ,these facts are       
confirmed by the remark of ,proclus that
 ,euclid1 in compiling his ,elements1   
collected many of the theorems of       
,eudoxus1 perfected many others by      
,theaetetus1 and brought to irrefragable

 demonstration the propositions which   
had only been somewhat loosely proved by
 his predecessors4  ,eudoxus 7about     
#djh-cee ,b4,c47 was perhaps the        
greatest of all ,archimedes's           
predecessors1 and it is his             
achievements1 especially the discovery  
of the method of exhaustion1 which      
interest us in connexion with           
,archimedes4  ,in astronomy ,eudoxus is 
famous for the beautiful theory of      
concentric spheres which he invented to 
explain the apparent motions of the     
planets1 and1 particularly1 their       
apparent stationary points and          
retrogradations4 ,the theory applied    
also to the sun and moon1 for which     
,eudoxus required only three spheres in 
each case4 ,he represented the motion of
 each planet as compounded of the       
rotations of four interconnected spheres
 about diameters1 all of which pass     
through the centre of the earth4 ,the   
outermost sphere represents the daily   
rotation1 the second a motion along the 

zodiac circle or ecliptic2 the poles of 
the third sphere1 about which that      
sphere revolves1 are fixed at two       
opposite points on the zodiac circle1   
and are carried round in the motion of  
the second sphere2 and on the surface of
 the third sphere the poles of the      
fourth sphere are fixed2 the fourth     
sphere1 revolving about the diameter    
joining its two poles1 carries the      
planet which is fixed at a point on its 
equator4 ,the poles and the speeds and  
directions of rotation are so chosen    
that the planet actually describes a    
hippopede1 or horse-fetter1 as it was   
called 7i4e4 a figure of eight71 which  
lies along and is longitudinally        
bisected by the zodiac circle1 and is   
carried round that circle4 ,as a tour de
 force of geometrical imagination it    
would be difficult to parallel this     
hypothesis4  ,in geometry ,eudoxus      
discovered the great theory of          
proportion1 applicable to               
incommensurable as well as commensurable

 magnitudes1 which is expounded in      
,euclid1 ,book ,v41 and which still     
holds its own and will do so for all    
time4 ,he also solved the problem of the
 two mean proportionals by means of     
certain curves1 the nature of which1 in 
the absence of any description of them  
in our sources1 can only be conjectured4
  ,last of all1 and most important for  
our purpose1 is his use of the famous   
method of exhaustion for the measurement
 of the areas of curves and the volumes 
of solids4 ,the example of this method  
which will be most familiar to the      
reader is the proof in ,euclid ,,xii41  
#b1 of the theorem that the areas of    
circles are to one another as the       
squares on their diameters4 ,the proof  
in this and in all cases depends on a   
lemma which forms ,prop4 #a of ,euclid's
 ,book ,x4 to the effect that1 if there 
are two unequal magnitudes of the same  
kind and from the greater you subtract  
not less than its half1 then from the   
remainder not less than its half1 and so

 on continually1 you will at length have
 remaining a magnitude less than the    
lesser of the two magnitudes set out1   
however small it is4 ,archimedes says   
that the theorem of ,euclid ,,xii41 #b1 
was proved by means of a certain lemma  
to the effect that1 if we have two      
unequal magnitudes 7i4e4 lines1         
surfaces1 or solids respectively71 the  
greater exceeds the lesser by such a    
magnitude as is capable1 if added       
continually to itself1 of exceeding any 
magnitude of the same kind as the       
original magnitudes4 ,this assumption is
 known as the ,axiom or ,postulate of   
,archimedes1 though1 as he states1 it   
was assumed before his time by those who
 used the method of exhaustion4 ,it is  
in reality used in ,euclid's lemma      
7,eucl4 ,x41 #a7 on which ,euclid       
,,xii41 #b1 depends1 and only differs in
 statement from ,def4 #d of ,euclid1    
,book ,v41 which is no doubt due to     
,eudoxus4  ,the method of exhaustion was
 not discovered all at once2 we find    

traces of gropings after such a method  
before it was actually evolved4 ,it was 
perhaps ,antiphon1 the sophist1 of      
,athens1 a contemporary of ,socrates    
7#dgj-cii ,b4,c471 who took the first   
step4 ,he inscribed a square 7or1       
according to another account1 an        
equilateral triangle7 in a circle1 then 
bisected the arcs subtended by the      
sides1 and so inscribed a polygon of    
double the number of sides2 he then     
repeated the process1 and maintained    
that1 by continuing it1 we should at    
last arrive at a polygon with sides so  
small as to make the polygon coincident 
with the circle4 ,though this was       
formally incorrect1 it nevertheless     
contained the germ of the method of     
exhaustion4  ,hippocrates1 as we have   
seen1 is said to have proved the theorem
 that circles are to one another as the 
squares on their diameters1 and it is   
difficult to see how he could have done 
this except by some form1 or            
anticipation1 of the method4 ,there is1 

however1 no doubt about the part taken  
by ,eudoxus2 he not only based the      
method on rigorous demonstration by     
means of the lemma or lemmas aforesaid1 
but he actually applied the method to   
find the volumes 7#a7 of any pyramid1   
7#b7 of the cone1 proving 7#a7 that any 
pyramid is one third part of the prism  
which has the same base and equal       
height1 and 7#b7 that any cone is one   
third part of the cylinder which has the
 same base and equal height4            
,archimedes1 however1 tells us the      
remarkable fact that these two theorems 
were first discovered by ,democritus    
7who flourished towards the end of the  
fifth century ,b4,c471 though he was not
 able to prove them 7which no doubt     
means1 not that he gave no sort of      
proof1 but that he was not able to      
establish the propositions by the       
rigorous method of ,eudoxus74           
,archimedes adds that we must give no   
small share of the credit for these     
theorems to ,democritus2 and this is    

another testimony to the marvellous     
powers1 in mathematics as well as in    
other subjects1 of the great man who1 in
 the words of ,aristotle1 seems to have 
thought of everything4 ,we know from    
other sources that ,democritus wrote on 
irrationals2 he is also said to have    
discussed the question of two parallel  
sections of a cone 7which were evidently
 supposed to be indefinitely close      
together71 asking whether we are to     
regard them as unequal or equal3 for if 
they are unequal they will make the cone
 irregular as having many indentations1 
like steps1 and unevennesses1 but1 if   
they are equal1 the cone will appear to 
have the property of the cylinder and to
 be made up of equal1 not unequal1      
circles1 which is very absurd4 ,this    
explanation shows that ,democritus was  
already close on the track of           
infinitesimals4  ,archimedes says       
further that the theorem that spheres   
are in the triplicate ratio of their    
diameters was proved by means of the    

same lemma4 ,the proofs of the          
propositions about the volumes of       
pyramids1 cones and spheres are1 of     
course1 contained in ,euclid1 ,book     
,,xii4 7,props4 #c-g ,cor41 #aj1 #af-ah 
respectively74  ,it is no doubt         
desirable to illustrate ,eudoxus's      
method by one example4 ,we will take one
 of the simplest1 the proposition       
7,eucl4 ,,xii41 #aj7 about the cone4    
,given ,,abcd1 the circular base of the 
cylinder which has the same base as the 
cone and equal height1 we inscribe the  
square ,,abcd2 we then bisect the arcs  
subtended by the sides1 and draw the    
regular inscribed polygon of eight      
sides1 then similarly we draw the       
regular inscribed polygon of sixteen    
sides1 and so on4 ,we erect on each     
regular polygon the prism which has the 
polygon for base1 thereby obtaining     
successive prisms inscribed in the      
cylinder1 and of the same height with   
it4 ,each time we double the number of  
sides in the base of the prism we take  

away more than half of the volume by    
which the cylinder exceeds the prism    
7since we take away more than half of   
the excess of the area of the circular  
base over that of the inscribed polygon1
 as in ,euclid ,,xii41 #b74 ,suppose now
 that ,v is the volume of the cone1 ,c  
that of the cylinder4 ,we have to prove 
that ,c  #c,v4 ,if ,c is not equal to   
#c,v1 it is either greater or less than 
#c,v4  ,suppose 7#a7 that ,c  #c,v1 and 
that ,c  #c,v  ,e4 ,continue the        
construction of prisms inscribed in the 
cylinder until the parts of the cylinder
 left over outside the final prism 7of  
volume ,p7 are together less than ,e4   
,then        ,c - ,p  ,e4   ,but        
,c - #c,v  ,e2   ,therefore   ,p  #c,v4 
,but it has been proved in earlier      
propositions that ,p is equal to three  
times the pyramid with the same base as 
the prism and equal height4  ,therefore 
that pyramid is greater than ,v1 the    
volume of the cone3 which is impossible1
 since the cone encloses the pyramid4   

,therefore ,c is not greater than #c,v4 
,next 7#b7 suppose that ,c  #c,v1 so    
that1 inversely1    ,v  #a_/c ,c4  ,this
 time we inscribe successive pyramids in
 the cone until we arrive at a pyramid  
such that the portions of the cone left 
over outside it are together less than  
the excess of ,v over #a_/c ,c4 ,it     
follows that the pyramid is greater than
 #a_/c ,c4 ,hence the prism on the same 
base as the pyramid and inscribed in the
 cylinder 7which prism is three times   
the pyramid7 is greater than ,c3 which  
is impossible1 since the prism is       
enclosed by the cylinder1 and is        
therefore less than it4  ,therefore ,v  
is not greater than #a_/c ,c1 or ,c is  
not less than #c,v4  ,accordingly ,c1   
being neither greater nor less than     
#c,v1 must be equal to it2 that is1 ,v  
#a_/c ,c4  ,it only remains to add that 
,archimedes is fully acquainted with the
 main properties of the conic sections4 
,these had already been proved in       
earlier treatises1 which ,archimedes    

refers to as the ,elements of ,conics4  
,we know of two such treatises1 7#a7    
,euclid's four ,books on ,conics1 7#b7 a
 work by one ,aristaeus called ,solid   
,loci1 probably a treatise on conics    
regarded as loci4 ,both these treatises 
are lost2 the former was1 of course1    
superseded by ,apollonius's great work  
on ,conics in eight ,books4             
,,chapter ,,iii4  ,,the ,,works ,,of    
,,archimedes4   ,the range of           
,archimedes's writings will be gathered 
from the list of his various treatises4 
,an extraordinarily large proportion of 
their contents represents entirely new  
discoveries of his own4 ,he was no      
compiler or writer of text-books1 and in
 this respect he differs from ,euclid   
and ,apollonius1 whose work largely     
consisted in systematising and          
generalising the methods used and the   
results obtained by earlier geometers4  
,there is in ,archimedes no mere        
working-up of existing material2 his    
objective is always something new1 some 

definite addition to the sum of         
knowledge4 ,confirmation of this is     
found in the introductory letters       
prefixed to most of his treatises4 ,in  
them we see the directness1 simplicity  
and humanity of the man4 ,there is full 
and generous recognition of the work of 
predecessors and contemporaries2 his    
estimate of the relation of his own     
discoveries to theirs is obviously just 
and free from any shade of egoism4 ,his 
manner is to state what particular      
discoveries made by his predecessors had
 suggested to him the possibility of    
extending them in new directions2 thus  
he says that1 in connexion with the     
efforts of earlier geometers to square  
the circle1 it occurred to him that no  
one had tried to square a parabolic     
segment2 he accordingly attempted the   
problem and finally solved it4          
,similarly he describes his discoveries 
about the volumes and surfaces of       
spheres and cylinders as supplementing  
the theorems of ,eudoxus about the      

pyramid1 the cone and the cylinder4 ,he 
does not hesitate to say that certain   
problems baffled him for a long time2 in
 one place he positively insists1 for   
the purpose of pointing a moral1 on     
specifying two propositions which he had
 enunciated but which on further        
investigation proved to be wrong4  ,the 
ordinary ,,mss4 of the ,greek text of   
,archimedes give his works in the       
following order3--    #a4 ,on the       
,sphere and ,cylinder 7two books74   #b4
 ,measurement of a ,circle4   #c4 ,on   
,conoids and ,spheroids4   #d4 ,on      
,spirals4   #e4 ,on ,plane ,equilibriums
 7two books74   #f4 ,the ,sandreckoner4 
#g4 ,quadrature of a ,parabola4  ,a most
 important addition to this list has    
been made in recent years through an    
extraordinary piece of good fortune4 ,in
 #aijf ,j4 ,l4 ,heiberg1 the most recent
 editor of the text of ,archimedes1     
discovered a palimpsest of mathematical 
content in the ,jerusalemic ,library of 
one ,papadopoulos ,kerameus at          

,constantinople4 ,this proved to contain
 writings of ,archimedes copied in a    
good hand of the tenth century4 ,an     
attempt had been made 7fortunately with 
only partial success7 to wash out the   
old writing1 and then the parchment was 
used again to write a ,euchologion upon4
 ,however1 on most of the leaves the    
earlier writing remains more or less    
legible4 ,the important fact about the  
,,ms4 is that it contains1 besides      
substantial portions of the treatises   
previously known1 7#a7 a considerable   
portion of the work1 in two books1 ,on  
,floating ,bodies1 which was formerly   
supposed to have been lost in ,greek and
 only to have survived in the           
translation by ,wilhelm of ,morbeke1 and
 7#b7 most precious of all1 the greater 
part of the book called ,the ,method1   
treating of ,mechanical ,problems and   
addressed to ,eratosthenes4 ,the        
important treatise so happily recovered 
is now included in ,heiberg's new       
7second7 edition of the ,greek text of  

,archimedes 7,teubner1 #aiaj-ae71 and   
some account of it will be given in the 
next chapter4  ,the order in which the  
treatises appear in the ,,mss4 was not  
the order of composition2 but from the  
various prefaces and from internal      
evidence generally we are able to       
establish the following as being        
approximately the chronological         
sequence3--     #a4 ,on ,plane          
,equilibriums1 ,i4    #b4 ,quadrature of
 a ,parabola4    #c4 ,on ,plane         
,equilibriums1 ,,ii4    #d4 ,the        
,method4    #e4 ,on the ,sphere and     
,cylinder1 ,i1 ,,ii4    #f4 ,on         
,spirals4    #g4 ,on ,conoids and       
,spheroids4    #h4 ,on ,floating        
,bodies1 ,i1 ,,ii4    #i4 ,measurement  
of a ,circle4   #aj4 ,the ,sandreckoner4
  ,in addition to the above we have a   
collection of geometrical propositions  
which has reached us through the ,arabic
 with the title ,liber assumptorum      
,archimedis4 ,they were not written by  
,archimedes in their present form1 but  

were probably collected by some later   
,greek writer for the purpose of        
illustrating some ancient work4 ,it is1 
however1 quite likely that some of the  
propositions1 which are remarkably      
elegant1 were of ,archimedean origin1   
notably those concerning the geometrical
 figures made with three and four       
semicircles respectively and called     
7from their shape7 7#a7 the shoemaker's 
knife and 7#b7 the ,salinon or          
salt-cellar1 and another theorem which  
bears on the trisection of an angle4    
,an interesting fact which we now know  
from ,arabian sources is that the       
formula for the area of any triangle in 
terms of its sides which we write in the
 form    ,delta  roots7s - a77s - b77s -
 c71  and which was supposed to be      
,heron's because ,heron gives the       
geometrical proof of it1 was really due 
to ,archimedes4  ,archimedes is further 
credited with the authorship of the     
famous ,cattle-,problem enunciated in a 
,greek epigram edited by ,lessing in    

#aggc4 ,according to its heading the    
problem was communicated by ,archimedes 
to the mathematicians at ,alexandria in 
a letter to ,eratosthenes2 and a        
scholium to ,plato's ,charmides speaks  
of the problem called by ,archimedes the
 ,cattle-,problem4 ,it is an            
extraordinarily difficult problem in    
indeterminate analysis1 the solution of 
which involves enormous figures4  ,of   
lost works of ,archimedes the following 
can be identified3--  #a4               
,investigations relating to polyhedra   
are referred to by ,pappus1 who1 after  
speaking of the five regular solids1    
gives a description of thirteen other   
polyhedra discovered by ,archimedes     
which are semi-regular1 being contained 
by polygons equilateral and equiangular 
but not similar4 ,one at least of these 
semi-regular solids was1 however1       
already known to ,plato4  #b4 ,a book of
 arithmetical content entitled          
,principles dealt1 as we learn from     
,archimedes himself1 with the naming of 

numbers1 and expounded a system of      
expressing large numbers which could not
 be written in the ordinary ,greek      
notation4 ,in setting out the same      
system in the ,sandreckoner 7see        
,chapter ,v4 below71 ,archimedes        
explains that he does so for the benefit
 of those who had not seen the earlier  
work4  #c4 ,on ,balances 7or perhaps    
levers74 ,pappus says that in this work 
,archimedes proved that greater circles 
overpower lesser circles when they      
rotate about the same centre4  #d4 ,a   
book ,on ,centres of ,gravity is alluded
 to by ,simplicius4 ,it is not1 however1
 certain that this and the              
last-mentioned work were separate       
treatises1 ,possibly ,book ,i4 ,on      
,plane ,equilibriums may have been part 
of a larger work 7called perhaps        
,elements of ,mechanics71 and ,on       
,balances may have been an alternative  
title4 ,the title ,on ,centres of       
,gravity may be a loose way of referring
 to the same treatise4  #e4 ,catoptrica1

 an optical work from which ,theon of   
,alexandria quotes a remark about       
refraction4  #f4 ,on ,sphere-making1 a  
mechanical work on the construction of a
 sphere to represent the motions of the 
heavenly bodies 7cf4 pp4 #e-f above74   
,arabian writers attribute yet further  
works to ,archimedes1 7#a7 ,on the      
circle1 7#b7 ,on a heptagon in a circle1
 7#c7 ,on circles touching one another1 
7#d7 ,on parallel lines1 7#e7 ,on       
triangles1 7#f7 ,on the properties of   
right-angled triangles1 7#g7 a book of  
,data2 but we have no confirmation of   
these statements4     ,,chapter ,,iv4   
,,geometry ,,in ,,archimedes4   ,the    
famous ,french geometer1 ,chasles1 drew 
an instructive distinction between the  
predominant features of the geometry of 
the two great successors of ,euclid1    
namely1 ,archimedes and ,apollonius of  
,perga 7the great geometer1 and author  
of the classical treatise on ,conics74  
,the works of these two men may1 says   
,chasles1 be regarded as the origin and 

basis of two great inquiries which seem 
to share between them the domain of     
geometry4 ,apollonius is concerned with 
the ,geometry of ,forms and ,situations1
 while in ,archimedes we find the       
,geometry of ,measurements1 dealing with
 the quadrature of curvilinear plane    
figures and with the quadrature and     
cubature of curved surfaces1            
investigations which gave birth to the  
calculus of the infinite conceived and  
brought to perfection by ,kepler1       
,cavalieri1 ,fermat1 ,leibniz and       
,newton4  ,in geometry ,archimedes      
stands1 as it were1 on the shoulders of 
,eudoxus in that he applied the method  
of exhaustion to new and more difficult 
cases of quadrature and cubature4       
,further1 in his use of the method he   
introduced an interesting variation of  
the procedure as we know it from        
,euclid4 ,euclid 7and presumably        
,eudoxus also7 only used inscribed      
figures1 exhausting the figure to be    
measured1 and had to invert the second  

half of the reductio ad absurdum to     
enable approximation from below 7so to  
speak7 to be applied in that case also4 
,archimedes1 on the other hand1         
approximates from above as well as from 
below2 he approaches the area or volume 
to be measured by taking closer and     
closer circumscribed figures1 as well as
 inscribed1 and thereby compressing1 as 
it were1 the inscribed and circumscribed
 figure into one1 so that they          
ultimately coincide with one another and
 with the figure to be measured4 ,but he
 follows the cautious method to which   
the ,greeks always adhered2 he never    
says that a given curve or surface is   
the limiting form of the inscribed or   
circumscribed figure2 all that he       
asserts is that we can approach the     
curve or surface as nearly as we please4
  ,the deductive form of proof by the   
method of exhaustion is apt to obscure  
not only the way in which the results   
were arrived at but also the real       
character of the procedure followed4    

,what ,archimedes actually does in      
certain cases is to perform what are    
seen1 when the analytical equivalents   
are set down1 to be real integrations2  
this remark applies to his investigation
 of the areas of a parabolic segment and
 a spiral respectively1 the surface and 
volume respectively of a sphere and a   
segment of a sphere1 and the volume of  
any segments of the solids of revolution
 of the second degree4 ,the result is1  
as a rule1 only obtained after a long   
series of preliminary propositions1 all 
of which are links in a chain of        
argument elaborately forged for the one 
purpose4 ,the method suggests the       
tactics of some master of strategy who  
foresees everything1 eliminates         
everything not immediately conducive to 
the execution of his plan1 masters every
 position in its order1 and then        
suddenly 7when the very elaboration of  
the scheme has almost obscured1 in the  
mind of the onlooker1 its ultimate      
object7 strikes the final blow4 ,thus we

 read in ,archimedes proposition after  
proposition the bearing of which is not 
immediately obvious but which we find   
infallibly used later on2 and we are led
 on by such easy stages that the        
difficulty of the original problem1 as  
presented at the outset1 is scarcely    
appreciated4 ,as ,plutarch says1 ,it is 
not possible to find in geometry more   
difficult and troublesome questions1 or 
more simple and lucid explanations4 ,but
 it is decidedly a rhetorical           
exaggeration when ,plutarch goes on to  
say that we are deceived by the easiness
 of the successive steps into the belief
 that any one could have discovered them
 for himself4 ,on the contrary1 the     
studied simplicity and the perfect      
finish of the treatises involve at the  
same time an element of mystery4        
,although each step depends upon the    
preceding ones1 we are left in the dark 
as to how they were suggested to        
,archimedes4 ,there is1 in fact1 much   
truth in a remark of ,wallis to the     

effect that he seems as it were of set  
purpose to have covered up the traces of
 his investigation as if he had grudged 
posterity the secret of his method of   
inquiry while he wished to extort from  
them assent to his results4  ,a partial 
exception is now furnished by the       
,method2 for here we have 7as it were7 a
 lifting of the veil and a glimpse of   
the interior of ,archimedes's workshop4 
,he tells us how he discovered certain  
theorems in quadrature and cubature1 and
 he is at the same time careful to      
insist on the difference between 7#a7   
the means which may serve to suggest the
 truth of theorems1 although not        
furnishing scientific proofs of them1   
and 7#b7 the rigorous demonstrations of 
them by approved geometrical methods    
which must follow before they can be    
finally accepted as established4        
,writing to ,eratosthenes he says3      
,seeing in you1 as ,i say1 an earnest   
student1 a man of considerable eminence 
in philosophy and an admirer of         

mathematical inquiry when it comes your 
way1 ,i have thought fit to write out   
for you and explain in detail in the    
same book the peculiarity of a certain  
method1 which1 when you see it1 will put
 you in possession of a means whereby   
you can investigate some of the problems
 of mathematics by mechanics4 ,this     
procedure is1 ,i am persuaded1 no less  
useful for the proofs of the actual     
theorems as well4 ,for certain things   
which first became clear to me by a     
mechanical method had afterwards to be  
demonstrated by geometry1 because their 
investigation by the said method did not
 furnish an actual demonstration4 ,but  
it is of course easier1 when we have    
previously acquired by the method some  
knowledge of the questions1 to supply   
the proof than it is to find the proof  
without any previous knowledge4 ,this is
 a reason why1 in the case of the       
theorems the proof of which ,eudoxus was
 the first to discover1 namely1 that the
 cone is a third part of the cylinder1  

and the pyramid a third part of the     
prism1 having the same base and equal   
height1 we should give no small share of
 the credit to ,democritus1 who was the 
first to assert this truth with regard  
to the said figures1 though he did not  
prove it4 ,i am myself in the position  
of having made the discovery of the     
theorem now to be published in the same 
way as ,i made my earlier discoveries2  
and ,i thought it desirable now to write
 out and publish the method1 partly     
because ,i have already spoken of it and
 ,i do not want to be thought to have   
uttered vain words1 but partly also     
because ,i am persuaded that it will be 
of no little service to mathematics2 for
 ,i apprehend that some1 either of my   
contemporaries or of my successors1     
will1 by means of the method when once  
established1 be able to discover other  
theorems in addition1 which have not    
occurred to me4  ,first then ,i will set
 out the very first theorem which became
 known to me by means of mechanics1     

namely1 that ,any segment of a section  
of a right-angled cone i4e4 a parabola  
is four-thirds of the triangle which has
 the same base and equal height2 and    
after this ,i will give each of the     
other theorems investigated by the same 
method4 ,then1 at the end of the book1  
,i will give the geometrical proofs of  
the propositions4  ,the following       
description will1 ,i hope1 give an idea 
of the general features of the          
mechanical method employed by           
,archimedes4 ,suppose that ,x is the    
plane or solid figure the area or       
content of which is to be found4 ,the   
method in the simplest case is to weigh 
infinitesimal elements of ,x against the
 corresponding elements of another      
figure1 ,b say1 being such a figure that
 its area or content and the position of
 its centre of gravity are already      
known4 ,the diameter or axis of the     
figure ,x being drawn1 the infinitesimal
 elements taken are parallel sections of
 ,x in general1 but not always1 at right

 angles to the axis or diameter1 so that
 the centres of gravity of all the      
sections lie at one point or other of   
the axis or diameter and their weights  
can therefore be taken as acting at the 
several points of the diameter or axis4 
,in the case of a plane figure the      
infinitesimal sections are spoken of as 
parallel straight lines and in the case 
of a solid figure as parallel planes1   
and the aggregate of the infinite number
 of sections is said to make up the     
whole figure ,x4 7,although the sections
 are so spoken of as straight lines or  
planes1 they are really indefinitely    
narrow plane strips or indefinitely thin
 laminae respectively47 ,the diameter or
 axis is produced in the direction away 
from the figure to be measured1 and the 
diameter or axis as produced is imagined
 to be the bar or lever of a balance4   
,the object is now to apply all the     
separate elements of ,x at one point on 
the lever1 while the corresponding      
elements of the known figure ,b operate 

at different points1 namely1 where they 
actually are in the first instance4     
,archimedes contrives1 therefore1 to    
move the elements of ,x away from their 
original position and to concentrate    
them at one point on the lever1 such    
that each of the elements balances1     
about the point of suspension of the    
lever1 the corresponding element of ,b  
acting at its centre of gravity4 ,the   
elements of ,x and ,b respectively      
balance about the point of suspension in
 accordance with the property of the    
lever that the weights are inversely    
proportional to the distances from the  
fulcrum or point of suspension4 ,now the
 centre of gravity of ,b as a whole is  
known1 and it may then be supposed to   
act as one mass at its centre of        
gravity4 7,archimedes assumes as known  
that the sum of the moments1 as we call 
them1 of all the elements of the figure 
,b1 acting severally at the points where
 they actually are1 is equal to the     
moment of the whole figure applied as   

one mass at one point1 its centre of    
gravity47 ,moreover all the elements of 
,x are concentrated at the one fixed    
point on the bar or lever4 ,if this     
fixed point is ,h1 and ,g is the centre 
of gravity of the figure ,b1 while ,c is
 the point of suspension1    ,x 3 ,b    
,,cg 3 ,,ch4  ,thus the area or content 
of ,x is found4  ,conversely1 the method
 can be used to find the centre of      
gravity of ,x when its area or volume is
 known beforehand4 ,in this case the    
elements of ,x1 and ,x itself1 have to  
be applied where they are1 and the      
elements of the known figure or figures 
have to be applied at the one fixed     
point ,h on the other side of ,c1 and   
since ,x1 ,b and ,,ch are known1 the    
proportion    ,b 3 ,x  ,,cg 3 ,,ch      
determines ,,cg1 where ,g is the centre 
of gravity of ,x4  ,the mechanical      
method is used for finding 7#a7 the area
 of any parabolic segment1 7#b7 the     
volume of a sphere and a spheroid1 7#c7 
the volume of a segment of a sphere and 

the volume of a right segment of each of
 the three conicoids of revolution1 7#d7
 the centre of gravity 7a7 of a         
hemisphere1 7b7 of any segment of a     
sphere1 7c7 of any right segment of a   
spheroid and a paraboloid of revolution1
 and 7d7 of a half-cylinder1 or1 in     
other words1 of a semicircle4           
,archimedes then proceeds to find the   
volumes of two solid figures1 which are 
the special subject of the treatise4    
,the solids arise as follows3--  7#a7   
,given a cylinder inscribed in a        
rectangular parallelepiped on a square  
base in such a way that the two bases of
 the cylinder are circles inscribed in  
the opposite square faces1 suppose a    
plane drawn through one side of the     
square containing one base of the       
cylinder and through the parallel       
diameter of the opposite base of the    
cylinder4 ,the plane cuts off a solid   
with a surface resembling that of a     
horse's hoof4 ,archimedes proves that   
the volume of the solid so cut off is   

one sixth part of the volume of the     
parallelepiped4  7#b7 ,a cylinder is    
inscribed in a cube in such a way that  
the bases of the cylinder are circles   
inscribed in two opposite square faces4 
,another cylinder is inscribed which is 
similarly related to another pair of    
opposite faces4 ,the two cylinders      
include between them a solid with all   
its angles rounded off2 and ,archimedes 
proves that the volume of this solid is 
two-thirds of that of the cube4  ,having
 proved these facts by the mechanical   
method1 ,archimedes concluded the       
treatise with a rigorous geometrical    
proof of both propositions by the method
 of exhaustion4 ,the ,,ms4 is           
unfortunately somewhat mutilated at the 
end1 so that a certain amount of        
restoration is necessary4  ,i shall now 
attempt to give a short account of the  
other treatises of ,archimedes in the   
order in which they appear in the       
editions4 ,the first is--   ,on the     
,sphere and ,cylinder4  ,book ,i4 begins

 with a preface addressed to ,dositheus 
7a pupil of ,conon71 which reminds him  
that on a former occasion he had        
communicated to him the treatise proving
 that any segment of a section of a     
right-angled cone 7i4e4 a parabola7 is  
four-thirds of the triangle with the    
same base and height1 and adds that he  
is now sending the proofs of certain    
theorems which he has since discovered1 
and which seem to him to be worthy of   
comparison with ,eudoxus's propositions 
about the volumes of a pyramid and a    
cone4 ,the theorems are 7#a7 that the   
surface of a sphere is equal to four    
times its greatest circle 7i4e4 what we 
call a great circle of the sphere72 7#b7
 that the surface of any segment of a   
sphere is equal to a circle with radius 
equal to the straight line drawn from   
the vertex of the segment to a point on 
the circle which is the base of the     
segment2 7#c7 that1 if we have a        
cylinder circumscribed to a sphere and  
with height equal to the diameter1 then 

7a7 the volume of the cylinder is       
#a-a_/b times that of the sphere and 7b7
 the surface of the cylinder1 including 
its bases1 is #a-a_/b times the surface 
of the sphere4  ,next come a few        
definitions1 followed by certain        
,assumptions1 two of which are well     
known1 namely3--  #a4 ,of all lines     
which have the same extremities the     
straight line is the least 7this has    
been made the basis of an alternative   
definition of a straight line74  #b4 ,of
 unequal lines1 unequal surfaces and    
unequal solids the greater exceeds the  
less by such a magnitude as1 when       
7continually7 added to itself1 can be   
made to exceed any assigned magnitude   
among those which are comparable with it
 and with one another 7i4e4 are of the  
same kind74 ,this is the ,postulate of  
,archimedes4  ,he also assumes that1 of 
pairs of lines 7including broken lines7 
and pairs of surfaces1 concave in the   
same direction and bounded by the same  
extremities1 the outer is greater than  

the inner4 ,these assumptions are       
fundamental to his investigation1 which 
proceeds throughout by means of figures 
inscribed and circumscribed to the      
curved lines or surfaces that have to be
 measured4  ,after some preliminary     
propositions ,archimedes finds 7,props4 
#ac1 #ad7 the area of the surfaces 7#a7 
of a right cylinder1 7#b7 of a right    
cone4 ,then1 after quoting certain      
,euclidean propositions about cones and 
cylinders1 he passes to the main        
business of the book1 the measurement of
 the volume and surface of a sphere and 
a segment of a sphere4 ,by              
circumscribing and inscribing to a great
 circle a regular polygon of an even    
number of sides and making it revolve   
about a diameter connecting two opposite
 angular points he obtains solids of    
revolution greater and less respectively
 than the sphere4 ,in a series of       
propositions he finds expressions for   
7a7 the surfaces1 7b7 the volumes1 of   
the figures so inscribed and            

circumscribed to the sphere4 ,next he   
proves 7,prop4 #cb7 that1 if the        
inscribed and circumscribed polygons    
which1 by their revolution1 generate the
 figures are similar1 the surfaces of   
the figures are in the duplicate ratio1 
and their volumes in the triplicate     
ratio1 of their sides4 ,then he proves  
that the surfaces and volumes of the    
inscribed and circumscribed figures     
respectively are less and greater than  
the surface and volume respectively to  
which the main propositions declare the 
surface and volume of the sphere to be  
equal 7,props4 #be1 #bg1 #cj1 #ca       
,cor474 ,he has now all the material for
 applying the method of exhaustion and  
so proves the main propositions about   
the surface and volume of the sphere4   
,the rest of the book applies the same  
procedure to a segment of the sphere4   
,surfaces of revolution are inscribed   
and circumscribed to a segment less than
 a hemisphere1 and the theorem about the
 surface of the segment is finally      

proved in ,prop4 #db4 ,prop4 #dc deduces
 the surface of a segment greater than a
 hemisphere4 ,prop4 #dd gives the volume
 of the sector of the sphere which      
includes any segment4  ,book ,,ii begins
 with the problem of finding a sphere   
equal in volume to a given cone or      
cylinder2 this requires the solution of 
the problem of the two mean             
proportionals1 which is accordingly     
assumed4 ,prop4 #b deduces1 by means of 
#a41 #dd1 an expression for the volume  
of a segment of a sphere1 and ,props4   
#c1 #d solve the important problems of  
cutting a given sphere by a plane so    
that 7a7 the surfaces1 7b7 the volumes1 
of the segments may have to one another 
a given ratio4 ,the solution of the     
second problem 7,prop4 #d7 is difficult4
 ,archimedes reduces it to the problem  
of dividing a straight line ,,ab into   
two parts at a point ,m such that       
,,mb 3 7a given length7  7a given area7 
3 ,,am#b4  ,the solution of this problem
 with a determination of the limits of  

possibility are given in a fragment by  
,archimedes1 discovered and preserved   
for us by ,eutocius in his commentary on
 the book2 they are effected by means of
 the points of intersection of two      
conics1 a parabola and a rectangular    
hyperbola4 ,three problems of           
construction follow1 the first two of   
which are to construct a segment of a   
sphere similar to one given segment1 and
 having 7a7 its volume1 7b7 its surface1
 equal to that of another given segment 
of a sphere4 ,the last two propositions 
are interesting4 ,prop4 #h proves that1 
if ,v1 ,v' be the volumes1 and ,s1 ,s'  
the surfaces1 of two segments into which
 a sphere is divided by a plane1 ,v and 
,s belonging to the greater segment1    
then    ,s#b 3 ,s'#b  ,v 3 ,v'          
,s7#c_/b7 3 ,s'7#c_/b74  ,prop4 #i      
proves that1 of all segments of spheres 
which have equal surfaces1 the          
hemisphere is the greatest in volume4   
,the ,measurement of a ,circle4  ,this  
treatise1 in the form in which it has   

come down to us1 contains only three    
propositions2 the second1 being an easy 
deduction from ,props4 #a and #c1 is out
 of place in so far as it uses the      
result of ,prop4 #c4  ,in ,prop4 #a     
,archimedes inscribes and circumscribes 
to a circle a series of successive      
regular polygons1 beginning with a      
square1 and continually doubling the    
number of sides2 he then proves in the  
orthodox manner by the method of        
exhaustion that the area of the circle  
is equal to that of a right-angled      
triangle1 in which the perpendicular is 
equal to the radius1 and the base equal 
to the circumference1 of the circle4    
,prop4 #c is the famous proposition in  
which ,archimedes finds by sheer        
calculation upper and lower arithmetical
 limits to the ratio of the             
circumference of a circle to its        
diameter1 or what we call pi2 the result
 obtained is #c-a_/g pi  #c-aj_/ga4     
,archimedes inscribes and circumscribes 
successive regular polygons1 beginning  

with hexagons1 and doubling the number  
of sides continually1 until he arrives  
at inscribed and circumscribed regular  
polygons with #if sides2 seeing then    
that the length of the circumference of 
the circle is intermediate between the  
perimeters of the two polygons1 he      
calculates the two perimeters in terms  
of the diameter of the circle4 ,his     
calculation is based on two close       
approximations 7an upper and a lower7 to
 the value of root#c1 that being the    
cotangent of the angle of #cj deg41 from
 which he begins to work4 ,he assumes as
 known that #bfe_/aec  root#c           
#acea_/ghj4 ,in the text1 as we have it1
 only the results of the steps in the   
calculation are given1 but they involve 
the finding of approximations to the    
square roots of several large numbers3  
thus #aagb-a_/h is given as the         
approximate value of                    
root7#acgcidc-cc_/fd71 #cjac-c_/d as    
that of root7#ijhbcba7 and #ahch-i_/aa  
as that of root7#cchjibi74 ,in this way 

,archimedes arrives at                  
#adfhh_/7dfgc-a_/b7 as the ratio of the 
perimeter of the circumscribed polygon  
of #if sides to the diameter of the     
circle2 this is the figure which he     
rounds up into #c-a_/g4 ,the            
corresponding figure for the inscribed  
polygon is #fccf_/7bjag-a_/d71 which1 he
 says1 is  #c-aj_/ga4 ,this example     
shows how little the ,greeks were       
embarrassed in arithmetical calculations
 by their alphabetical system of        
numerals4   ,on ,conoids and ,spheroids4
  ,the preface addressed to ,dositheus  
shows1 as we may also infer from        
internal evidence1 that the whole of    
this book also was original4 ,archimedes
 first explains what his conoids and    
spheroids are1 and then1 after each     
description1 states the main results    
which it is the aim of the treatise to  
prove4 ,the conoids are two4 ,the first 
is the right-angled conoid1 a name      
adapted from the old name 7section of a 
right-angled cone7 for a parabola2 this 

conoid is therefore a paraboloid of     
revolution4 ,the second is the          
obtuse-angled conoid1 which is a        
hyperboloid of revolution described by  
the revolution of a hyperbola 7a section
 of an obtuse-angled cone7 about its    
transverse axis4 ,the spheroids are two1
 being the solids of revolution         
described by the revolution of an       
ellipse 7a section of an acute-angled   
cone7 about 7#a7 its major axis and 7#b7
 its minor axis2 the first is called the
 oblong 7or oblate7 spheroid1 the second
 the flat 7or prolate7 spheroid4 ,as the
 volumes of oblique segments of conoids 
and spheroids are afterwards found in   
terms of the volume of the conical      
figure with the base of the segment as  
base and the vertex of the segment as   
vertex1 and as the said base is thus an 
elliptic section of an oblique circular 
cone1 ,archimedes calls the conical     
figure with an elliptic base a segment  
of a cone as distinct from a cone4  ,as 
usual1 a series of preliminary          

propositions is required4 ,archimedes   
first sums1 in geometrical form1 certain
 series1 including the arithmetical     
progression1 a1 #b;a1 #c;a1 444 na1 and 
the series formed by the squares of     
these terms 7in other words the series  
#ab1 #bb1 #cb1 444 n#b72 these          
summations are required for the final   
addition of an indefinite number of     
elements of each figure1 which amounts  
to an integration4 ,next come two       
properties of conics 7,prop4 #c71 then  
the determination by the method of      
exhaustion of the area of an ellipse    
7,prop4 #d74 ,three propositions follow1
 the first two of which 7,props4 #g1 #h7
 show that the conical figure above     
referred to is really a segment of an   
oblique circular cone2 this is done by  
actually finding the circular sections4 
,prop4 #i gives a similar proof that    
each elliptic section of a conoid or    
spheroid is a section of a certain      
oblique circular cylinder 7with axis    
parallel to the axis of the segment of  

the conoid or spheroid cut off by the   
said elliptic section74 ,props4 #aa-ah  
show the nature of the various sections 
which cut off segments of each conoid   
and spheroid and which are circles or   
ellipses according as the section is    
perpendicular or obliquely inclined to  
the axis of the solid2 they include also
 certain properties of tangent planes1  
etc4  ,the real business of the treatise
 begins with ,props4 #ai1 #bj2 here it  
is shown how1 by drawing many plane     
sections equidistant from one another   
and all parallel to the base of the     
segment of the solid1 and describing    
cylinders 7in general oblique7 through  
each plane section with generators      
parallel to the axis of the segment and 
terminated by the contiguous sections on
 either side1 we can make figures       
circumscribed and inscribed to the      
segment1 made up of segments of         
cylinders with parallel faces and       
presenting the appearance of the steps  
of a staircase4 ,adding the elements of 

the inscribed and circumscribed figures 
respectively and using the method of    
exhaustion1 ,archimedes finds the       
volumes of the respective segments of   
the solids in the approved manner       
7,props4 #ba1 #bb for the paraboloid1   
,props4 #be1 #bf for the hyperboloid1   
and ,props4 #bg-cj for the spheroids74  
,the results are stated in this form3   
7#a7 ,any segment of a paraboloid of    
revolution is half as large again as the
 cone or segment of a cone which has the
 same base and axis2 7#b7 ,any segment  
of a hyperboloid of revolution or of a  
spheroid is to the cone or segment of a 
cone with the same base and axis in the 
ratio of ,,ad  #c;,,ca to ,,ad  #b;,,ca 
in the case of the hyperboloid1 and of  
#c;,,ca - ,,ad to #b;,,ca - ,,ad in the 
case of the spheroid1 where ,c is the   
centre1 ,a the vertex of the segment1   
and ,,ad the axis of the segment        
7supposed in the case of the spheroid to
 be not greater than half the spheroid74
   ,on ,spirals4  ,the preface addressed

 to ,dositheus is of some length and    
contains1 first1 a tribute to the memory
 of ,conon1 and next a summary of the   
theorems about the sphere and the       
conoids and spheroids included in the   
above two treatises4 ,archimedes then   
passes to the spiral which1 he says1    
presents another sort of problem1 having
 nothing in common with the foregoing4  
,after a definition of the spiral he    
enunciates the main propositions about  
it which are to be proved in the        
treatise4 ,the spiral 7now known as the 
,spiral of ,archimedes7 is defined as   
the locus of a point starting from a    
given point 7called the origin7 on a    
given straight line and moving along the
 straight line at uniform speed1 while  
the line itself revolves at uniform     
speed about the origin as a fixed point4
 ,props4 #a-aa are preliminary1 the last
 two amounting to the summation of      
certain series required for the final   
addition of an indefinite number of     
element-areas1 which again amounts to   

integration1 in order to find the area  
of the figure cut off between any       
portion of the curve and the two radii  
vectores drawn to its extremities4      
,props4 #ac-bj are interesting and      
difficult propositions establishing the 
properties of tangents to the spiral4   
,props4 #ba-bc show how to inscribe and 
circumscribe to any portion of the      
spiral figures consisting of a multitude
 of elements which are narrow sectors of
 circles with the origin as centre2 the 
area of the spiral is intermediate      
between the areas of the inscribed and  
circumscribed figures1 and by the usual 
method of exhaustion ,archimedes finds  
the areas required4 ,prop4 #bd gives the
 area of the first complete turn of the 
spiral 7 #a_/cpi7#bpia7#b1 where the    
spiral is r  atheta71 and of any portion
 of it up to ,,op where ,p is any point 
on the first turn4 ,props4 #be1 #bf deal
 similarly with the second turn of the  
spiral and with the area subtended by   
any arc 7not being greater than a       

complete turn7 on any turn4 ,prop4 #bg  
proves the interesting property that1 if
 ,r#a be the area of the first turn of  
the spiral bounded by the initial line1 
,r#b the area of the ring added by the  
second complete turn1 ,r#c the area of  
the ring added by the third turn1 and so
 on1 then ,r#c  #b,r#b1 ,r#d  #c,r#b1   
,r#e  #d,r#b1 and so on to ,rn  7n -    
#a7,r#b1 while ,r#b1  #f,r#a4           
,quadrature of the ,parabola4  ,the     
title of this work seems originally to  
have been ,on the ,section of a         
,right-angled ,cone and to have been    
changed after the time of ,apollonius1  
who was the first to call a parabola by 
that name4 ,the preface addressed to    
,dositheus was evidently the first      
communication from ,archimedes to him   
after the death of ,conon4 ,it begins   
with a feeling allusion to his lost     
friend1 to whom the treatise was        
originally to have been sent4 ,it is in 
this preface that ,archimedes alludes to
 the lemma used by earlier geometers as 

the basis of the method of exhaustion   
7the ,postulate of ,archimedes1 or the  
theorem of ,euclid ,x41 #a74 ,he        
mentions as having been proved by means 
of it 7#a7 the theorems that the areas  
of circles are to one another in the    
duplicate ratio of their diameters1 and 
that the volumes of spheres are in the  
triplicate ratio of their diameters1 and
 7#b7 the propositions proved by        
,eudoxus about the volumes of a cone and
 a pyramid4 ,no one1 he says1 so far as 
he is aware1 has yet tried to square the
 segment bounded by a straight line and 
a section of a right-angled cone 7a     
parabola72 but he has succeeded in      
proving1 by means of the same lemma1    
that the parabolic segment is equal to  
four-thirds of the triangle on the same 
base and of equal height1 and he sends  
the proofs1 first as investigated by    
means of mechanics and secondly as      
demonstrated by geometry4 ,the          
phraseology shows that here1 as in the  
,method1 ,archimedes regarded the       

mechanical investigation as furnishing  
evidence rather than proof of the truth 
of the proposition1 pure geometry alone 
furnishing the absolute proof required4 
,the mechanical proof with the necessary
 preliminary propositions about the     
parabola 7some of which are merely      
quoted1 while two1 evidently original1  
are proved1 ,props4 #d1 #e7 extends down
 to ,prop4 #ag2 the geometrical proof   
with other auxiliary propositions       
completes the book 7,props4 #ah-bd74    
,the mechanical proof recalls that of   
the ,method in some respects1 but is    
more elaborate in that the elements of  
the area of the parabola to be measured 
are not straight lines but narrow       
strips4 ,the figures inscribed and      
circumscribed to the segment are made up
 of such narrow strips and have a       
saw-like edge2 all the elements are     
trapezia except two1 which are          
triangles1 one in each figure4 ,each    
trapezium 7or triangle7 is weighed where
 it is against another area hung at a   

fixed point of an assumed lever2 thus   
the whole of the inscribed and          
circumscribed figures respectively are  
weighed against the sum of an indefinite
 number of areas all suspended from one 
point on the lever4 ,the result is      
obtained by a real integration1         
confirmed as usual by a proof by the    
method of exhaustion4  ,the geometrical 
proof proceeds thus4 ,drawing in the    
segment the inscribed triangle with the 
same base and height as the segment1    
,archimedes next inscribes triangles in 
precisely the same way in each of the   
segments left over1 and proves that the 
sum of the two new triangles is #a_/d of
 the original inscribed triangle4       
,again1 drawing triangles inscribed in  
the same way in the four segments left  
over1 he proves that their sum is #a_/d 
of the sum of the preceding pair of     
triangles and therefore 7#a_/d7b of the 
original inscribed triangle4 ,proceeding
 thus1 we have a series of areas        
exhausting the parabolic segment4 ,their

 sum1 if we denote the first inscribed  
triangle by ,delta1 is    ,delta#a      
#a_/d  7#a_/d7b  7#a_/d7c  4 4 4 4      
,archimedes proves geometrically in     
,prop4 #bc that the sum of this infinite
 series is #d_/c;,delta1 and then       
confirms by reductio ad absurdum the    
equality of the area of the parabolic   
segment to this area4     ,,chapter ,v4 
,,the ,,sandreckoner4   ,the            
,sandreckoner deserves a place by       
itself4 ,it is not mathematically very  
important2 but it is an arithmetical    
curiosity which illustrates the         
versatility and genius of ,archimedes1  
and it contains some precious details of
 the history of ,greek astronomy which1 
coming from such a source and at first  
hand1 possess unique authority4 ,we will
 begin with the astronomical data4 ,they
 are contained in the preface addressed 
to ,king ,gelon of ,syracuse1 which     
begins as follows3--  ,there are some1  
,king ,gelon1 who think that the number 
of the sand is infinite in multitude2   

and ,i mean by the sand not only that   
which exists about ,syracuse and the    
rest of ,sicily but also that which is  
found in every region whether inhabited 
or uninhabited4 ,again1 there are some  
who1 without regarding it as infinite1  
yet think that no number has been named 
which is great enough to exceed its     
multitude4 ,and it is clear that they   
who hold this view1 if they imagined a  
mass made up of sand in other respects  
as large as the mass of the earth1      
including in it all the seas and the    
hollows of the earth filled up to a     
height equal to that of the highest of  
the mountains1 would be many times      
further still from recognising that any 
number could be expressed which exceeded
 the multitude of the sand so taken4    
,but ,i will try to show you1 by means  
of geometrical proofs which you will be 
able to follow1 that1 of the numbers    
named by me and given in the work which 
,i sent to ,zeuxippus1 some exceed not  
only the number of the mass of sand     

equal in size to the earth filled up in 
the way described1 but also that of a   
mass equal in size to the universe4     
,now you are aware that 'universe' is   
the name given by most astronomers to   
the sphere the centre of which is the   
centre of the earth1 while the radius is
 equal to the straight line between the 
centre of the sun and the centre of the 
earth4 ,this is the common account1 as  
you have heard from astronomers4 ,but   
,aristarchus of ,samos brought out a    
book consisting of some hypotheses1 in  
which the premises lead to the          
conclusion that the universe is many    
times greater than that now so called4  
,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 centre 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 ratio to the distance of the fixed   
stars as the centre of the sphere bears 
to its surface4  ,here then is absolute 
and practically contemporary evidence   
that the ,greeks1 in the person of      
,aristarchus of ,samos 7about #caj-bcj  
,b4,c471 had anticipated ,copernicus4   
,by the last words quoted ,aristarchus  
only meant to say that the size of the  
earth is negligible in comparison with  
the immensity of the universe4 ,this1   
however1 does not suit ,archimedes's    
purpose1 because he has to assume a     
definite size1 however large1 for the   
universe4 ,consequently he takes a      
liberty with ,aristarchus4 ,he says that
 the centre 7a mathematical point7 can  
have no ratio whatever to the surface of
 the sphere1 and that we must therefore 
take ,aristarchus to mean that the size 
of the earth is to that of the so-called
 universe as the size of the so-called  
universe is to that of the real universe
 in the new sense4  ,next1 he has to    
assume certain dimensions for the earth1

 the moon and the sun1 and to estimate  
the angle subtended at the centre of the
 earth by the sun's diameter2 and in    
each case he has to exaggerate the      
probable figures so as to be on the safe
 side4 ,while therefore 7he says7 some  
have tried to prove that the perimeter  
of the earth is #cjj1jjj stadia         
7,eratosthenes1 his contemporary1 made  
it #beb1jjj stadia1 say #bd1ffb miles1  
giving a diameter of about #g1hej       
miles71 he will assume it to be ten     
times as great or #c1jjj1jjj stadia4    
,the diameter of the earth1 he          
continues1 is greater than that of the  
moon and that of the sun is greater than
 that of the earth4 ,of the diameter of 
the sun he observes that ,eudoxus had   
declared it to be nine times that of the
 moon1 and his own father1 ,phidias1 had
 made it twelve times1 while            
,aristarchus had tried to prove that the
 diameter of the sun is greater than    
eighteen times but less than twenty     
times the diameter of the moon 7this was

 in the treatise of ,aristarchus ,on the
 ,sizes and ,distances of the ,sun and  
,moon1 which is still extant1 and is an 
admirable piece of geometry1 proving    
rigorously1 on the basis of certain     
assumptions1 the result stated74        
,archimedes again intends to be on the  
safe side1 so he takes the diameter of  
the sun to be thirty times that of the  
moon and not greater4 ,lastly1 he says  
that ,aristarchus discovered that the   
diameter of the sun appeared to be about
 #a_/gbjth part of the zodiac circle1   
i4e4 to subtend an angle of about half a
 degree2 and he describes a simple      
instrument by which he himself found    
that the angle subtended by the diameter
 of the sun at the time when it had just
 risen was less than #a_/afdth part and 
greater than #a_/bjjth part of a right  
angle4 ,taking this as the size of the  
angle subtended at the eye of the       
observer on the surface of the earth1 he
 works out1 by an interesting           
geometrical proposition1 the size of the

 angle subtended at the centre of the   
earth1 which he finds to be  #a_/bjcrd  
part of a right angle4 ,consequently the
 diameter of the sun is greater than the
 side of a regular polygon of #hab sides
 inscribed in a great circle of the     
so-called universe1 and a fortiori      
greater than the side of a regular      
chiliagon 7polygon of #ajjj sides7      
inscribed in that circle4  ,on these    
assumptions1 and seeing that the        
perimeter of a regular chiliagon 7as of 
any other regular polygon of more than  
six sides7 inscribed in a circle is more
 than #c times the length of the        
diameter of the circle1 it easily       
follows that1 while the diameter of the 
earth is less than #a1jjj1jjj stadia1   
the diameter of the so-called universe  
is less than #aj1jjj times the diameter 
of the earth1 and therefore less than   
#aj1jjj1jjj1jjj stadia4  ,lastly1       
,archimedes assumes that a quantity of  
sand not greater than a poppy-seed      
contains not more than #aj1jjj grains1  

and that the diameter of a poppy-seed is
 not less than #a_/djth of a dactylus   
7while a stadium is less than #aj1jjj   
dactyli74  ,archimedes is now ready to  
work out his calculation1 but for the   
inadequacy of the alphabetic system of  
numerals to express such large numbers  
as are required4 ,he1 therefore1        
develops his remarkable terminology for 
expressing large numbers4  ,the ,greek  
has names for all numbers up to a myriad
 7#aj1jjj72 there was1 therefore1 no    
difficulty in expressing with the       
ordinary numerals all numbers up to a   
myriad myriads 7#ajj1jjj1jjj74 ,let us1 
says ,archimedes1 call all these numbers
 numbers of the first order4 ,let the   
second order of numbers begin with      
#ajj1jjj1jjj1 and end with              
#ajj1jjj1jjjb4 ,let #ajj1jjj1jjjb be the
 first number of the third order1 and   
let this extend to #ajj1jjj1jjjc2 and so
 on1 to the myriad-myriadth order1      
beginning with #ajj1jjj1jjj7ii1iii1iii7 
and ending with                         

#ajj1jjj1jjj7ajj1jjj1jjj71 which for    
brevity we will call ,p4 ,let all the   
numbers of all the orders up to ,p form 
the first period1 and let the first     
order of the second period begin with ,p
 and end with #ajj1jjj1jjj ,p2 let the  
second order begin with this1 the third 
order with #ajj1jjj1jjjb ,p1 and so on  
up to the #ajj1jjj1jjjth order of the   
second period1 ending with              
#a1jjj1jjj1jjj7ajj1jjj1jjj7 ,p or ,p#b4 
,the first order of the third period    
begins with ,p#b1 and the orders proceed
 as before4 ,continuing the series of   
periods and orders of each period1 we   
finally arrive at the #ajj1jjj1jjjth    
period ending with ,p7#ajj1jjj1jjj74    
,the prodigious extent of this scheme is
 seen when it is considered that the    
last number of the first period would   
now be represented by #a followed by    
#hjj1jjj1jjj ciphers1 while the last    
number of the #ajj1jjj1jjjth period     
would require #ajj1jjj1jjj times as many
 ciphers1 i4e4 #hj1jjj million million  

ciphers4  ,as a matter of fact1         
,archimedes does not need1 in order to  
express the number of the sand1 to go   
beyond the eighth order of the first    
period4 ,the orders of the first period 
begin respectively with #a1 #ajh1 #ajaf1
 #ajbd1 444 7#ajh77ii1iii1iii72 and we  
can express all the numbers required in 
powers of #aj4  ,since the diameter of a
 poppy-seed is not less than #a_/djth of
 a dactylus1 and spheres are to one     
another in the triplicate ratio of their
 diameters1 a sphere of diameter #a     
dactylus is not greater than #fd1jjj    
poppy-seeds1 and1 therefore1 contains   
not more than #fd1jjj ,x #aj1jjj grains 
of sand1 and a fortiori not more than   
#a1jjj1jjj1jjj1 or #aji grains of sand4 
,archimedes multiplies the diameter of  
the sphere continually by #ajj1 and     
states the corresponding number of      
grains of sand4 ,a sphere of diameter   
#aj1jjj dactyli and a fortiori of one   
stadium contains less than #ajba grains2
 and proceeding in this way to spheres  

of diameter #ajj stadia1 #aj1jjj stadia 
and so on1 he arrives at the number of  
grains of sand in a sphere of diameter  
#aj1jjj1jjj1jjj stadia1 which is the    
size of the so-called universe2 the     
corresponding number of grains of sand  
is #ajea4 ,the diameter of the real     
universe being #aj1jjj times that of the
 so-called universe1 the final number of
 grains of sand in the real universe is 
found to be #ajfc1 which in             
,archimedes's terminology is a          
myriad-myriad units of the eighth order 
of numbers4     ,,chapter ,,vi4         
,,mechanics4   ,it is said that         
,archytas was the first to treat        
mechanics in a systematic way by the aid
 of mathematical principles2 but no     
trace survives of any such work by him4 
,in practical mechanics he is said to   
have constructed a mechanical dove which
 would fly1 and also a rattle to amuse  
children and keep them from breaking    
things about the house 7so says         
,aristotle1 adding for it is impossible 

for children to keep still74  ,in the   
,aristotelian ,mechanica we find a      
remark on the marvel of a great weight  
being moved by a small force1 and the   
problems discussed bring in the lever in
 various forms as a means of doing this4
 ,we are told also that practically all 
movements in mechanics reduce to the    
lever and the principle of the lever    
7that the weight and the force are in   
inverse proportion to the distances from
 the point of suspension or fulcrum of  
the points at which they act1 it being  
assumed that they act in directions     
perpendicular to the lever74 ,but the   
lever is merely referred to the circle2 
the force which acts at the greater     
distance from the fulcrum is said to    
move a weight more easily because it    
describes a greater circle4  ,there is1 
therefore1 no proof here4 ,it was       
reserved for ,archimedes to prove the   
property of the lever or balance        
mathematically1 on the basis of certain 
postulates precisely formulated and     

making no large demand on the faith of  
the learner4 ,the treatise ,on ,plane   
,equilibriums in two books is1 as the   
title implies1 a work on statics only2  
and1 after the principle of the lever or
 balance has been established in ,props4
 #f1 #g of ,book ,i41 the rest of the   
treatise is devoted to finding the      
centre of gravity of certain figures4   
,there is no dynamics in the work and   
therefore no room for the parallelogram 
of velocities1 which is given with a    
fairly adequate proof in the            
,aristotelian ,mechanica4  ,archimedes's
 postulates include assumptions to the  
following effect3 7#a7 ,equal weights at
 equal distances are in equilibrium1 and
 equal weights at unequal distances are 
not in equilibrium1 but the system in   
that case inclines towards the weight   
which is at the greater distance1 in    
other words1 the action of the weight   
which is at the greater distance        
produces motion in the direction in     
which it acts2 7#b7 and 7#c7 ,if when   

weights are in equilibrium something is 
added to or subtracted from one of the  
weights1 the system will incline towards
 the weight which is added to or the    
weight from which nothing is taken      
respectively2 7#d7 and 7#e7 ,if equal   
and similar figures be applied to one   
another so as to coincide throughout1   
their centres of gravity also coincide2 
if figures be unequal but similar1 their
 centres of gravity are similarly       
situated with regard to the figures4    
,the main proposition1 that two         
magnitudes balance at distances         
reciprocally proportional to the        
magnitudes1 is proved first for         
commensurable and then for              
incommensurable magnitudes4 ,preliminary
 propositions have dealt with equal     
magnitudes disposed at equal distances  
on a straight line and odd or even in   
number1 and have shown where the centre 
of gravity of the whole system lies4    
,take first the case of commensurable   
magnitudes4 ,if ,a1 ,b be the weights   

acting at ,e1 ,d on the straight line   
,,ed respectively1 and ,,ed be divided  
at ,c so that ,a 3 ,b  ,,dc 3 ,,ce1     
,archimedes has to prove that the system
 is in equilibrium about ,c4 ,he        
produces ,,ed to ,k1 so that ,,dk  ,,ec1
 and ,,de to ,l so that ,,el  ,,cd2 ,,lk
 is then a straight line bisected at ,c4
 ,again1 let ,h be taken on ,,lk such   
that ,,lh  #b,,le or #b;,,cd1 and it    
follows that the remainder ,,hk  #b;,,dk
 or #b;,,ec4 ,since ,a1 ,b are          
commensurable1 so are ,,ec1 ,,cd4 ,let x
 be a common measure of ,,ec1 ,,cd4     
,take a weight w such that w is the same
 part of ,a that x is of ,,lh4 ,it      
follows that w is the same part of ,b   
that x is of ,,hk4 ,archimedes now      
divides ,,lh1 ,,hk into parts equal to  
x1 and ,a ,b into parts equal to w1 and 
places the w's at the middle points of  
the x's respectively4 ,all the w's are  
then in equilibrium about ,c4 ,but all  
the w's acting at the several points    
along ,,lh are equivalent to ,a acting  

as a whole at the point ,e4 ,similarly  
the w's acting at the several points on 
,,hk are equivalent to ,b acting at ,d4 
,therefore ,a1 ,b placed at ,e1 ,d      
respectively balance about ,c4  ,prop4  
#g deduces by reductio ad absurdum the  
same result in the case where ,a1 ,b are
 incommensurable4 ,prop4 #h shows how to
 find the centre of gravity of the      
remainder of a magnitude when the centre
 of gravity of the whole and of a part  
respectively are known4 ,props4 #i-ae   
find the centres of gravity of a        
parallelogram1 a triangle and a         
parallel-trapezium respectively4  ,book 
,,ii41 in ten propositions1 is entirely 
devoted to finding the centre of gravity
 of a parabolic segment1 an elegant but 
difficult piece of geometrical work     
which is as usual confirmed by the      
method of exhaustion4     ,,chapter     
,,vii4  ,,hydrostatics4   ,the science  
of hydrostatics is1 even more than that 
of statics1 the original creation of    
,archimedes4 ,in hydrostatics he seems  

to have had no predecessors4 ,only one  
of the facts proved in his work ,on     
,floating ,bodies1 in two books1 is     
given with a sort of proof in           
,aristotle4 ,this is the proposition    
that the surface of a fluid at rest is  
that of a sphere with its centre at the 
centre of the earth4  ,archimedes founds
 his whole theory on two postulates1 one
 of which comes at the beginning and the
 other after ,prop4 #g of ,book ,i4     
,postulate #a is as follows3--  ,let us 
assume that a fluid has the property    
that1 if its parts lie evenly and are   
continuous1 the part which is less      
compressed is expelled by that which is 
more compressed1 and each of its parts  
is compressed by the fluid above it     
perpendicularly1 unless the fluid is    
shut up in something and compressed by  
something else4  ,postulate #b is3 ,let 
us assume that any body which is borne  
upwards in water is carried along the   
perpendicular to the surface which      
passes through the centre of gravity of 

the body4  ,in ,prop4 #b ,archimedes    
proves that the surface of any fluid at 
rest is the surface of a sphere the     
centre of which is the centre of the    
earth4 ,props4 #c-g deal with the       
behaviour1 when placed in fluids1 of    
solids 7#a7 just as heavy as the fluid1 
7#b7 lighter than the fluid1 7#c7       
heavier than the fluid4 ,it is proved   
7,props4 #e1 #f7 that1 if the solid is  
lighter than the fluid1 it will not be  
completely immersed but only so far that
 the weight of the solid will be equal  
to that of the fluid displaced1 and1 if 
it be forcibly immersed1 the solid will 
be driven upwards by a force equal to   
the difference between the weight of the
 solid and that of the fluid displaced4 
,if the solid is heavier than the fluid1
 it will1 if placed in the fluid1       
descend to the bottom and1 if weighed in
 the fluid1 the solid will be lighter   
than its true weight by the weight of   
the fluid displaced 7,prop4 #g74  ,the  
last-mentioned theorem naturally        

connects itself with the story of the   
crown made for ,hieron4 ,it was         
suspected that this was not wholly of   
gold but contained an admixture of      
silver1 and ,hieron put to ,archimedes  
the problem of determining the          
proportions in which the metals were    
mixed4 ,it was the discovery of the     
solution of this problem when in the    
bath that made ,archimedes run home     
naked1 shouting ,greek3 eureka1 eureka4 
,one account of the solution makes      
,archimedes use the proposition last    
quoted2 but on the whole it seems more  
likely that the actual discovery was    
made by a more elementary method        
described by ,vitruvius4 ,observing1 as 
he is said to have done1 that1 if he    
stepped into the bath when it was full1 
a volume of water was spilt equal to the
 volume of his body1 he thought of      
applying the same idea to the case of   
the crown and measuring the volumes of  
water displaced respectively 7#a7 by the
 crown itself1 7#b7 by the same weight  

of pure gold1 and 7#c7 by the same      
weight of pure silver4 ,this gives an   
easy means of solution4 ,suppose that   
the weight of the crown is ,w1 and that 
it contains weights w#a and w#b1 of gold
 and silver respectively4 ,now          
experiment shows 7#a7 that the crown    
itself displaces a certain volume of    
water1 ,v say1 7#b7 that a weight ,w of 
gold displaces a certain other volume of
 water1 ,v#a say1 and 7#c7 that a weight
 ,w of silver displaces a volume ,v#b4  
,from 7#b7 it follows1 by proportion1   
that a weight w#a of gold will displace 
w#a_/,w 4 ,v#a of the fluid1 and from   
7#c7 it follows that a weight w#b of    
silver displaces w#b_/,w 4 ,v#b of the  
fluid4    ,hence       ,v  w#a_/,w 4    
,v#a  w#b_/,w 4 ,v#b2    therefore      
,,wv  w#a,v#a  w#b,v#b1    that is1     
7w#a  w#b7,v  w#a,v#a  w#b,v#b1    so   
that     w#a_/w#b  7,v#b - ,v7_/7,v -   
,v#a71  which gives the required ratio  
of the weights of gold and silver       
contained in the crown4  ,the last two  

propositions of ,book ,i4 investigate   
the case of a segment of a sphere       
floating in a fluid when the base of the
 segment is 7#a7 entirely above and 7#b7
 entirely below the surface of the      
fluid2 and it is shown that the segment 
will in either case be in equilibrium in
 the position in which the axis is      
vertical1 the equilibrium being in the  
first case stable4  ,book ,,ii4 is a    
geometrical tour de force4 ,here1 by the
 methods of pure geometry1 ,archimedes  
investigates the positions of rest and  
stability of a right segment of a       
paraboloid of revolution floating with  
its base upwards or downwards 7but      
completely above or completely below the
 surface7 for a number of cases         
differing 7#a7 according to the relation
 between the length of the axis of the  
paraboloid and the principal parameter  
of the generating parabola1 and 7#b7    
according to the specific gravity of the
 solid in relation to the fluid2 where  
the position of rest and stability is   

such that the axis of the solid is not  
vertical1 the angle at which it is      
inclined to the vertical is fully       
determined4  ,the idea of specific      
gravity appears all through1 though this
 actual term is not used4 ,archimedes   
speaks of the solid being lighter or    
heavier than the fluid or equally heavy 
with it1 or when a ratio has to be      
expressed1 he speaks of a solid the     
weight of which 7for an equal volume7   
has a certain ratio to that of the      
fluid4     ,,bibliography4   ,the editio
 princeps of the works of ,archimedes   
with the commentaries of ,eutocius was  
brought out by ,hervagius 7,herwagen7 at
 ,basel in #aedd4 ,d4 ,rivault 7,paris1 
#afae7 gave the enunciations in ,greek  
and the proofs in ,latin somewhat       
retouched4 ,the ,arenarius              
7,sandreckoner7 and the ,dimensio       
circuli with ,eutocius's commentary were
 edited with ,latin translation and     
notes by ,wallis in #afgh 7,oxford74    
,torelli's monumental edition 7,oxford1 

#agib7 of the ,greek text of the        
complete works and of the commentaries  
of ,eutocius1 with a new ,latin         
translation1 remained the standard text 
until recent years2 it is now superseded
 by the definitive text with ,latin     
translation of the complete works1      
,eutocius's commentaries1 the fragments1
 scholia1 etc41 edited by ,heiberg in   
three volumes 7,teubner1 ,leipzig1 first
 edition1 #ahhj-a2 second edition1      
including the newly discovered ,method1 
etc41 #aiaj-ae74  ,of translations the  
following may be mentioned4 ,the ,aldine
 edition of #aeeh1 #dto1 contains the   
,latin translation by ,commandinus of   
the ,measurement of a ,circle1 ,on      
,spirals1 ,quadrature of the ,parabola1 
,on ,conoids and ,spheroids1 ,the       
,sandreckoner4 ,isaac ,barrow's version 
was contained in ,opera ,archimedis1    
,apollonii ,pergoei conicorum libri1    
,theodosii ,sphoerica1 methodo novo     
illustrata et demonstrata 7,london1     
#afge74 ,the first ,french version of   

the works was by ,peyrard in two volumes
 7second edition1 #ahjh74 ,a valuable   
,german translation1 with notes1 by ,e4 
,nizze1 was published at ,stralsund in  
#ahbd4 ,there is a complete edition in  
modern notation by ,t4 ,l4 ,heath 7,the 
,works of ,archimedes1 ,cambridge1      
#ahig1 supplemented by ,the ,method of  
,archimedes1 ,cambridge1 #aiab74        
,,chronology4   7,,approximate ,,in     
,,some ,,cases47            ,b4,c4      
#fbd-edg    ,thales         #egb-dig    
,pythagoras         #ejj-dbh            
,anaxagoras         #dgj-djj  _/        
,hippocrates of ,chios                  
,hippias of ,elis         #dgj-chj      
,democritus         #dfj-che            
,theodorus of ,cyrene         #dcj-cfj  
,archytas of ,taras 7,tarentum7         
#dbg-cdg    ,plato         #dae-cfi     
,theaetetus         #djh-cee    ,eudoxus
 of ,cnidos                  _/ ,leon   
fl4 about #cej   ,menaechmus            
,dinostratus                   ,theudius
         fl4 #cjj    ,euclid            

#caj-bcj    ,aristarchus of ,samos      
#bhg-bab    ,archimedes         #bhd-bjc
    ,eratosthenes         #bfe-aij      
,apollonius of ,perga