,,m5o ,,ii    ,a ,,cont9uation ,,(      
,,socrates' ,,dialogue ,,) ,,m5o  ,,9   
,,:i* ,,! ,,boy ,,proves ,,root #b ,,is 
,,irrational    ,by ,socrates      ,a   
,mill5nium ,fulcrum ,edition ,copyright 
#aiie        ,socrates3  ,well1 "h we   
are at ! appo9t$ "t1 ,m5o4    ,m5o3     
,yes1 & x looks l a f9e day = x1 too4   
,socrates3  ,& ,i see our s]v+ boy is   
also "h4    ,boy3  ,yes1 ,i am1 & ready 
to d y\r bidd+4    ,socrates3           
,wond]ful4  ,n[1 ,m5o1 ,i want y to be  
on y\r gu>d1  z y 7 ! o!r day1 to insure
 t ,i tea* no?+ to ! boy1  b r pull out 
( 8 m9d ! premises :i* are already !re4 
,m5o3  ,i %all d my be/1 ,socrates4     
,socrates3  ,i c ask m ( no man1 ,m5o1 &
 ,i am c]ta9 t  y w d well1 & ,i hope ,i
 w give y no call to halt me  9 my say+ 
if ,i %\ld say too mu*1 9 :i* y w\ld    
feel ,i  0 actually tea*+ ! boy ! answ] 
to ?is riddle4    ,m5o3  ,no1 ,socrates1
 ,i don't ?9k ,i w h to call y on  any?+
 y might say today1 = ! mo/ wondr\sly   
le>n$ m5  ( ! gr\p ( ,py?agoras h sp5t  

_m h\rs1 weeks1 & ev5  mon?s & ye>s     
toil+ 9 _! mann] to arrive at ! my/ic   
solutions to ! puzzles =m$ by ! simple  
squ>es ) :i*  we work$ ! o!r day4       
,!re=e1 ,i am c]ta9 to rega9 my  virtue1
 :i* ,i lo/ ! o!r day1 :5 ,i 0 s        
/eadfa/ly  prov5 by y to be 9 error 9 my
 /atem5t t ! root ( a  squ>e ) an area (
 two squ>e feet 0 beyond ?is boy1  :o is
 a f9e boy1 :om we mu/ make to und]/& t 
he %\ld  d 8 be/ "h1 & n feel t he has  
done any wrongness by  caus+ me to lose 
my virtue to y ! o!r day4    ,socrates3 
,m5o1 my fri5d1 x is my op9ion1 & ,i    
hope x w  soon be y\rs1 t y\r virtue 0  
increas$ ! o!r day1  r ?an decreas$4    
,m5o3  ,i fail to see h[1 :5 ,i 0       
humiliat$ by see+  ?is "y boy1 ( mode/  
education1 arrive 9 m9utes at !  highe/ 
my/ic levels ( ! magic ( ! ,py?agoreans4
  ,mo/ ( all :5 ,i wag]$ z _m d9n]s z y 
c\ld eat  at my h\se t ?is c\ld n be !  
case4    ,socrates3  ,fir/1 fri5d ,m5o1 
let me assure y t ,i w  promise nev] to 
eat y out ( h\se & home1 n t ,i c\ld  if

 ,i tri$1 = my ta/es are simple & y\r   
wallet is l>ge4  ,nev]!less1 ,m5o1 my   
fri5d1 ,i w\ld ha/5 to add t ,i w       
promise1 if y l1 n to ev] come to y\r   
table un9vit$4    ,z a second reason y &
 y\r virtue %\ld feel bett] aft]  !     
ev5ts ( ! o!r day1 because y 7 9 error  
be=e1  b are less 9 error n[4  ,& ! pa? 
to virtue1 at lea/ "o  aspect ( ! pa? to
 virtue1 is 9 f9d+ & correct+ error4    
,m5o3  ,socrates1 y "k y are always     
welcome at my table1  except :5 ,i am   
suff]+ f my ulc]1 :i* y aggravate       
greatly1 or at times :5 ,i am ent]ta9+ !
 highe/ nobles  ( ! l&1 & y w\ld appe>  
out ( place 9 y\r clo?+4  7,socrates 0  
kn[n = 8 simple attire1 & = we>+ 8      
g>m5ts ov] & ov] till !y wore out4      
,h[ev]1 ! only  surviv+ example ( 8     
writ+ is a laundry li/1 s we "k he  kept
 8 clo!s clean & some:at pres5table1    
?\gh simple7    ,socrates3  ,i w\ld hope
 y w\ld h me ov] because ,i 0  a good   
influ5ce on y\r developm5t1 ?an = any   
o!r reason4  ,i notice y did n respond  

to my claim to h increas$  y\r virtue1  
?r\gh ! exorcism ( y\r error4    ,m5o3  
,well ,socrates1 y "k t x is n always ! 
easie/ ?+ to give up "o's ways1 ev5 ?\gh
 "o has f\nd  !m to be 9 error4  ,!re=e1
 please =give me if ,i am n  s\nd+ z    
grateful z y w\ld l = y\r lessons4      
,socrates3  ,! easi] "o f9ds x to give  
up ! ways ( error1  ! easi] x is to     
replace ! error ) t :i* we hope  is n 9 
error4  ,is ?is n ! way to virtue8      
,m5o3  ,yes1 ,socrates1 & y "k ! pa? is 
h>d1  & t we (t5 /umble & fall4         
,socrates3  ,yes1 b is x n true t we    
/umble & fall  ov] ! ob/acles :i* we    
make = ourselves to trip ov]8    ,m5o3  
,c]ta9ly t is mo/ true1 ,socrates1 9 "s 
cases4    ,socrates3  ,well1 !n1 let u  
proce$1 = ,i see ! h\r is ^u u  :5 ,i d 
my be/ ?9k+1 & t h\r %all be pass$ soon1
  & hopefully ) x %all pass a bit ( y\r 
ignorance4    ,m5o3  ,well said1        
,socrates4  ,i am ) y4    ,socrates3  ,&
 %all h we a wag] on ! ev5ts ( today8   
,m5o3  ,c]ta9ly1 ,socrates4             

,socrates3  ,& :at %all y wag] aga9/ ?is
 boy prov+  t ! l5g? ( ! root ( a squ>e 
) an area ( two  squ>e feet1 _c be made 
by ! ratio ( two :ole numb]s8    ,m5o3  
,y may h any?+ x is 9 my p[] to give1   
unless x cause h>m to myself or to ano!r
 to give x4    ,socrates3  ,well said1  
my fri5d ,m5o1 & ,i %all leave x at  t4 
,& :at %all ,i (f] y z a return wag]8   
,m5o3  ,well1 ! easie/ ?+ :i* comes to  
m9d is to  wag] all ^? d9n]s y won f me 
! o!r day4    ,socrates3  ,v well1 s be 
x4    ,m5o3  ,n[ ,socrates1 s9ce y are  
my fri5d1 ,i mu/ give y  ?is fri5dly    
w>n+3  y "k t ! ,py?agoreans jeal\sly   
gu>d _! secrets ) secret meet+s1        
protect$ by secret  h&%akes1 secret     
signs1 passwords1 & all t1 d y n8       
,socrates3  ,i h he>d z mu*1 fri5d ,m5o4
    ,m5o3  ,!n be sure t !y w seek      
rev5ge ^u y =  demy/ify+ ! ideas &      
concepts :i* !y work$ s long &  h>d &   
secretly to create & protect2 = !y are a
 jeal\s lot  9 ! extreme1 hid+ 9 m\nta9 
caves1 :i* are h>dly fit  to be call$   

mona/]ies by ev5 ! mo/ h>d5$ monk4      
,socrates3  ,i take y\r mean+1 fri5d    
,m5o1 & ?ank y  = y\r consid]ation1 b ,i
 ?9k t if ,i lose1 t !y  w n bo!r me1 & 
if ,i w91 x w appe> s simple to         
ev]yone1 t if w\ld be %e]e/ folly =     
anyone to make  ev5 ! smalle/ ge/ure to 
protect its fall5 my/ic secrecy4        
,besides1 ,i h a citiz5's responsibility
 to ,a!ns & to all  ,a!nians to d my be/
 to protect !m & enlight5 !m4    ,m5o3  
,v well1 ,socrates4  ,please d n ev] say
 t ,i did n  try to w>n y1 especially   
aft] !y h nail$ y to a cross  9 a public
 place1 :]e anyone & ev]yone c\ld he> y 
say  t ! fault ( ?is lay 9 my "n4       
,socrates3  ,d n worry1 fri5d ,m5o1 = if
 ,i 7 n  to %[ ?is simple feat ( logic  
to y1 ,i %\ld j walk  d[n ! /reet & f9d 
someone else1 ?\gh n someone ^:  company
 & conv]sation ,i %\ld enjoy z mu* z    
y\rs4    ,m5o3  ,?ank y1 fri5d          
,socrates4    ,socrates3  ,n[1 boy1 d y 
rememb] me1 & ! squ>es )  :i* we work$ &
 play$ ! o!r day8    ,boy3  ,yes1 sir1  

,socrates4    ,socrates3  ,please1 ,m5o1
 in/ruct ! boy to m]ely call me  by my  
"n1 z does ev]yone else4  ,call+ me sir 
m]ely puts  me (f my m5tal /ride1 &1    
besides1 x w create a great]  di/ance   
betwe5 me & ! boy4    ,m5o3  ,y he>d :at
 ,socrates1 said1 boy4  ,c y d x8       
,boy3  ,yes1 sir4  7,turn+ to ,socrates7
  ,y "k ,i l y  v mu*1 & t ,i call y sir
 n only out ( relation (  our positions 
9 society1 b also because ( my true     
respect &  admiration1 especially aft] !
 ev5ts ( ! o!r day4    ,socrates3  ,yes1
 boy4  ,& ,i w try to live up to y\r    
expectations4  7,turn+ to ,m5o7  ,w\ld y
 all[ "s rew>d =  ! boy1 z well z t :i* 
is = myself1 if he %\ld prove  to y\r   
satisfaction t ! squ>e root ( two is    
irrational8    ,m5o3  ,c]ta9ly1         
,socrates4    ,socrates3  7tak+ ! boy   
aside7  ,:at w\ld y l  ! mo/ 9 ! :ole   
_w1 boy8    ,boy3  ,y mean any?+8       
,socrates3  ,well1 ,i c't gu>antee to   
get x = y1 b at  lea/ ,i c ask x1 & x   
%\ldn't hurt to ask2 & besides1  z y    

%\ld "k1 x is v h>d to expect someone to
 give  y :at y want1 if y nev] let !m "k
 y want x4    ,boy3  ,well1 ,socrates444
 y "k :at ,i w\ld want4    ,socrates3   
,d ,i8    ,boy3  ,bett] ?an ,i knew !   
squ>e root ( two ! o!r day4             
,socrates3  ,y want to be a free man1   
!n1 & a citiz54    ,boy3  7look+ d[n7   
,yes4    ,socrates3  ,don't look d[n1   
!n1 = t is an admirable  desire = "o to 
h1 & speaks highly ( him :o has x4  ,i w
 speak to ,m5o1 :ile y hold y\r tongue4 
,boy3  ,yes1 ,socrates4  7b[s to kiss 8 
h&1 ,socrates turns7    ,socrates3      
,fri5d ,m5o1 h[ h>d d y ?9k x w be =    
?is boy to prove ! irrationality ( !    
squ>e root ( two8    ,m5o3  ,y "k t ,i  
?9k x is impossible1 ,socrates4         
,socrates3  ,well1 h[ long did x take ! 
,py?agoreans8    ,m5o3  ,i %\ld ?9k x   
took !m ye>s4    ,socrates3  ,& h[ _m ( 
!m 7 !re8    ,m5o3  ,q a few1 ?\gh n all
 work$ equally1 & "s  h>dly at all1 = !y
 7 mo/ int]e/$ 9 triangles ( !  "r &    
virtu\s v>iety1 & n 9 squ>es & _! roots4

    ,socrates3  ,c y give me an e/imate8
    ,m5o3  ,no1 ,i c't say t ,i c4  ,i  
am sorry1 ,socrates4    ,socrates3  ,no 
problem1 w\ld y accept five ?9k]s z an  
e/imate4    ,m5o3  ,i ?9k t %\ld be     
fair4    ,socrates3  ,& %all we assume  
!y work$ = two ye>s1 t is !  smalle/    
numb] :i* reta9s ! plural1 & our        
assumption 0 t  !y work$ = ye>s4        
,m5o3  ,two ye>s is inde$ acceptable to 
me1 ,socrates4    ,socrates3  ,v well   
!n1 ,m5o1 x w\ld appe> t !  ,py?agoreans
 sp5t #aj total ye>s ( ?9k+ "t to solve 
!  riddles ( ! squ>e root ( two4        
,m5o3  ,i agree4    ,socrates3  ,& w\ld 
y l to hire ! ,py?agoreans  to run y\r  
h\sehold1 ,m5o8    ,m5o3  ,surely ,i    
w\ld1 ,socrates1 if !y 7 only = hire1   
b1 z y well "k1 !y are a secret lot1 &  
hire to no "o4    ,socrates3  ,well1 if 
,i c\ld get y "o1 p]haps "o ( !  be/ (  
!m1 9 fact ! lead] ( ! gr\p t solv$ !   
squ>e root ( two1 w\ld y n hire him1 &  
at high wages8    ,m5o3  ,c]ta9ly1      
,socrates4  ,i'd be a fool n to4        

,socrates3  ,& y w\ld put him 9 *>ge (  
y\r h\se4    ,m5o3  ,& all my l&s1 too1 
,socrates4    ,socrates3  ,possibly4    
,!n ,i w\ld l to propose1 t if  ?is boy 
%\ld solve ! pro( ( ! squ>e root ( two  
be+  irrational1 9 ! next few h\rs ( our
 discussion1 t he be  giv5 wages equal  
to ^? due to y\r mo/ highly plac$ s]vant
  = t5 ye>s ( s]vice1 z he %all p]=m t5 
ye>s s]vice  = y 9 ! next few h\rs1 %\ld
 he succe$4    ,m5o3  ,t s\nds q fair1  
,socrates1 ,i l y\r logic4    ,socrates3
  7! boy tugs 8 tunic1 to compla9 t he  
wants  8 fre$om1 n a m]e bucket ( gold7 
,hu%1 boy1 did y n  promise to hold y\r 
tongue8    ,boy3  ,yes1 ,socrates1 b4444
    ,socrates3  7turn+ to ,m5o7  ,&1 (  
c\rse1 ) !  monet>y rew>ds = su* a      
position1 g all ! re/ ( x4    ,m5o3  ,( 
c\rse1 ,socrates4  ,i nev] ?\ght to *eat
 y4    ,socrates3  ,i "k t1 fri5d ,m5o1 
b ,i m]ely ask = !  boy's sake1 :o is n 
us$ to he>+ ab\t high f9ance &  ! p[]s &
 rank :i* accompany su* ?+s4    ,m5o3   
,( c\rse1 ,socrates4  ,%all ,i tell !   

boy :at he  %all receive8    ,socrates3 
,y are v k9d to d s1 my fri5d ,m5o4     
,m5o3  7turns to ! boy7  ,y are aw>e t a
 s]vant may n own  ! am\nt ( gold ,i    
w\ld h to give y1 %\ld y w9 ! day8      
,boy3  ,yes1 sir4    ,m5o3  ,!re=e1 ,i  
w\ld h to give to y ! fre$om to  own !  
money1 be=e ,i c\ld give y ! money1 w\ld
 ,i n8    ,boy3  ,yes1 sir4    ,m5o3  ,&
 9 giv+ y fre$om1 ,i w\ld be remiss if  
,i did  n give y a job & a com+ out p>ty
 ( equal position  ) y\r weal?1 w\ld ,i 
n8    ,boy3  ,i c't really say1 sir1    
?\gh ,i suppose s4    ,m5o3  ,y suppose 
correctly4  ,i w fe$ y = a week (       
p>ty+1 & dress y 9 ! f9e/ g>m5ts1 :ile y
 are  introduc$ to ! f9e/ ladies &      
g5tlem5 ( ,a!ns1 f  :om y are free to   
select = y\r int]e/s z fri5ds1  bus9ess 
p>tn]s1 social acqua9tances1            
connections1 & p]haps  ev5 a wife1 %\ld 
y f9d someone y l = t4  ,d y  n[ und]/& 
t !re is no?+ ,i w\ld leave out t y     
w\ld h to ask =1 or t if y did h to ask1
 ,i w\ld  give x imm$iately1 & ask y\r  

=giv5ess = my error8    ,boy3  ,x is h>d
 to und]/&1 b ,i take y\r ^w4           
,socrates3  ,n[ don't let ?is all g to  
y\r head1 boy4  ,?is is some?+ y c\ld h 
figur$ out = y\rself1  if y _h appli$   
y\r m9d to x z y did to squ>es  ! o!r   
day4  ,c y d z well1 today8    ,boy3  ,i
 %\ld ?9k & hope s1 fri5d ,socrates1 =  
,i see y  are inde$ my fri5d1 & ,i %\ld 
hope ,i am m capable today1  = hav+     
le>n$ "s ! o!r day4    ,socrates3  ,we  
%all see1 boy4  ,let u on to ! te/4  ,n[
 y rememb] ! squ>es we dealt ) ! o!r    
day4    ,boy3  ,yes1 ,socrates4         
,socrates3  ,& ! "o p>ticul> squ>e on ! 
diagonal we made1  ^: area 0 two1 d y   
rememb] t "o8    ,boy3  ,yes1 ,socrates4
    ,socrates3  ,& y rememb] t ! l5g? ( 
! side ( a squ>e1  :5 multipli$ by      
itself1 yields ! area ( ! squ>e4        
,boy3  ,ev]yone at s*ool kn[s t1        
,socrates4    ,socrates3  ,well1 maybe4 
,h[ev]1 x is ab\t t side1  :i* :5       
multipli$ "t itself yields an area (    
two1  t ,i w\ld l to speak fur!r today4 

,h[ is t ) y8    ,boy3  ,t is f9e1      
,socrates4  ,i rememb] t l9e1  & ,i sort
 ( lik$ x ! be/1 if y "k :at ,i mean4   
,socrates3  ,good1 !n we %\ld h a great 
"t4  ,d y "k h[ long t l9e is1 boy8     
,boy3  ,well1 ,i "k t y bo? ?\ght x wise
 :5 ,i said  x 0 ( a l5g? :i* :5 made a 
squ>e (1 yield$ a squ>e  ) an area (    
two1 s ,i suppose ,i %\ld answ] t way4  
,socrates3  ,& a good answ] x is1 too4  
,we are go+ to  make x an ev5 bett]     
answ] z we proce$4    ,boy3  ,good4     
,socrates3  ,d y rememb] :5 y tripp$ up 
& fell on  y\r face ! o!r day1 :5 y     
?\ght t ! squ>e ( area  n9e 0 actually a
 squ>e ( area eight8    ,boy3  ,oh yes1 
,socrates6  ,& ,i am sorely a%am$1      
because ,i /ill  d n "k en\gh to make   
sure ,i nev] make su* an error aga91  & 
!re=e ,i "k my virtue & rightness are   
lack+4    ,socrates3  ,!y are n lack+ s 
mu* t !y _c be improv$1  are !y boy8    
,boy3  ,i %\ld hope & pray n4           
,socrates3  ,well today1 y are go+ to   
tell u "s ?+s  ab\t t numb]1 :i* :5     

multipli$ by itself gives u two4        
,boy3  ,i w tell y ev]y?+ ,i "k1 or ?9k 
,i "k1  ,socrates1 & hope t ,i am       
correct or c be correct$4    ,socrates3 
,to ,m5o1 surely he is a f9e boy1 eh    
,m5o8    ,m5o3  ,yes1 ,i am pr\d to own 
him1 b ,i don't see h[ he  c be sm>t    
en\gh to d ! "w today t w\ld take a     
,py?agorean monk t5 ye>s ( cloi/]$ life 
to accompli%4    ,socrates3  ,we %all   
see4  ,boy1 y are do+ f9e4  ,i ?9k ,i   
c\ld ev5 make a s*ol> ( y1 ?\gh ,i fe> y
 might turn to  w9e & wom5 ) y\r new    
f\nd weal?1 if y succe$1 r  ?an cont9ue 
to poli% ! wit :i* %\ld get y t rew>d4  
,boy3  ,i don't ?9k ,i w\ld want to sp5d
 t mu* "t )  wom5 or ) w9e1 ,socrates4  
,socrates3  ,y w f9d some?+1 no d\bt4   
,s1 back to  ! numb] :i* :5 squ>e gives 
u two4  ,:at c we say ab\t  su* a numb]8
  ,is x odd or ev58  ,well x w\ld h to  
be a  :ole numb] to be "o ( ^?1 w\ld x  
n1 & we saw !  o!r day :at happ5s to    
:ole numb]s :5 !y are squ>$8  ,!y give u
 #a1d1i & #af z squ>e areas1 did !y n8  

,boy3  ,yes1 ,socrates1 ?\gh ,i rememb] 
?9k+ t !re  %\ld h be5 a numb] :i* w\ld 
give eight1 ,socrates8    ,socrates3  ,i
 ?9k we %all f9d "o1 if we keep se>*+4  
,n[1 ?is numb]1 d y rememb] if x _h to  
be l>g] or  small] ?an "o8    ,boy3     
,l>g]1 ,socrates4  ,= "o squ>$ gives    
only an area  ( "o1 & we ne$ & area (   
two1 :i* is l>g]4    ,socrates3  ,good4 
,& :at ( two8    ,boy3  ,two gives a    
squ>e ( f\r1 :i* is too l>ge4           
,socrates3  ,f9e4  ,s ! squ>e root ( two
 is small] ?an  ! side two :i* is ! root
 ( f\r1 & l>g] ?an ! side  "o :i* yields
 "o8    ,boy3  ,yes1 ,socrates4         
,socrates3  7,turn+ to ,m5o7  ,s n[ he  
is z f> z mo/ (  u get 9 det]m9+ !      
magnitude ( ! squ>e root ( two8  ,&     
gett+ f>!r is l>gely a matt] (          
guesswork1 is x n8    ,m5o3  ,yes1      
,socrates1 b ,i don't see h[ he w d x4  
,socrates3  ,nei!r does he4  ,b ,i d4   
,wat*6  7turn+ to  ! boy7  ,n[ ,i am go+
 to tell y some?+ y don't "k1  s ,m5o w 
li/5 v closely to make sure he agrees t 

,i c  tell y4  ,y "k multiplication1    
boy8    ,boy3  ,i ?\ght ,i _h demon/rat$
 t1 ,socrates8    ,socrates3  ,s y h1 my
 boy1 has he n ,m5o8    ,m5o3  ,yes1    
,socrates1 ,i recall he did ! o!r day4  
,socrates3  ,& y "k ! way to undo       
multiplication8    ,boy3  ,x is call$   
division1 b ,i d n "k x z well z        
multiplication1 s9ce we h n /udi$ x z   
long4    ,socrates3  ,well1 ,i w n ask y
 to d mu* division1 b  r ,i w ask y only
 :e!r c]ta9 answ]s may be  call$ odd or 
ev51 & ! l4  ,does t suit y8    ,boy3   
,x suits me well1 ,socrates4            
,socrates3  ,!n y "k :at odd & ev5 are1 
boy8    ,boy3  ,yes1 %all ,i tell y8    
,socrates3  ,please d4  ,i w\ld love to 
he> :at !y tea*4    ,boy3 7! boy        
recites7 ,a numb] c only be odd or ev5  
if x  is a :ole numb]1 t is has no p>ts 
b only :oles ( :at  x measures4  ,ev5   
numb]s are special 9 t !y h only  :ole  
twos 9 !m1 ) no ones left ov]1 :ile odd 
numb]s  always h a "o left ov] :5 all ! 
twos are tak5 out4    ,socrates3  ,an   

int]e/+1 & some:at effective def9ition4 
,d y agree1 ,m5o4    ,m5o3  ,yes1       
,socrates4  ,please cont9ue4            
,socrates3  ,n[ boy1 :at d y get :5 y   
divide ^!  odd & ev5 numb]s by o!r odd &
 ev5 numb]s4    ,boy3  ,sometimes y get 
:ole numb]s1 especially :5 y  divide an 
ev5 numb] by an ev5 numb]1 b odd numb]s 
sometimes give :ole numb]s1 bo? odd &   
ev51 & sometimes  !y give numb]s :i* are
 n :ole numb]s1 b h p>ts4    ,socrates3 
,v good1 & h y\r tea*]s ev] call$  ^!   
numb]s ratios8    ,boy3  ,sometimes1    
,socrates1 b usually only ) simple      
numb]s  :i* make "o-half1 "o-?ird1      
two-?irds & ! l4    ,socrates3  ,yes1 t 
is usually :at p mean by ratios4  ,!    
le>n$ p call numb]s made f ! ratios1    
rational4  ,does ! "n rational numb]    
suit y to call a numb] :i* c  be        
express$ z ! ratio ( two :ole numb]s1   
:e!r !y be  odd or ev5 :ole numb]s8     
,boy3  ,y want me to call ! numb]s made 
f ratios (  :ole numb]s some?+ call$    
rational8  ,a ratio makes a  rational   

numb]8    ,socrates3  ,yes boy1 c y d t8
    ,boy3  ,c]ta9ly1 ,socrates4         
,socrates3  ,d y agree ) ! way ,i told  
him ?is1 ,m5o8  ,does x violate our     
agreem5t8    ,m5o3  ,y add$ -nal to ! ^w
 ratio1 j z we add -nal  to ! ,fr5* ^w  
j\r to create ! ^w j\rnal :i* means     
some?+ t conta9s words ( ! j\r or (     
today4  ,s we  n[ h a ^w :i* means a    
numb] made f a ratio4  ,?is is  m ?an   
acceptable to me1 ,socrates4  ,a sort ( 
lesson 9  l+ui/ics1 p]haps1 b c]ta9ly n 
9 ma!matics4  ,no1 ,i d  n see t y h    
told him h[ to solve any?+ ab\t !  squ>e
 root ( two1 b ?ank y = ask+4  ,i give y
 y\r  j\rnali/ic lic5se to d s4         
,socrates3  ,good4  ,n[ boy1 ,i ne$ y\r 
att5tion4  ,please get  up & /ret*1 if x
 w help y /ay & ?9k = a:ile4    ,boy3   
7/ret*es only a little7  ,i am f9e1     
,socrates4    ,socrates3  ,n[ ?9k       
c>efully1 boy1 :at k9d ( ratios c  we   
make f ev5 numb]s & odd numb]s8    ,boy3
  ,we c\ld make ev5 numb]s divid$ by odd
 numb]s1 &  odd numb]s divid$ by ev5    

numb]s4    ,socrates3  ,yes1 we c\ld4   
,c\ld we make any o!r k9d8    ,boy3     
,well444 we c\ld make ev5 numb]s divid$ 
by ev5  numb]s1 or odd numb]s divid$ by 
odd4    ,socrates3  ,v good4  ,any o!r  
k9d8    ,boy3  ,i'm n sure1 ,i c't ?9k (
 any1 b ,i might h to  ?9k a :ile to be 
sure4    ,socrates3  7to ,m5o7  ,are y  
/ill satisfi$4    ,m5o3  ,yes1          
,socrates4  ,he kn[s ev5 & odd numb]s1  
& ratios2 z d all ! s*ool *ildr5 8 age4 
,socrates3  ,v well1 boy4  ,y h nam$ f\r
 k9ds (  ratios3  ,ev5 ov] odd1 odd ov] 
ev51 ev5 ov] ev51 odd ov]  odd1 & all ! 
ratios make numb]s we call rational     
numb]s4    ,boy3  ,t's :at x looks l1   
,socrates4    ,socrates3  ,m5o1 h y     
any?+ to contribute "h8    ,m5o3  ,no1  
,socrates1 ,i am f9e4    ,socrates3  ,v 
well4  ,n[1 boy1 we are (f 9 se>* ( m   
ab\t ! squ>e root ( two4  ,we h divid$ !
 rational numb]s  into f\r gr\ps1       
odd_/ev51 ev5_/odd1 ev5_/ev51 odd_/odd8 
,boy3  ,yes4    ,socrates3  ,& if we f9d
 ano!r gr\p we c include !m4  ,n[1 we   

want to f9d :i* "o ( ^! gr\ps1 if any1  
conta9s  ! numb] y f\nd ! o!r day1 ! "o 
:i* squ>$ is two4    ,w\ld t be fun to  
try8    ,boy3  ,yes1 ,socrates1 & also  
educational4    ,socrates3  ,i ?9k we c 
n>r[ ^! f\r gr\ps d[n to  ?ree1 & ?us   
make ! se>* easi]4  ,w\ld y l t8        
,boy3  ,c]ta9ly1 ,socrates4             
,socrates3  ,let's take ev5 ov] ev5     
ratios4  ,:at are !y8    ,boy3  ,we "k t
 bo? p>ts ( ! ratio h two 9 !m4         
,socrates3  ,excell5t4  ,see1 ,m5o1 h[  
well he has le>n$ 8  lessons 9 s*ool4   
,8 tea*] mu/ be pr\d1 = ,i h taught  him
 no?+ ( ?is1 h ,i8    ,m5o3  ,no1 ,i h n
 se5 y tea* x to him1  !re=e he mu/ h   
be5 expos$ to x else:]e4    ,socrates3  
7back to ! boy7  ,& :at h y le>n$ ab\t  
ratios ( ev5 numb]s1 boy8    ,boy3  ,t  
bo? p>ts c be divid$ by two1 to get !   
twos  out1 ov] & ov]1 until "o "p       
becomes odd4    ,socrates3 ,v good4  ,d 
all s*ool *ildr5 "k t1 ,m5o8    ,m5o3   
,all ! ones :o /ay awake 9 class4  7he  
/ret*es7    ,socrates3  ,s1 boy1 we c   

*ange ! p>ts ( ! ratios1  )\t *ang+ !   
real mean+ ( ! ratio itself8    ,boy3   
,yes1 ,socrates4  ,i w demon/rate1 z we 
d 9 class4  ,suppose ,i use #af & #h1 z 
we did ! o!r day4  ,if ,i make a  ratio 
( #af divid$ by #h1 ,i c divide bo? !   
#af & ! #h by  two & get #h divid$ by   
#d4  ,we c see t #h divid$ by #d is  !  
same z #af divid$ by #h1 ea* "o is twice
 ! o!r1 z x  %\ld be4  ,we c !n divide  
by two aga9 & get #d ov] #b1 &  aga9 to 
get #b ov] #a4  ,we c't d x aga91 s we  
say t ?is  fraction has be5 r$uc$ z f> z
 x w g1 & ev]y?+  t is true ( ! o!r ways
 ( express+ x is true ( ?is4            
,socrates3  ,y\r demon/ration is        
effective4  ,c y divide  by o!r numb]s  
?an two8    ,boy3  ,yes1 ,socrates4  ,we
 c divide by any numb] :i* goes  z :oles
 into ! p>ts :i* make up ! ratio4  ,we  
c\ld h  />t$ by divid+ by #h be=e1 b ,i 
divid$ by ?ree times1  ea* "t by two1 to
 %[ y ! process1 ?\gh n[ ,i feel  a%am$ 
because ,i realize y are bo? ma/]s (    
?is1  & t ,i spoke to y 9 too simple a  

mann]4    ,socrates3  ,bett] to speak   
too simply1 ?an 9 a mann] 9  :i* "p or  
all ( y\r audi5ce gets lo/1 l !         
,sophi/s4    ,boy3  ,i agree1 b please  
/op me if ,i get too simple4            
,socrates3  ,i am sure we c survive a   
simple explanation4  7nudges ,m5o1 :o   
has be5 gaz+ else:]e7  ,b back to y\r   
simple pro(3  we "k t a ratio ( two ev5 
numb]s c be  divid$ until r$uc$ until "o
 or bo? its p>ts are odd8    ,boy3      
,yes1 ,socrates4  ,!n x is a prop]      
ratio4    ,socrates3  ,s we c elim9ate  
"o ( our f\r gr\ps1 !  "o :]e ev5 0     
divid$ by ev51 & n[ we h odd_/odd1      
odd_/ev5 & ev5_/odd8    ,boy3  ,yes1    
,socrates4    ,socrates3  ,let's try odd
 ov] ev5 next1 %all we8    ,boy3  ,f9e4 
,socrates3  ,:at happ5s :5 y multiply an
 ev5 numb] by  an ev5 numb]1 :at k9d (  
numb] d y get1 ev5 or odd8    ,boy3     
,ev51 ( c\rse4  ,an ev5 multiple ( any  
:ole numb]  gives ano!r ev5 numb]4      
,socrates3  ,wond]ful1 y h answ]$ two   
que/ions1 b we  ne$ only "o at ! mom5t4 

,we %all save ! o!r4  ,s1 )  odd ov]    
ev51 if we multiply any ( ^! times      
!mselves1 we  well get odd times odd ov]
 ev5 times ev51 & !re=e odd  ov] ev51   
s9ce odd times odd is odd & ev5 ( ev5 is
 ev54    ,boy3  ,yes4  ,a ratio ( odd   
ov] ev51 :5 multipli$ times  itself1    
yields odd ov] ev54    ,socrates3  ,& c 
our squ>e root ( two be 9 t gr\p8       
,boy3  ,i don't "k1 ,socrates4  ,h ,i   
fail$8    ,socrates3  ,oh1 y "k1 y j    
don't "k t y "k4    ,try ?is3  aft] we  
multiply our numb] times itself1  :i* ! 
le>n$ call squ>+ ! numb] :i* is ! root1 
we ne$ to get a ratio 9 :i* ! fir/ or   
top numb] is twice  z l>ge z ! second or
 bottom numb]4  ,is ?is mu* correct8    
,boy3  ,a ratio :i* :5 squ>$ z y call$  
x1 yields  an area ( two1 mu/ !n yield  
"o "p :i* is two times !  o!r "p4  ,t is
 ! def9ition ( a ratio ( two to "o4     
,socrates3  ,s y agree t ?is is correct8
    ,boy3  ,c]ta9ly4    ,socrates3  ,n[ 
if a numb] is to be twice z great z     
ano!r1  x mu/ be two times t numb]8     

,boy3  ,c]ta9ly4    ,socrates3  ,& if a 
numb] is two times any :ole numb]1  x   
mu/ !n be an ev5 numb]1 mu/ x n8        
,boy3  ,yes1 ,socrates4    ,socrates3   
,s1 9 our ratio we want to squ>e to get 
two1  ! top numb] _c be odd1 c x8       
,boy3  ,no1 ,socrates4  ,!re=e1 ! gr\p (
 odd ov] ev5  rational numb]s _c h !    
squ>e root ( two 9 x6  ,nor c ! gr\p    
ratios ( odd numb]s ov] odd numb]s4     
,socrates3  ,wond]ful4  ,we h j elim9at$
 ?ree ( !  f\r gr\ps ( rational numb]s1 
fir/ we elim9at$ !  gr\p ( ev5 ov] ev5  
numb]s1 !n ! ones ) odd numb]s  divid$  
by o!r numb]s4  ,h[ev]1 ^! 7 ! easi] "p1
  & we are n[ mo/ ( ! way up ! m\nta91 s
 we mu/ re/  & prep>e to try ev5 h>d] to
 conqu] ! re/1 :]e !  altitude is       
highe/1 & ! t]ra9 is rockie/4  ,s let u 
sit  & re/ a m9ute1 & look ov] :at we h 
done1 if y w4    ,boy3  ,c]ta9ly1       
,socrates1 ?\gh ,i am mu* invigorat$ by 
! solution ( two p>ts ( ! puzzle ) "o   
?\ght4  ,x 0 truly wond]ful to see su*  
simple effectiv5ess4  ,are all great    

?\ghts z simple z ^!1 once y see !m     
cle>ly8    ,socrates3  ,:at d y say1    
,m5o8  ,d ?\ghts get simpl] z  !y get   
great]8    ,m5o3  ,well1 x w\ld appe> t 
!y d1 = z ! ma/]  ( a great h\se1 ,i c j
 ord] some?+ be done1 & x is2  b if ,i 7
 a ma/] 9 a less] h\se1 ,i w\ld h to    
wat*  ov] x mu* m closely to insure x   
got done4  ,! bigg] !  decisions ,i h to
 make1 ! m help & advice ,i get 9 !     
mak+ ( !m1 s ,i w\ld h to agree4        
,socrates3  ,glad to see t y are /ill   
agreeable1 ,m5o1  ?\gh ,i ?9k !re are "s
 slight diff]5ces 9 ! way ea*  ( u view 
! simplicity ( great ?\ght4  ,%all we g 
on8    ,m5o3  ,yes1 q4    ,boy3  ,yes1  
,socrates4  ,i am ready = ! la/ gr\p1 ! 
ratios ( ev5 numb]s divid$ by ! odd1    
?\gh1 ,i _c yet  see h[ we w figure ^!  
out1 yet1 someh[ ,i h confid5ce  t !    
walls ( ^! numb]s %all tumble be=e u1 z 
did  ! ?ree gr\ps be=e !m4    ,socrates3
  ,let u review ! ?ree e>li] gr\ps1 to  
prep>e  u = ! f\r?1 & to make sure t we 
h n already  brok5 ! rules & !re=e      

=feit$ our wag]4  ,! f\r  gr\ps 7 ev5   
ov] ev5 ratios1 :i* we decid$ c\ld be   
r$uc$ 9 v>i\s mann]s to ! o!r gr\ps by  
divid+ until  "o numb] ( ! ratio 0 no   
long] ev52 !n we elim9at$  ! two o!r    
gr\ps :i* _h odd numb]s divid$ by ei!r  
odd  or ev5 numb]s1 because ! fir/ or   
top numb] _h to be twice  ! second or   
bottom numb]1 & !re=e c\ld n be odd2    
?is left ! la/ gr\p we are n[ to greet1 
ev5 divid$ by odd4    ,boy3  ,wond]fully
 put1 ,socrates4  ,x is amaz+ h[ neatly 
y put an h\r ( ?9k+ into a m9ute4       
,p]haps we c1  inde$1 put t5 ye>s ( ?9k+
 into ?is "o day4  ,please  cont9ue 9   
?is mann]1 if y "k h[ x c be done4      
,socrates3  ,w\ld y h me cont9ue1 ,m5o8 
,y "k :at  %all h to happ5 if we solve  
?is next gr\p & d n  f9d ! squ>e root ( 
two 9 x4    ,m5o3  ,socrates1 y are my  
fri5d1 & my tea*]1 & a  good companion4 
,i w n %irk my duty to y or to ?is f9e  
boy1 :o appe>s to be gr[+ beyond my     
head1 ev5 z we speak4  ,h[ev]1 ,i /ill d
 n see t 8 head has rea*$ ! cl\ds  :]e9 

lie ! m9ds ( ! ,py?agoreans4            
,socrates3  ,v well1 on !n1 to ev5 ov]  
odd4  ,if we multiply  ^! numb]s times  
!mselves1 :at d we get1 boy8    ,boy3   
,we w get a ratio ( ev5 ov] odd1        
,socrates4    ,socrates3  ,& c\ld an ev5
 numb] be d\ble an odd numb]8    ,boy3  
,yes1 ,socrates4    ,socrates3  ,s1     
inde$1 ?is c\ld be :]e we f9d a numb]   
su* t :5 multipli$ times itself yields  
an area ( two8    ,boy3  ,yes1          
,socrates4  ,x c\ld v well be 9 ?is     
gr\p4    ,socrates3  ,s1 ! fir/1 or top 
numb]1 is ! result ( an  ev5 numb] times
 itself8    ,boy3  ,yes4    ,socrates3  
,& ! second1 or bottom numb]1 is !      
result (  an odd numb] times itself8    
,boy3  ,yes4    ,socrates3  ,& an ev5   
numb] is two times "o :ole numb]8       
,boy3  ,( c\rse4    ,socrates3  ,s if we
 use ?is ev5 numb] twice 9              
multiplication1  z we h on top1 we h two
 twos times two :ole numb]s8    ,boy3   
,yes1 ,socrates4    ,socrates3  7nudges 
,m5o7 & !re=e ! top numb] is f\r  times 

"s :ole numb] times t :ole numb] aga98  
,boy3  ,yes1 ,socrates4    ,socrates3   
,& ?is numb] on top has to be twice !   
numb]  on ! bottom1 if ! ev5 ov] odd    
numb] we began ) is to  give u two :5   
multipli$ by itself1 or squ>$1 z we call
 x8    ,boy3  ,yes1 ,socrates4          
,socrates3  ,& if ! top numb] is f\r    
times "s :ole numb]1 !n  a numb] half z 
l>ge w\ld h to be two times t same :ole 
numb]8    ,boy3  ,( c\rse1 ,socrates4   
,socrates3  ,s ! numb] on ! bottom is   
two times t :ole numb]1  :atev] x is8   
,boy3  ,yes1 ,socrates4    ,socrates3   
7/&+7  ,& if x is two times a :ole      
numb]1  !n x mu/ be an ev5 numb]1 mu/ x 
n8    ,boy3  ,yes4    ,socrates3  ,!n is
 _c be a memb] ( ! gr\p :i* has  an odd 
numb] on ! bottom1 c x8    ,boy3  ,no1  
,socrates4    ,socrates3  ,s c x be a   
memb] ( ! ratios creat$ by an  ev5 numb]
 divid$ by an odd numb] & !n us$ z a    
root  to create a squ>e8    ,boy3  ,no1 
,socrates4  ,& t mu/ mean x c't be a    
memb]  ( ! la/ gr\p1 doesn't x8         

,socrates3  ,yes1 my boy1 al?\gh ,i     
don't see h[ we c  cont9ue call+ y boy1 
s9ce y h n[ won y\r fre$om1  & are f>   
ri*] ?an ,i w ev] be4    ,boy3  ,are y  
sure we h prov$ ?is prop]ly8  ,let me g 
ov] x aga91 s ,i c see x 9 my head4     
,socrates3  ,yes1 my boy1 er1 ah1 sir4  
,boy3  ,we want to see if ?is squ>e root
 ( two we discov]$  ! o!r day is a memb]
 ( ! rational numb]s8    ,socrates3     
,yes4    ,boy3  ,s we def9e ! rational  
numb]s z numb]s made f  ! division into 
ratios ( :ole numb]s1 :e!r ^? :ole      
numb]s are ev5 or odd4    ,socrates3    
,yes4    ,boy3  ,we get f\r gr\ps1 ev5  
ov] ev51 :i* we don't use1  odd ov] ev51
 odd ov] odd1 & ev5 ov] odd4            
,socrates3  ,cont9ue4    ,boy3  ,we "k !
 fir/ numb] 9 ! squ>$ ratio _c be odd   
because x mu/ be twice ! value ( !      
second numb]1  & !re=e is mu/ be an ev5 
numb]1 two times a :ole numb]4  ,!re=e x
 _c be a memb] ( ei!r ( ! next gr\ps1   
because !y bo? h :ole numb]s ov] odd    
numb]s4    ,socrates3  ,wond]ful6       

,boy3  ,s we are left ) "o gr\p1 ! ev5s 
ov] odds4    ,socrates3  ,yes4    ,boy3 
,:5 we squ>e an ev5 ov] odd ratio1 !    
fir/ numb]  becomes ev5 times ev51 :i*  
is two times two times "s o!r  :ole     
numb]1 :i* means x is f\r times ! :ole  
numb]1  & ?is numb] mu/ be d\ble !      
second numb]1 :i* is odd1  z x 0 made ( 
odd times odd4  ,b ! top numb] _c be    
d\ble  "s bottom odd numb] because ! top
 numb] is f\r times "s  :ole numb]1 & ! 
bottom numb] is odd--b a numb] :i* is   
f\r times ano!r :ole numb]1 _c be odd :5
 cut 9 half1  s an ev5 numb] times an   
ev5 numb] c nev] be d\ble :at  y w\ld   
get f any odd numb] times ano!r odd     
numb]444  !re=e none ( ^! rational      
numb]s1 :5 multipli$ times  !mselves1   
c\ld possibly yield a ratio 9 :i* ! top 
numb]  0 twice ! bottom numb]4  ,amaz+4 
,we h prov$ t !  squ>e root ( two is n a
 rational numb]4  ,fanta/ic6    7,he    
cont9ues to w&] up & d[n ! /age1 recit+ 
v>i\s  portions ( ! pro( to himself1    
look+ up1 !n d[n1 !n all  ar\nd4  ,he   

comes to ,m5o7    ,boy3  ,d y see8  ,x's
 s simple1 s cle>4  ,?is is really      
wond]ful6  ,?is is fanta/ic6            
,socrates3  7lays an arm on ,m5o's arm7 
,tell him h[ happy  y are = 8 new f\nd  
?\ghts1 ,m5o1 = y c easily tell  he is n
 ?9k+ at all ( 8 newly won fre$om &     
weal?4    ,m5o3  ,i q agree ) y1 son1 ! 
cl>ity ( y\r  reason+ is truly a/\nd+4  
,i w leave y "h )  ,socrates1 z ,i g to 
prep>e my h\sehold4  ,i tru/ y w  bo? be
 happy = ! re/ ( ! day )\t my assi/ance4
    ,! p>ty1 ! pres5tation ( #aj ye>s   
sal>y to ! newly  fre$ "y man1 is ano!r 
/ory1 z is ! orig9al /ory  ( ! draw+ 9 !
 s& ! squ>e ) an area ( two4