,,! ,,tea*+ ,,(   ,,geometry    ,,by    
,,david ,,eug5e ,,smi?    ,,g9n ,,&     
,,company   ,,bo/on 4 ,,new ,,york 4    
,,*icago 4 ,,london       ,,copyright1  
#aiaa1 ,,by ,,david ,,eug5e ,,smi?      
,,all ,,rights ,,res]v$   #iaa4f    ,!  
,a!naeum ,press   ,,g9n ,,& ,,company 4 
,,proprietors   ,,bo/on 4 ,u4,s4,a4     
,,preface   ,a book ^u ! tea*+ (        
geometry may be plann$ 9 div]s ways4 ,x 
may be writt5 to exploit a new !ory (   
geometry1 or a new me?od ( pres5t+ !    
sci5ce z we already h x4 ,on ! o!r h&1 x
 may be ultracons]vative1 mak+ a plea = 
! anci5t tea*+ & ! anci5t geometry4 ,x  
may be prep>$ = ! purpose ( sett+ =? !  
"w z x n[ is1 or ) ! tempt+ b dang]\s   
idea ( prophecy4 ,x may appeal to !     
iconocla/ by its _s ( de/ruction1 or to 
! disciples ( laissez faire by its _s ( 
cons]v+ :at ! pa/ has bequea!d4 ,x may  
be writt5 = ! few :o always lead1 or ?9k
 !y lead1 or = ! _m :o are rank$ by !   
few z foll[]s4 ,& 9 view ( ^! v>i$      
pa?ways into ! jo9t doma9 ( geometry &  

education1 a writ] may well af=d to     
pause be=e he sets 8 p5 to pap]1 & to   
decide ) c>e ! r\te t he w take4  ,at   
pres5t 9 ,am]ica we h a fairly          
well-def9$ body ( matt] 9 geometry1 &   
?is occupies a fairly well-def9$ place 9
 ! curriculum4 ,!re are n want+ _m e>ne/
 tea*]s :o w\ld *ange bo? ! matt] & !   
place 9 a v radical fa%ion4 ,!re are n  
want+ o!rs1 also _m 9 numb]1 :o are     
cont5t ) ?+s z !y f9d !m4 ,b by f> !    
l>ge/ "p ( ! tea*+ body is ( a m9d to   
welcome ! natural & gradual evolution ( 
geometry t[>d bett] ?+s1 contribut+ to  
?is evolution z mu* z x c1 glad to "k ! 
be/ t o!rs h to (f]1 receptive ( ideas t
 make = bett] tea*+1 b out ( sympa?y )  
ei!r ! extreme ( revolution or ! extreme
 ( /agnation4  ,x is = ?is l>g] class1 !
 great body ( progressive tea*]s1 t ?is 
book is writt54 ,x /&s = vitaliz+       
geometry 9 e legitimate way2 = improv+ !
 subject matt] 9 su* mann] z n to de/roy
 ! pupil's int]e/2 = s tea*+ geometry z 
to make x appeal to pupils z /rongly z  

any o!r subject 9 ! curriculum2 b = !   
recognition ( geometry = geometry's sake
 & n = ! sake ( a fanci$ utility t h>dly
 exi/s4 ,express+ full appreciation ( ! 
desirability ( e/abli%+ a motive = all  
/udies1 s z to h ! "w proce$ ) int]e/ & 
vigor1 x does n hesitate to express d\bt
 z to c]ta9 motives t h be5 exploit$1   
nor to /& = su* a g5u9e1 ?\ght-compell+ 
developm5t ( ! sci5ce z is 9 h>mony ) ! 
m5tal p[]s ( ! pupils 9 ! ,am]ican high 
s*ool4  ,= ?is class ( tea*]s ! au?or   
hopes t ! book w prove ( s]vice1 & t    
?r\gh its p]usal !y w come to admire !  
subject m & m1 & to tea* x ) great]     
int]e/4 ,x (f]s no panacea1 x *ampions  
no s+le me?od1 b x seeks to set =?      
pla9ly ! reasons = tea*+ a geometry ( ! 
k9d t we h inh]it$1 & = hop+ = a gradual
 b def9ite improvem5t 9 ! sci5ce & 9 !  
me?ods ( its pres5tation4      ,,david  
,,eug5e ,,smi?     ,,cont5ts            
,,*apt]                                 
,,page           ,i4 ,,c]ta9 ,,que/ions 
,,n[ ,,at ,,issue               #a      

,,ii4 ,,:y ,,geometry ,,is ,,/udi$      
#g         ,,iii4 ,a ,,brief ,,hi/ory   
,,( ,,geometry                 #bf      
,,iv4 ,,developm5t ,,( ,,! ,,tea*+ ,,(  
,,geometry     #dj           ,v4        
,,euclid                                
#dg          ,,vi4 ,,ef=ts ,,at         
,,improv+ ,,euclid                 #eg  
,,vii4 ,,! ,,textbook ,,9 ,,geometry    
#gj        ,,viii4 ,,! ,,relation ,,(   
,,algebra ,,to ,,geometry         #hd   
,,ix4 ,,! ,,introduction ,,to ,,geometry
                #ic           ,x4 ,,!   
,,conduct ,,( ,a ,,class ,,9 ,,geometry 
#ajh          ,,xi4 ,,! ,,axioms ,,&    
,,po/ulates                  #aaf       
,,xii4 ,,! ,,def9itions ,,( ,,geometry  
#acb        ,,xiii4 ,,h[ ,,to ,,attack  
,,! ,,ex]cises                #afj      
,,xiv4 ,,book ,i ,,& ,,its              
,,propositions                #afe      
,,xv4 ,,! ,,lead+ ,,propositions ,,(    
,,book ,,ii        #bja         ,,xvi4  
,,! ,,lead+ ,,propositions ,,( ,,book   
,,iii       #bbg        ,,xvii4 ,,!     

,,lead+ ,,propositions ,,( ,,book ,,iv  
#beb       ,,xviii4 ,,! ,,lead+         
,,propositions ,,( ,,book ,v            
#bfi         ,,xix4 ,,! ,,lead+         
,,propositions ,,( ,,book ,,vi          
#bhi          ,,xx4 ,,! ,,lead+         
,,propositions ,,( ,,book ,,vii         
#cjc         ,,xxi4 ,,! ,,lead+         
,,propositions ,,( ,,book ,,viii        
#cba        ,,index                     
#cce     ,,! ,,tea*+ ,,( ,,geometry     
,,*apt] ,i  ,,c]ta9 ,,que/ions ,,n[ ,,at
 ,,issue   ,x is commonly said at !     
pres5t "t t ! op5+ ( ! tw5tie? c5tury is
 a p]iod ( unusual advancem5t 9 all t   
has to d ) ! s*ool4 ,x w\ld be pleasant 
to feel t we are liv+ 9 su* an age1 b x 
is d\btful if ! future hi/orian (       
education w f9d ?is to be ! case1 or t  
biograph]s w rank ! lead]s ( our        
g5]ation relatively z high z _m :o h    
pass$ away1 or t any great movem5ts ( ! 
pres5t w be f\nd t measure up to c]ta9  
ones t ! _w n[ recognizes z epo*-mak+4  
,e g5]ation s9ce ! inv5tion ( pr9t+ has 

be5 a p]iod ( agitation 9 educational   
matt]s1 b out ( all ! noise &           
self-ass]tion1 out ( all ! pret5se ( !  
*ronic revolutioni/1 out ( all ! %am t  
leads to dogmatism1 s little is rememb]$
 t we are apt to feel t ! pa/ _h no     
problems & 0 cont5t simply to accept its
 inh]itance4 ,9 "o s5se x is n a        
mis=tune ?us to be bl9d$ by ! du/ (     
pres5t agitation & to be deaf5$ by !    
noisy clamor ( ! agitator1 s9ce x /irs u
 to action at f9d+ ourselves 9 ! mid/ ( 
! skirmi%2 b 9 ano!r s5se x is          
detrim5tal to our progress1 s9ce we     
!reby t5d to lose ! idea ( p]spective1 &
 ! co9 comes to appe> to our vision z   
l>ge z ! moon4  ,9 consid]+ a "q l !    
tea*+ ( geometry1 we at once f9d        
ourselves 9 ! mid/ ( a skirmi% ( ?is    
nature4 ,if we jo9 ?\ghtlessly 9 !      
noise1 we may easily p]suade ourselves t
 we are wag+ a mighty battle1 fight+ =  
"s /up5d\s pr9ciple1 do+ de$s ( great   
valor & ( p]sonal sacrifice4 ,if1 on !  
o!r h&1 we /& alo( & ?9k ( ! pres5t     

movem5t z m]ely a *ronic eff]vesc5ce1   
fo/]$ by ! pr(essional educator at !    
exp5se ( ! practical tea*]1 we are      
equally %ortsight$4 ,sir ,conan ,doyle  
express$ ?is s5tim5t mo/ delightfully 9 
^! words3       ,! dead are su* good    
company t "o may come to ?9k too        
little ( ! liv+4 ,x is a real & press+  
dang] )      _m ( u t we %\ld nev] f9d  
our own ?\ghts & our      own s\ls1 b be
 ev] obsess$ by ! dead4  ,9 e g5]ation x
 behooves ! op5-m9d$1 e>ne/1 progressive
 tea*] to seek = ! be/ 9 ! way (        
improvem5t1 to endeavor to sift ! few   
gra9s ( gold out ( ! common du/1 to     
weigh ! values ( propos$ re=ms1 & to put
 =? 8 ef=ts to "k & to use ! be/ t !    
sci5ce ( education has to (f]4 ,?is has 
be5 ! attitude ( m9d ( ! real lead]s 9 !
 s*ool life ( ! pa/1 & x w be t ( !     
lead]s ( ! future4  ,) ^! rem>ks to     
guide u1 x is n[ propos$ to take up !   
issues ( ! pres5t day 9 ! tea*+ (       
geometry1 9 ord] t we may consid] !m    
calmly & dispassionately1 & may see :]e 

! opportunities = improvem5t lie4  ,at !
 pres5t "t1 9 ! educational circles ( ! 
,unit$ ,/ates1 que/ions ( ! foll[+ type 
are caus+ ! *ief discussion among tea*]s
 ( geometry3  #a4 ,%all geometry cont9ue
 to be taught z an application ( logic1 
or %all x be treat$ solely ) ref]5ce to 
its applications8  #b4 ,if ! latt] is ! 
purpose 9 view1 %all ! propositions (   
geometry be limit$ to ^? t (f] an       
opportunity = real application1 ?us     
contract+ ! :ole subject to v n>r[      
dim5sions8  #c4 ,%all a subject call$   
geometry be ext5d$ ov] sev]al ye>s1 z is
 ! case 9 ,europe1#a or %all ! "n be    
appli$ only to s]i\s demon/rative       
geometry#b z giv5 9 ! second ye> ( !    
f\r-ye> high s*ool c\rse 9 ! ,unit$     
,/ates at pres5t8  #d4 ,%all geometry be
 taught by itself1 or %all x be ei!r    
mix$ ) algebra 7say a day ( "o subject  
foll[$ by a day ( ! o!r7 or fus$ ) x 9 !
 =m ( a comb9$ ma!matics8  #e4 ,%all a  
textbook be us$ 9 :i* ! basal           
propositions are prov$ 9 full1 !        

ex]cises furni%+ ! opportunity = orig9al
 "w & be+ look$ ^u z ! mo/ important    
feature1 or %all "o be employ$ 9 :i* !  
pupil is expect$ to inv5t ! pro(s = !   
basal propositions z well z = !         
ex]cises8  #f4 ,%all ! t]m9ology & ! _s 
( a modifi$ ,euclid & ,leg5dre prevail 9
 ! future z !y h 9 ! pa/1 or %all !re be
 a revolution 9 ! use ( t]ms & 9 ! g5]al
 /atem5ts ( ! propositions8  #g4 ,%all  
geometry be made a /rong elective       
subject1 to be tak5 only by ^? ^: m9ds  
are capable ( s]i\s "w8 ,%all x be a    
requir$ subject1 dilut$ to !            
compreh5sion ( ! weake/ m9ds8 ,or is x  
n[1 by prop] tea*+1 z suitable = all    
pupils z is any o!r requir$ subject 9 ! 
s*ool curriculum8 ,& 9 any case1 w !    
v>i\s di/9ct types ( high s*ools n[     
aris+ call = di/9ct types ( geometry8   
,?is brief li/ might easily be amplifi$1
 b x is suffici5tly ext5d$ to set =? !  
tr5d ( ?\ght at ! pres5t "t1 & to %[ t !
 que/ions be=e ! tea*]s ( geometry are  
nei!r p>ticul>ly novel nor p>ticul>ly   

s]i\s4 ,^! que/ions & o!rs ( simil>     
nature are really side issues ( two l>g]
 que/ions ( f> great] significance3 7#a7
 ,are ! reasons = tea*+ demon/rative    
geometry su* t x %\ld be a requir$      
subject1 or at lea/ a subject t is      
/rongly recomm5d$ to all1 :atev] ! type 
( high s*ool8 7#b7 ,if s1 h[ c x be made
 int]e/+8  ,! pres5t "w is writt5 ) ^!  
two l>g] que/ions 9 m9d1 al?\gh x       
consid]s f "t to "t ! m9or ones already 
m5tion$1 toge!r ) o!rs ( a simil>       
nature4 ,x recognizes t ! rec5t gr[? 9  
popul> education has br\ght into ! high 
s*ool a less c>efully select$ type ( m9d
 ?an 0 =m]ly ! case1 & t = ?is type a   
diff]5t k9d ( ma!matical tra9+ w        
naturally be develop$4 ,x proce$s ^u !  
!ory1 h[ev]1 t = ! normal m9d1--= ! boy 
or girl :o is prep>+ to w9 out 9 ! long 
run1--geometry w cont9ue to be taught z 
demon/rative geometry1 z a vigor\s      
?\ght-compell+ subject1 & along ! g5]al 
l9es t ! exp]i5ce ( ! _w has %[n to be !
 be/4 ,s(t ma!matics is n int]e/+ to ?is

 normal m9d1 & a %am treatm5t w nev]    
appeal to ! pupil2 & ?is book is writt5 
= tea*]s :o believe 9 ?is pr9ciple1 :o  
believe 9 geometry = ! sake ( geometry1 
& :o e>ne/ly seek to make ! subject s   
int]e/+ t pupils w wi% to /udy x :e!r x 
is requir$ or elective4 ,! "w /&s = !   
great basal propositions t h come d[n to
 u1 z logically arrang$ & z             
sci5tifically prov$ z ! p[]s ( ! pupils 
9 ! ,am]ican high s*ool w p]mit2 & x    
seeks to tell ! /ory ( ^! propositions &
 to %[ _! possible & _! probable        
applications 9 su* a way z to furni%    
tea*]s ) a fund ( int]e/+ mat]ial ) :i* 
to supplem5t ! book "w ( _! classes4    
,aft] all1 ! problem ( tea*+ any subject
 comes d[n to ?is3 ,get a subject wor?  
tea*+ & !n make e m9ute ( x int]e/+4    
,pupils d n object to "w if !y l a      
subject1 b !y d object to aimless &     
un9t]e/+ tasks4 ,geometry is p>ticul>ly 
=tunate 9 t ! feel+ ( accompli%m5t comes
 ) e proposition prov$2 &1 giv5 a class 
( fair intellig5ce1 a tea*] mu/ be lack+

 9 k & en?usiasm :o _c fo/] an int]e/ t 
w make geometry /& =? z ! subject t br+s
 ! mo/ pleasure1 & t seems ! mo/        
pr(itable ( all t are /udi$ 9 ! fir/    
ye>s ( ! high s*ool4  ,cont9ually to    
advance1 cont9ually to attempt to make  
ma!matics fasc9at+1 always to cons]ve ! 
be/ ( ! old & to sift out & use ! be/ ( 
! new1 to believe t mank9d is bett] s]v$
 by nature's quiet & progressive *anges 
?an by e>?quakes1#c to believe t        
geometry z geometry is s valuable & s   
int]e/+ t ! normal m9d may rightly dem& 
x1--?is is to ally ourselves ) progress4
 ,cont9ually to de/roy1 cont9ually to   
foll[ /range gods1 always to decry ! be/
 ( ! old1 & to h no well-consid]$ aim 9 
! tea*+ ( a subject1--?is is to jo9 !   
=ces ( reaction1 to wa/e our "t1 to be  
recreant to our tru/1 to bl9d ourselves 
to ! failures ( ! pa/1 & to confess our 
weakness z tea*]s4 ,x is ) ! desire to  
aid 9 ! progressive movem5t1 to assi/ ^?
 :o believe t real geometry %\ld be     
recomm5d$ to all1 & to %[ t geometry is 

bo? attractive & valuable t ?is book is 
writt54   ,,footnotes3  #a ,& really1   
?\gh n nom9ally1 9 ! ,unit$ ,/ates1 :]e 
! fir/ concepts are f\nd 9 ! k9d]g>t51 &
 :]e an excell5t c\rse 9 m5suration is  
giv5 9 any ( our bett] class (          
ari?metics4 ,t we are wise 9 n attempt+ 
s]i\s demon/rative geometry mu* e>li]   
seems to be g5]ally conc$$4  #b ,! ?ird 
/age ( geometry z def9$ 9 ! rec5t       
circul> 7,no4 #gaa7 ( ! ,briti% ,bo>d ( 
,education1 ,london1 #aiji4  #c ,! clos+
 words ( a s5sible review ( ! ,briti%   
,bo>d ( ,education circul> 7,no4 #gaa71 
on ,! ,tea*+ ( ,geometry 7,london1      
#aiji71 by ,h4 ,s4 ,hall 9 ! ,s*ool ,_w1
 #aiji1 p4 #bbb4     ,,*apt] ,,ii  ,,:y 
,,geometry ,,is ,,/udi$   ,) geometry1 z
 ) o!r subjects1 x is easi] to set =?   
:at are n ! reasons = /udy+ x ?an to    
proce$ positively & enum]ate !          
advantages4 ,al?\gh su* a negative c\rse
 is n satisfy+ to ! m9d z a f9ality1 x  
possesses def9ite advantages 9 ! beg9n+ 
( su* a discussion z ?is4 ,:5ev] false  

prophets arise1 & ) an attitude ( pa9$  
sup]iority proclaim unwor?y aims 9 human
 life1 x is well to %[ ! fallacy ( _!   
position be=e proce$+ to a con/ructive  
philosophy4 ,tak+ = a mom5t ?is negative
 c\rse1 let u inquire z to :at are n !  
reasons = /udy+ geometry1 or1 to be m   
emphatic1 z to :at are n ! wor?y        
reasons4  ,9 view ( a p]iodic activity 9
 favor ( ! utilities ( geometry1 x is   
well to und]/&1 9 ! fir/ place1 t       
geometry is n /udi$1 & nev] has be5     
/udi$1 because ( its positive utility 9 
comm]cial life or ev5 9 ! work%op4 ,9   
,am]ica we commonly all[ at lea/ a ye>  
to plane geometry & a half ye> to solid 
geometry2 b all ( ! facts t a skill$    
me*anic or an eng9e] w\ld ev] ne$ c\ld  
be taught 9 a few lessons4 ,all ! re/ is
 ei!r obvi\s or is comm]cially &        
te*nically useless4 ,we prove1 =        
example1 t ! angles opposite ! equal    
sides ( a triangle are equal1 a fact t  
is probably q z obvi\s z ! po/ulate t b 
"o l9e c be drawn ?r\gh a giv5 po9t     

p>allel to a giv5 l9e4 ,we !n prove1    
sometimes by ! unsatisfactory process ( 
r$uctio ad absurdum1 ! conv]se ( ?is    
proposition1--a fact t is z obvi\s z mo/
 o!r facts t come to our consci\sness1  
at lea/ aft] ! prec$+ proposition has   
be5 prov$4 ,& ^! two !orems are p]fectly
 fair types ( upw>ds ( "o hundr$ sixty  
or sev5ty propositions compris+         
,euclid's books on plane geometry4 ,!y  
are g5]ally n useful 9 daily life1 & !y 
7 nev] int5d$ to be s4 ,!re is an       
(t-repeat$ b n well-au!nticat$ /ory (   
,euclid t illu/rates ! feel+ ( ! f\nd]s 
( geometry z well z ( its mo/ wor?y     
tea*]s4 ,a ,greek writ]1 ,/obaeus1      
relates ! /ory 9 ^! words3       ,"s "o 
:o _h begun to read geometry ) ,euclid1 
:5 he      _h le>n$ ! fir/ !orem1 ask$1 
,b :at %all ,i get by      le>n+ ^! ?+s8
 ,euclid call$ 8 slave & said1 ,give    
him ?ree obols1 s9ce he mu/ make ga9 out
 ( :at he      le>ns4  ,:e!r true or n1 
! /ory expresses ! s5tim5t t runs ?r\gh 
,euclid's "w1 & n improbably we h "h a  

bit ( real biography1--practically all (
 ! p]sonal ,euclid t has come d[n to u f
 ! _w's fir/ great textbook mak]4 ,x is 
well t we read ! /ory occasionally1 &   
also su* words z ! foll[+1 rec5tly      
utt]$#d by ,sir ,conan ,doyle1--words   
be>+ ! same lesson1 al?\gh ^u a diff]5t 
!me3       ,9 ! pres5t utilit>ian age "o
 frequ5tly he>s !      "q ask$1 ,:at is 
! use ( x all8 z if e noble      de$ 0 n
 its own ju/ification4 ,z if e action   
:i*      makes = self-d5ial1 = h>dihood1
 & = endurance 0      n 9 itself a mo/  
preci\s lesson to mank9d4 ,t p      c be
 f\nd to ask su* a "q %[s h[ f>         
mat]ialism      has gone1 & h[ ne$ful x 
is t we insi/ ^u ! value      ( all t is
 nobl] & high] 9 life4  ,an ,am]ican    
/atesman & juri/1 speak+ ^u a simil>    
occasion#e1 gave utt]ance to ! same     
s5tim5ts 9 ^! words3       ,:5 ! "t     
comes t k w n be s\ght = its      own   
sake1 & m5 w n press =w>d simply 9 a    
desire (      a*ievem5t1 )\t hope ( ga91
 to ext5d ! limits (      human k &     

in=mation1 !n1 inde$1 w ! race      ent]
 ^u its decad5ce4  ,!re h n be5 want+1  
h[ev]1 9 e age1 ^? ^: zeal is 9 inv]se  
proportion to _! exp]i5ce1 :o 7 possess$
 ) ! idea t x is ! duty ( ! s*ools to   
make geometry practical4 ,we h !m       
to-day1 & ! _w _h !m ye/]day1 & ! future
 %all see !m z active z ev]4  ,^! p d   
good to ! _w1 & _! labors %\ld always be
 welcome1 = out ( ! myriad ( sugge/ions 
t !y make a few h value1 & ^! are       
helpful bo? to ! ma!matician & !        
artisan4 ,n infrequ5tly !y h contribut$ 
mat]ial t s]ves to make geometry some:at
 m int]e/+1 b x mu/ be confess$ t mo/ ( 
_! "w is m]ely ! ?re%+ ( old /raw1 l !  
"w ( ^? :o foll[ ! w-o'-!-wisp ( !      
circle squ>]s4 ,! m$ieval a/rolog]s wi%$
 to make geometry m practical1 & s !y   
c>ri$ to a consid]able l5g? ! /udy ( !  
/> polygon1 a figure t !y c\ld use 9 _! 
pr(ession4 ,! ca!dral build]s1 z _! art 
progress$1 f\nd t ar*itectural draw+s 7 
m exact if made ) a s+le op5+ ( !       
compasses1 & x is probable t _! influ5ce

 l$ to ! developm5t ( ?is phase (       
geometry 9 ! ,middle ,ages z a practical
 application ( ! sci5ce4 ,lat]1 & ab\t !
 beg9n+ ( ! sixte5? c5tury1 ! revival ( 
art1 & p>ticul>ly ! great developm5t (  
pa9t+1 l$ to ! practical application (  
geometry to ! /udy ( p]spective & ( ^?  
curves#f t occur mo/ frequ5tly 9 !      
graphic arts4 ,! sixte5? & sev5te5?     
c5turies witness$ ! publication ( a l>ge
 numb] ( treatises on practical         
geometry1 usually relat+ to ! measur+ ( 
di/ances & p>tly answ]+ ! purposes ( our
 pres5t trigonometry4 ,su* 7 ! well-kn[n
 treatises ( ,belli 7#aefi71 ,cataneo   
7#aefg71 & ,b>toli 7#aehi74g  ,! p]iod (
 two c5turies f ab\t #afjj to ab\t #ahjj
 0 q z mu* giv5 to exp]im5ts 9 !        
creation ( a practical geometry z is !  
pres5t "t1 & x 0 no d\bt z mu* by way ( 
prote/ aga9/ ?is false idea ( ! subject 
z a desire to improve ^u ,euclid t l$ ! 
great ,fr5* ma!matician1 ,leg5dre1 to   
publi% 8 geometry 9 #agid1--;a "w t soon
 replac$ ,euclid 9 ! s*ools ( ,am]ica4  

,x ?us appe>s t ! ef=t to make geometry 
practical is by no means new4 ,euclid   
knew ( x1 ! ,middle ,ages contribut$ to 
x1 t p]iod vaguely /yl$ ! ,r5aissance   
jo9$ 9 ! movem5t1 & ! fir/ ?ree c5turies
 ( pr9t+ contribut$ a l>ge lit]ature to 
! subject4 ,out ( all ?is ef=t "s g5u9e 
good rema9s1 b relatively n v mu*4#h ,& 
s x w be ) ! pres5t movem5t2 x w s]ve   
its greate/ purpose 9 mak+ tea*]s ?9k & 
read1 & 9 add+ to _! int]e/ & en?usiasm 
& to ! int]e/ ( _! pupils2 b x w n      
greatly *ange geometry1 because no s]i\s
 p]son ev] believ$ t geometry 0 taught  
*iefly = practical purposes1 or 0 made m
 int]e/+ or valuable ?r\gh su* a        
pret5se4 ,*anges 9 sequ5ce1 9           
def9itions1 & 9 pro(s w come little by  
little2 b t !re w be any su* radical    
*ange 9 ^! matt]s 9 ! imm$iate future1 z
 "s writ]s h anticipat$1 is n           
probable4#i  ,a rec5t writ] ( mu*       
acum5#aj has summ$ up ?is ?\ght 9 ^!    
words3       ,n "o t5? ( ! graduates (  
our high s*ools ev] ent]      pr(essions

 9 :i* _! algebra & geometry are appli$ 
to      concrete realities2 n "o day 9  
?ree hundr$ sixty-five is      a high   
s*ool graduate call$ ^u to apply1 z x is
 call$1      an algebraic or a          
geometrical proposition4444 ,:y1 !n1 d  
we      tea* ^! subjects1 if ?is alone  
is ! s5se ( ! ^w      practical6444 ,to 
me ! solution ( ?is p>adox consi/s 9    
boldly confront+ ! dilemma1 & 9 say+ t  
our      conception ( ! practical       
utility ( ^? /udies mu/ be              
readju/$1 & t we h frankly to face !    
tru? t !      practical ends we seek are
 9 a s5se ideal practical ends1      yet
 su* z h1 aft] all1 an em95tly          
utilit>ian value 9      ! intellectual  
sph]e4  ,he quotes f ,c4 ,s4 ,jackson1 a
 progressive contempor>y tea*] (        
me*anics 9 ,engl&1 :o speaks ( pupils   
confus+ millimet]s & c5timet]s 9 "s     
simple computation1 & :o adds3          
,!re is ! enemy6 ,! real enemy we h to  
fight aga9/1      :atev] we tea*1 is    
c>elessness1 inaccuracy1 =getfulness1   

& slov5l9ess4 ,t battle has be5 f\ght & 
won )      div]se weapons4 ,x has1 =    
in/ance1 be5 f\ght ) ,lat9      gramm>  
be=e n[1 & won4 ,i say t because we mu/ 
be v      c>eful to gu>d aga9/ ! notion 
t !re is any "o      panacea = ?is sort 
( ?+4 ,x bord]s on quack]y to say      t
 elem5t>y physics w cure ev]y?+4  ,& (  
c\rse ! same ?+ may be said = ma!matics4
 ,nev]!less x is d\btful if we h any o!r
 subject t does s mu* to br+ to ! front 
?is dang] ( c>elessness1 ( slov5ly      
reason+1 ( inaccuracy1 & ( =getfulness z
 ?is sci5ce ( geometry1 :i* has be5 s   
poli%$ & p]fect$ z ! c5turies h gone on4
  ,!re h be5 ^? :o did n proclaim !     
utilit>ian value ( geometry1 b :o fell  
into z s]i\s an error1 namely1 !        
advocat+ ( geometry z a means ( tra9+ ! 
memory4 ,9 times n s v f> pa/1 & to "s  
ext5t to-day1 ! memoriz+ ( pro(s has be5
 ju/ifi$ on ?is gr\nd4 ,?is error has1  
h[ev]1 be5 fully expos$ by our mod]n    
psy*ologi/s4 ,!y h %[n t ! p]son :o     
memorizes ! propositions ( ,euclid by   

numb] is no m capable ( memoriz+ o!r    
facts ?an he 0 be=e1 & t ! le>n+ ( pro(s
 v]batim is ( no assi/ance :atev] 9     
reta9+ matt] t is helpful 9 o!r l9es (  
"w4 ,geometry1 !re=e1 z a tra9+ ( !     
memory is ( no m value ?an any o!r      
subject 9 ! curriculum4  ,if geometry is
 n /udi$ *iefly because x is practical1 
or because x tra9s ! memory1 :at reasons
 c be adduc$ = its pres5ce 9 ! c\rses ( 
/udy ( e civiliz$ c\ntry8 ,is x n1 aft] 
all1 a m]e feti%1 & are n ^? virul5t    
writ]s correct :o see no?+ good 9 !     
subject save only its utilities8#aa ,(  
?is type "o ( ! mo/ ent]ta9+ is ,william
 ,j4 ,locke1#ab ^: words ^u ! subject   
are well wor? read+3       444 ,i e>n$  
my liv+ at s*ool slav]y1 tea*+ to *ildr5
      ! mo/ useless1 ! mo/ disa/r\s1 !  
mo/ s\l-cramp+      bran* ( k :]e)      
p$agogues 9 _! ins5sate      folly h    
crippl$ ! m9ds & bla/$ ! lives (        
?\s&s ( _! fell[ creatures--elem5t>y    
ma!matics4      ,!re is no m reason =   
any human be+ on ,god's e>? to      be  

acqua9t$ ) ! b9omial !orem or ! solution
 (      triangles1 unless he is a       
pr(essional sci5ti/1--:5 he c      beg9 
to specialize 9 ma!matics at ! same age 
z !      lawy] beg9s to specialize 9 law
 or ! surgeon 9      anatomy1--?an = him
 to be exp]t 9 ,*octaw1 ! ,cabala1 or   
! ,book ( ,mormon4 ,i look back ) feel+s
 ( %ame &      degradation to ! days :51
 = a cru/ ( bread1 ,i      pro/itut$ my 
intellig5ce to wa/+ ! preci\s h\rs (    
impressionable *ildhood1 :i* c\ld h be5 
fill$ ) s      _m beautiful & mean+ful  
?+s1 ov] ?is utt]ly futile      &       
inhuman subject4 ,x tra9s ! m9d1--x     
tea*es boys to      ?9k1 !y say4 ,x     
doesn't4 ,9 reality x is a cut-&-dri$   
subject1 easy to fit into a s*ool       
curriculum4 ,its      sacrosanctity     
saves educationali/s an enorm\s am\nt ( 
tr\ble1 & its *ief use is to enable     
m9dless "y m5 f      ! univ]sities to   
make a di%one/ liv+ by tea*+ x to       
o!rs1 :o 9 _! turn may tea* x to a      
future g5]ation4  ,to be fair we mu/    

face j su* attacks1 & we mu/ recognize t
 !y set =? ! feel+s ( _m hone/ p4 ,"o is
 tempt$ to inquire if ,mr4 ,locke c\ld h
 writt5 9 su* an incisive /yle if he _h 
n1 z 0 ! case1 graduat$ ) honors 9      
ma!matics at "o ( ! great univ]sities4  
,b he might reply t if 8 m9d _h n be5   
w>p$ by ma!matics1 he w\ld h writt5 m   
temp]ately1 s ! honors 9 ! argum5t w\ld 
be ev54 ,mu* m to ! po9t is ! fact t    
,mr4 ,locke taught ma!matics 9 ! s*ools 
( ,engl&1 & t ^! s*ools d n seem to !   
re/ ( ! _w to furni% a good type ( !    
tea*+ ( elem5t>y ma!matics4 ,no c\ntry  
goes to ,engl& = its model 9 ?is        
p>ticul> bran* ( education1 al?\gh ! "w 
is rapidly *ang+ !re1 & ,mr4 ,locke     
pictures a local condition 9 tea*+ r ?an
 a g5]al condition 9 ma!matics4 ,few    
visitors to ! s*ools ( ,engl& w\ld c>e  
to tea* ma!matics z !y see x taught !re1
 9 spite ( _! recognition ( ! ?or\ghness
 ( ! "w & ! e>ne/ness ( _m ( ! tea*]s4  
,x is also ( int]e/ to note t ! greate/ 
prote/s aga9/ =mal ma!matics h come f   

,engl&1 z witness ! utt]ances ( su* m5 z
 ,sir ,william ,hamilton & ,pr(essors   
,p]ry1 ,m9*91 ,h5rici1 & ,alfr$ ,lodge4 
,x may !re=e be que/ion$ :e!r ^! s*ol>s 
are n unconsci\sly prote/+ aga9/ !      
,engli% me?ods & curriculum r ?an aga9/ 
! subject itself4 ,:5 ,pr(essor ,m9*9   
says t he _h be5 ?r\gh ! six books (    
,euclid )\t really und]/&+ an angle1 x  
is ,euclid's text & 8 own tea*] t are at
 fault1 & n geometry4  ,be=e consid]+   
directly ! "q z to :y geometry %\ld be  
taught1 let u turn = a mom5t to ! o!r   
subjects 9 ! second>y curriculum4 ,:y1 =
 example1 d we /udy lit]ature8 ,x does n
 l[] ! price ( bread1 z ,malh]be rem>k$ 
9 speak+ ( ! comm5t>y ( ,ba*et on !     
great "w ( ,diophantus4 ,is x = !       
purpose ( mak+ au?ors8 ,n "o p]son out (
 t5 ?\s& :o /udy lit]ature ev] writes = 
publication4 ,& :y d we all[ pupils to  
wa/e _! "t 9 physical education8 ,x uses
 valuable h\rs1 x wa/es money1 & x is   
dang]\s to life & limb4 ,w\ld x n be    
bett] to set pupils at saw+ wood8 ,& :y 

d we /udy music8 ,to give pleasure by   
our p]=mances8 ,h[ _m :o attempt to play
 ! piano or to s+ give mu* pleasure to  
any b !mselves1 & possibly _! p>5ts8 ,! 
/udy ( gramm> does n make an accurate   
writ]1 nor ! /udy ( rhetoric an orator1 
nor ! /udy ( met] a poet1 nor ! /udy (  
p$agogy a tea*]4 ,! /udy ( geography 9 !
 s*ool does n make travel p>ticul>ly    
easi]1 nor does ! /udy ( biology t5d to 
populate ! e>?4 ,s we might pass 9      
review ! v>i\s subjects t we /udy &     
ought to /udy1 & 9 no case w\ld we f9d  
utility ! mov+ cause1 & 9 e case w\ld we
 f9d x difficult to /ate ! "o great     
reason = ! pursuit ( ! subject 9 "q1--& 
s x is ) geometry4  ,:at positive       
reasons c n[ be adduc$ = ! /udy ( a     
subject t occupies upw>ds ( a ye> 9 !   
s*ool c\rse1 & t is1 p]haps unwisely1   
requir$ ( all pupils8 ,probably ! prim>y
 reason1 if we d n attempt to deceive   
ourselves1 is pleasure4 ,we /udy music  
because music gives u pleasure1 n       
necess>ily our own music1 b good music1 

:e!r ours1 or1 z is m probable1 t (     
o!rs4 ,we /udy lit]ature because we     
d]ive pleasure f books2 ! bett] ! book !
 m subtle & la/+ ! pleasure4 ,we /udy   
art because we receive pleasure f !     
great works ( ! ma/]s1 & probably we    
appreciate !m ! m because we h dabbl$ a 
little 9 pigm5ts or 9 clay4 ,we d n     
expect to be compos]s1 or poets1 or     
sculptors1 b we wi% to appreciate music 
& lett]s & ! f9e arts1 & to d]ive       
pleasure f !m & to be uplift$ by !m4 ,at
 any rate1 ^! are ! nobl] reasons = _!  
/udy4  ,s x is ) geometry4 ,we /udy x   
because we d]ive pleasure f contact ) a 
great & an anci5t body ( le>n+ t has    
occupi$ ! att5tion ( ma/] m9ds dur+ !   
?\s&s ( ye>s 9 :i* x has be5 p]fect$1 & 
we are uplift$ by x4 ,to d5y t our      
pupils d]ive ?is pleasure f ! /udy is to
 confess ourselves poor tea*]s1 = mo/   
pupils d h positive enjoym5t 9 ! pursuit
 ( geometry1 9 spite ( ! tradition t    
leads !m to proclaim a g5]al dislike =  
all /udy4 ,?is enjoym5t is p>tly t ( !  

game1--! play+ ( a game t c always be   
won1 b t _c be won too easily4 ,x is    
p>tly t ( ! ae/hetic1 ! pleasure (      
symmetry ( =m1 ! delight ( fitt+ ?+s    
toge!r4 ,b probably x lies *iefly 9 !   
m5tal uplift t geometry br+s1 ! contact 
) absolute tru?1 & ! approa* t "o makes 
to ! ,inf9ite4 ,we are n q sure ( any "o
 ?+ 9 biology2 our k ( geology is       
relatively v slight1 & ! economic laws (
 society are unc]ta9 to e "o except "s  
individual :o attempts to set !m =?2 b  
be=e ! _w 0 fa%ion$ ! squ>e on !        
hypot5use 0 equal to ! sum ( ! squ>es on
 ! o!r two sides ( a "r triangle1 & x w 
be s aft] ?is _w is dead2 & ! inhabitant
 ( ,m>s1 if he exi/s1 probably kn[s its 
tru? z we "k x4 ,! uplift ( ?is contact 
) absolute tru?1 ) tru? et]nal1 gives   
pleasure to humanity to a great] or less
 degree1 dep5d+ ^u ! m5tal equipm5t ( ! 
p>ticul> individual2 b x probably gives 
an appreciable am\nt ( pleasure to e    
/ud5t ( geometry :o has a tea*] wor?y ( 
! "n4 ,fir/1 !n1 & =emo/ z a reason =   

/udy+ geometry has always /ood1 & w     
always /&1 ! pleasure & ! m5tal uplift t
 comes f contact ) su* a great body (   
human le>n+1 & p>ticul>ly ) ! exact tru?
 t x conta9s4 ,! tea*] :o is imbu$ ) ?is
 feel+ is on ! road to success1 :atev]  
me?od ( pres5tation he may use2 ! "o :o 
is n imbu$ ) x is on ! road to failure1 
h[ev] logical 8 pres5tation or h[ev]    
l>ge 8 supply ( practical applications4 
,subord9ate to ^! reasons = /udy+       
geometry are _m o!rs1 exactly z ) all   
o!r subjects ( ! curriculum4 ,geometry1 
= example1 (f]s ! be/ develop$          
application ( logic t we h1 or are      
likely to h1 9 ! s*ool c\rse4 ,?is does 
n mean t x always exemplifies p]fect    
logic1 = x does n2 b to ! pupil :o is n 
ready = logic1 p] se1 x (f]s an example 
( close reason+ su* z 8 o!r subjects d n
 (f]4 ,we may say1 & possibly ) tru?1 t 
"o :o /udies geometry w n reason m      
cle>ly on a f9ancial proposition ?an "o 
:o does n2 b 9 spite ( ! results ( ! v  
meag] exp]im5ts ( ! psy*ologi/s1 x is   

probable t ! man :o has _h "s drill 9   
syllogisms1 & :o has le>n$ to select !  
ess5tials & to neglect ! noness5tials 9 
rea*+ 8 conclusions1 has acquir$ habits 
9 reason+ t w help him 9 e l9e ( "w4 ,z 
"p ( ?is equipm5t !re is also a t]s5ess 
( /atem5t & a cle>ness 9 arrangem5t (   
po9ts 9 an argum5t t has be5 ! subject (
 comm5t by _m writ]s4  ,^u ?is same     
topic an ,engli% writ]1 9 "o ( ! sane/ (
 rec5t monographs ^u ! subject1#ac has  
express$ 8 views 9 ! foll[+ words3      
,! /atem5t t a giv5 individual has      
receiv$ a s\nd      geometrical tra9+   
implies t he has segregat$ f !      :ole
 ( 8 s5se impressions a c]ta9 set ( ^!  
impressions1 t he has !n elim9at$ f _!  
consid]ation all irrelevant impressions 
79 o!r words1      acquir$ a subjective 
comm& ( ^! impressions71 t he      has  
develop$ on ! basis ( ^! impressions an 
ord]$ &      cont9u\s sy/em ( logical   
d$uction1 & f9ally t he is      capable 
( express+ ! nature ( ^! impressions & 8
      d$uctions !refrom 9 t]ms simple & 

free f ambiguity4      ,n[ ! slighte/   
consid]ation w conv9ce any "o n         
already conv]sant ) ! idea1 t ! same    
sequ5ce (      m5tal processes und]lies 
! :ole c>e] ( any individual      9 any 
walk ( life if only he is n conc]n$     
entirely )      manual labor2           
consequ5tly a full tra9+ 9 ! p]=mance   
( su* sequ5ces mu/ be reg>d$ z =m+ an   
ess5tial "p      ( any education wor?y (
 ! "n4 ,moreov]1 ! full                 
appreciation ( su* processes has a high]
 value ?an is      conta9$ 9 ! m5tal    
tra9+ involv$1 great ?\gh ?is      be1 =
 x induces an appreciation (            
intellectual unity &      beauty :i*    
plays = ! m9d t "p :i* !                
appreciation ( s*emes ( %ape & color    
plays = !      arti/ic faculties2 or1   
aga91 t "p :i* ! appreciation      ( a  
body ( religi\s doctr9e plays = ! e?ical
      aspirations4 ,n[ geometry is n !  
sole possible basis =      inculcat+ ?is
 appreciation4 ,logic is an alt]native =
      adults1 provid$ t ! individual is 

possess$ ( suffici5t      wide1 ?\gh    
r\gh1 exp]i5ce on :i* to base 8 reason+4
      ,geometry is1 h[ev]1 highly       
desirable 9 t ! objective      bases are
 s simple & precise t !y c be grasp$ at 
an      e>ly age1 t ! am\nt ( tra9+ = ! 
imag9ation is      v l>ge1 t ! d$uctive 
processes are n beyond !      scope (   
ord9>y boys1 & f9ally t x af=ds a bett] 
basis = ex]cise 9 ! art ( simple & exact
 expression      ?an any o!r possible   
subject ( a s*ool c\rse4  ,are ^!       
results really secur$ by tea*]s1 h[ev]1 
or are !y m]ely imag9$ by ! p$agogue z a
 ju/ification = 8 exi/5ce8 ,d tea*]s h  
any su* appreciation ( geometry z has   
be5 sugge/$1 & ev5 if !y h x1 d !y imp>t
 x to _! pupils8 ,9 reply x may be said1
 probably ) p]fect safety1 t tea*]s (   
geometry appreciate _! subject & lead _!
 pupils to appreciate x to q z great a  
degree z obta9s 9 any o!r bran* (       
education4 ,:at tea*] appreciates fully 
! beauties ( ,9 ,memoriam1 or ( ,hamlet1
 or ( ,p>adise ,lo/1 & :at "o inspires 8

 pupils ) all ! nobility ( ^! _w        
classics8 ,:at tea*] sees 9 biology all 
! gr&eur ( ! evolution ( ! race1 or     
imp>ts to 8 pupils ! noble lessons (    
life t ! /udy ( ?is subject %\ld sugge/8
 ,:at tea*] ( ,lat9 br+s 8 pupils to    
read ! anci5t lett]s ) full appreciation
 ( ! dignity ( /yle & ! nobility ( ?\ght
 t !y conta98 ,& :at tea*] ( ,fr5*      
succe$s 9 br++ a pupil to c>ry on a     
conv]sation1 to read a ,fr5* magaz9e1 to
 see ! hi/ory imb$d$ 9 ! words t are    
us$1 to realize ! *>m & p[] ( !         
language1 or to appreciate to ! full a  
s+le classic8 ,9 o!r words1 none ( u    
fully appreciates 8 subject1 & none ( u 
c hope to br+ 8 pupils to ! ideal       
attitude t[>d any "p ( x4 ,b x is       
probable t ! tea*] ( geometry succe$s   
relatively bett] ?an ! tea*] ( o!r      
subjects1 because ! sci5ce has rea*$ a  
relatively high] /ate ( p]fection4 ,!   
body ( tru? 9 geometry has be5 m cle>ly 
m>k$ out1 x has be5 m successfully fitt$
 toge!r1 its lesson is m pat5t1 & !     

exp]i5ce ( c5turies has br\ght x into a 
%ape t is m usable 9 ! s*ool4 ,:ile1    
!re=e1 we h all k9ds ( tea*+ 9 all k9ds 
( subjects1 ! v nature ( ! case leads to
 ! belief t ! class 9 geometry receives 
q z mu* f ! tea*] & ! subject z ! class 
9 any o!r bran* 9 ! s*ool curriculum4   
,b is ?is n m]e conjecture8 ,:at are !  
results ( sci5tific inve/igation ( !    
tea*+ ( geometry8 ,un=tunately !re is   
little hope f ! results ( su* an        
inquiry1 ei!r "h or 9 o!r fields4 ,we _c
 fir/ weigh a pupil 9 an intellectual or
 moral balance1 !n fe$ him geometry1 &  
!n weigh him aga91 & !n set back 8 clock
 ( "t & beg9 all ov] aga9 ) ! same      
individual4 ,!re is no be=e tak+ & aft] 
tak+ ( a subject t ext5ds ov] a ye> or  
two ( a pupil's life4 ,we c weigh       
utilities r\ghly1 we c e/imate !        
pleasure ( a subject relatively1 b we _c
 say t geometry is wor? s _m doll>s1 &  
hi/ory s _m1 & s on ?r\gh ! curriculum4 
,! be/ we c d is to ask ourselves :at ! 
v>i\s subjects1 ) tea*]s ( fairly equal 

m]it1 h done = u1 & to inquire :at has  
be5 ! exp]i5ce ( o!r p]sons4 ,su* an    
inve/igation results 9 %[+ t1 ) few     
exceptions1 p :o h /udi$ geometry       
receiv$ z mu* ( pleasure1 ( inspiration1
 ( satisfaction1 ( :at !y call tra9+ f  
geometry z f any o!r subject (          
/udy1--giv5 tea*]s ( equal m]it1--& t !y
 w\ld n will+ly give up ! some?+ :i*    
geometry br\ght to !m4 ,if ?is 7 n !    
feel+1 & if humanity believ$ t geometry 
is :at ,mr4 ,locke's words w\ld seem to 
indicate1 x w\ld long ago h bani%$ x f !
 s*ools1 s9ce ^u ?is gr\nd r ?an ^u !   
gr\nd ( utility ! subject has always    
/ood4  ,^! seem to be ! great reasons = 
! /udy ( geometry1 & to se>* = o!rs w\ld
 t5d to weak5 ! argum5t4 ,at fir/ sight 
!y may n seem to ju/ify ! exp5diture (  
"t t geometry dem&s1 & !y may seem      
unduly to neglect ! argum5t t geometry  
is a /epp+-/one to high] ma!matics4 ,ea*
 ( ^! po9ts1 h[ev]1 has be5 neglect$    
purposely4 ,a pupil has a numb] ( s*ool 
ye>s at 8 disposal2 to :at %all !y be   

devot$8 ,to lit]ature8 ,:at claim has   
lett]s t is su* z to ju/ify ! exclusion 
( geometry8 ,to music1 or natural       
sci5ce1 or language8 ,^! are all        
valuable1 & all %\ld be /udi$ by "o     
seek+ a lib]al education2 b = ! same    
reason geometry %\ld h its place4 ,:at  
subject1 9 f9e1 c supply exactly :at    
geometry does8 ,& if none1 !n h[ c !    
pupil's "t be bett] exp5d$ ?an 9 ! /udy 
( ?is sci5ce8#ad ,z to ! second po9t1 t 
a claim %\ld be set =? t geometry is a  
s9e qua non to high] ma!matics1 ?is     
belief is consid]ably exagg]at$ because 
!re are relatively few :o proce$ f      
geometry to a high] bran* ( ma!matics4  
,?is argum5t w\ld ju/ify its /atus z an 
elective r ?an z a requir$ subject4     
,let u !n /& ^u ! gr\nd already m>k$    
out1 hold+ t ! pleasure1 ! culture1 !   
m5tal poise1 ! habits ( exact reason+ t 
geometry br+s1 & ! g5]al exp]i5ce (     
mank9d ^u ! subject are suffici5t to    
ju/ify u 9 dem&+ = x a reasonable am\nt 
( "t 9 ! fram+ ( a curriculum4 ,let u be

 fair 9 our appreciation ( all o!r      
bran*es1 b let u urge t e /ud5t may h an
 opportunity to "k ( real geometry1 say 
= a s+le ye>1 !reaft] pursu+ x or n1    
accord+ z we succe$ 9 mak+ its value    
app>5t1 or fail 9 our attempt to pres5t 
wor?ily an anci5t & noble sci5ce to !   
m9d confid$ to our in/ruction4  ,!      
%ortsight$ness ( a n>r[ education1 ( an 
education t tea*es only ma*9es to a     
prospective me*anic1 & agriculture to a 
prospective f>m]1 & cook+ & dressmak+ to
 ! girl1 & t w\ld exclude all ma!matics 
t is n utilit>ian 9 ! n>r[ s5se1 _c     
endure4       ,! community has f\nd out 
t su* s*emes may be well      fitt$ to  
give ! *ildr5 a good "t 9 s*ool1 b lead 
!m to a bad "t aft]w>d4 ,life is h>d "w1
 & if !y      h nev] le>n$ 9 s*ool to   
give _! conc5trat$      att5tion to t   
:i* does n appeal to !m & :i* does      
n int]e/ !m imm$iately1 !y h miss$ ! mo/
      valuable lesson ( _! s*ool ye>s4  
,! little practical      in=mation !y   
c\ld h le>n$ at any "t2 ! en]gy (       

att5tion & conc5tration c no long] be   
le>n$ if !      e>ly ye>s are wa/$4     
,h[ev] n>r[ & comm]cial !      /&po9t   
:i* is *os5 may be1 x c always be f\nd t
      x is ! g5]al education :i* pays   
be/1 & ! m !      p]iod ( cultural "w c 
be exp&$ ! m effici5t w      be !       
s]vices ( ! s*ool = ! practical s]vices 
( !      nation4#ae  ,( c\rse no "o %\ld
 con/rue ^! rem>ks z oppos+ 9 ! slighte/
 degree ! laudable ef=ts t are con/antly
 be+ put =? to make geometry m int]e/+ &
 to vitalize x by e/abli%+ z /rong      
motives z possible = its /udy4 ,let !   
home1 ! work%op1 physics1 art1          
play1--all contribute _! quota ( motive 
to geometry z to all ma!matics & all o!r
 bran*es4 ,b let u nev] =get t geometry 
has a raison d'etre beyond all ?is1 & t 
^! applications are s\ght prim>ily = !  
sake ( geometry1 & t geometry is n      
taught prim>ily = ! sake ( ^!           
applications4  ,:5 we consid] h[ (t5    
geometry is attack$ by ^? :o pr(ess to  
be its fri5ds1 & h[ tea*]s :o h be5     

tra9$ 9 ma!matics occasionally seem to  
make ( ! subject little besides a       
mongrel c\rse 9 draw+ & measur+1 all !  
"t insi/+ t !y are progressive :ile !   
*ampions ( real geometry are reaction>y1
 x is well to read "s ( ! op9ions ( !   
ma/]s4 ,! foll[+ quotations may be giv5 
occasionally 9 geometry classes z %[+ ! 
e/eem 9 :i* ! subject has be5 held 9    
v>i\s ages1 & at any rate !y %\ld s]ve  
to inspire ! tea*] to great] love = 8   
subject4       ,! enemies ( geometry1 ^?
 :o "k x only imp]fectly1      look ^u !
 !oretical problems1 :i* con/itute ! mo/
      difficult "p ( ! subject1 z m5tal 
games :i* consume      "t & en]gy t     
might bett] be employ$ 9 o!r ways4      
,su* a belief is false1 & x w\ld block !
 progress (      sci5ce if x 7 cr$ible4 
,b aside f ! fact t !      speculative  
problems1 :i* at fir/ sight seem b>r51 c
      (t5 be appli$ to useful purposes1 
!y always /& z among      ! be/ means to
 develop & to express all ! =ces ( !    
human intellig5ce4--,,abbe ,,bossut4    

,! sailor :om an exact obs]vation (     
longitude saves f      %ipwreck owes 8  
life to a !ory develop$ two ?\s&        
ye>s ago by m5 :o _h 9 m9d m]ely !      
speculations (      ab/ract             
geometry4--,,condorcet4       ,if       
ma!matical heights are h>d to climb1 !  
fundam5tal      pr9ciples lie at e      
?re%old1 & ?is fact all[s !m to      be 
compreh5d$ by t common s5se :i*         
,desc>tes decl>$      0 apportion$      
equally among all m54--,,collet4        
,x may seem /range t geometry is unable 
to def9e ! t]ms      :i* x uses mo/     
frequ5tly1 s9ce x def9es nei!r          
movem5t1 nor numb]1 nor space1---! ?ree 
?+s ) :i*      x is *iefly conc]n$4 ,b  
we %all n be surpris$ if we      /op to 
consid] t ?is admirable sci5ce conc]ns  
only !      mo/ simple ?+s1 & ! v       
quality t r5d]s ^!      ?+s wor?y ( /udy
 r5d]s !m incapable ( be+ def9$4        
,?us ! v lack ( def9ition is r an       
evid5ce (      p]fection ?an a defect1  
s9ce x comes n f ! obscurity      ( !   

t]ms1 b f ! fact t !y are s v well      
kn[n4--,,pascal4       ,god et]nally    
geometrizes4--,,plato4       ,god is a  
circle ( :i* ! c5t] is ev]y:]e & !      
circumf]5ce n[h]e4--,,rabelais4         
,)\t ma!matics no "o c fa?om ! dep?s (  
philosophy4      ,)\t philosophy no "o c
 fa?om ! dep?s ( ma!matics4      ,)\t ! 
two no "o c fa?om ! dep?s (             
any?+4--,,bordas-,,dem\l94       ,we may
 look ^u geometry z a practical logic1 =
 ! tru?s      :i* x /udies1 be+ ! mo/   
simple & mo/ cle>ly      und]/ood ( all 
tru?s1 are on ?is acc\nt ! mo/          
susceptible ( ready application 9       
reason+4--,d',,alemb]t4       ,! advance
 & ! p]fect+ ( ma!matics are closely    
jo9$ to ! prosp]ity ( !                 
nation4--,,napoleon4       ,hold no?+ z 
c]ta9 save :at c be                     
demon/rat$4--,,newton4       ,to measure
 is to "k4--,,kepl]4       ,! me?od (   
mak+ no mi/ake is s\ght by e "o4 ,!     
logicians pr(ess to %[ ! way1 b !       
geomet]s alone ev]      rea* x1 & aside 

f _! sci5ce !re is no g5u9e             
demon/ration4--,,pascal4       ,! ta/e =
 exactness1 ! impossibility ( cont5t+   
"o's      self ) vague notions or (     
lean+ ^u m]e hypo!ses1 !      necessity 
= p]ceiv+ cle>ly ! connection betwe5    
c]ta9      propositions & ! object 9    
view1--^! are ! mo/      preci\s fruits 
( ! /udy ( ma!matics4--,,lacroix4       
,bibliography4 ,smi?1 ,! ,tea*+ (       
,elem5t>y ,ma!matics1      p4 #bcd1 ,new
 ,york1 #aijj2 ,h5rici1 ,presid5tial    
,address be=e      ! ,briti%            
,association1 ,nature1 ,vol4 ,,xxviii1  
p4 #dig2 ,hill1      ,educational ,value
 ( ,ma!matics1 ,educational ,review1    
,vol4      ,,ix1 p4 #cdi2 ,"y1 ,! ,tea*+
 ( ,ma!matics1 p4 #i1 ,new ,york1       
#aijg4 ,! clos+ quotations are f        
,rebi]e1 ,ma!matiques et                
,ma!matici5s1 ,p>is1 #ahic4             
,,footnotes3  #d ,9 an address 9        
,london1 ,june #ae1 #aiji1 at a d9n] to 
,sir ,erne/ ,%ackelton4  #e ,gov]nor    
,hughes1 n[ ,ju/ice ,hughes1 ( ,new     

,york1 at ! ,pe>y te/imonial on ,febru>y
 #h1 #aiaj1 at ,new ,york ,city4  #f ,! 
fir/ "w ^u ?is subject1 & inde$ ! fir/  
pr9t$ treatise on curves 9 g5]al1 0     
writt5 by ! fam\s arti/ ( ,nu]nb]g1     
,albre*t ,du]]4  #g ,sev]al ( ^! writ]s 
are m5tion$ 9 ,*apt] ,,iv4  #h ,if any  
read] *ances ^u ,george ,birkbeck's     
,engli% translation ( ,*>les ,dup9's    
,ma!matics ,practically ,appli$1        
,halifax1 #ahed1 he w f9d t ,dup9 gave m
 good applications ( geometry ?an all ( 
our ,am]ican advocates ( practical      
geometry comb9$4  #i ,see1 = example1   
,h5rici's ,congru5t ,figures1 ,london1  
#ahgi1 & ! review ( ,borel's ,elem5ts ( 
,ma!matics1 by ,pr(essor ,sisam 9 !     
,bullet9 ( ! ,am]ican ,ma!matical       
,society1 ,july1 #aiaj1 a matt] discuss$
 lat] 9 ?is "w4  #aj ,t4 ,j4            
,mc,cormack1 ,:y d we /udy ,ma!matics3 a
 ,philosophical & ,hi/orical            
,retrospect1 p4 #i1 ,c$> ,rapids1 ,i[a1 
#aiaj4  #aa ,( ! fair & c&id argum5ts   
aga9/ ! culture value ( ma!matics1 "o ( 

! be/ ( ! rec5t ones is t by ,g4 ,f4    
,swa91 9 ! ,atti del ,,iv ,congresso    
,int]nazionale dei ,matematici1 ,rome1  
#aiji1 ,vol4 ,,iii1 p4 #cfa4 ,!         
lit]ature ( ?is s*ool is q ext5sive1 b  
,p]ry's ,engl&'s ,neglect ( ,sci5ce1    
,london1 #aijj1 & ,discussion on !      
,tea*+ ( ,ma!matics1 ,london1 #aija1 are
 typical4  #ab ,9 8 novel1 ,! ,morals ( 
,m>cus ,ordeyne4  #ac ,g4 ,w4 ,l4       
,c>son1 ,! ,functions ( ,geometry z a   
,subject ( ,education1 p4 #c1           
,tonbridge1 #aiaj4  #ad ,x may well be1 
h[ev]1 t ! gr[+ curriculum may ju/ify "s
 r$uction 9 ! "t =m]ly assign$ to       
geometry1 & any reasonable proposition (
 ?is nature %\ld be fairly met by tea*]s
 ( ma!matics4  #ae ,pr(essor ,mu5/]b]g1 
9 ! ,metropolitan ,magaz9e = ,july1     
#aiaj4     ,,*apt] ,,iii  ,a ,,brief    
,,hi/ory ,,( ,,geometry   ,! geometry ( 
v anci5t peoples 0 l>gely ! m5suration (
 simple areas & solids1 su* z is taught 
to *ildr5 9 elem5t>y ari?metic to-day4  
,!y e>ly le>n$ h[ to f9d ! area ( a     

rectangle1 & 9 ! olde/ ma!matical       
records t h come d[n to u !re is "s     
discussion ( ! area ( triangles & !     
volume ( solids4  ,! e>lie/ docum5ts t  
we h relat+ to geometry come to u f     
,babylon & ,egypt4 ,^? f ,babylon are   
writt5 on small clay tablets1 "s ( !m   
ab\t ! size ( ! h&1 ^! tablets aft]w>ds 
hav+ be5 bak$ 9 ! sun4 ,!y %[ t !       
,babylonians ( t p]iod knew some?+ ( l& 
measures1 & p]haps _h advanc$ f> en\gh  
to compute ! area ( a trapezoid4 ,= !   
m5suration ( ! circle !y lat] us$1 z did
 ! e>ly ,hebrews1 ! value pi  #c4 ,a    
tablet 9 ! ,briti% ,museum %[s t !y also
 us$ su* geometric =ms z triangles &    
circul> segm5ts 9 a/rology or z         
talismans4  ,! ,egyptians mu/ h _h a    
fair k ( practical geometry long be=e ! 
date ( any ma!matical treatise t has    
come d[n to u1 = ! build+ ( ! pyramids1 
betwe5 #cjjj & #bdjj ,b4,c41 requir$ !  
application ( sev]al geometric          
pr9ciples4 ,"s k ( survey+ mu/ also h   
be5 necess>y to c>ry out ! ext5sive     

plans = irrigation t 7 execut$ "u       
,am5emhat ,,iii1 ab\t #bbjj ,b4,c4  ,!  
fir/ def9ite k t we h ( ,egyptian       
ma!matics comes to u f a manuscript     
copi$ on papyrus1 a k9d ( pap] us$ ab\t 
! ,m$it]ranean 9 e>ly times4 ,?is copy 0
 made by "o ,aah-mesu 7,! ,moon-born71  
commonly call$ ,ahmes1 :o probably      
fl\ri%$ ab\t #agjj ,b4,c4 ,! orig9al f  
:i* he copi$1 writt5 ab\t #bcjj ,b4,c41 
has be5 lo/1 b ! papyrus ( ,ahmes1      
writt5 ne>ly f\r ?\s& ye>s ago1 is /ill 
pres]v$1 & is n[ 9 ! ,briti% ,museum4 ,9
 ?is manuscript1 :i* is devot$ *iefly to
 fractions & to a crude algebra1 is f\nd
 "s "w on m5suration4 ,among ! curi\s   
rules are ! incorrect ones t ! area ( an
 isosceles triangle equals half !       
product ( ! base & "o ( ! equal sides2 &
 t ! area ( a trapezoid hav+ bases b1   
b'1 & ! nonp>allel sides ea* equal to a1
 is #a_/b;a7b  b'74 ,"o notewor?y       
advance appe>s1 h[ev]4 ,ahmes gives a   
rule = f9d+ ! area ( a circle1          
sub/antially z foll[s3 ,multiply ! squ>e

 on ! radius by 7#af_/i7b1 :i* is       
equival5t to tak+ = pi ! value #c4afje4 
,?is papyrus also conta9s "s treatm5t ( 
! m5suration ( solids1 p>ticul>ly )     
ref]5ce to ! capacity ( gran>ies4 ,!re  
is also "s slight m5tion ( simil>       
figures1 & an ext5sive treatm5t ( unit  
fractions1--fractions t 7 q univ]sal    
among ! anci5ts4 ,9 ! l9e ( algebra x   
conta9s a brief treatm5t ( ! equation ( 
! fir/ degree ) "o unkn[n1 & (          
progressions4#af  ,h]odotus tells u t   
,seso/ris1 k+ ( ,egypt1#ag divid$ ! l&  
among 8 p & m>k$ out ! b\nd>ies aft] !  
ov]fl[ ( ! ,nile1 s t survey+ mu/ h be5 
well kn[n 9 8 day4 ,inde$1 !            
h>p$onaptae1 or rope /ret*]s1 acquir$ _!
 "n because !y /ret*$ cords1 9 :i* 7    
knots1 s z to make ! "r triangle #c1 #d1
 #e1 :5 !y wi%$ to erect a p]p5dicul>4  
,?is is a plan occasionally us$ by      
surveyors to-day1 & x %[s t ! practical 
application ( ! ,py?agorean ,!orem 0    
kn[n long be=e ,py?agoras gave :at seems
 to h be5 ! fir/ g5]al pro( ( !         

proposition4  ,f ,egypt1 & possibly f   
,babylon1 geometry pass$ to ! %ores (   
,asia ,m9or & ,greece4 ,! sci5tific /udy
 ( ! subject beg9s ) ,?ales1 "o ( !     
,sev5 ,wise ,m5 ( ! ,grecian            
civilization4 ,born at ,miletus1 n f> f 
,smyrna & ,ephesus1 ab\t #fdj ,b4,c41 he
 di$ at ,a!ns 9 #edh ,b4,c4 ,he sp5t 8  
e>ly manhood z a m]*ant1 accumulat+ !   
weal? t enabl$ him to sp5d 8 lat] ye>s 9
 /udy4 ,he visit$ ,egypt1 & is said to h
 le>n$ su* elem5ts ( geometry z 7 kn[n  
!re4 ,he f\nd$ a s*ool ( ma!matics &    
philosophy at ,miletus1 kn[n f ! c\ntry 
z ! ,ionic ,s*ool4 ,h[ elem5t>y ! k (   
geometry !n 0 may be und]/ood f ! fact t
 tradition attributes only ab\t f\r     
propositions to ,?ales1--7#a7 t v]tical 
angles are equal1 7#b7 t equal angles   
lie opposite ! equal sides ( an         
isosceles triangle1 7#c7 t a triangle is
 det]m9$ by two angles & ! includ$ side1
 7#d7 t a diamet] bisects ! circle1 &   
possibly ! propositions ab\t ! angle-sum
 ( a triangle = special cases1 & ! angle

 inscrib$ 9 a semicircle4#ah  ,! greate/
 pupil ( ,?ales1 & "o ( ! mo/ rem>kable 
m5 ( antiquity1 0 ,py?agoras4 ,born     
probably on ! isl& ( ,samos1 j (f ! coa/
 ( ,asia ,m9or1 ab\t ! ye> #ehj ,b4,c41 
,py?agoras set =? z a "y man to travel4 
,he w5t to ,miletus & /udi$ "u ,?ales1  
probably sp5t sev]al ye>s 9 ,egypt1 v   
likely w5t to ,babylon1 & possibly w5t  
ev5 to ,india1 s9ce tradition ass]ts ?is
 & ! nature ( 8 "w 9 ma!matics sugge/s  
x4 ,9 lat] life he w5t to a ,greek      
colony 9 s\!rn ,italy1 & at ,crotona1 9 
! s\!a/]n "p ( ! p59sula1 he f\nd$ a    
s*ool & e/abli%$ a secret society to    
propagate 8 doctr9es4 ,9 geometry he is 
said to h be5 ! fir/ to demon/rate !    
proposition t ! squ>e on ! hypot5use is 
equal to ! sum ( ! squ>es ^u ! o!r two  
sides ( a "r triangle4 ,! proposition 0 
kn[n 9 ,india & ,egypt be=e 8 "t1 at any
 rate = special cases1 b he seems to h  
be5 ! fir/ to prove x4 ,to him or to 8  
s*ool seems also to h be5 due !         
con/ruction ( ! regul> p5tagon & ( !    

five regul> polyh$rons4 ,! con/ruction (
 ! regul> p5tagon requires ! divid+ ( a 
l9e into extreme & mean ratio1 & ?is    
problem is commonly assign$ to !        
,py?agoreans1 al?\gh x play$ an         
important "p 9 ,plato's s*ool4          
,py?agoras is also said to h kn[n t six 
equilat]al triangles1 ?ree regul>       
hexagons1 or f\r squ>es1 c be plac$ ab\t
 a po9t s z j to fill ! #cfj deg41 b t  
no o!r regul> polygons c be s plac$4 ,to
 8 s*ool is also due ! pro( = ! g5]al   
case t ! sum ( ! angles ( a triangle    
equals two "r angles1 ! fir/ k ( ! size 
( ea* angle ( a regul> polygon1 & !     
con/ruction ( at lea/ "o />-polygon1 !  
/>-p5tagon1 :i* became ! badge ( 8      
frat]nity4 ,! bro!rhood f\nd$ by        
,py?agoras prov$ s (f5sive to ! gov]nm5t
 t x 0 disp]s$ be=e ! dea? ( ! ma/]4    
,py?agoras fl$ to ,megapontum1 a seaport
 ly+ to ! nor? ( ,crotona1 & !re he di$ 
ab\t #eja ,b4,c4#ai  ,illu/ration3      
,,fanciful ,,portrait ,,( ,,py?agoras   
,cal&ri's ,ari?metic1 #adia  ,= two     

c5turies aft] ,py?agoras geometry pass$ 
?r\gh a p]iod ( discov]y ( propositions4
 ,! /ate ( ! sci5ce may be se5 f ! fact 
t ,o5opides ( ,*ios1 :o fl\ri%$ ab\t    
#dfe ,b4,c41 & :o _h /udi$ 9 ,egypt1 0  
celebrat$ because he %[$ h[ to let fall 
a p]p5dicul> to a l9e1 & h[ to make an  
angle equal to a giv5 angle4 ,a few ye>s
 lat]1 ab\t #ddj ,b4,c41 ,hippocrates ( 
,*ios wrote ! fir/ ,greek textbook on   
ma!matics4 ,he knew t ! areas ( circles 
are proportional to ! squ>es on _!      
radii1 b 0 ignorant ( ! fact t equal    
c5tral angles or equal inscrib$ angles  
int]cept equal arcs4  ,antiphon &       
,bryson1 two ,greek s*ol>s1 fl\ri%$ ab\t
 #dcj ,b4,c4 ,! =m] attempt$ to f9d !   
area ( a circle by d\bl+ ! numb] ( sides
 ( a regul> inscrib$ polygon1 & ! latt] 
by do+ ! same = bo? inscrib$ &          
circumscrib$ polygons4 ,!y ?us          
approximately exhau/$ ! area betwe5 !   
polygon & ! circle1 & h5ce ?is me?od is 
kn[n z ! me?od ( exhau/ions4  ,ab\t #dbj
 ,b4,c4 ,hippias ( ,elis inv5t$ a c]ta9 

curve call$ ! quadratrix1 by means ( :i*
 he c\ld squ>e ! circle & trisect any   
angle4 ,?is curve _c be con/ruct$ by !  
unm>k$ /raight$ge & ! compasses1 & :5 we
 say t x is impossible to squ>e ! circle
 or to trisect any angle1 we mean t x is
 impossible by ! help ( ^! two in/rum5ts
 alone4  ,dur+ ?is p]iod ! great        
philosophic s*ool ( ,plato 7#dbi-cdh    
,b4,c47 fl\ri%$ at ,a!ns1 & to ?is s*ool
 is due ! fir/ sy/ematic attempt to     
create exact def9itions1 axioms1 &      
po/ulates1 & to di/+ui% betwe5 elem5t>y 
& high] geometry4 ,x 0 at ?is "t t      
elem5t>y geometry became limit$ to ! use
 ( ! compasses & ! unm>k$ /raight$ge1   
:i* took f ?is doma9 ! possibility (    
con/ruct+ a squ>e equival5t to a giv5   
circle 7squ>+ ! circle71 ( trisect+ any 
giv5 angle1 & ( con/ruct+ a cube t %\ld 
h twice ! volume ( a giv5 cube          
7duplicat+ ! cube71 ^! be+ ! ?ree fam\s 
problems ( antiquity4 ,plato & 8 s*ool  
int]e/$ !mselves ) ! s-call$ ,py?agorean
 numb]s1 t is1 ) numb]s t w\ld repres5t 

! ?ree sides ( a "r triangle & h5ce     
fulfill ! condition t a#b  b#b  c#b4    
,py?agoras _h already giv5 a rule t w\ld
 be express$ 9 mod]n =m1 z #a_/d7m#b    
#a7b  m#b  #a_/d7m#b - #a7b4 ,! s*ool ( 
,plato f\nd t 7#a_/bm7#b  #ab  m#b      
7#a_/bm7#b - #ab4 ,by giv+ v>i\s values 
to m1 diff]5t ,py?agorean numb]s may be 
f\nd4 ,plato's nephew1 ,speusippus 7ab\t
 #cej ,b4,c471 wrote ^u ?is subject4    
,su* numb]s 7 kn[n1 h[ev]1 bo? 9 ,india 
& 9 ,egypt1 long be=e ?is "t4  ,"o (    
,plato's pupils 0 ,philippus ( ,m5de1 9 
,egypt1 :o fl\ri%$ ab\t #chj ,b4,c4 ,x  
is said t he discov]$ ! proposition     
relat+ to ! ext]ior angle ( a triangle4 
,8 int]e/1 h[ev]1 0 *iefly 9 a/ronomy4  
,ano!r ( ,plato's pupils 0 ,eudoxus (   
,cnidus 7#djh-cee ,b4,c474 ,he elaborat$
 ! !ory ( proportion1 plac+ x ^u a      
?or\ghly sci5tific f\ndation4 ,x is     
probable t ,book ,v ( ,euclid1 :i* is   
devot$ to proportion1 is ess5tially ! "w
 ( ,eudoxus4 ,by means ( ! me?od (      
exhau/ions ( ,antiphon & ,bryson he     

prov$ t ! pyramid is "o ?ird ( a prism1 
& ! cone is "o ?ird ( a cyl9d]1 ea* ( ! 
same base & ! same altitude4 ,he wrote !
 fir/ textbook kn[n on solid geometry4  
,! subject ( conic sections />ts ) ano!r
 pupil ( ,plato's1 ,m5ae*mus1 :o liv$   
ab\t #cej ,b4,c4 ,he cut ! ?ree =ms (   
conics 7! ellipse1 p>abola1 & hyp]bola7 
out ( ?ree diff]5t =ms ( cone1--!       
acute-angl$1 "r-angl$1 &                
obtuse-angl$1--n notic+ t he c\ld h     
obta9$ all ?ree f any =m ( "r circul>   
cone4 ,x is int]e/+ to see ! f>-rea*+   
influ5ce ( ,plato4 ,:ile prim>ily       
int]e/$ 9 philosophy1 he laid ! fir/    
sci5tific f\ndations = a sy/em (        
ma!matics1 & 8 pupils 7 ! lead]s 9 ?is  
sci5ce 9 ! g5]ation foll[+ 8 greate/    
activity4  ,! great successor ( ,plato  
at ,a!ns 0 ,ari/otle1 ! tea*] ( ,alex&] 
! ,great4 ,he also 0 m int]e/$ 9        
philosophy ?an 9 ma!matics1 b 9 natural 
r ?an m5tal philosophy4 ,) him comes !  
fir/ application ( ma!matics to physics 
9 ! h&s ( a great man1 & ) notewor?y    

results4 ,he seems to h be5 ! fir/ to   
repres5t an unkn[n quantity by lett]s4  
,he set =? ! !ory ( ! p>allelogram (    
=ces1 us+ only rectangul> compon5ts1    
h[ev]4 ,to "o ( 8 pupils1 ,eudemus (    
,rhodes1 we are indebt$ = a hi/ory (    
anci5t geometry1 "s fragm5ts ( :i* h    
come d[n to u4  ,! fir/ great textbook  
on geometry1 & ! greate/ "o t has ev]   
appe>$1 0 writt5 by ,euclid1 :o taught  
ma!matics 9 ! great univ]sity at        
,alex&ria1 ,egypt1 ab\t #cjj ,b4,c4     
,alex&ria 0 !n practically a ,greek     
city1 hav+ be5 nam$ 9 honor ( ,alex&] ! 
,great1 & be+ rul$ by ! ,greeks4  ,9 8  
"w ,euclid plac$ all ( ! lead+          
propositions ( plane geometry !n kn[n1 &
 arrang$ !m 9 a logical ord]4 ,mo/      
geometries ( any importance writt5 s9ce 
8 "t h be5 bas$ ^u ,euclid1 improv+ !   
sequ5ce1 symbols1 & word+ z occasion    
dem&$4 ,he also wrote ^u o!r bran*es (  
ma!matics besides elem5t>y geometry1    
includ+ a "w on optics4 ,he 0 n a great 
creator ( ma!matics1 b 0 r a compil] ( !

 "w ( o!rs1 an (fice q z difficult to   
fill & q z honorable4  ,euclid did n    
give mu* solid geometry because n mu* 0 
kn[n !n4 ,x 0 to ,ar*im$es 7#bhg-bab    
,b4,c471 a fam\s ma!matician (          
,syracuse1 on ! isl& ( ,sicily1 t "s ( !
 mo/ important propositions ( solid     
geometry are due1 p>ticul>ly ^? relat+  
to ! sph]e & cyl9d]4 ,he also %[$ h[ to 
f9d ! approximate value ( pi by a me?od 
simil> to ! "o we tea* to-day1 prov+ t !
 real value lay betwe5 #c #a_/g & #c    
#aj_/ga4 ,! /ory goes t ! sph]e & cyl9d]
 7 engrav$ ^u 8 tomb1 & ,cic]o1 visit+  
,syracuse _m ye>s aft] 8 dea?1 f\nd !   
tomb by look+ = ^! symbols4 ,ar*im$es 0 
! greate/ ma!matical physici/ ( anci5t  
times4  ,! ,greeks contribut$ little m  
to elem5t>y geometry1 al?\gh ,apollonius
 ( ,p]ga1 :o taught at ,alex&ria betwe5 
#bej & #bjj ,b4,c41 wrote ext5sively on 
conic sections1 & ,hypsicles (          
,alex&ria1 ab\t #aij ,b4,c41 wrote on   
regul> polyh$rons4 ,hypsicles 0 ! fir/  
,greek writ] :o is kn[n to h us$        

sexagesimal fractions1--! degrees1      
m9utes1 & seconds ( our angle measure4  
,z5odorus 7#ahj ,b4,c47 wrote on        
isop]imetric figures1 & 8 contempor>y1  
,nicom$es ( ,g]asa1 inv5t$ a curve kn[n 
z ! con*oid1 by means ( :i* he c\ld     
trisect any angle4 ,ano!r contempor>y1  
,diocles1 inv5t$ ! cissoid1 or ivy-%ap$ 
curve1 by means ( :i* he solv$ ! fam\s  
problem ( duplicat+ ! cube1 t is1       
con/ruct+ a cube t %\ld h twice ! volume
 ( a giv5 cube4  ,! greate/ ( ! ,greek  
a/ronom]s1 ,hipp>*us 7#ahj-abe ,b4,c471 
liv$ ab\t ?is p]iod1 & ) him beg9s      
sph]ical trigonometry z a def9ite       
sci5ce4 ,a k9d ( plane trigonometry _h  
be5 kn[n to ! anci5t ,egyptians4 ,!     
,greeks usually employ$ ! *ord ( an     
angle in/ead ( ! half *ord 7s9e71 !     
latt] hav+ be5 pref]r$ by ! lat] ,arab  
writ]s4  ,! mo/ celebrat$ ( ! lat]      
,greek physici/s 0 ,h]on ( ,alex&ria1   
=m]ly suppos$ to h liv$ ab\t #ajj       
,b4,c41 b n[ assign$ to ! fir/ c5tury   
,a4,d4 ,8 contribution to geometry 0 !  

=mula = ! area ( a triangle 9 t]ms ( its
 sides a1 b1 & c1 ) s /&+ = !           
semip]imet] #a_/b7;a  b  c74 ,! =mula is
 sqrt7s7s-a77s-b77s-c774  ,probably     
ne>ly contempor>y ) ,h]on 0 ,m5elaus (  
,alex&ria1 :o wrote a sph]ical          
trigonometry4 ,he gave an int]e/+       
proposition relat+ to plane & sph]ical  
triangles1 _! sides be+ cut by a        
transv]sal4 ,= ! plane triangle ,,abc1 !
 sides a1 b1 & c be+ cut respectively 9 
,x1 ,y1 & ,z1 ! !orem ass]ts            
sub/antially t      7,,az_/,,bz7 4      
7,,bx_/,,cx7 4 7,,cy_/,,ay7  #a4  ,! mo/
 popul> writ] on a/ronomy among !       
,greeks 0 ,ptolemy 7,claudius           
,ptolemaeus1 #hg-afe ,a4,d471 :o liv$ at
 ,alex&ria4 ,he wrote a "w entitl$      
,megale ,syntaxis 7,! ,great            
,collection71 :i* 8 foll[]s designat$ z 
,megi/os 7greate/71 on :i* acc\nt !     
,arab translators gave x ! "n ,almage/  
7al mean+ !74 ,he advanc$ ! sci5ce (    
trigonometry1 b did n contribute to     
geometry4  ,at ! close ( ! ?ird c5tury  

,pappus ( ,alex&ria 7#bie ,a4,d47 wrote 
on geometry1 & "o ( 8 !orems1 a g5]aliz$
 =m ( ! ,py?agorean proposition1 is     
m5tion$ 9 ,*apt] ,,xvi ( ?is "w4 ,only  
two o!r ,greek writ]s on geometry ne$ be
 m5tion$4 ,!on ( ,alex&ria 7#cgj        
,a4,d471 ! fa!r ( ! ,hypatia :o is !    
h]o9e ( ,*>les ,k+sley's well-kn[n      
novel1 wrote a comm5t>y on ,euclid to   
:i* we are indebt$ = "s hi/orical       
in=mation4 ,proclus 7#daj-dhe ,a4,d47   
also wrote a comm5t>y on ,euclid1 & mu* 
( our in=mation conc]n+ ! fir/ ,book (  
,euclid is due to him4  ,! ,ea/ did     
little = geometry1 al?\gh contribut+    
consid]ably to algebra4 ,! fir/ great   
,h9du writ] 0 ,aryabhatta1 :o 0 born 9  
#dgf ,a4,d4 ,he gave ! v close          
approximation = pi1 express$ 9 mod]n    
notation z #c4adaf4 ,he also gave rules 
= f9d+ ! volume ( ! pyramid & sph]e1 b  
!y 7 incorrect1 %[+ t ! ,greek ma!matics
 _h n yet rea*$ ! ,ganges4 ,ano!r ,h9du 
writ]1 ,brahmagupta 7born 9 #eih        
,a4,d471 wrote an encyclop$ia (         

ma!matics4 ,he gave a rule = f9d+       
,py?agorean numb]s1 express$ 9 mod]n    
symbols z foll[s3      #a_/d77p#b_/q7   
q7#b  #a_/d77p#b_/q7 - q7#b  p#b4  ,he  
also g5]aliz$ ,h]on's =mula by ass]t+ t 
! area ( an inscrib$ quadrilat]al (     
sides a1 b1 c1 d1 & semip]imet] s1 is   
sqrt77s - a77s - b77s - c77s - d774  ,! 
,arabs1 ab\t ! "t ( ! ,arabian ,nights  
,tales 7#hjj ,a4,d471 did mu* =         
ma!matics1 translat+ ! ,greek au?ors    
into _! language & also br++ le>n+ f    
,india4 ,inde$1 x is to !m t mod]n      
,europe ow$ its fir/ k ( ,euclid4 ,!y   
contribut$ no?+ ( importance to elem5t>y
 geometry1 h[ev]4  ,! greate/ ( ! ,arab 
writ]s 0 ,mohamm$ ibn ,musa al-,kh[>azmi
 7#hbj ,a4,d474 ,he liv$ at ,bagdad &   
,damascus4 ,al?\gh *iefly int]e/$ 9     
a/ronomy1 he wrote ! fir/ book be>+ ! "n
 algebra 7,al-jabr wa'l-muqabalah1      
,re/oration & ,equation71 compos$ an    
ari?metic us+ ! ,h9du num]als1#bj & paid
 mu* att5tion to geometry &             
trigonometry4  ,euclid 0 translat$ f !  

,arabic into ,lat9 9 ! twelf? c5tury1   
,greek manuscripts n be+ !n at h&1 or   
be+ neglect$ because ( ignorance ( !    
language4 ,! lead+ translators 7 ,a!lh>d
 ( ,ba? 7#aabj71 an ,engli% monk2 ,gh]>d
 ( ,cremona 7#aafj71 an ,italian monk2 &
 ,johannes ,campanus 7#abej71 *apla9 to 
,pope ,urban ,,iv4  ,! greate/ ,european
 ma!matician ( ! ,middle ,ages 0        
,leon>do ( ,pisa#ba 7ca4 #aagj-abej74   
,he 0 v influ5tial 9 mak+ !             
,h9du-,arabic num]als kn[n 9 ,europe1   
wrote ext5sively on algebra1 & 0 ! au?or
 ( "o book on geometry4 ,he contribut$  
no?+ to ! elem5t>y !ory1 h[ev]4 ,! fir/ 
edition ( ,euclid 0 pr9t$ 9 ,lat9 9     
#adhb1 ! fir/ "o 9 ,engli% appe>+ 9     
#aegj4  ,our symbols are mod]n1  & -    
fir/ appe>+ 9 a ,g]man "w 9 #adhi2  9   
,recorde's ,:et/one ( ,witte 9 #aeeg2  &
  9 ! works ( ,h>riot 7#aefj-afba72 & x 
9 a publication by ,oughtr$             
7#aegd-affj74  ,! mo/ notewor?y advance 
9 geometry 9 mod]n times 0 made by !    
great ,fr5* philosoph] ,desc>tes1 :o    

publi%$ a small "w entitl$ ,la          
,geometrie 9 #afcg4 ,f ?is spr+s ! mod]n
 analytic geometry1 a subject t has     
revolutioniz$ ! me?ods ( all ma!matics4 
,mo/ ( ! subsequ5t discov]ies 9         
ma!matics h be5 9 high] bran*es4 ,to !  
great ,swiss ma!matician ,eul]          
7#agjg-aghc7 is due1 h[ev]1 "o          
proposition t has f\nd its way into     
elem5t>y geometry1 ! "o %[+ ! relation  
betwe5 ! numb] ( edges1 v]tices1 & faces
 ( a polyh$ron4  ,!re has ( late aris5 a
 mod]n elem5t>y geometry devot$ *iefly  
to special po9ts & l9es relat+ to !     
triangle & ! circle1 & _m int]e/+       
propositions h be5 discov]$4 ,! subject 
is s ext5sive t x _c f9d any place 9 our
 cr[d$ curriculum1 & mu/ necess>ily be  
left to ! speciali/4#bb ,"s idea ( !    
nature ( ! "w may be obta9$ f a m5tion (
 a few propositions3  ,! m$ians ( a     
triangle are concurr5t 9 ! c5troid1 or  
c5t] ( gravity ( ! triangle4  ,!        
bisectors ( ! v>i\s int]ior & ext]ior   
angles ( a triangle are concurr5t by    

?rees 9 ! inc5t] or 9 "o ( ! ?ree       
exc5t]s ( ! triangle4  ,! common *ord ( 
two int]sect+ circles is a special case 
( _! radical axis1 & tang5ts to !       
circles f any po9t on ! radical axis are
 equal4  ,if ,o is ! or?oc5t] ( !       
triangle ,,abc1 & ,x1 ,y1 ,z are ! feet 
( ! p]p5dicul>s f ,a1 ,b1 ,c            
respectively1 & ,p1 ,q1 ,r are !        
mid-po9ts ( a1 b1 c respectively1 & ,l1 
,m1 ,n are ! mid-po9ts ( ,,oa1 ,,ob1    
,,oc respectively2 !n ! po9ts ,l1 ,m1   
,n2 ,p1 ,q1 ,r2 ,x1 ,y1 ,z all lie on a 
circle1 ! n9e po9ts circle4  ,9 ! tea*+ 
( geometry x adds a human int]e/ to !   
subject to m5tion occasionally "s ( !   
hi/orical facts connect$ ) x4 ,= ?is    
reason ?is brief sket* w be supplem5t$  
by _m notes ^u ! v>i\s important        
propositions z !y occur 9 ! sev]al books
 describ$ 9 ! lat] *apt]s ( ?is "w4     
,,footnotes3  #af ,x 0 publi%$ 9 ,g]man 
translation by ,a4 ,eis5lohr1 ,e9       
ma!matis*es ,h&bu* d] alt5 ,aegypt]1    
,leipzig1 #ahgg1 & 9 facsimile by !     

,briti% ,museum1 "u ! title1 ,! ,rh9d   
,papyrus1 9 #ahih4  #ag ,g5]ally kn[n z 
,rameses ,,ii4 ,he reign$ 9 ,egypt ab\t 
#acej ,b4,c4  #ah ,two excell5t works on
 ,?ales & 8 successors1 & inde$ ! be/ 9 
,engli%1 are ! foll[+3 ,g4 ,j4 ,allman1 
,greek ,geometry f ,?ales to ,euclid1   
,dubl91 #ahhi2 ,j4 ,g[1 ,a ,hi/ory (    
,greek ,ma!matics1 ,cambridge1 #ahhd4   
,on all ma!matical subjects ! be/ g5]al 
hi/ory is t ( ,m4 ,cantor1 ,ges*i*te d] 
,ma!matik1 #d vols1 ,leipzig1           
#ahhj-aijh4  #ai ,ano!r good "w on      
,greek geometry1 ) consid]able mat]ial  
on ,py?agoras1 is by ,c4 ,a4            
,brets*neid]1 ,die ,geometrie und die   
,geomet] vor ,eukleides1 ,leipzig1      
#ahgj4  #bj ,smi? & ,k>p9ski1 ,!        
,h9du-,arabic ,num]als1 ,bo/on1 #aiaa4  
#ba ,= a sket* ( 8 life see ,smi? &     
,k>p9ski1 loc4 cit4  #bb ,^? :o c>e = a 
brief description ( ?is phase ( !       
subject may consult ,j4 ,casey1 ,a      
,sequel to ,euclid1 ,dubl91 fif?        
edition1 #ahhh2 ,w4 ,j4 ,m',clell&1 ,a  

,treatise on ! ,geometry ( ! ,circle1   
,new ,york1 #ahia2 ,m4 ,simon1 ,ueb] die
 ,entwicklung d] ,elem5t>-,geometrie im 
,,xix4 ,jahrhund]t1 ,leipzig1 #aijf4    
,,*apt] ,,iv  ,,developm5t ,,( ,,!      
,,tea*+ ,,( ,,geometry   ,we "k little (
 ! tea*+ ( geometry 9 v anci5t times1 b 
we c inf] its nature f ! tea*+ t is /ill
 se5 9 ! native s*ools ( ! ,ea/4 ,"h a  
man1 le>n$ 9 any sci5ce1 w h a gr\p (   
volunt>y /ud5ts sitt+ ab\t him1 & to !m 
he w exp\nd ! tru?4 ,su* s*ools may /ill
 be se5 9 ,india1 ,p]sia1 & ,*9a1 ! ma/]
 sitt+ on a mat plac$ on ! gr\nd or on !
 floor ( a v]&a1 & ! pupils read+ al\d  
or li/5+ to 8 words ( exposition4  ,9   
,egypt geometry seems to h be5 9 e>ly   
times m]e m5suration1 conf9$ l>gely to !
 prie/ly ca/e4 ,x 0 taught to novices :o
 gave promise ( success 9 ?is subject1 &
 n to o!rs1 ! idea ( g5]al culture1 (   
tra9+ 9 logic1 ( ! cultivation ( exact  
expression1 & ( com+ 9 contact ) tru?   
be+ :olly want+4  ,9 ,greece x 0 taught 
9 ! s*ools ( philosophy1 (t5 z a g5]al  

prep>ation = philosophic /udy4 ,?us     
,?ales introduc$ x into 8 ,ionic s*ool1 
,py?agoras made x v prom95t 9 8 great   
s*ool at ,crotona 9 s\!rn ,italy 7,magna
 ,graecia71 & ,plato plac$ above ! door 
( 8 ,academia ! words1 ,let no "o       
ignorant ( geometry ent] "h1--a k9d (   
entrance exam9ation = 8 s*ool (         
philosophy4 ,9 ^! ga!r+s ( /ud5ts x is  
probable t geometry 0 taught 9 mu* ! way
 already m5tion$ = ! s*ools ( ! ,ea/1 a 
small gr\p ( /ud5ts be+ in/ruct$ by a   
ma/]4 ,pr9t+ 0 unkn[n1 papyrus 0 de>1   
p>*m5t 0 only 9 process ( inv5tion4     
,pap] su* z we "k _h n yet appe>$1 s t  
in/ruction 0 l>gely oral1 & geometric   
figures 7 drawn by a po9t$ /ick on a    
bo>d cov]$ ) f9e s&1 or on a tablet (   
wax4  ,b ) ^! crude mat]ials !re w5t an 
abundance ( "t1 s t a numb] ( great     
results 7 accompli%$ 9 spite ( !        
difficulties att5d+ ! /udy ( ! subject4 
,x is said t ,hippocrates ( ,*ios 7ca4  
#ddj ,b4,c47 wrote ! fir/ elem5t>y      
textbook on ma!matics & inv5t$ ! me?od (

 geometric r$uction1 ! replac+ ( a      
proposition to be prov$ by ano!r :i*1 :5
 prov$1 all[s ! fir/ "o to be           
demon/rat$4 ,a little lat] ,eudoxus (   
,cnidus 7ca4 #cge ,b4,c471 a pupil (    
,plato's1 us$ ! r$uctio ad absurdum1 &  
,plato is said to h inv5t$ ! me?od (    
pro( by analysis1 an elaboration ( !    
plan us$ by ,hippocrates4 ,?us ^! e>ly  
philosoph]s taught _! pupils n facts    
alone1 b me?ods ( pro(1 giv+ !m p[] z   
well z k4 ,fur!rmore1 !y taught !m h[ to
 discuss _! problems1 inve/igat+ !      
conditions "u :i* !y are capable (      
solution4 ,?is feature ( ! "w !y call$ !
 diorismus1 & x seems to h />t$ ) ,leon1
 a foll[] ( ,plato4  ,betwe5 ! "t (     
,plato 7ca4 #djj ,b4,c47 & ,euclid 7ca4 
#cjj ,b4,c47 sev]al attempts 7 made to  
arrange ! accumulat$ mat]ial ( elem5t>y 
geometry 9 a textbook4 ,plato _h laid ! 
f\ndations = ! sci5ce1 9 ! =m ( axioms1 
po/ulates1 & def9itions1 & he _h limit$ 
! in/rum5ts to ! /raight$ge & !         
compasses4 ,ari/otle 7ca4 #cej ,b4,c47  

_h paid special att5tion to ! hi/ory ( !
 subject1 ?us f9d+ out :at _h already   
be5 accompli%$1 & _h also made mu* ( !  
applications ( geometry4 ,! _w 0 !re=e  
ready = a good tea*] :o %\ld ga!r !     
mat]ial & arrange x sci5tifically4 ,aft]
 sev]al attempts to f9d ! man = su* a   
task1 he 0 discov]$ 9 ,euclid1 & to 8 "w
 ! next *apt] is devot$4  ,aft] ,euclid1
 ,ar*im$es 7ca4 #bej ,b4,c47 made 8     
great contributions4 ,he 0 n a tea*] l 8
 illu/ri\s pr$ecessor1 b he 0 a great   
discov]]4 ,he has left u1 h[ev]1 a      
/atem5t ( 8 me?ods ( inve/igation :i* is
 helpful to ^? :o tea*4 ,^! me?ods 7    
l>gely exp]im5tal1 ev5 ext5d+ to !      
weigh+ ( geometric =ms to discov] c]ta9 
relations1 ! pro( be+ giv5 lat]4 ,"h 0  
born1 p]haps1 :at has be5 call$ !       
laboratory me?od ( ! pres5t4  ,( ! o!r  
,greek tea*]s we h b little in=mation z 
to me?ods ( imp>t+ in/ruction4 ,x is n  
until ! ,middle ,ages t !re is mu* kn[n 
9 ?is l9e4 ,:atev] ( geometry 0 taught  
seems to h be5 imp>t$ by ^w ( m\? 9 !   

way ( exp\nd+ ,euclid1 & ?is 0 done 9 ! 
anci5t fa%ion4  ,! e>ly ,*ur* lead]s    
usually paid no att5tion to geometry1 b 
z "t progress$ ! quadrivium1 or f\r     
sci5ces ( ari?metic1 music1 geometry1 & 
a/ronomy1 came to rank ) ! trivium      
7gramm>1 rhetoric1 dialectics71 ! two   
mak+ up ! sev5 lib]al arts4 ,all t !re 0
 ( geometry 9 ! fir/ ?\s& ye>s (        
,*ri/ianity1 h[ev]1 at lea/ 9 ! great   
majority ( ,*ur* s*ools1 0 summ$ up 9 a 
few def9itions & rules ( m5suration4    
,g]b]t1 :o became ,pope ,sylve/] ,,ii 9 
#iii ,a4,d41 gave a new impetus to      
geometry by discov]+ a manuscript ( !   
old ,roman surveyors & a copy ( !       
geometry ( ,boe?ius1 :o p>aphras$       
,euclid ab\t #ejj ,a4,d4 ,he !reupon    
wrote a brief geometry1 & 8 elevation to
 ! papal *air t5d$ to br+ ! /udy (      
ma!matics aga9 into prom95ce4  ,geometry
 n[ began to h "s place 9 ! ,*ur*       
s*ools1 naturally ! only s*ools ( high  
rank 9 ! ,middle ,ages4 ,! /udy ( !     
subject1 h[ev]1 seems to h be5 m]ely a  

matt] ( memoriz+4 ,geometry receiv$     
ano!r impetus 9 ! book writt5 by        
,leon>do ( ,pisa 9 #abbj1 ! ,practica   
,geometriae4 ,euclid 0 also translat$   
into ,lat9 ab\t ?is "t 7/rangely en\gh1 
z already /at$1 f ! ,arabic in/ead ( !  
,greek71 & ?us ! treasury ( elem5t>y    
geometry 0 op5$ to s*ol>s 9 ,europe4 ,f 
n[ on1 until ! inv5tion ( pr9t+ 7ca4    
#adej71 num]\s writ]s on geometry appe>1
 b1 s f> z we "k1 ! me?od ( in/ruction  
rema9$ mu* z x _h always be54 ,!        
univ]sities began to appe> ab\t !       
?irte5? c5tury1 & ,sacrobosco1 a        
well-kn[n m$ieval ma!matician1 taught   
ma!matics ab\t #abej 9 ! ,univ]sity (   
,p>is4 ,9 #accf ?is univ]sity decre$ t  
ma!matics %\ld be requir$ = a degree4 ,9
 ! ?irte5? c5tury ,ox=d requir$ six     
books ( ,euclid = "o :o 0 to tea*1 b ?is
 am\nt ( "w seems to h be5 m]ely nom9al1
 = 9 #adej only two books 7 actually    
read4 ,! univ]sities ( ,prague 7f\nd$ 9 
#acej7 & ,vi5na 7/atutes ( #achi7       
requir$ mo/ ( plane geometry = ! tea*]'s

 lic5se1 al?\gh ,vi5na dem&$ b "o book =
 ! ba*elor's degree4 ,s1 9 g5]al1 !     
univ]sities ( ! ?irte5?1 f\rte5?1 &     
fifte5? c5turies requir$ less = ! degree
 ( ma/] ( arts ?an we n[ require f a    
pupil 9 our ,am]ican high s*ools4 ,on ! 
o!r h&1 ! univ]sity /ud5ts 7 y\ng] ?an  
n[1 & 7 really do+ only high s*ool "w4  
,! inv5tion ( pr9t+ made possible ! /udy
 ( geometry 9 a new fa%ion4 ,x n[ became
 possible = any "o to /udy f a book1    
:]eas be=e ?is "t in/ruction 0 *iefly by
 ^w ( m\?1 consi/+ ( an explanation (   
,euclid4 ,! fir/ ,euclid 0 pr9t$ 9      
#adhb1 at ,v5ice1 & new editions &      
v>iations ( ?is text came out frequ5tly 
9 ! next c5tury4 ,practical geometries  
became v popul>1 & ! reaction aga9/ !   
idea ( m5tal discipl9e ?reat5$ to aboli%
 ! old /yle ( text4 ,x 0 argu$ t        
geometry 0 un9t]e/+1 t x 0 n suffici5t 9
 itself1 t boys ne$$ to see ! practical 
uses ( ! subject1 t only ^? propositions
 t 7 capable ( application %\ld be      
reta9$1 t !re mu/ be a fusion betwe5 !  

dem&s ( culture & ! dem&s ( bus9ess1 & t
 e man :o /ood = ma!matical ideals      
repres5t$ an obsolete type4 ,su* writ]s 
z ,f9aeus 7#aeef71 ,b>toli 7#aehi71     
,belli 7#aefi71 & ,cataneo 7#aefg71 9 ! 
sixte5? c5tury1 & ,capra 7#afgh71       
,g>giolli 7#afee71 & _m o!rs 9 !        
sev5te5? c5tury1 ei!r directly or       
inf]5tially1 took ?is attitude t[>ds !  
subject1--exactly ! attitude t is be+   
tak5 at ! pres5t "t by a numb] ( tea*]s 
9 ! ,unit$ ,/ates4 ,z is always ! case1 
to su* an extreme did ?is movem5t lead t
 !re 0 a reaction t br\ght ! ,euclid    
type ( book aga9 to ! front1 & x has    
ma9ta9$ its prom95ce ev5 to ! pres5t4   
,! /udy ( geometry 9 ! high s*ools is   
relatively rec5t4 ,! ,gymnasium         
7classical s*ool prep>atory to !        
univ]sity7 at ,nu]nb]g1 f\nd$ 9 #aebf1 &
 ! ,ca!dral s*ool at ,wu]ttemb]g 7z %[n 
by ! curriculum ( #aeef7 seem to h _h no
 geometry be=e #afjj1 al?\gh !          
,gymnasium at ,/rassburg includ$ "s (   
?is bran* ( ma!matics 9 #aegh1 & an     

elective c\rse 9 geometry 0 (f]$ at     
,zwickau1 9 ,saxony1 9 #aeba4 ,9 !      
sev5te5? c5tury geometry is f\nd 9 a    
consid]able numb] ( second>y s*ools1 z  
at ,coburg 7#afje71 ,kurfalz 7#afae1    
elective71 ,erfurt 7#afdc71 ,go?a       
7#afje71 ,giess5 7#afje71 & num]\s o!r  
places 9 ,g]many1 al?\gh x appe>$ b     
r>ely 9 ! second>y s*ools ( ,france be=e
 ! eighte5? c5tury4 ,9 ,g]many !        
,reals*ul5--s*ools ) m sci5ce & less    
classics ?an are f\nd 9 !               
,gymnasium--came into be+ 9 ! eighte5?  
c5tury1 & consid]able ef=t 0 made to    
con/ruct a c\rse 9 geometry t %\ld be m 
practical ?an t ( ! modifi$ ,euclid4 ,at
 ! op5+ ( ! n9ete5? c5tury ! ,prussian  
s*ools 7 reorganiz$1 & f t "t on        
geometry has _h a firm position 9 !     
second>y s*ools ( all ,g]many4 ,9 !     
eighte5? c5tury "s excell5t textbooks on
 geometry appe>$ 9 ,france1 among ! be/ 
be+ t ( ,leg5dre 7#agid71 :i* influ5c$ 9
 su* a m>k$ degree ! geometries (       
,am]ica4 ,soon aft] ! op5+ ( ! n9ete5?  

c5tury ! lycees ( ,france became /rong  
in/itutions1 & geometry1 *iefly bas$ on 
,leg5dre1 0 well taught 9 ! ma!matical  
divisions4 ,a wor?y rival ( ,leg5dre's  
geometry 0 ! "w ( ,lacroix1 :o call$    
att5tion cont9ually to ! analogy betwe5 
! !orems ( plane & solid geometry1 & ev5
 w5t s f> z to sugge/ treat+ ! relat$   
propositions toge!r 9 c]ta9 cases4  ,9  
,engl& ! prep>atory s*ools1 su* z       
,rugby1 ,h>r[1 & ,eton1 did n commonly  
tea* geometry until q rec5tly1 leav+ ?is
 "w = ! univ]sities4 ,9 ,*ri/'s         
,hospital1 ,london1 h[ev]1 geometry 0   
taught z e>ly z #afha1 f a "w writt5 by 
sev]al tea*]s ( prom95ce4 ,! highe/     
class at ,h>r[ /udi$ ,euclid & vulg>    
fractions "o p]iod a week 9 #ahbi1 b    
geometry 0 n s]i\sly /udi$ be=e #ahcg4  
,9 ! ,ed9burgh ,academy z e>ly z #ahhe1 
& 9 ,rugby by #ahci1 plane geometry 0   
complet$4  ,n until #ahdd did ,h>v>d    
require any plane geometry = entrance4  
,9 #ahee ,yale requir$ only two books ( 
,euclid4 ,x 0 !re=e f #ahej to #ahge t  

plane geometry took a def9ite place 9 ! 
,am]ican high s*ool4 ,solid geometry has
 n be5 g5]ally requir$ = entrance to any
 ea/]n college1 al?\gh 9 ! ,we/ ?is is n
 ! case4 ,! ,ea/ tea*es plane geometry m
 ?or\ghly1 b all[s a pupil to ent]      
college or to g into bus9ess ) no solid 
geometry4 ,giv5 a ye> to ! subject1 x is
 possible to d little m ?an cov] plane  
geometry2 ) a ye> & a half ! solid      
geometry ought easily to be cov]$ also4 
,bibliography4 ,/amp]1 ,a ,hi/ory ( !   
,tea*+ (      ,elem5t>y ,geometry1 ,new 
,york1 #aiji1 ) a v full                
bibliography ( ! subject2 ,cajori1 ,!   
,tea*+ (      ,ma!matics 9 ! ,unit$     
,/ates1 ,wa%+ton1 #ahij2 ,cantor1       
,ges*i*te d] ,ma!matik1 ,vol4 ,,iv1 p4  
#cba1 ,leipzig1 #aijh2      ,s*ott51    
,inhalt und ,me?ode des planimetris*5   
,unt]ri*ts1      ,leipzig1 #ahij4       
,,*apt] ,v  ,,euclid   ,x is fitt+ t a  
*apt] 9 a book ^u ! tea*+ ( ?is subject 
%\ld be devot$ to ! life & labors ( !   
greate/ ( all textbook writ]s1          

,euclid1--a man ^: "n has be51 = m ?an  
two ?\s& ye>s1 a synonym = elem5t>y     
plane geometry :]ev] ! subject has be5  
/udi$4 ,& yet :5 an ef=t is made to pick
 up ! scatt]$ fragm5ts ( 8 biography1 we
 are surpris$ to f9d h[ little is kn[n (
 "o ^: fame is s univ]sal4 ,al?\gh m    
editions ( 8 "w h be5 pr9t$ ?an ( any   
o!r book save ! ,bible1#bc we d n "k :5 
he 0 born1 or 9 :at city1 or ev5 9 :at  
c\ntry1 nor d we "k 8 race1 8 p>5tage1  
or ! "t ( 8 dea?4 ,we %\ld n feel t we  
knew mu* ( ! life ( a man :o liv$ :5 !  
,magna ,*>ta 0 wre/$ f ,k+ ,john1 if our
 fir/ & only s\rce ( in=mation 0 a      
p>agraph 9 ! works ( "s hi/orian (      
to-day2 & yet ?is is ab\t ! situation 9 
respect to ,euclid4 ,proclus (          
,alex&ria1 philosoph]1 tea*]1 &         
ma!matician1 liv$ f #daj to #dhe ,a4,d41
 & wrote a comm5t>y on ! works (        
,euclid4 ,9 8 writ+s1 :i* seem to set =?
 9 amplifi$ =m 8 lectures to ! /ud5ts 9 
! ,neoplatoni/ ,s*ool ( ,alex&ria1      
,proclus makes ?is /atem5t1 & (         

,euclid's life we h little else3        
,n mu* y\ng] ?an ^!#bd is ,euclid1 :o   
put toge!r !      ,elem5ts1 collect+ _m 
( ! !orems ( ,eudoxus1      p]fect+ _m (
 ^? ( ,!aetetus1 & also demon/rat+      
) p]fect c]ta9ty :at 8 pr$ecessors _h b 
insuffici5tly prov$4 ,he fl\ri%$ 9 ! "t 
( ! fir/      ,ptolemy1 = ,ar*im$es1 :o 
closely foll[$ ?is rul]1#be      speaks 
( ,euclid4 ,fur!rmore x is relat$ t     
,ptolemy "o      "t dem&$ ( him if !re 0
 9 geometry no %ort] way      ?an t ( ! 
,elem5ts1 to :om he repli$ t !re 0      
no royal road to geometry4#bf ,he 0     
!re=e y\ng] ?an      ! pupils ( ,plato1 
b old] ?an ,erato/h5es &      ,ar*im$es2
 = ! latt] 7 contempor>y ) "o ano!r1    
z ,erato/h5es some:]e says4#bg  ,?us we 
h 9 a few l9es1 f "o :o liv$ p]haps sev5
 or eight hundr$ ye>s aft] ,euclid1     
ne>ly all t is kn[n ( ! mo/ fam\s tea*] 
( geometry t ev] liv$4 ,nev]!less1 ev5  
?is little tells u ab\t :5 he fl\ri%$1 =
 ,h]motimus & ,philippus 7 pupils (     
,plato1 :o di$ 9 #cdg ,b4,c41 :]eas     

,ar*im$es 0 born ab\t #bhg ,b4,c4 & 0   
writ+ ab\t #bej ,b4,c4 ,fur!rmore1 s9ce 
,ptolemy ,i reign$ f #cjf to #bhc       
,b4,c41 ,euclid mu/ h be5 tea*+ ab\t    
#cjj ,b4,c41 & ?is is ! date t is       
g5]ally assign$ to him4  ,euclid        
probably /udi$ at ,a!ns1 = until he     
himself assi/$ 9 transf]r+ ! c5t] (     
ma!matical culture to ,alex&ria1 x _h   
long be5 9 ! ,grecian capital1 inde$    
s9ce ! "t ( ,py?agoras4 ,moreov]1 num]\s
 attempts _h be5 made at ,a!ns to d     
exactly :at ,euclid succe$$ 9 do+1--to  
con/ruct a logical sequ5ce (            
propositions2 9 o!r words1 to write a   
textbook on plane geometry4 ,x 0 at     
,a!ns1 !re=e1 t he c\ld be/ h receiv$ ! 
inspiration to compose 8 ,elem5ts4#bh   
,aft] f9i%+ 8 education at ,a!ns x is q 
probable t he1 l o!r savants ( ! p]iod1 
0 call$ to ,alex&ria by ,ptolemy ,sot]1 
! k+1 to assi/ 9 e/abli%+ ! great s*ool 
:i* made t city ! c5t] ( ! _w's le>n+ = 
sev]al c5turies4 ,9 ?is s*ool he taught1
 & "h he wrote ! ,elem5ts & num]\s o!r  

works1 p]haps t5 9 all4  ,al?\gh !      
,greek writ]s :o may h kn[n some?+ ( !  
life ( ,euclid h little to say ( him1 ! 
,arab writ]s1 :o c\ld h kn[n no?+ save f
 ,greek s\rces1 h all[$ _! imag9ations !
 usual latitude 9 speak+ ( him & ( 8    
labors4 ,?us ,al-,qift4i1 :o wrote 9 !  
?irte5? c5tury1 has ?is to say 9 8      
biographical treatise ,ta'rikh          
al-,h4ukama3       ,euclid1 son (       
,naucrates1 gr&son ( ,z5>*us1 call$ !   
au?or ( geometry1 a ,greek by           
nationality1 domicil$ at      ,damascus1
 born at ,tyre1 mo/ le>n$ 9 ! sci5ce (  
geometry1 publi%$ a mo/ excell5t & mo/  
useful "w      entitl$ ,! ,f\ndation or 
,elem5ts ( ,geometry1 a subject 9       
:i* no m g5]al treatise exi/$ be=e among
 ! ,greeks2      nay1 !re 0 no "o ev5 ( 
lat] date :o did n walk 9      8        
foot/eps & frankly pr(ess 8 doctr9e4    
,?is is r a specim5 ( ! ,arab t5d5cy to 
manufacture hi/ory ?an a s]i\s          
contribution to ! biography ( ,euclid1 (
 ^: p]sonal hi/ory we h only ! in=mation

 giv5 by ,proclus4  ,illu/ration3       
,,euclid  ,f an old pr9t  ,euclid's     
works at once took high rank1 & !y are  
m5tion$ by v>i\s classical au?ors4      
,cic]o knew ( !m1 & ,capella 7ca4 #dgj  
,a4,d471 ,cassiodorius 7ca4 #eae        
,a4,d471 & ,boe?ius 7ca4 #dhj-ebd       
,a4,d47 7 all m or less famili> ) !     
,elem5ts4 ,) ! advance ( ! ,d>k ,ages1  
h[ev]1 le>n+ 0 held 9 less & less e/eem1
 s t ,euclid 0 f9ally =gott51 &         
manuscripts ( 8 works 7 ei!r de/roy$ or 
buri$ 9 "s remote cloi/]4 ,! ,arabs1    
h[ev]1 ^: civilization assum$ prom95ce f
 ab\t #gej ,a4,d4 to ab\t #aejj1        
translat$ ! mo/ important treatises ( ! 
,greeks1 & ,euclid's ,elem5ts among !   
re/4 ,"o ( ^! ,arabic editions an       
,engli% monk ( ! twelf? c5tury1 "o      
,a!lh>d 7,a,ethelhard7 ( ,ba?1 f\nd &   
translat$ into ,lat9 7ca4 #aabj ,a4,d474
 ,a little lat] ,gh]>d ( ,cremona       
7#aaad-aahg7 made a new translation f ! 
,arabic1 diff]+ 9 ess5tial features f t 
( ,a!lh>d1 & ab\t #abfj ,johannes       

,campanus made /ill a ?ird translation1 
also f ,arabic into ,lat94#bi ,!re is   
reason to believe t ,a!lh>d1 ,campanus1 
& ,gh]>d may all h _h access to an e>li]
 ,lat9 translation1 s9ce all are q alike
 9 "s p>ticul>s :ile div]g+ noticeably 9
 o!rs4 ,inde$1 !re is an old ,engli%    
v]se t relates3      ,! cl]k ,euclide on
 ?is wyse hit fonde     ,?ys craft (    
gemetry yn ,egypte londe 444     ,?ys   
craft com into ,engl&1 z y y[ say1      
,yn tyme ( good ,kyng ,adel/one's day4  
,if ?is be true1 ,euclid 0 kn[n 9 ,engl&
 z e>ly z #ibd-idj ,a4,d4  ,)\t go+ into
 p>ticul>s fur!r1 x suffices to say t ! 
mod]n k ( ,euclid came fir/ ?r\gh !     
,arabic into ! ,lat91 & ! fir/ pr9t$    
edition ( ! ,elem5ts 7,v5ice1 #adhb7 0 !
 ,campanus translation4 ,greek          
manuscripts n[ began to appe>1 & at !   
pres5t "t sev]al are kn[n4 ,!re is a    
manuscript ( ! n9? c5tury 9 ! ,bodleian 
libr>y at ,ox=d1 "o ( ! t5? c5tury 9 !  
,vatican1 ano!r ( ! t5? c5tury 9        
,flor5ce1 "o ( ! elev5? c5tury at       

,bologna1 & two ( ! twelf? c5tury at    
,p>is4 ,!re are also fragm5ts conta9+   
bits ( ,euclid 9 ,greek1 & go+ back z f>
 z ! second & ?ird c5tury ,a4,d4 ,! fir/
 mod]n translation f ! ,greek into !    
,lat9 0 made by ,zamb]ti 7or            
,zamb]to71#cj & 0 pr9t$ at ,v5ice 9     
#aeac4 ,! fir/ translation into ,engli% 
0 made by ,sir ,h5ry ,bill+sley & 0     
pr9t$ 9 #aegj1 sixte5 ye>s be=e he      
became ,"l ,mayor ( ,london4  ,proclus1 
9 8 comm5t>y ^u ,euclid's "w1 rem>ks3   
,9 ! :ole ( geometry !re are c]ta9 lead+
 !orems1      be>+ to ^? :i* foll[ !    
relation ( a pr9ciple1      all-p]vad+1 
& furni%+ pro(s ( _m prop]ties4 ,su*    
!orems are call$ by ! "n ( elem5ts1 & _!
      function may be comp>$ to t ( !   
lett]s ( ! alphabet      9 relation to  
language1 lett]s be+ inde$ call$ by !   
same "n 9 ,greek ,greek3 /oi*eia1       
/oi*eia4#ca  ,?is *>act]izes ! "w (     
,euclid1 a collection ( ! basic         
propositions ( geometry1 & *iefly (     
plane geometry1 arrang$ 9 logical       

sequ5ce1 ! pro( ( ea* dep5d+ ^u "s      
prec$+ proposition1 def9ition1 or       
assumption 7axiom or po/ulate74 ,! numb]
 ( ! propositions ( plane geometry      
includ$ 9 ! ,elem5ts is n entirely      
c]ta91 ow+ to "s disagreem5t 9 !        
manuscripts1 b x 0 betwe5 "o hundr$     
sixty & "o hundr$ sev5ty-five4 ,x is    
possible to r$uce ?is numb] by ab\t     
?irty or =ty1 because ,euclid includ$ a 
c]ta9 am\nt ( geometric algebra2 b      
beyond ?is we _c safely g 9 ! way (     
elim9ation1 s9ce f ! v nature ( !       
,elem5ts ^! propositions are basic4 ,!  
ef=ts at revis+ ,euclid h be5 g5]ally   
conf9$1 !re=e1 to re>rang+ 8 mat]ial1 to
 r5d]+ m mod]n 8 phraseology1 & to mak+ 
a book t is m usable ) beg9n]s if n m   
logical 9 its pres5tation ( ! subject4  
,:ile !re has be5 an improvem5t ^u      
,euclid 9 ! art ( bookmak+1 & 9 m9or    
matt]s ( phraseology & sequ5ce1 !       
educational ga9 has n be5 comm5surate ) 
! ef=t put =?4 ,) a little modification 
( ,euclid's semi-algebraic ,book ,,ii & 

( 8 treatm5t ( proportion1 ) "s scatt]+ 
( ! def9itions & ! inclusion (          
well-grad$ ex]cises at prop] places1 & )
 att5tion to ! mod]n sci5ce ( bookmak+1 
! ,elem5ts w\ld answ] q z well = a      
textbook to-day z mo/ ( our mod]n       
sub/itutes1 & mu* bett] ?an "s ( !m4 ,x 
w\ld1 moreov]1 h ! advantage ( be+ a    
classic1--some:at ! same advantage t    
comes f read+ ,hom] 9 ! orig9al in/ead (
 f ,pope's metrical translation4 ,?is is
 n a plea = a return to ! ,euclid text1 
b = a recognition ( ! excell5ce (       
,euclid's "w4  ,! di/9ctive feature (   
,euclid's ,elem5ts1 comp>$ ) ! mod]n    
,am]ican textbook1 is p]haps ?is3       
,euclid beg9s a book ) :at seems to him 
! easie/ proposition1 be x !orem or     
problem2 ^u ?is he builds ano!r2 ^u ^! a
 ?ird1 & s on1 conc]n+ himself b little 
) ! classification ( propositions4      
,fur!rmore1 he arranges 8 propositions s
 z to con/ruct 8 figures be=e us+ !m4   
,we1 on ! o!r h&1 make "s little attempt
 to classify our propositions )9 ea*    

book1 & we make no attempt to con/ruct  
our figures be=e us+ !m1 or at lea/ to  
prove t ! con/ructions are correct4     
,inde$1 we g s f> z to /udy ! prop]ties 
( figures t we _c con/ruct1 z :5 we ask 
= ! size ( ! angle ( a regul> heptagon4 
,?us ,euclid beg9s ,book ,i by a        
problem1 to con/ruct an equilat]al      
triangle on a giv5 l9e4 ,8 object is to 
foll[ ?is by problems on draw+ a /raight
 l9e equal to a giv5 /raight l9e1 &     
cutt+ (f f ! great] ( two /raight l9es a
 l9e equal to ! less4 ,he n[ introduces 
a !orem1 :i* might equally well h be5 8 
fir/ proposition1 namely1 ! case ( !    
congru5ce ( two triangles1 hav+ giv5 two
 sides & ! includ$ angle4 ,by means ( 8 
?ird & f\r? propositions he is n[ able  
to prove ! pons as9orum1 t ! angles at !
 base ( an isosceles triangle are equal4
 ,we1 on ! o!r h&1 seek to gr\p our     
propositions :]e ?is c conv5i5tly be    
done1 putt+ ! congru5t propositions     
toge!r1 ^? ab\t inequalities by         
!mselves1 & ! propositions ab\t p>allels

 9 "o set4 ,! results ( ! two           
arrangem5ts are n radically diff]5t1 & !
 effect ( ei!r ^u ! pupil's m9d does n  
seem p>ticul>ly bett] ?an t ( ! o!r4    
,tea*]s :o h us$ bo? plans q commonly   
feel t1 ap>t f ,books ,,ii & ,v1 ,euclid
 is ne>ly z easily und]/ood z our mod]n 
texts1 if pres5t$ 9 z satisfactory      
dress4  ,! topics treat$ & ! numb] (    
propositions 9 ! plane geometry ( !     
,elem5ts are z foll[s3      ,book ,i4   
,rectil9e> figures         #dh     ,book
 ,,ii4  ,geometric algebra           #ad
     ,book ,,iii4 ,circles              
#cg     ,book ,,iv4  ,problems ab\t     
circles      #af     ,book ,v4          
,proportion                  #be        
,book ,,vi4  ,applications ( proportion 
#cc                                     
----                                    
#agc  ,( ^! we n[ omit ,euclid's ,book  
,,ii1 because we h an algebraic         
symbolism t 0 unkn[n 9 8 "t1 al?\gh he  
w\ld n h us$ x 9 geometry ev5 _h x be5  
kn[n4 ,?us 8 fir/ proposition 9 ,book   

,,ii is z foll[s3       ,if !re be two  
/raight l9es1 & "o ( !m be cut into any 
numb] ( segm5ts :atev]1 ! rectangle     
conta9$ by ! two      /raight l9es is   
equal to ! rectangles conta9$ by !      
uncut /raight l9e & ea* ( ! segm5ts4    
,?is am\nts to say+ t if x  p  q  r     
4441 !n ax  ap  aq  ar  4444 ,we also   
mat]ially simplify ,euclid's ,book ,v4  
,he1 = example1 proves t ,if f\r        
magnitudes be proportional1 !y w also be
 proportional alt]nately4 ,?is he proves
 g5]ally = any k9d ( magnitude1 :ile we 
m]ely prove x = numb]s hav+ a common    
measure4 ,we say t we may sub/itute = ! 
old] =m ( proportion1 namely1      a 3 b
  c 3 d1      ! fractional =m   a_/b    
c_/d4      ,f ?is we h     ad  bc4      
,:5ce                a_/c  b_/d4  ,9 ?is
 "w we assume t we may multiply equals  
by b & d4 ,b suppose b & d are cubes1 ( 
:i*1 inde$1 we d n ev5 "k ! approximate 
num]ical measure2 :at %all we d8 ,to    
,euclid ! multiplication by a cube or a 
polygon or a sph]e w\ld h be5 entirely  

mean+less1 z x always is f ! /&po9t (   
pure geometry4 ,h5ce x is t our treatm5t
 ( proportion has no s]i\s /&+ 9        
geometry z comp>$ ) ,euclid's1 & our    
only ju/ification = x lies 9 ! fact t x 
is easi]4 ,euclid's treatm5t is mu* m   
rigor\s ?an ours1 b x is adapt$ to !    
compreh5sion ( only advanc$ /ud5ts1 :ile
 ours is m]ely a confession1 & x %\ld be
 a frank confession1 ( ! weakness ( our 
pupils1 & possibly1 at times1 (         
ourselves4  ,if we %\ld take ,euclid's  
,books ,,ii & ,v = grant$1 or z         
suffici5tly evid5t f our /udy ( algebra1
 we %\ld h rema9+ only "o hundr$        
?irty-f\r propositions1 mo/ ( :i* may be
 designat$ z basal propositions ( plane 
geometry4 ,revise ,euclid z we w1 we    
%all n be able to elim9ate any l>ge     
numb] ( 8 fundam5tal tru?s1 :ile we     
might d mu* worse ?an to adopt ^! "o    
hundr$ ?irty-f\r propositions 9 toto z !
 bases1 & inde$ z ! def9ition1 (        
elem5t>y plane geometry4                
,bibliography4 ,hea?1 ,! ,?irte5 ,books 

( ,euclid's ,elem5ts1      #c vols41    
,cambridge1 #aijh2 ,frankl&1 ,! ,fir/   
,book ( ,euclid1      ,cambridge1 #aijf2
 ,smi?1 ,diction>y ( ,greek & ,roman    
,biography1 article ,eukleides2 ,simon1 
,euclid und die se*s      planimetris*5 
,bue*]1 ,leipzig1 #aija2 ,g[1 ,hi/ory ( 
,greek      ,ma!matics1 ,cambridge1     
#ahhd1 & any ( ! /&>d hi/ories      (   
ma!matics4 ,bo? ,hea? & ,simon give     
ext5sive      bibliographies4 ,! late/  
/&>d ,greek & ,lat9 texts are           
,heib]g's1 publi%$ by ,teubn] (         
,leipzig4  ,,footnotes3  #bc ,ricc>di1  
,saggio di una bibliografia ,euclidea1  
,"p ,i1 p4 #c1 ,bologna1 #ahhg4 ,ricc>di
 li/s well t[>ds two ?\s& editions4  #bd
 ,h]motimus ( ,colophon & ,philippus (  
,m5de4  #be ,lit]ally1 ,:o closely      
foll[$ ! fir/1 i4e4 ! fir/ ,ptolemy4    
#bf ,m5ae*mus is said to h repli$ to a  
simil> "q ( ,alex&] ! ,great3 ,o ,k+1   
?r\gh ! c\ntry !re are royal roads &    
roads = common citiz5s1 b 9 geometry !re
 is "o road = all4  #bg ,?is is also %[n

 9 a lett] f ,ar*im$es to ,erato/h5es1  
rec5tly discov]$ by ,heib]g4  #bh ,on   
?is phase ( ! subject1 & inde$ ^u       
,euclid & 8 propositions & works 9      
g5]al1 consult ,t4 ,l4 ,hea?1 ,! ,?irte5
 ,books ( ,euclid's ,elem5ts1 #c vols41 
,cambridge1 #aijh1 a ma/]ly treatise (  
:i* frequ5t use has be5 made 9 prep>+   
?is "w4  #bi ,a contempor>y copy ( ?is  
translation is n[ 9 ! libr>y ( ,george  
,a4 ,plimpton1 ,esq41 ( ,new ,york4 ,see
 ! au?or's ,r>a ,ari?metica1 p4 #dcc1   
,bo/on1 #aiji4  #cj ,a beautiful vellum 
manuscript ( ?is translation is 9 !     
libr>y ( ,george ,a4 ,plimpton1 ,esq41 (
 ,new ,york4 ,see ! au?or's ,r>a        
,ari?metica1 p4 #dha1 ,bo/on1 #aiji4    
#ca ,hea?1 loc4 cit41 ,vol4 ,i1 p4 #aad4
     ,,*apt] ,,vi  ,,ef=ts ,,at         
,,improv+ ,,euclid   ,f "t to "t an ef=t
 is made by "s tea*]1 or association (  
tea*]s1 animat$ by a s]i\s desire to    
improve ! in/ruction 9 geometry1 to     
prep>e a new syllabus t %all m>k out "s 
royal road1 & x !re=e becomes ^? :o are 

int]e/$ 9 tea*+ to consid] ) c>e !      
results ( simil> ef=ts 9 rec5t ye>s4    
,!re are _m que/ions :i* su* an attempt 
sugge/s3 ,:at is ! real purpose ( !     
movem5t8 ,:at w ! tea*+ _w say ( !      
result8 ,%all a reckless1 ill-consid]$  
radicalism dom9ate ! ef=t1 br++ 9 a     
di/a/eful t]m9ology & symbolism m]ely = 
its novelty1 insi/+ ^u an ultralogical  
treatm5t t is beyond ! p[]s ( ! le>n]1  
re>rang+ ! subject matt] to fit "s n>r[ 
notion ( ! projectors1 seek+ to         
emasculate ma!matics by look+ only to ! 
applications1 rid+ "s little hobby 9 !  
way ( "s p>ticul> class ( ex]cises1 &   
cutt+ ! numb] ( propositions to a m9imum
 t w satisfy ! m]e dem&s ( ! artisan8   
,su* are "s ( ! que/ions t naturally    
arise 9 ! m9d ( e "o :o wi%es well = !  
anci5t sci5ce ( geometry4  ,x is n      
propos$ 9 ?is *apt] to attempt to answ] 
^! que/ions1 b r to assi/ 9 und]/&+ !   
problem by consid]+ ! results ( simil>  
attempts4 ,if x %all be f\nd t syllabi h
 be5 prep>$ "u circum/ances q z         

favorable z ^? t obta9 at pres5t1 & if  
^! syllabi h _h little or no real       
influ5ce1 !n x becomes our duty to see  
if new plans may be work$ out s z to be 
m successful ?an _! pr$ecessors4 ,if !  
old] attempts h l$ to "s good1 x is well
 to "k :at is ! nature ( ?is good1 to ! 
end t new ef=ts may also result 9 some?+
 ( b5efit to ! s*ools4  ,x is propos$ 9 
?is *apt] to call att5tion to f\r       
important syllabi1 sett+ =? briefly _!  
di/+ui%+ features & draw+ "s conclusions
 t may be helpful 9 o!r ef=ts ( ?is     
nature4  ,9 ,engl& two notewor?y        
attempts h be5 made )9 a c5tury1 look+  
to a m satisfactory sequ5ce & selection 
( propositions ?an is f\nd 9 ,euclid4   
,ea* began ) a li/ ( propositions       
arrang$ 9 prop] sequ5ce1 & ea* 0 !reaft]
 elaborat$ into a textbook4 ,nei!r      
accompli%$ fully ! purpose int5d$1 b ea*
 0 in/rum5tal 9 provok+ heal?y          
discussion & 9 improv+ ! texts f :i*    
geometry is /udi$4  ,! fir/ ( ^!        
attempts 0 made by ,pr(essor ,augu/us de

 ,morgan1 "u ! auspices ( ! ,society = !
 ,diffusion ( ,useful ,k1 & x result$ 9 
a textbook1 includ+ plane1 solid1 &     
sph]ical geometry1 9 six books4 ,accord+
 to ,de ,morgan's plan1 plane geometry  
consi/$ ( ?ree books1 ! numb] (         
propositions be+ z foll[s3      ,book   
,i4    ,rectil9e> figures               
#fj     ,book ,,ii4   ,ratio1           
proportion1 applications     #fi        
,book ,,iii4  ,! circle                 
#fe                                     
----     ,total = plane geometry        
#aid  ,( ! #aid propositions ,de ,morgan
 select$ #aad ) _! coroll>ies z necess>y
 = a beg9n] :o is tea*+ himself4  ,9    
solid geometry ! plan 0 z foll[s3       
,book ,,iv4 ,l9es 9 diff]5t planes1     
solids     conta9$ by planes            
#eb     ,book ,v4 ,cyl9d]1 cone1 sph]e  
#be     ,book ,,vi4 ,figures on a sph]e 
#db                                     
----     ,total = solid geometry        
#aai  ,( ^! #aai propositions ,de       
,morgan select$ #gf ) _! coroll>ies z   

necess>y = a beg9n]1 ?us mak+ #aij      
necess>y propositions out ( #cje        
desirable ones1 besides ! coroll>ies 9  
plane & solid geometry4 ,9 o!r words1 ( 
! desirable propositions he consid]$ t  
ab\t two ?irds are absolutely necess>y4 
,x is int]e/+ to note1 h[ev]1 t he summ$
 up ! results ( 8 labors by say+3       
,x w be f\nd t ! c\rse j laid d[n1      
except+ !      six? book ( x only1 is n 
( mu* great] ext5t1 nor v      diff]5t 9
 po9t ( matt] f t ( ,euclid1 ^:         
,elem5ts h at all times be5 ju/ly e/eem$
 a model n      only ( easy &           
progressive in/ruction 9 geometry1 b (  
accuracy & p]spicuity 9 reason+4  ,de   
,morgan's ef=t1 ess5tially t ( a        
syllabus-mak] r ?an a textbook writ]1   
al?\gh x 0 publi%$ "u ! patronage ( a   
prom95t society ) :i* 7 associat$ !     
names ( m5 l ,h5ry ,hallam1 ,r[l& ,hill1
 ,"l ,john ,russell1 & ,george ,peacock1
 _h no app>5t influ5ce on geometry ei!r 
9 ,engl& or abroad4 ,nev]!less !        
syllabus 0 9 _m respects excell5t2 x    

re>rang$ ! matt]1 x classifi$ !         
propositions1 x improv$ "s ( !          
t]m9ology1 & x r$uc$ ! numb] ( ess5tial 
propositions2 x _h ! assi/ance ( ,de    
,morgan's en?usiasm & ( ! society ) :i* 
he 0 s prom95tly connect$1 & x 0        
circulat$ ) consid]able g5]osity ?r\gh\t
 ! ,engli%-speak+ _w2 b 9 spite ( all   
?is x is to-day practically unkn[n4  ,a 
second notewor?y attempt 9 ,engl& 0 made
 ab\t a qu>t] ( a c5tury ago by a       
society t 0 organiz$ practically = ?is v
 purpose1 ! ,association = ! ,improvem5t
 ( ,geometrical ,tea*+4 ,?is society 0  
compos$ ( _m ( ! mo/ progressive tea*]s 
9 ,engl&1 & x includ$ 9 its memb]%ip m5 
( high /&+ 9 ma!matics 9 ! univ]sities4 
,z a result ( _! labors a syllabus 0    
prep>$1 :i* 0 elaborat$ into a textbook1
 & 9 #ahhi a revis$ syllabus 0 issu$4   
,z to ! arrangem5t ( matt]1 ! syllabus  
dep>ts f ,euclid *iefly by sep>at+ !    
problems f ! !orems1 z is ! case 9 our  
,am]ican textbooks1 & 9 improv+ !       
phraseology4 ,! c\rse is prec$$ by "s   

simple ex]cises 9 ! use ( ! compasses & 
rul]1 a valuable plan t is foll[$ by _m 
( ! be/ tea*]s ev]y:]e4 ,consid]able    
att5tion is paid to logical processes   
be=e beg9n+ ! "w1 su* t]ms z            
contrapositive & obv]se1 & su* rules z !
 rule ( conv]sion & ! rule ( id5tity be+
 introduc$ be=e any propositions are    
consid]$4  ,! arrangem5t ( ! "w & !     
numb] ( propositions 9 plane geometry   
are z foll[s3      ,book ,i4    ,!      
/raight l9e        #ea     ,book ,,ii4  
,equality ( areas        #ai     ,book  
,,iii4  ,! circle               #db     
,book ,,iv4   ,ratio & proportion       
#cb     ,book ,v4    ,proportion        
#bd                                     
----     ,total = plane geometry        
#afh  ,"h1 !n1 is ! result ( sev]al ye>s
 ( labor by a some:at radical           
organization1 fo/]$ by excell5t         
ma!maticians1 & c>ri$ on 9 a c\ntry :]e 
elem5t>y geometry is held 9 highe/      
e/eem1 & :]e ,euclid 0 ?\ght unsuit$ to 
! ne$s ( ! beg9n]4 ,! numb] (           

propositions rema9s sub/antially ! same 
z 9 ,euclid1 & ! introduction ( "s      
unusable logic t5ds to c\nt]balance !   
improvem5t 9 sequ5ce ( ! propositions4  
,! report provok$ ?\ght2 x %ook !       
,euclid /ronghold2 x 0 probably         
in/rum5tal 9 br++ ab\t ! pres5t upheaval
 9 geometry 9 ,engl&1 b z a work+       
syllabus x has n appeal$ to ! _w z !    
great improvem5t ^u ,euclid's ,elem5ts t
 0 hop$ by _m ( its e>ly advocates4  ,! 
same association publi%$ lat]1 &        
republi%$ 9 #aije1 a ,report on ! ,tea*+
 ( ,geometry1 9 :i* x return$ to        
,euclid1 modify+ ! ,elem5ts by omitt+   
c]ta9 propositions1 *ang+ ! ord] & pro( 
( o!rs1 & introduc+ a few new !orems4 ,x
 seems to r$uce ! propositions to be    
prov$ 9 plane geometry to ab\t "o hundr$
 fifte51 & x recomm5ds ! omission ( !   
incomm5surable case4 ,?is numb] is1     
h[ev]1 some:at mislead+1 = ,euclid      
frequ5tly puts 9 "o proposition :at we 9
 ,am]ica1 = educational reasons1 f9d x  
bett] to treat 9 two1 or ev5 ?ree1      

propositions4 ,?is report1 !re=e1 rea*es
 ab\t ! same conclusion z to ! geometric
 facts to be ma/]$ z is rea*$ by our    
lat] textbook writ]s 9 ,am]ica4 ,x is n 
extreme1 & x /&s = good ma!matics4  ,9 !
 ,unit$ ,/ates ! influ5ce ( our e>ly w>s
 ) ,engl&1 & ! sympa?y ( ,france at t   
"t1 turn$ ! att5tion ( our s*ol>s ( a   
c5tury ago f ,cambridge to ,p>is z a    
ma!matical c5t]4 ,! influx ( ,fr5*      
ma!matics br\ght ) x su* works z        
,leg5dre's geometry 7#agid7 & ,b\rdon's 
algebra1 & made kn[n ! texts ( ,lacroix1
 ,b]tr&1 & ,bez\t4 ,leg5dre's geometry 0
 ! result ( ! ef=ts ( a great           
ma!matician at syllabus-mak+1 a natural 
?+ 9 a c\ntry t _h e>ly brok5 away f    
,euclid4 ,leg5dre *ang$ ! ,greek        
sequ5ce1 s\ght to select only           
propositions t are necess>y to a good   
und]/&+ ( ! subject1 & add$ a good c\rse
 9 solid geometry4 ,8 arrangem5t1 ) !   
numb] ( propositions z giv5 9 ! ,davies 
translation1 is z foll[s3      ,book ,i4
    ,rectil9e> figures                  

#ca     ,book ,,ii4   ,ratio &          
proportion                #ad     ,book 
,,iii4  ,! circle                       
#dh     ,book ,,iv4   ,proportions (    
figures & areas    #ea     ,book ,v4    
,polygons & circles                #ag  
----     ,total = plane geometry        
#afa  ,leg5dre made1 !re=e1 practically 
no r$uction 9 ! numb] ( ,euclid's       
propositions1 & 8 improvem5t on ,euclid 
consi/$ *iefly 9 8 sep>ation ( problems 
& !orems1 & 9 a less rigor\s treatm5t ( 
proportion :i* boys & girls c\ld        
compreh5d4 ,d',alemb]t _h dem&$ t !     
sequ5ce ( propositions %\ld be det]m9$  
by ! ord] 9 :i* !y _h be5 discov]$1 b   
,leg5dre wisely ignor$ su* an extreme & 
gave ! _w a v usable book4  ,! pr9cipal 
effect ( ,leg5dre's geometry 9 ,am]ica 0
 to make e textbook writ] 8 own         
syllabus-mak]1 & to put solid geometry  
on a m satisfactory foot+4 ,! m9ute we  
dep>t f a /&>d text l ,euclid's1 & h no 
recogniz$ exam9+ body1 e "o is free to  
set up 8 own /&>d1 always )9 ! some:at  

unc]ta9 b\nd>y prescrib$ by public      
op9ion & by ! colleges4 ,! ef=ts ( ! pa/
 few ye>s at syllabus-mak+ h be5 m]ely  
attempts to def9e ?is b\nd>y m cle>ly4  
,( ^! attempts two are especially wor?y 
( consid]ation z hav+ be5 v c>efully    
plann$ & hav+ br\ght =? su* def9ite     
results z to appeal to a l>ge numb] (   
tea*]s4 ,o!r syllabi h be5 made & are   
famili> to _m tea*]s1 b 9 po9t (        
cle>ness ( purpose1 concis5ess (        
expression1 & =m ( publication !y h n   
be5 su* z to comp>e ) ! two 9 "q4  ,!   
fir/ ( ^! is ! ,h>v>d syllabus1 :i* is  
plac$ 9 ! h&s ( /ud5ts = ref]5ce :5 try+
 ! entrance exam9ations ( t univ]sity1 a
 plan n foll[$ else:]e4 ,x sets =? !    
basal propositions t %\ld =m ! ess5tial 
"p ( ! /ud5t's prep>ation1 & t are      
necess>y & suffici5t = prov+ any orig9al
 proposition 7to take ! common          
expression7 t may be set on !           
exam9ation4 ,! propositions are arrang$ 
by books z foll[s3      ,book ,i4       
,angles1 triangles1 p>allels         #be

     ,book ,,ii4   ,! circle1 angle     
measure            #ah     ,book ,,iii4 
,simil> polygons                     #aj
     ,book ,,iv4   ,area ( polygons     
#h     ,book ,v4    ,polygons & circle  
measure          #aa     ,con/ructions  
#ba     ,ratio & proportion             
#f                                      
----     ,total = plane geometry        
#ii  ,! total = solid geometry is #gi   
propositions1 or #agh = bo? plane &     
solid geometry4 ,?is is p]haps ! mo/    
successful attempt t has be5 made at    
rea*+ a m9imum numb] ( propositions4 ,x 
might well be fur!r r$uc$1 s9ce x       
includes ! proposition ab\t two adjac5t 
angles =m$ by "o l9e meet+ ano!r1 & ! "o
 ab\t ! circle z ! limit ( ! inscrib$ & 
circumscrib$ regul> polygons4 ,! fir/ ( 
^! leads a beg9n] to d\bt ! value (     
geometry1 & ! second is beyond ! p[]s ( 
! majority ( /ud5ts4 ,z comp>$ ) !      
syllabus report$ by a ,wiscons9         
committee 9 #aijd1 = example1 "h are #ii
 propositions aga9/ #acb4 ,on ! o!r h&1 

a committee appo9t$ by ! ,c5tral        
,association ( ,sci5ce & ,ma!matics     
,tea*]s report$ 9 #aiji a syllabus ) :at
 seems at fir/ sight to be a li/ ( only 
#ei propositions 9 plane geometry4 ,?is 
numb] is fictiti\s1 h[ev]1 = ! reason t 
num]\s conv]ses are indicat$ ) !        
propositions1 & are n includ$ 9 ! c\nt1 
& directions are giv5 to include relat$ 
!orems & problems deal+ ) ! l5g? & area 
( a circle1 s t 9 "s cases "o           
proposition is evid5tly int5d$ to cov]  
sev]al o!rs4 ,?is syllabus is !re=e     
lack+ 9 def9it5ess1 s t ! ,h>v>d li/ /&s
 out z p]haps ! be/ ( its type4  ,!     
second notewor?y rec5t attempt 9 ,am]ica
 is t made by a committee ( !           
,association ( ,ma!matical ,tea*]s 9    
,new ,engl&4 ,?is committee 0 organiz$ 9
 #aijd4 ,x held sixte5 meet+s & c>ri$ on
 a great deal ( correspond5ce4 ,z a     
result1 x prep>$ a syllabus arrang$ by  
topics1 ! propositions ( solid geometry 
be+ gr\p$ imm$iately aft] ! correspond+ 
ones ( plane geometry4 ,= example1 ! n9e

 propositions on congru5ce 9 a plane are
 foll[$ by n9e on congru5ce 9 space4 ,z 
a result1 ! foll[+ summ>izes ! "w 9     
plane geometry3      ,congru5ce 9 a     
plane              #i     ,equival5ce   
#c     ,p>allels & p]p5dicul>s       #i 
,symmetry                          #bj  
,angles                            #ae  
,tang5ts                           #d   
,simil> figures                   #ah   
,inequalities                       #h  
,l5g?s & areas                 #ag      
,loci                               #b  
,concurr5t l9es                   #e    
----     ,total = plane geometry        
#aaj  ,n s conv5tional 9 arrangem5t z ! 
,h>v>d syllabus1 & ) a few propositions 
t are evid5tly n basal to ! same ext5t z
 ! re/1 ! li/ is nev]!less a v          
satisfactory "o1 & ! p>allelism %[n     
betwe5 plane & solid geometry is        
sugge/ive to bo? /ud5t & tea*]4  ,on !  
:ole1 h[ev]1 ! ,h>v>d selection ( basal 
propositions is p]haps z satisfactory z 
any t has be5 made1 ev5 ?\gh x appe>s to

 lack a factor ( safety1 & x is probable
 t any fur!r r$uction w\ld be unwise4   
,:at1 n[1 has be5 ! effect ( all ^!     
ef=ts8 ,:at tea*] or s*ool w\ld be      
cont5t to foll[ any "o ( ^! syllabi     
exactly8 ,:at textbook writ] w\ld feel x
 safe to limit 8 regul> propositions to 
^? 9 any "o syllabus8 ,^! que/ions      
sugge/ _! own answ]s1 & ! effect ( all  
?is ef=t seems at fir/ ?\ght to h be5 s 
slight z to be entirely out ( proportion
 to ! end 9 view4 ,?is dep5ds1 h[ev]1 on
 :at ?is end is conceiv$ to be4 ,if !   
purpose has be5 to cut out a v l>ge     
numb] ( ! propositions t are f\nd 9     
,euclid's plane geometry1 ! ef=t has n  
be5 successful4 ,we may r$uce ?is numb] 
to ab\t "o hundr$ ?irty1 b 9 g5]al1     
:atev] a syllabus may give z a m9imum1  
tea*]s w favor a l>g] numb] ?an is      
sugge/$ by ! ,h>v>d li/1 = ! purpose (  
ex]cise 9 ! read+ ( ma!matics if = no   
o!r reason4 ,! ,fr5* geomet]1 ,lacroix1 
:o wrote m ?an a c5tury ago1 propos$ to 
limit ! propositions to ^? ne$$ to prove

 o!r important ones1 & ^? ne$$ 9        
practical ma!matics4 ,if to ?is we %\ld 
add ^? t are us$ 9 treat+ a consid]able 
range ( ex]cises1 we %\ld h a li/ ( ab\t
 "o hundr$ ?irty4  ,b ?is is n ! real   
purpose ( ^! syllabi1 or at mo/ x seems 
l a relatively unimportant "o4 ,!       
purpose t has be5 atta9$ is to /op !    
indef9ite increase 9 ! numb] (          
propositions t w\ld foll[ f ! rec5t     
developm5ts 9 ! geometry ( ! triangle & 
circle1 & ( simil> mod]n topics1 if "s  
su* c\nt]-movem5t z ?is did n take      
place4 ,if ! result is1 z x probably w  
be1 to let ! basal propositions (       
,euclid rema9 ab\t z !y always h be51 z 
! /&>ds = beg9n]s1 ! syllabi w h        
accompli%$ a wor?y a*ievem5t4 ,if1 9    
addition1 !y furni% an irr$ucible m9imum
 ( propositions to :i* a /ud5t may h    
access if he desires x1 on an           
exam9ation1 z 0 int5d$ 9 ! case ( !     
,h>v>d & ! ,new ,engl& ,association     
syllabi1 ! a*ievem5t may possibly be    
/ill m wor?y4  ,9 prep>+ a syllabus1    

!re=e1 no "o %\ld hope to br+ ! tea*+ _w
 at once to agree to any great r$uction 
9 ! numb] ( basal propositions1 nor to  
agree to any radical *ange ( t]m9ology1 
symbolism1 or sequ5ce4 ,r %\ld x be !   
purpose to %[ t we h en\gh topics 9     
geometry at pres5t1 & t ! numb] (       
propositions is really great] ?an is    
absolutely necess>y1 s t tea*]s %all n  
be l$ to introduce any consid]able numb]
 ( propositions out ( ! l>ge am\nt ( new
 mat]ial t has rec5tly be5 accumulat+4  
,su* a syllabus w always accompli% a    
good purpose1 = at lea/ x w provoke     
?\ght & ar\se int]e/1 b any o!r k9d is  
b\nd to be ephem]al4#cb  ,besides !     
evolution>y attempts at re>rang+ & r$uc+
 9 numb] ! propositions ( ,euclid1 !re h
 be5 v _m revolution>y ef=ts to *ange 8 
treatm5t ( geometry entirely4 ,! great  
,fr5* ma!matician1 ,d',alemb]t1 =       
example1 9 ! eighte5? c5tury1 wi%$ to   
divide geometry into ?ree bran*es3 7#a7 
t deal+ ) /raight l9es & circles1       
app>5tly n limit$ to a plane2 7#b7 t    

deal+ ) surfaces2 & 7#c7 t deal+ )      
solids4 ,s ,m]ay 9 ,france & ,de        
,paolis#cc 9 ,italy h attempt$ to fuse  
plane & solid geometry1 b h n produc$ a 
sy/em t has be5 p>ticul>ly successful4  
,m rec5tly ,b\rlet1 ,grevy1 ,borel1 &   
o!rs 9 ,france h produc$ sev]al works on
 ! elem5ts ( ma!matics t may lead to    
some?+ ( value4 ,!y place intuition to !
 front1 favor z mu* appli$ ma!matics z  
is reasonable1 to all ( :i* ,am]ican    
tea*]s w\ld g5]ally agree1 b !y claim t 
! basis ( elem5t>y geometry 9 ! future  
mu/ be ! inve/igation ( ! gr\p (        
motions4 ,x is1 ( c\rse1 possible t     
c]ta9 ( ! notions ( ! high] ma!matical  
?\ght ( ! n9ete5? c5tury may be s       
simplifi$ z to be )9 ! compreh5sion ( ! 
tyro 9 geometry1 & we %\ld be ready to  
receive all ef=ts ( ?is k9d ) op5 m9d4  
,^! writ]s h n h[ev] produc$ ! ideal "w1
 & x may s]i\sly be que/ion$ :e!r a "w  
bas$ ^u _! ideas w prove to be          
educationally any m s\nd & usable ?an ! 
labors ( su* excell5t writ]s z ,h5rici &

 ,treutle91 & ,h4 ,muell]1 & ,s*legel a 
few ye>s ago 9 ,g]many1 & ( ,v]onese 9  
,italy4 ,all su* ef=ts1 h[ev]1 %\ld be  
welcom$ & tri$ out1 al?\gh s f> z at    
pres5t appe>s !re is no?+ 9 sight to    
replace a well-arrang$1 vitaliz$1       
simplifi$ textbook bas$ ^u ! labors (   
,euclid & ,leg5dre4  ,! mo/ broad-m9d$ (
 ! great ma!maticians :o h rec5tly giv5 
att5tion to second>y problems is        
,pr(essor ,kle9 ( ,goett+54 ,he has _h !
 good s5se to look at some?+ besides !  
m]e "q ( good ma!matics4#cd ,?us he     
insi/s ^u ! psy*ologic po9t ( view1 to !
 end t ! geometry %all be adapt$ to !   
m5tal developm5t ( ! pupil1--a ?+ t is  
app>5tly ignor$ by ,m]ay 7at lea/ = !   
av]age pupil71 &1 x is to be fe>$1 by ! 
o!r rec5t ,fr5* writ]s4 ,he !n dem&s a  
c>eful selection ( ! subject matt]1 :i* 
9 our ,am]ican s*ools w\ld mean !       
elim9ation ( propositions t are n basal1
 t is1 t are n us$ = mo/ ( ! ex]cises t 
"o naturally meets 9 elem5t>y geometry &
 9 appli$ "w4 ,he fur!r insi/s ^u a     

reasonable correlation ) practical "w to
 :i* e tea*] w agree s long z ! "w is   
really or ev5 pot5tially practical4 ,&  
f9ally he asks t we look ) favor ^u !   
union ( plane & solid geometry1 & (     
algebra & geometry4 ,he does n make any 
plea = extreme fusion1 b presumably he  
asks t to :i* e "o ( op5 m9d w\ld agree1
 namely1 t :5ev] ! opportunity (f]s 9   
tea*+ plane geometry1 to op5 ! vision to
 a g5]alization 9 space1 or to !        
measurem5t ( well-kn[n solids1 or to !  
use ( ! algebra t ! pupil has le>n$1 !  
opportunity %\ld be seiz$4  ,,footnotes3
  #cb ,! au?or is a memb] ( a committee 
t has = m ?an a ye> be5 consid]+ a      
syllabus 9 geometry4 ,?is committee w   
probably report sometime dur+ ! ye>     
#aiaa4 ,at ! pres5t writ+ x seems       
dispos$ to recomm5d ab\t ! usual li/ (  
basal propositions4  #cc ,elem5ti di    
,geometria1 ,milan1 #ahhd4  #cd ,see 8  
,elem5t>ma!matik vom hoeh]5 ,/&punkt    
aus1 ,"p ,,ii1 ,leipzig1 #aiji4         
,,*apt] ,,vii  ,,! ,,textbook ,,9       

,,geometry   ,9 consid]+ ! nature ( !   
textbook 9 geometry we ne$ to be> 9 m9d 
! fact t ! subject is be+ taught to-day 
9 ,am]ica to a class ( pupils t is n    
compos$ l ! classes f\nd 9 o!r c\ntries 
or 9 e>li] g5]ations4 ,9 g5]al1 9 o!r   
c\ntries1 geometry is n taught to mix$  
classes ( boys & girls4 ,fur!rmore1 x is
 g5]ally taught to a m select gr\p (    
pupils ?an 9 a c\ntry :]e ! high s*ool &
 college are s popul> ) p 9 all ! walks 
( life4 ,9 ,am]ica x is n alone ! boy :o
 is int]e/$ 9 education 9 g5]al1 or 9   
ma!matics 9 p>ticul>1 :o /udies         
geometry1 & :o jo9s ) o!rs ( l ta/es 9  
?is pursuit1 b x is (t5 ! boy & ! girl  
:o are n compell$ to g out & "w1 & :o   
fill ! ye>s ( y\? ) a n ov]-/r5u\s s*ool
 life4 ,x is !re=e cle> t we _c hold !  
int]e/ ( su* pupils by ! /udy ( ,euclid 
alone4 ,geometry mu/1 = !m1 be less =mal
 ?an x 0 half a c5tury ago4 ,we _c      
expect to make our classes en?usia/ic   
m]ely ov] a logical sequ5ce ( prov$     
propositions4 ,x becomes necess>y to    

make ! "w m concrete1 & to give a mu*   
l>g] numb] ( simple ex]cises 9 ord] to  
create ! int]e/ t comes f indep5d5t "w1 
f a feel+ ( conque/1 & f a desire to d  
some?+ orig9al4 ,if we w\ld ca/ a glamor
 ov] ! multiplication table1 z an admir]
 ( ,macaulay has said t ! latt] c\ld d1 
we mu/ h ! facilities = s do+4  ,x !re=e
 becomes necess>y 9 weigh+ ! m]its ( a  
textbook to consid]3 7#a7 if ! numb] (  
prov$ propositions is r$uc$ to a safe   
m9imum2 7#b7 if !re is reasonable       
opportunity to apply ! !ory1 ! actual   
applications com+ be/1 h[ev]1 f ! tea*] 
z an outside int]e/2 7#c7 if !re is an  
abundance ( mat]ial 9 ! way ( simple    
ex]cises1 s9ce su* mat]ial is n s       
readily giv5 by ! tea*] z ! seem+ly     
local applications ( ! propositions to  
outdoor measurem5ts2 7#d7 if ! book     
gives a reasonable am\nt ( introductory 
"w 9 ! use ( simple & inexp5sive        
in/rum5ts1 n at t "t emphasiz+ ! =mal   
side ( ! subject2 7#e7 if !re is af=d$  
"s opportunity to see ! recreative side 

( ! subject1 & to "k a little ( ! /ory (
 geometry z x has develop$ f anci5t to  
mod]n times4  ,b ?is does n mean t !re  
is to be a geometric cataclysm4 ,x means
 t we mu/ h ! same safe1 cons]vative    
evolution 9 geometry t we h 9 o!r       
subjects4 ,geometry is n go+ to deg5]ate
 into m]e measur+1 nor is ! anci5t      
sequ5ce go+ to become a m]e hodge-podge 
)\t sy/em & ) no inc5tive to /r5u\s     
ef=t4 ,x is n[ ab\t fifte5 hundr$ ye>s  
s9ce ,proclus laid d[n :at he consid]$ !
 ess5tial features ( a good textbook1 & 
9 all ( our ef=ts at re=m we _c improve 
v mu* ^u 8 /atem5t4 ,x is ess5tial1 he  
says1 t su* a treatise %\ld be rid (    
ev]y?+ sup]flu\s1 = ! sup]flu\s is an   
ob/acle to ! acquisition ( k2 x %\ld    
select ev]y?+ t embraces ! subject &    
br+s x to a focus1 = ?is is ( ! highe/  
s]vice to sci5ce2 x mu/ h great reg>d   
bo? to cle>ness & to concis5ess1 = _!   
opposites tr\ble our und]/&+2 x mu/ aim 
to g5]alize its !orems1 = ! division ( k
 into small elem5ts r5d]s x difficult ( 

compreh5sion4  ,x be+ prefac$ t we mu/  
make ! book m concrete 9 its            
applications1 ei!r directly or by       
sugge/+ seem+ly practical outdoor "w2 t 
we mu/ increase ! numb] ( simple        
ex]cises call+ = orig9al "w2 t we mu/   
reasonably r$uce ! numb] ( prov$        
propositions2 & t we mu/ n all[ ! good (
 ! anci5t geometry to dep>t1 let u      
consid] 9 detail "s ( ! features ( a    
good1 practical1 common-s5se textbook4  
,! e>ly textbooks 9 geometry conta9$    
only ! propositions1 ) ! pro(s 9 full1  
prec$$ by li/s ( def9itions &           
assumptions 7axioms & po/ulates74 ,!re 7
 no ex]cises1 & ! pro(s 7 giv5 9 essay  
=m4 ,!n came treatises ) ex]cises1 ^!   
ex]cises be+ gr\p$ at ! end ( ! "w or at
 ! close ( ! respective books4 ,! next  
/ep 0 to ! unit page1 arrang$ 9 /eps to 
aid ! eye1 "o proposition to a page     
:5ev] ?is 0 possible4 ,"s ef=t 0 made 9 
?is direction 9 ,france ab\t two hundr$ 
ye>s ago1 b ) no success4 ,! arrangem5t 
has s mu* to comm5d x1 h[ev]1 ! pro( be+

 s mu* m easily foll[$ by ! eye ?an 0 ! 
case 9 ! old-/yle works1 t x has ( late 
be5 reviv$4 ,9 ?is respect ! ,w5twor?   
geometry 0 a pione] 9 ,am]ica1 & s      
successful 0 ! ef=t t ?is type ( page   
has be5 adopt$1 z f> z ! v>i\s writ]s 7 
able to adopt x1 9 all successful       
geometries t h appe>$ ( late ye>s 9 ?is 
c\ntry4 ,z a result1 ! ,am]ican         
textbooks on ?is subject are m helpful &
 pleas+ to ! eye ?an ^? f\nd else:]e4   
,! late/ improvem5ts 9 textbook-mak+ h  
remov$ mo/ ( ! blemi%es ( arrangem5t t  
rema9$1 scatt]+ ! ex]cises ?r\gh ! book1
 grad+ !m ) great] c>e1 & mak+ !m m     
mod]n 9 "*4 ,b ! be/ ( ! late/ works d m
 ?an ?is4 ,!y r$uce ! numb] ( prov$     
!orems & increase ! numb] ( ex]cises1 & 
!y simplify ! pro(s :5ev] possible &    
elim9ate ! mo/ difficult ( ! ex]cises ( 
tw5ty-five ye>s ago4 ,x w\ld be possible
 to c>ry ?is *ange too f> by putt+ 9    
only half z _m1 or a qu>t] z _m1 regul> 
propositions1 b x %\ld n be ! object to 
see h[ ! "w c be cut d[n1 b to see h[ x 

c be improv$4  ,:at %\ld be ! basis (   
selection ( propositions & ex]cises8    
,evid5tly ! selection mu/ include !     
great basal propositions t are ne$$ 9   
m5suration & 9 lat] ma!matics1 toge!r ) 
o!rs t are necess>y to prove !m4        
,euclid's "o hundr$ sev5ty-?ree         
propositions ( plane geometry 7 really  
upw>ds ( "o hundr$ eighty1 because he   
sev]al times comb9$ two or m 9 "o4 ,^!  
we may r$uce to ab\t "o hundr$ ?irty )  
p]fect safety1 or less ?an "o a day = a 
s*ool ye>1 b to r$uce /ill fur!r is     
undesirable z well z unnecess>y4 ,x w\ld
 n be difficult to disp5se ) a few m2   
inde$1 we might disp5se ) ?irty m if we 
%\ld set ab\t x1 al?\gh we mu/ nev] =get
 t a goodly numb] 9 addition to ^? ne$$ 
= ! logical sequ5ce are necess>y = !    
wide range ( ex]cises t are (f]$4 ,b let
 x be cle> t if we tea* #ajj in/ead (   
#acj1 our results are liable to be ab\t 
#ajj_/acj z satisfactory4 ,we may !orize
 on p$agogy z we please1 b geometry w   
pay u ab\t 9 proportion to :at we give4 

,& z to ! ex]cises1 :at is ! basis (    
selection8 ,9 g5]al1 let x be said t any
 ex]cise t pret5ds to be real %\ld be s1
 & t words tak5 f sci5ce or measurem5ts 
d n necess>ily make ! problem g5u9e4 ,to
 take a proposition & apply x 9 a mann] 
t ! _w nev] sanctions is to indulge 9   
deceit4 ,on ! o!r h&1 :olly to neglect !
 common applications ( geometry to      
h&work ( v>i\s k9ds is to miss "o ( our 
great opportunities to make ! subject   
vital to ! pupil1 to ar\se new int]e/1 &
 to give a mean+ to x t is o!rwise      
want+4 ,x %\ld always be rememb]$ t     
m5tal discipl9e1 :atev] ! phrase may    
mean1 c z readily be obta9$ f a g5u9e   
application ( a !orem z f a m]e         
geometric puzzle4 ,on ! o!r h&1 x is    
evid5t t n m ?an #be p] c5t (           
propositions h any g5u9e applications   
outside ( geometry1 & t if we are to    
attempt any applications at all1 ^! mu/ 
be s\ght ma9ly 9 ! field ( pure         
geometry4 ,9 ! ex]cises1 !re=e1 we seek 
to-day a sane & a balanc$ book1 giv+    

equal weight to !ory & to practice1 to !
 dem&s ( ! artisan & to ^? ( !          
ma!matician1 to ! applications (        
concrete sci5ce & to ^? ( pure geometry1
 ?us mak+ a fusion ( pure & appli$      
ma!matics1 ) ! latt] z prom95t z !      
supply ( g5u9e problems p]mits4 ,! old  
is n all bad & ! new is n all good1 & a 
textbook is a success 9 s f> z x selects
 boldly ! good t is 9 ! old & rejects ) 
equal boldness ! bad t is 9 ! new4  ,le/
 ! nature ( ! ex]cises ( geometry may be
 misund]/ood1 x is well t we consid] = a
 mom5t :at con/itutes a g5u9e           
application ( ! subject4 ,x is !        
ephem]al fa%ion j at pres5t 9 ,am]ica to
 call ^! g5u9e applications by ! "n (   
real problems4 ,! "n is an un=tunate    
importation1 b t is n a matt] ( s]i\s   
mom5t4 ,! important ?+ is t we %\ld "k  
:at makes a problem real to ! pupil (   
geometry1 especially z ! :ole ?+ is com+
 rapidly into disrepute ?r\gh ! mi/ak5  
zeal ( "s ( its support]s4  ,a real     
problem is a problem t ! av]age citiz5  

may sometime be call$ ^u to solve2 t1 if
 s call$ ^u1 he w solve 9 ! mann]       
indicat$2 & t is express$ 9 t]ms t are  
famili> to ! pupil4  ,?is def9ition1 :i*
 seems fairly to /ate ! conditions "u   
:i* a problem c be call$ real 9 !       
s*oolroom1 involves ?ree po9ts3 7#a7 p  
mu/ be liable to meet su* a problem2    
7#b7 9 t case !y w solve x 9 ! way      
sugge/$ by ! book2 7#c7 x mu/ be clo!d 9
 language famili> to ! pupil4 ,=        
example1 let ! problem be to f9d !      
dim5sions ( a rectangul> field1 ! data  
be+ ! area ( ! field & ! area ( a road  
f\r rods wide t is cut f ?ree sides ( ! 
field4 ,z a real problem ?is is         
ridicul\s1 s9ce no "o w\ld ev] meet su* 
a case outside ! puzzle dep>tm5t ( a    
s*oolroom4 ,aga91 if by any /ret* ( a   
vigor\s imag9ation any human be+ %\ld   
c>e to f9d ! area ( a piece ( glass1    
b\nd$ by ! arcs ( circles1 9 a ,go?ic   
w9d[ 9 ,york ,m9/]1 x is fairly c]ta9 t 
he w\ld n g ab\t x 9 ! way sugge/$ 9 "s 
( ! e>ne/ attempts t h be5 made by      

sev]al successful tea*]s to add int]e/  
to geometry4 ,& = ! ?ird po9t1 a problem
 is n real to a pupil simply because x  
relates to mom5ts ( in]tia or ! t5sile  
/r5g? ( a /eel b>4 ,inde$1 x is unreal  
precisely because x does talk ( ^! ?+s  
at a "t :5 !y are unfamili>1 & prop]ly  
s1 to ! pupil4  ,x mu/ n be ?\ght t     
puzzle problems1 & unreal problems      
g5]ally1 h no value4 ,all t is insi/$ ^u
 is t su* problems z ! above are n real1
 & t ab\t #ij p] c5t ( problems t g by  
?is "n are equally lack+ 9 ! elem5ts t  
make = reality 9 ?is s5se ( ! ^w4 ,= !  
o!r #aj p] c5t ( su* problems we %\ld be
 ?ankful1 & we %\ld endeavor to add to !
 numb]4 ,z = ! great mass1 h[ev]1 !y are
 no bett] ?an ^? t h /ood ! te/ (       
g5]ations1 & by _! pret5se !y are       
di/9ctly worse4  ,x is prop]1 h[ev]1 to 
consid] :e!r a tea*] is n ju/ifi$ 9     
relat+ 8 "w to ^? geometric =ms t are   
f\nd 9 art1 let u say 9 floor patt]ns1 9
 domes ( build+s1 9 oilclo? designs1 & !
 l1 = ! purpose ( ar\s+ int]e/1 if = no 

o!r reason4 ,! answ] is app>5t to any   
tea*]3 ,x is c]ta9ly ju/ifiable to ar\se
 ! pupil's int]e/ 9 8 subject1 & to call
 8 att5tion to ! fact t geometric design
 plays an important "p 9 art2 b we mu/  
see to x t our ef=ts accompli% ?is      
purpose4 ,to make a c\rse 9 geometry "o 
on oilclo? design w\ld be absurd1 & no?+
 m unpr(itable or depress+ c\ld be      
imag9$ 9 connection ) ?is subject4 ,(   
c\rse no "o w\ld advocate su* an        
extreme1 b x sometimes seems z if we are
 gett+ pa9fully ne> x 9 c]ta9 s*ools4   
,a pupil has a pass+ int]e/ 9 geometric 
design4 ,he %\ld le>n to use ! in/rum5ts
 ( geometry1 & he le>ns ?is mo/ easily  
by draw+ a few su* patt]ns4 ,b to keep  
him week aft] week on que/ions relat+ to
 su* designs ( h[ev] great v>iety1 &    
especially to keep him ^u designs relat+
 to only "o or two types1 is nei!r s\nd 
educational policy nor ev5 common s5se4 
,t ?is en?usia/ic tea*] or t "o succe$s 
by su* a plan is ( no significance2 x is
 ! en?usiasm t succe$s1 n ! plan4  ,!   

exp]i5ce ( ! _w is t pupils ( geometry l
 to use ! subject practically1 b t !y   
are m int]e/$ 9 ! pure !ory ?an 9 any   
fictiti\s applications1 & ?is is :y pure
 geometry has endur$1 :ile ! great mass 
( appli$ geometry t 0 br\ght =w>d "s    
?ree hundr$ ye>s ago has long s9ce be5  
=gott54 ,! "q ( ! real applications ( ! 
subject is consid]$ 9 subsequ5t *apt]s4 
,9 ,*apt] ,,vi we consid]$ ! "q ( !     
numb] ( regul> propositions to be       
expect$ 9 ! text1 & we h j consid]$ !   
nature ( ! ex]cises :i* %\ld foll[ ^?   
propositions4 ,x is well to turn our    
att5tion next to ! nature ( ! pro(s ( ! 
basal !orems4 ,%all !y appe> 9 full8    
,%all !y be m]ely sugge/$ demon/rations8
 ,%all !y be only a s]ies ( que/ions t  
lead to ! pro(8 ,%all ! pro(s be omitt$ 
entirely8 ,or %all !re be "s comb9ation 
( ^! plans8  ,! natural temptation 9 !  
n]v\s atmosph]e ( ,am]ica is to li/5 to 
! voice ( ! mob & to proce$ at once to  
lyn* ,euclid & e "o :o /&s = t = :i* !  
,elem5ts has /ood ^! two ?\s& ye>s4 ,?is

 is :at "s :o wi% to be consid]$ z      
educators t5d to d2 9 ! language ( !    
mob1 to sma% ?+s2 to call reaction>y t  
:i* does n con=m to _! ephem]al views4  
,x is s easy to be an iconocla/1 to ?9k 
t cui bono is a conclusive argum5t1 to  
say s glibly t ,raphael 0 n a great     
pa9t]1--to d any?+ b con/ruct4 ,a few   
ye>s ago e "o mu/ take up ) ! heuri/ic  
me?od develop$ 9 ,g]many half a c5tury  
back & conta9+ mu* t 0 comm5dable4 ,a   
little lat] "o :o did n believe t !     
,culture ,epo* ,!ory 0 vital 9 education
 0 look$ ^u ) pity by a consid]able     
numb] ( s]i\s educators4 ,a little lat] 
! man :o did n ?9k t ! pr9ciple (       
,conc5tration 9 education 0 a regula    
aurea 0 ?\ght to be hopeless4 ,a little 
lat] x may h be5 t ,correlation 0 ! sav+
 factor1 to be look$ ^u 9 geometry tea*+
 z a guid+ beacon1 ev5 z ! fusion ( all 
ma!matics is ! tempor>y view ( a few    
en?usia/s to-day4#ce  ,& j n[ x is      
vocational tra9+ t is ! cat* phrase1 &  
to _m ?is phrase seems to s\nd ! fun]al 

knell ( ! /&>d textbook 9 geometry4 ,b  
does x d s8 ,does ?is pres5t cry ( !    
p$agogical circle really mean t we are  
no long] to h geometry = geometry's     
sake8 ,does x mean t a panacea has be5  
f\nd = ! ills ( memoriz+ )\t und]/&+ a  
pro( 9 ! class ( a tea*] :o is s        
ineffici5t z to all[ ?is k9d ( "w to g  
on8 ,does x mean t a tea*] :o does n see
 ! human side ( geometry1 :o does n "k !
 real uses ( geometry1 & :o has no      
faculty ( mak+ pupils en?usia/ic ov]    
geometry1--t ?is tea*] is to succe$ ) "s
 scrappy1 weak1 pret5d+ apology = a real
 "w on ! subject8  ,no "o believes 9    
/upid tea*+1 9 memoriz+ a textbook1 9   
hav+ a book t does all ! "w = a pupil1  
or 9 any ( ! o!r ills ( ineffici5t      
in/ruction4 ,on ! o!r h&1 no fair-m9d$  
p]son c condemn a type ( book t has /ood
 = g5]ations until some?+ besides ! m]e 
transi5t exp]im5ts ( ! mom5t has be5    
sugge/$ to replace x4 ,let u1 = example1
 consid] ! "q ( hav+ ! basal            
propositions prov$ 9 full1 a feature t  

is s easy to condemn z lead+ to         
memoriz+4  ,! argum5t 9 favor ( a book )
 e basal proposition prov$ 9 full1 or ) 
mo/ ( !m s prov$1 ! re/ hav+ only       
sugge/ions = ! pro(1 is t ! pupil has   
be=e him /&>d =ms exhibit+ ! be/1 mo/   
succ9ct1 mo/ cle>ly /at$ demon/rations t
 geometry conta9s4 ,! demon/rations /& =
 ! same ?+ t ! type problems /& = 9     
algebra1 & are g5]ally giv5 9 full 9 !  
same way4 ,! argum5t aga9/ ! plan is t x
 takes away ! pupil's orig9ality by do+ 
all ! "w = him1 all[+ him to m]ely      
memorize ! "w4 ,n[ if all !re is to     
geometry 7 9 ! basal propositions1 ?is  
argum5t might hold1 j z x w\ld hold 9   
algebra 9 case !re 7 only ^? ex]cises t 
are solv$ 9 full4 ,b j z ?is is n ! case
 9 algebra1 ! solv$ ex]cises /&+ z types
 or z bases = ! pupil's real "w1 s !    
demon/rat$ proposition =ms a relatively 
small "p ( geometry1 /&+ z a type1 a    
basis = ! m important "p ( ! "w4        
,moreov]1 a pupil :o uses a syllabus is 
expos$ to a dang] t %\ld be consid]$1   

namely1 t ( di%one/y4 ,any textbook 9   
geometry w furni% ! pro(s ( mo/ ( !     
propositions 9 a syllabus1 :atev] *anges
 !re may be 9 ! sequ5ce1 & x is n a     
heal?y condition ( m9d t is induc$ by   
gett+ ! pro(s surreptiti\sly4 ,unless a 
tea*] has m "t = ! c\rse ?an is usually 
all[$1 he _c develop ! new "w z mu* z is
 necess>y ) only a syllabus1 & ! result 
is t a pupil gets m ( 8 "w f o!r books &
 has less "t = ex]cises4 ,! "q !re=e    
comes to ?is3 ,is x bett] to use a book 
conta9+ /&>d =ms ( pro( = ! basal       
propositions1 & h "t = solv+ a l>ge     
numb] ( orig9al ex]cises & = seek+ !    
applications ( geometry8 ,or is x bett] 
to use a book t requires m "t on ! basal
 propositions1 ) ! dang] ( di%one/y1 &  
all[s less "t = solv+ orig9als8 ,to ^!  
que/ions ! great majority ( tea*]s answ]
 9 favor ( ! textbook ) mo/ ( ! basal   
propositions fully demon/rat$4 ,9 g5]al1
 !re=e1 x is a good rule to use ! pro(s 
( ! basal propositions z models1 & to   
get ! orig9al "w f ! ex]cises4 ,unless  

we pres]ve ^! model pro(s1 or unless we 
supply !m ) a syllabus1 ! habit (       
correct1 succ9ct self-expression1 :i* is
 "o ( ! *ief assets ( geometry1 w t5d to
 become atrophi$4 ,s important is ?is   
habit t no sy/em ( education 9 :i* its  
p]=mance is neglect$ c hope or pr(ess to
 evolve m5 & wom5 :o are compet5t 9 !   
full s5se ( ! ^w4 ,s long z tea*]s (    
geometry neglect ! possibilities ( !    
subject 9 ?is respect1 s long w ! "t    
devot$ to x be 9 l>ge "p wa/$1 & s long 
w _! pupils cont9ue to imbibe ! vici\s  
idea t x is mu* m important to be able  
to d a ?+ ?an to say h[ x c be done4#cf 
,x is "h t ! *ief dang] ( syllabus-tea*+
 lies1 & x is because ( ?is pat5t fact t
 a syllabus )\t a c>efully select$ set (
 model pro(s1 or )\t ! unnecess>y       
exp5diture ( "t by ! class1 is a dang]\s
 k9d ( textbook4  ,:at %all !n be said (
 ^? books t m]ely sugge/ ! pro(s1 or t  
give a s]ies ( que/ions t lead to !     
demon/rations8 ,!re is a c]ta9          
plausibility ab\t su* a plan at fir/    

sight4 ,b x is easily se5 to h only a   
fictiti\s claim to educational value4 ,9
 ! fir/ place1 x is m]ely an attempt on 
! "p ( ! book to take ! place ( ! tea*] 
& to develop e lesson by ! heuri/ic     
me?od4 ,! que/ions are s fram$ z to     
admit1 9 mo/ cases1 ( only a s+le answ]1
 s t ?is answ] might j z well be giv5   
in/ead ( ! "q4 ,! pupil has !re=e a pro(
 requir+ no m ef=t ?an is ! case 9 !    
/&>d =m ( textbook1 b n giv5 9 ! cle>   
language ( a c>eful writ]4 ,fur!rmore1 !
 pupil is los+ "h1 z :5 he uses only a  
syllabus1 "o ( ! v ?+s t he %\ld be     
acquir+1 namely1 ! habit ( read+        
ma!matics4 ,if he met only syllabi )\t  
pro(s1 or sugge/ive geometries1 or books
 t endeavor$ to "q e pro( out ( him1 he 
w\ld be 9 a sorry plight :5 he tri$ to  
read high] ma!matics1 or ev5 o!r        
elem5t>y treatises4 ,x is = reasons su* 
z ^! t ! heuri/ic textbook has nev]     
succe$$ = any great l5g? ( "t or 9 any  
wide t]ritory4  ,& f9ally1 ^u ?is po9t1 
%all ! demon/rations be omitt$ entirely1

 leav+ only ! li/ ( propositions1--9 o!r
 words1 a pure syllabus8 ,?is has be5   
suffici5tly answ]$ above4 ,b !re is a   
modification ( ! pure syllabus t has mu*
 to comm5d itself to tea*]s (           
exceptional /r5g? & ) m confid5ce 9     
!mselves ?an is usually f\nd4 ,?is is an
 arrangem5t t beg9s l ! ord9>y textbook 
&1 aft] ! pupil has acquir$ ! =m ( pro(1
 gradually m]ges into a syllabus1 s t   
!re is no temptation to g surreptiti\sly
 to o!r books = help4 ,su* a book1 if   
work$ out ) skill1 w\ld appeal to an    
en?usia/ic tea*]1 & w\ld accompli% !    
results claim$ = ! crud] =ms ( manual   
already describ$4 ,x w\ld n be 9 g5]al z
 safe a book z ! /&>d =m1 b ) ! "r tea*]
 x w\ld br+ good results4  ,9           
conclusion1 !re are two types ( textbook
 t h any hope ( success4 ,! fir/ is ! "o
 ) all or a l>ge "p ( ! basal           
propositions demon/rat$ 9 full1 & ) ^!  
propositions n unduly r$uc$ 9 numb]4    
,su* a book %\ld give a l>ge numb] (    
simple ex]cises scatt]$ ?r\gh ! "w1 ) a 

relatively small numb] ( difficult ones4
 ,x %\ld be mod]n 9 its _s1 ) figures   
sy/ematically lett]$1 ) ea* page a unit 
z f> z possible1 & ) e pro( a model (   
cle>ness ( /atem5t & neatness ( =m4     
,above all1 x %\ld n yield to ! dem& ( a
 few :o are always look+ m]ely = some?+ 
to *ange1 nor %\ld x 9 a reaction>y _s  
return to ! old essay =m ( pro(1 :i*    
h9d]s ! pupil at ?is /age4  ,! second   
type is ! semisyllabus1 o!rwise ) all ! 
_s ( ! fir/ type4 ,9 bo? !re %\ld be an 
hone/ fusion ( pure & appli$ geometry1 )
 no ex]cises t pret5d to be practical   
)\t be+ s1 ) no =c$ applications t lead 
! pupil to measure ?+s 9 a way t w\ld   
appeal to no practical man1 ) no m]ely  
n>r[ range ( applications1 & ) no array 
( difficult t]ms f physics & eng9e]+ t  
subm]ge all ?\ght ( ma!matics 9 ! sl\gh 
( despond ( an unkn[n te*nical          
vocabul>y4 ,outdoor ex]cises1 ev5 if    
some:at primitive1 may be introduc$1 b x
 %\ld be p]fectly und]/ood t su*        
ex]cises are giv5 = ! purpose ( increas+

 ! int]e/ 9 geometry1 & !y %\ld be      
ab&on$ if !y fail ( ?is purpose4        
,bibliography4 ,= a li/ ( /&>d textbooks
 issu$ prior      to ! pres5t g5]ation1 
consult ! bibliography 9 ,/amp]1        
,hi/ory ( ! ,tea*+ ( ,geometry1 ,new    
,york1 #aijh4  ,,footnotes3  #ce ,= "s  
classes ( s*ools & "u c]ta9 circum/ances
 c\rses 9 comb9$ ma!matics are v        
desirable4 ,all t is "h insi/$ ^u is t  
any g5]al fusion all along ! l9e w\ld   
result 9 weak1 insipid1 & un9t]e/+      
ma!matics4 ,a beg9n+1 inspirational     
c\rse 9 comb9$ ma!matics has a good     
reason = be+ 9 _m high s*ools 9 spite ( 
its manife/ disadvantages1 & su* a c\rse
 may be develop$ to cov] all ( ! requir$
 ma!matics giv5 9 c]ta9 s*ools4  #cf    
,c>son1 loc4 cit41 p4 #ae4     ,,*apt]  
,,viii  ,,! ,,relation ,,( ,,algebra    
,,to ,,geometry   ,f ! /&po9t ( !ory !re
 is or ne$ be no relation :atev] betwe5 
algebra & geometry4 ,algebra 0 orig9ally
 ! sci5ce ( ! equation1 z its "n#cg     
indicates4 ,?is means t x 0 ! sci5ce (  

f9d+ ! value ( an unkn[n quantity 9 a   
/atem5t ( equality4 ,lat] x came to mean
 mu* m ?an ?is1 & ,newton spoke ( x z   
univ]sal ari?metic1 & wrote an algebra )
 ?is title4 ,at pres5t ! t]m is appli$  
to ! elem5ts ( a sci5ce 9 :i* numb]s are
 repres5t$ by lett]s & 9 :i* c]ta9      
functions are /udi$1 functions :i* x is 
n necess>y to specify at ?is "t4 ,! "w  
relates *iefly to functions involv+ !   
idea ( numb]4 ,9 geometry1 on ! o!r h&1 
! "w relates *iefly to =m4 ,inde$1 9    
pure geometry numb] plays practically no
 "p1 :ile 9 pure algebra =m plays       
practically no "p4  ,9 #afhg ! great    
,fr5* philosoph]1 ,desc>tes1 wi%+ to    
picture c]ta9 algebraic functions1 wrote
 a "w ( ab\t a hundr$ pages1 entitl$ ,la
 ,geometrie1 & 9 ?is he %[$ a           
correspond5ce betwe5 ! numb]s ( algebra 
7:i* may be express$ by lett]s7 & !     
concepts ( geometry4 ,?is 0 ! fir/ great
 /ep 9 ! analytic geometry t f9ally gave
 u ! graph 9 algebra4 ,s9ce !n !re h be5
 br\ght out f "t to "t o!r analogies    

betwe5 algebra & geometry1 always to !  
advantage ( ea* sci5ce4 ,?is has l$ to a
 desire on ! "p ( "s tea*]s to unite    
algebra & geometry into "o sci5ce1 hav+ 
simply a class 9 ma!matics )\t ^!       
special names4  ,x is well to consid] ! 
advantages & ! disadvantages ( su* a    
plan1 & to decide z to ! rational       
attitude to be tak5 by tea*]s conc]n+ ! 
"q at issue4 ,on ! side ( advantages x  
is claim$ t !re is economy ( "t & (     
en]gy4 ,if a pupil is /udy+ =mulas1 let 
! =mulas ( geometry be /udi$2 if he is  
tak+ up ratio & proportion2 let him d s 
= algebra & geometry at ! same "t2 if he
 is solv+ quadratics1 let him apply !m  
at once to c]ta9 propositions conc]n+   
secants2 & if he is prov+ t 7a  b7#b    
equals a#b  #b;ab  b#b1 let him d s by  
algebra & by geometry simultane\sly4 ,x 
is claim$ t n only is !re economy 9 ?is 
arrangem5t1 b t ! pupil sees ma!matics z
 a :ole1 & ?us acquires m ( a ma/]y ?an 
comes by our pres5t t&em arrangem5t4    
,on ! side ( disadvantages x may be ask$

 if ! same argum5ts w\ld n lead u to    
tea* ,lat9 & ,greek toge!r1 or ,lat9 &  
,fr5*1 or all ?ree simultane\sly8 ,if   
pupils %\ld decl9e n\ns 9 all ?ree      
languages at ! same "t1 le>n to c\nt 9  
all at ! same "t1 & beg9 to translate 9 
all simultane\sly1 w\ld !re n be an     
economy ( "t & ef=t1 & w\ld !re n be    
develop$ a mu* broad] view ( language8  
,n[ ! fusioni/ ( algebra & geometry does
 n l ?is argum5t1 & he says t ! cases   
are n p>allel1 & he tries to tell :y !y 
are n4 ,he dem&s t 8 oppon5t ab&on      
argum5t by analogy & advance "s positive
 reason :y algebra & geometry %\ld n be 
fus$4 ,!n 8 oppon5t says t x is n = him 
to advance any reason = :at already     
exi/s1 ! tea*+ ( ! two sep>ately2 t he  
has only to refute ! fusioni/'s         
argum5ts1 & t he has done s4 ,he ass]ts 
t algebra & geometry are z di/9ct z     
*emi/ry & biology2 t !y h a few common  
po9ts1 b n en\gh to require tea*+ !m    
toge!r4 ,he claims t to beg9 ,lat9 &    
,greek at ! same "t has always prov$ to 

be confus+1 & t ! same is true ( algebra
 & geometry4 ,he grants t unifi$ k is   
desirable1 b he argues t :5 ! f9e arts (
 music & color "w fuse1 & :5 ! natural  
sci5ces ( *emi/ry & physics are taught 9
 ! same class1 & :5 we foll[ ! decl5sion
 ( a ,g]man n\n by t ( a ,fr5* n\n & a  
,lat9 n\n1 & :5 we tea* draw+ & p5man%ip
 toge!r1 !n x is well to talk ( mix+    
algebra & geometry4  ,x is well1 be=e   
decid+ su* a "q = ourselves 7= evid5tly 
we _c decide x = ! _w71 to consid] :at  
has be5 ! result ( exp]i5ce4 ,algebra & 
geometry 7 always taught toge!r 9 e>ly  
times1 z 7 trigonometry & a/ronomy4 ,!  
,ahmes papyrus conta9s bo? primitive    
algebra & primitive geometry4 ,euclid's 
,elem5ts conta9s n only pure geometry1 b
 also a geometric algebra & ! !ory (    
numb]s4 ,! e>ly works ( ! ,h9dus (t5    
fus$ geometry & ari?metic1 or geometry &
 algebra4 ,ev5 ! fir/ great pr9t$       
comp5dium ( ma!matics1 ! ,suma (        
,paciuolo 7#adid7 conta9$ all ( !       
bran*es ( ma!matics4 ,mu* ( ?is lat]    

attempt 0 n1 h[ev]1 an example ( p]fect 
fusion1 b r ( assign+ "o set ( *apt]s to
 algebra1 ano!r to geometry1 & ano!r to 
ari?metic4 ,s fusion1 m or less p]fect1 
has be5 tri$ ov] long p]iods1 & ab&on$ z
 ea* subject grew m complete 9 itself1 )
 its own language & its peculi> symbols4
  ,b x is ass]t$ t fusion is be+ c>ri$  
on successfully to-day by m ?an "o      
en?usia/ic tea*]1 & t ?is proves !      
cont5tion t ! plan is a good "o4 ,books 
are cit$ to %[ t ! arrangem5t is        
feasible1 & classes are indicat$ :]e !  
"w is progress+ along ?is l9e4  ,:at1   
!n1 is ! conclusion8 ,t is a "q = !     
tea*] to settle1 b x is "o ^u :i* a     
writ] on ! tea*+ ( ma!matics %\ld n fe> 
to express 8 c&id op9ion4  ,x is a fact 
t ! ,greek & ,lat9 fusion is a fair     
analogy4 ,!re are reasons = x1 b !re are
 _m m aga9/ x1 ! *ief "o be+ ! confusion
 ( beg9n+ two languages at once1 & !    
le>n+ simultane\sly ( two vocabul>ies t 
mu/ be kept sep>ate4 ,x is also a fact t
 algebra & geometry are fully z di/9ct z

 physics & *emi/ry1 or *emi/ry &        
biology4 ,life may be electricity1 & a  
brief cessation ( oxidization 9 ! lungs 
br+s dea?1 b ^! facts are no reasons =  
fus+ ! sci5ces ( physics1 biology1 &    
*emi/ry4 ,algebra is prim>ily a !ory (  
c]ta9 elem5t>y functions1 a g5]aliz$    
ari?metic1 :ile geometry is prim>ily a  
!ory ( =m ) a highly ref9$ logic to be  
us$ 9 its ma/]y4 ,!y h a few ?+s 9      
common1 z _m o!r subjects h1 b !y h v _m
 m features t are peculi> to ! "o or !  
o!r4 ,! exp]i5ce ( ! _w has l$ x away f 
a simultane\s treatm5t1 & ! contr>y     
exp]i5ce ( a few en?usia/ic tea*]s (    
to-day proves only _! own p[]s to succe$
 ) any me?od4 ,x is easy to tea*        
log>i?ms 9 ! sev5? s*ool ye>1 b x is n  
good policy to d s "u pres5t conditions4
 ,s ! exp]i5ce ( ! _w is aga9/ ! plan ( 
/rict fusion1 & no argum5ts h z yet be5 
advanc$ t are likely to *ange ! _w's    
view4 ,no "o has writt5 a book comb9+   
algebra & geometry 9 ?is fa%ion t has   
help$ ! cause ( fusion a p>ticle2 on !  

contr>y1 e su* "w t has appe>$ has      
damag$ t cause by %[+ h[ unsci5tific a  
result has come f ! labor ( an          
en?usia/ic support] ( ! movem5t4  ,b !re
 is "o feature t has n be5 consid]$     
above1 & t is a s]i\s h&icap to any ef=t
 at comb9+ ! two sci5ces 9 ! high s*ool1
 & ?is is ! "q ( relative difficulty4 ,x
 is sometimes said1 9 a doctr9aire      
fa%ion1 t geometry is easi] ?an algebra1
 s9ce =m is easi] to grasp ?an function1
 & t !re=e geometry %\ld prec$e algebra4
 ,b e tea*] ( ma!matics kn[s bett] ?an  
?is4 ,he kn[s t ! simple/ =m is easi] to
 grasp ?an ! simple/ function1 b        
nev]!less t plane geometry1 z we und]/& 
! t]m to-day1 is mu* m difficult ?an    
elem5t>y algebra = a pupil ( f\rte54 ,! 
*ild /udies =m 9 ! k9d]g>t5 be=e he     
/udies numb]1 & ?is is s\nd educational 
policy4 ,he /udies =m1 9 m5suration1    
?r\gh\t 8 c\rse 9 ari?metic1 & ?is1 too1
 is good educational policy4 ,?is k9d ( 
geometry v prop]ly prec$es algebra4 ,b !
 demon/rations ( geometry1 ! /udy by    

pupils ( f\rte5 ye>s ( a geometry t 0   
writt5 = college /ud5ts & always /udi$  
by !m until ab\t fifty ye>s ago1--t is  
by no means z easy z ! /udy ( a simple  
algebraic symbolism & its application to
 easy equations4 ,if geometry is to be  
taught = ! same reasons z at pres5t1 x  
_c advantage\sly be taught e>li] ?an n[ 
)\t mu* simplification1 & x _c          
successfully be fus$ ) algebra save by  
"s tea*] :o is will+ to sacrifice an    
undue am\nt ( en]gy to no really wor?y  
purpose4 ,:5 great ma!maticians l       
,pr(essor ,kle9 speak ( ! fusion ( all  
ma!matics1 !y speak f ! /&po9t ( advanc$
 /ud5ts1 n = ! tea*] ( elem5t>y         
geometry4  ,x is !re=e probable t simple
 m5suration w cont9ue1 z a "p (         
ari?metic1 to prec$e algebra1 z at      
pres5t2 & t algebra into or ?r\gh       
quadratics w prec$e geometry1#ch draw+  
^u ! m5suration ( ari?metic z may be    
ne$$2 & t geometry w foll[ ?is "p (     
algebra1 us+ its pr9ciples z f> z       
possible to assi/ 9 ! demon/rations & to

 express & manipulate its =mulas4 ,plane
 geometry1 or else a ye> ( plane & solid
 geometry1 w probably1 9 ?is c\ntry1 be 
foll[$ by algebra1 complet+ quadratics &
 /udy+ progressions2 & by solid         
geometry1 or a supplem5t>y c\rse 9 plane
 & solid geometry1 ?is "w be+ elective 9
 _m1 if n all1 s*ools4#ci ,x is also    
probable t a g5]al review ( ma!matics1  
:]e ! fusion idea may be c>ri$ out1 w   
prove to be a feature ( ! la/ ye> ( !   
high s*ool1 & "o t w gr[ 9 popul>ity z  
"t goes on4 ,su* a plan w keep algebra &
 geometry sep>ate1 b x w all[ ea* to use
 all ( ! o!r t has prec$$ x1 & w        
enc\rage e ef=t 9 ?is direction4 ,x w   
accompli% all t a m complete fusion     
really hopes to accompli%1 & x w give   
enc\ragem5t to all :o seek to mod]nize !
 _s ( ea* ( ^! great bran*es (          
ma!matics4  ,!re is1 h[ev]1 a *ance =   
fusion 9 two classes ( s*ool1 nei!r (   
:i* is z yet well develop$ 9 ?is c\ntry4
 ,! fir/ is ! te*nical high s*ool t is  
at pres5t com+ into "s prom95ce4 ,x is n

 probable ev5 "h t ! be/ results c be   
secur$ by elim9at+ all ma!matics save   
only :at is applicable 9 ! %op1 b if ?is
 view %\ld prevail = a "t1 !re w\ld be s
 little left ( ei!r algebra or geometry 
t ea* c\ld readily be jo9$ to ! o!r4 ,! 
actual am\nt ( algebra ne$$ by a =eman 9
 a ma*9e %op c be taught 9 ab\t f\r     
lessons1 & ! geometry or m5suration t he
 ne$s c be taught 9 eight lessons at !  
mo/4 ,! necess>y trigonometry may take  
eight m1 s t x is entirely feasible to  
unite ^! ?ree subjects4 ,! boy :o takes 
su* a c\rse w\ld "k z mu* ab\t ma!matics
 z a *ild :o _h read t5 pages 9 a prim] 
w\ld "k ab\t lit]ature1 b he w\ld h     
en\gh = 8 imm$iate ne$s1 ev5 ?\gh he _h 
no appreciation ( ma!matics z a sci5ce4 
,if any "o asks if ?is is n all t !     
s*ool %\ld give him1 x might be well to 
ask if ! s*ool %\ld give only ! ability 
to read1 )\t ! k ( any good lit]ature2  
if x %\ld give only ! ability to s+1 )\t
 ! k ( good music2 if x %\ld give only !
 ability to speak1 )\t any tra9+ 9 ! use

 ( good language2 & if x %\ld give a k (
 home geography1 )\t any intimation t ! 
_w is r\nd1--an atom 9 ! unfa?omable    
univ]se ab\t u4  ,! second opportunity =
 fusion is possibly 7= x is by no means 
c]ta97 to be f\nd 9 a type ( s*ool 9 :i*
 ! only requir$ c\rses are ! initial    
ones4 ,^! s*ools h "s /rong advocates1 x
 be+ claim$ t e pupil %\ld be introduc$ 
to ! l>ge bran*es ( k & !n all[$ to     
elect ! ones 9 :i* he f9ds himself ! mo/
 int]e/$4 ,:e!r or n ?is is s\nd        
educational policy ne$ n be discuss$ at 
?is "t2 b if su* a plan 7 develop$1 x   
might be well to (f] a some:at          
sup]ficial 79 ! s5se ( abridg$7 c\rse t 
%\ld embody a little ( algebra1 a little
 ( geometry1 & a little ( trigonometry4 
,?is w\ld unconsci\sly become a bait =  
/ud5ts1 & ! result w\ld probably be "s  
good tea*+ 9 ! class 9 "q4 ,x is to be  
hop$ t we may h "s /rong1 well-consid]$ 
textbooks ^u ?is phase ( ! "w4  ,z to ! 
fusion ( trigonometry & plane geometry  
little ne$ be said1 because ! subject is

 9 ! doctr9aire /age4 ,trigonometry     
naturally foll[s ! *apt] on simil>      
triangles1 b to put x !re means1 9 our  
cr[d$ curriculum1 to elim9ate some?+ f  
geometry4 ,:i*1 !n1 is bett]1--to give  
up ! latt] portion ( geometry1 or "p ( x
 at lea/1 or to give up trigonometry8   
,"s advocates h ent]$ a plea = two or   
?ree lessons 9 trigonometry at ?is po9t1
 & ?is is a feature t any tea*] may     
introduce z a bit ( int]e/1 z is sugge/$
 9 ,*apt] ,,xvi1 j z he may give a      
popul> talk to 8 class ^u ! f\r?        
dim5sion or ! non-,euclidean geometry4  
,! la/+ impression ^u ! pupil w be      
exactly ! same z t ( f\r lessons 9      
,sanskrit :ile he is /udy+ ,lat94 ,he   
might rememb] ea* ) pleasure1 ,lat9 be+ 
relat$1 z x is1 to ,sanskrit1 &         
trigonometry be+ an outcome ( ! !ory (  
simil> triangles4 ,b t ei!r ( ^!        
dep>tures f ! regul> sequ5ce is ( any   
s]i\s ma!matical or l+ui/ic significance
 no "o w\ld feel l ass]t+4 ,ea* is      
all[able on ! score ( int]e/1 b nei!r w 

add to ! pupil's p[] 9 any ess5tial     
feature4  ,ea* ( ^! subjects is bett]   
taught by itself1 ea* us+ ! o!r z f> z  
possible & be+ foll[$ by a review t %all
 make use ( all4 ,x is n improbable t we
 may 9 due "t h high s*ools t give less 
ext5d$ c\rses 9 algebra & geometry1 add+
 brief practical c\rses 9 trigonometry &
 ! elem5ts ( ! calculus2 b ev5 9 su*    
s*ools x is likely to be f\nd t geometry
 is be/ taught by itself1 mak+ use ( all
 ! ma!matics t has prec$$ x4  ,x w (    
c\rse be und]/ood t ! fusion ( algebra &
 geometry z "h und]/ood has no?+ to d ) 
! "q ( tea*+ ! two subjects             
simultane\sly1 say two days 9 ! week =  
"o & ?ree days = ! o!r4 ,?is plan has _m
 advocates1 al?\gh on ! :ole x has n be5
 well receiv$ 9 ?is c\ntry4 ,b :at is   
meant "h is ! actual fus+ ( algebra &   
geometry day aft] day1--a plan t has z  
yet met ) only a sporadic success1 b :i*
 may be develop$ = beg9n+ classes 9 due 
"t4  ,,footnotes3  #cg ,al-jabr         
wa'l-muqabalah3 re/oration & equation is

 a fairly good translation ( ! ,arabic4 
#ch ,or be c>ri$ along at ! same "t z a 
di/9ct topic4  #ci ,) a s+le ye> =      
requir$ geometry x w\ld be bett] f e    
po9t ( view to cut ! plane geometry     
en\gh to admit a fair c\rse 9 solid     
geometry4     ,,*apt] ,,ix  ,,!         
,,introduction ,,to ,,geometry   ,!re   
are two difficult crises 9 ! geometry   
c\rse1 bo? = ! pupil & = ! tea*]4 ,^!   
crises are met at ! beg9n+ ( ! subject &
 at ! beg9n+ ( solid geometry4 ,once a  
class has fairly got into ,book ,i1 if !
 int]e/ 9 ! subject c be ma9ta9$1 !re   
are only ! incid5tal difficulties (     
logical advance ?r\gh\t ! plane         
geometry4 ,:5 ! pupil :o has be5 see+   
figures 9 "o plane = a ye> attempts to  
visualize solids f a flat draw+1 !      
second difficult place is rea*$4 ,tea*]s
 go+ ov] solid geometry f ye> to ye> (t5
 =get ?is difficulty1 b mo/ ( !m c      
easily place !mselves 9 ! pupil's       
position by look+ at ! work+ draw+s (   
any artisan1--usually simple cases 9 !  

s-call$ descriptive geometry4 ,!y w !n  
realize h[ difficult x is to visualize a
 solid f an unfamili> k9d ( picture4 ,! 
tr\ble is usually avoid$ by ! help ( a  
c\ple ( pieces ( heavy c>dbo>d or box   
bo>d1 & a few knitt+ ne$les ) :i* to    
repres5t l9es 9 space4 ,if ^! are       
judici\sly us$ 9 class = a few days1    
until ! figures are und]/ood1 ! second  
crisis is easily pass$4 ,! cont9u$ use (
 su* mat]ial1 h[ev]1 or ! daily use (   
ei!r models or photographs1 weak5s !    
pupil1 ev5 z a *ild is weak5$ by be+    
kept too long 9 a p]ambulator4 ,su*     
devices h _! place2 !y are useful :5    
ne$$1 b !y are p]nici\s :5 unnecess>y4  
,j z ! me*anic mu/ be able to make & to 
visualize 8 work+ draw+s1 s ! /ud5t (   
solid geometry mu/ be able to get on )  
p5cil & pap]1 repres5t+ 8 solid figures 
9 ! flat4  ,b ! introduction to plane   
geometry is n s easily dispos$ (4 ,!    
pupil at t "t is ent]+ a field t is     
entirely unfamili>4 ,he is only f\rte5  
or fifte5 ye>s ( age1 & 8 ?\ghts are    

di/9ctly n on geometry4 ,( logic he kn[s
 little & c>es less4 ,he is n int]e/$ 9 
a subject ( :i* he kn[s no?+1 n ev5 !   
mean+ ( its "n4 ,he asks1 naturally &   
prop]ly1 :at x all signifies1 :at       
possible use !re is = /udy+ geometry1 & 
:y he %\ld h to prove :at seems to him  
evid5t )\t pro(4 ,to pass him           
successfully ?r\gh ?is /age has tax$ !  
+5uity ( e real tea*] f ! "t ( ,euclid  
to ! pres5t2 & j z ,euclid rem>k$ to ,k+
 ,ptolemy1 8 patron1 t !re is no royal  
road to geometry1 s we may affirm t !re 
is no royal road to ! tea*+ ( geometry4 
,nev]!less ! exp]i5ce ( tea*]s c\nts = a
 great deal1 & ?is exp]i5ce has %[n t1  
aside f ! matt] ( te*nic 9 h&l+ ! class1
 c]ta9 sugge/ions are ( value1 & a few (
 ^! w n[ be set =?4  ,fir/1 z to :y     
geometry is /udi$1 x is manife/ly       
impossible successfully to expla9 to a  
boy ( f\rte5 or fifte5 ! l>g] reasons = 
/udy+ any?+ :atev]4 ,:5 we confess      
ourselves hone/ly we f9d t ^! reasons1  
:e!r 9 ma!matics1 ! natural sci5ces1    

h&work1 lett]s1 ! vocations1 or ! f9e   
arts1 are none too cle> 9 our own m9ds1 
9 spite ( any pret5ti\s language t we   
may use4 ,x is !re=e mo/ satisfactory to
 anticipate ! "q at once1 & to set !    
pupils1 = a few days1 at us+ ! compasses
 & rul] 9 ! draw+ ( geometric designs & 
( ! mo/ common figures t !y w use4 ,?is 
s]ves sev]al purposes3 x excites _!     
int]e/1 x gu>ds aga9/ ! slov5ly figures 
t s (t5 lead !m to errone\s conclusions1
 x has a g5u9e value = ! future artisan1
 & x %[s t geometry is some?+ besides   
m]e !ory4 ,:e!r ! textbook provides = x 
or n1 ! tea*] w f9d a few days ( su* "w 
well sp5t1 x be+ a simple matt] to      
supplem5t ! book 9 ?is respect4 ,!re 0 a
 "t :5 "s =m ( me*anical draw+ 0 g5]ally
 taught 9 ! s*ools1 b ?is has giv5 place
 to m g5u9e art "w1 leav+ x to ! tea*] (
 geometry to imp>t su* k ( draw+ z is a 
necess>y prelim9>y to ! regul> /udy ( ! 
subject4  ,su* "w 9 draw+ %\ld g s f>1 &
 only s f>1 z to ar\se an int]e/ 9      
geometric =m )\t becom+ we>isome1 & to  

famili>ize ! pupil ) ! use ( !          
in/rum5ts4 ,he %\ld be c\nsel$ ab\t mak+
 f9e l9es1 ab\t be+ c>eful 9 sett+ !    
po9t ( 8 compasses on ! exact c5t] t he 
wi%es to use1 & ab\t repres5t+ a po9t by
 a v f9e dot1 or1 pref]ably at fir/1 by 
two cross$ l9es4 ,unless ^! details are 
c>efully consid]$1 ! pupil w soon f9d t 
! l9es ( 8 draw+s d n fit toge!r1 & t ! 
result is n pleas+ to ! eye4 ,! figures 
"h giv5 are good ones ^u :i* to beg91 ! 
dott$ con/ruction l9es be+ eras$ aft] ! 
"w is complet$4 ,!y may be con/ruct$ ) !
 compasses & rul] alone1 or !           
draftsman's ,t-squ>e1 triangle1 &       
protractor may be us$1 al?\gh ^! latt]  
in/rum5ts are n necess>y4 ,we %\ld      
con/antly rememb] t !re is a dang] 9 !  
slavi% use ( in/rum5ts & ( su* helps z  
squ>$ pap]4       ,j z ,euclid rode     
r\gh%od ov] ! gr[+ intellects (         
boys & girls1 s may in/rum5ts ride      
r\gh%od ov] _!      gr[+ p]ceptions by  
int]f]+ ) natural & heal?y              
intuitions1 & mak+ !m ! subject (       

labori\s      measurem5t4#dj            
,illu/ration  ,! pupil :o _c see !      
equality ( v]tical angles intuitively   
bett] ?an by ! use ( ! protractor is    
abnormal4 ,nev]!less x is ! pupil's     
int]e/ t is at /ake1 toge!r ) 8 ability 
to use ! in/rum5ts ( daily life4 ,if1   
!re=e1 he c readily be suppli$ )        
draftsm5's mat]ials1 & is n compell$ to 
use !m 9 a fooli% mann]1 s mu* ! bett]4 
,!y w n hurt 8 geometry if ! tea*] does 
n int]f]e1 & !y w help 8 practical      
draw+2 b = obvi\s reasons we _c dem& t !
 pupil pur*ase :at is n really ess5tial 
to 8 /udy ( ! subject4 ,! mo/ valuable  
s+le in/rum5t ( ! ?ree j m5tion$ is !   
protractor1 & s9ce a pap] "o co/s only a
 few c5ts & is (t5 helpful 9 ! draw+ (  
figures1 x %\ld be recomm5d$ to pupils4 
,!re is also ano!r l9e ( "w t (t5 ar\ses
 a good deal ( int]e/1 namely1 ! simple 
field measures t c easily be made ab\t !
 s*ool gr\nds4 ,gu>d+ aga9/ ! ev]-pres5t
 dang] ( do+ too mu* ( su* "w1 ( do+ "w 
t has no int]e/ = ! pupil1 ( requir+ x  

done 9 a way t seems unreal to a class1 
& ( neglect+ ! ess5ce ( geometry by a   
l9e ( "w t involves no new              
pr9ciples1--su* outdoor ex]cises 9      
measurem5t h a positive value1 & a      
pl5tiful supply ( sugge/ions 9 ?is l9e  
is giv5 9 ! subsequ5t *apt]s4 ,! object 
is *iefly to furni% a motive = geometry1
 & = _m pupils ?is is q unnecess>y4 ,=  
"s1 h[ev]1 & p>ticul>ly = ! en]getic1   
re/less boy1 su* "w has be5 successfully
 (f]$ by v>i\s tea*]s z an alt]native to
 "s ( ! book "w4 ,because ( ?is value a 
consid]able am\nt ( su* "w w be sugge/$ 
= tea*]s :o may c>e to use x1 ! textbook
 be+ manife/ly n ! place = occasional   
topics ( ?is nature4  ,= ! purposes ( an
 introduction only a tape l9e ne$ be    
pur*as$4 ,wood5 p9s & a plumb l9e c     
easily be made4 ,ev5 be=e he comes to ! 
propositions 9 m5suration 9 geometry !  
pupil kn[s1 f 8 ari?metic1 h[ to f9d    
ord9>y areas & volumes1 & he may !re=e  
be set at "w to f9d ! area ( ! s*ool    
gr\nd1 or ( a field1 or ( a city block4 

,! foll[+ are among ! simple ex]cises = 
a beg9n]3  ,illu/ration  #a4 ,drive     
/akes at two corn]s1 ,a & ,b1 ( ! s*ool 
gr\nds1 putt+ a cross on top ( ea*2 or  
make ! crosses on ! sidewalk1 s z to get
 two po9ts betwe5 :i* to measure4       
,measure f ,a to ,b by hold+ ! tape taut
 & level1 dropp+ p]p5dicul>s :5 necess>y
 by means ( ! plumb l9e1 z %[n 9 !      
figure4 ,*eck ! "w by measur+ f ,b back 
to ,a 9 ! same way4 ,pupils w f9d t _!  
"w %\ld always be *eck$1 & !y w be      
surpris$ to see h[ ! results w v>y 9 su*
 a simple measurem5t z ?is1 unless v    
great c>e is tak54 ,if !y le>n ! lesson 
( accuracy ?us e>ly1 !y w h ga9$ mu*4   
,illu/ration  #b4 ,take two /akes1 ,x1  
,y1 9 a field1 pref]ably two or ?ree    
hundr$ feet ap>t1 always m>k$ on top )  
crosses s z to h exact po9ts f :i* to   
"w4 ,let x !n be requir$ to /ake out or 
range ! l9e f ,x to ,y by plac+ /akes at
 specifi$ di/ances4 ,"o boy /&s at ,y & 
ano!r at ,x1 ea* ) a plumb l9e4 ,a ?ird 
"o takes a plumb l9e & /&s at ,p1 !     

obs]v] at ,x motion+ to him to move 8   
plumb l9e to ! "r or ! left until x is  
exactly 9 l9e ) ,x & ,y4 ,a /ake is !n  
driv5 at ,p1 & ! pupil at ,x moves on to
 ! /ake ,p4 ,!n ,q is locat$ 9 ! same   
way1 & !n ,r1 & s on4 ,! "w is *eck$ by 
rang+ back f ,y to ,x4 ,9 "s ( ! simple 
ex]cises sugge/$ lat] x is necess>y to  
range a l9e s t ?is "w is useful 9 mak+ 
measurem5ts4 ,! geometric pr9ciple      
involv$ is t two po9ts det]m9e a /raight
 l9e4  ,illu/ration  ,illu/ration  #c4  
,to te/ a p]p5dicul> or to draw "o l9e  
p]p5dicul> to ano!r 9 a field1 we may   
take a /\t cord twelve feet long1 hav+ a
 knot at ! end ( e foot4 ,if ?is is laid
 along f\r feet1 ! ends ( ?is "p be+    
fix$1 & x is /ret*$ z "h %[n1 s t ! next
 v]tex is five feet f "o ( ^! ends &    
?ree feet f ! o!r end1 a "r angle w be  
=m$4 ,a "r angle c also be run by mak+ a
 simple in/rum5t1 su* z is describ$ 9   
,*apt] ,,xv4 ,/ill ano!r plan ( draw+ a 
l9e p]p5dicul> to ano!r l9e ,,ab1 f a   
po9t ,p1 consi/s 9 sw++ a tape f ,p1    

cutt+ ,,ab at ,x & ,y1 & !n bisect+ ,,xy
 by d\bl+ ! tape4 ,?is fixes ! foot ( ! 
p]p5dicul>4  ,illu/ration  #d4 ,x is n[ 
possible to f9d ! area ( a field (      
irregul> %ape by divid+ x into triangles
 & trapezoids1 z %[n 9 ! figure4 ,pupils
 "k f _! "w 9 ari?metic h[ to f9d ! area
 ( a triangle or a trapezoid1 s t ! area
 ( ! field is easily f\nd4 ,! "w may be 
*eck$ by comp>+ ! results ( diff]5t     
gr\ps ( pupils1 or by draw+ ano!r       
diagonal & divid+ ! field into o!r      
triangles & trapezoids4  ,^! are ab\t z 
_m types ( field "w z !re is any        
advantage 9 und]tak+ = ! purpose (      
secur+ ! int]e/ ( pupils z a prelim9>y  
to ! "w 9 geometry4 ,:e!r any ( x is    
necess>y1 & = :at pupils x is necess>y1 
& h[ mu* x %\ld trespass ^u ! "t (      
sci5tific geometry are matt]s t c be    
decid$ only by ! tea*] ( a p>ticul>     
class4  ,illu/ration  ,illu/ration      
,illu/ration  ,illu/ration  ,illu/ration
  ,illu/ration  ,illu/ration  ,a second 
difficulty ( ! pupil is se5 9 8 attitude

 ( m9d t[>ds pro(s 9 g5]al4 ,he does n  
see :y v]tical angles %\ld be prov$     
equal :5 he kn[s t !y are s by look+ at 
! figure4 ,?is difficulty %\ld also be  
anticipat$ by giv+ him "s opportunity to
 "k ! weakness ( 8 judgm5t1 & = ?is     
purpose figures l ! foll[+ %\ld be plac$
 be=e him4 ,he %\ld be ask$ :i* ( ^!    
l9es is long]1 ,,ab or ,,xy4 ,two equal 
l9es %\ld !n be arrang$ 9 ! =m ( a lett]
 ,t1 z "h %[n1 & he %\ld be ask$ :i* is 
! long]1 ,,ab or ,,cd4 ,a figure t is v 
deceptive1 p>ticul>ly if drawn l>g] & ) 
heavy cross l9es1 is ?is "o 9 :i* ,,ab &
 ,,cd are really p>allel1 b d n seem to 
be s4 ,o!r int]e/+ deceptions h to d )  
produc+ l9es1 z 9 ^! figures1 :]e x is q
 difficult 9 advance to tell :e!r ,,ab &
 ,,cd are 9 ! same l9e1 & simil>ly =    
,,wx & ,,yz4 ,equally deceptive is ?is  
figure1 9 :i* x is difficult to tell :i*
 l9e ,,ab w lie along :5 produc$4 ,9 !  
next figure ,,ab appe>s to be curv$ :5 9
 reality x is /raight1 & ,,cd appe>s    
/raight :5 9 reality x is curv$4 ,! fir/

 ( ! foll[+ circles seems to be slightly
 flatt5$ at ! po9ts ,p1 ,q1 ,r1 ,s1 & 9 
! second "o ! di/ance ,,bd seems great] 
?an ! di/ance ,,ac4 ,!re are _m equally 
deceptive figures1 & a few ( !m w       
conv9ce ! beg9n] t ! pro(s are necess>y 
features ( geometry4  ,x is int]e/+1 9  
connection ) ! t5d5cy to feel t a       
/atem5t is app>5t )\t pro(1 to recall an
 anecdote relat$ by ! ,fr5* ma!matician1
 ,biot1 conc]n+ ! great sci5ti/1        
,laplace3       ,once ,laplace1 hav+ be5
 ask$ ab\t a c]ta9 po9t 9 8             
,cele/ial ,me*anics1 sp5t ne>ly an h\r 9
 try+ to recall      ! *a9 ( reason+ :i*
 he _h c>elessly conceal$ by !          
words ,x is easy to see4  ,a ?ird       
difficulty lies 9 ! necessity = putt+ a 
consid]able numb] ( def9itions at !     
beg9n+ ( geometry1 9 ord] to get a work+
 vocabul>y4 ,al?\gh practically all     
writ]s scatt] ! def9itions z mu* z      
possible1 !re mu/ necess>ily be "s      
vocabul>y at ! beg9n+4 ,9 ord] to       
m9imize ! difficulty ( rememb]+ s _m new

 t]ms1 x is helpful to m+le ) !m a      
consid]able numb] ( ex]cises 9 :i* ^!   
t]ms are employ$1 s t !y may become fix$
 9 m9d ?r\gh actual use4 ,?us x is (    
value to h a class f9d ! complem5ts (   
#bg deg41 #cb deg4 #bj'1 #da deg4 #cb'  
#dh1 #bf4ge deg41 #cc #a_/c deg41 & #j  
deg44 ,x is true t into ! pure geometry 
( ,euclid ! measur+ ( angles 9 degrees  
does n ent]1 b x has place 9 ! practical
 applications1 & x s]ves at ?is juncture
 to fix ! mean+ ( a new t]m l complem5t4
  ,! tea*] :o ?us anticipates ! "q z to 
! reason = /udy+ geometry1 ! m5tal      
opposition to prov+ /atem5ts1 & !       
=getfulness ( ! mean+ ( common t]ms w   
f9d t mu* ( ! initial difficulty is     
avoid$4 ,if1 n[1 great c>e is giv5 to ! 
fir/ half doz5 propositions1 ! pupil w  
be well on 8 way 9 geometry4 ,z to ^!   
propositions1 two plans ( selection are 
employ$4 ,! fir/ takes a few prelim9>y  
propositions1 easily demon/rat$1 & seeks
 ?us to introduce ! pupil to ! nature ( 
a pro(4 ,?is has ! advantage ( inspir+  

confid5ce & ! disadvantage ( appe>+ to  
prove ! obvi\s4 ,! second plan disc>ds  
all su* app>5tly obvi\s propositions z  
^? ab\t ! equality ( "r angles1 & ! sum 
( two adjac5t angles =m$ by "o l9e meet+
 ano!r1 & beg9s at once on ?+s t seem to
 ! pupil z wor? ! prov+4 ,9 ?is latt]   
plan ! introduction is usually made ) ! 
proposition conc]n+ v]tical angles1 & ! 
two simple/ cases ( congru5t triangles4 
,:i*ev] plan ( selection is tak51 x is  
important to introduce a consid]able    
numb] ( "o-/ep ex]cises imm$iately1 t   
is1 ex]cises t require only "o          
significant /ep 9 ! pro(4 ,9 ?is way !  
pupil acquires confid5ce 9 8 own p[]s1  
he f9ds t geometry is n m]e memoriz+1 & 
he sees t ea* proposition makes him !   
ma/] ( a l>ge field4 ,to delay !        
ex]cises to ! end ( ea* book1 or ev5 to 
delay !m = sev]al lessons1 is to s[ se$s
 t w result 9 ! attempt to ma/] geometry
 by ! %e] process ( memoriz+4  ,z to !  
nature ( ^! ex]cises1 h[ev]1 ! mi/ake   
mu/ n be made ( feel+ t only ^? h any   

value t relate to football or ! lay+ out
 ( a t5nis c\rt4 ,su* ex]cises are      
valuable1 b su* ex]cises alone are      
"o-sid$4 ,moreov]1 any "o :o exam9es !  
hundr$s ( sugge/$ ex]cises t are        
con/antly appe>+ 9 v>i\s j\rnals1 or :o1
 9 ! prep>ation ( tea*]s1 looks ?r\gh ! 
?\s&s ( ex]cises t come to him 9 ! pap]s
 ( 8 /ud5ts1 comes v soon to see h[     
holl[ is ! pret5se ( mo/ ( !m4 ,z has   
already be5 said1 !re are relatively few
 propositions 9 geometry t h any        
practical applications1 applications t  
are ev5 hone/ 9 _! pret5se4 ,! pr9ciple 
t ! writ] has s (t5 laid d[n 9 o!r      
works1 t :atev] pret5ds to be practical 
%\ld really be s1 applies ) mu* =ce to  
^! ex]cises4 ,:5 we c f9d ! g5u9e       
application1 if x is )9 reasonable grasp
 ( ! pupil1 by all means let u use x4 ,b
 to put be=e a class ( girls "s         
te*nicality ( ! /eam eng9e t only a     
skill$ me*anic w\ld be expect$ to "k is 
n education1--x is m]e %am4 ,!re is a   
noble dignity to geometry1 a dignity t a

 l>ge majority ( any class comes to     
appreciate :5 guid$ by an e>ne/ tea*]2 b
 ! be/ way to de/roy ?is dignity1 to    
take away ! appreciation ( pure         
ma!matics1 & to furni% weak] c&idates   
?an n[ = advance 9 ?is field is to      
deceive our pupils & ourselves into     
believ+ t ! ultimate purpose ( ma!matics
 is to measure ?+s 9 a way 9 :i* no "o  
else measures !m or has ev] measur$ !m4 
,9 ! pro( ( ! e>ly propositions ( plane 
geometry1 & aga9 at ! beg9n+ ( solid    
geometry1 !re is a little advantage 9   
us+ color$ crayon to br+ out m di/9ctly 
! equal p>ts ( two figures1 or ! l9es   
outside ! plane1 or to diff]5tiate "o   
plane f ano!r4 ,?is device1 h[ev]1 l t (
 models 9 solid geometry1 c easily be   
abus$1 & h5ce %\ld be us$ sp>+ly1 & only
 until ! purpose is accompli%$4 ,! /ud5t
 ( ma!matics mu/ le>n to grasp ! mean+ (
 a figure drawn 9 black on :ite pap]1   
or1 m r>ely1 9 :ite on a blackbo>d1 & ! 
soon] he is able to d ?is ! bett] = him4
 ,! same ?+ may be said ( ! con/ruct+ ( 

models = any consid]able numb] ( figures
 9 solid geometry2 en\gh "w ( ?is k9d to
 enable a pupil cle>ly to visualize !   
solids is valuable1 b !reaft] ! value is
 usually m ?an (fset by ! "t consum$ & !
 weak5$ p[] to grasp ! mean+ ( a        
geometric draw+4  ,!re is (t5 a t5d5cy  
on ! "p ( tea*]s 9 _! fir/ ye>s ( "w to 
ov]e/imate ! logical p[]s ( _! pupils & 
to introduce =ms ( reason+ & te*nical   
t]ms t exp]i5ce has prov$ to be unsuit$ 
to "o :o is beg9n+ geometry4 ,usually b 
little h>m is done1 because ! en?usiasm 
( any tea*] :o w\ld use ?is "w w\ld c>ry
 ! pupils ov] ! difficulties )\t mu*    
wa/e ( en]gy on _! "p4 ,9 ! long run1   
h[ev]1 ! attempt is usually ab&on$ z n  
wor? ! ef=t4 ,su* a t]m z               
contrapositive1 su* di/9ctions z t      
betwe5 ! logical & ! geometric conv]se1 
or betwe5 p]fect & p>tial geometric     
conv]sion1 & su* pron\nc$ =malism z !   
syllogi/ic me?od1--all ^! are happily   
unkn[n to mo/ tea*]s & might pr(itably  
be unkn[n to all pupils4 ,! mod]n       

,am]ican textbook 9 geometry does n beg9
 to be z good a piece ( logic z         
,euclid's ,elem5ts1 & yet x is to be    
obs]v$ t none ( ^! t]ms is f\nd 9 ?is   
classic "w1 s t !y _c be ?\ght to be    
necess>y to a logical treatm5t ( !      
subject4 ,we ne$ ! ^w conv]se1 & "s     
ref]5ce to ! law ( conv]se is !re=e     
p]missible2 ! mean+ ( ! r$uctio ad      
absurdum1 ( a necess>y & suffici5t      
condition1 & ( ! t]ms syn!sis & analysis
 may prop]ly =m "p ( ! pupil's equipm5t 
because ( _! univ]sal use2 b any ext5d$ 
incursion into ! doma9 ( logic w be f\nd
 unpr(itable1 & x is liable to be       
positively h>mful to a beg9n] 9         
geometry4  ,a ^w %\ld be said z to !    
lett]+ ( ! figures 9 ! e>ly /ages (     
geometry4 ,9 g5]al1 x is a great aid to 
! eye if ?is is c>ri$ out ) "s sy/em1 & 
! foll[+ sugge/ions are giv5 z 9 accord 
) ! be/ au?ors :o h giv5 any att5tion to
 ! subject3  #a4 ,9 g5]al1 lett] a      
figure c\nt]clockwise1 = ! reason t we  
read angles 9 ?is way 9 high] ma!matics1

 & x is z easy to =m ?is habit n[ z to  
=m "o t may h to be *ang$4 ,:]e two     
triangles are congru5t1 h[ev]1 b h _!   
sides arrang$ 9 opposite ord]1 x is     
bett] to lett] !m s t _! correspond+    
p>ts appe> 9 ! same ord]1 al?\gh ?is    
makes "o read clockwise4  ,illu/ration  
#b4 ,= ! same reason1 read angles       
c\nt]clockwise4 ,?us ,l,a is read ,,bac1
 ! reflex angle on ! outside ( !        
triangle be+ read ,,cab4 ,( c\rse ?is is
 n vital1 & _m au?ors pay no att5tion to
 x2 b x is conv5i5t1 & if ! tea*]       
habitually does x1 ! pupils w also t5d  
to d x4 ,x is helpful 9 trigonometry1 & 
x saves confusion 9 ! case ( a reflex   
angle 9 a polygon4 ,designate an angle  
by a s+le lett] if ?is c conv5i5tly be  
done4  #c4 ,designate ! sides opposite  
angles ,a1 ,b1 ,c1 9 a triangle1 by a1  
b1 c1 & use ^! lett]s 9 writ+ pro(s4    
#d4 ,9 ! case ( two congru5t triangles  
use ! lett]s ,a1 ,b1 ,c & ,a'1 ,b'1 ,c'1
 or ,x1 ,y1 ,z1 in/ead ( lett]s *os5 at 
r&om1 l ,d1 ,k1 ,l4 ,x is easi] to foll[

 a pro( :]e "s sy/em is %[n 9 lett]+ !  
figures4 ,"s tea*]s insi/ t a pupil at !
 blackbo>d %\ld n use ! lett]s giv5 9 ! 
textbook1 hop+ !reby to avoid memoriz+4 
,:ile ! dang] is probably exagg]at$1 x  
is easy to *ange ) "s sy/em1 us+ ,p1 ,q1
 ,r & ,p'1 ,q'1 ,r'1 = example4  #e4    
,use small lett]s = l9es1 z above /at$1 
& also place !m )9 angles1 x be+ easi]  
to speak ( & to see ,lm ?an ,l,,def4 ,! 
,g]mans h a conv5i5t sy/em t "s ,am]ican
 tea*]s foll[ to advantage1 b t a       
textbook has no "r to require4 ,!y use1 
z 9 ! foll[+ figure1 ,a = ! po9t1 a = ! 
opposite side1 & ! ,greek lett] alpha   
7alpha7 = ! angle4 ,! le>n+ ( ! fir/    
?ree ,greek lett]s1 alpha 7alpha71 beta 
7beta71 & gamma 7gamma71 is n a h>d%ip1 
& !y are wor? us+1 al?\gh ,greek is s   
little kn[n 9 ?is c\ntry to-day t !     
alphabet _c be dem&$ ( tea*]s :o d n c>e
 to use x4  ,illu/ration  #f4 ,also use 
small lett]s to repres5t num]ical       
values4 ,= example1 write c  #bpir      
in/ead ( ,c  #bpi,r4 ,?is is 9 accord ) 

! usage 9 algebra to :i* ! pupil is     
accu/om$4  #g4 ,use initial lett]s :5ev]
 conv5i5t1 z 9 ! case ( a = area1 b =   
base1 c = circumf]5ce1 d = diamet]1 h = 
height 7altitude71 & s on4  ,_m ( ^!    
sugge/ions seem ( slight importance 9   
!mselves1 & "s tea*]s w be dispos$ to   
object to any attempt at lett]+ a figure
 ) any reg>d to sy/em4 ,if1 h[ev]1 !y w 
notice h[ a class /ruggles to foll[ a   
demon/ration giv5 ) ref]5ce to a figure 
on ! blackbo>d1 !y w see h[ helpful x is
 to h "s simple /&>ds ( lett]+4 ,x is   
h>dly necess>y to add t 9 demon/rat+ f a
 figure on a blackbo>d x is usually     
bett] to say ?is l9e1 or ! r$ l9e1 ?an  
to say1 )\t po9t+ to x1 ! l9e ,,ab4 ,x  
is by su* simplicity ( /atem5t & by su* 
ef=ts to help ! class to foll[          
demon/rations t pupils are l$ ?r\gh _m (
 ! initial disc\ragem5ts ( ! subject4   
,,footnotes3  #dj ,c>son1 loc4 cit41 p4 
#ac4     ,,*apt] ,x  ,,! ,,conduct ,,(  
,a ,,class ,,9 ,,geometry   ,no def9ite 
rules c be giv5 = ! detail$ conduct ( a 

class 9 any subject4 ,if x 7 possible to
 =mulate su* rules1 all ! p]sonal       
magnetism ( ! tea*]1 all ! en?usiasm1   
all ! orig9ality1 all ! _s ( ! class1   
w\ld dep>t1 & we %\ld h a dull1 dry     
me*anism4 ,!re is no "o be/ me?od (     
tea*+ geometry or any?+ else4 ,!        
exp]i5ce ( ! s*ools has %[n t a few     
great pr9ciples /& out z g5]ally        
accept$1 b z to ! c>ry+ out ( ^!        
pr9ciples !re c be no def9ite rules4    
,let u fir/ consid] ! g5]al "q ( !      
employm5t ( "t 9 a recitation 9         
geometry4 ,we might all agree on c]ta9  
g5]al pr9ciples1 & yet no two tea*]s ev]
 w\ld or ev] %\ld divide ! p]iod ev5    
approximately 9 ! same way4 ,fir/1 a    
class %\ld h an opportunity to ask      
que/ions4 ,a tea*] "h %[s 8 p[] at its  
be/1 li/5+ sympa!tically to any good "q1
 quickly see+ ! ess5tial po9t1 & ei!r   
answ]+ x or re/at+ x 9 su* a way t !    
pupil c answ] x = himself4 ,c]ta9       
que/ions %\ld be answ]$ by ! tea*]2 he  
is !re = t purpose4 ,o!rs c at once be  

put 9 su* a light t ! pupil c himself   
answ] !m4 ,o!rs may bett] be answ]$ by !
 class4 ,occasionally1 b m r>ely1 a     
pupil may be told to look t up =        
to-morr[1 a plan t is commonly consid]$ 
by /ud5ts z a confession ( weakness on !
 "p ( ! tea*]1 z x probably is4 ,( c\rse
 a class w wa/e "t 9 que/ion+ a weak    
tea*]1 b a /rong "o ne$ h no fe> on ?is 
acc\nt4 ,five m9utes giv5 at ! op5+ ( a 
recitation to brisk1 po9t$ que/ions by !
 class1 ) ! same cr$it giv5 to a good "q
 z to a good answ]1 w d a great deal to 
create a _s ( comrade%ip1 ( frankness1 &
 ( hone/y1 & w reveal to a sympa!tic    
tea*] ! difficulties ( a class mu* bett]
 ?an ! same am\nt ( "t devot$ to        
blackbo>d "w4 ,b !re mu/ be no dawdl+1 &
 ! class mu/ feel t x has only a limit$ 
"t1 say five m9utes at ! mo/1 to get !  
help x ne$s4  ,next 9 ord] ( ! division 
( ! "t may be ! tea*]'s report on any   
pap]s t ! class has h&$ 94 ,x is        
impossible to tell h[ mu* ( ?is pap] "w 
%\ld be dem&$4 ,! local s*ool           

conditions1 ! m5tal condition ( ! class1
 & ! "t at ! disposal ( ! tea*] are all 
factors 9 ! case4 ,9 g5]al1 x may be    
said t en\gh ( ?is k9d ( "w is necess>y 
to see t pupils are neat & accurate 9   
sett+ d[n _! demon/rations4 ,on ! o!r   
h&1 pap] "w gives an opportunity =      
di%one/y1 & x consumes a great deal ( ! 
tea*]'s "t t might bett] be giv5 to     
read+ good books on ! subject t he is   
tea*+4 ,if1 h[ev]1 any pap]s h be5      
submitt$1 ab\t five m9utes may well be  
giv5 to a rapid review ( ! failures & ! 
successes4 ,9 g5]al1 x is good          
educational policy to speak ( ! errors &
 failures imp]sonally1 b occasionally to
 m5tion by "n any "o :o has done a piece
 ( "w t is wor?y ( special comm5t4      
,pupils may bett] be prais$ 9 public &  
blam$ 9 private4 ,!re is su* a ?+1      
h[ev]1 z prais+ too mu*1 :5 no?+ wor?y (
 note has be5 done1 j z !re is dang] (  
blam+ too mu*1 result+ 9 m]e nagg+4  ,! 
?ird division ( ! recitation p]iod may  
pr(itably g to assign+ ! advance lesson4

 ,! class que/ions & ! tea*]'s report on
 writt5 "w h %[n ! m5tal /atus ( !      
pupils1 s t ! tea*] n[ kn[s :at he may  
expect = ! next lesson4 ,if he assigns 8
 lesson at ! beg9n+ ( ! p]iod1 he does n
 h ?is in=mation4 ,if he waits to ! end1
 he may be too hurri$ to give any       
developm5t t ! new lesson may require4  
,!re c be no rule z to h[ to assign a   
new lesson2 x all dep5ds ^u :at ! lesson
 is1 ^u ! m5tal /ate ( ! class1 & n a   
little ^u ! idiosyncrasy ( ! tea*]4 ,!  
,g]man educator1 ,h]b>t1 laid d[n c]ta9 
=mal /eps 9 develop+ a new lesson1 & 8  
successors h elaborat$ ^! some:at z     
foll[s3  #a4 ,aim4 ,always take a class 
into y\r confid5ce4 ,tell ! memb]s at ! 
outset ! goal4 ,no "o likes to be l$    
bl9dfold$4  #b4 ,prep>ation4 ,a few     
brief que/ions to br+ ! class to ?9k (  
:at is to be consid]$4  #c4 ,pres5tation
 ( ! new4 ,pref]ably ?is is done by     
que/ions1 ! answ]s lead+ ! memb]s ( !   
class to discov] ! new tru? = !mselves4 
#d4 ,app]ception4 ,call+ att5tion to !  

fact t ?is new fact 0 kn[n be=e1 9 "p1 &
 t x relates to a numb] ( ?+s already 9 
! m9d4 ,! m ! new c be ti$ up to ! old !
 m t5aci\sly x w be held4  #e4          
,g5]alization & application4  ,x is     
evid5t at once t a great deal ( "t may  
be wa/$ 9 always foll[+ su* a plan1     
p]haps 9 ev] foll[+ x consci\sly4 ,b1 on
 ! o!r h&1 probably e good tea*]1 :e!r  
he has he>d ( ,h]b>t or n1 naturally    
cov]s ^! po9ts 9 sub/antially ?is ord]4 
,= an inexp]i5c$ tea*] x is helpful to  
be famili> ) !m1 t he may call to m9d ! 
/eps1 arrang$ 9 a psy*ological sequ5ce1 
t he w\ld d well to foll[4 ,x mu/ always
 be rememb]$ t !re is q z mu* dang] 9   
develop+ too mu* z 9 tak+ ! opposite    
extreme4 ,a me*anical tea*] may develop 
a new lesson :]e !re is ne$ = only a "q 
or two or a m]e sugge/ion4 ,x %\ld also 
be recogniz$ t /ud5ts ne$ to le>n to    
read ma!matics = !mselves1 & t always to
 take away e difficulty by explanations 
giv5 9 advance is weak5+ to any "o4     
,!re=e1 9 assign+ ! new lesson we may   

say ,take ! next two pages1 & ?us       
disc\rage mo/ ( ! class4 ,on ! o!r h&1  
we may sp5d an unnecess>y am\nt ( "t &  
ov]develop ! "w ( ^? same pages1 & h !  
:ole lesson lose all its ze/4 ,x is "h t
 ! g5ius ( ! tea*] comes =? to f9d !    
happy mean4  ,! f\r? division ( ! h\r   
%\ld be rea*$1 9 g5]al1 9 ab\t t5       
m9utes4 ,?is includes ! recitation      
prop]4 ,b z to ! nature ( ?is "w no     
def9ite in/ructions %\ld be attempt$4   
,to a good tea*] !y w\ld be unnecess>y1 
to a poor "o !y w\ld be h>mful4 ,"p ( ! 
class may g to ! bo>d1 & z !y are work+1
 ! re/ may be recit+4 ,^? at ! bo>d %\ld
 be limit$ z to "t1 = o!rwise a premium 
is plac$ on m]e dawdl+4 ,!y %\ld be s   
arrang$ z to prev5t copy+1 b ! tea*]'s  
eye is ! be/ prev5tive ( ?is annoy+     
feature4 ,^? at _! seats may be call$ ^u
 "o at a "t to demon/rate at !          
blackbo>d1 ! re/ be+ call$ ^u = quick   
responses1 z occasion dem&s4 ,!         
,european plan ( hav+ small blackbo>ds  
is 9 _m respects bett] ?an ours1 s9ce   

pupils _c s easily wa/e "t4 ,!y h to "w 
rapidly & talk rapidly1 or else take _! 
seats4  ,:at %\ld be put on ! bo>d1 :e!r
 ! figure alone1 or ! figure & ! pro(1  
dep5ds ^u ! proposition4 ,9 g5]al1 !re  
%\ld be a c]ta9 numb] ( figures put on !
 bo>d = ! sake ( rapid "w & z a basis = 
! pro(s ( ! day4 ,!re %\ld also be a    
c]ta9 am\nt ( writt5 "w = ! sake (      
comm5d+ or ( criticiz+ adv]sely ! pro(  
us$4 ,!re are "s figures t are s        
complicat$ z to w>rant be+ put ^u %eets 
( pap] & hung be=e ! class4 ,?us !re is 
no rule ^u ! subject1 & ! tea*] mu/ use 
8 judgm5t accord+ to ! circum/ances & ! 
propositions4  ,if ! e>ly orig9als are  
"o-/ep ex]cises1 & a pupil is requir$ to
 recite rapidly1 a habit ( quick        
expression is easily acquir$ t leads to 
close att5tion on ! "p ( all ! class4   
,/ud5ts z a rule recite sl[] ?an !y ne$ 
to1 f m]e habit4 ,phlegmatic z we ?9k ! 
,g]man is1 & n]v\s z is ! ,am]ican      
temp]am5t1 a /ud5t 9 geometry 9 a ,g]man
 s*ool w usually recite m quickly & ) m 

vigor ?an "o ) u4 ,our ext5sive         
blackbo>ds h some?+ to d ) ?is1 all[+ s 
_m pupils to be work+ at ! bo>d t a     
tea*] _c att5d to !m all4 ,! result is a
 habit ( wa/+ ! m9utes t c only be      
ov]come by ! tea*] sett+ a def9ite b    
reasonable "t limit1 & hold+ ! pupil    
responsible if ! "w is n done 9 ! "t    
specifi$4 ,if ?is matt] is tak5 9 h& !  
fir/ day1 & special ef=t made 9 ! e>ly  
weeks ( ! ye>1 mu* ( ! difficulty c be  
ov]come4  ,z to ! nature ( ! recitation 
to be expect$ f ! pupil1 no def9ite rule
 c be laid d[n1 s9ce x v>ies s mu* ) !  
"w ( ! day4 ,9 g5]al1 h[ev]1 a pupil    
%\ld /ate ! !orem quickly1 /ate exactly 
:at is giv5 & :at is to be prov$1 )     
respect to ! figure1 & !n give ! pro(4  
,at fir/ x is desirable t he %\ld give !
 au?orities 9 full1 & lat] give only !  
ess5tial "p 9 a few words4 ,x is bett]  
to avoid ! expression by previ\s        
proposition1 = x soon comes to be abus$1
 & ( c\rse ! le>n+ ( section numb]s 9 a 
book is a b>b>ism4 ,x is only by        

cont9ually /at+ ! propositions us$ t a  
pupil comes to h well fix$ 9 8 memory ! 
basal !orems ( geometry1 & )\t ^! he _c 
make progress 9 8 subsequ5t ma!matics4  
,9 g5]al1 x is bett] to all[ a pupil to 
f9i% 8 pro( be=e ask+ him any que/ions1 
! con/ant int]ruptions indulg$ 9 by "s  
tea*]s be+ ! cause ( no little confusion
 & hesitancy on ! "p ( pupils4          
,sometimes x is well to h a figure drawn
 diff]5tly f ! "o 9 ! book1 or lett]$   
diff]5tly1 s z to make sure t ! pupil   
has n memoriz$ ! pro(1 b 9 g5]al su*    
devices are unnecess>y1 = a tea*] c     
easily discov] :e!r ! pro( is ?or\ghly  
und]/ood1 ei!r by ! mann] ( ! pupil or  
by "s slight que/ion+4 ,a good textbook 
has ! figures sy/ematically lett]$ 9 "s 
helpful way t is easily foll[$ by !     
class t is li/5+ to ! recitation1 & x is
 n advisable to ab&on ?is = a r&om set (
 lett]s arrang$ 9 no prop] ord]4  ,x is 
good educational policy = ! tea*] to    
comm5d at lea/ z (t5 z he f9ds fault :5 
criticiz+ a recitation at ! blackbo>d & 

:5 discuss+ ! pupils' pap]s4 ,optimism1 
enc\ragem5t1 sympa?y1 ! g5u9e desire to 
help1 ! putt+ ( "o's self 9 ! pupil's   
place1 ! do+ to ! pupil z ! tea*] w\ld t
 he %\ld d 9 return1--^! are educational
 policies t make = bett] geometry z !y  
make = bett] life4  ,! prime failure 9  
tea*+ geometry lies unque/ionably 9 !   
lack ( int]e/ on ! "p ( ! pupil1 & ?is  
has be5 br\ght ab\t by ! anci5t plan (  
simply read+ & memoriz+ pro(s4 ,x is to 
get away f ?is t tea*]s resort to "s su*
 developm5t ( ! lesson 9 advance1 z has 
be5 sugge/$ above4 ,x is usually a good 
plan to give ! easi] propositions z     
ex]cises be=e !y are rea*$ 9 ! text1 :]e
 ?is c be done4 ,an ,engli% writ] has   
rec5tly contribut$ ?is fur!r idea3      
,x might be m /imulat+ to enc\rage      
inve/igation ?an to      dem& pro(s (   
/at$ facts2 t is to say1 ,"h is a       
figure drawn 9 ?is way1 f9d out any?+ y 
c ab\t x4      ,"s su* ex]cises hav+ be5
 p]=m$ jo9tly by tea*]s      & pupils1 !
 lu/ ( inve/igation & heal?y competition

      :i* is pres5t 9 e normal boy or   
girl might be awak5$      s f> z to make
 su* little rese>*es really attractive2 
moreov]1 ! tra9+ ?us giv5 is ( f> m     
value ?an      t obta9$ by prov+ facts  
:i* are /at$ 9 advance1 =      x is     
seldom1 if ev]1 t ! problems ( adult    
life pres5t      !mselves 9 ?is mann]4  
,! _s ( ! "q1 ,:at is      true8 is     
positive & con/ructive1 b t involv$ 9   
,is      ?is true8 is negative &        
de/ructive4#da   ,:5 ! "q is ask$1 ,h[  
%all ,i tea*8 or ,:at is ! ,me?od8 !re  
is no answ] su* z ! que/ion] expects4 ,a
 ,japanese writ]1 ,mot[ori1 a great     
au?ority ^u ! ,%9to fai? ( 8 p1 once    
wrote ^! words3 ,to h le>n$ t !re is no 
way to be le>n$ & practic$ is really to 
h le>n$ ! way ( ! gods4  ,,footnotes3   
#da ,c>son1 loc4 cit41 p4 #ab4          
,,*apt] ,,xi  ,,! ,,axioms ,,&          
,,po/ulates   ,! int]e/ z well z ! value
 ( geometry lies *iefly 9 ! fact t f a  
small numb] ( assumptions x is possible 
to d$uce an unlimit$ numb] (            

conclusions4 ,) ! tru? ( ^! assumptions 
we are n s mu* conc]n$ z ) ! reason+ by 
:i* we draw ! conclusions1 al?\gh x is  
manife/ly desirable t ! assumptions %\ld
 n be false1 & t !y %\ld be z few z     
possible4  ,x w\ld be natural1 & 9 "s   
respects desirable1 to call ^!          
f\ndations ( geometry by ! "n           
assumptions1 s9ce !y are simply /atem5ts
 t are assum$ to be true4 ,! real       
f\ndation pr9ciples _c be prov$2 !y are 
! means by :i* we prove o!r /atem5ts4 ,b
 z ) mo/ names ( m5 or ?+s1 !y h receiv$
 c]ta9 titles t are "t-honor$1 & t x is 
n wor? ! :ile to attempt to *ange4 ,9   
,engli% we call !m axioms & po/ulates1 &
 !re is no m reason = attempt+ to *ange 
^! t]ms ?an !re is = attempt+ to *ange !
 names ( geometry#db & ( algebra4#dc    
,s9ce ^! t]ms are likely to cont9ue1 x  
is necess>y to di/+ui% betwe5 !m m      
c>efully ?an is (t5 done1 & to consid]  
:at assumptions we are ju/ifi$ 9 includ+
 "u ea*4 ,9 ! fir/ place1 ^! names d n g
 back to ,euclid1 z is ord9>ily suppos$1

 al?\gh ! ideas & ! /atem5ts are 84     
,po/ulate is a ,lat9 =m ( ! ,greek      
,greek3 aitema 7aitema71 & appe>s only 9
 late translations4 ,euclid /at$ 9      
sub/ance1 ,let ! foll[+ be assum$4      
,axiom 7,greek3 axioma1 axioma7 dates   
p]haps only f ,proclus 7fif? c5tury     
,a4,d471 ,euclid us+ ! words common     
notions 7,greek3 ko9ai ennoiai1 ko9ai   
ennoiai7 = axioms1 z ,ari/otle be=e him 
_h us$ common ?+s1 common pr9ciples1 &  
common op9ions4  ,! di/9ction betwe5    
axiom & po/ulate 0 n cle>ly made by     
anci5t writ]s4 ,probably :at 0 9        
,euclid's m9d 0 ! ,ari/otelian di/9ction
 t an axiom 0 a pr9ciple common to all  
sci5ces1 self-evid5t b incapable ( pro(1
 :ile ! po/ulates 7 ! assumptions       
necess>y = build+ up ! p>ticul> sci5ce  
"u consid]ation1 9 ?is case geometry4#dd
  ,we ?us come to ! mod]n di/9ction     
betwe5 axiom & po/ulate1 & say t a g5]al
 /atem5t admitt$ to be true )\t pro( is 
an axiom1 :ile a po/ulate 9 geometry is 
a geometric /atem5t admitt$ to be true1 

)\t pro(4 ,= example1 :5 we say ,if     
equals are add$ to equals1 ! sums are   
equal1 we /ate an assumption t is tak5  
also z true 9 ari?metic1 9 algebra1 & 9 
elem5t>y ma!matics 9 g5]al4 ,?is is     
!re=e an axiom4 ,at "o "t su* a /atem5t 
0 def9$ z a self-evid5t tru?1 b ?is has 
9 rec5t ye>s be5 ab&on$1 s9ce :at is    
evid5t to "o p]son is n necess>ily      
evid5t to ano!r1 & s9ce all su* /atem5ts
 are m]e matt]s ( assumption 9 any case4
 ,on ! o!r h&1 :5 we say1 ,a circle may 
be describ$ ) any giv5 po9t z a c5t] &  
any giv5 l9e z a radius1 we /ate a      
special assumption ( geometry1 & ?is    
assumption is !re=e a geometric         
po/ulate4 ,"s few writ]s h s\ght to     
di/+ui% betwe5 axiom & po/ulate by say+ 
t ! =m] 0 an assum$ !orem & ! latt] an  
assum$ problem1 b !re is no /&>d        
au?ority = su* a di/9ction1 & inde$ !   
diff]5ce betwe5 a !orem & a problem is v
 slight4 ,if we say1 ,a circle may be   
pass$ ?r\gh ?ree po9ts n 9 ! same       
/raight l9e1 we /ate a !orem2 b if we   

say1 ,requir$ to pass a circle ?r\gh    
?ree po9ts1 we /ate a problem4 ,! m5tal 
process ( h&l+ ! two propositions is1   
h[ev]1 practically ! same 9 spite ( !   
m9or detail ( word+4 ,s ) ! /atem5t1 ,a 
/raight l9e may be produc$ to any       
requir$ l5g?4 ,?is is /at$ 9 ! =m ( a   
!orem1 b x might equally well be /at$   
?us3 ,to produce a /raight l9e to any   
requir$ l5g?4 ,x is unreasonable to call
 ?is an axiom 9 "o case & a po/ulate 9 !
 o!r4 ,h[ev] /at$1 x is a geometric     
po/ulate & %\ld be s class$4  ,:at1 n[1 
are ! axioms & po/ulates t we are       
ju/ifi$ 9 assum+1 & :at det]m9es _!     
numb] & "*8 ,x seems reasonable to agree
 t !y %\ld be z few z possible1 & t =   
educational purposes !y %\ld be s cle> z
 to be intelligible to beg9n]s4 ,b "h we
 enc\nt] two conflict+ ideas4 ,to get ! 
irr$ucible m9imum ( assumptions is to   
get a set ( /atem5ts q un9telligible to 
/ud5ts beg9n+ geometry or any o!r bran* 
( elem5t>y ma!matics4 ,su* an ef=t is   
laudable :5 ! results are int5d$ =      

advanc$ /ud5ts 9 ! univ]sity1 b x is    
m]ely sugge/ive to tea*]s r ?an usable )
 pupils :5 x t\*es ^u ! prim>y /eps (   
any sci5ce4 ,9 rec5t ye>s sev]al su*    
attempts h be5 made4 ,9 p>ticul>1       
,pr(essor ,hilb]t has giv5 a sy/em#de ( 
congru5ce po/ulates1 b !y are r = !     
sci5ti/ ?an = ! /ud5t ( elem5t>y        
geometry4  ,9 view ( ^! ef=ts x is well 
to g back to ,euclid & see :at ?is great
 tea*] ( univ]sity m5#df _h to sugge/4  
,! foll[+ are ! five common notions t   
,euclid deem$ suffici5t = ! purposes (  
elem5t>y geometry4  #a4 ,?+s equal to ! 
same ?+ are also equal to ea* o!r4 ,?is 
axiom has p]si/$ 9 all elem5t>y         
textbooks4 ,( c\rse x is a simple matt] 
to attempt criticism1--to say t -#b is !
 squ>e root ( #d1 & #b is also ! squ>e  
root ( #d1 :5ce -#b  #b2 b x is evid5t t
 ! argum5t is n s\nd1 & t x does n      
invalidate ! axiom4 ,proclus tells u t  
,apollonius attempt$ to prove ! axiom by
 say+1 ,let a equal b1 & b equal c4 ,i  
say t a equals c4 ,=1 s9ce a equals b1 a

 occupies ! same space z b4 ,!re=e a    
occupies ! same space z c4 ,!re=e a     
equals c4 ,! pro( is ( no value1 h[ev]1 
save z a curiosity4  #b4 ,& if to equals
 equals are add$1 ! :oles are equal4    
#c4 ,if equals are subtract$ f equals1 !
 rema9d]s are equal4  ,axioms #b & #c   
are old] ?an ,euclid's "t1 & are ! only 
ones giv5 by him relat+ to ! solution ( 
! equation4 ,c]ta9 o!r axioms 7 add$ by 
lat] writ]s1 z1 ,?+s :i* are d\ble ( !  
same ?+ are equal to "o ano!r1 & ,?+s   
:i* are halves ( ! same ?+ are equal to 
"o ano!r4 ,^! two illu/rate ! anci5t use
 ( duplatio 7d\bl+7 & m$iatio 7halv+71 !
 primitive =ms ( multiplication &       
division4 ,euclid w\ld n admit !        
multiplication axiom1 s9ce to him ?is   
meant m]ely repeat$ addition4 ,!        
p>tition 7halv+7 axiom he did n ne$1 &  
if ne$$1 he w\ld h inf]r$ its tru?4 ,!re
 are also ! axioms1 ,if equals are add$ 
to unequals1 ! :oles are unequal1 & ,if 
equals are subtract$ f unequals1 !      
rema9d]s are unequal1 nei!r ( :i*       

,euclid w\ld h us$ because he did n     
def9e unequals4 ,! mod]n arrangem5t (   
axioms1 cov]+ addition1 subtraction1    
multiplication1 division1 p[]s1 & roots1
 sometimes ( unequals z well z equals1  
comes f ! developm5t ( algebra4 ,!y are 
n all ne$$ = geometry1 b 9 s f> z !y %[ 
! relation ( ari?metic1 algebra1 &      
geometry1 !y s]ve a useful purpose4 ,!re
 are also o!r axioms conc]n+ unequals t 
are ( advantage to beg9n]s1 ev5 ?\gh    
unnecess>y f ! /&po9t ( /rict logic4    
#d4 ,?+s t co9cide ) "o ano!r are equal 
to "o ano!r4 ,?is is no long] includ$ 9 
! li/ ( axioms4 ,x is r a def9ition (   
equal1 or ( congru5t1 to take ! mod]n   
t]m4 ,if n a def9ition1 x is c]ta9ly a  
po/ulate r ?an an axiom1 be+ purely     
geometric 9 "*4 ,x is probable t ,euclid
 includ$ x to %[ t sup]position is to be
 consid]$ a legitimate =m ( pro(1 b :y x
 0 n plac$ among ! po/ulates is n easily
 se54 ,at any rate x is un=tunately     
word$1 & mod]n writ]s g5]ally ins]t !   
po/ulate ( motion in/ead1--t a figure   

may be mov$ ab\t 9 space )\t alt]+ its  
size or %ape4 ,! ,g]man philosoph]1     
,s*op5hau] 7#ahdd71 criticiz$ ,euclid's 
axiom z foll[s3 ,co9cid5ce is ei!r m]e  
tautology or some?+ entirely empirical1 
:i* belongs n to pure intuition b to    
ext]nal s5su\s exp]i5ce4 ,x presupposes1
 9 fact1 ! mobility ( figures4  #e4 ,!  
:ole is great] ?an ! "p4 ,to ?is        
,clavius 7#aegd7 add$1 ,! :ole is equal 
to ! sum ( its p>ts1 :i* may be tak5 to 
be a def9ition ( :ole1 b :i* is helpful 
to beg9n]s1 ev5 if n logically necess>y4
 ,"s writ]s d\bt ! g5u95ess ( ?is axiom4
  ,hav+ consid]$ ! axioms ( ,euclid1 we 
%all n[ consid] ! axioms t are ne$$ 9 ! 
/udy ( elem5t>y geometry4 ,! foll[+ are 
sugge/$1 n f ! /&po9t ( pure logic1 b f 
t ( ! ne$s ( ! tea*] & pupil4  #a4 ,if  
equals are add$ to equals1 ! sums are   
equal4 ,in/ead ( ?is axiom1 ! "o numb]$ 
#h bel[ is (t5 giv5 fir/4 ,= conv5i5ce 9
 memoriz+1 h[ev]1 x is bett] to give !  
axioms 9 ! foll[+ ord]3 7#a7 addition1  
7#b7 subtraction1 7#c7 multiplication1  

7#d7 division1 7#e7 p[]s & roots1--all (
 equal quantities4  #b4 ,if equals are  
subtract$ f equals1 ! rema9d]s are      
equal4  #c4 ,if equals are multipli$ by 
equals1 ! products are equal4  #d4 ,if  
equals are divid$ by equals1 ! quoti5ts 
are equal4  #e4 ,l p[]s or l positive   
roots ( equals are equal4 ,=m]ly /ud5ts 
( geometry knew no?+ ( algebra1 & 9     
p>ticul> no?+ ( negative quantities4    
,n[1 h[ev]1 9 ,am]ican s*ools a pupil   
usually /udies algebra a ye> be=e he    
/udies demon/rative geometry4 ,x is     
!re=e bett]1 9 speak+ ( roots1 to limit 
!m to positive numb]s1 s9ce ! two squ>e 
roots ( #d 7#b & -#b71 = example1 are n 
equal4 ,if ! pupil _h /udi$ complex     
numb]s be=e he began geometry1 x w\ld h 
be5 advisable to limit ! roots /ill     
fur!r to real roots1 s9ce ! f\r f\r?    
roots ( #a 7#a1 -#a1 sqrt7-#a71         
-sqrt7-#a771 = example1 are n equal save
 9 absolute value4 ,x is well1 h[ev]1 to
 elim9ate ^! f9e di/9ctions z f> z      
possible1 s9ce _! pres5ce only cl\ds !  

vision ( ! beg9n]4  ,x %\ld also be not$
 t ^! five axioms might be comb9$ 9 "o1 
namely1 ,if equals are op]at$ on by     
equals 9 ! same way1 ! results are      
equal4 ,9 ,axiom #a ?is op]ation is     
addition1 9 ,axiom #b x is subtraction1 
& s on4 ,inde$1 9 ord] to r$uce ! numb] 
( axioms two are already comb9$ 9 ,axiom
 #e4 ,b !re is a good reason = n comb9+ 
! fir/ f\r ) ! fif?1 & !re is also a    
good reason = comb9+ two 9 ,axiom #e4 ,!
 reason is t ^! are ! axioms cont9ually 
us$ 9 equations1 & to comb9e !m all 9 "o
 w\ld be to enc\rage laxness ( ?\ght on 
! "p ( ! pupil4 ,he w\ld always say by  
,axiom #a :atev] he did to an equation1 
& ! tea*] w\ld n be c]ta9 :e!r ! pupil 0
 ?9k+ def9itely ( divid+ equals by      
equals1 or _h a hazy idea t he 0        
manipulat+ an equation 9 "s o!r way t l$
 to an answ]4 ,on ! o!r h&1 ,axiom #e is
 n us$ z (t5 z ! prec$+ f\r1 & !        
int]*ange ( integral & fractional       
expon5ts is relatively common1 s t !    
jo9+ ( ^! two axioms 9 "o = ! purpose ( 

r$uc+ ! total numb] is ju/ifiable4  #f4 
,if unequals are op]at$ on by positive  
equals 9 ! same way1 ! results are      
unequal 9 ! same ord]4 ,?is includes 9 a
 s+le /atem5t ! six op]ations m5tion$ 9 
! prec$+ axioms2 t is1 if a  b & if x   
y1 !n a  x  b  y1 a - x  b - y1 ax  by1 
etc4 ,! reason = ?us comb9+ six axioms 9
 "o 9 ! case ( inequalities is app>5t4  
,!y are r>ely us$ 9 geometry1 & if a    
tea*] is 9 d\bt z to ! pupil's k1 he c  
easily inquire 9 ! few cases t arise1   
:]eas x w\ld consume a great deal ( "t  
to d ?is = ! _m equations t are met4 ,! 
axiom is /at$ 9 su* a way z to exclude  
multiply+ or divid+ by negative numb]s1 
?is case n be+ ne$$4  #g4 ,if unequals  
are add$ to unequals 9 ! same ord]1 !   
sums are unequal 9 ! same ord]2 if      
unequals are subtract$ f equals1 !      
rema9d]s are unequal 9 ! rev]se ord]4   
,^! are ! only cases 9 :i* unequals are 
necess>ily comb9$ ) unequals1 or op]ate 
^u equals 9 geometry1 & ! axiom is      
easily expla9$ to ! class by ! use (    

numb]s4  #h4 ,quantities t are equal to 
! same quantity or to equal quantities  
are equal to ea* o!r4 ,9 ?is axiom ! ^w 
quantity is us$1 9 ! common mann] ( !   
pres5t "t1 to include numb] & all       
geometric magnitudes 7l5g?1 area1       
volume74  #i4 ,a quantity may be        
sub/itut$ = its equal 9 an equation or 9
 an inequality4 ,?is axiom is tacitly   
assum$ by all writ]s1 & is v useful 9 ! 
pro(s ( geometry4 ,x is really ! basis (
 sev]al o!r axioms1 & if we 7 seek+ !   
irr$ucible m9imum1 x w\ld replace !m4   
,s9ce1 h[ev]1 we are seek+ only a       
reasonably abridg$ li/ ( conv5i5t       
assumptions t beg9n]s w und]/& & use1   
?is axiom has mu* to comm5d x4 ,if we   
consid] ! equations 7#a7 a  x & 7#b7 b  
x1 we see t = x 9 equation 7#a7 we may  
sub/itute b f equation 7#b7 & h a  b2 9 
o!r words1 t ,axiom #h is includ$ 9     
,axiom #i4 ,fur!rmore1 if 7#a7 a  b &   
7#b7 x  y1 !n s9ce a  x is ! same z a   
x1 we may1 by sub/itut+1 say t a  x  a  
x  b  x  b  y4 ,9 o!r words1 ,axiom #a  

is includ$ 9 ,axiom #i4 ,?us an axiom t 
includes o!rs has a legitimate place1   
because a beg9n] w\ld be too mu* confus$
 by see+ its entire scope1 & because he 
w make frequ5t use ( x 9 8 ma!matical   
"w4  #aj4 ,if ! fir/ ( ?ree quantities  
is great] ?an ! second1 & ! second is   
great] ?an ! ?ird1 !n ! fir/ is great]  
?an ! ?ird4 ,?is axiom is ne$$ sev]al   
times 9 geometry4 ,! case 9 :i* a  b & b
  c1 !re=e a  c1 is provid$ = 9 ,axiom  
#i4  #aa4 ,! :ole is great] ?an any (   
its p>ts & is equal to ! sum ( all its  
p>ts4 ,! latt] "p ( ?is axiom is really 
only ! def9ition ( :ole1 & x w\ld be    
legitimate to /ate a def9ition accord+ly
 & ref] to x :]e ! ^w is employ$4 ,:]e1 
h[ev]1 we wi% to speak ( a polygon1 =   
example1 & wi% to say t ! area is equal 
to ! comb9$ areas ( ! triangles compos+ 
x1 x is m satisfactory to h ?is axiom to
 :i* to ref]4 ,x w be notic$ t two      
relat$ axioms are "h comb9$ 9 "o1 = a   
reason simil> to ! "o /at$ "u ,axiom #e4
  ,9 ! case ( ! po/ulates we are met by 

a problem simil> to ! "o confront+ u 9  
connection ) ! axioms1--! problem ( !   
irr$ucible m9imum z relat$ to ! "q (    
tea*+4 ,manife/ly ,euclid us$ po/ulates 
t he did n /ate1 & prov$ "s /atem5ts t  
he might h po/ulat$4#dg  ,! po/ulates   
giv5 by ,euclid "u ! "n ,greek3         
aitemata7aitemata7 7 reque/s made by !  
tea*] to 8 pupil t c]ta9 ?+s be conc$$4 
,!y 7 five 9 numb]1 z foll[s3  #a4 ,let 
! foll[+ be conc$$3 to draw a /raight   
l9e f any po9t to any po9t4  ,/rictly   
speak+1 ,euclid might h be5 requir$ to  
po/ulate t po9ts & /raight l9es exi/1 b 
he evid5tly consid]$ ?is /atem5t        
suffici5t4 ,ari/otle _h1 h[ev]1 already 
call$ att5tion to ! fact t a m]e        
def9ition 0 suffici5t only to %[ :at a  
concept is1 & t ?is mu/ be foll[$ by a  
pro( t ! ?+ exi/s4 ,we might1 = example1
 def9e x z a l9e t bisects an angle )\t 
meet+ ! v]tex1 b ?is w\ld n %[ t an x   
exi/s1 & inde$ x does n exi/4 ,euclid   
evid5tly int5d$ ! po/ulate to ass]t t   
?is l9e jo9+ two po9ts is unique1 :i* is

 only ano!r way ( say+ t two po9ts      
det]m9e a /raight l9e1 & really includes
 ! idea t two /raight l9es _c inclose   
space4 ,= purposes ( in/ruction1 !      
po/ulate w\ld be cle>] if x read1 ,"o   
/raight l9e1 & only "o1 c be drawn ?r\gh
 two giv5 po9ts4  #b4 ,to produce a     
f9ite /raight l9e cont9u\sly 9 a /raight
 l9e4  ,9 ?is po/ulate ,euclid          
practically assumes t a /raight l9e c be
 produc$ only 9 a /raight l9e2 9 o!r    
words1 t two diff]5t /raight l9es _c h a
 common segm5t4 ,sev]al attempts h be5  
made to prove ?is fact1 b )\t any m>k$  
success4  #c4 ,to describe a circle )   
any c5t] & radius4  #d4 ,t all "r angles
 are equal to "o ano!r4       ,:ile ?is 
po/ulate ass]ts ! ess5tial tru? t a "r  
angle is a det]m9ate magnitude s t x    
really s]ves z      an inv>iable /&>d by
 :i* o!r 7acute & obtuse7 angles        
may be measur$1 mu* m ?an ?is is impli$1
 z w easily      be se5 f ! foll[+      
consid]ation4 ,if ! /atem5t is      to  
be prov$1 x c only be prov$ by ! me?od (

 apply+      "o pair ( "r angles to     
ano!r & s argu+ _!      equality4 ,b ?is
 me?od w\ld n be valid unless on !      
assumption ( ! inv>iability ( figures1  
:i* w\ld h      to be ass]t$ z an       
antec$5t po/ulate4 ,euclid pref]r$ to   
ass]t z a po/ulate1 directly1 ! fact t  
all "r angles      are equal2 & h5ce 8  
po/ulate mu/ be tak5 z equival5t      to
 ! pr9ciple ( inv>iability ( figures1 or
 its      equival5t1 ! homog5eity (     
space4#dh  ,x is bett] educational      
policy1 h[ev]1 to ass]t ?is fact m      
def9itely1 & to /ate ! additional       
assumption t figures may be mov$ ab\t 9 
space )\t de=mation4 ,! f\r? ( ,euclid's
 po/ulates is (t5 giv5 z an axiom1      
foll[+ ! idea ( ! ,greek philosoph]     
,gem9us 7:o fl\ri%$ 9 ! fir/ c5tury     
,b4,c471 b ?is is because ,euclid's     
di/9ction betwe5 axiom & po/ulate is n  
always und]/ood4 ,proclus 7#daj-dhe     
,a4,d47 endeavor$ to prove ! po/ulate1 &
 a lat] & m sci5tific ef=t 0 made by !  
,italian geometrician ,sac*]i           

7#affg-agcc74 ,x is v commonly tak5 z a 
po/ulate t all /raight angles are equal1
 ?is be+ m evid5t to ! s5ses1 & !       
equality ( "r angles is d$uc$ z a       
coroll>y4 ,?is me?od ( proc$ure has !   
sanction ( _m ( our be/ mod]n s*ol>s4   
#e4 ,t1 if a /raight l9e fall+ on two   
/raight l9es make ! int]ior angle on !  
same side less ?an two "r angles1 ! two 
/raight l9es1 if produc$ indef9itely1   
meet on t side on :i* are ! angles less 
?an ! two "r angles4  ,?is fam\s        
po/ulate1 long s9ce ab&on$ 9 tea*+ !    
beg9n] 9 geometry1 is a rem>kable       
evid5ce ( ! cle> vision ( ,euclid4 ,=   
two ?\s& ye>s ma!maticians s\ght to     
prove x1 only to demon/rate ! wisdom (  
its au?or 9 plac+ x among !             
assumptions4#di ,e pro( adduc$ conta9s  
"s assumption t practically conceals !  
po/ulate itself4 ,?us ! great ,engli%   
ma!matician ,john ,wallis 7#afaf-agjc7  
gave a pro( bas$ ^u ! assumption t giv5 
a figure1 ano!r figure is possible :i*  
is simil> to ! giv5 "o1 & ( any size    

:atev]4 ,leg5dre 7#ageb-ahcc7 did       
sub/antially ! same at "o "t1 & (f]$    
sev]al o!r pro(s1 ea* dep5d+ ^u "s      
equally unprovable assumption4 ,!       
def9ite pro( t ! po/ulate _c be         
demon/rat$ is due to ! ,italian         
,beltrami 7#ahfh74  ,( ! alt]native =ms 
( ! po/ulate1 t ( ,proclus is g5]ally   
consid]$ ! be/ suit$ to beg9n]s4 ,z /at$
 by ,playfair 7#agie71 ?is is1 ,?r\gh a 
giv5 po9t only "o p>allel c be drawn to 
a giv5 /raight l9e2 & z /at$ by         
,proclus1 ,if a /raight l9e int]sect "o 
( two p>allels1 x w int]sect ! o!r also4
 ,playfair's =m is n[ ! common po/ulate 
( p>allels1 & is ! "o t seems de/9$ to  
endure4  ,posidonius & ,gem9us1 bo?     
,/oics ( ! fir/ c5tury ,b4,c41 gave z _!
 alt]native1 ,!re exi/ /raight l9es     
ev]y:]e equidi/ant f "o ano!r4 ,"o (    
,leg5dre's alt]natives is1 ,!re exi/s a 
triangle 9 :i* ! sum ( ! ?ree angles is 
equal to two "r angles4 ,"o ( ! late/   
attempts to sugge/ a sub/itute is t ( ! 
,italian ,+rami 7#aijd71 ,two p>allel   

/raight l9es int]cept1 on e transv]sal  
:i* passes ?r\gh ! mid-po9t ( a segm5t  
includ$ betwe5 !m1 ano!r segm5t !       
mid-po9t ( :i* is ! mid-po9t ( ! fir/4  
,( c\rse x is entirely possible to      
assume t ?r\gh a po9t m ?an "o l9e c be 
drawn p>allel to a giv5 /raight l9e1 9  
:i* case ano!r type ( geometry c be     
built up1 equally rigor\s ) ,euclid's4  
,?is 0 done at ! close ( ! fir/ qu>t] ( 
! n9ete5? c5tury by ,loba*evsky         
7#agic-ahef7 & ,bolyai 7#ahjb-ahfj71    
result+ 9 ! fir/ ( sev]al non-,euclidean
 geometries4#ej  ,tak+ ! problem to be t
 ( /at+ a reasonably small numb] (      
geometric assumptions t may =m a basis  
to supplem5t ! g5]al axioms1 t %all cov]
 ! mo/ important matt]s to :i* ! /ud5t  
mu/ ref]1 & t %all be s simple z easily 
to be und]/ood by a beg9n]1 ! foll[+ are
 recomm5d$3  #a4 ,"o /raight l9e1 & only
 "o1 c be drawn ?r\gh two giv5 po9ts4   
,?is %\ld also be /at$ = conv5i5ce 9 !  
=m1 ,two po9ts det]m9e a /raight l9e4 ,f
 x may also be drawn ?is coroll>y1 ,two 

/raight l9es c int]sect 9 only "o po9t1 
s9ce two po9ts w\ld det]m9e a /raight   
l9e4 ,su* obvi\s re/atem5ts ( or        
coroll>ies to a po/ulate are to be      
comm5d$1 s9ce a beg9n] is (t5 disc\rag$ 
by hav+ to prove :at is s obvi\s t a    
demon/ration fails to comm5d itself to 8
 m9d4  #b4 ,a /raight l9e may be produc$
 to any requir$ l5g?4 ,?is1 l ,po/ulate 
#a1 requires ! use ( a /raight$ge =     
draw+ ! physical figure4 ,! requir$ l5g?
 is atta9$ by us+ ! compasses to measure
 ! di/ance4 ,! /raight$ge & ! compasses 
are ! only two draw+ in/rum5ts recogniz$
 9 elem5t>y geometry4#ea ,:ile ?is      
involves m ?an ,euclid's po/ulate1 x is 
a bett] work+ assumption = beg9n]s4  #c4
 ,a /raight l9e is ! %orte/ pa? betwe5  
two po9ts4 ,?is is easily prov$ by !    
me?od ( ,euclid#eb = ! case :]e ! pa?s  
are brok5 l9es1 b x is ne$$ z a po/ulate
 = ! case ( curve pa?s4 ,x is a bett]   
/atem5t ?an ! common "o t a /raight l9e 
is ! %orte/ di/ance betwe5 two po9ts2 = 
di/ance is measur$ on a l9e1 b x is n   

itself a l9e4 ,fur!rmore1 !re are       
sci5tific objections to us+ ! ^w di/ance
 any m ?an is necess>y4  #d4 ,a circle  
may be describ$ ) any giv5 po9t z a c5t]
 & any giv5 l9e z a radius4 ,?is        
involves ! use ( ! second ( ! two       
geometric in/rum5ts1 ! compasses4  #e4  
,any figure may be mov$ f "o place to   
ano!r )\t alt]+ ! size or %ape4 ,?is is 
! po/ulate ( ! homog5eity ( space1 &    
ass]ts t space is su* t we may move a   
figure z we please )\t de=mation ( any  
k9d4 ,x is ! basis ( all cases (        
sup]position4  #f4 ,all /raight angles  
are equal4 ,x is possible to prove ?is1 
& !re=e1 f ! /&po9t ( /rict logic1 x is 
unnecess>y z a po/ulate4 ,on ! o!r h&1 x
 is poor educational policy = a beg9n]  
to attempt to prove a ?+ t is s obvi\s4 
,! attempt leads to a loss ( int]e/ 9 ! 
subject1 ! proposition be+ 7to /ate a   
p>adox7 h>d because x is s easy4 ,x is1 
( c\rse1 possible to po/ulate t /raight 
angles are equal1 & to draw ! conclusion
 t _! halves 7"r angles7 are equal2 or  

to proce$ 9 ! opposite direction1 &     
po/ulate t all "r angles are equal1 &   
draw ! conclusion t _! d\bles 7/raight  
angles7 are equal4 ,( ! two ! =m] has ! 
advantage1 s9ce x is probably m obvi\s t
 all /raight angles are equal4 ,x is    
well to /ate ! foll[+ def9ite coroll>ies
 to ?is po/ulate3 7#a7 ,all "r angles   
are equal2 7#b7 ,f a po9t 9 a l9e only  
"o p]p5dicul> c be drawn to ! l9e1 s9ce 
two p]p5dicul>s w\ld make ! :ole 7"r    
angle7 equal to its "p2 7#c7 ,equal     
angles h equal complem5ts1 equal        
supplem5ts1 & equal conjugates2 7#d7 ,! 
great] ( two angles has ! less          
complem5t1 ! less supplem5t1 & ! less   
conjugate4 ,all ( ^! f\r might appe> z  
propositions1 b1 z already /at$1 !y are 
s obvi\s z to be m h>mful ?an useful to 
beg9n]s :5 giv5 9 su* =m4  ,! po/ulate (
 p>allels may prop]ly appe> 9 connection
 ) t topic 9 ,book ,i1 & x is accord+ly 
treat$ 9 ,*apt] ,,xiv4  ,!re is also    
ano!r assumption t "s writ]s are n[ try+
 to =mulate 9 a simple fa%ion4 ,we take1

 = example1 a l9e segm5t ,,ab1 &        
describe circles ) ,a & ,b respectively 
z c5t]s1 & ) a radius ,,ab4 ,we say t ! 
circles w int]sect z at ,c & ,d4 ,b h[ d
 we "k t !y int]sect8 ,we assume x1 j z 
we assume t an indef9ite /raight l9e    
drawn f a po9t inclos$ by a circle w1 if
 produc$ f> en\gh1 cut ! circle twice4  
,( c\rse a pupil w\ld n ?9k ( ?is if 8  
att5tion 0 n call$ to x1 & ! h>m        
outweighs ! good 9 do+ ?is ) "o :o is   
beg9n+ ! /udy ( geometry4  ,) axioms & )
 po/ulates1 !re=e1 ! conclusion is !    
same3 f ! /&po9t ( sci5tific geometry   
!re is an irr$ucible m9imum (           
assumptions1 b f ! /&po9t ( practical   
tea*+ ?is li/ %\ld give place to a work+
 set ( axioms & po/ulates t meet ! ne$s 
( ! beg9n]4       ,bibliography4 ,smi?1 
,tea*+ ( ,elem5t>y ,ma!matics1 ,new     
,york1 #aijj2 ,"y1 ,! ,tea*+ (          
,ma!matics1 ,new ,york1 #aija2          
,moore1 ,on ! ,f\ndations ( ,ma!matics1 
,bullet9 ( !      ,am]ican ,ma!matical  
,society1 #aijc1 p4 #djb2 ,betz1        

,intuition      & ,logic 9 ,geometry1 ,!
 ,ma!matics ,tea*]1 ,vol4 ,,ii1 p4      
#c2 ,hilb]t1 ,! ,f\ndations ( ,geometry1
 ,*icago1 #aijb2 ,vebl51      ,a ,sy/em 
( ,axioms = ,geometry1 ,transactions ( !
 ,am]ican      ,ma!matical ,society1    
#aijd1 p4 #cdc4  ,,footnotes3  #db ,f ! 
,greek ,greek3 ge1 ge 7e>?71  ,greek3   
metre91 metre9 7to measure71 al?\gh !   
sci5ce has n _h to d directly ) !       
measure ( ! e>? = ov] two ?\s& ye>s4    
#dc ,f ! ,arabic al 7!7  jabr           
7re/oration71 ref]r+ to tak+ a quantity 
f "o side ( an equation & !n re/or+ !   
balance by tak+ x f ! o!r side 7see page
 #cg74  #dd ,"o ( ! cle>e/ discussions (
 ! subject is 9 ,w4 ,b4 ,frankl&1 ,!    
,fir/ ,book ( ,euclid's ',elem5ts1' p4  
#bf1 ,cambridge1 #aije4  #de ,grundlag5 
d] ,geometrie1 ,leipzig1 #ahii4 ,see    
,hea?'s ,euclid1 ,vol4 ,i1 p4 #bbi1 = an
 ,engli% v]sion2 also ,d4 ,e4 ,smi?1    
,tea*+ ( ,elem5t>y ,ma!matics1 p4 #bff1 
,new ,york1 #aijj4  #df ,we ne$         
frequ5tly to recall ! fact t ,euclid's  

,elem5ts 0 int5d$ = advanc$ /ud5ts :o   
w5t to ,alex&ria z "y m5 n[ g to        
college1 & t ! book 0 us$ only 9        
univ]sity in/ruction 9 ! ,middle ,ages &
 inde$ until rec5t times4  #dg ,=       
example1 he moves figures )\t de=mation1
 b /ates no po/ulate on ! subject2 & he 
proves t "o side ( a triangle is less   
?an ! sum ( ! o!r two sides1 :5 he might
 h po/ulat$ t a /raight l9e is ! %orte/ 
pa? betwe5 two po9ts4 ,inde$1 8 foll[]s 
7 laugh$ at = prov+ a fact s obvi\s z   
?is "o conc]n+ ! triangle4  #dh ,t4 ,l4 
,hea?1 ,euclid1 ,vol4 ,i1 p4 #bjj4  #di 
,= a resume ( ! be/ kn[n attempts to    
prove ?is po/ulate1 see ,hea?1 ,euclid1 
,vol4 ,i1 p4 #bjb2 ,w4 ,b4 ,frankl&1    
,!ories ( ,p>allelism1 ,cambridge1      
#aiaj4  #ej ,= ! e>ly hi/ory ( ?is      
movem5t see ,engel & ,/aeckel1 ,die     
,!orie d] ,p>allell9i5 von ,euklid bis  
auf ,gauss1 ,leipzig1 #ahie2 ,bonola1   
,sulla teoria delle p>allele e sulle    
geometrie non-euclidee1 9 8 ,que/ioni   
rigu>danti la geometria elem5t>e1 #aijj2

 ,k>agiannides1 ,die ni*teuklidis*e     
,geometrie vom ,alt]?um bis zur         
,geg5w>t1 ,b]l91 #ahic4  #ea ,?is       
limitation ^u elem5t>y geometry 0 plac$ 
by ,plato 7di$ #cdg ,b4,c471 z already  
/at$4  #eb ,book ,i1 ,proposition #bj4  
,,*apt] ,,xii  ,,! ,,def9itions ,,(     
,,geometry   ,:5 we consid] ! nature (  
geometry x is evid5t t m att5tion mu/ be
 paid to accuracy ( def9itions ?an is ! 
case 9 mo/ ( ! o!r sci5ces4 ,! ess5ce ( 
all geometry wor?y ( s]i\s /udy is n ! k
 ( "s fact1 b ! pro( ( t fact2 & ?is    
pro( is always bas$ ^u prec$+ pro(s1    
assumptions 7axioms or po/ulates71 or   
def9itions4 ,if we are to prove t "o l9e
 is p]p5dicul> to ano!r1 x is ess5tial t
 we h an exact def9ition ( p]p5dicul>1  
else we %all n "k :5 we h rea*$ !       
conclusion ( ! pro(4  ,! ess5tial       
features ( a def9ition are t ! t]m def9$
 %all be describ$ 9 t]ms t are simpl]   
?an1 or at lea/ bett] kn[n ?an1 ! ?+    
itself2 t ?is %all be done 9 su* a way z
 to limit ! t]m to ! ?+ def9$2 & t !    

description %all n be r$undant4 ,x w\ld 
n be a good def9ition to say t a "r     
angle is "o f\r? ( a p]igon & "o half ( 
a /raight angle1 because ! concept      
p]igon is n s simple1 & ! t]m p]igon is 
n s well kn[n1 z ! t]m & ! concept "r   
angle1 & because ! def9ition is         
r$undant1 conta9+ m ?an is necess>y4  ,x
 is evid5t t satisfactory def9itions are
 n always possible2 = s9ce ! numb] (    
t]ms is limit$1 !re mu/ be at lea/ "o t 
is at lea/ z simple z any o!r1 & ?is _c 
be describ$ 9 t]ms simpl] ?an itself4   
,su*1 = example1 is ! t]m angle4 ,we c  
easily expla9 ! mean+ ( ?is ^w1 & we c  
make ! concept cle>1 b ?is mu/ be done  
by a c]ta9 am\nt ( circumlocution &     
explanation1 n by a concise & p]fect    
def9ition4 ,unless a beg9n] 9 geometry  
kn[s :at an angle is be=e he reads !    
def9ition 9 a textbook1 he w n "k f !   
def9ition4 ,?is fact ( ! impossibility (
 def9+ "s ( ! fundam5tal concepts w be  
evid5t :5 we come to consid] c]ta9      
attempts t h be5 made 9 ?is direction4  

,x %\ld also be und]/ood 9 ?is          
connection t a def9ition makes no       
ass]tion z to ! exi/5ce ( ! ?+ def9$4   
,if we say t a tang5t to a circle is an 
unlimit$ /raight l9e t t\*es ! circle 9 
"o po9t1 & only "o1 we d n ass]t t x is 
possible to h su* a l9e2 t is a matt] = 
pro(4 ,n 9 all cases1 h[ev]1 c ?is pro( 
be giv51 z 9 ! exi/5ce ( ! simple/      
concepts4 ,we _c1 = example1 prove t a  
po9t or a /raight l9e exi/s aft] we h   
def9$ ^! concepts4 ,we !re=e tacitly or 
explicitly assume 7po/ulate7 ! exi/5ce (
 ^! fundam5tals ( geometry4 ,on ! o!r   
h&1 we c prove t a tang5t exi/s1 & ?is  
may prop]ly be consid]$ a legitimate    
proposition or coroll>y ( elem5t>y      
geometry4 ,9 relation to geometric pro( 
x is necess>y to be> 9 m9d1 !re=e1 t we 
are p]mitt$ to def9e any t]m we please2 
= example1 a sev5-edg$ polyh$ron or     
,leibnitz's t5-fac$ regul> polyh$ron1   
nei!r ( :i* exi/s2 b1 /rictly speak+1 we
 h no "r to make use ( a def9ition 9 a  
pro( until we h %[n or po/ulat$ t ! ?+  

def9$ has an exi/5ce4 ,?is is "o ( !    
/rong features ( ,euclid's textbook4 ,n 
be+ able to prove t a po9t1 a /raight   
l9e1 & a circle exi/s1 he practically   
po/ulates ^! facts2 b he uses no o!r    
def9ition 9 a pro( )\t %[+ t ! ?+ def9$ 
exi/s1 & ?is is 8 reason = m+l+ 8       
problems ) 8 !orems4 ,at ! pres5t "t we 
confess$ly sacrifice 8 logic 9 ?is      
respect = ! reason t we tea* geometry to
 pupils :o are too "y to appreciate t   
logic4  ,x 0 po9t$ out by ,ari/otle1    
long be=e ,euclid1 t x is n a           
satisfactory proc$ure to def9e a ?+ by  
means ( t]ms t are /rictly n prior to x1
 z :5 we attempt to def9e some?+ by     
means ( its opposite4 ,?us to def9e a   
curve z a l9e1 no "p ( :i* is /raight1  
w\ld be a bad def9ition unless /raight  
_h already be5 explicitly def9$2 & to   
def9e bad z n good is unsatisfactory = !
 reason t bad & good are concepts t are 
evolv$ simultane\sly4 ,b all ?is is only
 a detail "u ! g5]al pr9ciple t a       
def9ition mu/ employ t]ms t are bett]   

und]/ood ?an ! "o def9$4  ,x %\ld be    
und]/ood t "s def9itions are mu* m      
important ?an o!rs1 consid]$ f ! po9t ( 
view ( ! logic ( geometry4 ,^? t ent]   
into geometric pro(s are basal2 ^? t =m 
"p ( ! conv]sational language ( geometry
 are n4 ,euclid gave tw5ty-?ree         
def9itions 9 ,book ,i1 & did n make use 
( ev5 all ( ^! t]ms4 ,o!r t]ms1 ^? n    
employ$ 9 8 pro(s1 he assum$ to be kn[n1
 j z he assum$ a k ( any o!r words 9 8  
language4 ,su* proc$ure w\ld n be       
satisfactory "u mod]n conditions1 b x is
 ( great importance t ! tea*] %\ld      
recognize t c]ta9 def9itions are basal1 
:ile o!rs are m]ely in=mational4  ,x is 
n[ propos$ to consid] ! basal def9itions
 ( geometry1 fir/1 t ! tea*] may "k :at 
ones are to be emphasiz$ & le>n$2 &     
second1 t he may "k t ! idea t ! /&>d   
def9itions c easily be improv$ is       
incorrect4 ,x is hop$ t ! result w be ! 
br++ into prom95ce ( ! basal concepts1 &
 ! disc\rag+ ( attempts to *ange 9      
unimportant respects ! def9itions 9 !   

textbook us$ by ! pupil4  ,9 ord] to h a
 sy/ematic basis = "w1 ! def9itions (   
two books ( ,euclid w fir/ be           
consid]$4#ec  #a4 ,,po9t4 ,a po9t is t  
:i* has no "p4 ,?is 0 incorrectly       
translat$ by ,capella 9 ! fif? c5tury1  
,punctum e/ cuius p>s nihil e/ 7a po9t  
is t ( :i* a "p is no?+71 :i* is z mu* z
 to say t ! po9t itself is no?+4 ,x     
g5]ally appe>s1 h[ev]1 z 9 ! ,campanus  
edition1#ed ,punctus e/ cuius p>s non   
e/1 :i* is sub/antially ,euclid's word+4
 ,ari/otle tells ( ! def9itions ( po9t1 
l9e1 & surface t prevail$ 9 8 "t1 say+ t
 !y all def9$ ! prior by means ( !      
po/]ior4#ee ,?us a po9t 0 def9$ z an    
extremity ( a l9e1 a l9e z ! extremity (
 a surface1 & a surface z ! extremity ( 
a solid1--def9itions /ill 9 use & n )\t 
_! value4 ,= x mu/ n be assum$ t        
sci5tific priority is necess>ily        
priority 9 fact2 a *ild kn[s ( solid    
be=e he kn[s ( po9t1 s t x may be a v   
good way to expla91 if n to def9e1 by   
beg9n+ ) solid1 pass+ !nce to surface1  

!nce to l9e1 & !nce to po9t4  ,! fir/   
def9ition ( po9t ( :i* ,proclus c\ld    
le>n is attribut$ by him to !           
,py?agoreans1 namely1 a monad hav+      
position1 ! e>ly =m ( our pres5t popul> 
def9ition ( a po9t z position )\t       
magnitude4 ,plato def9$ x z ! beg9n+ ( a
 l9e1 ?us presuppos+ ! def9ition ( l9e2 
&1 /rangely en\gh1 he anticipat$ by two 
?\s& ye>s ,cavali]i1 ! ,italian geomet]1
 by speak+ ( po9ts z indivisible l9es4  
,to ,ari/otle1 :o prote/$ aga9/ ,plato's
 def9itions1 is due ! def9ition ( a po9t
 z some?+ indivisible b hav+ position4  
,euclid's def9ition is ess5tially t (   
,ari/otle1 & is foll[$ by mo/ mod]n     
textbook writ]s1 except z to its        
omission ( ! ref]5ce to position4 ,x has
 be5 criticiz$ z be+ negative1 :i* has  
no "p2 b x is g5]ally admitt$ t a       
negative def9ition is admissible 9 !    
case ( ! mo/ elem5t>y concepts4 ,=      
example1 bl9d mu/ be def9$ 9 t]ms ( a   
negation4  ,at pres5t n mu* att5tion is 
giv5 to ! def9ition ( po9t1 s9ce ! t]m  

is n us$ z ! basis ( a pro(1 b e ef=t is
 made to h ! concept cle>4 ,x is ! cu/om
 to />t f a small solid1 conceive x to  
decrease 9 size1 & ?9k ( ! po9t z !     
limit to :i* x is approa*+1 us+ ^! t]ms 
9 _! usual s5se )\t fur!r explanation4  
#b4 ,,l9e4 ,a l9e is bread?less l5g?4   
,?is is usually modifi$ 9 mod]n         
textbooks by say+ t a l9e is t :i* has  
l5g? )\t bread? or ?ickness1 a /atem5t t
 is bett] und]/ood by beg9n]s4 ,euclid's
 def9ition is ?\ght to be due to ,plato1
 & is only "o ( _m def9itions t h be5   
sugge/$4 ,! ,py?agoreans hav+ spok5 ( ! 
po9t z a monad naturally 7 l$ to speak (
 ! l9e z dyadic1 or relat$ to two4      
,proclus speaks ( ano!r def9ition1      
magnitude 9 "o dim5sion1 & he gives an  
excell5t illu/ration ( l9e z ! edge ( a 
%ad[1 ?us mak+ x real b n mat]ial4      
,ari/otle speaks ( a l9e z a magnitude  
divisible 9 "o way only1 z contra/$ ) a 
surface :i* is divisible 9 two ways1 & )
 a solid :i* is divisible 9 ?ree ways4  
,proclus also gives ano!r def9ition z ! 

flux ( a po9t1 :i* is sometimes r5d]$ z 
! pa? ( a mov+ po9t4 ,ari/otle _h       
sugge/$ ! idea :5 he wrote1 ,!y say t a 
l9e by its motion produces a surface1 & 
a po9t by its motion a l9e4  ,euclid did
 n deem x necess>y to attempt a         
classification ( l9es1 cont5t+ himself )
 def9+ only a /raight l9e & a circle1 & 
^! are really ! only l9es ne$$ 9        
elem5t>y geometry4 ,8 comm5tators1      
h[ev]1 made ! attempt4 ,= example4 ,h]on
 7fir/ c5tury ,a4,d47 probably foll[$ 8 
def9ition ( l9e by ?is classification3  
,/raight     ,l9es                      
,circul> circumf]5ces            ,n     
/raight  ,spiral %ap$                   
,curv$ 7g5]ally7  ,proclus relates t bo?
 ,plato & ,ari/otle divid$ l9es into    
/raight1 circul>1 & a mixture ( ! two1 a
 /atem5t :i* is n q exact1 b :i* %[s !  
orig9 ( a classification n infrequ5tly  
f\nd 9 rec5t textbooks4 ,gem9us 7ca4 #ej
 ,b4,c47 is said by ,proclus to h giv5  
two classifications1 ( :i* "o w suffice 
= our purposes3             ,composite  

7brok5 l9e =m+ an angle7                
,l9es               ,=m+ a figure1 or   
det]m9ate4 7,circle1                    
ellipse1 cissoid47                      
,incomposite  ,n =m+ a figure1 or       
indet]m9ate &                           
ext5d+ )\t a limit4 7,/raight           
l9e1 p>abola1 hyp]bola1 con*oid47  ,(   
c\rse 8 view ( ! cissoid1 ! curve       
repres5t$ by ! equation y#b7;a  x7  7a -
 x7#c1 is n ! mod]n view4  #c4 ,!       
extremities ( a l9e are po9ts4 ,?is is n
 a def9ition 9 ! s5se ( its two         
pr$ecessors4 ,a mod]n writ] w\ld put x z
 a note "u ! def9ition ( l9e4 ,euclid   
did n wi% to def9e a po9t z ! extremity 
( a l9e1 = ,ari/otle _h ass]t$ t ?is 0 n
 sci5tific2 s he def9$ po9t & l9e1 & !n 
add$ ?is /atem5t to %[ ! relation ( "o  
to ! o!r4 ,ari/otle _h improv$ ^u ?is by
 /at+ t ! division ( a l9e1 z well z an 
extremity1 is a po9t1 z is also !       
int]section ( two l9es4 ,^! /atem5ts1 if
 !y _h be5 made by ,euclid1 w\ld h      
avoid$ ! objection made by ,proclus1 t  

"s l9es h no extremities1 z1 = example1 
a circle1 & also a /raight l9e ext5d+   
inf9itely 9 bo? directions4  #d4        
,,/raight ,,l9e4 ,a /raight l9e is t :i*
 lies ev5ly ) respect to ! po9ts on     
itself4 ,?is is ! lea/ satisfactory (   
all ( ! def9itions ( ,euclid1 &         
emphasizes ! fact t ! /raight l9e is !  
mo/ difficult to def9e ( ! elem5t>y     
concepts ( geometry4 ,:at is meant by   
lies ev5ly8 ,:o w\ld "k :at a /raight   
l9e is1 f ?is def9ition1 if he did n "k 
9 advance8  ,! anci5ts sugge/$ _m       
def9itions ( /raight l9e1 & x is well to
 consid] a few 9 ord] to appreciate !   
difficulties involv$4 ,plato spoke ( x z
 t ( :i* ! middle cov]s ! ends1 mean+ t 
if look$ at endways1 ! middle w\ld make 
x impossible to see ! remote end4 ,?is  
is (t5 modifi$ to read t a /raight l9e  
:5 look$ at endways appe>s z a po9t1--an
 idea t involves ! po/ulate t our l9e ( 
sight is /raight4 ,ar*im$es made !      
/atem5t t ( all ! l9es :i* h ! same     
extremities1 ! /raight l9e is ! lea/1 & 

?is has be5 modifi$ by lat] writ]s into 
! /atem5t t a /raight l9e is ! %orte/   
di/ance betwe5 two po9ts4 ,?is is op5 to
 two objections z a def9ition3 7#a7 a   
l9e is n di/ance1 b di/ance is ! l5g? ( 
a l9e1--x is measur$ on a l9e2 7#b7 x is
 m]ely /at+ a prop]ty ( a /raight l9e to
 say t a /raight l9e is ! %orte/ pa?    
betwe5 two po9ts1--a prop] po/ulate b n 
a good def9ition4 ,equally objectionable
 is "o ( ! def9itions sugge/$ by bo?    
,h]on & ,proclus1 t a /raight l9e is a  
l9e t is /ret*$ to its utt]mo/2 = ev5 !n
 x is reasonable to ?9k ( x z a cat5>y1 
al?\gh ,proclus d\btless _h 9 m9d !     
,ar*im$es /atem5t4 ,he also /at$ t a    
/raight l9e is a l9e su* t if any "p ( x
 is 9 a plane1 ! :ole ( x is 9 !        
plane1--a def9ition t runs 9 a circle1  
s9ce plane is def9$ by means ( /raight  
l9e4 ,proclus also def9es x z a uni=m   
l9e1 capable ( slid+ along itself1 b ?is
 is also true ( a circle4  ,( ! v>i\s   
def9itions two ( ! be/ g back to ,h]on1 
ab\t ! beg9n+ ( our era4 ,proclus gives 

"o ( !m 9 ?is =m1 ,t l9e :i*1 :5 its    
ends rema9 fix$1 itself rema9s fix$4    
,h]on propos$ to add1 :5 x is1 z x 71   
turn$ r\nd 9 ! same plane4 ,?is has be5 
modifi$ into t :i* does n *ange its     
position :5 x is turn$ ab\t its         
extremities z poles1 & appe>s 9         
sub/antially ?is =m 9 ! works (         
,leibnitz & ,gauss4 ,! def9ition ( a    
/raight l9e z su* a l9e z1 ) ano!r      
/raight l9e1 does n inclose space1 is   
only a modification ( ?is "o4 ,! o!r    
def9ition ( ,h]on /ates t 9 a /raight   
l9e all its p>ts fit on all 9 all ways1 
& ?is 9 its mod]n =m is p]haps ! mo/    
satisfactory ( all4 ,9 ?is mod]n =m x   
may be /at$1 ,a l9e su* t any "p1 plac$ 
) its ends on any o!r "p1 mu/ lie :olly 
9 ! l9e1 is call$ a /raight l9e1 9 :i* !
 =ce ( ! ^w mu/ %\ld be not$4 ,?is :ole 
hi/orical discussion goes to %[ h[      
futile x is to attempt to def9e a       
/raight l9e4 ,:at is ne$$ is t we %\ld  
expla9 :at is meant by a /raight l9e1 t 
we %\ld illu/rate x1 & t pupils %\ld !n 

read ! def9ition und]/&+ly4  #e4        
,,surface4 ,a surface is t :i* has l5g? 
& bread?4 ,?is is sub/antially ! common 
def9ition ( our mod]n textbooks4 ,z )   
l9e1 s ) surface1 ! def9ition is n      
entirely satisfactory1 & ! *ief         
consid]ation is t ! mean+ ( ! t]m %\ld  
be made cle> by explanations &          
illu/rations4 ,! %ad[ ca/ on a table top
 is a good illu/ration1 s9ce all idea ( 
?ickness is want+4 ,x adds to ! und]/&+ 
( ! concept to introduce ,ari/otle's    
/atem5t t a surface is g5]at$ by a mov+ 
l9e1 modifi$ by say+ t x may be s       
g5]at$1 s9ce ! l9e might slide along its
 own trace1 or1 z is commonly said 9    
ma!matics1 along itself4  #f4 ,!        
extremities ( a surface are l9es4 ,?is  
is op5 to ! same explanation & objection
 z def9ition #c1 & is n usually giv5 9  
mod]n textbooks4 ,proclus calls att5tion
 to ! fact t ! /atem5t is h>dly true = a
 complete sph]ical surface4  #g4        
,,plane4 ,a plane surface is a surface  
:i* lies ev5ly ) ! /raight l9es on      

itself4 ,euclid "h foll[s 8 def9ition ( 
/raight l9e1 ) a result t is equally    
unsatisfactory4 ,= tea*+ purposes !     
translation f ! ,greek is n cle> to a   
beg9n]1 s9ce lies ev5ly is a t]m n      
simpl] ?an ! "o def9$4 ,z ) ! def9ition 
( a /raight l9e1 s ) t ( a plane1 num]\s
 ef=ts at improvem5t h be5 made4        
,proclus1 foll[+ a h9t ( ,h]on's1 def9es
 x z ! surface :i* is /ret*$ to ! utmo/1
 & also1 ?is "t influ5c$ by ,ar*im$es's 
assumption conc]n+ a /raight l9e1 z !   
lea/ surface among all ^? :i* h ! same  
extremities4 ,h]on gave "o ( ! be/      
def9itions1 ,a surface all ! p>ts ( :i* 
h ! prop]ty ( fitt+ on ea* o!r4 ,!      
def9ition t has met ) ! wide/           
acceptance1 h[ev]1 is a modification (  
"o due to ,proclus1 ,a surface su* t a  
/raight l9e fits on all p>ts ( x4       
,proclus else:]e says1 ,a plane surface 
is su* t ! /raight l9e fits on x all    
ways1 & ,h]on gives x 9 ?is =m1 ,a plane
 surface is su* t1 if a /raight l9e pass
 ?r\gh two po9ts on x1 ! l9e co9cides ) 

x at e spot1 all ways4 ,9 mod]n =m ?is  
appe>s z foll[s3 ,a surface su* t a     
/raight l9e jo9+ any two ( its po9ts    
lies :olly 9 ! surface is call$ a plane1
 & = tea*+ purposes we h no bett]       
def9ition4 ,x is (t5 kn[n z ,simson's   
def9ition1 hav+ be5 giv5 by ,rob]t      
,simson 9 #agef4  ,! ,fr5* ma!matician1 
,f\ri]1 propos$ to def9e a plane z =m$  
by ! aggregate ( all ! /raight l9es :i*1
 pass+ ?r\gh "o po9t on a /raight l9e 9 
space1 are p]p5dicul> to t l9e4 ,?is is 
cle>1 b x is n s usable = beg9n]s z     
,simson's def9ition4 ,x appe>s z a !orem
 9 _m rec5t geometries4 ,! ,g]man       
ma!matician1 ,crelle1 def9$ a plane z a 
surface conta9+ all ! /raight l9es      
7?r\gh\t _! :ole l5g?7 pass+ ?r\gh a    
fix$ po9t & also int]sect+ a /raight l9e
 9 space1 b ( c\rse ?is int]sect$       
/raight l9e mu/ n pass ?r\gh ! fix$     
po9t4 ,crelle's def9ition is            
occasionally se5 9 mod]n textbooks1 b x 
is n s cle> to ! pupil z ,simson's4 ,( !
 v>i\s ultrasci5tific def9itions ( a    

plane t h be5 sugge/$ ( late x is h>dly 
( use to speak 9 a book conc]n$ prim>ily
 ) practical tea*+4 ,no "o ( !m is      
adapt$ to ! ne$s & ! compreh5sion ( !   
beg9n]1 & x seems t we are n likely to  
improve ^u ! s-call$ ,simson =m4  #h4   
,,plane ,,angle4 ,a plane angle is !    
incl9ation to ea* o!r ( two l9es 9 a    
plane :i* meet ea* o!r & d n lie 9 a    
/raight l9e4 ,?is def9ition1 x w be     
notic$1 includes curvil9e> angles1 & !  
expression & d n lie 9 a /raight l9e    
/ates t ! l9es mu/ n be cont9u\s "o ) ! 
o!r1 t is1 t z]o & /raight angles are   
exclud$4 ,s9ce ,euclid does n use !     
curvil9e> angle1 & x is only ! rectil9e>
 angle ) :i* we are conc]n$1 we w pass  
to ! next def9ition & consid] ?is "o 9  
connection !re)4  #i4 ,,rectil9e>       
,,angle4 ,:5 ! l9es conta9+ ! angle are 
/raight1 ! angle is call$ rectil9e>4    
,?is def9ition1 tak5 ) ! prec$+ "o1 has 
always be5 a subject ( criticism4 ,9 !  
fir/ place x expressly excludes !       
/raight angle1 &1 inde$1 ! angles (     

,euclid are always less ?an #ahj deg41  
contr>y to our mod]n concept4 ,9 !      
second place x def9es angle by means ( !
 ^w incl9ation1 :i* is itself z         
difficult to def9e z angle4 ,to rem$y ^!
 defects _m sub/itutes h be5 propos$4   
,apollonius def9$ angle z a contract+ ( 
a surface or a solid at "o po9t "u a    
brok5 l9e or surface4 ,ano!r ( ! ,greeks
 def9$ x z a quantity1 namely1 a di/ance
 betwe5 ! l9es or surfaces conta9+ x4   
,s*ott5#ef says t ! def9itions ( angle  
g5]ally fall into ?ree gr\ps3  a4 ,an   
angle is ! diff]5ce ( direction betwe5  
two l9es t meet4 ,?is is no bett] ?an   
,euclid's1 s9ce diff]5ce ( direction is 
z difficult to def9e z incl9ation4  b4  
,an angle is ! am\nt ( turn+ necess>y to
 br+ "o side to ! position ( ! o!r side4
  c4 ,an angle is ! portion ( ! plane   
includ$ betwe5 its sides4  ,( ^!1 b is  
giv5 by way ( explanation 9 mo/ mod]n   
textbooks4 ,inde$1 we _c d bett] ?an    
simply to def9e an angle z ! op5+ betwe5
 two l9es :i* meet1 & !n expla9 :at is  

meant by size1 ?r\gh ! br++ 9 ( ! idea (
 rotation4 ,?is is a simple pres5tation1
 x is easily und]/ood1 & x is           
suffici5tly accurate = ! real purpose 9 
m9d1 namely1 ! grasp+ ( ! concept4 ,we  
%\ld frankly ackn[l$ge t ! concept (    
angle is su* a simple "o t a            
satisfactory def9ition is impossible1 & 
we %\ld !re=e conf9e our att5tion to    
hav+ ! concept und]/ood4  #aj4 ,:5 a    
/raight l9e set up on a /raight l9e     
makes ! adjac5t angles equal to "o      
ano!r1 ea* ( ! equal angles is "r1 & !  
/raight l9e /&+ on ! o!r is call$ a     
p]p5dicul> to t on :i* x /&s4 ,we at    
pres5t sep>ate ^! def9itions & simplify 
! language4  #aa4 ,an obtuse angle is an
 angle great] ?an a "r angle4  #ab4 ,an 
acute angle is an angle less ?an a "r   
angle4  ,! "q sometimes ask$ z to :e!r  
an angle ( #bjj deg4 is obtuse1 & :e!r a
 negative angle1 say -#ij deg41 is      
acute1 is answ]$ by say+ t ,euclid did n
 conceive ( angles equal to or great]   
?an #ahj deg4 & _h no notion ( negative 

quantities4 ,g5]ally to-day we def9e an 
obtuse angle z great] ?an "o & less ?an 
two "r angles4 ,an acute angle is def9$ 
z an angle less ?an a "r angle1 & is    
consid]$ z positive "u ! g5]al und]/&+ t
 all geometric magnitudes are positive  
unless ! contr>y is /at$4  #ac4 ,a      
b\nd>y is t :i* is an extremity ( any?+4
 ,! def9ition is n exactly satisfactory1
 = a circle is ! b\nd>y ( ! space       
inclos$1 b we h>dly consid] x z !       
extremity ( t space4 ,euclid wi%es !    
def9ition be=e ,no4 #ad4  #ad4 ,a figure
 is t :i* is conta9$ by any b\nd>y or   
b\nd>ies4 ,! def9ition is n             
satisfactory1 s9ce x excludes ! unlimit$
 /raight l9e1 ! angle1 an assemblage (  
po9ts1 & o!r comb9ations ( l9es & po9ts 
:i* we %\ld n[ consid] z figures4  #ae4 
,a circle is a plane figure conta9$ by  
"o l9e su* t all ! /raight l9es fall+ ^u
 x f "o po9t among ^? ly+ )9 ! figure   
are equal to "o ano!r4  #af4 ,& ! po9t  
is call$ ! c5t] ( ! circle4  ,"s        
comm5tators add aft] "o l9e1 def9ition  

#ae1 ! words :i* is call$ ! circumf]5ce1
 b ^! are n 9 ! olde/ manuscripts4 ,!   
,greek idea ( a circle 0 usually t ( "p 
( a plane :i* is b\nd$ by a l9e call$ 9 
mod]n times ! circumf]5ce1 al?\gh       
,ari/otle us$ circle z synonym\s ) !    
b\nd+ l9e4 ,) ! gr[? ( mod]n ma!matics1 
h[ev]1 & p>ticul>ly z a result ( !      
developm5t ( analytic geometry1 ! ^w    
circle has come to mean ! b\nd+ l9e1 z x
 did ) ,ari/otle1 a c5tury be=e         
,euclid's "t4 ,?is has gr[n out ( !     
equations ( ! v>i\s curves1 x#b  y#b    
r#b repres5t+ ! circle-l9e1 a#by#b      
b#bx#b  a#b;b#b repres5t+ ! ellipse-l9e1
 & s on4 ,x is natural1 !re=e1 t circle1
 ellipse1 p>abola1 & hyp]bola %\ld all  
be look$ ^u z l9es4 ,s9ce ?is is ! mod]n
 use ( circle 9 ,engli%1 x has naturally
 f\nd its way into elem5t>y geometry1 9 
ord] t /ud5ts %\ld n h to =m an entirely
 diff]5t idea ( circle on beg9n+        
analytic geometry4 ,! g5]al body (      
,am]ican tea*]s1 !re=e1 at pres5t favors
 us+ circle to mean ! b\nd+ l9e &       

circumf]5ce to mean ! l5g? ( t l9e4 ,?is
 requires r$ef9+ area ( a circle1 & ?is 
is done by say+ t x is ! area ( ! plane 
space inclos$4 ,! matt] is n ( greate/  
consequ5ce1 b tea*]s w probably pref] to
 jo9 9 ! mod]n ,am]ican usage ( ! t]m4  
#ag4 ,,diamet]4 ,a diamet] ( ! circle is
 any /raight l9e drawn ?r\gh ! c5t] &   
t]m9at$ 9 bo? directions by !           
circumf]5ce ( ! circle1 & su* a /raight 
l9e also bisects ! circle4 ,! ^w diamet]
 is f two ,greek words mean+ a ?r\gh    
measur]1 & x 0 also us$ by ,euclid = !  
diagonal ( a squ>e1 & m g5]ally = !     
diagonal ( any p>allelogram4 ,! ^w      
diagonal is a lat] t]m & means ! ?r\gh  
angle4 ,x w be notic$ t ,euclid adds to 
! usual def9ition ! /atem5t t a diamet] 
bisects ! circle4 ,he does ?is app>5tly 
to ju/ify 8 def9ition 7#ah71 ( a        
semicircle 7a half circle74  ,?ales is  
said to h be5 ! fir/ to prove t a       
diamet] bisects ! circle1 ?is be+ "o (  
?ree or f\r propositions def9itely      
attribut$ to him1 & x is sometimes giv5 

z a proposition to be prov$4 ,z a       
proposition1 h[ev]1 x is unsatisfactory1
 s9ce ! pro( ( :at is s evid5t usually  
in/ills m d\bt ?an c]ta9ty 9 ! m9ds (   
beg9n]s4  #ah4 ,,semicircle4 ,a         
semicircle is ! figure conta9$ by !     
diamet] & ! circumf]5ce cut (f by x4 ,& 
! c5t] ( ! semicircle is ! same z t ( ! 
circle4 ,proclus rem>k$ t ! semicircle  
is ! only plane figure t has its c5t] on
 its p]imet]4 ,"s writ]s object to def9+
 a circle z a l9e & !n speak+ ( ! area (
 a circle1 %[+ m9ds t h at lea/ "o      
*>act]i/ic ( t ( ,proclus4 ,! mod]n     
def9ition ( semicircle is half ( a      
circle1 t is1 an arc ( #ahj deg41 al?\gh
 ! t]m is commonly us$ to mean bo? ! arc
 & ! segm5t4  #ai4 ,,rectil9e>          
,,figures4 ,rectil9e> figures are ^? :i*
 are conta9$ by /raight l9es1 trilat]al 
figures be+ ^? conta9$ by ?ree1         
quadrilat]al ^? conta9$ by f\r1 &       
multilat]al ^? conta9$ by m ?an f\r1    
/raight l9es4  #bj4 ,( trilat]al        
figures1 an equilat]al triangle is t :i*

 has its ?ree sides equal1 an isosceles 
triangle t :i* has two ( its sides alone
 equal1 & a scal5e triangle t :i* has   
its ?ree sides unequal4  #ba4 ,fur!r1 ( 
trilat]al figures1 a "r-angl$ triangle  
is t :i* has a "r angle1 an obtuse-angl$
 triangle t :i* has an obtuse angle1 &  
an acute-angl$ triangle t :i* has its   
?ree angles acute4  ,^! ?ree def9itions 
may prop]ly be consid]$ toge!r4         
,rectil9e> is f ! ,lat9 translation ( ! 
,greek eu?ygrammos1 & means "r-l9$1 or  
/raight-l9$4 ,euclid's idea ( su* a     
figure is t ( ! space inclos$1 :ile !   
mod]n idea is t5d+ to become t ( !      
inclos+ l9es4 ,9 elem5t>y geometry1     
h[ev]1 ! ,euclidean idea is /ill held4  
,trilat]al is f ! ,lat9 translation ( ! 
,greek tripleuros 7?ree-sid$74 ,9       
elem5t>y geometry ! ^w triangle is m    
commonly us$1 al?\gh quadrilat]al is m  
common ?an quadrangle4 ,! use ( ^! two  
diff]5t =ms is ecc5tric & is m]ely a    
matt] ( fa%ion4 ,?us we speak ( a       
p5tagon b n ( a tetragon or a trigon1   

al?\gh bo? words are correct 9 =m4 ,! ^w
 multilat]al 7_m-sid$7 is a translation 
( ! ,greek polypleuros4 ,fa%ion has     
*ang$ ?is to polygonal 7_m-angl$71 ! ^w 
multilat]al r>ely be+ se54  ,( !        
triangles1 equilat]al means equal-sid$2 
isosceles is f ! ,greek isoskeles1 mean+
 ) equal legs1 & scal5e f skal5os1      
possibly f skazo 7to limp71 or f skolios
 7crook$74 ,euclid's limitation (       
isosceles to a triangle ) two1 & only   
two1 equal sides w\ld n n[ be accept$4  
,we are at pres5t m giv5 to g5]aliz+ ?an
 he 01 & :5 we h prov$ a proposition    
relat+ to ! isosceles triangle1 we wi%  
to say t we h !reby prov$ x = !         
equilat]al triangle4 ,we !re=e say t an 
isosceles triangle has two sides equal1 
leav+ x possible t all ?ree sides %\ld  
be equal4 ,! expression equal legs is n[
 be+ disc>d$ on ! score ( inelegance4 ,9
 place ( "r-angl$ triangle mod]n writ]s 
speak ( "r triangle1 & s = ! obtuse &   
acute triangles4 ,! t]ms are brief] &   
are z readily und]/ood4 ,x may add a    

little int]e/ to ! subject to "k t      
,plut>* tells u t ! anci5ts ?\ght t !   
p[] ( ! triangle is expressive ( !      
nature ( ,pluto1 ,bac*us1 & ,m>s4 ,he   
also /ates t ! ,py?agoreans call$ !     
equilat]al triangle ! head-born ,m9]va &
 ,tritog5eia 7born ( ,triton7 because x 
may be equally divid$ by ! p]p5dicul>   
l9es drawn f ea* ( its angles4  #bb4 ,( 
quadrilat]al figures a squ>e is t :i* is
 bo? equilat]al & "r-angl$2 an oblong t 
:i* is "r-angl$ b n equilat]al2 a       
rhombus t :i* is equilat]al & n         
"r-angl$2 & a rhomboid t :i* has its    
opposite sides & angles equal to "o     
ano!r1 b is nei!r equilat]al nor        
"r-angl$4 ,& let all quadrilat]als o!r  
?an ^! be call$ trapezia4 ,9 ?is        
def9ition ,euclid also specializes 9 a  
mann] n n[ g5]ally approv$4 ,?us we are 
m apt to-day to omit ! oblong & rhomboid
 z unnecess>y1 & to def9e rhombus 9 su* 
a mann] z to include a squ>e4 ,we use   
p>allelogram to cov] rhomboid1 rhombus1 
oblong1 & squ>e4 ,= oblong we use       

rectangle1 lett+ x include squ>e4       
,euclid's def9ition ( squ>e illu/rates 8
 fre$om 9 /at+ m attributes ?an are     
necess>y1 9 ord] to make sure t !       
concept is cle>2 = he might h said t x  
is t :i* is equilat]al & has "o "r      
angle4 ,we may pr(it by 8 me?od1        
sacrific+ logic to educational          
necessity4 ,euclid does n use oblong1   
rhombus1 rhomboid1 & trapezium 7plural1 
trapezia7 9 8 pro(s1 s t he might well h
 omitt$ ! def9itions1 z we (t5 d4  #bc4 
,,p>allels4 ,p>allel /raight l9es are   
/raight l9es :i*1 be+ 9 ! same plane &  
be+ produc$ indef9itely 9 bo?           
directions1 d n meet "o ano!r 9 ei!r    
direction4 ,?is def9ition ( p>allels1   
simplifi$ 9 its language1 is ! "o       
commonly us$ to-day4 ,o!r def9itions h  
be5 sugge/$1 b none has be5 s g5]ally   
us$4 ,proclus /ates t ,posidonius gave !
 def9ition bas$ ^u ! l9es always be+ at 
! same di/ance ap>t4 ,gem9us has ! same 
idea 9 8 def9ition4 ,!re are1 z ,s*ott5 
has po9t$ out1 ?ree g5]al types (       

def9itions ( p>allels1 namely3  a4 ,!y h
 no po9t 9 common4 ,?is may be express$ 
by say+ t 7#a7 !y d n int]sect1 7#b7 !y 
meet at inf9ity4  b4 ,!y are equidi/ant 
f "o ano!r4  c4 ,!y h ! same direction4 
,( ^!1 ! fir/ is ,euclid's1 ! idea ( !  
po9t at inf9ity be+ sugge/$ by ,kepl]   
7#afjd74 ,! second "p ( ?is def9ition   
is1 ( c\rse1 unusable = beg9n]s4 ,dr4   
7n[ ,sir ,?omas7 ,hea? says1 ,x seems   
be/1 !re=e1 to leave to high] geometry !
 conception ( inf9itely di/ant po9ts on 
a l9e & ( two /raight l9es meet+ at     
inf9ity1 l imag9>y po9ts ( int]section1 
&1 = ! purposes ( elem5t>y geometry1 to 
rely on ! pla9 di/9ction betwe5         
'p>allel' & 'cutt+1' :i* av]age human   
intellig5ce c readily grasp4  ,!        
direction def9ition seems to h orig9at$ 
) ,leibnitz4 ,x is op5 to ! s]i\s       
objection t direction is n easy (       
def9ition1 & t x is us$ v loosely4 ,if  
two p on diff]5t m]idians travel due    
nor?1 d !y travel 9 ! same direction8 on
 p>allel l9es8 ,! def9ition is z        

objectionable z t ( angle z ! diff]5ce (
 direction ( two int]sect+ l9es4  ,f ^! 
def9itions ( ! fir/ book ( ,euclid we   
see 7#a7 :at a small numb] ,euclid      
consid]$ z basal2 7#b7 :at a *ange has  
tak5 place 9 ! g5]alization ( concepts2 
7#c7 h[ ! language has v>i$4 ,nev]!less 
we are n to be comm5d$ if we adh]e to   
,euclid's small numb]1 because geometry 
is n[ taught to pupils ^: vocabul>y is  
limit$4 ,x is necess>y to def9e m t]ms1 
& to scatt] ! def9itions ?r\gh ! "w =   
use z !y are ne$$1 in/ead ( mass+ !m at 
! beg9n+1 z 9 a diction>y4 ,! mo/       
important lesson to be le>n$ f ,euclid's
 def9itions is t only ! basal ones1     
relatively few 9 numb]1 ne$ to be le>n$1
 & ^! because !y are us$ z ! f\ndations 
^u :i* pro(s are built4 ,x %\ld also be 
notic$ t ,euclid expla9s no?+ 9 ^!      
def9itions2 !y are h>d /atem5ts ( fact1 
mass$ at ! beg9n+ ( 8 treatise4 ,n      
always z /atem5ts1 & n at all 9 _!      
arrangem5t1 are !y suit$ to ! ne$s ( our
 boys & girls at pres5t4  ,hav+ consid]$

 ,euclid's def9itions ( ,book ,i1 x is  
prop] to turn to "s ( ^? t]ms t h be5   
add$ f "t to "t to 8 li/1 & are n[      
usually incorporat$ 9 ,am]ican          
textbooks4 ,x w be se5 t mo/ ( ^! 7     
assum$ by ,euclid to be kn[n by 8 mature
 read]s4 ,!y ne$ to be def9$ = "y p1 b  
mo/ ( !m are n basal1 t is1 !y are n us$
 9 ! pro(s ( propositions4 ,"s ( ^!     
t]ms1 su* z magnitudes1 curve l9e1 brok5
 l9e1 curvil9e> figure1 bisector1       
adjac5t angles1 reflex angles1 oblique  
angles & l9es1 & v]tical angles1 ne$    
m]ely a ^w ( explanation s t !y may be  
us$ intellig5tly4 ,if !y 7 num]\s en\gh 
to make x wor? ! :ile1 !y c\ld be       
classifi$ 9 our textbooks z ( m9or      
importance1 b su* a c\rse w\ld cause m  
tr\ble ?an x is wor?4  ,o!r t]ms h come 
into use 9 mod]n times t are n common   
expressions ) :i* /ud5ts are famili>4   
,su* a t]m is /raight angle1 a concept n
 us$ by ,euclid1 b "o t adds s mat]ially
 to ! int]e/ & value ( geometry z n[ to 
be g5]ally recogniz$4 ,!re is also ! ^w 

p]igon1 mean+ ! :ole angul> space ab\t a
 po9t4 ,?is 0 exclud$ by ! ,greeks      
because _! idea ( angle requir$ x to be 
less ?an a /raight angle4 ,! ^w means   
ar\nd angle1 & is ! be/ "o t has be5    
co9$ = ! purpose4 ,flat angle & :ole    
angle are among ! names sugge/$ = ^! two
 mod]n concepts4 ,! t]ms complem5t1     
supplem5t1 & conjugate1 mean+ ! diff]5ce
 betwe5 a giv5 angle & a "r angle1      
/raight angle1 & p]igon respectively1 h 
also ent]$ our vocabul>y & ne$ def9+4   
,!re are also c]ta9 t]ms express+       
relation%ip :i* ,euclid does n def9e1 & 
:i* h be5 s *ang$ 9 rec5t times z to    
require c>eful def9ition at pres5t4     
,*ief among ^! are ! words equal1       
congru5t1 & equival5t4 ,euclid us$ !    
s+le ^w equal = all ?ree concepts1      
al?\gh "s ( 8 rec5t editors h *ang$ x to
 id5tically equal 9 ! case ( congru5ce4 
,9 mod]n spee* we use ! ^w equal        
commonly to mean l-valu$1 hav+ ! same   
measure1 z :5 we say ! circumf]5ce ( a  
circle equals a /raight l9e ^: l5g? is  

#bpir1 al?\gh x c\ld n co9cide ) x4 ,(  
late1 !re=e1 9 ,europe & ,am]ica1 &     
:]ev] ,european influ5ce rea*es1 ! ^w   
congru5t is com+ into use to mean       
id5tically equal 9 ! s5se ( sup]posable4
 ,we !re=e speak ( congru5t triangles & 
congru5t p>allelograms z be+ ^? t are   
sup]posable4  ,x is a little un=tunate t
 equal has come to be s loosely us$ 9   
ord9>y conv]sation t we _c keep x to    
mean congru5t2 b our language w n p]mit 
x1 & we are =c$ to use ! new] ^w4 ,:5ev]
 x c be us$ )\t misund]/&+1 h[ev]1 x    
%\ld be reta9$1 z 9 ! case ( equal      
/raight l9es1 equal angles1 & equal arcs
 ( ! same circle4 ,! ma!matical &       
educational _w w nev] cons5t to use     
congru5t /raight l9es1 or congru5t      
angles1 = ! reason t ! t]ms are         
unnecess>ily long1 no misund]/&+ be+    
possible :5 equal is us$4  ,! ^w        
equival5t 0 introduc$ by ,leg5dre at !  
close ( ! eighte5? c5tury to indicate   
equality ( l5g?1 or ( area1 or ( volume4
 ,euclid _h said1 ,p>allelograms :i* are

 on ! same base & 9 ! same p>allels are 
equal to "o ano!r1 :ile ,leg5dre & 8    
foll[]s w\ld modify ! word+ some:at &   
introduce equival5t = equal4 ,?is usage 
has be5 reta9$4 ,congru5t polygons are  
!re=e necess>ily equival5t1 b equival5t 
polygons are n 9 g5]al congru5t4        
,congru5t polygons h mutually equal     
sides & mutually equal angles1 :ile     
equival5t polygons h no equality save t 
( area4  ,9 g5]al1 z already /at$1 ^! & 
o!r t]ms %\ld be def9$ j be=e !y are us$
 in/ead ( at ! beg9n+ ( geometry4 ,!    
reason = ?is1 f ! educational /&po9t &  
consid]+ ! pres5t position ( geometry 9 
! curriculum1 is app>5t4  ,we %all n[   
consid] ! def9itions ( ,euclid's ,book  
,,iii1 :i* is usually tak5 z ,book ,,ii 
9 ,am]ica4  #a4 ,,equal ,,circles4      
,equal circles are ^? ! diamet]s ( :i*  
are equal1 or ! radii ( :i* are equal4  
,manife/ly ?is is a !orem1 = x ass]ts t 
if ! radii ( two circles are equal1 !   
circles may be made to co9cide4 ,9 "s   
textbooks a pro( is giv5 by             

sup]position1 & ! pro( is legitimate1 b 
,euclid usually avoid$ sup]position if  
possible4 ,nev]!less he might z well h  
prov$ ?is z t two triangles are congru5t
 if two sides & ! includ$ angle ( ! "o  
are respectively equal to ! correspond+ 
p>ts ( ! o!r1 & he might z well h       
po/ulat$ ! latt] z to h sub/antially    
po/ulat$ ?is fact4 ,= 9 reality ?is     
def9ition is a po/ulate1 & x 0 s        
consid]$ by ! great ,italian ma!matician
 ,t>taglia 7ca4 #aejj-;ca4 #aeeg74 ,!   
plan usually foll[$ 9 ,am]ica to-day is 
to consid] ?is z "o ( _m unprov$        
propositions1 too evid5t1 inde$1 = pro(1
 accept$ by intuition4 ,! result is a   
loss 9 ! logic ( ,euclid1 b ! me?od is  
?\ght to be bett] adapt$ to ! m9d ( !   
y\?ful le>n]4 ,x is int]e/+ to note 9   
?is connection t ! ,greeks _h no ^w =   
radius1 & 7 !re=e compell$ to use "s su*
 phrase z ! /raight l9e f ! c5t]1 or1   
briefly1 ! f ! c5t]1 z if f ! c5t] 7 "o 
^w4  #b4 ,,tang5t4 ,a /raight l9e is    
said to t\* a circle :i*1 meet+ ! circle

 & be+ produc$1 does n cut ! circle4    
,tea*]s :o pref] to use circumf]5ce     
in/ead ( circle = ! l9e %\ld notice h[  
(t5 su* phrases z cut ! circle &        
int]sect+ circle are us$1--phrases t    
signify no?+ unless circle is tak5 to   
mean ! l9e4 ,s ,ari/otle uses an        
expression mean+ t ! locus ( a c]ta9    
po9t is a circle1 & he speaks ( a circle
 z pass+ ?r\gh all ! angles4 ,our ^w t\*
 is f ! ,lat9 tang]e1 f :i* comes       
tang5t1 & also tag1 an old t\*+ game4   
#c4 ,,tang5t ,,circles4 ,circles are    
said to t\* "o ano!r :i*1 meet+ "o      
ano!r1 d n cut "o ano!r4  ,! def9ition  
has n be5 look$ ^u z entirely           
satisfactory1 ev5 aside f its un=tunate 
phraseology4 ,x is n c]ta91 = in/ance1  
:e!r ,euclid meant t ! circles c\ld n   
cut at "s o!r po9t ?an t ( tang5cy4     
,fur!rmore1 no di/9ction is made betwe5 
ext]nal & int]nal contact1 al?\gh bo?   
=ms are us$ 9 ! propositions4 ,mod]n    
textbook mak]s f9d x conv5i5t to def9e  
tang5t circles z ^? t are tang5t to !   

same /raight l9e at ! same po9t1 & to   
def9e ext]nal & int]nal tang5cy by      
ref]5ce to _! position ) respect to !   
l9e1 al?\gh ?is may be *>act]iz$ z op5  
to ab\t ! same objection z ,euclid's4   
#d4 ,,di/ance4 ,9 a circle /raight l9es 
are said to be equally di/ant f ! c5t]1 
:5 ! p]p5dicul>s drawn to !m f ! c5t]   
are equal4  ,x is n[ cu/om>y to def9e   
di/ance f a po9t to a l9e z ! l5g? ( !  
p]p5dicul> f ! po9t to ! l9e1 & to d ?is
 9 ,book ,i4 ,9 high] ma!matics x is    
f\nd t di/ance is n a satisfactory t]m  
to use1 b ! objections to x h no        
p>ticul> significance 9 elem5t>y        
geometry4  #e4 ,,great] ,,di/ance4 ,& t 
/raight l9e is said to be at a great]   
di/ance on :i* ! great] p]p5dicul>      
falls4  ,su* a def9ition is n ?\ght     
ess5tial at ! pres5t "t4  #f4 ,,segm5t4 
,a segm5t ( a circle is ! figure conta9$
 by a /raight l9e & ! circumf]5ce ( a   
circle4  ,! ^w segm5t is f ! ,lat9 root 
sect1 mean+ cut4 ,s we h sector 7a      
cutt]71 section 7a cut71 int]sect1 & s  

on4 ,! ^w is n limit$ to a circle2 we h 
long spok5 ( a sph]ical segm5t1 & x is  
common to-day to speak ( a l9e segm5t1  
to :i* "s w\ld apply a new "n sect4 ,!re
 is little confusion 9 ! matt]1 h[ev]1 =
 ! context %[s :at k9d ( a segm5t is to 
be und]/ood1 s t ! ^w sect is r p$antic 
?an important4 ,x w be notic$ t ,euclid 
"h uses circumf]5ce to mean arc4  #g4   
,,angle ,,( ,a ,,segm5t4 ,an angle ( a  
segm5t is t conta9$ by a /raight l9e & a
 circumf]5ce ( a circle4  ,?is t]m has  
entirely dropp$ out ( geometry1 & few   
tea*]s w\ld "k :at x meant if !y %\ld   
he> x us$4 ,proclus call$ su* angles    
mix$4  #h4 ,,angle ,,9 ,a ,,segm5t4 ,an 
angle 9 a segm5t is ! angle :i*1 :5 a   
po9t is tak5 on ! circumf]5ce ( ! segm5t
 & /raight l9es are jo9$ f x to !       
extremities ( ! /raight l9e :i* is !    
base ( ! segm5t1 is conta9$ by ! /raight
 l9es s jo9$4  ,su* an involv$ def9ition
 w\ld n be usable to-day4 ,moreov]1 !   
words circumf]5ce ( ! segm5t w\ld n be  
us$4  #i4 ,& :5 ! /raight l9es conta9+ !

 angle cut (f a circumf]5ce1 ! angle is 
said to /& ^u t circumf]5ce4  #aj4      
,,sector4 ,a sector ( a circle is !     
figure :i*1 :5 an angle is con/ruct$ at 
! c5t] ( ! circle1 is conta9$ by !      
/raight l9es conta9+ ! angle & !        
circumf]5ce cut (f by !m4  ,!re is no   
reason = su* an ext5d$ def9ition1 our   
mod]n phraseology be+ bo? m exact 7z se5
 9 ! above use ( circumf]5ce = arc7 & m 
intelligible4 ,! ,greek ^w = sector is  
knife 7tomeus71 sector be+ ! ,lat9      
translation4 ,a sector is suppos$ to    
resemble a %oemak]'s knife1 & h5ce !    
significance ( ! t]m4 ,euclid foll[$ ?is
 by a def9ition ( simil> sectors1 a t]m 
n[ g5]ally ab&on$ z unnecess>y4  ,x w be
 notic$ t ,euclid did n use or def9e !  
^w polygon4 ,he uses rectil9e> figure   
in/ead4 ,polygon may be def9$ to be a   
b\nd+ l9e1 z a circle is n[ def9$1 or z 
! space inclos$ by a brok5 l9e1 or z a  
figure =m$ by a brok5 l9e1 ?us includ+  
bo? ! limit$ plane & its b\nd>y4 ,x is n
 ( any great consequ5ce geometrically   

:i* ( ^! ideas is adopt$1 s t ! usual   
def9ition ( a portion ( a plane b\nd$ by
 a brok5 l9e may be tak5 z suffici5t =  
elem5t>y purposes4 ,x is prop] to call  
att5tion1 h[ev]1 to ! fact t we may h   
cross polygons ( v>i\s types1 & t ! l9e 
t b\nds ! polygon mu/ be cont9u\s1 z !  
def9ition /ates4 ,t is1 9 ! second ( ^! 
figures ! %ad$ portion is n consid]$ a  
polygon4 ,su* special cases are n liable
 to arise1 b if que/ions relat+ to !m   
are sugge/$1 ! tea*] %\ld be prep>$ to  
answ] !m4 ,if sugge/$ to a class1 a note
 ( ?is k9d %\ld come out only           
incid5tally z a bit ( int]e/1 & %\ld n  
occupy mu* "t nor be unduly emphasiz$4  
,illu/ration  ,x may also be m5tion$ to 
a class at "s conv5i5t "t t ! old idea (
 a polygon 0 t ( a convex figure1 & t ! 
mod]n idea1 :i* is met 9 high]          
ma!matics1 leads to a modification (    
e>li] concepts4 ,= example1 "h is a     
quadrilat]al ) "o ( its diagonals1 ,,bd1
 outside ! figure4 ,fur!rmore1 if we    
consid] a quadrilat]al z a figure =m$ by

 f\r int]sect+ l9es1 ,,ac1 ,,cf1 ,,be1 &
 ,,ea1 x is app>5t t ?is g5]al          
quadrilat]al has six v]tices1 ,a1 ,b1   
,c1 ,d1 ,e1 ,f1 & ?ree diagonals1 ,,ad1 
,,bf1 & ,,ce4 ,su* broad] ideas (       
geometry =m ! basis ( :at is call$ mod]n
 elem5t>y geometry4  ,illu/ration  ,!   
o!r def9itions ( plane geometry ne$ n be
 discuss$1 s9ce all t h any hi/orical   
int]e/ h be5 consid]$4 ,on ! :ole x may 
be said t our def9itions to-day are n 9 
g5]al s c>efully consid]$ z ^? (        
,euclid1 :o weigh$ ea* ^w ) greate/     
skill1 b !y are m tea*able to beg9n]s1 &
 are1 on ! :ole1 m satisfactory f !     
educational /&po9t4 ,! greate/ lesson to
 be le>n$ f ?is discussion is t ! numb] 
( basal def9itions to be le>n$ =        
subsequ5t use is v small4  ,s9ce tea*]s 
are occasionally di/urb$ ov] ! =m 9 :i* 
def9itions are /at$1 x is well to say a 
few words ^u ?is subject4 ,!re are      
sev]al /&>d types t may be us$4 7#a7 ,we
 may use ! diction>y =m1 putt+ ! ^w     
def9$ fir/1 ?us3 ,"r triangle4 ,a       

triangle t has "o ( its angles a "r     
angle4 ,?is is sci5tifically correct1 b 
x is n a complete s5t5ce1 & h5ce x is n 
easily repeat$ :5 x has to be quot$ z an
 au?ority4 7#b7 ,we may put ! ^w def9$  
at ! end1 ?us3 ,a triangle t has "o (   
its angles a "r angle is call$ a "r     
triangle4 ,?is is m satisfactory4 7#c7  
,we may comb9e 7#a7 & 7#b71 ?us3 ,"r    
triangle4 ,a triangle t has "o ( its    
angles a "r angle is call$ a "r         
triangle4 ,?is is /ill bett]1 = x has ! 
cat*word at ! beg9n+ ( ! p>agraph4  ,!re
 is occasionally "s m5tal agitation ov] 
! trivial ?+s ( a def9ition1 su* z ! use
 ( ! words is call$4 ,x w\ld n be a v   
s]i\s matt] if !y 7 omitt$1 b x is bett]
 to h !m !re4 ,! reason is t !y m>k !   
/atem5t at once z a def9ition4 ,=       
example1 suppose we say t a triangle t  
has "o ( its angles a "r angle is a "r  
triangle4 ,we h also ! fact t a triangle
 ^: base is ! diamet] ( a semicircle &  
^: v]tex lies on ! semicircle is a "r   
triangle4 ,! /yle ( /atem5t is ! same1 &

 we h no?+ 9 ! phraseology to %[ t !    
fir/ is a def9ition & ! second a !orem4 
,?is may happ5 ) mo/ ( ! def9itions1 &  
h5ce ! mo/ c>eful writ]s h n cons5t$ to 
omit ! di/9ctive words 9 "q4  ,apropos (
 ! def9itions ( geometry1 ! great ,fr5* 
philosoph] & ma!matician1 ,pascal1 set  
=? c]ta9 rules relat+ to ?is subject1 z 
also to ! axioms employ$1 & ^! may      
prop]ly sum up ?is *apt]4  #a4 ,d n     
attempt to def9e t]ms s well kn[n 9     
!mselves t !re are no simpl] t]ms by :i*
 to express !m4  #b4 ,admit no obscure  
or equivocal t]ms )\t def9+ !m4  #c4    
,use 9 ! def9itions only t]ms t are     
p]fectly und]/ood or are !re expla9$4   
#d4 ,omit no necess>y pr9ciples )\t     
g5]al agreem5t1 h[ev] cle> & evid5t !y  
may be4  #e4 ,set =? 9 ! axioms only ^? 
?+s t are 9 !mselves p]fectly evid5t4   
#f4 ,d n attempt to demon/rate any?+ t  
is s evid5t 9 itself t !re is no?+ m    
simple by :i* to prove x4  #g4 ,prove   
:atev] is 9 ! lea/ obscure1 us+ 9 !     
demon/ration only axioms t are p]fectly 

evid5t 9 !mselves1 or propositions      
already demon/rat$ or all[$4  #h4 ,9    
case ( any unc]ta9ty aris+ f a t]m      
employ$1 always sub/itute m5tally !     
def9ition = ! t]m itself4               
,bibliography4 ,hea?1 ,euclid1 z cit$2  
,frankl&1 ,! ,fir/      ,book ( ,euclid1
 z cit$2 ,smi?1 ,tea*+ ( ,elem5t>y      
,ma!matics1 p4 #beg1 ,new ,york1 #aijj2 
,"y1 ,tea*+ (      ,ma!matics1 p4 #ahi1 
,new ,york1 #aijg2 ,vebl51 ,on          
,def9itions1 9      ! ,moni/1 #aijc1 p4 
#cjc4  ,,footnotes3  #ec ,free use has  
be5 made ( ,w4 ,b4 ,frankl&1 ,! ,fir/   
,book ( ,euclid's ',elem5ts1'           
,cambridge1 #aije2 ,t4 ,l4 ,hea?1 ,!    
,?irte5 ,books ( ,euclid's ',elem5ts1'  
,cambridge1 #aijh2 ,h4 ,s*ott51 ,inhalt 
und ,me?ode des planimetris*5           
,unt]ri*ts1 ,leipzig1 #ahic2 ,m4 ,simon1
 ,euclid und die se*s planimetris*5     
,bue*]1 ,leipzig1 #aija4  #ed ,= a      
facsimile ( a ?irte5?-c5tury ,,ms4      
conta9+ ?is def9ition1 see ! au?or's    
,r>a ,ari?metica1 ,plate ,,iv1 ,bo/on1  

#aiji4  #ee ,our slang expression ,! c>t
 be=e ! horse is sugge/ive ( ?is        
proc$ure4  #ef ,loc4 cit41 ,vol4 ,,ii1  
p4 #id4     ,,*apt] ,,xiii  ,,h[ ,,to   
,,attack ,,! ,,ex]cises   ,! old        
geometry1 say ( a c5tury ago1 usually   
consi/$1 z has be5 /at$1 ( a s]ies (    
!orems fully prov$ & ( problems fully   
solv$4 ,dur+ ! n9ete5? c5tury ex]cises 7
 gradually introduc$1 ?us develop+      
geometry f a sci5ce 9 :i* "o le>n$ by   
see+ ?+s done1 into "o 9 :i* he ga9$ p[]
 by actually do+ ?+s4 ,( ! nature ( ^!  
ex]cises 7orig9als1 rid]s71 & ( _!      
gradual *ange 9 ! pa/ few ye>s1 m5tion  
has be5 made 9 ,*apt] ,,vii4 ,x n[      
rema9s to consid] ! me?ods ( attack+ ^! 
ex]cises4  ,x is evid5t t !re is no s+le
 me?od1 & ?is is a =tunate fact1 s9ce if
 x 7 n s1 ! attack w\ld be too me*anical
 to be int]e/+4 ,!re is no "o rule =    
solv+ e problem nor ev5 = see+ h[ to    
beg94 ,on ! o!r h&1 a pupil is sav$ "s  
"t by hav+ 8 att5tion call$ to a few r  
def9ite l9es ( attack1 & he w und\bt$ly 

f>e ! bett] by n wa/+ 8 en]gies ov]     
attempts t are 9 advance doom$ to       
failure4  ,!re are two g5]al que/ions to
 be consid]$3 fir/1 z to ! discov]y (   
new tru?s1 & second1 z to ! pro(4 ,) !  
fir/ ! pupil w h little to d1 n hav+ z  
yet arriv$ at ?is /age 9 8 progress4 ,a 
bright /ud5t may take a little int]e/ 9 
see+ :at he c f9d out t is new 7at lea/ 
to him71 & if s1 he may be told t _m new
 propositions h be5 discov]$ by !       
accurate draw+ ( figures2 t "s h be5    
f\nd by actually weigh+ pieces ( %eet   
metal ( c]ta9 sizes2 & t /ill o!rs h    
made !mselves kn[n ?r\gh pap] fold+4 ,9 
all ( ^! cases1 h[ev]1 ! suppos$        
proposition mu/ be prov$ be=e x c be    
accept$4  ,z to ! pro(1 ! pupil usually 
w&]s ab\t m or less until he /rikes ! "r
 l9e1 & !n he foll[s ?is to !           
conclusion4 ,he %\ld n be blam$ = do+   
?is1 = he is pursu+ ! me?od t ! _w      
foll[$ 9 ! e>lie/ times1 & "o t has     
always be5 common & always w be4 ,?is is
 ! syn!tic me?od1 ! build+ up ( ! pro( f

 propositions previ\sly prov$4 ,if !    
proposition is a !orem1 x is usually n  
difficult to recall propositions t may  
lead to ! demon/ration1 & to select !   
ones t are really ne$$4 ,if x is a      
problem1 x is usually easy to look ahead
 & see :at is necess>y = ! solution & to
 select ! prec$+ propositions accord+ly4
  ,b pupils %\ld be told t if !y d n r  
easily f9d ! necess>y propositions = !  
con/ruction or ! pro(1 !y %\ld n delay 9
 resort+ to ano!r & m sy/ematic me?od4  
,?is is kn[n z ! me?od ( analysis1 & x  
is applicable bo? to !orems & to        
problems4 ,x has sev]al =ms1 b x is (   
little s]vice to a pupil to h ^!        
diff]5tiat$1 & x suffices t he be giv5 !
 ess5tial feature ( all ^! =ms1 a       
feature t goes back to ,plato & 8 s*ool 
9 ! fif? c5tury ,b4,c4  ,= a !orem1 !   
me?od ( analysis consi/s 9 reason+ z    
foll[s3 ,i c prove ?is proposition if ,i
 c prove ?is ?+2 ,i c prove ?is ?+ if ,i
 c prove t2 ,i c prove t if ,i c prove a
 ?ird ?+1 & s ! reason+ runs until !    

pupil comes to ! po9t :]e he is able to 
add1 b ,i c prove t4 ,?is does n prove !
 proposition1 b x enables him to rev]se 
! process1 beg9n+ ) ! ?+ he c prove &   
go+ back1 /ep by /ep1 to ! ?+ t he is to
 prove4 ,analysis is1 !re=e1 8 me?od (  
discov]y ( ! way 9 :i* he may arrange 8 
syn!tic pro(4 ,pupils (t5 wond] h[ any  
"o ev] came to "k h[ to arrange ! pro(s 
( geometry1 & ?is answ]s ! "q4 ,"s "o   
guess$ t a /atem5t 0 true2 he appli$    
analysis & f\nd t he c\ld prove x2 he !n
 appli$ syn!sis & did prove x4  ,= a    
problem1 ! me?od ( analysis is mu* !    
same z 9 ! case ( a !orem4 ,two ?+s are 
involv$1 h[ev]1 in/ead ( "o1 = "h we mu/
 make ! con/ruction & !n prove t ?is    
con/ruction is correct4 ,! pupil1 !re=e1
 fir/ supposes ! problem solv$1 & sees  
:at results foll[4 ,he !n rev]ses !     
process & sees if he c atta9 ^! results 
& ?us effect ! requir$ con/ruction4 ,if 
s1 he /ates ! process & gives ! result+ 
pro(4 ,= example3       ,9 a triangle   
,,abc1 to draw ,,pq p>allel to ! base   

,,ab1      cutt+ ! sides 9 ,p & ,q1 s t 
,,pq %all equal      ,,ap  ,,bq4        
,illu/ration       ,analysis4 ,assume ! 
problem solv$4       ,!n ,,ap mu/ equal 
"s "p ( ,,pq z ,,px1 & ,,bq mu/         
equal ,,qx4       ,b if ,,ap  ,,px1 :at 
mu/ ,l,,pxa equal8       because ,,pq is
  ,,ab1 :at does ,l,,pxa equal8         
,!n :y mu/ ,l,,bax  ,l,,xap8            
,simil>ly1 :at ab\t ,l,,qbx & ,l,,xba8  
,con/ruction4 ,n[ rev]se ! process4 ,:at
 may we d to ,ls      ,a & ,b 9 ord] to 
fix ,x8 ,!n h[ %all ,,pq be drawn8      
,n[ give ! pro(4  ,illu/ration          
,illu/ration  ,! ?ird g5]al me?od (     
attack applies *iefly to problems :]e "s
 po9t is to be det]m9$4 ,?is is ! me?od 
( ! int]section ( loci4 ,?us1 to locate 
an electric light at a po9t eighte5 feet
 f ! po9t ( int]section ( two /reets &  
equidi/ant f !m1 evid5tly "o locus is a 
circle ) a radius eighte5 feet & ! c5t] 
at ! v]tex ( ! angle made by ! /reets1 &
 ! o!r locus is ! bisector ( ! angle4 ,!
 me?od is also occasionally applicable  

to !orems4 ,= example1 to prove t !     
p]p5dicul> bisectors ( ! sides ( a      
triangle are concurr5t4 ,"h ! locus (   
po9ts equidi/ant f ,a & ,b is ,,pp'1 & !
 locus ( po9ts equidi/ant f ,b & ,c is  
,,qq'4 ,^! c easily be %[n to int]sect1 
z at ,o4 ,!n ,o1 be+ equidi/ant f ,a1   
,b1 & ,c1 is also on ! p]p5dicul>       
bisector ( ,,ac4 ,!re=e ^! bisectors are
 concurr5t 9 ,o4  ,^! are ! *ief me?ods 
( attack1 & are all t %\ld be giv5 to an
 av]age class = practical use4  ,besides
 ! me?ods ( attack1 !re are a few g5]al 
directions t %\ld be giv5 to pupils4    
#a4 ,9 attack+ ei!r a !orem or a        
problem1 take ! mo/ g5]al figure        
possible4 ,?us1 if a proposition relates
 to a quadrilat]al1 take "o ) unequal   
sides & unequal angles r ?an a squ>e or 
ev5 a rectangle4 ,! simpl] figures (t5  
deceive a pupil into feel+ t he has a   
pro(1 :5 9 reality he has "o only = a   
special case4  #b4 ,set =? v exactly !  
?+ t is giv51 us+ lett]s relat+ to !    
figure t has be5 drawn4 ,!n set =? ) !  

same exactness ! ?+ t is to be prov$4 ,!
 neglect to d ?is is ! cause ( a l>ge p]
 c5t ( ! failures4 ,! kn[+ ( exactly :at
 we h to d & exactly :at we h ) :i* to d
 x is half ! battle4  #c4 ,if !         
proposition seems hazy1 ! difficulty is 
probably ) ! word+4 ,9 ?is case try     
sub/itut+ ! def9ition = ! "n ( ! ?+     
def9$4 ,?us in/ead ( ?9k+ too long ab\t 
prov+ t ! m$ian to ! base ( an isosceles
 triangle is p]p5dicul> to ! base1 draw 
! figure & ?9k t !re is giv5      ,,ac  
,,bc1     ,,ad  ,,bd1  & t !re is to be 
prov$ t      ,l,,cda  ,l,,bdc4          
,illu/ration  ,"h we h replac$ m$ian1   
isosceles1 & p]p5dicul> by /atem5ts t   
express ! same idea 9 simpl] language4  
,bibliography4 ,pet]s51 ,me?ods &       
,!ories = ! ,solution      ( ,geometric 
,problems ( ,con/ruction1 ,cop5hag51    
#ahgi1 a      curi\s piece ( ,engli% &  
an extreme view ( ! subject1      b well
 wor? consult+2 ,alex&r(f1 ,problemes de
 geometrie      elem5taire1 ,p>is1      
#ahii1 ) a ,g]man translation 9 #aijc2  

,loomis1 ,orig9al ,inve/igation2 or1 ,h[
 to attack an ,ex]cise      9 ,geometry1
 ,bo/on1 #aija2 ,sauvage1 ,les ,lieux   
geometriques en      geometrie          
elem5taire1 ,p>is1 #ahic2 ,hadam>d1     
,lecons de      geometrie1 p4 #bfa1     
,p>is1 #ahih2 ,duhamel1 ,des ,me?odes   
dans les      sci5ces de raisonnem5t1   
#c;e ed41 ,p>is1 #ahhe2 ,h5rici &       
,treutle91 ,lehrbu* d]                  
,elem5t>-,geometrie1 ,leipzig1 #c4      
,aufl41      #ahig2 ,h5rici1 ,congru5t  
,figures1 ,london1 #ahgi4     ,,*apt]   
,,xiv  ,,book ,i ,,& ,,its              
,,propositions   ,hav+ consid]$ ! nature
 ( ! geometry t we h inh]it$1 & "s ( !  
opportunities = improv+ ^u ! me?ods (   
pres5t+ x1 ! next "q t arises is !      
all-important "o ( ! subject matt]1 ,:at
 %all geometry be 9 detail8 ,%all x be !
 text or ! sequ5ce ( ,euclid8 ,few      
tea*]s h any su* idea at ! pres5t "t4   
,%all x be a m]e dabbl+ ) =ms t are se5 
9 me*anics or ar*itecture1 ) no s]i\s   
logical sequ5ce8 ,?is is an equally     

dang]\s extreme4 ,%all x be an entirely 
new /yle ( geometry bas$ ^u gr\ps (     
motions8 ,?is may sometime be develop$1 
b z yet x exi/s 9 ! future if x exi/s at
 all1 s9ce ! rec5t ef=ts 9 ?is respect  
are g5]ally q z ill suit$ to a "y pupil 
z is ,euclid's ,elem5ts itself4  ,no "o 
c d5y ! tru? ( ,m4 ,b\rlet's rec5t      
ass]tion t ,indu/ry1 daught] ( ! sci5ce 
( ! n9ete5? c5tury1 reigns to-day !     
mi/ress ( ! _w2 %e has trans=m$ all     
anci5t me?ods1 & %e has absorb$ 9 h]self
 almo/ all human activity4#eg ,nei!r c  
"o d5y ! ju/ice ( 8 comp>ison ( ,euclid 
) a noble piece ( ,go?ic ar*itecture & (
 8 ass]tion t z mod]n life dem&s ano!r  
type ( build+1 s x dem&s ano!r type (   
geometry4  ,b :at does ?is mean8 ,t     
geometry is to exi/ m]ely z x t\*es     
indu/ry1 or t bad ar*itecture is to     
replace ! good8 ,by no means4 ,a build+ 
%\ld to-day h /eam heat & elevators &   
electric lights1 b x %\ld be con/ruct$ (
 j z endur+ mat]ials z ! ,p>!non1 & x   
%\ld h l9es z pleas+ z ^? ( a ,go?ic    

facade4 ,ar*itecture %\ld /ill be       
arti/ic & con/ruction %\ld /ill be      
sub/antial1 else a build+ c nev] endure4
 ,s geometry mu/ /ill exemplify good    
logic & mu/ /ill br+ to ! pupil a feel+ 
( exaltation1 or x w p]i% & become a m]e
 relic 9 ! museum ( human culture4      
,:at1 !n1 %all ! propositions ( geometry
 be1 & 9 :at mann] %all !y answ] to !   
*all5ge ( ! indu/rial epo* 9 :i* we     
live8 ,9 reply1 !y mu/ be bett] adapt$  
to "y m9ds & to all "y m9ds ?an ,euclid 
ev] int5d$ 8 own propositions to be4    
,fur!rmore1 !y mu/ h a ri*ness (        
application to pure geometry1 9 ! way ( 
c>efully *os5 ex]cises1 t ,euclid nev]  
attempt$4 ,& f9ally1 !y mu/ h           
application to ?is same life ( indu/ry (
 :i* we h spok51 :5ev] ?is c really be  
f\nd1 b !re mu/ be no %am & pret5se ab\t
 x1 else ! v hone/y t p]meat$ ! anci5t  
geometry w seem to ! pupil to be want+ 9
 ! :ole subject4#eh  ,until "s geometry 
on a radically diff]5t basis %all appe>1
 & ( ?is !re is no v hopeful sign at    

pres5t1 ! propositions w be ! ess5tial  
ones ( ,euclid1 exclud+ ^? t may be     
consid]$ m]ely intuitive1 & exclud+ all 
t are too difficult = ! pupil :o to-day 
takes up _! /udy4 ,! numb] w be limit$ 9
 a reasonable way1 & e g5u9e type (     
application w be plac$ be=e ! tea*] to  
be us$ z necessity requires4 ,b a fair  
am\nt ( logic w be reta9$1 & ! ef=t to  
make ( geometry an empty bauble ( a     
li/less m9d w be reject$ by e wor?y     
tea*]4 ,:at ! propositions %\ld be is a 
matt] ^u :i* op9ions may ju/ly diff]2 b 
9 ?is *apt] !re is set =? a reasonable  
li/ = ,book ,i1 arrang$ 9 a workable    
sequ5ce1 & ?is li/ may fairly be tak5 z 
typical ( :at ! ,am]ican s*ool w        
probably use = _m ye>s to come4 ,) ! li/
 is giv5 a set ( typical applications1 &
 "s ( ! g5]al in=mation t w add to !    
int]e/ 9 ! "w & t %\ld =m "p ( !        
equipm5t ( ! tea*]4  ,an anci5t treatise
 0 usually writt5 on a k9d ( pap] call$ 
papyrus1 made f ! pi? ( a l>ge re$ =m]ly
 common 9 ,egypt1 b n[ gr[+ luxuriantly 

only above ,kh>tum 9 ,upp] ,egypt1 & ne>
 ,syracuse 9 ,sicily2 or else x 0 writt5
 on p>*m5t1 s call$ f ,p]gamos 9 ,asia  
,m9or1 :]e sk9s 7 fir/ prep>$ 9 p>*m5t  
=m2 or occasionally !y 7 writt5 on      
ord9>y lea!r4 ,9 any case !y 7 g5]ally  
writt5 on long /rips ( ! mat]ial us$1 & 
^! 7 roll$ up & ti$4 ,h5ce we h su* an  
expression z keep+ ! roll 9 s*ool1 & su*
 a ^w z volume1 :i* has 9 x ! same root 
z involve 7to roll 971 & evolve 7to roll
 out74 ,sev]al ( ^! rolls 7 (t5 necess>y
 = a s+le treatise1 9 :i* case ea* 0    
ti$1 & all 7 kept toge!r 9 a receptacle 
resembl+ a pail1 or 9 a comp>tm5t on a  
%elf4 ,! ,greeks call$ ea* ( ! sep>ate  
p>ts ( a treatise biblion 7,greek3      
biblion71 a ^w mean+ book4 ,h5ce we h ! 
books ( ! ,bible1 ! books ( ,hom]1 & !  
books ( ,euclid4 ,f ! same root1 inde$1 
comes ,bible1 bibliophile 7booklov]71   
bibliography 7li/ ( books71 & k9dr$     
words4 ,?us ! books ( geometry are !    
l>ge *apt]s ( ! subject1 *apt] be+ f !  
,lat9 caput 7head71 a section "u a new  

head+4 ,!re h be5 ef=ts to *ange books  
to *apt]s1 b !y h n succe$$1 & !re is no
 reason :y !y %\ld succe$1 = ! t]m is   
cle> & has ! sanction ( long usage4     
,,!orem4 ,if two l9es int]sect1 !       
v]tical angles are equal4  ,?is 0       
,euclid's ,proposition #ae1 be+ put s   
late because he bas$ ! pro( ^u 8        
,proposition #ac1 n[ ?\ght to be be/    
tak5 )\t pro(1 namely1 ,if a /raight l9e
 set ^u a /raight l9e makes angles1 x w 
make ei!r two "r angles or angles equal 
to two "r angles4 ,x is f\nd to be bett]
 p$agogy to assume t ?is foll[s f !     
def9ition ( /raight angle1 ) ref]5ce1 if
 necess>y1 to ! mean+ ( ! sum ( two     
angles4 ,?is proposition on v]tical     
angles is probably ! be/ "o ) :i* to    
beg9 geometry1 s9ce x is n s evid5t z to
 seem to ne$ no pro(1 al?\gh "s pref] to
 rank x z semiobvi\s1 :ile ! pro( is s  
simple z easily to be und]/ood4         
,eudemus1 a ,greek :o wrote n long be=e 
,euclid1 attribut$ ! discov]y ( ?is     
proposition to ,?ales ( ,miletus 7ca4   

#fdj-edh ,b4,c471 "o ( ! ,sev5 ,wise ,m5
 ( ,greece1 ( :om ,proclus wrote3 ,?ales
 x 0 :o visit$ ,egypt & fir/ transf]r$  
to ,hell5ic soil ?is !ory ( geometry4   
,he himself1 inde$1 discov]$ mu*1 b /ill
 m did he introduce to 8 successors !   
pr9ciples ( ! sci5ce4  ,! proposition is
 ! only basal "o relat+ to ! int]section
 ( two l9es1 & h5ce !re are no o!rs )   
:i* x is necess>ily gr\p$4 ,?is is !    
reason = plac+ x by itself1 foll[$ by ! 
congru5ce !orems4  ,!re are _m famili>  
illu/rations ( ?is !orem4 ,inde$1 any   
two cross$ l9es1 z 9 a pair ( %e>s or ! 
legs ( a camp /ool1 br+ x to m9d4 ,! ^w 
/raight is "h omitt$ be=e l9es 9        
accordance ) ! mod]n conv5tion t ! ^w   
l9e unmodifi$ means a /raight l9e4 ,(   
c\rse 9 cases ( special emphasis !      
adjective %\ld be us$4  ,,!orem4 ,two   
triangles are congru5t if two sides & ! 
includ$ angle ( ! "o are equal          
respectively to two sides & ! includ$   
angle ( ! o!r4  ,?is is ,euclid's       
,proposition #d1 8 fir/ ?ree            

propositions be+ problems ( con/ruction4
 ,?is w\ld !re=e h be5 8 fir/           
proposition if he _h plac$ 8 problems   
lat]1 z we d to-day4 ,! words congru5t &
 equal are n us$ z 9 ,euclid1 = reasons 
already set =? on page #aea4 ,!re h be5 
_m attempts to re>range ! propositions (
 ,book ,i1 putt+ 9 sep>ate gr\ps ^?     
conc]n+ angles1 ^? conc]n+ triangles1 & 
^? conc]n+ p>allels1 b !y h all fail$1 &
 = ! cog5t reason t su* a s*eme de/roys 
! logical sequ5ce4 ,?is proposition may 
prop]ly foll[ ! "o on v]tical angles    
simply because ! latt] is easi] & does n
 involve sup]position4  ,z f> z         
possible1 ,euclid & all o!r good        
geomet]s avoid ! pro( by sup]position4  
,z a practical te/ sup]position is      
valuable1 b z a !oretical "o x is op5 to
 num]\s objections4 ,z ,peleti] po9t$   
out 9 8 7#aeeg7 edition ( ,euclid1 if ! 
sup]position ( l9es & figures c\ld      
freely be assum$ z a me?od (            
demon/ration1 geometry w\ld be full (   
su* pro(s4 ,!re w\ld be no reason1 =    

example1 :y an angle %\ld n be con/ruct$
 equal to a giv5 angle by sup]pos+ !    
giv5 angle on ano!r "p ( ! plane4       
,inde$1 x is possible t we might !n     
assume to bisect an angle by imag9+ !   
plane fold$ l a piece ( pap]4 ,hea?     
7#aijh7 has po9t$ out a subtle defect 9 
,euclid's pro(1 9 t x is said t because 
two l9es are equal1 !y c be made to     
co9cide4 ,euclid says1 practically1 t if
 two l9es c be made to co9cide1 !y are  
equal1 b he does n say t if two /raight 
l9es are equal1 !y c be made to co9cide4
 ,= ! purposes ( elem5t>y geometry !    
matt] is h>dly wor? br++ to ! att5tion (
 a pupil1 b x %[s t ev5 ,euclid did n   
cov] e po9t4  ,applications ( ?is       
proposition are easily f\nd1 b !y are   
all v mu* alike4 ,!re are doz5s (       
measurem5ts t c be made by simply       
con/ruct+ a triangle t %all be congru5t 
to ano!r triangle4 ,x seems h>dly wor? !
 :ile at ?is "t to d m ?an m5tion "o    
typical case1#ei leav+ x to tea*]s :o   
may f9d x desirable to sugge/ o!rs to _!

 pupils4  ,illu/ration       ,wi%+ to   
measure ! di/ance across a riv]1 "s boys
      sight$ f ,a to a po9t ,p4 ,!y !n  
turn$ & measur$      ,,ab at "r angles  
to ,,ap4 ,!y plac$ a /ake at ,o1        
halfway f ,a to ,b1 & drew a p]p5dicul> 
to ,,ab at      ,b4 ,!y plac$ a /ake at 
,c1 on ?is p]p5dicul>1 & 9      l9e ) ,o
 & ,p4 ,!y !n f\nd ! wid? by measur+    
,,bc4 ,prove t !y 7 "r4   ,?is involves 
! rang+ ( a l9e1 & ! runn+ ( a l9e at "r
 angles to a giv5 l9e1 bo? ( :i* h be5  
describ$ 9 ,*apt] ,,ix4 ,x is also      
fairly accurate to run a l9e at any     
angle to a giv5 l9e by sight+ along two 
p9s /uck 9 a protractor4  ,,!orem4 ,two 
triangles are congru5t if two angles & !
 includ$ side ( ! "o are equal          
respectively to two angles & ! includ$  
side ( ! o!r4  ,euclid comb9es ?is ) 8  
,proposition #bf3       ,if two         
triangles h ! two angles equal to two   
angles      respectively1 & "o side     
equal to "o side1 namely1 ei!r      !   
side adjo9+ ! equal angles1 or t subt5d+

 "o (      ! equal angles1 !y w also h !
 rema9+ sides equal      to ! rema9+    
sides1 & ! rema9+ angle to !      rema9+
 angle4  ,he proves ?is cumb]some       
/atem5t )\t sup]position1 desir+ to     
avoid ?is me?od1 z already /at$1 :5ev]  
possible4 ,! pro( by sup]position is    
old1 h[ev]1 = ,al-,nairizi#fj gives x & 
ascribes x to "s e>li] au?or ^: "n he   
did n "k4 ,proclus tells u t ,eudemus 9 
8 geometrical hi/ory ref]s ?is !orem to 
,?ales4 ,= he says t 9 ! me?od by :i* !y
 say t ,?ales prov$ ! di/ance ( %ips 9 !
 sea1 x 0 necess>y to make use ( ?is    
!orem4 ,h[ ,?ales did ?is is purely a   
matt] ( conjecture1 b he might h /ood on
 ! top ( a t[] ris+ f ! level %ore1 or (
 su* headl&s z ab\nd ne> ,miletus1 & by 
"s simple in/rum5t sight$ to ! %ip4 ,!n1
 turn+1 he might h sight$ along ! %ore  
to a po9t hav+ ! same angle (           
decl9ation1 & !n h measur$ ! di/ance f !
 t[] to ?is po9t4 ,?is seems m          
reasonable ?an any ( ! v>i\s plans      
sugge/$1 & x is f\nd 9 s _m practical   

geometries ( ! fir/ c5tury ( pr9t+ t x  
seems to h long be5 a common exp$i5t4 ,!
 /one a/rolabe f ,mesopotamia1 n[       
pres]v$ 9 ! ,briti% ,museum1 %[s t su*  
in/rum5ts = ! measur+ ( angles are v    
old1 & = ! purposes ( ,?ales ev5 a pair 
( l>ge compasses w\ld h answ]$ v well4  
,an illu/ration ( ! me?od is se5 9      
,belli's "w ( #aefi1 z "h %[n4 ,at ! top
 ( ! picture a man is gett+ ! angle by  
means ( ! visor ( 8 cap2 at ! bottom ( !
 picture a man is us+ a rul] screw$ to a
 /aff4#fa ,! /ory goes t "o (           
,napoleon's eng9e]s won ! imp]ial favor 
by quickly measur+ ! wid? ( a /ream t   
block$ ! progress ( ! army1 us+ ?is v   
me?od4  ,illu/ration3 ,,sixte5?-,,c5tury
 ,,m5suration  ,belli's ,del ,misur> con
 la ,vi/a1 ,v5ice1 #aefi  ,?is          
proposition is ! reciprocal or dual ( ! 
prec$+ "o4 ,! relation betwe5 ! two may 
be se5 f ! foll[+ arrangem5t3       ,two
 triangles are congru5t if two sides & !
 includ$      angle ( ! "o are equal    
respectively to two sides &      !      

includ$ angle ( ! o!r4       ,two       
triangles are congru5t if two angles & !
 includ$      side ( ! "o are equal     
respectively to two angles &      !     
includ$ side ( ! o!r4  ,9 g5]al1 to e   
proposition involv+ po9ts & l9es !re is 
a reciprocal proposition involv+ l9es & 
po9ts respectively t is (t5             
true1--inde$1 t is always true 9 a c]ta9
 l9e ( propositions4 ,?is relation is   
kn[n z ! ,pr9ciple ( ,reciprocity or (  
,duality4 ,in/ead ( po9ts & l9es we h "h
 angles 7sugge/$ by ! v]tex po9ts7 &    
l9es4 ,x is int]e/+ to a class to h     
att5tion call$ to su* relations1 b x is 
n ( suffici5t importance 9 elem5t>y     
geometry to ju/ify m ?an a ref]5ce "h & 
!re4 ,!re are o!r dual features t are   
se5 9 geometry besides ^? giv5 above4   
,,!orem4 ,9 an isosceles triangle !     
angles opposite ! equal sides are equal4
  ,?is is ,euclid's ,proposition #e1 !  
second ( 8 !orems1 b he adds1 & if !    
equal /raight l9es be produc$ fur!r1 !  
angles "u ! base w be equal to "o ano!r4

 ,s9ce1 h[ev]1 he does n use ?is second 
"p1 its g5u95ess is d\bt$4 ,he w\ld n   
admit ! common pro( ( to-day ( suppos+ !
 v]tical angle bisect$1 because !       
problem ab\t bisect+ an angle does n    
prec$e ?is proposition1 & !re=e 8 pro(  
is mu* m involv$ ?an ours4 ,he makes    
,,cx  ,,cy1 & proves triangles,,xbc &   
,,yac congru5t1#fb & also triangles,,xba
 & ,,yab congru5t4 ,!n f ,l,,yac he     
takes ,l,,yab1 leav+ ,l,,bac1 & s on !  
o!r side1 leav+ ,l,,cba1 ^! !re=e be+   
equal4  ,illu/ration  ,?is proposition  
has long be5 call$ ! pons as9orum1 or   
bridge ( asses1 b no "o kn[s :]e or :5 !
 "n arose4 ,x is usually /at$ t x came f
 ! fact t fools c\ld n cross ?is bridge1
 & x is a fact t 9 ! ,middle ,ages ?is 0
 (t5 ! limit ( ! /ud5t's progress 9     
geometry4 ,x has h[ev] be5 sugge/$ t !  
"n came f ,euclid's figure1 :i*         
resembles ! simple/ type ( a wood5 truss
 bridge4 ,! "n is appli$ by ! ,fr5* to !
 ,py?agorean ,!orem4  ,proclus          
attributes ! discov]y ( ?is proposition 

to ,?ales4 ,he also says t ,pappus 7?ird
 c5tury ,a4,d471 a ,greek comm5tator on 
,euclid1 prov$ ! proposition z foll[s3  
,let ,,abc be ! triangle1 ) ,,ab  ,,ac4 
,conceive ( ?is      z two triangles2 !n
 ,,ab  ,,ac1 ,,ac  ,,ab1 & ,l,a is      
common2 h5ce ! triangles,,abc & ,,acb   
are congru5t1 &      ,l,b ( ! "o equals 
,l,c ( ! o!r4  ,?is is a bett] plan ?an 
t foll[$ by "s textbook writ]s ( imag9+ 
triangle,,abc tak5 up & laid d[n on     
itself4 ,ev5 to lay x d[n on its trace  
is m objectionable ?an ! plan ( ,pappus4
  ,,!orem4 ,if two angles ( a triangle  
are equal1 ! sides opposite ! equal     
angles are equal1 & ! triangle is       
isosceles4  ,! /atem5t is1 ( c\rse1     
tautological1 ! la/ five words be+      
unnecess>y f ! ma!matical /&po9t1 b (   
value at ?is /age ( ! /ud5t's progress z
 emphasiz+ ! nature ( ! triangle4       
,euclid /at$ ! proposition ?us1 ,if 9 a 
triangle two angles be equal to "o      
ano!r1 ! sides :i* subt5d ! equal angles
 w also be equal to "o ano!r4 ,he did n 

def9e subt5d1 suppos+ su* words to be   
already und]/ood4 ,?is is ! fir/ case ( 
a conv]se proposition 9 geometry4 ,hea? 
di/+ui%es ! logical f ! geometric       
conv]se4 ,! logical conv]se ( ,euclid   
,i1 #e1 w\ld be t "s triangles ) two    
angles equal are isosceles1 :ile !      
geometric conv]se is ! proposition z    
/at$4 ,proclus call$ att5tion to two =ms
 ( conv]se 7& 9 ! c\rse ( ! "w1 b n at  
?is "t1 ! tea*] may h to d ! same73 7#a7
 ! complete conv]se1 9 :i* t :i* is giv5
 9 "o becomes t :i* is to be prov$ 9 !  
o!r1 & vice v]sa1 z 9 ?is & ! prec$+    
proposition2 7#b7 ! p>tial conv]se1 9   
:i* two 7or ev5 m7 ?+s may be giv51 & a 
c]ta9 ?+ is to be prov$1 ! conv]se be+ t
 "o 7or m7 ( ! prec$+ ?+s is n[ giv51   
toge!r ) :at 0 to be prov$1 & ! o!r giv5
 ?+ is n[ to be prov$4 ,symbolically1 if
 x is giv5 t a  b & c  d1 to prove t x  
y1 ! p>tial conv]se w\ld h giv5 a  b & x
  y1 to prove t c  d4  ,sev]al pro(s = !
 proposition h be5 sugge/$1 b a c>eful  
exam9ation ( all ( !m %[s t ! "o giv5   

bel[ is1 all ?+s consid]$1 ! be/ "o =   
pupils beg9n+ geometry & foll[+ !       
sequ5ce laid d[n 9 ?is *apt]4 ,x has !  
sanction ( "s ( ! mo/ em95t             
ma!maticians1 & :ile n z satisfactory 9 
"s respects z ! r$uctio ad absurdum1    
m5tion$ bel[1 x is m satisfactory 9 mo/ 
p>ticul>s4 ,! pro( is z foll[s3         
,illu/ration3  ,giv5 ! triangle ,,abc1 )
 ! angle ,a equal to ! angle ,b4        
,to prove t    ,,ac  ,,bc4  ,pro(4      
,suppose ! second triangle ,a',b',c' to 
be an exact reproduction ( ! giv5       
triangle ,,abc4  ,turn ! triangle       
,a',b',c' ov] & place x ^u ,,abc s t ,b'
 %all fall on ,a & ,a' %all fall on ,b4 
,!n ,b',a' w co9cide ) ,,ab4      ,s9ce 
,l,a'  ,l,b'1         ,giv5      &      
,l,a   ,l,a'1         ,hyp4             
!re=e,l,a   ,l,b'4        !re=e,b',c' w 
lie along ,,ac4      ,simil>ly1   ,a',c'
 w lie along ,,bc4  ,!re=e ,c' w fall on
 bo? ,,ac & ,,bc1 & h5ce at _!          
int]section4           !re=e,b',c'      
,,ac4      ,b ,b',c' 0 made equal to    

,,bc4            !re=e,,ac  ,,bc4       
,q4,e4,d4  ,if ! proposition %\ld be    
po/pon$ until aft] ! "o on ! sum ( !    
angles ( a triangle1 ! pro( w\ld be     
simpl]1 b x is advantage\s to c\ple x ) 
its imm$iate pr$ecessor4 ,?is simpl]    
pro( consi/s 9 bisect+ ! v]tical angle1 
& !n prov+ ! two triangles congru5t4    
,among ! o!r pro(s is t ( ! r$uctio ad  
absurdum1 :i* ! /ud5t might n[ meet1 b  
:i* may bett] be po/pon$4 ,! phrase     
r$uctio ad absurdum seems likely to     
cont9ue 9 spite ( ! ef=ts to f9d ano!r  
"o t is simpl]4 ,su* a pro( is also     
call$ an indirect pro(1 b ?is t]m is n  
altoge!r satisfactory4 ,probably bo?    
names %\ld be us$1 ! ,lat9 to expla9 !  
nature ( ! ,engli%4 ,! ,lat9 "n is m]ely
 a translation ( "o ( sev]al ,greek     
names us$ by ,ari/otle1 a second be+ 9  
,engli% pro( by ! impossible1 & a ?ird  
be+ pro( lead+ to ! impossible4 ,if     
tea*]s desire to introduce ?is =m ( pro(
 "h1 x mu/ be borne 9 m9d t only "o     
supposition c be made if su* a pro( is  

to be valid1 = if two are made1 !n an   
absurd conclusion simply %[s t ei!r or  
bo? mu/ be false1 b we d n "k :i* is    
false1 or if only "o is false4  ,,!orem4
 ,two triangles are congru5t if ! ?ree  
sides ( ! "o are equal respectively to !
 ?ree sides ( ! o!r4  ,x w\ld be        
desirable to place ?is aft] ! f\r?      
proposition m5tion$ 9 ?is li/ if x c\ld 
be done1 s z to get ! triangles 9 a     
gr\p1 b we ne$ ! f\r? "o = prov+ ?is1 s 
t ! arrangem5t _c be made1 at lea/ ) ?is
 me?od ( pro(4       ,?is proposition is
 a p>tial conv]se ( ! second            
proposition 9 ?is li/2 = if ! triangles 
are ,,abc &      ,a',b',c'1 ) sides a1  
b1 c & a'1 b'1 c'1 !n      ! second     
proposition ass]ts t if b  b'1 c  c'1   
& ,l,a  ,l,a'1 !n a  a' & ! triangles   
are      congru5t1 :ile ?is proposition 
ass]ts t if a  a'1      b  b'1 & c  c'1 
!n ,l,a  ,l,a' & !      triangles are   
congru5t4  ,! proposition 0 kn[n at lea/
 z e>ly z ,ari/otle's "t4 ,euclid prov$ 
x by ins]t+ a prelim9>y proposition to !

 effect t x is impossible to h on ! same
 base ,,ab & ! same side ( x two diff]5t
 triangles ,,abc & ,,abc'1 ) ,,ac       
,,ac'1 & ,,bc  ,,bc'4 ,! pro( ord9>ily  
giv5 to-day1 :]e9 ! two triangles are   
con/ruct$ on opposite sides ( ! base1 is
 due to ,philo ( ,byzantium1 :o liv$    
aft] ,euclid's "t b be=e ! ,*ri/ian era1
 & x is also giv5 by ,proclus4 ,!re are 
really ?ree cases1 if "o wi%es to be    
ov]p>ticul>1 correspond+ to ! ?ree pairs
 ( equal sides4 ,b if we are all[$ to   
take ! longe/ side = ! common base1 only
 "o case ne$ be consid]$4  ,( !         
applications ( ! proposition "o ( ! mo/ 
important relates to mak+ a figure rigid
 by means ( diagonals4 ,= example1 h[ _m
 diagonals mu/ be drawn 9 ord] to make a
 quadrilat]al rigid8 to make a p5tagon  
rigid8 a hexagon8 a polygon ( n sides4  
,9 p>ticul>1 ! foll[+ que/ions may be   
ask$ ( a class3  ,illu/ration       #a4 
,?ree iron rods are h+$ at !            
extremities1 z %[n 9      ?is figure4   
,is ! figure rigid8 ,:y8       #b4 ,f\r 

iron rods are h+$1 z %[n 9 ?is figure4  
,is !      figure rigid8 ,if n1 :]e w\ld
 y put 9 ! fif? rod to      make x      
rigid8 ,prove t ?is w\ld accompli% !    
result4  ,illu/ration  ,ano!r int]e/+   
application relates to ! mo/ anci5t =m (
 level+ in/rum5t kn[n to u4 ,?is k9d (  
level is pictur$ on v anci5t monum5ts1 &
 x is /ill us$ 9 _m p>ts ( ! _w4 ,pupils
 9 manual tra9+ may make su* an         
in/rum5t1 & inde$ "o is easily made out 
( c>dbo>d4 ,if ! plumb l9e passes ?r\gh 
! mid-po9t ( ! base1 ! two triangles are
 congru5t & ! plumb l9e is !n p]p5dicul>
 to ! base4 ,9 o!r words1 ! base is     
level4 ,) su* simple primitive          
in/rum5ts1 easily made by pupils1 a good
 deal ( practical ma!matical "w c be    
p]=m$4 ,! int]e/+ old illu/ration "h    
giv5 %[s h[ ?is =m ( level 0 us$ ?ree   
hundr$ ye>s ago4  ,illu/ration3 ,,e>ly  
,,me?ods ,,( ,,level+  ,pomodoro's ,la  
geometria prattica1 ,rome1 #afbd        
,illu/ration  ,tea*]s :o seek =         
geometric figures 9 practical me*anics w

 f9d ?is proposition illu/rat$ 9 !      
ord9>y hoi/+ app>atus ( ! k9d "h %[n4 ,f
 ! /udy ( su* =ms & ( simple ro( &      
bridge trusses1 a numb] ( ! usual       
prop]ties ( ! isosceles triangle may be 
d]iv$4  ,,!orem4 ,! sum ( two l9es drawn
 f a giv5 po9t to ! extremities ( a giv5
 l9e is great] ?an ! sum ( two o!r l9es 
simil>ly drawn1 b includ$ by !m4  ,x    
%\ld be not$ t ! words ! extremities (  
are necess>y1 = x is possible to draw f 
a c]ta9 po9t )9 a c]ta9 triangle two    
l9es to ! base su* t _! sum is great]   
?an ! sum ( ! o!r two sides4            
,illu/ration       ,?us1 9 ! "r triangle
 ,,abc draw any l9e ,,cx f ,c      to ! 
base4 ,make ,,xy  ,,ac1 & ,,cp  ,,py4   
,!n x is      easily %[n t ,,pb  ,,px   
,,cb  ,,ca4       ,illu/ration       ,x 
is int]e/+ to a class to h a tea*] po9t 
out t1      9 ?is figure1 ,,ap  ,,pb    
,,ac  ,,cb1 & ,,ap'  ,p',b       ,,ap   
,,pb1 & t ! ne>] ,p gets to ,,ab1 !     
%ort]      ,,ap  ,,pb becomes1 ! limit  
be+ ! l9e ,,ab4 ,f ?is      we may inf] 

7al?\gh we h n prov$7 t a /raight       
l9e 7,,ab7 is ! %orte/ pa? betwe5 two   
po9ts4  ,,!orem4 ,only "o p]p5dicul> c  
be drawn to a giv5 l9e f a giv5 ext]nal 
po9t4  ,,!orem4 ,two l9es drawn f a po9t
 9 a p]p5dicul> to a giv5 l9e1 cutt+ (f 
on ! giv5 l9e equal segm5ts f ! foot ( !
 p]p5dicul>1 are equal & make equal     
angles ) ! p]p5dicul>4  ,,!orem4 ,( two 
l9es drawn f ! same po9t 9 a p]p5dicul> 
to a giv5 l9e1 cutt+ (f on ! l9e unequal
 segm5ts f ! foot ( ! p]p5dicul>1 ! m   
remote is ! great]4  ,,!orem4 ,!        
p]p5dicul> is ! %orte/ l9e t c be drawn 
to a /raight l9e f a giv5 ext]nal po9t4 
,^! f\r propositions1 :ile kn[n to !    
anci5ts & incid5tally us$1 are n        
explicitly /at$ by ,euclid4 ,! reason   
seems to be t he int]sp]s$ 8 problems ) 
8 !orems1 & 9 8 ,propositions #aa & #ab1
 :i* treat ( draw+ a p]p5dicul> to a    
l9e1 ! ess5tial features ( ^! !orems are
 prov$4 ,fur!r m5tion w be made ( !m :5 
we come to consid] ! problems 9 "q4 ,_m 
textbook writ]s put ! second & ?ird ( ! 

f\r be=e ! fir/1 =gett+ t ! fir/ is     
assum$ 9 ! o!r two1 & h5ce %\ld prec$e  
!m4  ,,!orem4 ,two "r triangles are     
congru5t if ! hypot5use & a side ( ! "o 
are equal respectively to ! hypot5use & 
a side ( ! o!r4  ,,!orem4 ,two "r       
triangles are congru5t if ! hypot5use & 
an adjac5t angle ( ! "o are equal       
respectively to ! hypot5use & an adjac5t
 angle ( ! o!r4  ,z /at$ 9 ! notes on ! 
?ird proposition 9 ?is sequ5ce1         
,euclid's cumb]some ,proposition #bf    
cov]s sev]al cases1 & ^! two among !m4  
,( c\rse ?is pres5t proposition c\ld m  
easily be prov$ aft] ! "o conc]n+ ! sum 
( ! angles ( a triangle1 b ! pro( is s  
simple t x is bett] to leave !          
proposition "h 9 connection ) o!rs      
conc]n+ triangles4  ,,!orem4 ,two l9es 9
 ! same plane p]p5dicul> to ! same l9e  
_c meet1 h[ev] f> !y are produc$4  ,?is 
proposition is n 9 ,euclid1 & x is      
introduc$ = educational r ?an =         
ma!matical reasons4 ,euclid introduc$ ! 
subject by ! proposition t1 if alt]nate 

angles are equal1 ! l9es are p>allel4 ,x
 is1 h[ev]1 simpl] to beg9 ) ?is        
proposition1 & !re is "s advantage 9    
/at+ x 9 su* a way z to prove t p>allels
 exi/ be=e !y are def9$4 ,! proposition 
is prop]ly foll[$ by ! def9ition (      
p>allels & by ! po/ulate t has be5      
discuss$ on page #abg4  ,a good         
application ( ?is proposition is ! "o   
conc]n+ a me?od ( draw+ p>allel l9es by 
! use ( a c>p5t]'s squ>e4 ,"h two l9es  
are drawn p]p5dicul> to ! edge ( a bo>d 
or a rul]1 & ^! are p>allel4  ,,!orem4  
,if a l9e is p]p5dicul> to "o ( two     
p>allel l9es1 x is p]p5dicul> to ! o!r  
also4  ,?is1 l ! prec$+ proposition1 is 
a special case "u a lat] !orem4 ,x      
simplifies ! treatm5t ( p>allels1 h[ev]1
 & ! beg9n] f9ds x easi] to approa* !   
difficulties gradually1 ?r\gh ^! two    
cases ( p]p5dicul>s4 ,x %\ld be notic$ t
 ?is is an example ( a p>tial conv]se1 z
 expla9$ on page #age4 ,! prec$+        
proposition may be /at$ ?us3 ,if a is   
p]p to x & b is p]p to x1 !n a is  to b4

 ,?is proposition may be /at$ ?us3 ,if a
 is p]p to x & a is  to b1 !n b is p]p  
to x4 ,?is is1 !re=e1 a p>tial conv]se4 
,^! two propositions hav+ be5 prov$1 !  
usual def9itions ( ! angles made by a   
transv]sal ( two p>allels may be giv54  
,x is un=tunate t we h no "n = ea* ( !  
two gr\ps ( f\r equal angles1 & ! "n (  
transv]se angles has be5 sugge/$4 ,?is  
w\ld simplify ! /atem5ts ( c]ta9 o!r    
propositions2 ?us3 ,if two p>allel l9es 
are cut by a transv]sal1 ! transv]se    
angles are equal1 & ?is includes two    
propositions z usually giv54 ,!re is n z
 yet1 h[ev]1 any g5]al sanction = ! t]m4
  ,,!orem4 ,if two p>allel l9es are cut 
by a transv]sal1 ! alt]nate-int]ior     
angles are equal4  ,euclid gave ?is z   
half ( 8 ,proposition #bi4 ,inde$1 he   
gives only f\r !orems on p>allels1 z    
aga9/ five propositions & sev]al        
coroll>ies 9 mo/ ( our ,am]ican         
textbooks4 ,! reason = increas+ ! numb] 
is t ea* proposition may be less        
involv$4 ,?us1 in/ead ( hav+ "o         

proposition = bo? ext]ior & int]ior     
angles1 mod]n au?ors usually h "o = !   
ext]ior & "o = ! int]ior1 s z to make ! 
difficult subject ( p>allels easi] =    
beg9n]s4  ,,!orem4 ,:5 two /raight l9es 
9 ! same plane are cut by a transv]sal1 
if ! alt]nate-int]ior angles are equal1 
! two /raight l9es are p>allel4  ,?is is
 ! conv]se ( ! prec$+ !orem1 & is half (
 ,euclid ,i1 #bh1 8 !orem be+ divid$ = !
 reason above /at$4 ,!re are sev]al     
typical pairs ( equal or supplem5tal    
angles t w\ld lead to p>allel l9es1 (   
:i* ,euclid uses only "p1 leav+ ! o!r   
cases to be inf]r$4 ,?is acc\nts = !    
numb] ( coroll>ies 9 ?is connection 9   
lat] textbooks4  ,surveyors make use (  
?is proposition :5 !y wi%1 )\t us+ a    
transit in/rum5t1 to run "o l9e p>allel 
to ano!r4  ,illu/ration       ,=        
example1 suppose two boys are lay+ out a
 t5nis c\rt &      !y wi% to run a l9e  
?r\gh ,p p>allel to ,,ab4 ,take a       
#fj-;foot tape & sw+ x ar\nd ,p until ! 
o!r end re/s      on ,,ab1 z at ,m4 ,put

 a /ake at ,o1 #cj feet f ,p &      ,m4 
,!n take any conv5i5t po9t ,n on ,,ab1 &
 measure      ,,on4 ,suppose x equals   
#bj feet4 ,!n sight f ,n ?r\gh      ,o1 
& put a /ake at ,q j #bj feet f ,o4 ,!n 
,p &      ,q det]m9e ! p>allel1 accord+ 
to ! proposition j      m5tion$4        
,,!orem4 ,if two p>allel l9es are cut by
 a transv]sal1 ! ext]ior-int]ior angles 
are equal4  ,?is is also a "p ( ,euclid 
,i1 #bi4 ,x is usually foll[$ by sev]al 
coroll>ies1 cov]+ ! m9or & obvi\s cases 
omitt$ by ! old] writ]s4 ,:ile x w\ld be
 possible to disp5se ) ^! coroll>ies1 !y
 are helpful = def9ite ref]5ce 9 lat]   
propositions4  ,,!orem4 ,! sum ( ! ?ree 
angles ( a triangle is equal to two "r  
angles4  ,euclid /at$ ?is z foll[s3 ,9  
any triangle1 if "o ( ! sides be        
produc$1 ! ext]ior angle is equal to !  
two int]ior & opposite angles1 & ! ?ree 
int]ior angles ( ! triangle are equal to
 two "r angles4 ,?is /ates m ?an is     
necess>y = ! basal fact ( ! proposition1
 :i* is ! con/ancy ( ! sum ( ! angles4  

,! !orem is "o ( ! ?ree mo/ important   
propositions 9 plane geometry1 ! o!r two
 be+ ! s-call$ ,py?agorean ,!orem1 & a  
proposition relat+ to ! proportionality 
( ! sides ( two triangles4 ,^! ?ree =m !
 f\ndation ( trigonometry & ( !         
m5suration ( plane figures4  ,! hi/ory (
 ! proposition is ext5sive4 ,eutocius   
7ca4 #eaj ,a4,d471 9 8 comm5t>y on      
,apollonius1 says t ,gem9us 7fir/ c5tury
 ,b4,c47 te/ifi$ t ! anci5ts inve/igat$ 
! !orem ( ! two "r angles 9 ea*         
individual species ( triangle1 fir/ 9 ! 
equilat]al1 aga9 9 ! isosceles1 &       
aft]w>ds 9 ! scal5e triangle4 ,?is1     
inde$1 0 ! anci5t plan1 to proce$ f !   
p>ticul> to ! g5]al4 ,x is ! natural    
ord]1 x is ! _w's ord]1 & x is well to  
foll[ x 9 all cases ( difficulty 9 !    
classroom4  ,proclus 7#daj-dhe ,a4,d47  
tells u t ,eudemus1 :o liv$ j be=e      
,euclid 7or probably ab\t #cbe ,b4,c471 
affirm$ t ! !orem 0 due to !            
,py?agoreans1 al?\gh ?is does n         
necess>ily mean to ! actual pupils (    

,py?agoras4 ,! pro( z he gives x consi/s
 9 %[+ t a  a'1 b  b'1 & a'  c  b'  two 
"r angles4 ,s9ce ! proposition ab\t !   
ext]ior angle ( a triangle is attribut$ 
to ,philippus ( ,m5de 7ca4 #chj ,b4,c471
 ! figure giv5 by ,eudemus is probably !
 "o us$ by ! ,py?agoreans4  ,illu/ration
  ,!re is also "s reason = believ+ t    
,?ales 7ca4 #fjj ,b4,c47 knew ! !orem1 =
 ,diog5es ,la]tius 7ca4 #bjj ,a4,d47    
quotes ,pamphilius 7fir/ c5tury ,a4,d47 
z say+ t he1 hav+ le>n$ geometry f !    
,egyptians1 0 ! fir/ to inscribe a "r   
triangle 9 a circle1 & sacrific$ an ox4 
,! pro( ( ?is proposition requires ! k t
 ! sum ( ! angles1 at lea/ 9 a "r       
triangle1 is two "r angles4 ,!          
proposition is frequ5tly ref]r$ to by   
,ari/otle4  ,!re h be5 num]\s attempts  
to prove ! proposition )\t ! use (      
p>allel l9es4 ,( ^! a ,g]man "o1 fir/   
giv5 by ,?ibaut 9 ! e>ly "p ( ! eighte5?
 c5tury1 is among ! mo/ int]e/+4 ,?is1 9
 simplifi$ =m1 is z foll[s3             
,illu/ration       ,suppose an indef9ite

 l9e ,,xy to lie on ,,ab4 ,let x sw+    
ab\t ,a1 c\nt]clockwise1 ?r\gh ,l,a1 s z
 to lie on      ,,ac1 z ,x',y'4 ,!n let 
x sw+ ab\t ,c1 ?r\gh ,l,c1      s z to  
lie on ,,cb1 z ,x'',y''4 ,!n let x sw+  
ab\t ,b1      ?r\gh ,l,b1 s z to lie on 
,,ba1 z ,x''',y'''4 ,x n[      lies on  
,,ab1 b x is turn$ ov]1 ,x''' be+ :]e ,y
      01 & ,y''' :]e ,x 04 ,9 turn+     
?r\gh ,ls,a1 ,b1      & ,c x has !re=e  
turn$ ?r\gh two "r angles4  ,"o tr\ble )
 ! pro( is t ! rotation has n be5 ab\t !
 same po9t1 s t x has nev] be5 look$ ^u 
z o!r ?an an int]e/+ illu/ration4       
,proclus tri$ to prove ! !orem by say+  
t1 if we h two p]p5dicul>s to ! same    
l9e1 & suppose !m to revolve ab\t _!    
feet s z to make a triangle1 !n ! am\nt 
tak5 f ! "r angles is add$ to ! v]tical 
angle ( ! triangle1 & !re=e ! sum ( !   
angles cont9ues to be two "r angles4 ,b1
 ( c\rse1 to prove 8 /atem5t requires a 
p]p5dicul> to be drawn f ! v]tex to !   
base1 & ! !orem ( p>allels to be appli$4
  ,pupils w f9d x int]e/+ to cut (f !   

corn]s ( a pap] triangle & fit ! angles 
toge!r s z to make a /raight angle4     
,?is !orem furni%es an opportunity = _m 
int]e/+ ex]cises1 & 9 p>ticul> = det]m9+
 ! ?ird angle :5 two angles ( a triangle
 are giv51 or ! second acute angle ( a  
"r triangle :5 "o acute angle is giv54  
,( ! simple outdoor applications ( !    
proposition1 "o ( ! be/ is illu/rat$ 9  
?is figure4  ,illu/ration       ,to     
asc]ta9 ! height ( a tree or ( ! s*ool  
build+1      fold a piece ( pap] s z to 
make an angle ( #de deg44 ,!n walk      
back f ! tree until ! top is se5 at an  
angle ( #de deg4      ) ! gr\nd 7be+    
!re=e c>eful to h ! base (      !       
triangle level74 ,!n ! height ,,ac w    
equal ! base      ,,ab1 s9ce ,,abc is   
isosceles4 ,a pap] protractor may be us$
      = ! same purpose4  ,di/ances c    
easily be measur$ by con/ruct+ a l>ge   
equilat]al triangle ( heavy pa/ebo>d1 & 
/&+ p9s at ! v]tices = ! purpose (      
sight+4  ,illu/ration       ,to measure 
,,pc1 /& at "s conv5i5t po9t ,a & sight 

along ,,apc & also along ,,ab4 ,!n walk 
along ,,ab until a      po9t ,b is rea*$
 f :i* ,,bc makes ) ,,ba an angle      (
 ! triangle 7#fj deg474 ,!n ,,ac  ,,ab1 
& s9ce ,,ap c be      measur$1 we c f9d 
,,pc4  ,ano!r simple me?od ( measur+ a  
di/ance ,,ac across a /ream is %[n 9 ?is
 figure4  ,illu/ration       ,measure ! 
angle ,,cax1 ei!r 9 degrees1 ) a        
protractor1      or by sight+ along a   
piece ( pap] & m>k+ d[n !      angle4   
,!n g along ,,xa produc$ until a po9t ,b
 is rea*$      f :i* ,,bc makes ) ,a an 
angle equal to half ( angle      ,,cax4 
,!n x is easily %[n t ,,ab  ,,ac4  ,a   
navigator uses ! same pr9ciple :5 he    
d\bles ! angle on ! b[ to f9d 8 di/ance 
f a ligh?\se or o!r object4             
,illu/ration       ,if he is sail+ on ! 
c\rse ,,abc & notes a ligh?\se ,l       
:5 he is at ,a1 & takes ! angle ,a1 & if
 he notices      :5 ! angle t ! ligh?\se
 makes ) 8 c\rse is      j twice ! angle
 not$ at ,a1 !n ,,bl  ,,ab4 ,he has     
,,ab f 8 log 7an in/rum5t t tells h[ f> 

a %ip goes      9 a giv5 "t71 s he kn[s 
,,bl4 ,he has d\bl$ ! angle      on ! b[
 to get ?is di/ance4  ,x w\ld h be5     
possible = ,?ales1 if he knew ?is       
proposition1 to h measur$ ! di/ance ( ! 
%ip at sea by "s su* device z ?is3      
,illu/ration       ,make a l>ge         
isosceles triangle out ( wood1 &1 /&+ at
      ,t1 sight to ! %ip & along ! %ore 
on a l9e ,,ta1      us+ ! v]tical angle 
( ! triangle4 ,!n g along ,,ta          
until a po9t ,p is rea*$1 f :i* ,t & ,s 
c be      se5 along ! sides ( a base    
angle ( ! triangle4 ,!n      ,,tp  ,,ts4
 ,by measur+ ,,tb1 ,,bs c !n be f\nd4   
,,!orem4 ,! sum ( two sides ( a triangle
 is great] ?an ! ?ird side1 & _!        
diff]5ce is less ?an ! ?ird side4  ,if !
 po/ulate is assum$ t a /raight l9e is !
 %orte/ pa? betwe5 two po9ts1 !n ! fir/ 
"p ( ?is !orem requires no fur!r pro(1 &
 ! second "p foll[s at once f ! axiom ( 
inequalities4 ,?is seems ! bett] plan = 
beg9n]s1 & ! proposition may be consid]$
 z semiobvi\s4 ,euclid prov$ ! fir/ "p1 

n hav+ assum$ ! po/ulate4 ,proclus tells
 u t ! ,epicureans 7! foll[]s (         
,epicurus1 ! ,greek philosoph]1 #cdb-bgj
 ,b4,c47 us$ to ridicule ?is !orem1 say+
 t ev5 an ass knew x1 = if he wi%$ to   
get food1 he walk$ 9 a /raight l9e & n  
along two sides ( a triangle4 ,proclus  
repli$ t x 0 "o ?+ to "k ! tru? & ano!r 
?+ to prove x1 mean+ t ! value (        
geometry lay 9 ! pro( r ?an 9 ! m]e     
facts1 a ?+ t all :o seek to re=m !     
tea*+ ( geometry w\ld d well to keep 9  
m9d4 ,! !orem might simply appe> z a    
coroll>y "u ! po/ulate if x 7 ( any     
importance to r$uce ! numb] (           
propositions "o m4  ,if ! proposition is
 po/pon$ until aft] ^? conc]n+ !        
inequalities ( angles & sides ( a       
triangle1 !re are sev]al good pro(s4    
,illu/ration      ,= example1 produce   
,,ac to ,x1     mak+             ,,cx   
,,cb4      ,!n   ,l,x  ,l,,xbc4         
!re=e ,l,,xba  ,l,x4      !re=e ,,ax    
,,ab4      !re=e ,,ac  ,,cb  ,,ab4  ,!  
above pro( is due to ,euclid4 ,h]on (   

,alex&ria 7fir/ c5tury ,a4,d47 is said  
by ,proclus to h giv5 ! foll[+3         
,illu/ration      ,let ,,cx bisect ,l,c4
      ,!n ,l,,bxc  ,l,,acx4      !re=e  
,l,,bxc  ,l,,xcb4      !re=e ,,cb  ,,xb4
      ,simil>ly1 ,,ac  ,,ax4      ,add+1
 ,,ac  ,,cb  ,,ab4  ,,!orem4 ,if two    
sides ( a triangle are unequal1 ! angles
 opposite ^! sides are unequal1 & !     
angle opposite ! great] side is !       
great]4  ,euclid /at$ ?is m briefly by  
say+1 ,9 any triangle ! great] side     
subt5ds ! great] angle4 ,?is is n s     
satisfactory1 = !re may be no great]    
side4  ,,!orem4 ,if two angles ( a      
triangle are unequal1 ! sides opposite  
^! angles are unequal1 & ! side opposite
 ! great] angle is ! great]4  ,euclid   
also /at$ ?is m briefly1 b less         
satisfactorily1 ?us1 ,9 any triangle !  
great] angle is subt5d$ by ! great]     
side4 ,/ud5ts %\ld h _! att5tion call$  
to ! fact t ^! two !orems are reciprocal
 or dual !orems1 ! words sides & angles 
( ! "o correspond+ to ! words angles &  

sides respectively ( ! o!r4       ,x may
 also be notic$ t ! pro( ( ?is          
proposition      involves :at is kn[n z 
! ,law ( ,conv]se2 =      7#a7 if b  c1 
!n ,l,b  ,l,c2     7#b7 if b  c1 !n ,l,b
  ,l,c2     7#c7 if b  c1 !n ,l,b  ,l,c2
       !re=e ! conv]ses mu/ necess>ily  
be true z a matt] (      logic2 =       
if ,l,b  ,l,c1 !n b _c be great] ?an c  
)\t      violat+ 7#b71 & b _c be less   
?an c )\t      violat+ 7#c71 !re=e b  c2
       & if ,l,b  ,l,c1 !n b _c equal c 
)\t      violat+ 7#a71 & b _c be less   
?an c )\t      violat+ 7#c71 !re=e b  c2
       simil>ly1 if ,l,b  ,l,c1 !n b  c4
  ,?is ,law ( ,conv]se may readily be   
taught to pupils1 & x has sev]al        
applications 9 geometry4  ,,!orem4 ,if  
two triangles h two sides ( ! "o equal  
respectively to two sides ( ! o!r1 b !  
includ$ angle ( ! fir/ triangle great]  
?an ! includ$ angle ( ! second1 !n !    
?ird side ( ! fir/ is great] ?an ! ?ird 
side ( ! second1 & conv]sely4           
,illu/ration  ,9 ?is proposition !re are

 ?ree possible cases3 ! po9t ,y may fall
 bel[ ,,ab1 z "h %[n1 or on ,,ab1 or    
above ,,ab4 ,z an ex]cise = pupils all  
?ree may be consid]$ if desir$4 ,foll[+ 
,euclid & mo/ e>ly writ]s1 h[ev]1 only  
"o case really ne$ be prov$1 provid$ t  
is ! mo/ difficult "o1 & is typical4    
,proclus gave ! pro(s ( ! o!r two cases1
 & x is int]e/+ to pupils to "w !m out =
 !mselves4 ,9 su* "w x con/antly appe>s 
t e proposition sugge/s abundant        
opportunity = orig9ality1 & t ! complete
 =m ( pro( 9 a textbook is n a b> to    
indep5d5t ?\ght4  ,! ,law ( ,conv]se1   
m5tion$ on page #aij1 may be appli$ to !
 conv]se case if desir$4  ,,!orem4 ,two 
angles ^: sides are p>allel1 ea* to ea*1
 are ei!r equal or supplem5t>y4  ,?is is
 n an anci5t proposition1 al?\gh !      
,greeks 7 well aw>e ( ! pr9ciple4 ,x may
 be /at$ s z to include ! case ( ! sides
 be+ p]p5dicul>1 ea* to ea*1 b ?is is   
bett] left z an ex]cise4 ,x is possible1
 by "s circumlocution1 to s /ate ! !orem
 z to tell 9 :at cases ! angles are     

equal & 9 :at cases supplem5t>y4 ,x _c  
be t]sely /at$1 h[ev]1 & x seems bett]  
to leave ?is po9t z a subject = que/ion+
 by ! tea*]4  ,,!orem4 ,! opposite sides
 ( a p>allelogram are equal4  ,,!orem4  
,if ! opposite sides ( a quadrilat]al   
are equal1 ! figure is a p>allelogram4  
,illu/ration  ,?is proposition is a v   
simple te/ = a p>allelogram4 ,x is !    
pr9ciple involv$ 9 ! case ( ! common    
fold+ p>allel rul]1 an in/rum5t t has   
long be5 recogniz$ z "o ( ! valuable    
tools ( practical geometry4 ,x w be ( "s
 int]e/ to tea*]s to see "o ( ! e>ly =ms
 ( ?is p>allel rul]1 z %[n 9 !          
illu/ration4#fc ,if su* an in/rum5t is n
 available 9 ! s*ool1 "o suitable =     
illu/rative purposes c easily be made f 
c>dbo>d4  ,illu/ration3 ,,p>allel ,,rul]
 ,,( ,,! ,,sev5te5? ,,c5tury  ,san      
,giovanni's ,seconda squ>a mobile1      
,vic5za1 #afhf  ,a some:at m complicat$ 
=m ( ?is in/rum5t may also be made by   
pupils 9 manual tra9+1 z is %[n 9 ?is   
illu/ration f ,bion's great treatise4 ,!

 pr9ciple involv$ may be tak5 up 9      
class1 ev5 if ! in/rum5t is n us$4 ,x is
 evid5t t1 unless ! workman%ip is       
unusually good1 ?is =m ( p>allel rul] is
 n z accurate z ! common "o illu/rat$   
above4 ,! pr9ciple is sometimes us$ 9   
iron gates4  ,illu/ration3 ,,p>allel    
,,rul] ,,( ,,! ,,eighte5? ,,c5tury  ,n4 
,bion's ,traite de la con/ruction 444   
des in/rum5s de ma!matique1 ,! ,hague1  
#agbc  ,,!orem4 ,two p>allelograms are  
congru5t if two sides & ! includ$ angle 
( ! "o are equal respectively to two    
sides & ! includ$ angle ( ! o!r4  ,?is  
proposition is discuss$ 9 connection ) !
 "o t foll[s4  ,,!orem4 ,if ?ree or m   
p>allels int]cept equal segm5ts on "o   
transv]sal1 !y int]cept equal segm5ts on
 e transv]sal4  ,^! two propositions are
 n giv5 9 ,euclid1 al?\gh g5]ally       
requir$ by ,am]ican syllabi ( ! pres5t  
"t4 ,! la/ "o is p>ticul>ly useful 9    
subsequ5t "w4 ,nei!r "o (f]s any        
difficulty1 & nei!r has any int]e/+     
hi/ory4 ,!re are1 h[ev]1 num]\s int]e/+ 

applications to ! la/ "o4 ,"o t is us$ 9
 me*anical draw+ is "h illu/rat$4       
,illu/ration       ,if x is desir$ to   
divide a l9e ,,ab into five equal p>ts1 
we may take a piece ( rul$ trac+ pap] & 
lay x ov] !      giv5 l9e s t l9e #j    
passes ?r\gh ,a1 & l9e #e      ?r\gh ,b4
 ,we may !n prick ?r\gh ! pap] & ?us    
det]m9e ! po9ts on ,,ab4 ,simil>ly1 we  
may divide ,,ab      into any o!r numb] 
( equal p>ts4  ,among ! applications (  
^! propositions is an int]e/+ "o due to 
! ,arab ,al-,nairizi 7ca4 #ijj ,a4,d474 
,! problem is to divide a l9e into any  
numb] ( equal p>ts1 & he beg9s ) ! case 
( trisect+ ,,ab4 ,x may be giv5 z a case
 ( practical draw+ ev5 be=e ! problems  
are rea*$1 p>ticul>ly if "s prelim9>y "w
 ) ! compasses & /raight$ge has be5     
giv54       ,make ,,bq & ,,aq'          
p]p5dicul> to ,,ab1 & make ,,bp  ,,pq   
,,ap'  ,p',q'4 ,!n triangle,,xyz is     
congru5t to      triangle,,ybp1 & also  
to triangle,,xap'4 ,!re=e      ,,ax     
,,xy  ,,yb4 ,9 ! same way we might      

cont9ue to produce      ,,bq until x is 
made up ( n - #a l5g?s ,,bp1 & s =      
,,aq'1      & by prop]ly jo9+ po9ts we  
c\ld divide ,,ab into n      equal p>ts4
 ,9 p>ticul>1 if we jo9 ,p & ,p'1 we    
bisect      ! l9e ,,ab4  ,illu/ration   
,,!orem4 ,if two sides ( a quadrilat]al 
are equal & p>allel1 !n ! o!r two sides 
are equal & p>allel1 & ! figure is a    
p>allelogram4  ,?is 0 ,euclid's fir/    
proposition on p>allelograms1 & ,proclus
 speaks ( x z ! connect+ l9k betwe5 !   
!ory ( p>allels & t ( p>allelograms4 ,! 
anci5ts1 writ+ = mature /ud5ts1 did n   
add ! words & ! figure is a             
p>allelogram1 because t foll[s at once f
 ! fir/ "p & f ! def9ition (            
p>allelogram1 b x is helpful to y\ng]   
/ud5ts because x emphasizes ! fact t "h 
is a te/ = ?is k9d ( figure4  ,,!orem4  
,! diagonals ( a p>allelogram bisect ea*
 o!r4  ,?is proposition 0 n giv5 9      
,euclid1 b x is usually requir$ 9       
,am]ican syllabi4 ,!re is (t5 giv5 9    
connection ) x ! ex]cise 9 :i* x is     

prov$ t ! diagonals ( a rectangle are   
equal4 ,:5 ?is is tak51 x is well to    
/ate to ! class t c>p5t]s & build]s f9d 
?is "o ( ! be/ *ecks 9 lay+ out floors &
 o!r rectangles4 ,x is frequ5tly appli$ 
also 9 lay+ out t5nis c\rts4 ,if ! class
 is do+ any "w 9 m5suration1 su* z f9d+ 
! area ( ! s*ool gr\nds1 x is a good    
plan to *eck a few rectangles by ?is    
me?od4  ,an int]e/+ outdoor application 
( ! !ory ( p>allelograms is ! foll[+3   
,illu/ration       ,suppose y are on !  
side ( ?is /ream opposite to ,,xy1      
& wi% to measure ! l5g? ( ,,xy4 ,run a  
l9e ,,ab along      ! bank4 ,!n take a  
c>p5t]'s squ>e1 or ev5 a l>ge book1     
& walk along ,,ab until y rea* ,p1 a    
po9t f :i* y      c j see ,x & ,b along 
two sides ( ! squ>e4 ,d !      same =   
,y1 ?us fix+ ,p & ,q4 ,us+ ! tape1      
bisect      ,,pq at ,m4 ,!n walk along  
,,ym produc$ until y rea* a      po9t   
,y' t is exactly 9 l9e ) ,m & ,y1 & also
      ) ,p & ,x4 ,!n walk along ,,xm    
produc$ until y rea*      a po9t ,x' t  

is exactly 9 l9e ) ,m & ,x1 & also      
) ,q & ,y4 ,!n measure ,y',x' & y h !   
l5g?      ( ,,xy4 ,= s9ce ,,yx' is p]p  
to ,,pq1 & ,,xy' is also      p]p to    
,,pq1 ,,yx' is  to ,,xy'4 ,& s9ce ,,pm  
,,mq1      !re=e ,,xm  ,,mx' & ,y',m    
,,my4 ,!re=e ,y',x',,yx is      a       
p>allelogram4  ,! prop]ties ( !         
p>allelogram are (t5 appli$ to prov+    
figures ( v>i\s k9ds congru5t1 or to    
con/ruct+ !m s t !y w be congru5t4      
,illu/ration       ,= example1 if we    
draw ,a',b' equal & p>allel to ,,ab1    
,b',c' equal & p>allel to ,,bc1 & s on1 
x is easily      prov$ t ,,abcd &       
,a',b',c',d' are congru5t4 ,?is may be  
done by ord9>y sup]position1 or by slid+
 ,,abcd along !      dott$ p>allels4    
,!re are _m applications ( ?is pr9ciple 
( p>allel translation 9 practical       
con/ruction "w4 ,! pr9ciple is m        
f>-rea*+ ?an "h intimat$1 h[ev]1 & a few
 words z to its significance w n[ be 9  
place4  ,! ef=ts usually made to improve
 ! _s ( ,euclid are trivial4 ,!y        

ord9>ily relate to "s commonplace *ange 
( sequ5ce1 to "s slight *ange 9         
language1 or to "s n>r[ l9e (           
applications4 ,su* attempts require no  
p>ticul> ?\ght & yield no v noticeable  
result4 ,b !re is a possibility1 remote 
?\gh x may be at pres5t1 t a geometry w 
be develop$ t w be z s]i\s z ,euclid's &
 z effective 9 ! education ( ! ?9k+     
individual4 ,if s1 x seems probable t x 
w n be bas$ ^u ! congru5ce ( triangles1 
by :i* s _m propositions ( ,euclid are  
prov$1 b ^u c]ta9 po/ulates ( motion1 ( 
:i* "o is involv$ 9 ! above             
illu/ration1--! po/ulate ( p>allel      
translation4 ,if to ?is we jo9 ! two    
po/ulates ( rotation ab\t an axis1#fd   
lead+ to axial symmetry2 & rotation ab\t
 a po9t1#fe lead+ to symmetry ) respect 
to a c5t]1 we h a gr\p ( ?ree motions ^u
 :i* x is possible to base an ext5sive &
 rigid geometry4#ff ,x w be ?r\gh "s su*
 ef=t z ?is1 r ?an ?r\gh ! weak5+ ( !   
,euclid-,leg5dre /yle ( geometry1 t any 
improvem5t is likely to come4 ,at       

pres5t1 9 ,am]ica1 ! important "w =     
tea*]s is to vitalize ! geometry !y     
h1--an ef=t 9 :i* !re are great         
possibilities1--see+ to x t geometry is 
n r$uc$ to m]e fro?1 & recogniz+ !      
possibility ( ano!r geometry t may      
sometime replace x1--a geometry z rigid1
 z ?\ght-compell+1 z logical1 & z truly 
educational4  ,,!orem4 ,! sum ( !       
int]ior angles ( a polygon is equal to  
two "r angles1 tak5 z _m times less two 
z ! figure has sides4  ,?is int]e/+     
g5]alization ( ! proposition ab\t ! sum 
( ! angles ( a triangle is giv5 by      
,proclus4 ,!re are sev]al pro(s1 b all  
are bas$ ^u ! possibility ( dissect+ !  
polygon into triangles4 ,! po9t f :i*   
l9es are drawn to ! v]tices is usually  
tak5 at a v]tex1 s t !re are n - #b     
triangles4 ,x may h[ev] be tak5 )9 !    
figure1 mak+ n triangles1 f ! sum ( !   
angles ( :i* ! f\r "r angles ab\t ! po9t
 mu/ be subtract$4 ,! po9t may ev5 be   
tak5 on "o side1 or outside ! polygon1 b
 ! pro( is n s simple4 ,tea*]s :o desire

 to d s may sugge/ to p>ticul>ly good   
/ud5ts ! prov+ ( ! !orem = a concave    
polygon1 or ev5 = a cross polygon1      
al?\gh ! latt] requires negative angles4
  ,"s s*ools h transit in/rum5ts = ! use
 ( _! classes 9 trigonometry4 ,9 su* a  
case x is a good plan to measure !      
angles 9 "s piece ( l& s z to v]ify !   
proposition1 z well z %[ ! c>e t mu/ be 
tak5 9 read+ angles4 ,9 ! abs5ce ( ?is  
ex]cise x is well to take any irregul>  
polygon & measure ! angles by ! help ( a
 protractor1 & ?us accompli% ! same     
results4  ,,!orem4 ,! sum ( ! ext]ior   
angles ( a polygon1 made by produc+ ea* 
( its sides 9 succession1 is equal to   
f\r "r angles4  ,?is is also a          
proposition n giv5 by ! anci5t writ]s4  
,we h1 h[ev]1 no m valuable !orem = !   
purpose ( %[+ ! nature & significance ( 
! negative angle2 & tea*]s may ar\se a  
great deal ( int]e/ 9 ! negative        
quantity by %[+ to a class t :5 an      
int]ior angle becomes #ahj deg4 !       
ext]ior angle becomes #j1 & :5 ! polygon

 becomes concave ! ext]ior angle becomes
 negative1 ! !orem hold+ = all ^! cases4
 ,we h few bett] illu/rations ( !       
significance ( ! negative quantity1 &   
few bett] opportunities to use ! k ( ?is
 k9d ( quantity already acquir$ 9       
algebra4  ,illu/ration  ,9 ! hilly &    
m\nta9\s p>ts ( ,am]ica1 :]e            
irregul>-%ap$ fields are m common ?an 9 
! m level portions1 a common te/ = a    
survey is t ( f9d+ ! ext]ior angles :5 !
 transit in/rum5t is set at ! corn]s4 ,9
 ?is field ^! angles are giv51 & x w be 
se5 t ! sum is #cfj deg44 ,9 ! abs5ce ( 
any outdoor "w a protractor may be us$  
to measure ! ext]ior angles ( a polygon 
drawn on pap]4 ,if !re is an irregul>   
piece ( l& ne> ! s*ool1 ! ext]ior angles
 c be fairly well measur$ by an ord9>y  
pap] protractor4  ,! idea ( locus is    
usually introduc$ at ! end ( ,book ,i4  
,x is too ab/ract to be introduc$       
successfully any e>li]1 al?\gh au?ors   
repeat ! attempt f "t to "t1 unm9dful ( 
! fact t all exp]i5ce is oppos$ to x4 ,!

 loci propositions are n anci5t4 ,!     
,greeks us$ ! ^w locus 79 ,greek1       
topos71 h[ev]4 ,proclus1 = example1     
says1 ,i call ^? locus !orems 9 :i* !   
same prop]ty is f\nd to exi/ on ! :ole (
 "s locus4 ,tea*]s %\ld be c>eful to h !
 pupils recognize ! necessity = prov+   
two ?+s ) respect to any locus3 7#a7 t  
any po9t on ! suppos$ locus satisfies ! 
condition2 7#b7 t any po9t outside !    
suppos$ locus does n satisfy ! giv5     
condition4 ,! fir/ ( ^! is call$ !      
suffici5t condition1 & ! second !       
necess>y condition4 ,?us 9 ! case ( !   
locus ( po9ts 9 a plane equidi/ant f two
 giv5 po9ts1 x is suffici5t t ! po9t be 
on ! p]p5dicul> bisector ( ! l9e jo9+ ! 
giv5 po9ts1 & ?is is ! fir/ "p ( ! pro(2
 x is also necess>y t x be on ?is l9e1  
i4e4 x _c be outside ?is l9e1 & ?is is !
 second "p ( ! pro(4 ,! pro( ( loci     
cases1 !re=e1 involves a consid]ation ( 
! necess>y & suffici5t condition t is s 
(t5 spok5 ( 9 high] ma!matics4 ,?is     
expression might well be incorporat$    

into elem5t>y geometry1 & :5 x becomes  
bett] und]/ood by tea*]s1 x probably w  
be m (t5 us$4  ,9 tea*+ loci x is       
helpful to call att5tion to loci 9 space
 7mean+ !reby ! space ( ?ree dim5sions71
 )\t /opp+ to prove ! proposition       
involv$4 ,inde$1 x is desirable all     
?r\gh plane geometry to ref] incid5tally
 to solid geometry4 ,9 ! m5suration (   
plane figures1 :i* may be b\nd>ies (    
solid figures1 ?is is p>ticul>ly true4  
,x is a great defect 9 mo/ s*ool c\rses 
9 geometry t      !y are entirely conf9$
 to two dim5sions4 ,ev5 if solid        
geometry 9 ! usual s5se is n attempt$1 e
 occasion      %\ld be tak5 to lib]ate  
boys' m9ds f :at becomes !      tyranny 
( pap]4 ,?us ! que/ions3 ,:at is ! locus
 ( a      po9t equidi/ant f two giv5    
po9ts2 at a con/ant di/ance      f a    
giv5 /raight l9e or f a giv5 po9t8 %\ld 
be      ext5d$ to space4#fg  ,! two loci
 problems usually giv5 at ?is "t1 ref]r+
 to a po9t equidi/ant f ! extremities ( 
a giv5 l9e1 & to a po9t equidi/ant f two

 int]sect+ l9es1 bo? p]mit ( an int]e/+ 
ext5sion to ?ree dim5sions )\t any =mal 
pro(4 ,x is possible to give o!r loci at
 ?is po9t1 b x is pref]able m]ely to    
introduce ! subject 9 ,book ,i1 res]v+ !
 fur!r discussion until aft] ! circle   
has be5 /udi$4  ,x is well1 9 speak+ (  
loci1 to rememb] t x is entirely prop]  
to speak ( ! locus ( a po9t or ! locus (
 po9ts4 ,?us ! locus ( a po9t s mov+ 9 a
 plane z con/antly to be at a giv5      
di/ance f a fix$ po9t 9 ! plane is a    
circle4 ,9 analytic geometry we usually 
speak ( ! locus ( a po9t1 ?9k+ ( ! po9t 
z be+ any:]e on ! locus4 ,"s tea*]s (   
elem5t>y geometry1 h[ev]1 pref] to speak
 ( ! locus ( po9ts1 or ! locus ( all    
po9ts1 ?us t5d+ to make ! language (    
elem5t>y geometry diff] f t ( analytic  
geometry4 ,s9ce x is a trivial matt] (  
phraseology1 x is bett] to recognize bo?
 =ms ( expression & to let pupils use ! 
two int]*angeably4  ,,footnotes3  #eg   
,address at ,brussels1 ,augu/1 #aiaj4   
#eh ,= a rec5t discussion ( ?is g5]al   

subject1 see ,pr(essor ,hobson on ,!    
,t5d5cies ( ,mod]n ,ma!matics1 9 !      
,educational ,review1 ,new ,york1 #aiaj1
 ,vol4 ,,xl1 p4 #ebd4  #ei ,a m ext5d$  
li/ ( applications is giv5 lat] 9 ?is   
"w4  #fj ,abu'l-',abbas al-,fadl ibn    
,hatim al-,nairizi1 s call$ f 8         
bir?place1 ,nairiz1 0 a well-kn[n ,arab 
writ]4 ,he di$ ab\t #ibb ,a4,d4 ,he     
wrote a comm5t>y on ,euclid4  #fa ,?is  
illu/ration1 tak5 f a book 9 ! au?or's  
libr>y1 appe>$ 9 a valuable monograph by
 ,w4 ,e4 ,/>k1 ,measur+ ,in/rum5ts (    
,long ,ago1 publi%$ 9 ,s*ool ,sci5ce &  
,ma!matics1 ,vol4 ,x1 pp4 #dh1 #abf4 ,) 
o!rs ( ! same nature x is "h reproduc$  
by ! c\rtesy ( ,pr9cipal ,/>k & ( !     
editors ( ! j\rnal 9 :i* x appe>$4  #fb 
,9 speak+ ( two congru5t triangles x is 
some:at easi] to foll[ ! congru5ce if ! 
two are read 9 ! same ord]1 ev5 ?\gh !  
relatively unimportant c\nt]clockwise   
read+ is neglect$4 ,no "o %\ld be a     
slave to su* a =malism1 b %\ld foll[ !  
plan :5 conv5i5t4  #fc ,/>k1 loc4 cit4  

#fd ,( :i* s mu* 0 made by ,pr(essor    
,olaus ,h5rici 9 8 ,congru5t ,figures1  
,london1 #ahgi1--;a book t e tea*] (    
geometry %\ld own4  #fe ,mu* is made (  
?is 9 ! excell5t "w by ,h5rici &        
,treutle91 ,lehrbu* d] ,geometrie1      
,leipzig1 #ahha4  #ff ,m]ay did mu* =   
?is movem5t 9 ,france1 & ! rec5t works (
 ,b\rlet & ,borel h br\ght x to ! front 
9 t c\ntry4  #fg ,w4 ,n4 ,bruce1 ,tea*+ 
( ,geometry & ,graphic ,algebra 9       
,second>y ,s*ools1 ,bo>d ( ,education   
circul> 7,no4 #gaa71 p4 #h1 ,london1    
#aiji4     ,,*apt] ,,xv  ,,! ,,lead+    
,,propositions ,,( ,,book ,,ii   ,hav+  
tak5 up all ( ! propositions usually    
giv5 9 ,book ,i1 x seems unnecess>y to  
consid] z specifically all ^? 9         
subsequ5t books4 ,x is !re=e propos$ to 
select c]ta9 ones t h "s special int]e/1
 ei!r f ! /&po9t ( ma!matics or f t (   
hi/ory or application1 & to discuss !m z
 fully z ! circum/ances seem to w>rant4 
,,!orems4 ,9 ! same circle or 9 equal   
circles equal c5tral angles int]cept    

equal arcs2 & ( two unequal c5tral      
angles ! great] int]cepts ! great] arc1 
& conv]sely = bo? ( ^! cases4  ,euclid  
made ^! ! tw5ty-six? & tw5ty-sev5?      
propositions ( 8 ,book ,,iii1 b he      
limit$ !m z foll[s3 ,9 equal circles    
equal angles /& on equal circumf]5ces1  
:e!r !y /& at ! c5t]s or at !           
circumf]5ces1 & conv]sely4 ,he !re=e    
includ$ two ( our pres5t !orems 9 "o1   
?us mak+ ! proposition d\bly h>d = a    
beg9n]4 ,aft] ^! two propositions ! ,law
 ( ,conv]se1 already m5tion$ on page    
#aij1 may prop]ly be introduc$4         
,,!orems4 ,9 ! same circle or 9 equal   
circles1 if two arcs are equal1 !y are  
subt5d$ by equal *ords2 & if two arcs   
are unequal1 ! great] is subt5d$ by !   
great] *ord1 & conv]sely4  ,euclid      
dismisses all ?is ) ! simple !orem1 ,9  
equal circles equal circumf]5ces are    
subt5d$ by equal /raight l9es4 ,x w     
!re=e be notic$ t he has no special ^w =
 *ord & none = arc1 & t ! ^w            
circumf]5ce1 :i* "s tea*]s are s anxi\s 

to reta91 is us$ to mean bo? ! :ole     
circle & any arc4 ,x _c be d\bt$ t lat] 
writ]s h greatly improv$ ! language (   
geometry by ! use ( ^! mod]n t]ms4 ,! ^w
 arc is ! same1 etymologically1 z ar*1  
ea* be+ d]iv$ f ! ,lat9 arcus 7a b[74   
,*ord is f ! ,greek1 mean+ ! /r+ ( a    
musical in/rum5t4 ,subt5d is f ! ,lat9  
sub 7"u71 & t5d]e 7to /ret*74  ,x %\ld  
be notic$ t ,euclid speaks ( equal      
circles1 :ile we speak ( ! same circle  
or equal circles1 conf9+ our pro(s to ! 
latt]1 on ! supposition t ?is           
suffici5tly cov]s ! =m]4  ,,!orem4 ,a   
l9e ?r\gh ! c5t] ( a circle p]p5dicul>  
to a *ord bisects ! *ord & ! arcs       
subt5d$ by x4  ,?is is an improvem5t on 
,euclid1 ,,iii1 #c3 ,if 9 a circle a    
/raight l9e ?r\gh ! c5t] bisects a      
/raight l9e n ?r\gh ! c5t]1 x also cuts 
x at "r angles2 & if x cuts x at "r     
angles1 x also bisects x4 ,x is a v     
important proposition1 !oretically &    
practically1 = x enables u to f9d ! c5t]
 ( a circle if we "k any "p ( its arc4  

,a civil eng9e]1 = example1 :o wi%es to 
f9d ! c5t] ( ! circle ( :i* "s curve 7l 
t on a runn+ track1 on a railroad1 or 9 
a p>k7 is an arc1 takes two *ords1 say (
 "o hundr$ feet ea*1 & erects p]p5dicul>
 bisectors4 ,x is well to ask a class   
:y1 9 practice1 x is bett] to take ^!   
*ords "s di/ance ap>t4 ,eng9e]s (t5 *eck
 _! "w by tak+ ?ree *ords1 ! p]p5dicul> 
bisectors ( ! ?ree pass+ ?r\gh a s+le   
po9t4 ,illu/rations ( ?is k9d ( "w are  
giv5 lat] 9 ?is *apt]4  ,,!orem4 ,9 !   
same circle or 9 equal circles equal    
*ords are equidi/ant f ! c5t]1 & *ords  
equidi/ant f ! c5t] are equal4  ,?is    
proposition is practically us$ by       
eng9e]s 9 locat+ po9ts on an arc ( a    
circle t is too l>ge to be describ$ by a
 tape1 or t _c easily be rea*$ f ! c5t] 
on acc\nt ( ob/ructions4  ,illu/ration  
,if "p ( ! curve ,,apb is kn[n1 take ,p 
z ! mid-po9t4      ,!n /ret* ! tape f ,a
 to ,b & draw ,,pm      p]p5dicul> to x4
 ,!n sw+ ! l5g? ,,am ab\t ,p1 &         
,,pm ab\t ,b1 until !y meet at ,l1 &    

/ret* ! l5g?      ,,ab along ,,pl to ,q4
 ,?is fixes ! po9t ,q4 ,9 ! same        
way fix ! po9t ,c4 ,po9ts on ! curve c  
?us be fix$ z      ne> toge!r z we wi%4 
,! *ords ,,ab1 ,,pq1 ,,bc1 & s      on1 
are equal & are equally di/ant f ! c5t]4
  ,,!orem4 ,a l9e p]p5dicul> to a radius
 at its extremity is tang5t to ! circle4
  ,! enunciation ( ?is proposition by   
,euclid is v int]e/+4 ,x is z foll[s3   
,! /raight l9e drawn at "r angles to !  
diamet] ( a      circle at its extremity
 w fall outside ! circle1 & into      ! 
space betwe5 ! /raight l9e & !          
circumf]5ce      ano!r /raight l9e _c be
 int]pos$2 fur!r1 ! angle      ( !      
semicircle is great] & ! rema9+ angle   
less ?an      any acute rectil9eal      
angle4  ,! fir/ ass]tion is practically 
t ( tang5cy1--w fall outside ! circle4  
,! second "o /ates1 sub/antially1 t !re 
is only "o su* tang5t1 or1 z we say 9   
mod]n ma!matics1 ! tang5t is unique4 ,! 
?ird /atem5t relates to ! angle =m$ by !
 diamet] & ! circumf]5ce1--a mix$ angle1

 z ,proclus call$ x1 & a k9d ( angle no 
long] us$ 9 elem5t>y geometry4 ,! f\r?  
/atem5t practically ass]ts t ! angle    
betwe5 ! tang5t & circumf]5ce is less   
?an any assignable quantity4 ,?is gives 
rise to a difficulty t seems to h puzzl$
 _m ( ,euclid's comm5tators1 & t w      
int]e/ a pupil3 ,z ! circle dim9i%es ?is
 angle app>5tly increases1 :ile z !     
circle increases ! angle decreases1 &   
yet ! angle is always /at$ to be z]o4   
,vieta 7#aedj-afjc71 :o did mu* to      
improve ! sci5ce ( algebra1 attempt$ to 
expla9 away ! difficulty by adopt+ a    
notion ( circle t 0 preval5t 9 8 "t4 ,he
 said t a circle 0 a polygon ( an       
inf9ite numb] ( sides 7:i* x _c be1 by  
def9ition71 & t1 a tang5t simply co9cid$
 ) "o ( ! sides1 & !re=e made no angle )
 x2 & ?is view 0 also held by ,galileo  
7#aefd-afdb71 ! great physici/ &        
ma!matician :o fir/ /at$ ! law ( !      
p5dulum4  ,,!orem4 ,p>allel l9es        
int]cept equal arcs on a circle4  ,!    
conv]se ( ?is proposition has an int]e/+

 application 9 outdoor "w4  ,illu/ration
       ,suppose we wi% to run a l9e     
?r\gh ,p p>allel to a giv5      l9e     
,,ab4 ,) any conv5i5t po9t ,o z a c5t]1 
& ,,op      z a radius1 describe a      
circle cutt+ ,,ab 9 ,x & ,y4      ,draw 
,,px4 ,!n ) ,y z a c5t] & ,,px z a      
radius draw      an arc cutt+ ! circle 9
 ,q4 ,!n run ! l9e f ,p to      ,q4 ,,pq
 is p>allel to ,,ab by ! conv]se ( !    
above      !orem1 :i* is easily %[n to  
be true = ?is figure4  ,,!orem4 ,if two 
circles are tang5t to ea* o!r1 ! l9e (  
c5t]s passes ?r\gh ! po9t ( contact4    
,!re are _m illu/rations ( ?is !orem 9  
practical "w1 z 9 ! case ( cog:eels4 ,an
 int]e/+ application to eng9e]+ is se5 9
 ! case ( two p>allel /reets or l9es (  
track :i* are to be connect$ by a rev]s$
 curve4  ,illu/ration       ,if ! l9es  
are ,,ab & ,,cd1 & ! connection is to be
      made1 z %[n1 f ,b to ,c1 we may   
proce$ z foll[s3      ,draw ,,bc &      
bisect x at ,m4 ,erect ,,po1 !          
p]p5dicul>      bisector ( ,,bm2 & ,,bo1

 p]p5dicul> to ,,ab4 ,!n ,o is      "o  
c5t] ( curvature4 ,9 ! same way fix ,o'4
 ,!n to      *eck ! "w apply ?is !orem1 
,m be+ 9 ! l9e (      c5t]s ,,oo'4 ,!   
curves may n[ be drawn1 & !y w be       
tang5t to ,,ab1 to ,,cd1 & to ea* o!r4  
,at ?is po9t 9 ! ,am]ican textbooks x is
 ! cu/om to ins]t a brief treatm5t (    
measurem5t1 expla9+ :at is meant by     
ratio1 comm5surable & incomm5surable    
quantities1 con/ant & v>iable1 & limit1 
& introduc+ "o or m propositions relat+ 
to limits4 ,! object ( ?is dep>ture f ! 
anci5t sequ5ce1 :i* po/pon$ ?is subject 
to ! book on ratio & proportion1 is to  
treat ! circle m completely 9 ,book     
,,iii4 ,x mu/ be confess$ t ! treatm5t  
is n z sci5tific z t ( ,euclid1 z w be  
expla9$ "u ,book ,,iii1 b x is f> bett] 
suit$ to ! m9d ( a boy or girl4  ,x     
beg9s by def9+ measurem5t 9 a practical 
way1 z ! f9d+ ( ! numb] ( times a       
quantity ( any k9d conta9s a kn[n       
quantity ( ! same k9d4 ,( c\rse ?is     
gives a numb]1 b ?is numb] may be a     

surd1 l sqrt#b4 ,9 o!r words1 !         
magnitude measur$ may be incomm5surable 
) ! unit ( measure1 a seem+ p>adox4 ,)  
?is difficulty1 h[ev]1 ! pupil %\ld n be
 call$ ^u to cont5d at ?is /age 9 8     
progress4 ,! :ole subject (             
incomm5surables might safely be po/pon$1
 al?\gh x may be treat$ 9 an elem5t>y   
fa%ion at ?is "t4 ,! fact t ! measure ( 
! diagonal ( a squ>e1 ( :i* a side is   
unity1 is sqrt#b1 & t ?is measure is an 
incomm5surable numb]1 is n s p>adoxical 
z x seems1 ! p>adox be+ v]bal r ?an     
actual4  ,x is !n cu/om>y to def9e ratio
 z ! quoti5t ( ! num]ical measures ( two
 quantities 9 t]ms ( a common unit4 ,?is
 br+s all ratios to ! basis ( num]ical  
fractions1 & :ile x is n sci5tifically s
 satisfactory z ! anci5t concept :i*    
consid]$ ! t]ms z l9es1 surfaces1       
angles1 or solids1 x is m practical1 & x
 suffices = ! ne$s ( elem5t>y pupils4   
,comm5surable1 incomm5surable1 con/ant1 
& v>iable are !n def9$1 & ^! def9itions 
are foll[$ by a brief discussion (      

limit4 ,x simplifies ! treatm5t ( ?is   
subject to /ate at once t !re are two   
classes ( limits1--^? :i* ! v>iable     
actually rea*es1 & ^? :i* x c only      
approa* indef9itely ne>4 ,we f9d ! "o z 
frequ5tly z we f9d ! o!r1 al?\gh x is ! 
latt] t is ref]r$ to 9 geometry4 ,=     
example1 ! sup]ior limit ( a *ord is a  
diamet]1 & ?is limit ! *ord may rea*4 ,!
 inf]ior limit is z]o1 b we d n consid] 
! *ord z rea*+ ?is limit4 ,x is also    
well to call ! att5tion ( pupils to !   
fact t a quantity may decrease t[>ds its
 limit z well z increase t[>ds x4  ,su* 
fur!r def9itions z are ne$$ 9 ! !ory (  
limits are n[ introduc$4 ,among ^! is   
area ( a circle4 ,x might occur to "s   
pupil t s9ce a circle is a l9e 7z us$ 9 
mod]n ma!matics71 x c h no area4 ,?is   
is1 h[ev]1 a m]e quibble ov] words4 ,x  
is n pret5d$ t ! l9e has area1 b t area 
( a circle is m]ely a %ort5$ =m ( !     
expression area inclos$ by a circle4  ,!
 ,pr9ciple ( ,limits is n[ usually giv5 
z foll[s3 ,if1 :ile approa*+ _!         

respective limits1 two v>iables are     
always equal1 _! limits are equal4 ,?is 
0 express$ by ,d',alemb]t 9 ! eighte5?  
c5tury z ,magnitudes :i* are ! limits ( 
equal magnitudes are equal1 or ?is 9    
sub/ance4 ,x w\ld easily be possible to 
elaborate ?is !ory1 prov+1 = example1 t 
if x approa*es y z its limit1 !n ax     
approa*es ay z its limit1 & x_/a        
approa*es y_/a z its limit1 & s on4 ,v  
mu* ( ?is !ory1 h[ev]1 we>ies a pupil s 
t ! entire mean+ ( ! subject is lo/1 &  
at be/ ! treatm5t 9 elem5t>y geometry is
 n rigor\s4 ,x is ano!r case ( hav+ to  
sacrifice a /rictly sci5tific treatm5t  
to ! educational abilities ( ! pupil4   
,tea*]s wi%+ to f9d a sci5tific treatm5t
 ( ! subject %\ld consult a good "w on !
 calculus4  ,,!orem4 ,9 ! same circle or
 9 equal circles two c5tral angles h !  
same ratio z _! int]cept$ arcs4  ,?is is
 usually prov$ fir/ = ! comm5surable    
case & !n = ! incomm5surable "o4 ,!     
latt] is r>ely und]/ood by all ( !      
class1 & x may v prop]ly be requir$ only

 ( ^? :o %[ "s aptitude 9 geometry4 ,x  
is bett] to h ! o!rs und]/& fully !     
comm5surable case & see ! nature ( its  
applications1 possibly read+ !          
incomm5surable pro( ) ! tea*]1 ?an to   
/umble ab\t 9 ! d>kness ( !             
incomm5surable case & nev] rea* ! goal4 
,9 ,euclid !re 0 no di/9ction betwe5 !  
two because 8 def9ition ( ratio cov]$   
bo?2 b1 z we %all see 9 ,book ,,iii1 ?is
 def9ition is too difficult = our       
pupils4 ,!on ( ,alex&ria 7f\r? c5tury   
,a4,d471 ! fa!r ( ! ,hypatia :o is !    
h]o9e ( ,k+sley's well-kn[n novel1 wrote
 a comm5t>y on ,euclid1 & he adds t     
sectors also h ! same ratio z ! arcs1 a 
fact v easily prov$4 ,9 propositions (  
?is type1 ref]r+ to ! same circle or to 
equal circles1 x is n wor? :ile to ask  
pupils to take up bo? cases1 ! pro( =   
ei!r be+ obvi\sly a pro( = ! o!r4  ,_m  
writ]s /ate ?is proposition s t x reads 
t c5tral angles are measur$ by _!       
int]cept$ arcs4 ,?is1 ( c\rse1 is n     
lit]ally true1 s9ce we c measure any?+  

only by "s ?+1 ( ! same k9d4 ,?us we    
measure a volume by f9d+ h[ _m times x  
conta9s ano!r volume :i* we take z a    
unit1 & we measure a l5g? by tak+ "s o!r
 l5g? z a unit2 b we _c measure a giv5  
l5g? 9 qu>ts nor a giv5 weight 9 feet1 &
 x is equally impossible to measure an  
arc by an angle1 & vice v]sa4 ,nev]!less
 x is (t5 f\nd conv5i5t to def9e "s     
brief expression t has no mean+ if tak5 
lit]ally1 9 su* way t x %all acquire a  
mean+4 ,?us we def9e area ( a circle1   
ev5 :5 we use circle to mean a l9e2 & s 
we may def9e ! expression c5tral angles 
are measur$ by _! int]cept$ arcs to mean
 t c5tral angles h ! same num]ical      
measure z ^! arcs4 ,?is is done by mo/  
writ]s1 & is legitimate z expla9+ an    
abbreviat$ expression4  ,,!orem4 ,an    
inscrib$ angle is measur$ by half !     
int]cept$ arc4  ,9 ,euclid ?is          
proposition is comb9$ ) ! prec$+ "o 9 8 
,book ,,vi1 ,proposition #cc4 ,su* a    
proc$ure is n adapt$ to ! ne$s ( /ud5ts 
to-day4 ,euclid gave 9 ,book ,,iii1     

h[ev]1 ! proposition 7,no4 #bj7 t a     
c5tral angle is twice an inscrib$ angle 
/&+ on ! same arc4 ,s9ce ,euclid nev]   
consid]$ an angle great] ?an #ahj deg41 
8 inscrib$ angle 0 necess>ily less ?an a
 "r angle4 ,! fir/ "o :o is kn[n to h   
giv5 ! g5]al case1 tak+ ! c5tral angle z
 be+ also great] ?an #ahj deg41 0 ,h]on 
( ,alex&ria1 probably ( ! fir/ c5tury   
,a4,d4#fh ,9 ?is he 0 foll[$ by v>i\s   
lat] comm5tators1 includ+ ,t>taglia &   
,clavius 9 ! sixte5? c5tury4  ,"o ( ! _m
 int]e/+ ex]cises t may be d]iv$ f ?is  
!orem is se5 9 ! case ( ! horizontal    
dang] angle obs]v$ by %ips4             
,illu/ration       ,if "s dang]\s rocks 
lie (f ! %ore1 & ,l & ,l' are      two  
ligh?\ses1 ! angle ,a is det]m9$ by     
obs]vation1 s      t ,a w lie on a      
circle inclos+ ! dang]\s area4          
,angle ,a is call$ ! horizontal dang]   
angle4 ,%ips      pass+ 9 sight ( ! two 
ligh?\ses ,l & ,l' mu/ keep      out f> 
en\gh s t ! angle ,l',,sl %all be less  
?an      angle ,a4  ,to ?is proposition 

!re are sev]al important coroll>ies1    
includ+ ! foll[+3  #a4 ,an angle        
inscrib$ 9 a semicircle is a "r angle4  
,?is coroll>y is m5tion$ by ,ari/otle & 
is attribut$ to ,?ales1 be+ "o ( ! few  
propositions ) :i* 8 "n is connect$4 ,x 
enables u to describe a circle by lett+ 
! arms ( a c>p5t]'s squ>e slide along   
two nails driv5 9 a bo>d1 a p5cil be+   
held at ! v]tex4  ,illu/ration  ,a m    
practical use = x is made by ma*9i/s to 
det]m9e :e!r a ca/+ is a true           
semicircle4 ,tak+ a c>p5t]'s squ>e z "h 
%[n1 if ! v]tex t\*es ! curve at e po9t 
z ! squ>e slides ar\nd1 x is a true     
semicircle4 ,by a simil> me?od a circle 
may be describ$ by slid+ a draftsman's  
triangle s t two sides t\* two tacks    
driv5 9 a bo>d4  ,illu/ration           
,ano!r int]e/+ application ( ?is        
coroll>y may be se5      by tak+ an     
ord9>y pap] protractor ,,acb1 & fa/5+ a 
plumb l9e at ,b4 ,if ! protractor is s  
held t ! plumb      l9e cuts !          
semicircle at ,c1 !n ,,ac is level      

because x      is p]p5dicul> to !       
v]tical l9e ,,bc4 ,?us1 if a class      
wi%es to det]m9e ! horizontal l9e ,,ac1 
:ile sight+ up      a hill 9 ! direction
 ,,ab1 ?is is easily det]m9$ )\t      a 
_s level4  ,x foll[s f ?is coroll>y1 z !
 pupil has already f\nd1 t ! mid-po9t ( 
! hypot5use ( a "r triangle is          
equidi/ant f ! ?ree v]tices4 ,?is is    
useful 9 outdoor measur+1 =m+ ! basis ( 
"o ( ! be/ me?ods ( lett+ fall a        
p]p5dicul> f an ext]nal po9t to a l9e4  
,illu/ration       ,suppose ,,xy to be !
 edge ( a sidewalk1 & ,p a po9t 9      !
 /reet f :i* we wi% to lay a gas pipe   
p]p5dicul>      to ! walk4 ,f ,p sw+ a  
cord or tape1 say #fj feet long1        
until x meets ,,xy at ,a4 ,!n take ,m1 !
 mid-po9t (      ,,pa1 & sw+ ,,mp ab\t  
,m1 to meet ,,xy at ,b4 ,!n ,b      is !
 foot ( ! p]p5dicul>1 s9ce ,l,,pba c be 
inscrib$ 9 a semicircle4  #b4 ,angles   
inscrib$ 9 ! same segm5t are equal4     
,illu/ration       ,by driv+ two nails 9
 a bo>d1 at ,a & ,b1 & tak+ an          

angle ,p made ( rigid mat]ial 79        
p>ticul>1 z already      /at$1 a        
c>p5t]'s squ>e71 a p5cil plac$ at ,p w  
g5]ate an arc ( a circle if ! arms slide
 along ,a &      ,b4 ,?is is an int]e/+ 
ex]cise = pupils4  ,,!orem4 ,an angle   
=m$ by two *ords int]sect+ )9 ! circle  
is measur$ by half ! sum ( ! int]cept$  
arcs4  ,,!orem4 ,an angle =m$ by a      
tang5t & a *ord drawn f ! po9t ( tang5cy
 is measur$ by half ! int]cept$ arc4    
,,!orem4 ,an angle =m$ by two secants1 a
 secant & a tang5t1 or two tang5ts1     
drawn to a circle f an ext]nal po9t1 is 
measur$ by half ! diff]5ce ( ! int]cept$
 arcs4  ,^! ?ree !orems are all special 
cases ( ! g5]al proposition t ! angle   
includ$ betwe5 two l9es t cut 7or t\*7 a
 circle is measur$ by half ! sum ( !    
int]cept$ arcs4 ,if ! po9t passes f )9 !
 circle to ! circle itself1 "o arc      
becomes z]o & ! angle becomes an        
inscrib$ angle4 ,if ! po9t passes       
outside ! circle1 ! small] arc becomes  
negative1 hav+ pass$ ?r\gh z]o4 ,! po9t 

may ev5 g to inf9ity1 z is said 9 high] 
ma!matics1 ! l9es !n becom+ p>allel1 & !
 angle becom+ z]o1 be+ measur$ by half !
 sum ( "o arc & a negative arc ( ! same 
absolute value4 ,?is is "o ( ! be/      
illu/rations ( ! ,pr9ciple ( ,cont9uity 
to be f\nd 9 geometry4  ,,problem4 ,to  
let fall a p]p5dicul> ^u a giv5 l9e f a 
giv5 ext]nal po9t4  ,?is is ! fir/      
problem t a /ud5t meets 9 mo/ ,am]ican  
geometries4 ,! reason = treat+ !        
problems by !mselves in/ead ( m+l+ !m ) 
! !orems has already be5 discuss$4#fi ,!
 /ud5t n[ has a suffici5t body ( !orems1
 by :i* he c prove t 8 con/ructions are 
correct1 & ! advantage ( treat+ ^!      
con/ructions toge!r is great] ?an t (   
foll[+ ,euclid's plan ( introduc+ !m    
:5ev] ne$$4  ,proclus tells u t ?is     
problem 0 fir/ inve/igat$ by            
,o5opides1#gj :o ?\ght x useful =       
a/ronomy4 ,proclus speaks ( su* a l9e z 
a gnomon1 a common "n = ! p]p5dicul> on 
a sundial1 :i* ca/s ! %ad[ by :i* ! "t (
 day is kn[n4 ,he also speaks ( two k9ds

 ( p]p5dicul>s1 ! plane & solid1 ! =m]  
be+ a l9e p]p5dicul> to a l9e1 & ! latt]
 a l9e p]p5dicul> to a plane4  ,x is    
int]e/+ to notice t ! solution tacitly  
assumes t a c]ta9 arc is go+ to cut !   
giv5 l9e 9 two po9ts1 & only two4       
,/rictly speak+1 :y may x n cut x 9 only
 "o po9t1 or ev5 9 ?ree po9ts8 ,we      
really assume t if a /raight l9e is     
drawn ?r\gh a po9t )9 a circle1 ?is l9e 
mu/ get out ( ! circle on ea* ( two     
sides ( ! giv5 po9t1 & 9 gett+ out x mu/
 cut ! circle twice4 ,proclus notic$ ?is
 assumption & endeavor$ to prove x4 ,x  
is bett]1 h[ev]1 n to raise ! "q )      
beg9n]s1 s9ce x seems to !m l           
hair-splitt+ to no purpose4  ,! problem 
is ( mu* value 9 survey+1 & tea*]s w\ld 
d well to ask a class to let fall a     
p]p5dicul> to ! edge ( a sidewalk f a   
po9t #bj feet f ! walk1 us+ an ord9>y   
#ff-;foot or #ej-;foot tape4            
,practically1 ! be/ plan is to sw+ #cj  
feet ( ! tape ab\t ! po9t & m>k ! two   
po9ts ( int]section ) ! edge ( ! walk4  

,!n measure ! di/ance betwe5 ! po9ts &  
take half ( ?is di/ance1 ?us fix+ ! foot
 ( ! p]p5dicul>4  ,,problem4 ,at a giv5 
po9t 9 a l9e1 to erect a p]p5dicul> to t
 l9e4  ,?is might be po/pon$ until aft] 
! problem to bisect an angle1 s9ce x    
m]ely requires ! bisection ( a /raight  
angle2 b consid]+ ! immaturity ( !      
av]age pupil1 x is bett] giv5           
indep5d5tly4 ,! usual case consid]s !   
po9t n at ! extremity ( ! l9e1 & !      
solution is ess5tially t ( ,euclid4 ,9  
practice1 h[ev]1 z = example 9 survey+1 
! po9t may be at ! extremity1 & x may n 
be conv5i5t to produce ! l9e4           
,illu/ration       ,surveyors sometimes 
measure ,,pb  #c ft41 & !n take #i ft4  
( tape1 ! ends be+ held at ,b & ,p1 & ! 
tape be+      /ret*$ to ,a1 s t ,,pa  #d
 ft4 & ,,ab  #e ft4 ,!n      ,p is a "r 
angle by ! ,py?agorean ,!orem4 ,?is     
!orem      n hav+ yet be5 prov$1 x _c be
 us$ at ?is "t4  ,a solution = ! problem
 ( erect+ a p]p5dicul> f ! extremity ( a
 l9e t _c be produc$1 dep5d+1 h[ev]1 on 

! problem ( bisect+ an angle1 & !re=e to
 be giv5 aft] t problem1 is attribut$ by
 ,al-,nairizi 7t5? c5tury ,a4,d47 to    
,h]on ( ,alex&ria4 ,x is also giv5 by   
,proclus4  ,illu/ration       ,requir$  
to draw f ,p a p]p5dicul> to ,,ap4 ,take
 ,x      any:]e on ! l9e & erect ,,xy   
p]p to ,,ap 9 ! usual      mann]4       
,bisect ,l,,pxy by ! l9e ,,xm4 ,on ,,xy 
take ,,xn  ,,xp1      & draw ,,nm p]p to
 ,,xy4 ,!n draw ,,pm4 ,! pro(      is   
evid5t4  ,^! may at ! prop] "t be giv5 z
 int]e/+ v>iants ( ! usual solution4    
,,problem4 ,to bisect a giv5 l9e4       
,euclid said f9ite /raight l9e1 b ?is   
word+ is n commonly foll[$1 because x w 
be inf]r$ t ! l9e is f9ite if x is to be
 bisect$1 & we use l9e alone to mean a  
/raight l9e4 ,euclid's plan 0 to        
con/ruct an equilat]al triangle 7by 8   
,proposition #a ( ,book ,i7 on ! l9e z a
 base1 & !n to bisect ! v]tical angle4  
,proclus tells u t ,apollonius ( ,p]ga1 
:o wrote ! fir/ great "w on conic       
sections1 us$ a plan :i* is sub/antially

 t :i* is commonly f\nd 9 textbooks     
to-day1--con/ruct+ two isosceles        
triangles ^u ! l9e z a common base1 &   
connect+ _! v]tices4  ,,problem4 ,to    
bisect a giv5 angle4  ,x %\ld be notic$ 
t 9 ! usual solution two arcs int]sect1 
& ! po9t ?us det]m9$ is connect$ ) !    
v]tex4 ,n[ ^! two arcs int]sect twice1 &
 s9ce "o ( ! po9ts ( int]section may be 
! v]tex itself1 ! o!r po9t ( int]section
 mu/ be tak54 ,x is n1 h[ev]1 wor? :ile 
to make mu* ( ?is matt] ) pupils4       
,proclus calls att5tion to ! possible   
sugge/ion t ! po9t ( int]section may be 
imag9$ to lie outside ! angle1 & he     
proce$s to %[ ! absurdity2 b "h1 aga91 !
 subject is n "o ( value to beg9n]s4 ,he
 also contributes to ! hi/ory ( !       
trisection ( an angle4 ,any angle is    
easily trisect$ by means ( c]ta9 high]  
curves1 su* z ! con*oid ( ,nicom$es 7ca4
 #ahj ,b4,c471 ! quadratrix ( ,hippias (
 ,elis 7ca4 #dbj ,b4,c471 or ! spiral ( 
,ar*im$es 7ca4 #bej ,b4,c474 ,b s9ce ?is
 problem1 /at$ algebraically1 requires !

 solution ( a cubic equation1 & ?is     
involves1 geometrically1 f9d+ ?ree      
po9ts1 we _c solve ! problem by means ( 
/raight l9es & circles alone4 ,9 o!r    
words1 ! trisection ( any angle1 by !   
use ( ! /raight$ge & compasses alone1 is
 impossible4 ,special angles may h[ev]  
be trisect$4 ,?us1 to trisect an angle (
 #ij deg4 we ne$ only to con/ruct an    
angle ( #fj deg41 & ?is c be done by    
con/ruct+ an equilat]al triangle4 ,b    
:ile we _c trisect ! angle1 we may      
easily approximate trisection4 ,= s9ce1 
9 ! inf9ite geometric s]ies #a_/b  #a_/h
  #a_/cb  #a_/abh  4441 s  a _/ 7#a -   
r71 we h s  #a_/b _/ #c_/d  #b_/c4 ,9   
o!r words1 if we add #a_/b ( ! angle1   
#a_/h ( ! angle1 #a_/cb ( ! angle1 & s  
on1 we approa* z a limit #b_/c ( !      
angle2 b all ( ^! fractions c be obta9$ 
by repeat$ bisections1 & h5ce by        
bisections we may approximate !         
trisection4  ,! approximate bisection   
7or any o!r division7 ( an angle may (  
c\rse be effect$ by ! help ( !          

protractor & a /raight$ge4 ,! geometric 
me?od is1 h[ev]1 usually m accurate1 & x
 is advantage\s to h ! pupils try bo?   
plans1 say = bisect+ an angle ( ab\t #di
 #a_/b deg44  ,illu/ration              
,applications ( ?is problem are num]\s4 
,x may be desir$1 = example1 to set a   
lamp-po/ on a l9e bisect+ ! angle =m$ by
 two /reets t come toge!r a little      
unsymmetrically1 z "h %[n1 9 :i* case ! 
bisect+ l9e c easily be run by ! use ( a
 measur+ tape1 or ev5 ( a /\t cord4  ,a 
m int]e/+ illu/ration is1 h[ev]1 !      
foll[+3  ,illu/ration       ,let !      
pupils set a /ake1 say ab\t #e feet     
high1 at a po9t      ,n on ! s*ool      
gr\nds ab\t #i ,a4,m41 & c>efully       
measure      ! l5g? ( ! %ad[1 ,,nw1     
plac+ a small wood5 p9 at      ,w4 ,!n  
ab\t #c ,p4,m4 let !m wat* until ! %ad[ 
,,ne is      exactly ! same l5g? t x 0  
:5 ,w 0 fix$1 &      !n place a small   
wood5 p9 at ,e4 ,if ! "w has be5 v      
c>efully done1 & !y take ! tape & bisect
 ! l9e      ,,we1 ?us fix+ ! l9e ,,ns1  

!y w h a nor? &      s\? l9e4 ,if ?is is
 m>k$ out = a %ort di/ance f      ,n1 !n
 :5 ! %ad[ falls on ,,ns1 x w be noon by
 sun      "t 7n /&>d "t7 at ! s*ool4    
,,problem4 ,f a giv5 po9t 9 a giv5 l9e1 
to draw a l9e mak+ an angle equal to a  
giv5 angle4  ,proclus says t ,eudemus   
attribut$ to ,o5opides ! discov]y ( !   
solution :i* ,euclid gave1 & :i* is     
sub/antially ! "o n[ commonly se5 9     
textbooks4 ,! problem 0 probably solv$ 9
 "s fa%ion be=e ! "t ( ,o5opides1 h[ev]4
 ,! object ( ! problem is prim>ily to   
enable u to draw a l9e p>allel to a giv5
 l9e4  ,practically1 ! draw+ ( "o l9e   
p>allel to ano!r is usually effect$ by  
means ( a p>allel rul] 7see page #aia71 
or by ! use ( draftsm5's triangles1 z "h
 %[n1 or ev5 m commonly by ! use ( a    
,t-squ>e1 su* z is "h se54 ,?is         
illu/ration %[s two ,t-squ>es us$ =     
draw+ l9es p>allel to ! sides ( a bo>d  
^u :i* ! draw+ pap] is fa/5$4#ga        
,illu/ration  ,illu/ration  ,an +5i\s   
in/rum5t describ$ by ,b>on ,dup9 is     

illu/rat$ bel[4  ,illu/ration       ,to 
! b> ,a is fa/5$ ! slid+ *eck ,b4 ,a    
movable      *eck ,d may be fa/5$ by a  
screw ,c4 ,a %>p po9t is      fix$ 9 ,b1
 s t z ,d slides along ! edge ( a bo>d1 
! po9t m>ks a l9e p>allel to ! edge4    
,moreov]1 ,f &      ,g are two brass    
arms ( equal l5g? jo9$ by a po9t$       
screw ,h t m>ks a l9e midway betwe5 ,b &
 ,d4      ,fur!rmore1 x is evid5t t ,h w
 draw a l9e bisect+      any irregul>   
bo>d if ! *ecks ,b & ,d are kept 9      
contact ) ! irregul> edges4  ,book ,,ii 
(f]s two g5]al l9es ( application t may 
be introduc$ to advantage1 pref]ably z  
additions to ! textbook "w4 ,"o ( ^! has
 ref]5ce to topographical draw+ & relat$
 subjects1 & ! o!r to geometric design4 
,z long z ^! c be introduc$ to ! pupil )
 an air ( reality1 !y s]ve a good       
purpose1 b if made a "p ( textbook "w1  
!y soon come to h less int]e/ ?an !     
ex]cises ( a m ab/ract "*4 ,if a tea*] c
 relate ! problems 9 topographical draw+
 to ! pupil's home t[n1 & c occasionally

 set "s outdoor "w ( ! nature "h        
sugge/$1 ! results are usually salut>y2 
b if he reit]ates only a half-doz5      
simple propositions "t aft] "t1 ) only  
slight *anges 9 ! nature ( !            
application1 !n ! results w n lead to a 
cultivation ( p[] 9 geometry1--a po9t   
:i* ! writ]s on appli$ geometry usually 
fail to recognize4  ,illu/ration  ,"o ( 
! simple applications ( ?is book relates
 to ! r\nd+ ( corn]s 9 lay+ out /reets 9
 "s ( our mod]n t[ns :]e !re is a desire
 to dep>t f ! conv5tional squ>e corn]4  
,x is also us$ 9 lay+ out p>k walks &   
drives4  ,illu/ration       ,! figure 9 
! middle ( ! page repres5ts two /reets1 
,,ap & ,,bq1 t w\ld1 if prolong$1       
int]sect at ,c4 ,x      is requir$ to   
con/ruct an arc s t !y %all beg9 to     
curve at ,p & ,q1 :]e ,,cp  ,,cq1 & h5ce
 ! c5t]      ( curvature ,o mu/ be f\nd4
       ,! problem is a common "o 9      
railroad "w1 only "h ,,ap is            
usually oblique to ,,bq if !y are       
produc$ to meet at ,c1 z      9 ! second

 figure on page #bah4 ,x is requir$ to  
con/ruct      an arc s t ! tracks %all  
beg9 to curve at ,p & ,q1      :]e ,,cp 
,,cq4  ,illu/ration  ,! problem becomes 
a little m complicat$1 & correspond+ly m
 int]e/+1 :5 we h to f9d ! c5t] (       
curvature = a /reet railway track t mu/ 
turn a corn] 9 su* a way z to all[1 say1
 exactly #e feet f ! po9t ,p1 on acc\nt 
( a sidewalk4  ,illu/ration  ,! problem 
becomes /ill m difficult if we h two    
roads ( diff]5t wid?s t we wi% to jo9 on
 a curve4 ,"h ! two c5t]s ( curvature   
are n ! same1 & ! "o road n>r[s to ! o!r
 on ! curve4 ,! solutions w be und]/ood 
f a /udy ( ! figures4  ,! numb] (       
problems ( ?is k9d t c easily be made is
 limitless1 & x is well to avoid ! dang]
 ( hobby rid+ on ?is or any simil>      
topic4 ,!re=e a s+le "o w suffice to    
close ?is gr\p4  ,illu/ration       ,if 
a road ,,ab on an arc describ$ ab\t ,o1 
is to be jo9$      to road ,,cd1        
describ$ ab\t ,o'1 ! arc ,,bc %\ld      
evid5tly be int]nally tang5t to ,,ab &  

ext]nally tang5t      to ,,cd4 ,h5ce !  
c5t] is on ,,box & ,o',,cy1 & is        
!re=e at ,p4 ,! problem becomes m real  
if we give "s      wid? to ! roads 9    
mak+ ! draw+1 & imag9e !m 9 a      p>k t
 is be+ laid out ) drives4  ,x w be     
notic$ t ! above problems require !     
erect+ ( p]p5dicul>s1 ! bisect+ (       
angles1 & ! application ( ! propositions
 on tang5ts4  ,a some:at diff]5t l9e (  
problems is t relat+ to ! pass+ ( a     
circle ?r\gh ?ree giv5 po9ts4 ,x is v   
easy to manufacture problems ( ?is k9d t
 h a semblance ( reality4  ,illu/ration 
,= example1 let x be requir$ to plan a  
driveway f !      gate ,g to ! por* ,p s
 z to avoid a mass ( rocks ,r1      an  
arc ( a circle to be tak54 ,( c\rse1 if 
we all[ pupils      to use ! ,py?agorean
 ,!orem at ?is "t 7& = metrical         
purposes ?is is entirely prop]1 because 
!y h long be5      famili> ) x71 !n we  
may ask n only = ! draw+1      b we may1
 = example1 give ! l5g? f ,g to ! po9t  
on ,r 7:i* we may also call ,r71 & !    

angle ,,rgo z      #fj deg41 to f9d !   
radius4  ,a second g5]al l9e ( ex]cises 
adapt$ to ,book ,,ii is a cont9uation ( 
! geometric draw+ recomm5d$ z a         
prelim9>y to ! "w 9 demon/rative        
geometry4 ,! copy+ or ! mak+ ( designs  
requir+ ! describ+ ( circles1 _!        
inscription 9 or circumscription ab\t   
triangles1 & _! con/ruction 9 v>i\s     
positions ( tang5cy1 has "s value z     
apply+ ! v>i\s problems /udi$ 9 ?is     
book4 ,= a numb] ( ye>s pa/1 sev]al     
en?usia/ic tea*]s h made mu* ( ! designs
 f\nd 9 ,go?ic w9d[s1 hav+ _! pupils    
make ! outl9e draw+s by ! help (        
compasses & /raight$ge4 ,:ile su* "w has
 its value1 x is liable soon to deg5]ate
 into purposeless =malism1 & h5ce to    
lose int]e/ by tak+ ! vigor\s m9d ( y\? 
f ! /rong /udy ( geometry to ! weak     
manipulation ( in/rum5ts4 ,nev]!less its
 value %\ld be appreciat$ & cons]v$1 & a
 few illu/rations ( ^! =ms are giv5 9   
ord] t ! tea*] may h examples f :i* to  
select4 ,! be/ way ( us+ ?is mat]ial is 

to (f] x z supplem5t>y "w1 us+ mu* or   
little1 z may seem be/1 ?us giv+ to x a 
fre%ness & int]e/ t "s h tr\ble 9 imp>t+
 to ! regul> book "w4  ,! be/ plan is to
 sket* rapidly ! outl9e ( a w9d[ on !   
blackbo>d1 ask+ ! pupils to make a r\gh 
draw+1 & to br+ 9 a ma!matical draw+ on 
! foll[+ day4  ,illu/ration       ,x    
might be said1 = example1 t 9 plann+ a  
,go?ic w9d[      ?is draw+ is ne$$4 ,!  
arc ,,bc is drawn ) ,a z a      c5t] &  
,,ab z a radius4 ,! small ar*es are     
describ$      ) ,a1 ,d1 & ,b z c5t]s &  
,,ad z a radius4 ,!      c5t] ,p is f\nd
 by tak+ ,a & ,b z c5t]s & ,,ae z      a
 radius4 ,h[ may ! po9ts ,d1 ,e1 & ,f be
 f\nd8 ,draw      ! figure4 ,f ! /udy ( 
! rectil9e> figures sugge/$      by su* 
a simple patt]n ! prop]ties ( !         
equilat]al      triangle may be inf]r$4 
,! ,go?ic w9d[ also (f]s "s int]e/+     
possibilities 9 connection ) ! /udy ( ! 
squ>e4 ,= example1 ! illu/ration giv5 on
 page #bbc %[s a numb] ( trac]ies       
involv+ ! con/ruction ( a squ>e1 !      

bisect+ ( angles1 & ! describ+ (        
circles4#gb  ,illu/ration  ,! prop]ties 
( ! squ>e1 a figure n[ easily con/ruct$ 
by ! pupils1 are n num]\s4 ,:at few !re 
are may be br\ght out ?r\gh ! /udy ( art
 =ms1 if desir$4 ,9 case ^! =ms are %[n 
to a class1 x is important t !y %\ld be 
select$ f good designs4 ,we h en\gh poor
 art 9 ! _w1 s t geometry %\ld n        
contribute any m4 ,?is illu/ration is a 
type ( ! be/ m$ieval ,go?ic p>quetry4#gc
  ,illu/ration3 ,,go?ic ,,designs       
,,employ+ ,,circles ,,& ,,bisect$       
,,angles  ,ev5 simple designs ( a       
semipuzzl+ nature h _! advantage 9 ?is  
connection4 ,9 ! foll[+ example ! inn]  
squ>e conta9s all ( ! triangles1 !      
lett]s %[+ :]e !y may be fitt$4#gd      
,/ill m elaborate designs1 bas$ *iefly  
^u ! squ>e & circle1 are %[n 9 ! w9d[   
trac]ies on page #bbe1 & o!rs w be giv5 
9 connection ) ! /udy ( ! regul>        
polygons4  ,illu/ration  ,designs l !   
figure bel[ are typical ( ! simple =ms1 
bas$ on ! squ>e & circle1 t pupils may  

pr(itably incorporate 9 any "w 9 art    
design t !y may be do+ at ! "t !y are   
/udy+ ! circle & ! problems relat+ to   
p]p5dicul>s & squ>es4  ,illu/ration     
,among ! applications ( ! problem to    
draw a tang5t to a giv5 circle is ! case
 ( ! common tang5ts to two giv5 circles4
 ,"s au?ors give ?is z a basal problem1 
al?\gh x is m commonly giv5 z an ex]cise
 or a coroll>y4 ,"o ( ! mo/ obvi\s      
applications ( ! idea is t relat+ to !  
transmission ( circul> motion by means (
 a b& ov] two :eels1#ge ,a & ,b1 z %[n  
on page #bbf4  ,illu/ration3 ,,go?ic    
,,designs ,,employ+ ,,circles ,,&       
,,bisect$ ,,angles  ,! b& may ei!r n be 
cross$ 7! case ( ! two ext]ior tang5ts71
 or be cross$ 7! int]ior tang5ts71 !    
latt] all[+ ! :eels to turn 9 opposite  
directions4 ,9 case ! b& is liable to   
*ange its l5g?1 on acc\nt ( /ret*+ or   
v>iation 9 heat or moi/ure1 a ?ird :eel1
 ,d1 is us$4 ,we !n h ! case ( tang5ts  
to ?ree pairs ( circles4 ,illu/rations (
 ?is nature make ! ex]cise on ! draw+ ( 

common tang5ts to two circles assume an 
appe>ance ( g5u9e reality t is (        
advantage to ! "w4  ,illu/ration        
,,footnotes3  #fh ,?is is ! late/       
op9ion4 ,he is usually assign$ to ! fir/
 c5tury ,b4,c4  #fi ,see page #ed4  #gj 
,a ,greek philosoph] & ma!matician ( !  
fif? c5tury ,b4,c4  #ga ,?is illu/ration
 & ! foll[+ two are f ,c4 ,dup91        
,ma!matics ,practically ,appli$1        
translat$ f ! ,fr5* by ,g4 ,birkbeck1   
,halifax1 #ahed4 ,?is is probably ! mo/ 
s*ol>ly attempt ev] made at con/ruct+ a 
practical geometry4  #gb ,?is           
illu/ration & o!rs ( ! same type us$ 9  
?is "w are f ! excell5t draw+s by ,r4   
,w4 ,bill+s1 9 ,! ,inf9ity ( ,geometric 
,design ,exemplifi$1 ,london1 #ahdi4    
#gc ,f ,h4 ,kolb1 ,d] ,ornam5t5s*atz 444
 aus all5 ,kun/-,epo*51 ,/uttg>t1 #ahhc4
 ,! orig9al is 9 ! ,*ur* ( ,sa9t        
,ana/asia 9 ,v]ona4  #gd ,f ,j4 ,b5nett1
 ,! ,arcanum 444 ,a ,concise ,!ory (    
,practicable ,geometry1 ,london1 #ahch1 
"o ( ! _m books t h assum$ to           

revolutionize geometry by mak+ x        
practical4  #ge ,! figures are f ,dup91 
loc4 cit4     ,,*apt] ,,xvi  ,,! ,,lead+
 ,,propositions ,,( ,,book ,,iii   ,9 ! 
,am]ican textbooks ,book ,,iii is       
usually assign$ to proportion4 ,x is    
!re=e necess>y at ! beg9n+ ( ?is        
discussion to consid] :at is meant by   
ratio & proportion1 & to comp>e ! anci5t
 & ! mod]n !ories4 ,! subject is treat$ 
by ,euclid 9 8 ,book ,v1 & an anonym\s  
comm5tator has told u t x is ! discov]y 
( ,eudoxus1 ! tea*] ( ,plato4 ,n[       
proportion _h be5 kn[n long be=e ! "t ( 
,eudoxus 7#djh-cee ,b4,c471 b x 0       
num]ical proportion1 & z su* x _h be5   
/udi$ by ! ,py?agoreans4 ,!y 7 also !   
fir/ to /udy s]i\sly ! incomm5surable   
numb]1 & ) ?is /udy ! treatm5t (        
proportion f ! /&po9t ( rational numb]s 
lo/ its sci5tific position ) respect to 
geometry4 ,x 0 because ( ?is t ,eudoxus 
work$ out a !ory ( geometric proportion 
t 0 indep5d5t ( numb] z an expression ( 
ratio4  ,! foll[+ f\r def9itions f      

,euclid are ! basal ones ( ! anci5t     
!ory3       ,a ratio is a sort (        
relation 9 respect ( size betwe5 two    
magnitudes ( ! same k9d4                
,magnitudes are said to h a ratio to "o 
ano!r :i* are      capable1 :5          
multipli$1 ( exce$+ "o ano!r4           
,magnitudes are said to be 9 ! same     
ratio1 ! fir/ to !      second & ! ?ird 
to ! f\r?1 :51 if any equimultiples     
:atev] be tak5 ( ! fir/ & ?ird1 & any   
equimultiples :atev] ( ! second & f\r?1 
! =m]      equimultiples alike exce$1   
are alike equal to1 or alike fall       
%ort (1 ! latt] equimultiples           
respectively tak5 9      correspond+    
ord]4       ,let magnitudes :i* h ! same
 ratio be call$      proportional4#gf   
,( ^!1 ! fir/ is s loose 9 /atem5t z (t5
 to h be5 ?\ght to be an int]polation ( 
"s lat] writ]4 ,x 0 probably1 h[ev]1 put
 into ! orig9al = ! sake ( complet5ess1 
to h "s k9d ( /atem5t conc]n+ ratio z a 
prelim9>y to ! important def9ition (    
quantities 9 ! same ratio4 ,l !         

def9ition ( /raight l9e1 x 0 n int5d$ to
 be tak5 s]i\sly z a ma!matical /atem5t4
  ,! second def9ition is int5d$ to      
exclude z]o & inf9ite magnitudes1 & to  
%[ t incomm5surable magnitudes are      
includ$4  ,! ?ird def9ition is !        
ess5tial "o ( ! anci5t !ory4 ,x def9es  
:at is meant by say+ t magnitudes are 9 
! same ratio2 9 o!r words1 x def9es a   
proportion4 ,into ! m]its ( ! def9ition 
x is n propos$ to ent]1 = ! reason t x  
is no long] met 9 tea*+ 9 ,am]ica1 & is 
practically ab&on$ ev5 :]e ! re/ (      
,euclid's "w is 9 use4 ,x %\ld be said1 
h[ev]1 t x is sci5tifically correct1 t x
 cov]s ! case ( incomm5surable          
magnitudes z well z t ( comm5surable    
ones1 & t x is ! ,greek =]unn] ( ! mod]n
 !ories ( irrational numb]s4  ,z comp>$ 
) ! above treatm5t1 ! "o n[ giv5 9      
textbooks is unsci5tific4 ,we def9e     
ratio z ! quoti5t ( ! num]ical measures 
( two quantities ( ! same k9d1 &        
proportion z an equality ( ratios4  ,b  
:at d we mean by ! quoti5t1 say ( sqrt#b

 by sqrt#c8 ,& :5 we multiply a ratio by
 sqrt#e1 :at is ! mean+ ( ?is op]ation8 
,if we say t sqrt#b 3 sqrt#c means a    
quoti5t1 :at mean+ %all we assign to    
quoti5t8 ,if x is ! numb] t %[s h[ _m   
times "o numb] is conta9$ 9 ano!r1 h[ _m
 times is sqrt#c conta9$ 9 sqrt#b8 ,if  
to multiply is to take a numb] a c]ta9  
numb] ( times1 h[ _m times d we take x  
:5 we multiply by sqrt#e8 ,we c]ta9ly   
take x m ?an #b times & less ?an #c     
times1 b :at mean+ c we assign to sqrt#e
 times8 ,x w ?us be se5 t our treatm5t (
 proportion assumes t we already "k !   
!ory ( irrationals & c apply x to       
geometric magnitudes1 :ile ! anci5t     
treatm5t is indep5d5t ( ?is !ory4       
,educationally1 h[ev]1 we are =c$ to    
proce$ z we d4 ,j z ,d$ek9d's !ory (    
numb]s is a simple "o = college /ud5ts1 
s is ! anci5t !ory ( proportion2 b z !  
=m] is n suit$ to pupils 9 ! high s*ool1
 s ! latt] mu/ be relegat$ to ! college 
classes4 ,& 9 ?is we m]ely h>monize     
educational progress ) _w progress1 = ! 

num]ical !ory ( proportion long prec$$ !
 !ory ( ,eudoxus4  ,! anci5ts made mu* (
 su* t]ms z duplicate1 triplicate1      
alt]nate1 & inv]se ratio1 & also su* z  
composition1 sep>ation1 & conv]sion (   
ratio4 ,^! ent]$ into su* propositions  
z1 ,if f\r magnitudes are proportional1 
!y w also be proportional alt]nately4 ,9
 lat] works !y appe> 9 ! =m ( proportion
 by composition1 by division1 & by      
composition & division4 ,none ( ^! is   
to-day ( mu* importance1 s9ce mod]n     
symbolism has greatly simplifi$ ! anci5t
 expressions1 & 9 p>ticul> ! proposition
 conc]n+ composition & division is no   
long] a basal !orem 9 geometry4 ,inde$1 
if our c\rse ( /udy 7 prop]ly arrang$1  
we might well relegate ! :ole !ory (    
proportion to algebra1 all[+ ?is to     
prec$e ! "w 9 geometry4  ,we %all n[    
consid] a few ( ! pr9cipal propositions 
( ,book ,,iii4  ,,!orem4 ,if a l9e is   
drawn ?r\gh two sides ( a triangle      
p>allel to ! ?ird side1 x divides ^?    
sides proportionally4  ,9 addition to ! 

usual pro( x is in/ructive to consid] 9 
class ! cases 9 :i* ! p>allel is drawn  
?r\gh ! two sides produc$1 ei!r bel[ !  
base or above ! v]tex1 & also 9 :i* !   
p>allel is drawn ?r\gh ! v]tex4         
,,!orem4 ,! bisector ( an angle ( a     
triangle divides ! opposite side into   
segm5ts :i* are proportional to !       
adjac5t sides4  ,! proposition relat+ to
 ! bisector ( an ext]ior angle may be   
consid]$ z a "p ( ?is "o1 b x is usually
 treat$ sep>ately 9 ord] t ! pro( %all  
appe> less involv$1 al?\gh ! two are    
discuss$ toge!r at ?is "t4 ,!           
proposition relat+ to ! ext]ior angle 0 
recogniz$ by ,pappus ( ,alex&ria4       
,if ,,abc is ! giv5 triangle1 & ,,cp#a1 
,,cp#b are      respectively ! int]nal &
 ext]nal bisectors1 !n ,,ab is          
divid$ h>monically by ,p#a & ,p#b4      
!re=e,,ap#a 3 ,p#a;,b  ,,ap#b 3 ,p#b;,b4
       !re=e,,ap#b 3 ,p#b;,b            
,,ap#b - ,p#a,p#b 3 ,p#a,p#b - ,p#b;,b1 
& ?is is ! crit]ion = ! h>monic         
progression /ill      se5 9 _m algebras4

 ,=1 lett+ ,,ap#b  a1      ,p#a,p#b  b1 
,p#b;,b  c1 we h       a_/c  7a - b7_/7b
 - c71       :i* is also d]iv$ f tak+ ! 
reciprocals ( a1 b1      c1 & plac+ !m 9
 an ari?metical progression1 ?us3       
#a_/;b - #a_/;a   #a_/;c - #a_/;b1      
:5ce 7a - b7_/ab  7b - c7_/bc1       or 
7a - b7_/7b - c7  ab_/bc  a_/c4         
,?is is ! reason :y ! l9e ,,ab is said  
to be divid$      h>monically4 ,! l9e   
,p#a,p#b is also call$ !      h>monic   
mean betwe5 ,,ap#b & ,p#b;,b1 & ! po9ts 
,a1 ,p#a1 ,b1 ,p#b are said to =m an    
h>monic      range4       ,illu/ration  
,x may be not$ t ,l,p#b;,,cp#a1 be+ made
 up ( halves      ( two supplem5t>y     
angles1 is a "r angle4 ,fur!rmore1 if   
! ratio ,,ca 3 ,,cb is giv51 & ,,ab is  
giv51 !n ,p#a      & ,p#b are bo? fix$4 
,h5ce ,c mu/ lie on a semicircle      ) 
,p#a,p#b z a diamet]1 & !re=e ! locus ( 
a      po9t su* t its di/ances f two    
giv5 po9ts are 9 a      giv5 ratio is a 
circle4 ,?is fact1 ,pappus tells u1 0   
kn[n      to ,apollonius4  ,at ?is po9t 

x is cu/om>y to def9e simil> polygons z 
su* z h _! correspond+ angles equal & _!
 correspond+ sides proportional4        
,ari/otle gave sub/antially ?is         
def9ition1 say+ t su* figures h _! sides
 proportional & _! angles equal4 ,euclid
 improv$ ^u ?is by say+ t !y mu/ h _!   
angles sev]ally equal & ! sides ab\t !  
equal angles proportional4 ,our pres5t  
phraseology seems cle>]4 ,in/ead (      
correspond+ angles we may say homolog\s 
angles1 b !re seems to be no reason =   
us+ ! less famili> ^w4  ,illu/ration    
,illu/ration  ,illu/ration  ,x is m     
g5]al to proce$ by fir/ consid]+ simil> 
figures in/ead ( simil> polygons1 ?us   
includ+ ! mo/ obvi\sly simil> ( all     
figures1--two circles2 b su* a proc$ure 
is felt to be too difficult by _m       
tea*]s4 ,by ?is plan we fir/ def9e      
simil> sets ( po9ts1 ,a#a1 ,a#b1 ,a#c1  
4441 & ,b#a1 ,b#b1 ,b#c1 4441 z su* t   
,a#a;,a#b1 ,b#a;,b#b1 ,c#a;,c#b1 444 are
 concurr5t 9 ,o1 & ,a#a,o 3 ,a#b,o      
,b#a,o 3 ,b#b,o  ,c#a,o 3 ,c#b,o  444   

,"h ! con/ant ratio ,a#a,o 3 ,a#b,o is  
call$ ! ratio ( similitude1 & ,o is     
call$ ! c5t] ( similitude4 ,hav+ def9$  
simil> sets ( po9ts1 we !n def9e simil> 
figures z ^? figures ^: po9ts =m simil> 
sets4 ,!n ! two circles1 ! f\r          
triangles1 & ! ?ree quadrilat]als       
respectively are simil> figures4 ,if !  
ratio ( similitude is #a1 ! simil>      
figures become symmetric figures1 & !y  
are !re=e congru5t4 ,all ( !            
propositions relat+ to simil> figures c 
be prov$ f ?is def9ition1 b x is cu/om>y
 to use ! ,greek "o in/ead4             
,illu/ration  ,among ! int]e/+          
applications ( simil>ity is ! case ( a  
%ad[1 z "h %[n1 :]e ! light is ! c5t] ( 
similitude4 ,x is also well kn[n to mo/ 
high s*ool pupils t 9 a cam]a ! l5s     
rev]ses ! image4 ,! ma!matical          
arrangem5t is "h %[n1 ! l5s inclos+ !   
c5t] ( similitude4 ,! proposition may   
also be appli$ to ! enl>gem5t ( maps &  
work+ draw+s4  ,! propositions conc]n+  
simil> figures h no p>ticul>ly int]e/+  

hi/ory1 nor d !y pres5t any difficulties
 t call = discussion4 ,9 s*ools :]e !re 
is a little "t = trigonometry1 tea*]s   
sometimes f9d x helpful to beg9 su* "w  
at ?is "t1 s9ce all ( ! trigonometric   
functions dep5d ^u ! prop]ties ( simil> 
triangles1 & a brief explanation ( !    
simple/ trigonometric functions may add 
a little int]e/ to ! "w4 ,9 ! pres5t    
/ate ( our curriculum we _c d m ?an     
m5tion ! matt] z a topic ( g5]al int]e/ 
9 ?is connection4  ,x is a mi/ak5 idea t
 geometry is a pr]equisite to           
trigonometry4 ,we c get along v well 9  
tea*+ trigonometry if we h ?ree         
propositions3 7#a7 ! "o ab\t ! sum ( !  
angles ( a triangle2 7#b7 ! ,py?agorean 
,!orem2 7#c7 ! "o t ass]ts t two "r     
triangles are simil> if an acute angle (
 ! "o equals an acute angle ( ! o!r4 ,= 
tea*]s :o may c>e to make a little      
digression at ?is "t1 ! foll[+ brief    
/atem5t ( a few ( ! facts ( trigonometry
 may be ( value3  ,illu/ration       ,9 
! "r triangle ,,oab we %all let ,,ab  y1

      ,,oa  x1 ,,ob  r1 ?us adopt+ !    
lett]s ( high]      ma!matics4 ,!n1 s   
long z ,l,o rema9s ! same1 su*          
ratios z y_/x1 y_/r1 etc41 w rema9 !    
same1      :atev] is ! size ( !         
triangle4 ,"s ( ^! ratios h      special
 names4 ,= example1 we call       y_/r !
 s9e ( ,o1 & we write s9 ,o  y_/r2      
x_/r ! cos9e ( ,o1 & we write cos ,o    
x_/r2       y_/x ! tang5t ( ,o1 & we    
write tan ,o  y_/x4       ,n[ because   
s9 ,o  y_/r1 !re=e r s9 ,o  y2       &  
because  cos ,o  x_/r1 !re=e r cos ,o   
x2       & because  tan ,o  y_/x1 !re=e 
x tan ,o  y4       ,h5ce1 if we knew !  
values ( s9 ,o1 cos ,o1 & tan ,o      = 
! v>i\s angles1 we c\ld f9d x1 y1 or r  
if we      knew any "o ( !m4       ,n[ !
 values ( ! s9e1 cos9e1 & tang5t        
7functions (      ! angles1 z !y are    
call$7 h be5 comput$ = !      v>i\s     
angles1 & "s int]e/ may be develop$ by  
obta9+      !m by actual measurem5t1 us+
 ! protractor & squ>$      pap]4 ,"s (  
^? ne$$ = su* angles z a pupil 9        

geometry is likely to use are z foll[s3 
,,angle  ,,s9e ,,cos9e,,tang5t ,,angle  
,,s9e ,,cos9e,,tang5t                   
----------------------------------------
------------       #e deg4   4#jhg      
4#iif   4#jhg   #ej deg4   4#gff  4#fdc 
#a4aib                                  
----------------------------------------
------------      #aj deg4   4#agd      
4#ihe   4#agf   #ee deg4   4#hai  4#egd 
#a4dbh                                  
----------------------------------------
------------      #ae deg4   4#bei      
4#iff   4#bfh   #fj deg4   4#hff  4#ejj 
#a4gcb                                  
----------------------------------------
------------      #bj deg4   4#cdb      
4#idj   4#cfd   #fe deg4   4#ijf  4#dbc 
#b4ade                                  
----------------------------------------
------------      #be deg4   4#dbc      
4#ijf   4#dff   #gj deg4   4#idj  4#cdb 
#b4gdh                                  
----------------------------------------
------------      #cj deg4   4#ejj      

4#hff   4#egg   #ge deg4   4#iff  4#bei 
#c4gcb                                  
----------------------------------------
------------      #ce deg4   4#egd      
4#hai   4#gjj   #hj deg4   4#ihe  4#agd 
#e4fga                                  
----------------------------------------
------------      #dj deg4   4#fdc      
4#gff   4#hci   #he deg4   4#iif  4#jhg 
#aa4dcj                                 
----------------------------------------
------------      #de deg4   4#gjg      
4#gjg  #a4jjj   #ij deg4   #a4jj  4#jjj 
inf9ity            ,x w ( c\rse be      
und]/ood t ! values are correct         
only to ! ne>e/ ?\s&?4 ,?us ! cos9e ( #e
 deg4 is      #j4iifai1 & ! s9e ( #he   
deg4 is #j4iifai4 ,! entire table c     
be copi$ by a class 9 five m9utes if a  
tea*] wi%es to      introduce ?is phase 
( ! "w1 & ! au?or has frequ5tly         
assign$ ! comput+ ( a simpl] table z a  
class ex]cise4       ,ref]r+ to !       
figure1 if we "k t r  #cj &      ,l,o   
#dj deg41 !n s9ce y  r s9 ,o1 we h      

y  #cj x #j4fdc  #ai4bi4 ,if we "k t x  
#fj &      ,l,o  #ce deg41 !n s9ce y  x 
tan ,o1 we h      y  #fj x #j4g  #db4   
,we may also f9d r1 = cos ,o  x_/r1     
:5ce r  x_/7cos ,o7  #fj_/j4hai  #gc4bf4
  ,!re=e1 if we c\ld easily measure ,l,o
 & c\ld measure ! di/ance x1 we c\ld f9d
 ! height ( a build+ y4 ,9 trigonometry 
we use a transit = measur+ angles1 b x  
is easy to measure !m ) suffici5t       
accuracy = illu/rative purposes by plac+
 an ord9>y pap] protractor ^u some?+    
level1 s t ! c5t] comes at ! edge1 & !n 
sight+ along a rul] held aga9/ x1 s z to
 f9d ! angle ( elevation ( a build+4 ,we
 may !n measure ! di/ance to ! build+ & 
apply ! =mula y  x tan ,o4              
,illu/ration3 ,a ,,quadrant ,,( ,,!     
,,sixte5? ,,c5tury  ,f9aeus's ,de re et 
praxi geometrica1 ,p>is1 #aeef  ,x %\ld 
always be und]/ood t exp5sive app>atus  
is n necess>y = su* illu/rative "w4 ,!  
telescope us$ on ! transit is only ?ree 
hundr$ ye>s old1 & ! _w got along v well
 ) its trigonometry be=e t 0 inv5t$4 ,s 

a little +5uity w enable any "o to make 
f *eap protractors ab\t z satisfactory  
in/rum5ts z ! _w us$ be=e #afjj4 ,9 ord]
 t ?is may be ! m fully appreciat$1 a   
few illu/rations are "h giv51 %[+ ! old 
in/rum5ts & me?ods us$ 9 practical      
survey+ be=e ! eighte5? c5tury4         
,illu/ration3 ,a ,,quadrant ,,( ,,!     
,,sev5te5? ,,c5tury  ,! illu/ration on  
page #bcf %[s a simple =m ( ! quadrant1 
an in/rum5t easily made by a pupil :o   
may be int]e/$ 9 outdoor "w4 ,x 0 !     
common survey+ in/rum5t ( ! e>ly days4  
,a m elaborate example is se5 9 !       
illu/ration1 on page #bcg1 ( a          
sev5te5?-c5tury brass specim5 9 !       
au?or's collection4#gg  ,illu/ration3 ,a
 ,,quadrant ,,( ,,! ,,sev5te5? ,,c5tury 
,b>toli's ,del modo di misur>e1 ,v5ice1 
#afhi  ,ano!r type1 easily made by      
pupils1 is %[n 9 ! above illu/ration f  
,b>toli1 #afhi4 ,su* in/rum5ts 7 usually
 made ( wood1 brass1 or ivory4#gh       
,in/rum5ts = ! runn+ ( l9es p]p5dicul>  
to o!r l9es 7 =m]ly common1 & are easily

 made4 ,!y suffice1 z ! foll[+          
illu/ration %[s1 = survey+ an ord9>y    
field4  ,illu/ration3 ,,survey+         
,,in/rum5t ,,( ,,! ,,eighte5? ,,c5tury  
,n4 ,bion's ,traite de la con/ruction   
444 des in/rum5s de ma!matique1 ,!      
,hague1 #agbc  ,illu/ration3 ,,!        
,,quadrant ,,us$ ,,= ,,altitudes        
,f9aeus's ,de re et praxi geometrica1   
,p>is1 #aeef  ,! quadrant 0 practically 
us$ = all sorts ( outdoor measur+4 ,=   
example1 ! illu/ration f ,f9aeus1 on ?is
 page1 %[s h[ x 0 us$ = altitudes1 & !  
"o reproduc$ on page #bdj %[s h[ x 0 us$
 = measur+ dep?s4  ,a simil> in/rum5t f 
! "w ( ,bett9us is giv5 on page #bda1 ! 
di/ance ( a %ip be+ f\nd by con/ruct+ an
 isosceles triangle4 ,a m elaborate =m1 
) a p5dulum atta*m5t1 is se5 9 !        
illu/ration f ,de ,judaeis1 :i* also    
appe>s on page #bda4  ,illu/ration3 ,,! 
,,quadrant ,,us$ ,,= ,,dep?s  ,f9aeus's 
,protoma!sis1 ,p>is1 #aecb              
,illu/ration3 ,a ,,quadrant ,,( ,,!     
,,sixte5? ,,c5tury  ,de ,judaeis's ,de  

quadrante geometrico1 ,nu]nb]g1 #aeid   
,illu/ration3 ,,! ,,quadrant ,,us$ ,,=  
,,di/ances  ,bett9us's ,api>ia univ]sae 
philosophiae ma!maticae1 ,bologna1 #afde
  ,! quadrant f9ally develop$ into !    
octant1 z %[n 9 ! foll[+ illu/ration f  
,h(fmann1 & ?is 9 turn develop$ into !  
sextant1 :i* is n[ us$ by all           
navigators4  ,illu/ration3 ,,! ,,octant 
,h(fmann's ,de ,octantis1 ,j5a1 #afab   
,9 connection ) ?is g5]al subject ! use 
( ! speculum 7mirror7 9 measur+ heights 
%\ld be m5tion$4 ,! illu/ration giv5 on 
page #bdc %[s h[ 9 e>ly days a simple   
device 0 us$ = ?is purpose4 ,two simil> 
triangles are =m$ 9 ?is way1 & we h only
 to measure ! height ( ! eye above !    
gr\nd1 & ! di/ances ( ! mirror f ! t[] &
 ! obs]v]1 to h ?ree t]ms ( a           
proportion4  ,all ( ^! in/rum5ts are    
easily made4 ,! mirror is always at h&1 
& a pap] protractor on a piece ( bo>d1 )
 a plumb l9e atta*$1 s]ves z a quadrant4
 ,= a few c5ts1 & by ! exp5diture ( an  
h\r or s1 a s*ool c h almo/ z good      

in/rum5ts z ! ord9>y surveyor _h be=e ! 
n9ete5? c5tury4  ,illu/ration3 ,,!      
,,speculum  ,f9aeus's ,de re et praxi   
geometrica1 ,p>is1 #aeef  ,a well-kn[n  
me?od ( measur+ ! di/ance across a /ream
 is illu/rat$ 9 ! figure bel[1 :]e !    
di/ance f ,a to "s po9t ,p is requir$4  
,illu/ration       ,run a l9e f ,a to ,c
 by /&+ at ,c 9 l9e ) ,a      & ,p4 ,!n 
run two p]p5dicul>s f ,a & ,c by any (  
! me?ods already giv51--sight+ on a     
protractor or along      ! edge ( a book
 if no bett] means are at h&4 ,!n sight 
f "s po9t ,d1 on ,,cd1 to ,p1 putt+ a   
/ake at ,b4      ,!n run ! p]p5dicul>   
,,be4 ,s9ce ,,de 3 ,,eb  ,,ba 3         
,,ap1 & s9ce we c measure ,,de1 ,,eb1 & 
,,ba ) !      tape1 we c compute !      
di/ance ,,ap4  ,!re are _m v>iations (  
?is s*eme ( measur+ di/ances by means ( 
simil> triangles1 & pupils may be       
enc\rag$ to try "s ( !m4 ,o!r figures   
are sugge/$ on page #bdd1 & ! triangles 
ne$ n be conf9$ to ^? hav+ a "r angle4  
,a v simple illu/ration ( ! use ( simil>

 triangles is f\nd 9 "o ( ! /ories told 
( ,?ales4 ,x is relat$ t he f\nd !      
height ( ! pyramids by measur+ _! %ad[  
at ! in/ant :5 8 own %ad[ j equal$ 8    
height4 ,he ?us _h ! case ( two simil>  
isosceles triangles4 ,?is is an int]e/+ 
ex]cise :i* may be tri$ ab\t ! "t t     
pupils are leav+ s*ool 9 ! aft]noon4    
,illu/ration  ,ano!r application ( !    
same pr9ciple is se5 9 a me?od (t5 tak5 
= measur+ ! height ( a tree4            
,illu/ration       ,! obs]v] has a l>ge 
"r triangle made ( wood4 ,su* a         
triangle is %[n 9 ! picture1 9 :i* ,,ab 
,,bc4 ,he      holds ,,ab level & walks 
t[>d ! tree until he j sees      ! top  
along ,,ac4 ,!n because       ,,ab      
,,bc1      & ,,ab 3 ,,bc  ,,ad 3 ,,de1  
! height above ,d w equal ! di/ance     
,,ad4       ,que/ions l ! foll[+ may be 
giv5 to ! class3       #a4 ,:at is !    
height ( ! tree 9 ! picture if !        
triangle is #e ft4 #d 94 f ! gr\nd1 &   
,,ad is #bc ft4 #h      948       #b4   
,suppose a triangle is us$ :i* has ,,ab 

twice ,,bc4 ,:at      is ! height if    
,,ad  #ge ft48  ,!re are _m v>iations ( 
?is pr9ciple4 ,"o consi/s 9 measur+ !   
%ad[s ( a tree & a /aff at ! same "t4 ,!
 height ( ! /aff be+ kn[n1 ! height ( ! 
tree is f\nd by proportion4 ,ano!r      
consi/s 9 sight+ f ! gr\nd1 across a m>k
 on an upright /aff1 to ! top ( ! tree4 
,! height ( ! m>k be+ kn[n1 & ! di/ances
 f ! eye to ! /aff & to ! tree be+      
measur$1 ! height ( ! tree is f\nd4     
,illu/ration  ,an in/rum5t sold by      
deal]s = ! measur+ ( heights is kn[n z !
 hypsomet]4 ,x is made ( brass1 & is ( !
 =m "h %[n4 ,! base is graduat$ 9 equal 
divisions1 say #ej1 & ! upright b> is   
simil>ly divid$4 ,at ! ends ( ! h+$     
radius are two sights4 ,if ! obs]v] /&s 
#ej feet f a tree & sights at ! top1 s t
 ! h+$ radius cuts ! upright b> at #bg1 
!n he kn[s at once t ! tree is #bg feet 
high4 ,x is easy = a class to make a    
fairly good in/rum5t ( ?is k9d out (    
/iff pa/ebo>d4  ,an int]e/+ application 
( ! !orem relat+ to simil> triangles is 

?is3 ,ext5d y\r arm & po9t to a di/ant  
object1 clos+ y\r left eye & sight+     
across y\r f+] tip ) y\r "r eye4 ,n[    
keep ! f+] 9 ! same position & sight )  
y\r left eye4 ,! f+] w !n seem to be    
po9t+ to an object "s di/ance to ! "r ( 
! "o at :i* y 7 po9t+4 ,if y c e/imate !
 di/ance betwe5 ^! two objects1 :i* c   
(t5 be done ) a fair degree ( accuracy  
:5 !re are h\ses int]v5+1 !n y w be able
 to tell approximately y\r di/ance f !  
objects1 = x w be t5 times ! e/imat$    
di/ance betwe5 !m4 ,! f9d+ ( ! reason = 
?is by measur+ ! di/ance betwe5 ! pupils
 ( ! two eyes1 & ! di/ance f ! eye to ! 
f+] tip1 & !n draw+ ! figure1 is an     
int]e/+ ex]cise4  ,p]haps "s pupil :o   
has read ,?oreau's descriptions (       
outdoor life may be int]e/$ 9 :at he    
says ( 8 crude ma!matics4 ,he writes1 ,i
 borr[$ ! plane & squ>e1 level &        
divid]s1 ( a c>p5t]1 & ) a %+le contriv$
 a rude sort ( a quadrant1 ) p9s =      
sights & pivots4 ,) ?is he measur$ !    
heights ( a cliff on ! ,massa*usetts    

coa/1 & ) simil> home-made or s*ool-made
 in/rum5ts a pupil 9 geometry c measure 
mo/ ( ! heights & di/ances 9 :i* he is  
int]e/$4  ,,!orem4 ,if 9 a "r triangle a
 p]p5dicul> is drawn f ! v]tex ( ! "r   
angle to ! hypot5use3  #a4 ,! triangles 
?us =m$ are simil> to ! giv5 triangle1 &
 are simil> to ea* o!r4  #b4 ,!         
p]p5dicul> is ! mean proportional betwe5
 ! segm5ts ( ! hypot5use4  #c4 ,ea* ( ! 
o!r sides is ! mean proportional betwe5 
! hypot5use & ! segm5t ( ! hypot5use    
adjac5t to t side4  ,to ?is important   
proposition !re is "o coroll>y (        
p>ticul> int]e/1 namely1 ,! p]p5dicul> f
 any po9t on a circle to a diamet] is ! 
mean proportional betwe5 ! segm5ts ( !  
diamet]4 ,by means ( ?is coroll>y we c  
easily con/ruct a l9e ^: num]ical value 
is ! squ>e root ( any numb] we please4  
,?us we may make ,,ad  #b 941 ,,db  #c  
941 & erect ,,dc      p]p to ,,ab4 ,!n !
 l5g? ( ,,dc w be sqrt#f 941      & we  
may f9d sqrt#f approximately by measur+ 
,,dc4  ,illu/ration       ,fur!rmore1 if

 we introduce negative magnitudes into  
geometry1      & let ,,db  #c & ,,da    
-#b1 !n ,,dc w equal sqrt7-#f74      ,9 
o!r words1 we h a ju/ification =        
repres5t+      imag9>y quantities by    
l9es p]p5dicul> to ! l9e on      :i* we 
repres5t real quantities1 z is done 9 ! 
graphic      treatm5t ( imag9>ies 9     
algebra4  ,x is an int]e/+ ex]cise to h 
a class f9d1 to "o decimal place1 by    
measur+ z above1 ! value ( sqrt#b1      
sqrt#c1 sqrt#e1 & sqrt#i1 ! la/ be+     
integral4 ,if1 z is n usually ! case1 ! 
class has /udi$ ! complex numb]1 !      
absolute value ( sqrt7-#f71 sqrt7-#g71  
4441 may be f\nd 9 ! same way4  ,a      
practical illu/ration ( ! value ( !     
above !orem is se5 9 a me?od = f9d+     
di/ances t is frequ5tly describ$ 9 e>ly 
pr9t$ books4 ,x seems to h come f !     
,roman surveyors4  ,illu/ration         
,if a c>p5t]'s squ>e is put on top ( an 
upright /ick1 z      "h %[n1 & an obs]v]
 sights along ! arms to a di/ant        
po9t ,b & a po9t ,a ne> ! /ick1 !n ! two

      triangles are simil>4 ,h5ce ,,ad 3
 ,,dc  ,,dc 3 ,,db4 ,h5ce1      if ,,ad 
& ,,dc are measur$1 ,,db c be f\nd4 ,!  
exp]im5t is an int]e/+ & in/ructive "o =
 a class1      especially z ! squ>e c   
easily be made out ( heavy              
pa/ebo>d4  ,,!orem4 ,if two *ords       
int]sect )9 a circle1 ! product ( !     
segm5ts ( ! "o is equal to ! product ( !
 segm5ts ( ! o!r4  ,,!orem4 ,if f a po9t
 )\t a circle a secant & a tang5t are   
drawn1 ! tang5t is ! mean proportional  
betwe5 ! secant & its ext]nal segm5t4   
,,coroll>y4 ,if f a po9t )\t a circle a 
secant is drawn1 ! product ( ! secant & 
its ext]nal segm5t is con/ant 9 :atev]  
direction ! secant is drawn4  ,^! two   
propositions & ! coroll>y are all p>ts (
 "o g5]al proposition3 ,if ?r\gh a po9t 
a l9e is drawn cutt+ a circle1 ! product
 ( ! segm5ts ( ! l9e is con/ant4        
,illu/ration       ,if ,p is )9 !       
circle1 !n xx'  yy'2 if ,p is on      ! 
circle1 !n x & y become #j1 & #j 4 x'   
#j 4 y'       #j2 if ,p is at ,p#c1 !n x

 & y1 hav+ pass$      ?r\gh #j1 may be  
consid]$ negative if we wi%1 al?\gh !   
two negative signs w\ld cancel out 9 !  
equation2 if ,p is      at ,p#d1 !n y   
y' & we h xx'  y#b1      or x 3 y  y 3  
x'1 z /at$ 9 ! proposition4  ,we ?us h  
an excell5t example ( ! ,pr9ciple (     
,cont9uity1 & classes are always int]e/$
 to consid] ! result ( lett+ ,p assume  
v>i\s positions4 ,among ! possible cases
 is ! "o ( two tang5ts f an ext]nal     
po9t1 & ! "o :]e ,p is at ! c5t] ( !    
circle4  ,/ud5ts %\ld frequ5tly be      
que/ion$ z to ! mean+ ( product ( l9es4 
,! ,greeks always us$ rectangle ( l9es1 
b x is entirely legitimate to speak (   
product ( l9es1 provid$ we def9e !      
expression consi/5tly4 ,mo/ writ]s d    
?is1 say+ t by ! product ( l9es is meant
 ! product ( _! num]ical values1 a      
subject already discuss$ at ! beg9n+ (  
?is *apt]4  ,,!orem4 ,! squ>e on !      
bisector ( an angle ( a triangle is     
equal to ! product ( ! sides ( ?is angle
 dim9i%$ by ! product ( ! segm5ts made  

by ! bisector ^u ! ?ird side ( !        
triangle4  ,?is proposition enables u to
 compute ! l5g? ( a bisector ( a        
triangle if ! l5g?s ( ! sides are kn[n4 
,illu/ration      ,=1 9 ?is figure1 let 
a  #c1 b  #e1 & c  #f4      ,!n because 
x 3 y  b 3 a1 & y  #f - x1      we h    
x_/7#f - x7  #e_/c4           !re=e #cx 
#cj - #ex4            !re=e x  #c #c_/d1
 y  #b #a_/d4      ,by ! !orem1 z#b  ab 
- xy                            #ae -   
7#h #g_/af7  #f #i_/af4           !re=e 
z  sqrt7#f #i_/af7  #a_/d sqrt#aje      
#b4e4   ,,!orem4 ,9 any triangle !      
product ( two sides is equal to !       
product ( ! diamet] ( ! circumscrib$    
circle by ! altitude ^u ! ?ird side4    
,?is enables u1 aft] ! ,py?agorean      
,!orem has be5 /udi$1 to compute ! l5g? 
( ! diamet] ( ! circumscrib$ circle 9   
t]ms ( ! ?ree sides4  ,illu/ration      
,= if we designate ! sides by a1 b1 & c1
 z usual1      & let ,,cd  d & ,,pb  x1 
!n            7,,cp7#b  a#b - x#b       
b#b - 7c - x7#b4     !re=e a#b - x#b    

b#b - c#b  #b;cx - x#b4                 
!re=e x  7a#b - b#b  c#b7 _/ #b;c4      
!re=e 7,,cp7#b  a#b - 77a#b - b#b  c#b7 
_/ #b;c7#b4      ,b            ,,cp 4 d 
ab4 !re=e d  #b;abc _/ sqrt7#d;a#b;c#b -
 7a#b - b#b  c#b7b74   ,?is is n        
available at ?is "t1 h[ev]1 because !   
,py?agorean ,!orem has n be5 prov$4  ,^!
 two propositions are m]ely special     
cases ( ! foll[+ g5]al !orem1 :i* may be
 giv5 z an int]e/+ ex]cise3  ,if ,,abc  
is an inscrib$ triangle1 & ?r\gh ,c !re 
are drawn two /raight l9es ,,cd1 meet+  
,,ab 9 ,d1 & ,,cp1 meet+ ! circle 9 ,p1 
) angles ,,acd & ,,pcb equal1 !n ,,ac x 
,,bc w equal ,,cd x ,,cp4  ,illu/ration3
 ,,fig4 #a  ,illu/ration3 ,,fig4 #b     
,illu/ration3 ,,fig4 #c  ,illu/ration3  
,,fig4 #d       ,fig4 #a is ! g5]al case
 :]e ,d falls betwe5 ,a & ,b4      ,if  
,,cp is a diamet]1 x r$uces to ! second 
figure giv5 on      page #bdi4 ,if ,,cp 
bisects ,l,,acb1 we h ,fig4 #c1 f :i*   
may be prov$ ! proposition giv5 at !    
foot ( page #bdh4 ,if      ,d lies on   

,,ba produc$1 we h ,fig4 #b4 ,if ,d lies
 on ,,ab      produc$1 we h ,fig4 #d4   
,?is g5]al proposition is prov$ by %[+ t
      triangles,,adc & ,,pbc are simil>1
 exactly z 9 !      second proposition  
giv5 on page #bdi4  ,^! !orems are      
usually foll[$ by problems (            
con/ruction1 ( :i* only "o has great    
int]e/1 namely1 ,to divide a giv5 l9e 9 
extreme & mean ratio4  ,! purpose ( ?is 
problem is to prep>e = ! con/ruction ( !
 regul> decagon & p5tagon4 ,! division (
 a l9e 9 extreme & mean ratio is call$ !
 gold5 section1 & is probably ! section 
m5tion$ by ,proclus :5 he says t        
,eudoxus greatly add$ to ! numb] ( !    
!orems :i* ,plato orig9at$ reg>d+ !     
section4 ,! expression gold5 section is 
n old1 h[ev]1 & its orig9 is unc]ta94   
,if a l9e ,,ab is divid$ 9 gold5 section
 at ,p1 we h      ,,ab x ,,pb  7,,ap7#b4
       ,!re=e1 if ,,ab  a1 & ,,ap  x1 we
 h               a7a - x7  x#b1     or  
x#b  ax - a#b  #j2     :5ce             
x  - a_/#b - a_/#bsqrt#e                

a7#a4aah - #j4e7                        
#j4fah;a1        ! o!r root repres5t+ ! 
ext]nal po9t4       ,t is1 x  ab\t      
#j4f;a1 & a - x  ab\t #j4d;a1 &      a  
is !re=e divid$ 9 ab\t ! ratio ( #b 3   
#c4  ,!re has be5 a great deal writt5 ^u
 ! ae/hetic features ( ! gold5 section4 
,x is claim$ t a l9e is mo/ h>moni\sly  
divid$ :5 x is ei!r bisect$ or divid$ 9 
extreme & mean ratio4 ,a pa9t+ has !    
/rong feature 9 ! c5t]1 or m (t5 at a   
po9t ab\t #j4d ( ! di/ance f "o side1 t 
is1 at ! gold5 section ( ! wid? ( !     
picture4 ,x is said t 9 nature ?is same 
h>mony is f\nd1 z 9 ! division ( ! ve9s 
( su* leaves z ! ivy & f]n4             
,,footnotes3  #gf ,= a v full discussion
 ( ^! f\r def9itions see ,hea?'s        
,euclid1 ,vol4 ,,ii1 p4 #aaf1 &         
au?orities !re cit$4  #gg ,^! two &     
sev]al :i* foll[ are f ,/>k1 loc4 cit4  
#gh ,! au?or has a beautiful ivory      
specim5 ( ! ,sixte5? c5tury4     ,,*apt]
 ,,xvii  ,,! ,,lead+ ,,propositions ,,( 
,,book ,,iv   ,book ,,iv treats ( ! area

 ( polygons1 & (f]s a l>ge numb] (      
practical applications4 ,s9ce ! numb] ( 
applications to ! measur+ ( areas (     
v>i\s k9ds ( polygons is unlimit$1 :ile 
9 ! fir/ ?ree books ^! applications are 
n s obvi\s1 less ef=t is made 9 ?is     
*apt] to sugge/ practical problems to ! 
tea*]s4 ,! survey ( ! s*ool gr\nds or ( 
vacant lots 9 ! vic9ity (f]s all !      
outdoor "w t is ne$$ to make ,book ,,iv 
seem v important4  ,,!orem4 ,two        
rectangles hav+ equal altitudes are to  
ea* o!r z _! bases4  ,euclid's /atem5t  
7,book ,,vi1 ,proposition #a7 0 z       
foll[s3 ,triangles & p>allelograms :i*  
are "u ! same height are to "o ano!r z  
_! bases4 ,our plan ( treat+ ! two      
figures sep>ately is manife/ly bett] f !
 educational /&po9t4  ,9 ! mod]n        
treatm5t by limits ! pro( is divid$ into
 two p>ts3 fir/1 = comm5surable bases2 &
 second1 = incomm5surable ones4 ,( ^! ! 
second may well be omitt$1 or m]ely be  
read ov] by ! tea*] & class & ! reasons 
expla9$4 ,9 g5]al1 x is d\btful if !    

majority ( an ,am]ican class 9 geometry 
get mu* out ( ! incomm5surable case4 ,( 
c\rse1 ) a bright class a tea*] may well
 af=d to take x z x is giv5 9 !         
textbook1 b ! important ?+ is t !       
comm5surable case %\ld be prov$ & !     
incomm5surable "o recogniz$4  ,euclid's 
treatm5t ( proportion 0 s rigor\s t no  
special treatm5t ( ! incomm5surable 0   
necess>y4 ,! ,fr5* geomet]1 ,leg5dre1   
gave a rigor\s pro( by r$uctio ad       
absurdum4 ,9 ,am]ica ! pupils are h>dly 
ready = ^! pro(s1 & s our treatm5t by   
limits is less rigor\s ?an ^! e>li]     
ones4  ,,!orem4 ,! area ( a rectangle is
 equal to ! product ( its base by its   
altitude4  ,! easie/ way to introduce   
?is is to m>k a rectangle1 )            
comm5surable sides1 on squ>$ pap]1 &    
c\nt up ! squ>es2 or1 :at is m conv5i5t1
 to draw ! rectangle & m>k ! area (f 9  
squ>es4  ,x is int]e/+ & valuable to a  
class to h its att5tion call$ to ! fact 
t ! p]imet] ( a rectangle is no crit]ion
 z to ! area4 ,?us1 if a rectangle has  

an area ( #a squ>e foot & is only       
#a_/ddj ( an in* high1 ! p]imet] is ov] 
#b miles4 ,! /ory ( h[ ,indians 7 induc$
 to sell _! l& by measur+ ! p]imet] is a
 v old "o4 ,proclus speaks ( travel]s :o
 describ$ ! size ( cities by ! p]imet]s1
 & ( m5 :o *eat$ o!rs by pret5d+ to give
 !m z mu* l& z !y !mselves _h1 :5 really
 !y made only ! p]imet]s equal4         
,?ucydides e/imat$ ! size ( ,sicily by !
 "t x took to sail r\nd x4 ,pupils w be 
int]e/$ to "k 9 ?is connection t (      
polygons hav+ ! same p]imet] & ! same   
numb] ( sides1 ! "o hav+ equal sides &  
equal angles is ! greate/1 & t ( plane  
figures hav+ ! same p]imet]1 ! circle is
 ! greate/4 ,^! facts 7 kn[n to ! ,greek
 writ]s1 ,z5odorus 7ca4 #aej ,b4,c47 &  
,proclus 7#daj-dhe ,a4,d474  ,! surfaces
 ( rectangul> solids may n[ be f\nd1 !re
 be+ an advantage 9 ?us incid5tally     
connect+ plane & solid geometry :]ev] x 
is natural to d s4  ,,!orem4 ,! area ( a
 p>allelogram is equal to ! product (   
its base by its altitude4  ,! be/ way to

 introduce ?is !orem is to cut a        
p>allelogram f pap]1 & !n1 ) ! class1   
sep>ate x into two p>ts by a cut        
p]p5dicul> to ! base4 ,! two p>ts may !n
 be fitt$ toge!r to make a rectangle4 ,9
 p>ticul>1 if we cut (f a triangle f "o 
end & fit x on ! o!r1 we h ! basis = !  
pro( ( ! textbooks4 ,! use ( squ>$ pap] 
= su* a proposition is n wise1 s9ce x   
makes ! measurem5t appe> to be m]ely an 
approximation4 ,! cutt+ ( ! pap] is 9 e 
way m satisfactory4  ,,!orem4 ,! area ( 
a triangle is equal to half ! product ( 
its base by its altitude4  ,( c\rse1 !  
,greeks w\ld nev] h us$ ! word+ ( ei!r (
 ^! two propositions4 ,euclid1 =        
example1 gives ?is "o z foll[s3 ,if a   
p>allelogram h ! same base ) a triangle 
& be 9 ! same p>allels1 ! p>allelogram  
is d\ble ( ! triangle4 ,z to !          
p>allelogram1 he simply says x is equal 
to a p>allelogram ( equal base & 9 !    
same p>allels1 :i* makes x equal to a   
rectangle ( ! same base & ! same        
altitude4  ,! numb] ( applications ( ^! 

two !orems is s great t ! tea*] w n be  
at a loss to f9d g5u9e ones t appeal to 
! class4 ,tea*]s may n[ introduce       
pyramids1 requir+ ! areas ( ! triangul> 
faces to be f\nd4  ,! ,ahmes papyrus    
7ca4 #agjj ,b4,c47 gives ! area ( an    
isosceles triangle z #a_/b bs1 :]e s is 
"o ( ! equal sides1 ?us tak+ s = !      
altitude4 ,?is %[s ! primitive /ate (   
geometry at t "t4  ,,!orem4 ,! area ( a 
trapezoid is equal to half ! sum ( its  
bases multipli$ by ! altitude4          
,illu/ration       ,an int]e/+ v>iation 
( ! ord9>y pro( is made by      plac+ a 
trapezoid ,t'1 congru5t to ,t1 9 !      
position      "h %[n4 ,! p>allelogram   
=m$ equals a7b  b'71      & !re=e       
,t  a 4 7b  b'7_/#b4       ,!           
proposition %\ld be discuss$ = ! case b 
b'1      :5 x r$uces to ! "o ab\t ! area
 ( a p>allelogram4      ,if b' #j1 !    
trapezoid r$uces to a triangle1 &       
,t  a 4 b_/#b4  ,?is proposition is !   
basis ( ! !ory ( l& survey+1 a piece (  
l& be+1 = purposes ( measurem5t1 divid$ 

into trapezoids & triangles1 ! latt]    
be+1 z we h se51 a k9d ( special        
trapezoid4  ,! proposition is n 9       
,euclid1 b is giv5 by ,proclus 9 ! fif? 
c5tury4  ,! t]m isosceles trapezoid is  
us$ to mean a trapezoid ) two opposite  
sides equal1 b n p>allel4 ,! area ( su* 
a figure 0 incorrectly giv5 by ! ,ahmes 
papyrus z #a_/b7;b  b'7s1 :]e s is "o ( 
! equal sides4 ,?is am\nts to tak+ s  a4
  ,! proposition is p>ticul>ly important
 9 ! survey+ ( an irregul> field su* z  
is f\nd 9 hilly di/ricts4 ,x is cu/om>y 
to consid] ! field z a polygon1 & to    
draw a m]idian l9e1 lett+ fall          
p]p5dicul>s ^u x f ! v]tices1 ?us =m+   
triangles & trapezoids t c easily be    
measur$4 ,an old] plan1 b "o bett] suit$
 to ! use ( pupils :o may be work+ only 
) ! tape1 is giv5 on page #ii4  ,,!orem4
 ,! areas ( two triangles :i* h an angle
 ( ! "o equal to an angle ( ! o!r are to
 ea* o!r z ! products ( ! sides includ+ 
! equal angles4  ,?is proposition may be
 omitt$ z f> z its use 9 plane geometry 

is conc]n$1 = we c prove ! next         
proposition "h giv5 )\t us+ x4 ,9 solid 
geometry x is us$ only 9 a proposition  
relat+ to ! volumes ( two triangul>     
pyramids hav+ a common trih$ral angle1 &
 ?is is usually omitt$4 ,b ! !orem is s 
simple t x takes b little "t1 & x adds  
greatly to ! /ud5t's appreciation (     
simil> triangles4 ,x n only simplifies !
 next "o "h giv51 b tea*]s c at once    
d$uce ! latt] f x z a special case by   
ask+ to :at x r$uces if a second angle (
 "o triangle is also equal to a second  
angle ( ! o!r triangle4  ,x is helpful  
to give num]ical values to ! sides ( a  
few triangles hav+ su* equal angles1 &  
to f9d ! num]ical ratio ( ! areas4      
,,!orem4 ,! areas ( two simil> triangles
 are to ea* o!r z ! squ>es on any two   
correspond+ sides4  ,illu/ration        
,?is may be prov$ indep5d5tly ( ! prec$+
 proposition      by draw+ ! altitudes p
 & p'4 ,!n                              
triangle,,abc_/triangle,a',b',c'        
cp_/c'p'4      ,b                       

c_/c'  p_/p'1       by simil> triangles4
      !re=e                             
triangle,,abc_/triangle,a',b',c'        
c#b_/;c'#b1       & s = o!r sides4      
,?is pro( is unnecess>ily long1 h[ev]1  
because ( !      introduction ( !       
altitudes4  ,9 ?is & sev]al o!r         
propositions 9 ,book ,,iv occurs !      
expression ! squ>e on a l9e4 ,we h1 9   
our dep>ture f ,euclid1 treat$ a l9e    
ei!r z a geometric figure or z a numb]  
7! l5g? ( ! l9e71 z 0 ! m conv5i5t4 ,(  
c\rse if we are speak+ ( a l9e1 !       
pref]able expression is squ>e on ! l9e1 
:]eas if we speak ( a numb]1 we say     
squ>e ( ! numb]4 ,9 ! case ( a rectangle
 ( two l9es we h come to speak ( !      
product ( ! l9es1 mean+ ! product ( _!  
num]ical values4 ,we are !re=e n z      
accurate 9 our phraseology z ,euclid1 & 
we d n pret5d to be1 = reasons already  
giv54 ,b :5 x comes to squ>e on a l9e or
 squ>e ( a l9e1 ! =m] is ! "o dem&+ no  
explanation or apology1 & x is ev5 bett]
 und]/ood ?an ! latt]4  ,,!orem4 ,!     

areas ( two simil> polygons are to ea*  
o!r z ! squ>es on any two correspond+   
sides4  ,?is is a proposition ( great   
importance1 & 9 due "t ! pupil sees t x 
applies to circles1 ) ! necess>y *ange (
 ! ^w sides to l9es4 ,x is well to ask a
 few que/ions l ! foll[+3 ,if "o squ>e  
is twice z high z ano!r1 h[ d ! areas   
comp>e8 ,if ! side ( "o equilat]al      
triangle is ?ree times z long z t (     
ano!r1 h[ d ! p]imet]s comp>e8 h[ d !   
areas comp>e8 ,if ! area ( "o squ>e is  
tw5ty-five times ! area ( ano!r squ>e1 !
 side ( ! fir/ is h[ _m times z long z !
 side ( ! second8 ,if a photograph is   
enl>g$ s t a tree is f\r times z high z 
x 0 be=e1 :at is ! ratio ( correspond+  
dim5sions8 ,! area ( ! enl>g$ photograph
 is h[ _m times z great z ! area ( !    
orig9al8  ,,!orem4 ,! squ>e on !        
hypot5use ( a "r triangle is equival5t  
to ! sum ( ! squ>es on ! o!r two sides4 
,( all ! propositions ( geometry ?is is 
! mo/ fam\s & p]haps ! mo/ valuable4    
,trigonometry is bas$ *iefly ^u two     

facts ( plane geometry3 7#a7 9 simil>   
triangles ! correspond+ sides are       
proportional1 & 7#b7 ?is proposition4 ,9
 m5suration1 9 g5]al1 ?is proposition   
ent]s m (t5 ?an any o!rs1 except ^? on !
 measur+ ( ! rectangle & triangle4 ,x is
 propos$1 !re=e1 to devote consid]able  
space to speak+ ( ! hi/ory ( ! !orem1 & 
to c]ta9 pro(s t may pr(itably be       
sugge/$ f "t to "t to diff]5t classes = 
! purpose ( add+ int]e/ to ! "w4        
,proclus1 ! old ,greek comm5tator on    
,euclid1 has ?is to say ( ! hi/ory3 ,if 
we li/5 to ^? :o wi% to rec\nt anci5t   
hi/ory1 we may f9d "s ( !m ref]r+ ?is   
!orem to ,py?agoras & say+ t he         
sacrific$ an ox 9 honor ( 8 discov]y4 ,b
 = my "p1 :ile ,i admire ^? :o fir/     
obs]v$ ! tru? ( ?is !orem1 ,i m>vel m at
 ! writ] ( ! ',elem5ts' 7,euclid71 n    
only because he made x fa/ by a mo/     
lucid demon/ration1 b because he        
compell$ ass5t to ! /ill m g5]al !orem  
by ! irrefragable argum5ts ( sci5ce 9   
,book ,,vi4 ,= 9 t book he proves1      

g5]ally1 t 9 "r triangles ! figure on ! 
side subt5d+ ! "r angle is equal to !   
simil> & simil>ly plac$ figures describ$
 on ! sides ab\t ! "r angle4 ,n[ x      
appe>s f ?is t ,proclus1 9 ! fif? c5tury
 ,a4,d41 ?\ght t ,py?agoras discov]$ !  
proposition 9 ! six? c5tury ,b4,c41 t ! 
usual pro(1 z giv5 9 mo/ ( our ,am]ican 
textbooks1 0 due to ,euclid1 & t !      
g5]aliz$ =m 0 also due to ! latt]4 ,= x 
%\ld be made kn[n to /ud5ts t !         
proposition is true n only = squ>es1 b =
 any simil> figures1 su* z equilat]al   
triangles1 p>allelograms1 semicircles1 &
 irregul> figures1 provid$ !y are       
simil>ly plac$ on ! ?ree sides ( ! "r   
triangle4  ,besides ,proclus1 ,plut>*   
te/ifies to ! fact t ,py?agoras 0 !     
discov]]1 say+ t ,py?agoras sacrific$ an
 ox on ! /r5g? ( 8 proposition z        
,apollodotus says1 b say+ t !re 7 two   
possible propositions to :i* ?is ref]s4 
,?is ,apollodotus 0 probably            
,apollodorus1 surnam$ ,logi/icus 7!     
,calculator71 ^: date is q unc]ta91 & :o

 speaks 9 "s v]ses ( a fam\s proposition
 discov]$ by ,py?agoras1 & all tradition
 makes ?is ! "o4 ,cic]o1 :o comm5ts ^u  
^! v]ses1 does n "q ! discov]y1 b d\bts 
! /ory ( ! sacrifice ( ! ox4 ,( o!r e>ly
 writ]s1 ,diog5es ,la]tius1 ^: date is  
entirely unc]ta9 7p]haps ! second c5tury
 ,a4,d471 & ,a!naeus 7?ird c5tury       
,a4,d47 may be m5tion$ z attribut+ !    
!orem to ,py?agoras1 :ile ,h]on 7fir/   
c5tury ,a4,d47 says t he gave a rule =  
=m+ "r triangles ) rational integ]s = ! 
sides1 l #c1 #d1 #e1 :]e #cb  #db  #eb4 
,x %\ld be said1 h[ev]1 t ! ,py?agorean 
orig9 has be5 d\bt$1 notably 9 an       
article by ,h4 ,vogt1 publi%$ 9 !       
,biblio!ca ,ma!matica 9 #aijh 7,vol4    
,,ix 7#c71 p4 #ae71 entitl$ ,die        
,geometrie des ,py?agoras1 & by ,g4     
,junge1 9 8 "w entitl$ ,wann hab5 die   
,grie*5 das ,irrationale entdeckt8      
7,halle1 #aijg74 ,^! writ]s claim t all 
! au?orities attribut+ ! proposition to 
,py?agoras are c5turies lat] ?an 8 "t1 &
 are op5 to grave suspicion4 ,nev]!less 

x is h>dly possible t su* a g5]al       
tradition1 & "o s univ]sally accept$1   
%\ld h aris5 )\t good f\ndation4 ,!     
evid5ce has be5 c>efully /udi$ by ,hea? 
9 8 ,euclid1 :o concludes ) ^! words3   
,on ! :ole1 !re=e1 ,i see no suffici5t  
reason to "q ! tradition t1 s f> z      
,greek geometry is conc]n$ 4441         
,py?agoras 0 ! fir/ to introduce ! !orem
 444 & to give a g5]al pro( ( x4 ,t !   
fact 0 kn[n e>li]1 probably )\t ! g5]al 
pro(1 is recogniz$ by all mod]n writ]s4 
,illu/ration  ,py?agoras _h /udi$ 9     
,egypt & possibly 9 ! ,ea/ be=e he      
e/abli%$ 8 s*ool at ,crotona1 9 s\!rn   
,italy4 ,9 ,egypt1 at any rate1 he c\ld 
easily h f\nd t a triangle ) ! sides #c1
 #d1 #e1 is a "r triangle1 & ,vitruvius 
7fir/ c5tury ,b4,c47 tells u t he taught
 ?is fact4 ,! ,egyptian h>p$onaptae     
7rope /ret*]s7 /ret*$ ropes ab\t pegs s 
z to make su* a triangle = ! purpose (  
lay+ out a "r angle 9 _! survey+1 j z   
our surveyors d to-day4 ,! great        
pyramids h an angle ( slope su* z is    

giv5 by ?is triangle4 ,inde$1 a papyrus 
( ! twelf? dyna/y1 lately discov]$ at   
,kahun1 9 ,egypt1 ref]s to f\r ( ^!     
triangles1 su* z #ab  7#c_/d7b  7#a     
#a_/d7b4 ,?is prop]ty seems to h be5 a  
matt] ( common k long be=e ,py?agoras1  
ev5 z f> ea/ z ,*9a4 ,he 01 !re=e1      
naturally l$ to attempt to prove ! g5]al
 prop]ty :i* _h already be5 recogniz$ = 
special cases1 & 9 p>ticul> = !         
isosceles "r triangle4  ,h[ ,py?agoras  
prov$ ! proposition is n kn[n4 ,x has   
be5 ?\ght t he us$ a pro( by proportion1
 because ,proclus says t ,euclid gave a 
new /yle ( pro(1 & ,euclid does n use   
proportion = ?is purpose1 :ile !        
subject1 9 incomplete =m1 0 highly      
e/eem$ by ! ,py?agoreans4 ,hea? sugge/s 
t ?is is among ! possibilities3         
,illu/ration      triangles,,abc & ,,apc
 are simil>4          !re=e ,,ab x ,,ap 
7,,ac7#b4           ,simil>ly1 ,,ab x   
,,pb  7,,bc7#b4      !re=e ,,ab7,,ap    
,,pb7  7,,ac7#b  7,,bc7#b1      or      
7,,ab7#b  7,,ac7#b  7,,bc7#b4  ,o!rs h  

?\ght t ,py?agoras d]iv$ 8 pro( f       
dissect+ a squ>e & %[+ t ! squ>e on !   
hypot5use mu/ equal ! sum ( ! squ>es on 
! o!r two sides1 9 "s su* mann] z ?is3  
,illu/ration3 ,,fig4 #a  ,illu/ration3  
,,fig4 #b       ,"h ,fig4 #a is evid5tly
 h#b  #d triangles4       ,fig4 #b is   
evid5tly a#b  b#b  #d triangles4        
!re=e h#b  #d triangles  a#b  b#b  #d   
triangles1 ! triangles all be+ congru5t4
       !re=e h#b  a#b  b#b4  ,! great   
,h9du ma!matician1 ,bhask>a 7born #aaad 
,a4,d471 proce$s 9 a some:at simil>     
mann]4 ,he draws ?is figure1 b gives no 
pro(4 ,x is evid5t t he _h 9 m9d ?is    
relation3  ,illu/ration      h#b  #d 4  
ab_/#b  7b - a7#b  a#b  b#b4  ,a some:at
 simil> pro( c be bas$ ^u ! foll[+      
figure3  ,illu/ration       ,if ! f\r   
triangles1 #a  #b  #c  #d1 are tak5     
away1 !re      rema9s ! squ>e on !      
hypot5use4 ,b if we take away !      two
 %ad$ rectangles1 :i* equal ! f\r       
triangles1 !re      rema9 ! squ>es on ! 
two sides4 ,!re=e ! squ>e on      !     

hypot5use mu/ equal ! sum ( ^! two      
squ>es4  ,illu/ration  ,x has long be5  
?\ght t ! tru? ( ! proposition 0 fir/   
obs]v$ by see+ ! tiles on ! floors (    
anci5t temples4 ,if !y 7 arrang$ z "h   
%[n1 ! proposition w\ld be evid5t = !   
special case ( an isosceles "r triangle4
  ,! ,h9dus knew ! proposition long be=e
 ,bhask>a1 h[ev]1 & possibly be=e       
,py?agoras4 ,x is ref]r$ to 9 ! old     
religi\s poems ( ! ,brahmans1 !         
,sulvasutras1 b ! date ( ^! poems is s  
unc]ta9 t x is impossible to /ate t !y  
prec$$ ! six? c5tury ,b4,c41#gi 9 :i*   
,py?agoras liv$4 ,! ,sulvasutra (       
,apa/amba has a collection ( rules1 )\t 
pro(s1 = con/ruct+ v>i\s figures4 ,among
 ^! is "o = con/ruct+ "r angles by      
/ret*+ cords ( ! foll[+ l5g?s3 #c1 #d1  
#e2 #ab1 #af1 #bj2 #ae1 #bj1 #be 7! two 
latt] be+ multiples ( ! fir/72 #e1 #ab1 
#ac2 #ae1 #cf1 #ci2 #h1 #ae1 #ag2 #ab1  
#ce1 #cg4 ,:atev] ! date ( ^!           
,sulvasutras1 !re is no evid5ce t !     
,indians _h a def9ite pro( ( ! !orem1   

ev5 ?\gh !y1 l ! e>ly ,egyptians1       
recogniz$ ! g5]al fact4  ,x is always   
int]e/+ to a class to see m ?an "o pro( 
( a fam\s !orem1 & _m tea*]s f9d x      
pr(itable to ask _! pupils to "w out    
pro(s t are 7to !m7 orig9al1 (t5 sugge/+
 ! figure4 ,two ( ! be/ kn[n hi/oric    
pro(s are "h giv54  ,! fir/ makes !     
,py?agorean ,!orem a special case ( a   
proposition due to ,pappus 7f\r? c5tury 
,a4,d471 relat+ to any k9d ( a triangle4
  ,illu/ration       ,some:at simplifi$1
 ?is proposition ass]ts t if ,,abc is   
any k9d ( triangle1 & ,,mc1 ,,nc are    
p>allelograms on      ,,ac1 ,,bc1 !     
opposite sides be+ produc$ to meet at   
,p2      & if ,,pc is produc$ mak+ ,,qr 
,,pc2 & if !      p>allelogram ,,at is  
con/ruct$1 !n ,,at  ,,mc  ,,nc4       ,=
 ,,mc  ,,ap  ,,ar1 hav+ equal bases &   
equal altitudes4      ,simil>ly1        
,,nc  ,,qt4      ,add+1     ,,mc  ,,nc  
,,at4       ,if1 n[1 ,,abc is a "r      
triangle1 & if ,,mc & ,,nc are          
squ>es1 x is easy to %[ t ,,at is a     

squ>e1 & !      proposition r$uces to ! 
,py?agorean ,!orem4  ,! ,arab writ]1    
,al-,nairizi 7di$ ab\t #ibb ,a4,d471    
attributes to ,?abit b5 ,qurra 7#hbf-ija
 ,a4,d47 a pro( sub/antially z foll[s3  
,illu/ration       ,! f\r triangles ,t c
 be prov$ congru5t4 ,!n if we take      
f ! :ole figure ,t & ,t'1 we h left !   
squ>es on      ! two sides ( ! "r angle4
 ,if we take away ! o!r two             
triangles in/ead1 we h left ! squ>e on !
 hypot5use4      ,!re=e ! =m] is        
equival5t to ! latt]4  ,a pro( attribut$
 to ! great arti/1 ,leon>do da ,v9ci    
7#adeb-aeai71 is z foll[s3  ,illu/ration
       ,! con/ruction ( ! foll[+ figure 
is evid5t4 ,x is      easily %[n t ! f\r
 quadrilat]als ,,abmx1 ,,xnca1          
,,sbcp1 & ,,srqp are congru5t4          
!re=e ,,abmxnca equals ,,sbcpqrs b is n 
congru5t to      x1 ! congru5t          
quadrilat]als be+ diff]5tly arrang$4    
,subtract ! congru5t triangles ,,mxn1   
,,abc1 ,,raq1 & !      proposition is   
prov$4#hj  ,! foll[+ is an int]e/+ pro( 

( ! proposition3      ,let ,,abc be !   
orig9al triangle1 ) ,,ab  ,,bc4 ,turn ! 
triangle ab\t ,b1 ?r\gh #ij deg41 until 
x comes into ! position     ,a',,bc'4   
,!n because x has be5 turn$ ?r\gh #ij   
deg41 ,c',a',p w     be p]p5dicul> to   
,,ac4 ,!n                               
#a_/b7;,,ab7#b  triangle,,aba'1      &  
#a_/b7;,,bc'7#b  triangle,,bc',c1       
because                          ,,bc   
,,bc'4      !re=e #a_/b77;,,ab7#b       
7,,bc7#b7          triangle,,aba'       
triangle,,bc',c4      !re=e             
#a_/b77;,,ab7#b  7,,bc7#b7              
triangle,,ac',a'  triangle,a',c',c      
,illu/ration       7,= triangle,,aba'   
triangle,,bc',a'  triangle,a',c',c      
is ! second memb] ( bo? equations47     
#a_/b;,a',c' 4 ,,ap  #a_/b;,a',c' 4 ,,pc
                                        
#a_/b;,a',c' 4 ,,ac                     
#a_/b7;,,ac7#b4      !re=e 7,,ab7#b     
7,,bc7#b  7,,ac7#b4   ,! ,py?agorean    
,!orem1 z x is g5]ally call$1 has _h o!r
 names4 ,x is n uncommonly call$ ! pons 

as9orum 7see page #agd7 9 ,france4 ,!   
,arab writ]s call$ x ! ,figure ( !      
,bride1 al?\gh ! reason = ?is "n is     
unkn[n2 possibly two be+ jo9$ 9 "o has  
some?+ to d ) x4 ,x has also be5 call$ !
 ,bride's ,*air1 & ! %ape ( ! ,euclid   
figure is n unlike ! *air t a slave     
c>ries on 8 back1 9 :i* ! ,ea/]n bride  
is sometimes transport$ to ! w$d+       
c]emony4 ,s*op5hau]1 ! ,g]man           
philosoph]1 ref]r+ to ! figure1 speaks (
 x z a pro( walk+ on /ilts1 & z a       
m\se-trap pro(4  ,an int]e/+ !ory       
sugge/$ by ! proposition is t ( comput+ 
! sides ( "r triangles s t !y %all be   
repres5t$ by rational numb]s4 ,py?agoras
 seems to h be5 ! fir/ to take up ?is   
!ory1 al?\gh su* numb]s 7 appli$ to ! "r
 triangle be=e 8 "t1 & ,proclus tells u 
t ,plato also contribut$ to x4 ,! rule (
 ,py?agoras1 put 9 mod]n symbols1 0 z   
foll[s3       n#b  77n#b - #a7_/b7b     
77n#b  #a7_/b7b1       ! sides be+ n1   
7n#b - #a7_/b1 & 7n#b  #a7_/b4 ,if =    
n we put #c1 we h #c1 #d1 #e4 ,if we    

take ! v>i\s odd      numb]s1 we h      
n  #a1 #c1 #e1 #g1 #i1 4441      7n#b - 
#a7_/b  #j1 #d1 #ab1 #bd1 #dj1 4441     
7n#b  #a7_/b  #a1 #e1 #ac1 #be1 #da1    
4444   ,( c\rse n may be ev51 giv+      
fractional values4 ,?us1 = n  #b we h = 
! ?ree sides1 #b1 #a #a_/b1 #b #a_/b4   
,o!r =mulas are also kn[n4 ,plato's1 =  
example1 is z foll[s3                   
7#bn7#b  7n#b - #a7b  7n#b  #a7b4       
,if          #bn  #b1 #d1 #f1 #h1 #aj1  
4441      !n   n#b - #a  #j1 #c1 #h1    
#ae1 #bd1 4441      &    n#b  #a  #b1   
#e1 #aj1 #ag1 #bf1 4444  ,?is =mula     
evid5tly comes f t ( ,py?agoras by d\bl+
 ! sides ( ! squ>es4#ha  ,,!orem4 ,9 any
 triangle ! squ>e ( ! side opposite an  
acute angle is equal to ! sum ( ! squ>es
 ( ! o!r two sides dim9i%$ by twice !   
product ( "o ( ^? sides by ! projection 
( ! o!r ^u t side4  ,,!orem4 ,a simil>  
/atem5t = ! obtuse triangle4  ,^! two   
propositions are usually prov$ by ! help
 ( ! ,py?agorean ,!orem4 ,"s writ]s1    
h[ev]1 actually con/ruct ! squ>es & give

 a pro( simil> to ! "o 9 t proposition4 
,?is plan goes back at lea/ to ,gregoire
 de ,/4 ,v9c5t 7#afdg74  ,illu/ration   
,x %\ld be obs]v$ t      a#b  b#b  c#b -
 #b;b'c4       ,if ,l,a  #ij deg41 !n b'
  #j1 & ?is becomes      a#b  b#b  c#b4 
,if ,l,a is obtuse1 !n b' passes ?r\gh  
#j & becomes      negative1 & a#b  b#b  
c#b  #b;b'c4       ,?us we h ?ree       
propositions 9 "o4  ,illu/ration  ,at ! 
close ( ,book ,,iv _m geometries give z 
an ex]cise1 & "s give z a regul>        
proposition1 ! celebrat$ problem t be>s 
! "n ( ,h]on ( ,alex&ria1 namely1 to    
compute ! area ( a triangle 9 t]ms ( its
 sides4 ,! result is ! important =mula  
,area  sqrt7s7s - a77s - b77s - c771    
:]e a1 b1 & c are ! sides1 & s is !     
semip]imet] #a_/b7;a  b  c74 ,z a       
practical application ! class may be    
able to f9d a triangul> piece ( l&1 z "h
 %[n1 & to measure ! sides4 ,if ! piece 
is cle>1 ! result may be *eck$ by       
measur+ ! altitude & apply+ ! =mula a   
#a_/b;bh4  ,x may be /at$ to ! class t  

,h]on's =mula is only a special case ( !
 m g5]al "o develop$ ab\t #fdj ,a4,d41  
by a fam\s ,h9du ma!matician1           
,brahmagupta4 ,?is =mula gives ! area ( 
an inscrib$ quadrilat]al z sqrt77s -    
a77s - b77s - c77s - d771 :]e a1 b1 c1 &
 d are ! sides & s is ! semip]imet]4 ,if
 d  #j1 ! quadrilat]al becomes a        
triangle & we h ,h]on's =mula4#hb  ,at !
 close ( ,book ,,iv1 also1 ! geometric  
equival5ts ( ! algebraic =mulas = 7a    
b7#b1 7a - b7#b1 & 7a  b77a - b7 are    
giv54 ,! class may l to "k t ,euclid _h 
no algebra & 0 compell$ to prove su*    
relations z ^! by geometry1 :ile we d x 
n[ mu* m easily by algebraic            
multiplication4  ,,footnotes3  #gi ,see1
 = example1 ,g4 ,b4 ,kaye1 ,! ,s\rce (  
,h9du ,ma!matics1 9 ! ,j\rnal ( ! ,royal
 ,asiatic ,society1 ,july1 #aiaj4  #hj  
,an int]e/+ ,japanese pro( ( ?is g5]al  
"* may be se5 9 ,y4 ,mikami1 ,ma!matical
 ,pap]s f ! ,f> ,ea/1 p4 #abg1 ,leipzig1
 #aiaj4  #ha ,special recognition (     
indebt$ness to ,h4 ,a4 ,nab]'s ,das     

,!orem des ,py?agoras 7,ha>lem1 #aijh71 
,hea?'s ,euclid1 ,g['s ,hi/ory ( ,greek 
,ma!matics1 & ,cantor's ,ges*i*te is due
 9 connection ) ! ,py?agorean ,!orem4   
#hb ,! rule 0 s ill und]/ood t ,bhask>a 
7twelf? c5tury7 said t ,brahmagupta 0 a 
blund]+ devil = giv+ x 7,lilavati1 ,sec4
 #agb74     ,,*apt] ,,xviii  ,,! ,,lead+
 ,,propositions ,,( ,,book ,v           
,illu/ration  ,book ,v treats ( regul>  
polygons & circles1 & includes !        
computation ( ! approximate value ( pi4 
,x op5s ) a def9ition ( a regul> polygon
 z "o t is bo? equilat]al & equiangul>4 
,:ile 9 elem5t>y geometry ! only regul> 
polygons /udi$ are convex1 x is int]e/+ 
to a class to see t !re are also regul> 
cross polygons4 ,inde$1 ! regul> cross  
p5tagon 0 ! badge ( ! ,py?agoreans1 z   
,lucian 7ca4 #ajj ,b4,c47 & an unkn[n   
comm5tator on ,ari/ophanes 7ca4 #djj    
,b4,c47 tell u4 ,at ! v]tices ( ?is     
polygon ! ,py?agoreans plac$ ! ,greek   
lett]s signify+ heal?4  ,euclid 0 n     
int]e/$ 9 ! measure ( ! circle1 & !re is

 no?+ 9 8 ,elem5ts on ! value ( pi4     
,inde$1 he expressly avoid$ num]ical    
measures ( all k9ds 9 8 geometry1 wi%+ !
 sci5ce to be kept di/9ct f t =m (      
ari?metic kn[n to ! ,greeks z logi/ic1  
or calculation4 ,8 ,book ,,iv is devot$ 
to ! con/ruction ( c]ta9 regul>         
polygons1 & 8 propositions on ?is       
subject are n[ embodi$ 9 ,book ,v z x is
 usually taught 9 ,am]ica4  ,if we      
consid] ,book ,v z a :ole1 we are /ruck 
by ?ree features4 ,( ^! ! fir/ is ! pure
 geometry involv$1 & ?is is ! ess5tial  
feature to be emphasiz$4 ,! second is ! 
m5suration ( ! circle1 a relatively     
unimportant piece ( !ory 9 view ( ! fact
 t ! pupil is n ready = incomm5surables1
 & a feature t imp>ts no in=mation t !  
pupil did n f9d 9 ari?metic4 ,! ?ird is 
! some:at int]e/+ b ma!matically        
unimportant application ( ! regul>      
polygons to geometric design4  ,z to !  
m5suration ( ! circle x is well = u to  
take a broad view be=e com+ d[n to      
details4 ,!re are only f\r lead+        

propositions necess>y = ! m5suration ( !
 circle & ! det]m9ation ( ! value ( pi4 
,^! are z foll[s3 7#a7 ,! inscrib+ ( a  
regul> hexagon1 or any o!r regul>       
polygon ( :i* ! side is easily comput$ 9
 t]ms ( ! radius4 ,we may />t ) a squ>e1
 = example1 b ?is is n s good z !       
hexagon because its side is             
incomm5surable ) ! radius1 & its p]imet]
 is n z ne> ! circumf]5ce4 7#b7 ,!      
p]imet]s ( simil> regul> polygons are   
proportional to _! radii1 & _! areas to 
! squ>es ( ! radii4 ,x is n[ necess>y to
 /ate1 9 ! =m ( a po/ulate if desir$1 t 
! circle is ! limit ( regul> inscrib$ & 
circumscrib$ polygons z ! numb] ( sides 
increases indef9itely1 & h5ce t 7#b7    
holds = circles4 7#c7 ,! proposition    
relat+ to ! area ( a regul> polygon1 & !
 result+ proposition relat+ to ! circle4
 7#d7 ,giv5 ! side ( a regul> inscrib$  
polygon1 to f9d ! side ( a regul>       
inscrib$ polygon ( d\ble ! numb] (      
sides4 ,x w ?us be se5 t if we 7 m]ely  
desir\s ( approximat+ ! value ( pi1 & ( 

f9d+ ! two =mulas c  #bpir & a  pir#b1  
we %\ld ne$ only f\r propositions 9 ?is 
book ^u :i* to base our "w4 ,x is also  
app>5t t ev5 if ! incomm5surable cases  
are g5]ally omitt$1 ! notion ( limit is 
ne$$ at ?is "t1 & t x mu/ briefly be    
review$ be=e proce$+ fur!r4  ,!re is1   
h[ev]1 a mu* m wor?y int]e/ ?an ! m]e   
m5suration ( ! circle1 namely1 !        
con/ruction ( su* polygons z c readily  
be =m$ by ! use ( compasses & /raight$ge
 alone4 ,! pleasure ( con/ruct+ su*     
figures & ( prov+ t ! con/ruction is    
correct is ( itself suffici5t           
ju/ification = ! "w4 ,z to ! use ( su*  
figures 9 geometric design1 "s          
discussion w be (f]$ at ! close ( ?is   
*apt]4  ,! fir/ few propositions include
 ^? t lead up to ! m5suration ( !       
circle4 ,aft] !y are prov$ x is assum$ t
 ! circle is ! limit ( ! regul> inscrib$
 & circumscrib$ polygons z ! numb] (    
sides increases indef9itely4 ,?is may   
(t5 be prov$ ) "s approa* to rigor by a 
few memb]s ( an elem5t>y class1 b x is !

 exp]i5ce ( tea*]s t ! pro( is too      
difficult = mo/ beg9n]s1 & s !          
assumption is usually made 9 ! =m ( an  
unprov$ !orem4  ,! foll[+ are "s ( !    
lead+ propositions ( ?is book3  ,,!orem4
 ,two circumf]5ces h ! same ratio z _!  
radii4  ,?is leads to def9+ ! ratio ( ! 
circumf]5ce to ! diamet] z pi4 ,al?\gh  
?is is a ,greek lett]1 x 0 n us$ by !   
,greeks to repres5t ?is ratio4 ,inde$1 x
 0 n until #agjf t an ,engli% writ]1    
,william ,jones1 9 8 ,synopsis          
,palm>iorum ,ma!seos1 us$ x 9 ?is way1 x
 be+ ! initial lett] ( ! ,greek ^w =    
p]iph]y4 ,aft] e/abli%+ ! prop]ties t c 
#bpir1 & a  pir#b1 ! textbooks foll[ !  
,greek cu/om & proce$ to %[ h[ to       
inscribe & circumscribe v>i\s regul>    
polygons1 ! purpose be+ to use ^! 9     
comput+ ! approximate num]ical value (  
pi4 ,( ^! regul> polygons two are (     
special int]e/1 & ^! w n[ be consid]$4  
,,problem4 ,to inscribe a regul> hexagon
 9 a circle4  ,t ! side ( a regul>      
inscrib$ hexagon equals ! radius mu/ h  

be5 recogniz$ v e>ly4 ,! common         
divisions ( ! circle 9 anci5t art are   
into f\r1 six1 & eight equal p>ts4 ,no  
draftsman c\ld h work$ ) a pair (       
compasses )\t quickly le>n+ h[ to effect
 ^! divisions1 & t compasses 7 e>ly us$ 
is atte/$ by ! specim5s ( ^! in/rum5ts  
(t5 se5 9 museums4 ,!re is a tradition t
 ! anci5t ,babylonians consid]$ ! circle
 ( ! ye> z made up ( #cfj days1 :5ce !y 
took ! circle z compos$ ( #cfj /eps or  
grades 7degrees74 ,?is tradition is )\t 
hi/oric f\ndation1 h[ev]1 !re be+ no    
au?ority 9 ! inscriptions = ?is         
assumption ( ! #cfj-;division by !      
,babylonians1 :o seem r to h pref]r$ #h1
 #ab1 #abj1 #bdj1 & #dhj z _! division  
numb]s4 ,! /ory ( #cfj deg4 9 !         
,babylonian circle seems to />t )       
,a*illes ,tatius1 an ,alex&rian         
gramm>ian ( ! second or ?ird c5tury     
,a4,d4 ,x is possible1 h[ev]1 t !       
,babylonians got _! favorite numb] #fj  
7z 9 #fj seconds make a m9ute1 #fj      
m9utes make an h\r or degree7 f !       

hexagon 9 a circle 7#a_/f ( #cfj deg4   
#fj deg471 al?\gh ! probabilities seem  
to be t !re is no su* connection4#hc  ,!
 applications ( ?is problem to          
m5suration are num]\s4 ,! fact t we may 
use = tiles on a floor ?ree regul>      
polygons--! triangle1 squ>e1 &          
hexagon--is notewor?y1 a fact t ,proclus
 tells u 0 recogniz$ by ,py?agoras4 ,!  
measurem5t ( ! regul> hexagon1 giv5 "o  
side1 may be us$ 9 comput+ sections (   
hexagonal columns1 9 f9d+ areas ( fl[]  
b$s1 & 9 o!r simil> cases4  ,?is review 
( ! names ( ! polygons (f]s an          
opportunity to impress _! etymology aga9
 on ! m9d4 ,9 ?is case1 = example1 we h 
hexagon f ! ,greek words = six & angle4 
,,problem4 ,to inscribe a regul> decagon
 9 a giv5 circle4  ,euclid /ates !      
problem ?us3 ,to con/ruct an isosceles  
triangle hav+ ea* ( ! angles at ! base  
d\ble ( ! rema9+ "o4 ,?is makes ea* base
 angle #gb deg4 & ! v]tical angle #cf   
deg41 ! latt] be+ ! c5tral angle ( a    
regul> decagon1--ess5tially our pres5t  

me?od4  ,?is proposition seems und\bt$ly
 due to ! ,py?agoreans1 z tradition has 
always ass]t$4 ,proclus tells u t       
,py?agoras discov]$ ! con/ruction ( !   
cosmic figures1 or ! five regul>        
polyh$rons1 & "o ( ^! 7! dodecah$ron7   
involves ! con/ruction ( ! regul>       
p5tagon4  ,iambli*us 7ca4 #cbe ,a4,d47  
tells u t ,hippasus1 a ,py?agorean1 0   
said to h be5 dr[n$ = d>+ to claim cr$it
 = ! con/ruction ( ! regul> dodecah$ron1
 :5 by ! rules ( ! bro!rhood all cr$it  
%\ld h be5 assign$ to ,py?agoras4  ,if a
 regul> polygon ( s sides c be inscrib$1
 we may bisect ! c5tral angles1 & !re=e 
inscribe "o ( #bs sides1 & !n ( #ds     
sides1 & !n ( #hs sides1 & 9 g5]al (    
#bns sides4 ,?is includes ! case ( s  #b
 & n  #j1 = we c inscribe a regul>      
polygon ( two sides1 ! angles be+1 by ! 
usual =mula1 #b7b - #b7_/b  #j1 al?\gh1 
( c\rse1 we nev] ?9k ( two equal &      
co9cid5t l9es z =m+ :at we might call a 
digon4  ,we !re=e h ! foll[+ regul>     
polygons3      ,f ! equilat]al triangle1

 regul> polygons ( #bn 4 #c sides2      
,f ! squ>e1 regul> polygons ( #bn sides2
     ,f ! regul> p5tagon1 regul>        
polygons ( #bn 4 #e sides2     ,f !     
regul> p5t$ecagon1 regul> polygons ( #bn
 4 #ae sides4  ,?is gives u1 =          
successive values ( n1 ! foll[+ regul>  
polygons ( less ?an #ajj sides3      ,f 
#bn 4 #c1   #c1 #f1 #ab1 #bd1 #dh1 #if2 
,f #bn1       #b1 #d1 #h1 #af1 #cb1 #fd2
     ,f #bn 4 #e1   #e1 #aj1 #bj1 #dj1  
#hj2     ,f #bn 4 #ae1  #ae1 #cj1 #fj4  
,illu/ration3 ,,roman ,,mosaic ,,f\nd   
,,at ,,pompeii  ,gauss 7#aggg-ahee71 a  
celebrat$ ,g]man ma!matician1 prov$ 79  
#agif7 t x is possible also to inscribe 
a regul> polygon ( #ag sides1 & h5ce    
polygons ( #bn 4 #ag sides1 or #ag1 #cd1
 #fh1 4441 sides1 & also #c 4 #ag  #ea &
 #e 4 #ag  #he sides1 by ! use ( !      
compasses & /raight$ge1 b ! pro( is n   
adapt$ to elem5t>y geometry4 ,9         
connection ) ! /udy ( ! regul> polygons 
"s int]e/ atta*es to ! ref]5ce to v>i\s 
=ms ( decorative design4 ,! mosaic      

floor1 p>quetry1 ,go?ic w9d[s1 & patt]ns
 ( v>i\s k9ds (t5 involve ! regul>      
figures4 ,if ! tea*] uses su* mat]ial1  
c>e %\ld be tak5 to exemplify good art4 
,= example1 ! equilat]al triangle & its 
relation to ! regul> hexagon is %[n 9 ! 
picture ( an anci5t ,roman mosaic floor 
on page #bgd4hd ,9 ! next illu/ration "s
 *>act]i/ic ,moori% mosaic "w appe>s1 9 
:i* x w be se5 t ! basal figure is !    
squ>e1 al?\gh at fir/ sight ?is w\ld n  
seem to be ! case4#he ,?is is foll[$ by 
a beautiful ,byzant9e mosaic1 ! orig9al 
( :i* 0 9 five colors ( m>ble4 ,"h x w  
be se5 t ! equilat]al triangle & !      
regul> hexagon are ! basal figures1 & a 
few ( ! prop]ties ( ^! polygons might be
 d]iv$ f ! /udy ( su* a design4 ,9 !    
,arabic patt]n on page #bgf ! dodecagon 
appe>s z ! basis1 & ! rem>kable p[]s ( !
 ,arab design] are %[n 9 ! use (        
symmetry )\t employ+ regul> figures4    
,illu/ration3 ,,mosaic ,,f ,,damascus   
,illu/ration3 ,,mosaic ,,f ,,an ,,anci5t
 ,,byzant9e ,,*ur*  ,,problem4 ,giv5 !  

side & ! radius ( a regul> inscrib$     
polygon1 to f9d ! side ( ! regul>       
inscrib$ polygon ( d\ble ! numb] (      
sides4  ,illu/ration3 ,,arabic ,,patt]n 
,! object ( ?is proposition is1 ( c\rse1
 to prep>e ! way = f9d+ ! p]imet] ( a   
polygon ( #bn sides1 kn[+ t ( n sides4  
,! ,greek plan 0 g5]ally to use bo? an  
inscrib$ & a circumscrib$ polygon1 ?us  
approa*+ ! circle z a limit bo? f )\t & 
)94 ,?is is m conclusive f !            
ultrasci5tific po9t ( view1 b x is1 if  
any?+1 less conclusive to a beg9n]1     
because he does n s readily foll[ !     
pro(4 ,! plan ( us+ ! two polygons 0    
c>ri$ out by ,ar*im$es ( ,syracuse      
7#bhg-bab ,b4,c47 9 8 fam\s me?od (     
approximat+ ! value ( pi1 al?\gh be=e   
him ,antiphon 7fif? c5tury ,b4,c47 _h   
inscrib$ a squ>e 7or equilat]al         
triangle7 z a basis = ! "w1 & ,bryson 78
 contempor>y7 _h attack$ ! problem by   
circumscrib+ z well z inscrib+ a regul> 
polygon4  ,,problem4 ,to f9d ! num]ical 
value ( ! ratio ( ! circumf]5ce ( a     

circle to its diamet]4  ,z already /at$1
 ! usual plan ( ! textbooks is 9 "p !   
me?od foll[$ by ,ar*im$es4 ,x is        
possible to />t ) any regul> polygon (  
:i* ! side c conv5i5tly be f\nd 9 t]ms (
 ! radius4 ,9 p>ticul> we might beg9 )  
an inscrib$ squ>e in/ead ( a regul>     
hexagon4 ,9 ?is case we %\ld h          
,l5g? ( ,side ,p]imet]      s#d         
#a4dad444                          #a4da
        #e4ff     s#h  sqrt7#b - sqrt7#d
 - #a4dadb77   #j4gb        #e4gf  & s  
on4  ,x is a little easi] to />t ) !    
hexagon1 h[ev]1 = we are already ne>] ! 
circle1 & ! side & p]imet] are bo?      
comm5surable ) ! radius4 ,x is n1 (     
c\rse1 int5d$ t pupils %\ld make ! long 
num]ical calculations4 ,!y may be       
requir$ to compute s#ab & possibly s#bd1
 b aside f ?is !y are expect$ m]ely to  
"k ! process4  ,if x 7 possible to f9d !
 value ( pi exactly1 we c\ld f9d !      
circumf]5ce exactly 9 t]ms ( ! radius1  
s9ce c  #bpir4 ,if we c\ld f9d !        
circumf]5ce exactly1 we c\ld f9d ! area 

exactly1 s9ce a  pir#b4 ,if we c\ld f9d 
! area exactly 9 ?is =m1 pi times a     
squ>e1 we %\ld h a rectangle1 & x is    
easy to con/ruct a squ>e equival5t to   
any rectangle4 ,!re=e1 if we c\ld f9d ! 
value ( pi exactly1 we c\ld con/ruct a  
squ>e ) area equival5t to ! area ( !    
circle2 9 o!r words1 we c\ld squ>e !    
circle4 ,we c\ld also1 z already /at$1  
con/ruct a /raight l9e equival5t to !   
circumf]5ce2 9 o!r words1 we c\ld       
rectify ! circumf]5ce4 ,^! two problems 
h attract$ ! att5tion ( ! _w = ov] two  
?\s& ye>s1 b on acc\nt ( _! int]relation
 !y are usually spok5 ( z a s+le        
problem1 to squ>e ! circle4  ,s9ce we c 
con/ruct sqrta by means ( ! /raight$ge &
 compasses1 x w\ld be possible = u to   
squ>e ! circle if we c\ld express pi by 
a f9ite numb] ( squ>e roots4 ,conv]sely1
 e geometric con/ruction r$uces to !    
int]section ( two /raight l9es1 ( a     
/raight l9e & a circle1 or ( two        
circles1 & is !re=e equival5t to a      
rational op]ation or to ! extract+ ( a  

squ>e root4 ,h5ce a geometric           
con/ruction _c be effect$ by !          
/raight$ge & compasses unless x is      
equival5t to a s]ies ( rational         
op]ations or to ! extract+ ( a f9ite    
numb] ( squ>e roots4 ,x 0 prov$ by a    
,g]man pr(essor1 ,l9demann1 9 #ahhb1 t  
pi _c be express$ z an algebraic numb]1 
t is1 z ! root ( an equation ) rational 
coeffici5ts1 & h5ce x _c be f\nd by !   
above op]ations1 &1 fur!rmore1 t !      
solution ( ?is fam\s problem is         
impossible by elem5t>y geometry4#hf  ,x 
%\ld also be po9t$ out to ! /ud5t t = _m
 practical purposes "o ( ! limits ( pi  
/at$ by ,ar*im$es1 namely1 #c #a_/g1 is 
suffici5t4 ,= m accurate "w #c4adaf is  
usually a satisfactory approximation4   
,inde$1 ! late ,pr(essor ,newcomb /at$ t
 t5 decimal places are suffici5t to give
 ! circumf]5ce ( ! e>? to ! fraction (  
an in*1 & ?irty decimal places w\ld give
 ! circumf]5ce ( ! :ole visible univ]se 
to a quantity imp]ceptible ) ! mo/      
p[]ful microscope4  ,probably ! e>lie/  

approximation ( ! value ( pi 0 #c4 ,?is 
appe>s v commonly 9 antiquity1 z 9 ,i   
,k+s vii1 #bc1 & #b ,*ronicles iv1 #b4  
,9 ! ,ahmes papyrus 7ca4 #agjj ,b4,c47  
!re is a rule = f9d+ ! area ( ! circle1 
express$ 9 mod]n symbols z 7#h_/i7b;d#b1
 :i* makes pi  #bef_/ha or #c4afjd4444  
,ar*im$es1 us+ a plan some:at simil> to 
ours1 f\nd t pi lay betwe5 #c #a_/g & #c
 #aj_/ga4 ,ptolemy1 ! great ,greek      
a/ronom]1 express$ ! value z #c         
#ag_/abj1 or #c4adaff4444 ,! fact t     
,ptolemy divid$ 8 diamet] into #abj     
units & 8 circumf]5ce into #cfj units   
probably %[s1 h[ev]1 ! influ5ce ( !     
anci5t value #c4  ,9 ,india an          
approximate value appe>s 9 a c]ta9 poem 
writt5 be=e ! ,*ri/ian era1 b ! date is 
unc]ta94 ,ab\t #ejj ,a4,d4 ,aryabhatta  
7or possibly a lat] writ] ( ! same "n7  
gave ! value #fbhcb_/bjjjj1 or #c4adaf4 
,brahmagupta1 ano!r ,h9du 7born #eih    
,a4,d471 gave sqrt7#aj71 & ?is also     
appe>s 9 ! writ+s ( ! ,*9ese ma!matician
 ,*ang ,h5g 7#gh-aci ,a4,d474 ,a little 

lat] 9 ,*9a1 ,wang ,fan 7#bbi-bfg7 gave 
#adb _/ #de1 or #c4aeee4442 & "o ( 8    
contempor>ies1 ,lui ,hui1 gave #aeg _/  
#ej1 or #c4ad4 ,9 ! fif? c5tury         
,*'ung-*ih gave z ! limits ( pi1        
#c4adaeibg & #c4adaeibf1 f :i* he inf]r$
 t #bb_/g & #cee_/aac 7 good            
approximations1 al?\gh he does n /ate h[
 he came to ?is conclusion4  ,9 !       
,middle ,ages ! greate/ ma!matician (   
,italy1 ,leon>do ,fibonacci1 or ,leon>do
 ( ,pisa 7ab\t #abjj ,a4,d471 f\nd z    
limits #c4adbg444 & #c4adaj4444 ,ab\t   
#afjj ! ,*9ese value #cee_/aac 0        
r$iscov]$ by ,adria5 ,an?onisz          
7#aebg-afjg71 be+ publi%$ by 8 son1 :o  
is kn[n z ,metius 7#aega-afce71 9 ! ye> 
#afbe4 ,ab\t ! same p]iod ! ,fr5*       
ma!matician ,vieta 7#aedj-afjc7 f\nd !  
value ( pi to #i decimal places1 &      
,adria5 van ,rooman 7#aefa-afae7 c>ri$ x
 to #ag decimal places1 & ,ludolph van  
,ceul5 7#aedj-afaj7 to #ce decimal      
places4 ,x 0 c>ri$ to #adj decimal      
places by ,georg ,vega 7di$ 9 #agic71 to

 #bjj by ,za*>ias ,dase 7di$ 9 #ahdd71  
to #ejj by ,ri*t] 7di$ 9 #ahed71 & m    
rec5tly by ,%anks to #gjg decimal       
places4  ,!re h be5 _m int]e/+ =mulas = 
pi1 among !m be+ ! foll[+3      pi_/#b  
#b_/a 4 #b_/c 4 #d_/c 4 #d_/e 4 #f_/e 4 
#f_/g 4 #h_/g 4 #h_/i 4 4444            
7,wallis1 #afaf-agjc7      #d_/pi  #a   
#a_/b                       #i_/b       
#be_/b                                  
#di_/b                                  
4444 7,br\nck]1 #afbj-afhd7      pi_/#d 
#a - #a_/c  #a_/e - #a_/g  4444         
7,gregory1 #afch-afge7      pi_/#f      
sqrt7#a_/c7 4 7#a - #a_/7c 4 #c7        
#a_/7cb 4 #e7 - #a_/7cc 4 #g7  44474    
pi_/#b  7log i7 _/ i4 7,b]n\lli7        
pi_/7#bsqrt7#c77  #a - #a_/e  #a_/g -   
#a_/aa  #a_/ac - #a_/ag  #a_/ai4441     
?us connect+ ! primes4      pi#b_/af  #a
 - #a_/bb - #a_/cb  #a_/db - #a_/eb     
#a_/fb - #a_/gb -       #a_/hb  #a_/ib  
4444      pi_/#b  x_/#b  s9 x  7s9#b x7 
_/ #b  7s9#c x7 _/ #c  4444         7#j 
x  #bpi7      pi_/#d  #c_/d  #a_/7b 4 #c

 4 #d7 - #a_/7d 4 #e 4 #f7  #a_/7f 4 #g 
4 #h7 - 4444      #bpi#b_/c  #g -       
7#a_/7a 4 #c7  #a_/7c 4 #f7  #a_/7f 4   
#aj7  44474      pi        #bnsqrt7#b - 
sqrt7#b  sqrt7#b  sqrt7#b               
sqrt7#b444777774  ,/ud5ts ( elem5t>y    
geometry are n prep>$ to appreciate x1 b
 tea*]s w be int]e/$ 9 ! rem>kable =mula
 discov]$ by ,eul] 7#agjg-aghc71 ! great
 ,swiss ma!matician1 namely1 #a  eipi   
#j4 ,9 ?is relation are includ$ ! five  
mo/ int]e/+ quantities 9                
ma!matics1--z]o1 ! unit1 ! base ( !     
s-call$ ,napi]ian log>i?ms1 i           
sqrt7-#a71 & pi4 ,x 0 by means ( ?is    
relation t ! transc5d5ce ( e 0 prov$ by 
! ,fr5* ma!matician ,h]mite1 & !        
transc5d5ce ( pi by ! ,g]man ,l9demann4 
,illu/ration  ,!re %\ld be introduc$ at 
?is "t1 if x has n already be5 done1 !  
proposition ( ! lunes ( ,hippocrates    
7ca4 #dgj ,b4,c471 :o prov$ a !orem t   
ass]ts1 9 some:at m g5]al =m1 t if ?ree 
semicircles be describ$ on ! sides ( a  
"r triangle z diamet]s1 z %[n1 ! lunes  

,l  ,l' are toge!r equival5t to !       
triangle ,t4  ,illu/ration  ,9 ! use ( !
 circle 9 design "o ( ! simple/ =ms     
sugge/$ by ,book ,v is ! trefoil        
7?ree-leaf71 z "h %[n1 ) ! necess>y     
con/ruction l9es4 ,?is is a v common    
ornam5t 9 ar*itecture1 bo? ) r\nd$ ends 
& ) ! ends slightly po9t$4  ,! trefoil  
is closely connect$ ) hexagonal designs1
 s9ce ! regul> hexagon is =m$ f !       
inscrib$ equilat]al triangle by d\bl+ ! 
numb] ( sides4 ,! foll[+ are designs t  
are easily made3  ,illu/ration  ,x is n 
v pr(itable1 because x is manife/ly     
unreal1 to measure ! p>ts ( su* figures1
 b x (f]s pl5ty ( practice 9 num]ical   
"w4  ,illu/ration3 ,,*oir ,,( ,,l9coln  
,,ca!dral  ,illu/ration3 ,,por* ,,(     
,,l9coln ,,ca!dral  ,9 ! illu/rations ( 
! ,go?ic w9d[s giv5 9 ,*apt] ,,xv only !
 squ>e & circle 7 g5]ally involv$4      
,tea*]s :o feel x necess>y or advisable 
to g outside ! regul> "w ( geometry = ! 
purpose ( increas+ ! pupil's int]e/ or (
 tra9+ 8 h& 9 ! draw+ ( figures w f9d   

pl5ty ( designs giv5 9 any pictures (   
,go?ic ca!drals4 ,= example1 ?is picture
 ( ! noble w9d[ 9 ! *oir ( ,l9coln      
,ca!dral %[s ! use ( ! squ>e1 hexagon1 &
 p5tagon4 ,9 ! por* ( ! same ca!dral1   
%[n 9 ! next illu/ration1 ! ar*itect has
 made use ( ! triangle1 squ>e1 & p5tagon
 9 plann+ 8 ornam5tal /onework4 ,x is   
possible to add to ! "w 9 pure geometry 
"s "w 9 ! m5suration ( ! curvil9e>      
figures %[n 9 ^! designs4 ,?is =m (     
m5suration is n ( mu* value1 h[ev]1 s9ce
 x places be=e ! pupil a problem t he   
sees at once is fictiti\s1 & t has no   
human int]e/4  ,illu/ration3 ,,go?ic    
,,designs ,,employ+ ,,circles ,,&       
,,bisect$ ,,angles  ,illu/ration3       
,,go?ic ,,designs ,,employ+ ,,circles   
,,& ,,squ>es  ,illu/ration3 ,,go?ic     
,,designs ,,employ+ ,,circles ,,& ,,!   
,,equilat]al ,,triangle  ,illu/ration3  
,,go?ic ,,designs ,,employ+ ,,circles   
,,& ,,! ,,regul> ,,hexagon  ,! designs  
giv5 on page #bhc involve *iefly ! squ>e
 z a basis1 b x w be se5 f "o ( !       

figures t ! equilat]al triangle & !     
hexagon also ent]4 ,! possibilities (   
endless v>iation ( a s+le design are %[n
 9 ! illu/ration on page #bhd1 ! basis 9
 ?is case be+ ! squ>e4 ,! v>iations 9 ! 
use ( ! triangle & hexagon h be5 !      
object ( /udy ( _m design]s ( ,go?ic    
w9d[s1 & "s examples ( ^! =ms are %[n on
 page #bhe4 ,9 m simple =m ?is r++ ( !  
*anges on elem5t>y figures is %[n on    
page #bhf4 ,"s tea*]s h us$ color "w )  
su* designs = ! purpose ( increas+ !    
int]e/ ( _! pupils1 b ! dang] ( ?us us+ 
! "t ) no s]i\s end 9 view w be app>5t4 
,illu/ration  ,9 ! matt] ( ! m5suration 
( ! circle ! annex$ design has "s       
int]e/4 ,! figure is n uncommon 9       
decoration1 & x is int]e/+ to %[1 z a   
matt] ( pure geometry1 t ! area ( !     
circle is divid$ into ?ree equal        
portions by means ( ! f\r int]ior       
semicircles4  ,illu/ration  ,an         
important application ( ! =mula a  pir#b
 is se5 9 ! area ( ! annulus1 or r+1 !  
=mula be+ a  pir#b - pir'#b  pi7r#b -   

r'#b7  pi7r  r'77r - r'74 ,x is us$ 9   
f9d+ ! area ( ! cross section ( pipes1 &
 ?is is ne$$ :5 we wi% to compute !     
volume ( ! iron us$4  ,ano!r excell5t   
application is t ( f9d+ ! area ( !      
surface ( a cyl9d]1 !re be+ no reason :y
 su* simple cases f solid geometry %\ld 
n furni% work+ mat]ial = plane geometry1
 p>ticul>ly z !y h already be5 met by ! 
pupils 9 ari?metic4  ,a little problem t
 always has "s int]e/ = pupils is "o t  
,napoleon is said to h sugge/$ to 8 /aff
 on 8 voyage to ,egypt3 ,to divide a    
circle into f\r equal p>ts by ! use (   
circles alone4  ,illu/ration       ,"h !
 circles ,b are tang5t to ! circle ,a at
 !      po9ts ( division4 ,fur!rmore1   
consid]+ areas1 & tak+      r z ! radius
 ( ,a1 we h ,a  pir#b1 &      ,b        
pi7r_/#b7b4 ,h5ce ,b  #a_/d;,a1 or ! sum
 ( ! areas (      ! f\r circles ,b      
equals ! area ( ,a4 ,h5ce ! f\r         
,d's mu/ equal ! f\r ,c's1 & ,d  ,c4 ,! 
re/ ( !      argum5t is evid5t4 ,!      
problem has "s int]e/ to pupils         

aside f ! orig9al "q sugge/$ by         
,napoleon4  ,at ! close ( plane geometry
 tea*]s may f9d x helpful to h ! class  
make a li/ ( ! propositions t are       
actually us$ 9 prov+ o!r propositions1 &
 to h x appe> :at ones are prov$ by !m4 
,?is =ms a k9d ( g5ealogical tree t     
s]ves to fix ! p>5t propositions 9 m9d4 
,su* a "w may also be c>ri$ on at !     
close ( ea* book1 if desir$4 ,x %\ld be 
und]/ood1 h[ev]1 t c]ta9 propositions   
are us$ 9 ! ex]cises1 ev5 ?\gh !y are n 
ref]r$ to 9 subsequ5t propositions1 s t 
_! omission mu/ n be con/ru$ to mean t  
!y are n important4  ,an ex]cise (      
di/9ctly less value is ! classification 
( ! def9itions4 ,= example1 !           
classification ( polygons or (          
quadrilat]als1 once s popul> 9 textbook 
mak+1 has g5]ally be5 ab&on$ z t5d+ to  
create or p]petuate unnecess>y t]ms4    
,su* "w is !re=e n recomm5d$4           
,,footnotes3  #hc ,bosanquet & ,sayre1  
,! ,babylonian ,a/ronomy1 ,mon?ly       
,notices ( ! ,royal ,asiatic ,society1  

,vol4 ,,xl1 p4 #ajh4  #hd ,?is & ! ?ree 
illu/rations foll[+ are f ,kolb1 loc4   
cit4  #he ,?is 0 9 five colors ( m>ble4 
#hf ,! pro( is too involv$ to be giv5   
"h4 ,! writ] has set x =? 9 a *apt] on !
 transc5d5cy ( pi 9 a "w soon to be     
publi%$ by ,pr(essor ,"y ( ,! ,univ]sity
 ( ,*icago4     ,,*apt] ,,xix  ,,!      
,,lead+ ,,propositions ,,( ,,book ,,vi  
,!re h be5 num]\s sugge/ions ) respect  
to solid geometry1 to ! effect t x %\ld 
be m closely connect$ ) plane geometry4 
,! attempt has be5 made1 notably by     
,m]ay 9 ,france & de ,paolis 9 ,italy1  
to treat ! correspond+ propositions (   
plane & solid geometry toge!r2 z1 =     
example1 ^? relat+ to p>allelograms &   
p>allelepip$s1 & ^? relat+ to plane &   
sph]ical triangles4 ,:atev] ! m]its (   
?is plan1 x is n feasible 9 ,am]ica at  
pres5t1 p>tly because ( ! nature ( !    
college-entrance requirem5ts4 ,:ile x is
 true t to a boy or girl a solid is m   
concrete ?an a plane1 x is n true t a   
geometric solid is m concrete ?an a     

geometric plane4 ,j z ! _w develop$ its 
solid geometry1 z a sci5ce1 long aft] x 
_h develop$ its plane geometry1 s !     
human m9d grasps ! ideas ( plane figures
 e>li] ?an ^? ( ! geometric solid4  ,!re
 is1 h[ev]1 e reason = ref]r+ to !      
correspond+ proposition ( plane geometry
 :5 any giv5 proposition ( solid        
geometry is "u consid]ation1 & frequ5t  
ref]5ces ( ?is k9d w be made 9 speak+ ( 
! propositions 9 ?is & ! two succe$+    
*apt]s4 ,su* ref]5ce has value 9 !      
app]ception ( ! v>i\s laws ( solid      
geometry1 & x also adds an int]e/ to !  
subject & creates "s approa* to p[] 9 ! 
discov]y ( new facts 9 relation to      
figures ( ?ree dim5sions4  ,!           
introduction to solid geometry %\ld be  
made sl[ly4 ,! pupil has be5 accu/om$ to
 see+ only plane figures1 & !re=e !     
draw+ ( a solid figure 9 ! flat is      
confus+4 ,! be/ way = ! tea*] to        
anticipate ?is difficulty is to h a few 
pieces ( c>dbo>d1 a few knitt+ ne$les   
fil$ to %>p po9ts1 a p9e bo>d ab\t a    

foot squ>e1 & "s small corks4 ,) !      
c>dbo>d he c illu/rate planes1 :e!r     
alone1 int]sect+ obliquely or at "r     
angles1 or p>allel1 & he c easily       
illu/rate ! figures giv5 9 ! textbook 9 
use4 ,!re are models ( ?is k9d = sale1 b
 ! simple ones made 9 a few seconds by !
 tea*] or ! pupil h mu* m mean+4 ,!     
knitt+ ne$les may be /uck 9 ! bo>d to   
illu/rate p]p5dicul> or oblique l9es1 & 
if two or m are to meet 9 a po9t1 !y may
 be held toge!r by /ick+ !m 9 "o ( !    
small corks4 ,su* homely app>atus1 co/+ 
almo/ no?+1 to be put toge!r 9 class1   
seems mu* m real & is mu* m satisfactory
 ?an ! ,g]man models4#hg  ,an ext5sive  
use ( models is1 h[ev]1 unwise4 ,! pupil
 mu/ le>n v e>ly h[ to visualize a solid
 f ! flat outl9e picture1 j z a build]  
or a me*anic le>ns to read 8 work+      
draw+s4 ,to h a model = ea* proposition1
 or ev5 to h a photograph or a          
/]eoscopic picture1 is a v poor         
educational policy4 ,a textbook may     
prop]ly illu/rate a few propositions by 

photographic aids1 b aft] t ! pupil %\ld
 use ! k9d ( figures t he mu/ meet 9 8  
ma!matical "w4 ,a *ild %\ld n be kept 9 
a p]ambulator all 8 life1--he mu/ le>n  
to walk if he is to be /rong & gr[ to   
maturity2 & x is s ) a pupil 9 ! use (  
models 9 solid geometry4#hh  ,! case is 
some:at simil> ) respect to color$      
crayons4 ,!y h _! value & _! prop]      
place1 b !y also h _! /rict limitations4
 ,x is difficult to keep _! use )9      
b\nds2 pupils come to use !m to make    
pleas+ pictures1 & tea*]s unconsci\sly  
fall into ! same habit4 ,! value (      
color$ crayons is two-fold3 7#a7 !y     
sometimes make two planes /& out m      
cle>ly1 or !y s]ve to diff]5tiate "s l9e
 t is "u consid]ation f o!rs t are n2   
7#b7 !y enable a class to foll[ a       
demon/ration m easily by he>+ ( ! r$    
plane p]p5dicul> to ! blue "o1 in/ead ( 
! plane ,,mn p]p5dicul> to ! plane ,,pq4
 ,b x %\ld always be borne 9 m9d t 9    
practical "w we d n h color$ ink or     
color$ p5cils commonly at h&1 nor d we  

g5]ally h color$ crayons4 ,pupils %\ld  
!re=e become accu/om$ to ! p5cil & !    
:ite crayon z ! regulation tools1 & 9   
g5]al !y %\ld use !m4 ,! figures may n  
be z /rik+1 b !y are m quickly made & !y
 are m practical4  ,! def9ition ( plane 
has already be5 discuss$ 9 ,*apt] ,,xii1
 & ! o!r def9itions ( ,book ,,vi are n (
 en\gh int]e/ to call = special rem>k4  
,! axioms are ! same z 9 plane geometry1
 b !re is at lea/ "o po/ulate t ne$s to 
be add$1 al?\gh x w\ld be possible to   
/ate v>i\s analogues ( ! po/ulates (    
plane geometry if we c>$ unnecess>ily to
 enl>ge ! numb]4  ,! mo/ important      
po/ulate ( solid geometry is z foll[s3  
,"o plane1 & only "o1 c be pass$ ?r\gh  
two int]sect+ /raight l9es4 ,?is is     
easily illu/rat$1 z 9 mo/ textbooks1 z  
also are ?ree important coroll>ies d]iv$
 f x3  #a4 ,a /raight l9e & a po9t n 9 !
 l9e det]m9e a plane4 ,( c\rse ?is may  
be made ! po/ulate1 z may also ! next   
"o1 ! po/ulate be+ plac$ among !        
coroll>ies1 b ! arrangem5t "h adopt$ is 

probably ! mo/ satisfactory =           
educational purposes4  #b4 ,?ree po9ts n
 9 a /raight l9e det]m9e a plane4 ,!    
common "q z to :y a ?ree-legg$ /ool /&s 
firmly1 :ile a f\r-legg$ table (t5 does 
n1 w add "s int]e/ at ?is po9t4  #c4    
,two p>allel l9es det]m9e a plane4 ,?is 
requires a slight b in=mal pro( to %[ t 
x prop]ly foll[s z a coroll>y f !       
po/ulate1 b a s+le s5t5ce suffices4     
,:ile /udy+ ?is book que/ions ( ! foll[+
 nature may arise ) an advanc$ class1 or
 may be sugge/$ to ^? :o h _h high]     
algebra3  ,h[ _m /raight l9es are 9     
g5]al 7t is1 at ! mo/7 det]m9$ by n     
po9ts 9 space8 ,two po9ts det]m9e #a    
l9e1 a ?ird po9t adds 79 g5]al1 9 all ^!
 cases7 #b m1 a f\r? po9t adds #c m1 &  
an n? po9t n - #a m4 ,h5ce ! maximum is 
#a  #b  #c  444  7n - #a71 or           
n7n-#a7_/b1 :i* ! pupil w und]/& if he  
has /udi$ ari?metical progression4 ,!   
maximum numb] ( int]section po9ts ( n   
/raight l9es 9 ! same plane is also n7n 
- #a7_/b4  ,h[ _m /raight l9es are 9    

g5]al det]m9$ by n planes8 ,! answ] is !
 same1 n7n - #a7_/b4  ,h[ _m planes are 
9 g5]al det]m9$ by n po9ts 9 space8 ,"h 
! answ] is #a  #c  #f  #aj  444  7n -   
#b77n - #a7_/b1 or n7n - #a77n - #b7_/7a
 x #b x #c74 ,! same numb] ( po9ts is   
det]m9$ by n planes4  ,,!orem4 ,if two  
planes cut ea* o!r1 _! int]section is a 
/raight l9e4  ,among ! simple           
illu/rations are ! back edges ( ! pages 
( a book1 ! corn]s ( ! room1 & ! simple 
te/ z to :e!r ! edge ( a c>d is /raight 
by te/+ x on a plane4 ,x is well to call
 att5tion to ! fact t if two int]sect+  
/raight l9es move p>allel to _! orig9al 
position1 & s t _! int]section re/s on a
 /raight l9e n 9 ! plane ( ^? l9es1 !   
figure g5]at$ w be t ( ?is proposition4 
,9 g5]al1 if we cut ?r\gh any figure (  
solid geometry 9 "s p>ticul> way1 we are
 liable to get ! figure ( a proposition 
9 plane geometry1 z w frequ5tly be se54 
,,!orem4 ,if a /raight l9e is p]p5dicul>
 to ea* ( two o!r /raight l9es at _!    
po9t ( int]section1 x is p]p5dicul> to !

 plane ( ! two l9es4  ,if /ud5ts h      
tr\ble 9 visualiz+ ! figure 9 ?ree      
dim5sions1 "s knitt+ ne$les ?r\gh a     
piece ( c>dbo>d w make x cle>4 ,tea*]s  
%\ld call att5tion to ! simple device = 
det]m9+ if a rod is p]p5dicul> to a bo>d
 7or a pipe to a floor1 ceil+1 or wall71
 by te/+ x twice1 only1 ) a c>p5t]'s    
squ>e4 ,simil>ly1 x may be ask$ ( a     
class1 ,h[ %all we te/ to see if ! corn]
 7l9e7 ( a room is p]p5dicul> to !      
floor1 or if ! edge ( a box is          
p]p5dicul> to "o ( ! sides8  ,9 "s      
elem5t>y & 9 mo/ high] geometries !     
p]p5dicul> is call$ a normal to ! plane4
  ,,!orem4 ,all ! p]p5dicul>s t c be    
drawn to a /raight l9e at a giv5 po9t   
lie 9 a plane :i* is p]p5dicul> to ! l9e
 at ! giv5 po9t4  ,?us ! h&s ( a clock  
pass ?r\gh a plane z ! h&s revolve1 if  
!y are1 z is usual1 p]p5dicul> to !     
axis2 & ! same is true ( ! spokes ( a   
:eel1 & ( a /r+ ) a /one atta*$1 swung z
 rapidly z possible ab\t a boy's arm z  
an axis4 ,a clock p5dulum too sw+s 9 a  

plane1 z does ! lev] 9 a pair ( scales4 
,,!orem4 ,?r\gh a giv5 po9t )9 or )\t a 
plane !re c be "o p]p5dicul> to a giv5  
plane1 & only "o4  ,?is !orem is bett]  
/at$ to a class z two !orems4  ,?us a   
plumb l9e hang+ f a po9t 9 ! ceil+1 )\t 
sw++1 det]m9es "o def9ite po9t 9 !      
floor2 &1 conv]sely1 if x t\*es a giv5  
po9t 9 ! floor1 x mu/ hang f "o def9ite 
po9t 9 ! ceil+4 ,x %\ld be notic$ t if  
we cut ?r\gh ?is figure1 on ! p]p5dicul>
 l9e1 we %all h ! figure ( ! correspond+
 proposition 9 plane geometry1 namely1 t
 !re c be1 "u simil> circum/ances1 only 
"o p]p5dicul> to a l9e4  ,,!orem4       
,oblique l9es drawn f a po9t to a plane1
 meet+ ! plane at equal di/ances f !    
foot ( ! p]p5dicul>1 are equal1 etc4    
,!re is no objection to speak+ ( a "r   
circul> cone 9 connection ) ?is         
proposition1 & say+ t ! slant height is 
?us prov$ to be con/ant4 ,! usual       
coroll>y1 t if ! obliques are equal !y  
meet ! plane 9 a circle1 (f]s a new plan
 ( draw+ a circle4 ,a plumb l9e t is a  

little too long to rea* ! floor w1 if   
swung s z j to t\* ! floor1 describe a  
circle4 ,a #aj-;foot pole /&+ 9 a       
#i-;foot room w1 if x moves s z to t\*  
con/antly a fix$ po9t on ei!r ! floor or
 ! ceil+1 describe a circle on ! ceil+  
or floor respectively4  ,"o ( !         
coroll>ies /ates t ! locus ( po9ts 9    
space equidi/ant f ! extremities ( a    
/raight l9e is ! plane p]p5dicul> to ?is
 l9e at its middle po9t4 ,?is has be5   
tak5 by "s writ]s z ! def9ition ( a     
plane1 b x is too ab/ract to be usable4 
,x is advisable to cut ?r\gh ! figure   
along ! giv5 /raight l9e1 & see t we    
come back to ! correspond+ proposition 9
 plane geometry4  ,a good _m %ips h be5 
sav$ f be+ wreck$ by ! pr9ciple involv$ 
9 ?is proposition4  ,illu/ration        
,if a dang]\s %oal ,a is ne> a headl&   
,h1 ! angle      ,,hax is measur$ & is  
put d[n ^u ! *>ts z !      v]tical dang]
 angle4 ,%ips com+ ne> ! headl& are     
c>eful to keep f> en\gh away1 say at ,s1
 s t ! angle      ,,hsx %all be less ?an

 ?is dang] angle4 ,!y are !n sure      t
 !y w avoid ! dang]\s %oal4  ,relat$ to 
?is proposition is ! problem ( support+ 
a tall iron smoke/ack by wire /ays4     
,evid5tly ?ree /ays are ne$$1 & !y are  
pref]ably plac$ at ! v]tices ( an       
equilat]al triangle1 ! smoke/ack be+ 9 !
 c5t]4 ,! practical problem may be giv5 
( locat+ ! v]tices ( ! triangle & ( f9d+
 ! l5g? ( ea* /ay4  ,,!orem4 ,two       
/raight l9es p]p5dicul> to ! same plane 
are p>allel4  ,"h aga9 we may cut ?r\gh 
! figure by ! plane ( ! two p>allels1 & 
we get ! figure ( plane geometry relat+ 
to l9es t are p]p5dicul> to ! same l9e4 
,! proposition %[s t ! opposite corn]s (
 a room are p>allel1 & t !re=e !y lie 9 
! same plane1 or are coplan>1 z is said 
9 high] geometry4  ,x is int]e/+ to a   
class to h att5tion call$ to ! coroll>y 
t if two /raight l9es are p>allel to a  
?ird /raight l9e1 !y are p>allel to ea* 
o!r2 & to h ! "q ask$ :y x is necess>y  
to prove ?is :5 ! same ?+ 0 prov$ 9     
plane geometry4 ,9 case ! reason is n   

cle>1 let "s /ud5t try to apply ! pro(  
us$ 9 plane geometry4  ,,!orem4 ,two    
planes p]p5dicul> to ! same /raight l9e 
are p>allel4  ,besides call+ att5tion to
 ! correspond+ proposition ( plane      
geometry1 x is well n[ to speak ( ! fact
 t 9 propositions involv+ planes & l9es 
we may (t5 int]*ange ^! words4 ,=       
example1 us+ l9e = /raight l9e1 =       
brevity1 we h3      ,"o l9e does n      
det]m9e       ,"o plane does n det]m9e  
a plane4                          a l9e4
      ,two int]sect+ l9es               
,two int]sect+ planes det]m9e           
det]m9e a plane4                a l9e4  
,two l9es p]p5dicul> to        ,two     
planes p]p5dicul> to     a plane are    
p>allel4             a l9e are p>allel4 
,if "o ( two p>allel l9es      ,if "o ( 
two p>allel planes     is p]p5dicul> to 
a plane1 !  is p]p5dicul> to a l9e1 !   
o!r is also p]p5dicul> to      o!r is   
also p]p5dicul> to     ! plane4         
! l9e4      ,if two l9es are p>allel1 e 
,if two planes are p>allel1     plane   

conta9+ "o ( !       e l9e 9 "o ( !     
planes     l9es is p>allel to ! o!r     
is p>allel to ! o!r plane4     l9e4     
,,!orem4 ,! int]sections ( two p>allel  
planes by a ?ird plane are p>allel l9es4
  ,?us "o ( ! edges ( a box is p>allel  
to ! next succe$+ edge if ! opposite    
faces are p>allel1 & 9 saw+ diagonally  
?r\gh an ord9>y bo>d 7) rectangul> cross
 section7 ! section is a p>allelogram4  
,,!orem4 ,a /raight l9e p]p5dicul> to "o
 ( two p>allel planes is p]p5dicul> to !
 o!r also4  ,notice 7#a7 ! correspond+  
proposition 9 plane geometry2 7#b7 !    
proposition t results f int]*ang+ plane 
& 7/raight7 l9e4  ,,!orem4 ,if two      
int]sect+ /raight l9es are ea* p>allel  
to a plane1 ! plane ( ^! l9es is p>allel
 to t plane4  ,int]*ang+ plane &        
7/raight7 l9e1 we h3 ,if two int]sect+  
planes are ea* p>allel to a l9e1 ! l9e (
 7int]section (7 ^! planes is p>allel to
 t l9e4 ,is ?is true8  ,,!orem4 ,if two 
angles n 9 ! same plane h _! sides      
respectively p>allel & ly+ on ! same    

side ( ! /raight l9e jo9+ _! v]tices1 !y
 are equal & _! planes are p>allel4     
,que/ions l ! foll[+ may be ask$ 9      
connection ) ! proposition3 ,:at is !   
correspond+ proposition 9 plane         
geometry8 ,:y d we ne$ ano!r pro( "h8   
,try ! plane-geometry pro( "h4  ,,!orem4
 ,if two /raight l9es are cut by ?ree   
p>allel planes1 _! correspond+ segm5ts  
are proportional4  ,"h1 aga91 x is      
desirable to ask = ! correspond+        
proposition ( plane geometry1 & to ask  
:y ! pro( ( t proposition w n suffice = 
?is "o4 ,! usual figure may be v>i$ 9 an
 int]e/+ mann] by hav+ ! two l9es meet  
on "o ( ! planes1 or outside ! planes1  
or by hav+ !m p>allel1 9 :i* cases !    
pro( ( ! plane-geometry proposition     
holds "h4 ,?is proposition is n ( great 
importance f ! practical /&po9t1 & x is 
omitt$ f "s ( ! /&>d syllabi at pres5t1 
al?\gh includ$ 9 c]ta9 o!rs4 ,x is easy1
 h[ev]1 to frame "s int]e/+ que/ions    
dep5d+ ^u x = _! answ]s1 su* z ! foll[+3
 ,9 a gymnasium swimm+ tank ! wat] is #d

 feet deep & ! ceil+ is #h feet above ! 
surface ( ! wat]4 ,a pole #ae feet long 
t\*es ! ceil+ & ! bottom ( ! tank4      
,requir$ to "k :at l5g? ( ! pole is 9 ! 
wat]4  ,at ?is po9t 9 ,book ,,vi x is   
cu/om>y to introduce ! dih$ral angle4 ,!
 ^w dih$ral is f ! ,greek1 di- mean+    
two1 & h$ra mean+ seat4 ,we h ! root    
h$ra also 9 trih$ral 7?ree-seat$71      
polyh$ral 7_m-seat$71 & ca!dral 7a *ur* 
hav+ a bi%op's seat74 ,! ^w is also1 b  
less prop]ly1 spell$ )\t ! h1 di$ral1 a 
spell+ n favor$ by mod]n usage4 ,x is n 
necess>y to dwell at l5g? ^u ! dih$ral  
angle1 except to %[ ! analogy betwe5 x &
 ! plane angle4 ,a few illu/rations1 z (
 an op5 book1 ! wall & floor ( a room1 &
 a sw++ door1 s]ve to make ! concept    
cle>1 :ile a plane at "r angles to !    
edge %[s ! measur+ plane angle4 ,s      
manife/ is ?is relation%ip betwe5 !     
dih$ral angle & its measur+ plane angle 
t "s tea*]s omit ! proposition t two    
dih$ral angles h ! same ratio z _! plane
 angles4  ,,!orem4 ,if two planes are   

p]p5dicul> to ea* o!r1 a /raight l9e    
drawn 9 "o ( !m p]p5dicul> to _!        
int]section is p]p5dicul> to ! o!r4     
,?is & ! relat$ propositions all[ (     
num]\s illu/rations tak5 f ! s*oolroom1 
z ( door edges be+ p]p5dicul> to !      
floor4 ,! pret5d$ applications ( ^!     
propositions are usually fictiti\s1 & ! 
propositions are ( value *iefly = _! own
 int]e/ & because !y are ne$$ 9         
subsequ5t pro(s4  ,,!orem4 ,! locus ( a 
po9t equidi/ant f ! faces ( a dih$ral   
angle is ! plane bisect+ ! angle4  ,by  
*ang+ plane to l9e1 & by mak+ o!r obvi\s
 *anges to correspond1 ?is r$uces to !  
analog\s proposition ( plane geometry4  
,! figure =m$ by ! plane p]p5dicul> to !
 edge is also ! figure ( t analog\s     
proposition4 ,?is at once sugge/s t !re 
are two planes 9 ! locus1 provid$ !     
planes ( ! dih$ral angle are tak5 z     
indef9ite 9 ext5t1 & t ^! planes are    
p]p5dicul> to ea* o!r4 ,x may int]e/ "s 
( ! pupils to draw ?is g5]al figure1    
analog\s to ! "o 9 plane geometry4      

,,!orem4 ,! projection ( a /raight l9e n
 p]p5dicul> to a plane ^u t plane is a  
/raight l9e4  ,9 high] ma!matics x w\ld 
simply be said t ! projection is a      
/raight l9e1 ! special case ( !         
projection ( a p]p5dicul> be+ consid]$ z
 a l9e-segm5t ( z]o l5g?4 ,!re is no    
advantage1 h[ev]1 ( br++ 9 z]o & inf9ity
 9 ! c\rse 9 elem5t>y geometry4 ,!      
legitimate reason = ! mod]n use ( ^!    
t]ms is seldom und]/ood by beg9n]s4     
,?is subject ( projection 7,lat9 pro-1  
=?1 & jac]e1 to ?r[7 is ext5sively us$ 9
 mod]n ma!matics & also 9 ! elem5t>y "w 
( ! draftsman1 & x w be ref]r$ to a     
little lat]4 ,at ?is "t1 h[ev]1 x is    
well to call att5tion to ! fact t !     
projection ( a /raight l9e on a plane is
 a /raight l9e or a po9t2 ! projection (
 a curve may be a curve or x may be     
/raight2 ! projection ( a po9t is a     
po9t2 & ! projection ( a plane 7:i* is  
easily und]/ood )\t def9+ x7 may be a   
surface or x may be a /raight l9e4 ,an  
artisan repres5ts a solid by draw+ its  

projection ^u two planes at "r angles to
 ea* o!r1 & a map mak] 7c>tograph]7     
repres5ts ! surface ( ! e>? by project+ 
x ^u a plane4 ,a photograph ( ! class is
 m]ely ! projection ( ! class ^u a      
photographic plate 7plane71 & :5 we draw
 a figure 9 solid geometry1 we m]ely    
project ! solid ^u ! plane ( ! pap]4    
,!re are o!r projections ?an ^? =m$ by  
l9es t are p]p5dicul> to ! plane4 ,!    
l9es may be oblique to ! plane1 & ?is is
 ! case ) mo/ projections4 ,a           
photograph1 = example1 is n =m$ by l9es 
p]p5dicul> to a plane1 = !y all conv]ge 
9 ! cam]a4 ,if ! l9es ( projection are  
all p]p5dicul> to ! plane1 ! projection 
is said to be or?ographic1 f ! ,greek   
or?o- 7/raight7 & graphe9 7to draw74 ,a 
good example ( or?ographic projection   
may be se5 9 ! %ad[ ca/ by an object ^u 
a piece ( pap] t is held p]p5dicul> to !
 sun's rays4 ,a good example ( oblique  
projection is a %ad[ on ! floor ( !     
s*oolroom4  ,,!orem4 ,betwe5 two /raight
 l9es n 9 ! same plane !re c be "o      

common p]p5dicul>1 & only "o4  ,! usual 
coroll>y /ates t ?is p]p5dicul> is !    
%orte/ l9e jo9+ !m4 ,x is int]e/+ to    
comp>e ?is ) ! case ( two l9es 9 ! same 
plane4 ,if !y are p>allel1 !re may be   
any numb] ( common p]p5dicul>s4 ,if !y  
int]sect1 !re is /ill a common          
p]p5dicul>1 b ?is c h>dly be said to be 
betwe5 !m1 except = its z]o segm5t4     
,!re are _m simple illu/rations ( ?is   
case4 ,= example1 :at is ! %orte/ l9e   
betwe5 any giv5 edge ( ! ceil+ & ! v>i\s
 edges ( ! floor ( ! s*oolroom8 ,if two 
gall]ies 9 a m9e are to be connect$ by  
an air %aft1 h[ %all x be plann$ s z to 
save labor8 ,make a draw+ ( ! plan4  ,at
 ?is po9t ! polyh$ral angle is          
introduc$4 ,! ^w is f ! ,greek polys    
7_m7 & h$ra 7seat74 ,/ud5ts h m         
difficulty 9 grasp+ ! mean+ ( ! size ( a
 polyh$ral angle ?an is ! case ) dih$ral
 & plane angles4 ,= ?is reason x is n   
good policy to dwell mu* ^u ?is subject 
unless ! "q arises1 s9ce x is bett]     
und]/ood :5 ! relation ( ! polyh$ral    

angle & ! sph]ical polygon is met4      
,tea*]s w naturally see t j z we may    
measure ! plane angle by tak+ ! ratio ( 
an arc to ! :ole circle1 & ( a dih$ral  
angle by tak+ ! ratio ( t "p ( !        
cyl9dric surface t is cut out by !      
planes to ! :ole surface1 s we may      
measure a polyh$ral angle by tak+ !     
ratio ( ! sph]ical polygon to ! :ole    
sph]ical surface4 ,x %\ld also be obs]v$
 t j z we may h cross polygons 9 a      
plane1 s we may h sph]ical polygons t   
are simil>ly tangl$1 & t to ^! w        
correspond polyh$ral angles t are also  
cross1 _! repres5tation by draw+s be+   
too complicat$ = class use4  ,! idea (  
symmetric solids may be illu/rat$ by a  
pair ( gloves1 all _! p>ts be+ mutually 
equal b arrang$ 9 opposite ord]4 ,our   
h&s1 feet1 & e>s af=d o!r illu/rations (
 symmetric solids4  ,,!orem4 ,! sum ( ! 
face angles ( any convex polyh$ral angle
 is less ?an f\r "r angles4  ,!re are   
sev]al int]e/+ po9ts ( discussion 9     
connection ) ?is proposition4 ,=        

example1 suppose ! v]tex ,v to approa* !
 plane t cuts ! edges 9 ,a1 ,b1 ,c1 ,d1 
4441 ! edges cont9u+ to pass ?r\gh ^! z 
fix$ po9ts4 ,! sum ( ! angles ab\t ,v   
approa*es :at limit8 ,on ! o!r h&1      
suppose ,v rec$es indef9itely2 !n ! sum 
approa*es :at limit8 ,!n :at are ! two  
limits ( ?is sum8 ,suppose ! polyh$ral  
angle 7 concave1 :y w\ld ! pro( n hold8 
,,footnotes3  #hg ,^! may be pur*as$    
?r\gh ! ,leipzig] ,lehrmittelan/alt1    
,leipzig1 ,g]many1 :i* w s5d catalogues 
to int5d+ buy]s4  #hh ,an excell5t set (
 /]eoscopic views ( ! figures ( solid   
geometry1 prep>$ by ,e4 ,m4 ,langley (  
,b$=d1 ,engl&1 is publi%$ by ,und]wood  
,und]wood1 ,new ,york4 ,su* a set may   
prop]ly h place 9 a s*ool libr>y or 9 a 
classroom 9 geometry1 to be us$ :5 x    
seems advantage\s4     ,,*apt] ,,xx  ,,!
 ,,lead+ ,,propositions ,,( ,,book ,,vii
   ,book ,,vii relates to polyh$rons1   
cyl9d]s1 & cones4 ,x op5s ) ! necess>y  
def9itions relat+ to polyh$rons1 !      
etymology ( ! t]ms (t5 prov+ int]e/+ &  

valuable :5 br\ght into ! "w incid5tally
 by ! tea*]4 ,polyh$ron is f ! ,greek   
polys 7_m7 & h$ra 7seat74 ,! ,greek     
plural1 polyh$ra1 is us$ 9 e>ly ,engli% 
works1 b polyh$rons is ! =m n[ m        
commonly se5 9 ,am]ica4 ,prism is f !   
,greek prisma 7some?+ saw$1 l a piece ( 
wood saw$ f a beam74 ,lat]al is f !     
,lat9 latus 7side74 ,p>allelepip$ is f !
 ,greek p>allelos 7p>allel7 & epip$on 7a
 plane surface71 f epi 7on7 & p$on      
7gr\nd74 ,by analogy to p>allelogram !  
^w is (t5 spell$ p>allelopip$1 b ! be/  
ma!matical works n[ adopt ! etymological
 spell+ above giv54 ,truncate is f !    
,lat9 trunc>e 7to cut (f74  ,a few ( !  
lead+ propositions are n[ consid]$4     
,,!orem4 ,! lat]al area ( a prism is    
equal to ! product ( a lat]al edge by ! 
p]imet] ( ! "r section4  ,x %\ld be not$
 t al?\gh "s syllabi d n give !         
proposition t p>allel sections are      
congru5t1 ?is is necess>y = ?is         
proposition1 because x %[s t ! "r       
sections are all congru5t & h5ce t any  

"o ( !m may be tak54  ,x is1 ( c\rse1   
possible to con/ruct a prism s oblique &
 s l[ t a "r section1 t is1 a section   
cutt+ all ! lat]al edges at "r angles1  
is impossible4 ,9 ?is case ! lat]al     
faces mu/ be ext5d$1 ?us =m+ :at is     
call$ a prismatic space4 ,?is t]m may or
 may n be introduc$1 dep5d+ ^u ! nature 
( ! class4  ,?is proposition is "o ( !  
mo/ important 9 ,book ,,vii1 because x  
is ! basis ( ! m5suration ( ! cyl9d] z  
well z ! prism4 ,practical applications 
are easily sugge/$ 9 connection ) beams1
 corridors1 & prismatic columns1 su* z  
are (t5 se5 9 s*ool build+s4 ,mo/       
geometries supply suffici5t mat]ial 9   
?is l9e1 h[ev]4  ,,!orem4 ,an oblique   
prism is equival5t to a "r prism ^: base
 is equal to a "r section ( ! oblique   
prism1 & ^: altitude is equal to a      
lat]al edge ( ! oblique prism4  ,?is is 
a fundam5tal !orem lead+ up to !        
m5suration ( ! prism4 ,att5tion %\ld be 
call$ to ! analog\s proposition 9 plane 
geometry relat+ to ! area ( !           

p>allelogram & rectangle1 & to ! fact t 
if we cut ?r\gh ! solid figure by a     
plane p>allel to "o ( ! lat]al edges1 ! 
result+ figure w be t ( ! proposition   
m5tion$4 ,z 9 ! prec$+ proposition1 s 9 
?is case1 !re may be a "q rais$ t w make
 x helpful to introduce ! idea (        
prismatic space4  ,,!orem4 ,! opposite  
lat]al faces ( a p>allelepip$ are       
congru5t & p>allel4  ,x is desirable to 
ref] to ! correspond+ case 9 plane      
geometry1 & to note aga9 t ! figure is  
obta9$ by pass+ a plane ?r\gh !         
p>allelepip$ p>allel to a lat]al edge4  
,! same may be said = ! proposition ab\t
 ! diagonal plane ( a p>allelepip$4 ,^! 
two propositions are fundam5tal 9 !     
m5suration ( ! prism4  ,,!orem4 ,two    
rectangul> p>allelepip$s are to ea* o!r 
z ! products ( _! ?ree dim5sions4  ,?is 
leads at once to ! coroll>y t ! volume (
 a rectangul> p>allelepip$ equals !     
product ( its ?ree dim5sions1 !         
fundam5tal law 9 ! m5suration ( all     
solids4 ,x is prec$$ by ! proposition   

ass]t+ t rectangul> p>allelepip$s hav+  
congru5t bases are proportional to _!   
altitudes4 ,?is includes !              
incomm5surable case1 b ?is case may be  
omitt$4  ,! numb] ( simple applications 
( ?is proposition is practically        
unlimit$4 ,9 all su* cases x is         
advisable to take a consid]able numb] ( 
num]ical ex]cises 9 ord] to fix 9 m9d ! 
real nature ( ! proposition4 ,any good  
geometry furni%es a c]ta9 numb] ( ^!    
ex]cises4  ,! foll[+ is an int]e/+      
prop]ty ( ! rectangul> p>allelepip$1 (t5
 call$ ! rectangul> solid3       ,if !  
edges are a1 b1 & c1 & ! diagonal is d1 
!n 7a_/d7#b  7b_/d7#b  7c_/d7#b  #a4    
,?is prop]ty      is easily prov$ by !  
,py?agorean ,!orem1 =      d#b  a#b  b#b
  c#b1 :5ce      7a#b  b#b  c#b7 _/ d#b 
#a4       ,9 case c  #j1 ?is r$uces to !
 ,py?agorean ,!orem4 ,!      prop]ty is 
! fundam5tal "o ( solid analytic        
geometry4  ,,!orem4 ,! volume ( any     
p>allelepip$ is equal to ! product ( its
 base by its altitude4  ,?is is "o ( !  

few propositions 9 ,book ,,vii :]e a    
model is ( any advantage4 ,x is easy to 
make "o out ( pa/ebo>d1 or to cut "o f  
wood4 ,if a wood5 "o is made1 x is      
advisable to take an oblique            
p>allelepip$ &1 by prop]ly saw+ x1 to   
trans=m x into a rectangul> "o in/ead ( 
us+ ?ree diff]5t solids4  ,on acc\nt (  
its awkw>d =m1 ?is figure is sometimes  
call$ ! ,devil's ,c(f91 b x is a "n t x 
w\ld be well n to p]petuate4  ,,!orem4  
,! volume ( any prism is equal to !     
product ( its base by its altitude4     
,?is is also "o ( ! basal propositions (
 solid geometry1 & x has _m applications
 9 practical m5suration4 ,a fir/-class  
textbook w give a suffici5t li/ (       
problems involv+ num]ical measurem5t1 to
 fix ! law 9 m9d4 ,= outdoor "w1 involv+
 measurem5ts ne> ! s*ool or )9 ! k ( !  
pupils1 ! foll[+ problem is a type3     
,illu/ration       ,if ?is repres5ts !  
cross section ( a railway embankm5t     
t is l feet long1 h feet high1 b feet   
wide at !      bottom1 & b' feet wide at

 ! top1 f9d ! numb] ( cubic      feet 9 
! embankm5t4 ,f9d ! volume if l  #cjj1 h
  #h1      b  #fj1 & b'  #bh4  ,!       
m5suration ( ! volume ( ! prism1 includ+
 ! rectangul> p>allelepip$ & cube1 0    
kn[n to ! anci5ts4 ,euclid 0 n conc]n$ )
 practical measurem5t1 s t none ( ?is "p
 ( geometry appe>s 9 8 ,elem5ts4 ,we    
f9d1 h[ev]1 9 ! papyrus ( ,ahmes1       
directions = ! measur+ ( b9s1 & !       
,egyptian build]s1 long be=e 8 "t1 mu/ h
 kn[n ! m5suration ( ! rectangul>       
p>allelepip$4 ,among ! ,h9dus1 long be=e
 ! ,*ri/ian era1 rules 7 kn[n = !       
con/ruction ( alt>s1 & among ! ,greeks !
 problem ( con/ruct+ a cube ) twice !   
volume ( a giv5 cube 7! duplication ( ! 
cube7 0 attack$ by _m ma!maticians4 ,!  
solution ( ?is problem is impossible by 
elem5t>y geometry4       ,if e equals ! 
edge ( ! giv5 cube1 !n e#c is its       
volume & #b;e#c is ! volume ( ! requir$ 
cube4 ,!re=e      ! edge ( ! requir$    
cube is e#croot#b4 ,n[ if e is          
giv51 x is n possible ) ! /raight$ge &  

compasses      to con/ruct a l9e equal  
to e#croot#b1 al?\gh x is easy      to  
con/ruct "o equal to esqrt#b4  ,! /udy (
 ! pyramid beg9s at ?is po9t4 ,9        
practical measurem5t we usually meet !  
regul> pyramid4 ,x is1 h[ev]1 a simple  
matt] to consid] ! oblique pyramid z    
well1 & 9 measur+ volumes we sometimes  
f9d ^! =ms4  ,,!orem4 ,! lat]al area ( a
 regul> pyramid is equal to half !      
product ( its slant height by ! p]imet] 
( its base4  ,?is leads to ! coroll>y   
conc]n+ ! lat]al area ( ! fru/um ( a    
regul> pyramid4 ,x %\ld be notic$ t !   
regul> pyramid may be consid]$ z a      
fru/um ) ! upp] base z]o1 & !           
proposition z a special case "u !       
coroll>y4 ,x is also possible1 if we    
*oose1 to let ! upp] base ( ! fru/um    
pass ?r\gh ! v]tex & cut ! lat]al edges 
above t po9t1 al?\gh ?is is too         
complicat$ = mo/ pupils4 ,if ?is case is
 consid]$1 x is well to br+ 9 ! g5]al   
idea ( pyramidal space1 ! inf9ite space 
b\nd$ on sev]al sides by ! lat]al faces1

 ( ! pyramid4 ,?is pyramidal space is   
d\ble1 ext5d+ on two sides ( ! v]tex4   
,,!orem4 ,if a pyramid is cut by a plane
 p>allel to ! base3      #a4 ,! edges & 
altitude are divid$ proportionally4     
#b4 ,! section is a polygon simil> to ! 
base4  ,to get ! analog\s proposition ( 
plane geometry1 pass a plane ?r\gh !    
v]tex s z to cut ! base4 ,we %all !n h !
 sides & altitude ( ! triangle divid$   
proportionally1 & ( c\rse ! section w   
m]ely be a l9e-segm5t1 & !re=e x is     
simil> to ! base l9e4  ,! cutt+ plane   
may pass ?r\gh ! v]tex1 or x may cut !  
pyramidal space above ! v]tex4 ,9 ei!r  
case ! pro( is ess5tially ! same4       
,,!orem4 ,! volume ( a triangul> pyramid
 is equal to "o ?ird ( ! product ( its  
base by its altitude1 & ?is is also true
 ( any pyramid4  ,?is is /at$ z two     
!orems 9 all textbooks1 & prop]ly s4 ,x 
is expla9$ to *ildr5 :o are /udy+       
ari?metic by means ( a holl[ pyramid & a
 holl[ prism ( equal base & equal       
altitude4 ,! pyramid is fill$ ) s& or   

gra91 & ! cont5ts is p\r$ into ! prism4 
,?is is repeat$1 & aga9 repeat$1 %[+ t !
 volume ( ! prism is ?ree times ! volume
 ( ! pyramid4 ,x sometimes v>ies ! "w to
 %[ ?is to a class 9 geometry4  ,?is    
proposition 0 fir/ prov$1 s ,ar*im$es   
ass]ts1 by ,eudoxus ( ,cnidus1 fam\s z  
an a/ronom]1 geomet]1 physician1 &      
lawgiv]1 born 9 humble circum/ances ab\t
 #djg ,b4,c4 ,he /udi$ at ,a!ns & 9     
,egypt1 & f\nd$ a fam\s s*ool ( geometry
 at ,cyzicus4 ,8 discov]y also ext5d$ to
 ! volume ( ! cone1 & x 0 8 "w t gave ! 
beg9n+ to ! sci5ce ( /]eometry1 !       
m5suration "p ( solid geometry4         
,,!orem4 ,! volume ( ! fru/um ( any     
pyramid is equal to ! sum ( ! volumes ( 
?ree pyramids ^: common altitude is !   
altitude ( ! fru/um1 & ^: bases are !   
l[] base1 ! upp] base1 & ! mean         
proportional betwe5 ! bases ( ! fru/um4 
,att5tion %\ld be call$ to ! fact t ?is 
=mula v  #a_/c a7b  b'  sqrt7bb'77      
applies to ! pyramid by lett+ b'  #j1 to
 ! prism by lett+ b  b'1 & also to !    

p>allelepip$ & cube1 ^! be+ special =ms 
( ! prism4 ,?is =mula is1 !re=e1 a v    
g5]al "o1 relat+ to all ! polyh$rons t  
are commonly met 9 m5suration4  ,,!orem4
 ,!re _c be m ?an five regul> convex    
polyh$rons4  ,eudemus ( ,rhodes1 "o ( ! 
pr9cipal pupils ( ,ari/otle1 9 8 hi/ory 
( geometry ( :i* ,proclus pres]ves "s   
fragm5ts1 tells u t ,py?agoras discov]$ 
! con/ruction ( ! mundane figures1 mean+
 ! five regul> polyh$rons4 ,iambli*us   
speaks ( ! discov]y ( ! dodecah$ron 9 ^!
 words3       ,z to ,hippasus1 :o 0 a   
,py?agorean1 !y say t he      p]i%$ 9 ! 
sea on acc\nt ( 8 impiety1 inasmu* z he 
boa/$ t he fir/ divulg$ ! k ( ! sph]e ) 
! twelve p5tagons4 ,hippasus assum$ !   
glory ( !      discov]y to himself1     
:]eas ev]y?+ belongs to ,him1 =      ?us
 !y designate ,py?agoras1 & d n call    
,him by "n4  ,iambli*us "h ref]s to !   
dodecah$ron inscrib$ 9 ! sph]e4 ,!      
,py?agoreans look$ ^u ^! five solids z  
fundam5tal =ms 9 ! /ructure ( ! univ]se4
 ,9 p>ticul> ,plato tells u t !y ass]t$ 

t ! f\r elem5ts ( ! real _w 7 !         
tetrah$ron1 octah$ron1 icosah$ron1 &    
cube1 & ,plut>* ascribes ?is doctr9e to 
,py?agoras himself4 ,philolaus1 :o liv$ 
9 ! fif? c5tury ,b4,c41 held t !        
elem5t>y nature ( bodies dep5d$ on _!   
=m4 ,! tetrah$ron 0 assign$ to fire1 !  
octah$ron to air1 ! icosah$ron to wat]1 
& ! cube to e>?1 x be+ ass]t$ t !       
smalle/ con/itu5t "p ( ea* ( ^!         
sub/ances _h ! =m "h assign$ to x4      
,al?\gh ,eudemus attributes all five to 
,py?agoras1 x is c]ta9 t ! tetrah$ron1  
cube1 & octah$ron 7 kn[n to !           
,egyptians1 s9ce !y appe> 9 _!          
ar*itectural decorations4 ,^! solids 7  
/udi$ s ext5sively 9 ! s*ool ( ,plato t 
,proclus also speaks ( !m z ! ,platonic 
bodies1 say+ t ,euclid propos$ to       
himself ! con/ruction ( ! s-call$       
,platonic bodies z ! f9al aim ( 8       
arrangem5t ( ! ',elem5ts4' ,ari/aeus1   
probably a little old] ?an ,euclid1     
wrote a book ^u ^! solids4  ,z an       
int]e/+ amplification ( ?is proposition1

 ! c5t]s ( ! faces 7squ>es7 ( a cube may
 be connect$1 an inscrib$ octah$ron be+ 
!reby =m$4 ,fur!rmore1 if ! v]tices ( ! 
cube are ,a1 ,b1 ,c1 ,d1 ,a'1 ,b'1 ,c'1 
,d'1 !n by draw+ ,,ac1 ,,cd'1 ,d',a1    
,d',b'1 ,b',a1 & ,b',c1 a regul>        
tetrah$ron w be =m$4 ,s9ce ! con/ruction
 ( ! cube is a simple matt]1 ?is %[s h[ 
?ree ( ! five regul> solids may be      
con/ruct$4 ,! actual con/ruction ( !    
solids is n suit$ to elem5t>y           
geometry4#hi  ,x is n difficult = a     
class to f9d ! relative areas ( ! cube &
 ! inscrib$ tetrah$ron & octah$ron4 ,if 
s is ! side ( ! cube1 ^! areas are      
#fs#b1 7#a_/b7s#bsqrt#c1 & s#bsqrt#c2 t 
is1 ! area ( ! octah$ron is twice t ( ! 
tetrah$ron inscrib$ 9 ! cube4  ,some:at 
relat$ to ! prec$+ p>agraph is ! fact t 
! edges ( ! five regul> solids are      
incomm5surable ) ! radius ( !           
circumscrib$ sph]e4 ,?is fact seems to h
 be5 kn[n to ! ,greeks1 p]haps to       
,!aetetus 7ca4 #djj ,b4,c47 & ,ari/aeus 
7ca4 #cjj ,b4,c471 bo? ( :om wrote on   

incomm5surables4  ,j z we may produce ! 
sides ( a regul> polygon & =m a regul>  
cross polygon or /ell> polygon1 s we may
 h /ell> polyh$rons4 ,kepl]1 ! great    
a/ronom]1 con/ruct$ "s ( ^! solids 9    
#afai1 & ,po9sot1 a ,fr5* ma!matician1  
c>ri$ ! con/ructions s f> 9 #ahja t     
sev]al ( ^! /ell> polyh$rons are kn[n z 
,po9sot solids4 ,!re is a v ext5sive    
lit]ature ^u ?is subject4  ,! foll[+    
table may be ( "s s]vice 9 assign+      
problems 9 m5suration 9 connection ) !  
regul> polyh$rons1 al?\gh "s ( ! =mulas 
are too difficult = beg9n]s to prove4 ,9
 ! table e  edge ( ! polyh$ron1 r       
radius ( circumscrib$ sph]e1 r'  radius 
( inscrib$ sph]e1 a  total area1 v      
volume4       ,,numb]                   
,,( ,,faces     #d                #f    
#h                                      
----------------------------------------
---------------     r    esqrt7#c_/h7   
7e_/#b7sqrt#c esqrt7#a_/b7              
r'   esqrt7#a_/bd7     e_/#b            
esqrt7#a_/f7                            

a     e#bsqrt#c       #f;e#b            
#b;e#bsqrt#c                            
v   7e#c_/ab7sqrt#b    e#c              
7e#c_/c7sqrt#b                          
----------------------------------------
------------------       ,,numb]        
,,( ,,faces             #ab             
#bj                                     
----------------------------------------
------------------------------     r    
7e_/#d7sqrt#c7sqrt#e  #a7               
esqrt77#e  sqrt#e7_/h7                  
r'  7e_/#b7sqrt77#be                    
#aasqrt#e7_/aj77;esqrt#c7_/ab7sqrt#e    
#c7                                     
a   #c;e#bsqrt7#e7e  #bsqrt#e77         
7#e;e#b7sqrt#c                          
v   77e#c7_/d77ae  #gsqrt#e7            
77#e;e#c7_/ab77sqrt#e  #c7              
----------------------------------------
--------------------------------  ,"s   
int]e/ is add$ to ! /udy ( polyh$rons by
 call+ att5tion to _! occurr5ce 9       
nature1 9 ! =m ( cry/als4 ,! computation
 ( ! surfaces & volumes ( ^! =ms (f]s an

 opportunity = apply+ ! rules (         
m5suration1 & ! con/ruction ( ! solids  
by pap] fold+ or by ! cutt+ ( crayon or 
"s o!r sub/ance (t5 ar\ses a consid]able
 int]e/4 ,! foll[+ are =ms ( cry/als t  
are occasionally f\nd3  ,illu/ration    
,!y %[ h[ ! cube is modifi$ by hav+ its 
corn]s cut (f4 ,a cube may be inscrib$ 9
 an octah$ron1 its v]tices be+ at !     
c5t]s ( ! faces ( ! octah$ron4 ,if we   
?9k ( ! cube z exp&+1 ! faces ( !       
octah$ron w cut (f ! corn]s ( ! cube z  
se5 9 ! fir/ figure1 leav+ ! cube z %[n 
9 ! second figure4 ,if ! corn]s are cut 
(f /ill m1 we h ! ?ird figure4          
,simil>ly1 an octah$ron may be inscrib$ 
9 a cube1 & by lett+ x exp& a little1 ! 
faces ( ! cube w cut (f ! corn]s ( !    
octah$ron4 ,?is is se5 9 ! foll[+       
figures3  ,illu/ration  ,?is is a =m t  
is f\nd 9 cry/als1 & ! computation ( !  
surface & volume is an int]e/+ ex]cise4 
,! qu>tz cry/al1 an hexagonal pyramid on
 an hexagonal prism1 is f\nd 9 _m p>ts (
 ! c\ntry1 or is to be se5 9 ! s*ool    

museum1 & ?is also =ms an int]e/+ object
 ( /udy 9 ?is connection4  ,! prop]ties 
( ! cyl9d] are next /udi$4 ,! ^w is f ! 
,greek kyl9dros1 f kylie9 7to roll74 ,9 
anci5t ma!matics circul> cyl9d]s 7 !    
only ones /udi$1 b s9ce "s ( ! prop]ties
 are z easily prov$ = ! case ( a        
noncircul> directrix1 x is n n[ cu/om>y 
to limit !m 9 ?is way4 ,x is conv5i5t to
 beg9 by a /udy ( ! cyl9dric surface1 & 
a piece ( pap] may be curv$ or roll$ up 
to illu/rate ?is concept4 ,if ! pap] is 
br\ght ar\nd s t ! edges meet1 :atev]   
curve may =m a cross section ! surface  
is said to inclose a cyl9dric space4    
,?is concept is sometimes conv5i5t1 b x 
ne$ be introduc$ only z necessity = us+ 
x arises4 ,! o!r def9itions conc]n+ !   
cyl9d] are s simple z to require no     
comm5t4  ,! m5suration ( ! volume ( a   
cyl9d] dep5ds ^u ! assumption t ! cyl9d]
 is ! limit ( a c]ta9 inscrib$ or       
circumscrib$ prism z ! numb] ( sides ( !
 base is indef9itely increas$4 ,x is    
possible to give a fairly satisfactory &

 simple pro( ( ?is fact1 b = pupils ( ! 
age ( beg9n]s 9 geometry 9 ,am]ica x is 
bett] to make ! assumption outright4    
,?is is "o ( sev]al cases 9 geometry :]e
 a pro( is less conv9c+ ?an ! assum$    
/atem5t4  ,,!orem4 ,! lat]al area ( a   
circul> cyl9d] is equal to ! product ( !
 p]imet] ( a "r section ( ! cyl9d] by an
 elem5t4  ,= practical purposes ! cyl9d]
 ( revolution 7"r circul> cyl9d]7 is !  
"o mo/ frequ5tly us$1 & ! important     
=mula is !re=e l  #bpirh :]e l  ! lat]al
 area1 r  ! radius1 & h  ! altitude4    
,applications ( ?is =mula are easily    
f\nd4  ,,!orem4 ,! volume ( a circul>   
cyl9d] is equal to ! product ( its base 
by its altitude4  ,"h aga9 ! important  
case is t ( ! cyl9d] ( revolution1 :]e v
  pir#b;h4  ,! numb] ( applications (   
?is proposition is1 ( c\rse1 v great4 ,9
 ar*itecture & 9 me*anics ! cyl9d] is   
con/antly se51 & ! m5suration ( !       
surface & ! volume is important4 ,a s+le
 illu/ration ( ?is type ( problem w     
suffice4       ,a ma*9i/ is mak+ a crank

 p9 7a k9d ( bolt7 = an      eng9e1     
accord+ to ?is draw+4 ,he consid]s x z  
weigh+      ! same z ?ree /eel cyl9d]s  
hav+ ! diamet]s &      l5g?s 9 in*es z  
"h %[n1 :]e #g #c_/d means #g #c_/d     
in*es4 ,he has ?is =mula = ! weight 7w7 
( a /eel      cyl9d] :]e d is ! diamet] 
& l is ! l5g?3      w  #j4jgpid#bl4     
,tak+ pi  #c #a_/g1 f9d ! weight (      
! p94  ,! mo/ elaborate /udy ( ! cyl9d]1
 cone1 & sph]e 7! ?ree r\nd bodies7 9 ! 
,greek lit]ature is t ( ,ar*im$es (     
,syracuse 7on ! isl& ( ,sicily71 :o liv$
 9 ! ?ird c5tury ,b4,c4 ,ar*im$es tells 
u1 h[ev]1 t ,eudoxus 7born ca4 #djg     
,b4,c47 discov]$ t any cone is "o ?ird (
 a cyl9d] ( ! same base & ! same        
altitude4 ,tradition says t ,ar*im$es   
reque/$ t a sph]e & a cyl9d] be c>v$ ^u 
8 tomb1 & t ?is 0 done4 ,cic]o relates t
 he discov]$ ! tomb by means ( ^!       
symbols4 ,! tomb n[ %[n to visitors 9   
anci5t ,syracuse z t ( ,ar*im$es _c be  
81 = x be>s no su* figures1 & is n      
outside ! gate ( ,agrig5tum1 z ,cic]o   

describes4  ,! cone is n[ introduc$4 ,a 
conic surface is easily illu/rat$ to a  
class by tak+ a piece ( pap] & roll+ x  
up into a cornucopia1 ! space inclos$   
be+ a conic space1 a t]m t is sometimes 
conv5i5t4 ,! g5]ation ( a conic surface 
may be %[n by tak+ a blackbo>d po9t] &  
sw++ x ar\nd by its tip s t ! o!r end   
moves 9 a curve4 ,if we consid] a       
/raight l9e z ! limit ( a curve1 !n !   
po9t] may sw+ 9 a plane1 & s a plane is 
! limit ( a conic surface4 ,if we sw+ ! 
po9t] ab\t a po9t 9 ! middle1 we %all   
g5]ate ! two nappes ( ! cone1 ! conic   
space n[ be+ d\ble4  ,9 practice ! "r   
circul> cone1 or cone ( revolution1 is !
 important type1 & special att5tion %\ld
 be giv5 to ?is =m4  ,,!orem4 ,e section
 ( a cone made by a plane pass+ ?r\gh   
its v]tex is a triangle4  ,at ?is "t1 or
 9 speak+ ( ! prelim9>y def9itions1     
ref]5ce %\ld be made to ! conic         
sections4 ,( ^! !re are ?ree great      
types3 7#a7 ! ellipse1 :]e ! cutt+ plane
 int]sects all ! elem5ts on "o side ( ! 

v]tex2 a circle is a special =m ( !     
ellipse2 7#b7 ! p>abola1 :]e ! plane is 
p>allel to an elem5t2 7#c7 ! hyp]bola1  
:]e ! plane cuts "s ( ! elem5ts on "o   
side ( ! v]tex1 & ! re/ on ! o!r side2 t
 is1 :]e x cuts bo? nappes4 ,x is to be 
obs]v$ t ! ellipse may v>y greatly 9    
%ape1 f a circle to a v long ellipse1 z 
! cutt+ plane *anges f be+ p]p5dicul> to
 ! axis to be+ ne>ly p>allel to an      
elem5t4 ,! in/ant x becomes p>allel to  
an elem5t ! ellipse *anges sudd5ly to a 
p>abola4 ,if ! plane tips ! slighte/    
am\nt m1 ! section becomes an hyp]bola4 
,:ile ^! conic sections are n /udi$ 9   
elem5t>y geometry1 ! t]ms %\ld be kn[n =
 g5]al in=mation1 p>ticul>ly ! ellipse &
 p>abola4 ,! /udy ( ! conic sections =ms
 a l>ge "p ( ! "w ( analytic geometry1 a
 subject 9 :i* ! figures resemble !     
graphic "w 9 algebra1 ?is hav+ be5 tak5 
f analytics1 z ! high] subject is       
commonly call$4 ,! planets move ab\t !  
sun 9 elliptic orbits1 & ,halley's comet
 t return$ to view 9 #aiji-aiaj has =   

its pa? an enorm\s ellipse4 ,mo/ comets 
seem to move 9 p>abolas1 & a body ?r[n  
into ! air w\ld take a p>abolic pa? if x
 7 n = ! resi/ance ( ! atmosph]e4 ,two (
 ! sides ( ! triangle 9 ?is proposition 
con/itute a special =m ( ! hyp]bola4  ,!
 /udy ( conic sections 0 br\ght to a    
high /ate by ! ,greeks4 ,!y 7 n kn[n to 
! ,py?agoreans1 b 7 discov]$ by         
,m5ae*mus 9 ! f\r? c5tury ,b4,c4 ,?is   
discov]y is m5tion$ by ,proclus1 :o     
says1 ,fur!r1 z to ^! sections1 ! conics
 7 conceiv$ by ,m5ae*mus4  ,s9ce if !   
cutt+ plane is p]p5dicul> to ! axis !   
section is a circle1 & if oblique x is  
an ellipse1 a p>abola1 or an hyp]bola1 x
 foll[s t if light proce$s f a po9t1 !  
%ad[ ( a circle is a circle1 an ellipse1
 a p>abola1 or an hyp]bola1 dep5d+ on ! 
position ( ! plane on :i* ! %ad[ falls4 
,x is int]e/+ & in/ructive to a class to
 see ^! %ad[s1 b ( c\rse n mu* "t c be  
all[$ = su* "w4 ,at ?is po9t ! *ief ?+  
is to h ! names ellipse & p>abola1 s (t5
 met 9 read+1 und]/ood4  ,x is also (   

int]e/ to pupils to see at ?is "t !     
me?od ( draw+ an ellipse by means ( a   
p5cil /ret*+ a /r+ b& t moves ab\t two  
p9s fa/5$ 9 ! pap]4 ,?is is a practical 
me?od1 & is famili> to all tea*]s :o h  
/udi$ analytic geometry4 ,9 design+     
elliptic ar*es1 h[ev]1 ?ree circul> arcs
 are (t5 jo9$1 z "h %[n1 ! result be+   
approximately an elliptic arc4          
,illu/ration       ,"h ,o is ! c5t] (   
arc ,,bc1 ,m ( arc ,,ab1 & ,n (      arc
 ,,cd4 ,s9ce ,,xy is p]p5dicul> to ,,bm 
& ,,bo1 x is      tang5t to arcs ,,ab & 
,,bc1 s !re is no abrupt turn+ at       
,b1 & simil>ly = ,c4#ij  ,,!orem4 ,!    
volume ( a circul> cone is equal to "o  
?ird ! product ( its base by its        
altitude4  ,x is easy to prove ?is =    
noncircul> cones z well1 b s9ce !y are n
 met commonly 9 practice1 !y may be     
omitt$ 9 elem5t>y geometry4 ,! important
 =mula at ?is "t is v  #a_/cpir#b;h4 ,z 
already /at$1 ?is proposition 0 discov]$
 by ,eudoxus ( ,cnidus 7born ca4 #djg   
,b4,c41 di$ ca4 #ced ,b4,c471 a man :o1 

z already /at$1 0 born poor1 b :o became
 "o ( ! mo/ illu/ri\s & mo/ highly      
e/eem$ ( all ! ,greeks ( 8 "t4  ,,!orem4
 ,! lat]al area ( a fru/um ( a cone (   
revolution is equal to half ! sum ( !   
circumf]5ces ( its bases multipli$ by ! 
slant height4  ,an int]e/+ case = a     
class to notice is t 9 :i* ! upp] base  
becomes z]o & ! fru/um becomes a cone1 !
 proposition be+ /ill true4 ,if ! upp]  
base is equal to ! l[] base1 ! fru/um   
becomes a cyl9d]1 & /ill ! proposition  
rema9s true4 ,! proposition ?us (f]s an 
excell5t illu/ration ( ! elem5t>y       
,pr9ciple ( ,cont9uity4  ,!n foll[s1 9  
mo/ textbooks1 a !orem relat+ to !      
volume ( a fru/um4       ,9 ! case ( a  
cone ( revolution      v  7#a_/c7pih7r#b
  r'#b  rr'74 ,"h if r'  #j1 we      h v
  7#a_/c7pir#b;h1 ! volume ( a cone4 ,if
 r'  r1      we h v  7#a_/c7pih7r#b  r#b
  r#b7  pihr#b1 !      volume ( a       
cyl9d]4  ,if "o ne$s examples 9         
m5suration beyond ^? giv5 9 a fir/-class
 textbook1 !y are easily f\nd4 ,!       

monum5t to ,sir ,*ri/oph] ,wr51 !       
pr(essor ( geometry 9 ,cambridge        
,univ]sity1 :o became ! great ar*itect (
 ,/4 ,paul's ,ca!dral 9 ,london1 has a  
,lat9 inscription :i* means1 ,read]1 if 
y w\ld see 8 monum5t1 look ab\t y4 ,s x 
is ) practical examples 9 ,book ,,vii4  
,app5d$ to ?is ,book1 or m (t5 to !     
c\rse 9 solid geometry1 is frequ5tly    
f\nd a proposition kn[n z ,eul]'s       
,!orem4 ,?is is (t5 consid]$ too        
difficult = ! av]age pupil & is !re=e   
omitt$4 ,on acc\nt ( its importance1    
h[ev]1 9 ! !ory ( polyh$rons1 "s ref]5ce
 to x at ?is "t may be helpful to !     
tea*]4 ,! !orem ass]ts t 9 any convex   
polyh$ron ! numb] ( edges increas$ by   
two is equal to ! numb] ( v]tices       
increas$ by ! numb] ( faces4 ,9 o!r     
words1 t e  #b  v  f4 ,on acc\nt ( its  
importance a pro( w be giv5 t diff]s f !
 "o ord9>ily f\nd 9 textbooks4          
,let s#a1 s#b1 4441 sn be ! numb] (     
sides (      ! v>i\s faces1 & f ! numb] 
( faces4 ,n[ s9ce !      sum ( ! angles 

( a polygon ( s sides is 7s - #b7ahj    
deg41      !re=e ! sum ( ! angles ( all 
! faces is      7s#a  s#b  s#c  444  sn 
- #b;f7#ahj deg44       ,b s#a  s#b  s#c
  444  sn is twice !      numb] ( edges1
 because ea* edge belongs to two faces4 
!re=e ! sum ( ! angles ( all ! faces is 
7#b;e - #b;f7#ahj deg41 or 7e - f7#cfj  
deg44       ,s9ce ! polyh$ron is convex1
 x is possible to f9d "s      outside   
po9t ( view1 ,p1 f :i* "s face1 z       
,,abcde1      cov]s up ! :ole figure1 z 
9 ?is illu/ration4 ,if we      ?9k ( all
 ! v]tices project$ on ,,abcde1 by l9es 
?r\gh ,p1 ! sum ( ! angles ( all ! faces
 w be !      same z ! sum ( ! angles (  
all _! projections on      ,,abcde4     
,call+ ,,abcde s#a1 & ?9k+ ( !          
projections z trac$ by dott$ l9es on !  
opposite side (      s#a1 ?is sum is    
evid5tly equal to       7#a7 ! sum ( !  
angles 9 s#a1 or 7s#a - #b7 #ahj deg41  
plus       7#b7 ! sum ( ! angles on !   
o!r side ( s#a1 or      7s#a - #b7ahj   
deg41 plus       7#c7 ! sum ( ! angles  

ab\t ! v>i\s po9ts %[n z      inside (  
s#a1 ( :i* !re are v - s#a po9ts1       
ab\t ea* ( :i* ! sum ( ! angles is #cfj 
deg41 mak+      7v - s#a7cfj deg4 9 all4
  ,illu/ration       ,add+1 we h        
7s#a - #b7ahj deg4  7s#a - #b7ahj deg4  
7v - s#a7cfj deg4       7s#a - #b7  7v -
 s#a7cfj deg4       7v - #b7cfj deg44   
,equat+ ! two sums already f\nd1 we h   
7e - f7#cfj deg4  7v - #b7cfj deg41     
or e - f  v - #b1       or e  #b  v  f4 
,?is pro( is too ab/ract = mo/ pupils 9 
! high s*ool1 b x is m sci5tific ?an ^? 
f\nd 9 any ( ! elem5t>y textbooks1 &    
tea*]s w f9d x ( s]vice 9 reliev+ _! own
 m9ds ( any "q z to ! legitimacy ( !    
!orem4  ,al?\gh ?is proposition is      
g5]ally attribut$ to ,eul]1 & 01 inde$1 
r$iscov]$ by him & publi%$ 9 #ageb1 x 0 
kn[n to ! great ,fr5* geomet] ,desc>tes1
 a fact t ,leibnitz m5tions4#ia  ,?is   
!orem has a v practical application 9 ! 
/udy ( cry/als1 s9ce x (f]s a conv5i5t  
*eck on ! c\nt ( faces1 edges1 &        
v]tices4 ,"s use ( cry/als1 or ev5 (    

polyh$rons cut f a piece ( crayon1 is   
desirable :5 /udy+ ,eul]'s proposition4 
,! foll[+ illu/rations ( common =ms (   
cry/als may be us$ 9 ?is connection3    
,illu/ration  ,! fir/ repres5ts two     
truncat$ pyramids plac$ base to base4   
,"h e  #bj1 f  #aj1 v  #ab1 s t e  #b  f
  v4 ,! second repres5ts a cry/al =m$ by
 replac+ ea* edge ( a cube by a plane1 )
 ! result t e  #dj1 f  #ah1 & v  #bd4 ,!
 ?ird repres5ts a cry/al =m$ by replac+ 
ea* edge ( an octah$ron by a plane1 x   
be+ easy to see t ,eul]'s law /ill holds
 true4  ,,footnotes3  #hi ,! actual     
con/ruction ( ^! solids is giv5 by      
,pappus4 ,see 8 ,ma!maticae             
,collectiones1 p4 #dh1 ,bologna1 #affj4 
#ij ,! illu/ration is f ,dup91 loc4 cit4
  #ia ,= ! hi/orical bibliography       
consult ,g4 ,holzmuell]1 ,elem5te d]    
,/]eometrie1 ,vol4 ,i1 p4 #aha1         
,leipzig1 #aijj4     ,,*apt] ,,xxi  ,,! 
,,lead+ ,,propositions ,,( ,,book ,,viii
   ,book ,,viii treats ( ! sph]e4 ,j z !
 circle may be def9$ ei!r z a plane     

surface or z ! b\nd+ l9e :i* is ! locus 
( a po9t 9 a plane at a giv5 di/ance f a
 fix$ po9t1 s a sph]e may be def9$ ei!r 
z a solid or z ! b\nd+ surface :i* is ! 
locus ( a po9t 9 space at a giv5 di/ance
 f a fix$ po9t4 ,9 high] ma!matics !    
circle is def9$ z ! b\nd+ l9e & ! sph]e 
z ! b\nd+ surface2 t is1 ea* is def9$ z 
a locus4 ,?is view ( ! circle z a l9e is
 becom+ q g5]al 9 elem5t>y geometry1 x  
be+ ! desire t /ud5ts may n h to *ange  
def9itions 9 pass+ f elem5t>y to high]  
ma!matics4 ,! sph]e is less frequ5tly   
look$ ^u 9 geometry z a surface1 & 9    
popul> usage x is always tak5 z a solid4
  ,analog\s to ! po/ulate t a circle may
 be describ$ ) any giv5 po9t z a c5t] & 
any giv5 l9e z a radius1 is ! po/ulate =
 con/ruct+ a sph]e ) any giv5 c5t] & any
 giv5 radius4 ,?is po/ulate is n s      
ess5tial1 h[ev]1 z ! "o ab\t ! circle1  
because we are n s conc]n$ )            
con/ructions "h z we are 9 plane        
geometry4  ,a good opportunity is (f]$ =
 illu/rat+ sev]al ( ! def9itions        

connect$ ) ! /udy ( ! sph]e1 su* z great
 circle1 axis1 small circle1 & pole1 by 
ref]r+ to geography4 ,inde$1 ! fir/ ?ree
 propositions usually giv5 9 ,book      
,,viii h a direct be>+ ^u ! /udy ( !    
e>?4  ,,!orem4 ,a plane p]p5dicul> to a 
radius at its extremity is tang5t to !  
sph]e4  ,! /ud5t %\ld always h 8        
att5tion call$ to ! analogue 9 plane    
geometry1 :]e !re is "o4 ,if "h we pass 
a plane ?r\gh ! radius 9 "q1 ! figure   
=m$ on ! plane w be t ( a l9e tang5t to 
a circle4 ,if we revolve ?is ab\t ! l9e 
( ! radius 9 "q1 z an axis1 ! circle w  
g5]ate ! sph]e aga91 & ! tang5t l9e w   
g5]ate ! tang5t plane4  ,,!orem4 ,a     
sph]e may be inscrib$ 9 any giv5        
tetrah$ron4  ,"h aga9 we may =m a       
correspond+ proposition ( plane geometry
 by pass+ a plane ?r\gh any ?ree po9ts (
 contact ( ! sph]e & ! tetrah$ron4 ,we  
%all !n =m ! figure ( a circle inscrib$ 
9 a triangle4 ,& j z 9 ! case ( !       
triangle we may h escrib$ circles by    
produc+ ! sides1 s 9 ! case ( !         

tetrah$ron we may h escrib$ sph]es by   
produc+ ! planes indef9itely & proce$+ 9
 ! same way z = ! inscrib$ sph]e4 ,!    
figure is difficult to draw1 b x is n   
difficult to und]/&1 p>ticul>ly if we   
con/ruct ! tetrah$ron out ( pa/ebo>d4   
,,!orem4 ,a sph]e may be circumscrib$   
ab\t any giv5 tetrah$ron4  ,by produc+  
"o ( ! faces indef9itely x w cut ! sph]e
 9 a circle1 & ! result+ figure1 on !   
plane1 w be t ( ! analog\s proposition (
 plane geometry1 ! circle circumscrib$  
ab\t a triangle4 ,x is easily prov$ f ! 
proposition t ! f\r p]p5dicul>s erect$  
at ! c5t]s ( ! faces ( a tetrah$ron meet
 9 a po9t 7are concurr5t71 ! analogue ( 
! proposition ab\t ! p]p5dicul>         
bisectors ( ! sides ( a triangle4       
,,!orem4 ,! int]section ( two sph]ical  
surfaces is a circle ^: plane is        
p]p5dicul> to ! l9e jo9+ ! c5t]s ( !    
surfaces & ^: c5t] is 9 t l9e4  ,!      
figure sugge/s ! case ( two circles 9   
plane geometry4 ,9 ! case ( two circles 
t d n int]sect or t\*1 "o n be+ )9 !    

o!r1 !re are f\r common tang5ts4 ,if !  
circles t\*1 two close up into "o4 ,if  
"o circle is :olly )9 ! o!r1 ?is la/    
tang5t disappe>s4 ,! same ?+ exi/s 9    
relation to two sph]es1 & ! analog\s    
cases are =m$ by revolv+ ! circles &    
tang5ts ab\t ! l9e ?r\gh _! c5t]s4  ,9  
plane geometry x is easily prov$ t if   
two circles int]sect1 ! tang5ts f any   
po9t on _! common *ord produc$ are      
equal4 ,= if ! common *ord is ,,ab & !  
po9t ,p is tak5 on ,,ab produc$1 !n !   
squ>e on any tang5t f ,p is equal to    
,,pb x ,,pa4 ,! l9e ,,pba is sometimes  
call$ ! radical axis4  ,simil>ly 9 ?is  
proposition conc]n+ sph]es1 if f any    
po9t 9 ! plane ( ! circle =m$ by !      
int]section ( ! two sph]ical surfaces   
l9es are drawn tang5t to ei!r sph]e1 ^! 
tang5ts are equal4 ,= x is easily prov$ 
t all tang5ts to ! same sph]e f an      
ext]nal po9t are equal1 & x c be prov$ z
 9 plane geometry t two tang5ts to ! two
 sph]es are equal4  ,among ! int]e/+    
analogies betwe5 plane & solid geometry 

is ! "o relat+ to ! f\r common tang5ts  
to two circles4 ,if ! figure be revolv$ 
ab\t ! l9e ( c5t]s1 ! circles g5]ate    
sph]es & ! tang5ts g5]ate conical       
surfaces4 ,to /udy ?is case = v>i\s     
sizes & positions ( ! two sph]es is "o (
 ! mo/ int]e/+ g5]alizations ( solid    
geometry4       ,an application ( !     
proposition is se5 9 ! case ( an        
eclipse1 :]e ! sph]e ,o' repres5ts !    
moon1 ,o !      e>?1 & ,s ! sun4 ,x is  
also se5 9 ! case ( ! full      moon1 :5
 ,s is on ! o!r side ( ! e>?4 ,9 ?is    
case      ! "p ,,m9 is fully illum9at$  
by ! moon1 b ! zone      ,,abnm is only 
p>tly illum9at$1 z ! figure %[s4#ib     
,illu/ration  ,,!orem4 ,! sum ( ! sides 
( a sph]ical polygon is less ?an #cfj   
deg44  ,9 all su* cases ! relation to ! 
polyh$ral angle %\ld be made cle>4 ,?is 
is done 9 ! pro(s usually giv5 9 !      
textbooks4 ,x is easily se5 t ?is is    
true only ) ! limitation set =? 9 mo/   
textbooks1 t ! sph]ical polygons        
consid]$ are convex4 ,?us we might h a  

sph]ical triangle t is concave1 ) its   
base #cei deg41 & its o!r two sides ea* 
#ij deg41 ! sum ( ! sides be+ #eci deg44
  ,,!orem4 ,! sum ( ! angles ( a        
sph]ical triangle is great] ?an #ahj    
deg4 & less ?an #edj deg44  ,x is = !   
purpose ( prov+ ?is important fact t    
pol> triangles are introduc$4 ,?is      
proposition %[s ! relation ( ! sph]ical 
to ! plane triangle4 ,if our planes 7 9 
reality slightly curv$1 be+ small       
portions ( enorm\s sph]ical surfaces1 !n
 ! sum ( ! angles ( a triangle w\ld n be
 exactly #ahj deg41 b w\ld exce$ #ahj   
deg4 by "s am\nt dep5d+ on ! curvature (
 ! surface4 ,j z a be+ may be imag9$ z  
hav+ only two dim5sions1 & liv+ always  
on a plane surface 79 a space ( two     
dim5sions71 & hav+ no conception ( a    
space ( ?ree dim5sions1 s we may ?9k (  
ourselves z liv+ 9 a space ( ?ree       
dim5sions b surr\nd$ by a space ( f\r   
dim5sions4 ,! flat be+ c\ld n po9t to a 
?ird dim5sion because he c\ld n get out 
( 8 plane1 & we _c po9t to ! f\r?       

dim5sion because we _c get out ( our    
space4 ,n[ :at ! flat be+ ?9ks is 8     
plane may be ! surface ( an enorm\s     
sph]e 9 our ?ree dim5sions2 9 o!r words1
 ! space he lives 9 may curve ?r\gh "s  
high] space )\t 8 be+ consci\s ( x4 ,s  
our space may also curve ?r\gh "s high] 
space )\t our be+ consci\s ( x4 ,if our 
planes h really "s curvature1 !n ! sum (
 ! angles ( our triangles has a slight  
excess ov] #ahj deg44 ,all ?is is m]e   
speculation1 b x may int]e/ "s /ud5t to 
"k t ! idea ( f\r? & high] dim5sions    
ent]s l>gely into ma!matical            
inve/igation to-day4  ,,!orem4 ,two     
symmetric sph]ical triangles are        
equival5t4  ,:ile x is n a subject t has
 any place 9 a s*ool1 save p]haps =     
incid5tal conv]sation ) "s gr\p (       
en?usia/ic /ud5ts1 x may int]e/ ! tea*] 
to consid] ?is proposition 9 connection 
) ! f\r? dim5sion j m5tion$4 ,consid] ^!
 triangles1 :]e ,l,a  ,l,a'1 ,,ab       
,a',b'1 ,,ac  ,a',c'4 ,we prove !m      
congru5t by sup]position1 turn+ "o ov] &

 plac+ x ^u ! o!r4 ,b suppose we 7 be+s 
9 ,flatl&1 be+s ) only two dim5sions &  
)\t ! p[] to po9t 9 any direction except
 9 ! plane we liv$ 94 ,we %\ld !n be    
unable to turn triangle,a',b',c' ov] s t
 x c\ld co9cide ) triangle,,abc1 & we   
%\ld h to prove ^! triangles equival5t 9
 "s o!r way1 probably by divid+ !m into 
isosceles triangles t c\ld be sup]pos$4 
,illu/ration  ,illu/ration  ,n[ x is !  
same ?+ ) symmetric sph]ical triangles2 
we _c sup]pose !m4 ,b might x n be      
possible to d s if we c\ld turn !m ?r\gh
 ! f\r? dim5sion exactly z we turn !    
,flatl&]'s triangle ?r\gh our ?ird      
dim5sion8 ,x is int]e/+ to ?9k ab\t ?is 
possibility ev5 ?\gh we c>ry x no fur!r1
 & 9 ^! side lights on ma!matics lies   
mu* ( ! fasc9ation ( ! subject4         
,,!orem4 ,! %orte/ l9e t c be drawn on !
 surface ( a sph]e betwe5 two po9ts is !
 m9or arc ( a great circle jo9+ ! two   
po9ts4  ,x is always int]e/+ to a class 
to apply ?is practically4 ,by tak+ a    
t]re/rial globe & draw+ a great circle  

betwe5 ! s\!rn po9t ( ,irel& & ,new     
,york ,city1 we repres5t ! %orte/ r\te =
 %ips cross+ to ,engl&4 ,n[ if we notice
 :]e ?is great-circle arc cuts ! v>i\s  
m]idians & m>k ?is on an ord9>y         
,m]cator's projection map1 su* z is f\nd
 9 any s*oolroom1 we %all f9d t ! pa? ( 
! %ip does n make a /raight l9e4        
,pass5g]s at sea (t5 d n und]/& :y !    
%ip's c\rse on ! map is n a /raight l9e2
 b ! *ief reason is t ! %ip is tak+ a   
great-circle arc1 & ?is is n1 9 g5]al1 a
 /raight l9e on a ,m]cator projection4  
,! small circles ( latitude are /raight 
l9es1 & s are ! m]idians & ! equator1 b 
o!r great circles are repres5t$ by curv$
 l9es4  ,,!orem4 ,! area ( ! surface ( a
 sph]e is equal to ! product ( its      
diamet] by ! circumf]5ce ( a great      
circle4  ,?is leads to ! rem>kable      
=mula1 a  #dpir#b4 ,t ! area ( ! sph]e1 
a curv$ surface1 %\ld exactly equal !   
sum ( ! areas ( f\r great circles1 plane
 surfaces1 is ! rem>kable feature4 ,?is 
0 "o ( ! greate/ discov]ies ( ,ar*im$es 

7ca4 #bhg-bab ,b4,c471 :o gives x z !   
?irty-fif? proposition ( 8 treatise on !
 ,sph]e & ! ,cyl9d]1 & :o m5tions x     
specially 9 a lett] to 8 fri5d ,dosi!us1
 a ma!matician ( "s prom95ce4 ,ar*im$es 
also /ates t ! surface ( a sph]e is two 
?irds t ( ! circumscrib$ cyl9d]1 or !   
same z ! curv$ surface ( ?is cyl9d]4    
,?is is evid5t1 s9ce ! cyl9dric surface 
( ! cyl9d] is #bpir x #br1 or #dpir#b1 &
 ! two bases h an area pir#b  pir#b1    
mak+ ! total area #fpir#b4  ,,!orem4 ,! 
area ( a sph]ical triangle is equal to !
 area ( a lune ^: angle is half !       
triangle's sph]ical excess4  ,?is !orem1
 s important 9 f9d+ areas on ! e>?'s    
surface1 %\ld be foll[$ by a consid]able
 am\nt ( computation ( triangul> areas1 
else x w be r mean+less4 ,/ud5ts t5d to 
memorize a pro( ( ?is "*1 & 9 ord] to h 
! proposition mean :at x %\ld to !m1 !y 
%\ld at once apply x4 ,! same is true ( 
! foll[+ proposition on ! area ( a      
sph]ical polygon4 ,x is probable t nei!r
 ( ^! propositions is v old2 at any     

rate1 !y d n seem to h be5 kn[n to !    
writ]s on elem5t>y ma!matics among !    
,greeks4  ,,!orem4 ,! volume ( a sph]e  
is equal to ! product ( ! area ( its    
surface by "o ?ird ( its radius4  ,?is  
gives ! =mula v  7#d_/c7pir#c4 ,?is is  
"o ( ! greate/ discov]ies ( ,ar*im$es4  
,he also f\nd z a result t ! volume ( a 
sph]e is two ?irds ! volume ( !         
circumscrib$ cyl9d]4 ,?is is easily se51
 s9ce ! volume ( ! cyl9d] is pir#b x    
#br1 or #bpir#c1 & 7#d_/c7pir#c is #b_/c
 ( #bpir#c4 ,x 0 because ( ^! discov]ies
 on ! sph]e & cyl9d] t ,ar*im$es wi%$ ^!
 figures engrav$ ^u 8 tomb1 z has       
already be5 /at$4 ,! ,roman g5]al       
,m>cellus conqu]$ ,syracuse 9 #bab      
,b4,c41 & at ! sack ( ! city ,ar*im$es 0
 kill$ by an ignorant soldi]4 ,m>cellus 
c>ri$ out ! wi%es ( ,ar*im$es ) respect 
to ! figures on 8 tomb4  ,! volume ( a  
sph]e c also be v elegantly f\nd by     
means ( a proposition kn[n z ,cavali]i's
 ,!orem4 ,?is ass]ts t if two solids lie
 betwe5 p>allel planes1 & are su* t !   

two sections made by any plane p>allel  
to ! giv5 planes are equal 9 area1 !    
solids are !mselves equal 9 volume4     
,?us1 if ^! solids h ! same altitude1 a1
 & if ,s & ,s' are equal sections made  
by a plane p>allel to ,,mn1 !n ! solids 
h ! same volume4 ,! pro( is simple1 s9ce
 prisms ( ! same altitude1 say a_/n1 &  
on ! bases ,s & ,s' are equival5t1 & !  
sums ( n su* prisms are ! giv5 solids2 &
 z n increases1 ! sums ( ! prisms       
approa* ! solids z _! limits2 h5ce !    
volumes are equal4  ,illu/ration  ,?is  
proposition1 :i* w n[ be appli$ to f9d+ 
! volume ( ! sph]e1 0 discov]$ by       
,bonav5tura ,cavali]i 7#aeia or         
#aeih-afdg74 ,he 0 a ,jesuit pr(essor 9 
! ,univ]sity ( ,bologna1 & 8 be/ kn[n "w
 is 8 ,geometria ,indivisilibus1 :i* he 
wrote 9 #afbf1 at lea/ 9 "p1 & publi%$ 9
 #afce 7second edition1 #afdg74 ,by     
means ( ! proposition x is also possible
 to prove sev]al o!r !orems1 z t !      
volumes ( triangul> pyramids ( equival5t
 bases & equal altitudes are equal4     

,illu/ration       ,to f9d ! volume ( a 
sph]e1 take ! quadrant ,,opq1 9 !       
squ>e ,,oprq4 ,!n if ?is figure is      
revolv$ ab\t ,,op1      ,,opq w g5]ate a
 hemisph]e1 ,,opr w g5]ate a cone (     
volume 7#a_/c7pir#c1 & ,,oprq w g5]ate a
 cyl9d] (      volume pir#c4 ,h5ce !    
figure g5]at$ by ,,orq w h      a volume
 pir#c - 7#a_/c7pir#c1 or 7#b_/c7pir#c1 
:i*      we w call x4      ,n[ ,,oa     
,,ab1 & ,,oc  ,,ad2 also 7,,oc7#b -     
7,,oa7#b  7,,ac7#b1     s t             
7,,ad7#b - 7,,ab7#b  7,,ac7#b1     &    
pi7,,ad7#b - pi7,,ab7#b  pi7,,ac7#b4    
,b pi7,,ad7#b - pi7,,ab7#b is ! area ( !
 r+      g5]at$ by ,,bd1 a section ( x1 
& pi7,,ac7#b is !      correspond+      
section ( ! hemisph]e4 ,h5ce1 by        
,cavali]i's      ,!orem1                
7#b_/c7pir#c  ! volume ( ! hemisph]e4   
!re=e 7#d_/c7pir#c  ! volume ( ! sph]e4 
,9 connection ) ! sph]e "s easy "w 9    
quadratics may be introduc$ ev5 if !    
class has _h only a ye> 9 algebra4      
,= example1 suppose a cube is inscrib$ 9

 a hemisph]e (      radius r & we wi% to
 f9d its edge1 & !reby its      surface 
& its volume4       ,if x  ! edge ( !   
cube1 ! diagonal ( ! base mu/ be        
xsqrt#b1 & ! projection ( r 7drawn f !  
c5t] (      ! base to "o ( ! v]tices7 on
 ! base is half ( ?is      diagonal1 or 
7xsqrt#b7_/b4       ,h5ce1 by !         
,py?agorean ,!orem1               r#b   
x#b  77xsqrt#b7_/b7b  7#c_/b7x#b        
!re=e x  rsqrt7#b_/c71      & ! total   
surface is  #fx#b  #dr#b1      & !      
volume is          x#c                  
7#b_/c7r#csqrt7#b_/c74   ,,footnotes3   
#ib ,! illu/ration is f ,dup91 loc4 cit4
     ,l',,envoi   ,9 ! ,valley ( ,y\?1  
?r\gh :i* all wayf>]s mu/ pass on _!    
j\rney f ! ,l& ( ,my/]y to ! ,l& ( !    
,inf9ite1 !re is a village :]e ! pilgrim
 re/s & indulges 9 v>i\s excursions =   
:i* ! valley is celebrat$4 ,!re also    
ga!r _m guides 9 ?is spot1 "s ( :om %[ !
 /rang] all ! v>i\s po9ts ( common      
int]e/1 & o!rs ( :om take visitors to   
special po9ts f :i* ! views are (       

peculi> significance4 ,z "t has gone on 
new pa?s h op5$1 & new re/+ places h be5
 made f :i* ^! views are be/ obta9$4 ,"s
 ( ! m\nta9 peaks h be5 neglect$ 9 !    
pa/1 b ( late !y too h be5 scal$1 & pa?s
 h be5 hewn out t approa* ! summits1 &  
_m pilgrims asc5d !m & f9d t ! result is
 abundantly wor? ! ef=t & ! "t4  ,!     
effect ( ^! sev]al improvem5ts has be5 a
 natural & usually fri5dly rivalry 9 !  
body ( guides t %[ ! way4 ,! m\nta9s h n
 *ang$1 & ! views are :at !y h always   
be54 ,b !re are n want+ ^? :o say1 ,my  
m\nta9 may n be z l(ty z y\rs1 b x is   
easi] to asc5d2 or ,!re are qu>ries on  
my peak1 & po9ts ( view f :i* a build+  
may be se5 9 process ( erection1 or a   
mill 9 op]ation1 or a canal1 :ile y\r   
m\nta9 %[s only a /ret* ( hills &       
valleys1 & ?us y w see t m9e is ! m     
pr(itable to visit4 ,!n !re are guides  
:o are !mselves (t5 weak ( limb1 & :o   
are atta*$ to num]\s s& dunes1 & ^! say 
to ! weak] pilgrims1 ,:y tire y\rselves 
climb+ a rocky m\nta9 :5 "h are peaks ^:

 summits y c rea* ) ease & f :i* ! view 
is j z good z t f ! mo/ fam\s precipice8
 ,! result is n :olly disadvantage\s1 = 
_m :o pass ?r\gh ! valley are able to   
approa* ! summits ( ! s& dunes only1 &  
w\ld make progress ) greate/ difficulty 
%\ld !y attempt to scale a real m\nta91 
al?\gh ev5 = !m x w\ld be bett] to climb
 a little way :]e x is really wor? !    
ef=t in/ead ( sp5d+ all _! ef=ts on !   
dunes4  ,!n1 too1 !re h ( late come     
guides :o h %[n mu* +5uity by digg+     
tunnels into "s ( ! greate/ m\nta9s4 ,^!
 !y h pav$ ) smoo? concrete1 & h arrang$
 = rubb]-tir$ c>s t run )\t j> to ! he>t
 ( "s m\nta94 ,arriv$ !re ! pilgrim has 
a glance1 z ! c> swiftly turns 9 a blaze
 ( electric light1 at a r\ghly pa9t$    
panorama ( ! view f ! summit1 & he is   
assur$ by ! guide t he has accompli%$   
all t he w\ld h done1 _h he labori\sly  
climb$ ! peak itself4  ,9 ! mid/ ( all !
 advocacy ( s&-dune climb+1 & (         
rubb]-tir$ c>s to see a pa9t$ view1 !   
great body ( guides /ill climb _!       

m\nta9s ) _! little gr\ps ( foll[]s1 & !
 vigor ( ! asc5t & ! magnific5ce ( !    
view /ill attract all :o are /rong &    
e>ne/1 dur+ _! soj\rn 9 ! ,valley (     
,y\?4 ,among ! m\nta9s t h = ages       
attract$ ! pilgrims is ,mons ,lat9us1   
usually call$ 9 ! valley by ! m pleas+  
"n ,lat9a4 ,ma!matica1 & ,rhetorica1 &  
,grammatica are also among ! be/ kn[n4  
,a gr\p kn[n z ,montes ,naturales       
comprises ,physica1 ,biologica1 &       
,*emica1 & "o great peak ) m9or peaks   
ab\t x is call$ by ! p ,philosophia4    
,!re are ^? :o claim t ^! great masses (
 rock are too old to be climb$1 z if t  
affect$ ! view2 :ile o!rs claim t !     
asc5t is too difficult & t all :o d n   
favor ! s& dunes are reaction>y4 ,b ?is 
affects only a few :o belong to ! real  
m\nta9s1 & ! o!rs labor dilig5tly to    
improve ! pa?s & to less5 unnecess>y    
toil1 b !y seek n to te> (f ! summits   
nor d !y att5d to ! amus+ attempts ( ^? 
:o sit by ! hillocks & ?r[ pebbles at ! 
rocky sides ( ! m\nta9s ^u :i* !y "w4   

99       99       99       99       99  
,geometry is a m\nta94 ,vigor is ne$$ = 
its asc5t4 ,! views all along ! pa?s are
 magnific5t4 ,! ef=t ( climb+ is        
/imulat+4 ,a guide :o po9ts out !       
beauties1 ! gr&eur1 & ! special places (
 int]e/ comm&s ! admiration ( 8 gr\p (  
pilgrims4 ,"o :o fails to d ?is1 :o does
 n "k ! pa?s1 :o puts unnecess>y burd5s 
^u ! pilgrim1 or :o bl9dfolds him 9 8   
progress1 is unwor?y ( 8 position4 ,!   
pret5d$ guide :o says t ! pa9t$         
panorama1 se5 f ! rubb]-tir$ c>1 is z   
good z ! view f ! summit is simply a    
fakir & is g5]ally recogniz$ z su*4 ,!  
m\nta9 w /&2 x w n be us$ z a m]e       
comm]cial qu>ry = build+ /one2 x w n be 
affect$ by pellets ?r[n f ! little      
hillocks ab\t2 b its pa?s w be fre$ f   
unnecess>y fl9ts1 !y w be /raight5$ :]e 
?is c advantage\sly be done1 & new pa?s 
on entirely novel plans w be made z "t  
goes on1 b ^! pa?s w be hew$ out ( rock1
 n made out ( ! dreams ( a day4 ,e wor?y
 guide w assi/ 9 all ^! ef=ts at        

bett]m5t1 & w urge ! pilgrim at lea/ to 
asc5d a little way because ( ! fact t ! 
same view _c be obta9$ f o!r peaks2 b he
 w n take s]i\sly ! ef=ts ( ! fakir1 nor
 w he li/5 ) m ?an pass+ int]e/ to him  
:o proclaims ! s& heap to be a          
,matt]horn4     ,,index       ,ahmes1   
#bg1 #bed1 #bgh1 #cjf      ,alex&r(f1   
#afd      ,algebra1 #cg1 #hd            
,al-,kh[>azmi1 #cg      ,allman1 ,g4    
,j41 #bi      ,almage/1 #ce             
,al-,nairizi1 #aga1 #aic1 #bad1 #bfd    
,al-,qift4i1 #di      ,analysis1 #da1   
#afa      ,angle1 #adb1 #aee2           
trisection (1 #ca1 #bae      ,an?onisz1 
,adria51 #bgi      ,antiphon1 #ca1 #cb1 
#bgf      ,apollodotus 7,apollodorus71  
#bei      ,apollonius1 #cd1 #bad1 #bca  
,appli$ problems1 #ge1 #ajc1 #agh1 #ahf1
 #aib1 #aie1 #bjc1 #bjd1 #bji1 #bae1    
#bag1       #bdb1 #bfg1 #bie1 #cag      
,appreciation ( geometry1 #ai      ,arab
 geometry1 #cg1 #ea      ,ar*im$es1 #cd1
 #db1 #dh1 #aci1 #ada1 #bae1 #bgf1 #bgh1
 #cad1 #cbg1 #cbh      ,ari/aeus1 #caj  

,ari/otle1 #cc1 #db1 #acd1 #ace1 #acg1  
#ade1 #aed1 #agg1 #bji      ,aryabhatta1
 #cf1 #bgi      ,associations1 syllabi  
(1 #eh1 #fj1 #fd      ,assumptions1 #aaf
      ,a/rolabe1 #agb      ,a!lh>d (    
,ba?1 #cg1 #ea      ,a!naeus1 #bei      
,axioms1 #ca1 #da1 #aaf       ,babylon1 
#bf1 #bgb      ,b>toli1 #aj1 #dd1 #bch  
,belli1 #aj1 #dd1 #agb      ,belt9us1   
#bci1 #bda      ,beltrami1 #abg         
,b5nett1 ,j41 #bbd      ,b]n\lli1 #bhj  
,b]tr&1 #fb      ,betz1 #aca            
,bez\t1 #fb      ,bhask>a1 #bcb1 #bfh   
,bill+s1 ,r4 ,w41 #bbb      ,bill+sley1 
#eb      ,bion1 #aib1 #bci              
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#afe1 #afg1 #bja1 #bbg1 #beb1 #bfi1     
#bhi1 #cjc1 #cba      ,bordas-,dem\l91  
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,brets*neid]1 ,c4 ,a41 #cj              
,br\nck]1 #bhj      ,bruce1 ,w4 ,n41    

#aii      ,bryson1 #ca1 #cb1 #bgf       
,cajori1 #df      ,cal&ri1 #cj          
,campanus1 #cg1 #ea1 #ace      ,cantor1 
,m41 #bi1 #df      ,capella1 #ej1 #ace  
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#aba      ,colleges1 geometry 9 !1 #df  
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#cae      ,congru5t1 #aea      ,conic   
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#bab      ,conv]se proposition1 #age1   
#aij1 #aia      ,crelle1 #adb           
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,gem9us1 #abf1 #abh1 #adi               
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