,BOOK ,,VI
                   #A

  ,N[ IF ! T]MS ,83T9U\S0'1 ,8IN
3TACT0'1 & ,8IN SU3ES.N0' >E "U/OOD Z
DEF9$ ABV ?+S 2+ ,83T9U\S0' IF _!
EXTREMITIES >E "O1 ,8IN 3TACT0' IF _!
EXTREMITIES >E TGR1 & ,8IN SU3ES.N0' IF
"! IS NO?+ ( _! [N K9D 9T]M$IATE 2T !M-
NO?+ T IS 3T9U\S C 2 -POS$ ,8(
9DIVISI#S0'3 E4G4 A L9E _C 2 -POS$ (
PO9TS1 ! L9E 2+ 3T9U\S &! PO9T 9DIVISI#4
,=! EXTREMITIES ( TWO PO9TS C NEI 2 "O
7S9CE ( AN 9DIVISI# "! C 2 NO EXTREM;Y Z
4T9CT F "S O!R "P7 NOR TGR 7S9CE T : HAS
NO "PS C H NO EXTREM;Y1 ! EXTREM;Y &! ?+
( : X IS ! EXTREM;Y 2+ 4T9CT74
  ,MOREOV]1 IF T : IS 3T9U\S IS -POS$ (
PO9TS1 ^! PO9TS M/ 2 EI 3T9U\S OR 9
3TACT ) "O ANO!R3 &! SAME R1SON+ APPLIES
9 ! CASE ( ALL 9DIVISI#S4 ,N[ =! R1SON
GIV5 ABV !Y _C 2 3T9U\S3 & "O ?+ C 2 9
3TACT ) ANO!R ONLY IF :OLE IS 9 3TACT )
:OLE OR "P ) "P OR "P ) :OLE4 ,B S9CE
9DIVISI#S H NO "PS1 !Y M/ 2 9 3TACT ) "O
ANO!R Z :OLE ) :OLE4 ,& IF !Y >E 9 3TACT
) "O ANO!R Z :OLE ) :OLE1 !Y W N 2
3T9U\S3 = T : IS 3T9U\S HAS 4T9CT "PS3 &
^! "PS 96: X IS DIVISI# >E DI6]5T 9 ?
WAY1 I4E4 SPATI,Y SEP>ATE4
  ,NOR1 AG1 C A PO9T 2 9 SU3ES.N 6A PO9T
OR A MO;T 6A MO;T 9 S* A WAY T L5G? C 2
-POS$ ( PO9TS OR "T ( MO;TS3 = ?+S >E 9
SU3ES.N IF "! IS NO?+ ( _! [N K9D
9T]M$IATE 2T !M1 ":AS T : IS 9T]M$IATE
2T PO9TS IS ALW A L9E & T : IS 9T]M$IATE
2T MO;TS IS ALW A P]IOD ( "T4
  ,AG1 IF L5G? & "T CD ?US 2 -POS$ (
9DIVISI#S1 !Y CD 2 DIVID$ 969DIVISI#S1
S9CE EA* IS DIVISI# 96! "PS ( : X IS
-POS$4 ,B1 Z WE SAW1 NO 3T9U\S ?+ IS
DIVISI# 96?+S )\T "PS4 ,NOR C "! 2 ANY?+
( ANY O!R K9D 9T]M$IATE 2T ! "PS OR 2T !
MO;TS3 = IF "! CD 2 ANY S* ?+ X IS CLE>
T X M/ 2 EI 9DIVISI# OR DIVISI#1 & IF X
IS DIVISI#1 X M/ 2 DIVISI# EI
969DIVISI#S OR 96DIVISI#S T >E 9F9ITELY
DIVISI#1 9 : CASE X IS 3T9U\S4
  ,MOREOV]1 X IS PLA9 T "EY?+ 3T9U\S IS
DIVISI# 96DIVISI#S T >E 9F9ITELY
DIVISI#3 = IF X 7 DIVISI# 969DIVISI#S1
WE %D H AN 9DIVISI# 9 3TACT ) AN
9DIVISI#1 S9CE ! EXTREMITIES ( ?+S T >E
3T9U\S ) "O ANO!R >E "O & >E 9 3TACT4
  ,! SAME R1SON+ APPLIES EQU,Y
6MAGNITUDE1 6"T1 & 6MO;N3 EI ALL ( ^! >E
-POS$ ( 9DIVISI#S & >E DIVISI#
969DIVISI#S1 OR N"O4 ,? MAY 2 MADE CLE>
Z FOLL[S4 ,IF A MAGNITUDE IS -POS$ (
9DIVISI#S1 ! MO;N OV] T MAGNITUDE M/ 2
-POS$ ( CORRESPOND+ 9DIVISI# MO;NS3 E4G4
IF ! MAGNITUDE ,,ABG IS -POS$ (!
9DIVISI#S ;,A1 ;,B1 ;,G1 EA* CORRESPOND+
"P (! MO;N ,,DEZ ( ,O OV] ,,ABG IS
9DIVISI#4 ,"!=E1 S9CE ": "! IS MO;N "!
M/ 2 "S?+ T IS 9 MO;N1 & ": "! IS "S?+ 9
MO;N "! M/ 2 MO;N1 "!=E ! 2+-MOV$ W AL 2
-POS$ ( 9DIVISI#S4 ,S ,O TRAV]S$ ,A :5
XS MO;N 0 ;,D1 ;,B :5 XS MO;N 0 ;,E1 &
;,G SIMIL>LY :5 XS MO;N 0 ,Z4 ,N[ A ?+ T
IS 9 MO;N F "O PLACE 6ANO!R _C AT ! MO;T
:5 X 0 9 MO;N BO? 2 9 MO;N & AT ! SAME
"T H -PLET$ XS MO;N AT ! PLACE 6: X 0 9
MO;N3 E4G4 IF A MAN IS WALK+ 6,!BES1 HE
_C 2 WALK+ 6,!BES & AT ! SAME "T H
-PLET$ 8 WALK 6,!BES3 &1 Z WE SAW1 ,O
TRAV]SES ;A ! "P.S SEC;N ,A 9 VIRTUE (!
PRES;E (! MO;N ,D4 ,3SEQU5TLY1 IF ,O
ACTU,Y PASS$ "? ,A AF 2+ 9 PROCESS (
PASS+ "?1 ! MO;N M/ 2 DIVISI#3 = AT ! "T
:5 ,O 0 PASS+ "?1 X NEI 0 AT RE/ NOR _H
-PLET$ XS PASSAGE B 0 9 AN 9T]M$IATE
/ATE3 :ILE IF X IS PASS+ "? & HAS -PLET$
XS PASSAGE AT ! SAME MO;T1 !N T : IS
WALK+ W AT ! MO;T :5 X IS WALK+ H -PLET$
XS WALK & W 2 9 ! PLACE 6: X IS WALK+2 T
IS 6SAY1 X W H -PLET$ XS MO;N AT ! PLACE
6: X IS 9 MO;N4 ,& IF A ?+ IS 9 MO;N OV]
! :OLE ,,KBG & XS MO;N IS ! ?REE ;,D1
;,E1 & ;,Z1 & IF X IS N 9 MO;N AT ALL
OV] ! "P.S SEC;N ,A B HAS -PLET$ XS MO;N
OV] X1 !N ! MO;N W 3SI/ N ( MO;NS B (
/>TS1 & W TAKE PLACE 0A ?+'S HAV+ -PLET$
A MO;N )\T 2+ 9 MO;N3 = ON ? ASSUMP;N X
HAS -PLET$ XS PASSAGE "? ,A )\T PASS+ "?
X4 ,S X W 2 POSSI# =A ?+ 6H -PLET$ A
WALK )\T "E WALK+3 = ON ? ASSUMP;N X HAS
-PLET$ A WALK OV] A "PICUL> 4T.E )\T
WALK+ OV] T 4T.E4 ,S9CE1 !N1 "EY?+ M/ 2
EI AT RE/ OR 9 MO;N1 & ,O IS "!=E AT RE/
9 EA* (! SEC;NS ;,A1 ;,B1 & ;,G1 X
FOLL[S T A ?+ C 2 3T9U\SLY AT RE/ & AT !
SAME "T 9 MO;N3 =1 Z WE SAW1 ,O IS 9
MO;N OV] ! :OLE ,,ABG & AT RE/ 9 ANY "P
7& 3SEQU5TLY 9 ! :OLE7 ( X4 ,MOREOV]1 IF
! 9DIVISI#S -POS+ ,,DEZ >E MO;NS1 X WD 2
POSSI# =A ?+ 9 SPITE (! PRES;E 9 X (
MO;N 6BE N 9 MO;N B AT RE/1 :ILE IF !Y
>E N MO;NS1 X WD 2 POSSI# = MO;N 6BE
-POS$ ( "S?+ O!R ?AN MO;NS4
  ,& IF L5G? & MO;N >E ?US 9DIVISI#1 X
IS NEI M NOR LESS NEC T "T AL 2 SIMIL>LY
9DIVISI#1 T IS 6SAY 2 -POS$ ( 9DIVISI#
MO;TS3 = IF ! :OLE 4T.E IS DIVISI# & AN
EQUAL VELOC;Y W CAUSE A ?+ 6PASS "? LESS
( X 9 LESS "T1 ! "T M/ AL 2 DIVISI#1 &
3V]SELY1 IF ! "T 9 : A ?+ IS C>RI$ OV] !
SEC;N ,A IS DIVISI#1 ? SEC;N ,A M/ AL 2
DIVISI#4

                   #B

  ,& S9CE E MAGNITUDE IS DIVISI#
96MAGNITUDES-= WE H %[N T X IS IMPOSSI#
= ANY?+ 3T9U\S 6BE -POS$ ( 9DIVISI# "PS1
& E MAGNITUDE IS 3T9U\S-X NECESS>ILY
FOLL[S T ! QK] ( TWO ?+S TRAV]SES A GRT]
MAGNITUDE 9 AN EQUAL "T1 AN EQUAL
MAGNITUDE 9 LESS "T1 &A GRT] MAGNITUDE 9
LESS "T1 9 3=M;Y )! DEF9I;N "S"TS GIV5 (
,8! QK]0'4 ,SUPPOSE T ,A IS QK] ?AN ,B4
,N[ S9CE ( TWO ?+S T : *ANGES SOON] IS
QK]1 9 ! "T ,,ZH1 9 : ,A HAS *ANG$ F ;,G
TO ;,D1 ;,B W N YET H >RIV$ AT ;,D B W 2
%ORT ( X3 S T 9 AN EQUAL "T ! QK] W PASS
OV] A GRT] MAGNITUDE4 ,M ?AN ?1 X W PASS
OV] A GRT] MAGNITUDE 9 LESS "T3 = 9 ! "T
9 : ,A HAS >RIV$ AT ;,D1 ;,B 2+ ! SL[]
HAS >RIV$1 LET U SAY1 AT ,E4 ,!N S9CE ,A
HAS O3UPI$ ! :OLE "T ,,ZH 9 >RIV+ AT
;,D1 W H >RIV$ AT ,O 9 LESS "T ?AN ?1
SAY ,,ZK4 ,N[ ! MAGNITUDE ,,G T ,A HAS
PASS$ OV] IS GRT] ?AN ! MAGNITUDE ,,GE1
&! "T ,,ZK IS LESS ?AN ! :OLE "T ,,ZH3 S
T ! QK] W PASS OV] A GRT] MAGNITUDE 9
LESS "T4 ,& F ? X IS AL CLE> T ! QK] W
PASS OV] AN EQUAL MAGNITUDE 9 LESS "T
?AN ! SL[]4 ,= S9CE X PASSES OV] ! GRT]
MAGNITUDE 9 LESS "T ?AN ! SL[]1 &
7REG>D$ 0XF7 PASSES OV] ,,LM ! GRT] 9 M
"T ?AN ,,LX ! LESS]1 ! "T ,,PR,'H 9 : X
PASSES OV] ,,LM W 2 M ?AN ! "T ,,PS1 : X
PASSES OV] ,,LX3 S T1 ! "T ,,PR,'H 2+
LESS ?AN ! "T ,,PC,'H 9 : ! SL[] PASSES
OV] ,,LX1 ! "T ,,PS W AL 2 LESS ?AN ! "T
,,PX3 = X IS LESS ?AN ! "T ,,PR,'H1 & T
: IS LESS ?AN "S?+ ELSE T IS LESS ?AN A
?+ IS AL XF LESS ?AN T ?+4 ,H;E X FOLL[S
T ! QK] W TRAV]SE AN EQUAL MAGNITUDE 9
LESS "T ?AN ! SL[]4 ,AG1 S9CE ! MO;N (
ANY?+ M/ ALW O3UPY EI AN EQUAL "T OR
LESS OR M "T 9 -P>ISON ) T ( ANO!R ?+1 &
S9CE1 ":AS A ?+ IS SL[] IF XS MO;N
O3UPIES M "T &( EQUAL VELOC;Y IF XS MO;N
O3UPIES AN EQUAL "T1 ! QK] IS NEI (
EQUAL VELOC;Y NOR SL[]1 X FOLL[S T !
MO;N (! QK] C O3UPY NEI AN EQUAL "T NOR
M "T4 ,X C ONLY BE1 !N1 T X O3UPIES LESS
"T1 & ?US WE GET ! NEC 3SEQU;E T ! QK] W
PASS OV] AN EQUAL MAGNITUDE 7Z WELL Z A
GRT]7 9 LESS "T ?AN ! SL[]4
  ,& S9CE E MO;N IS 9 "T &A MO;N MAY
O3UPY ANY "T1 &! MO;N ( "EY?+ T IS 9
MO;N MAY 2 EI QK] OR SL[]1 BO? QK] MO;N
& SL[] MO;N MAY O3UPY ANY "T3 & ? 2+ S1
X NECESS>ILY FOLL[S T "T AL IS 3T9U\S4
,03T9U\S ,I M1N T : IS DIVISI#
96DIVISI#S T >E 9F9ITELY DIVISI#3 & IF
WE TAKE ? Z ! DEF9I;N ( 3T9U\S1 X FOLL[S
NECESS>ILY T "T IS 3T9U\S4 ,= S9CE X HAS
BE5 %[N T ! QK] W PASS OV] AN EQUAL
MAGNITUDE 9 LESS "T ?AN ! SL[]1 SUPPOSE
T ,A IS QK] & ;,B SL[]1 & T ! SL[] HAS
TRAV]S$ ! MAGNITUDE ;,,GD 9 ! "T ,,ZH4
,N[ X IS CLE> T ! QK] W TRAV]SE ! SAME
MAGNITUDE 9 LESS "T ?AN ?3 LET U SAY 9 !
"T ,,ZO4 ,AG1 S9CE ! QK] HAS PASS$ OV] !
:OLE ;,D 9 ! "T ,,ZO1 ! SL[] W 9 ! SAME
"T PASS OV] ,,GK1 SAY1 : IS LESS ?AN
;,,GD4 ,& S9CE ;,B1 ! SL[]1 HAS PASS$
OV] ,,GK 9 ! "T ,,ZO1 ! QK] W PASS OV] X
9 LESS "T3 S T ! "T ,,ZO W AG 2 DIVID$4
,& IF ? IS DIVID$ ! MAGNITUDE ,,GK W AL
2 DIVID$ J Z ;,,GD WAS3 & AG1 IF !
MAGNITUDE IS DIVID$1 ! "T W AL 2 DIVID$4
,& WE C C>RY ON ? PROCESS = "E1 TAK+ !
SL[] AF ! QK] &! QK] AF ! SL[]
ALT]NATELY1 & US+ :AT HAS BE5 DEMON/RAT$
AT EA* /AGE Z A NEW PO9T ( DE"PURE3 =!
QK] W DIVIDE ! "T &! SL[] W DIVIDE !
L5G?4 ,IF1 !N1 ? ALT]N,N ALW HOLDS GD1 &
AT E TURN 9VOLVES A DIVI.N1 X IS EVID5T
T ALL "T M/ 2 3T9U\S4 ,& AT ! SAME "T X
IS CLE> T ALL MAGNITUDE IS AL 3T9U\S2 =!
DIVI.NS ( : "T & MAGNITUDE RESPECTIVELY
>E SUSCEPTI# >E ! SAME & EQUAL4
  ,MOREOV]1 ! CURR5T POPUL> >GU;TS MAKE
X PLA9 T1 IF "T IS 3T9U\S1 MAGNITUDE IS
3T9U\S AL1 9ASM* Z A ?+ ASSES OV] HALF A
GIV5 MAGNITUDE 9 HALF ! "T TAK5 6COV] !
:OLE3 9 FACT )\T QUALIFIC,N X PASSES OV]
A LESS MAGNITUDE 9 LESS "T2 =! DIVI.NS (
"T &( MAGNITUDE W 2 ! SAME4 ,& IF EI IS
9F9ITE1 S IS ! O!R1 &! "O IS S 9 ! SAME
WAY Z ! O!R2 I4E4 IF "T IS 9F9ITE 9
RESPECT ( XS EXTREMITIES1 L5G? IS AL
9F9ITE 9 RESPECT ( XS EXTREMITIES3 IF "T
IS 9F9ITE 9 RESPECT ( DIVISIBIL;Y1 L5G?
IS AL 9F9ITE 9 RESPECT ( DIVISIBIL;Y3 &
IF "T IS 9F9ITE 9 BO? RESPECTS1
MAGNITUDE IS AL 9F9ITE 9 BO? RESPECTS4
  ,H;E ,Z5O'S >GU;T MAKES A FALSE
ASSUMP;N 9 ASS]T+ T X IS IMPOSSI# =A ?+
6PASS OV] OR S"E,Y 6-E 9 3TACT ) 9F9ITE
?+S 9 A F9ITE "T4 ,= "! >E TWO S5SES 9 :
L5G? & "T & G5],Y ANY?+ 3T9U\S >E CALL$
,89F9ITE0'3 !Y >E CALL$ S EI 9 RESPECT (
DIVISIBIL;Y OR 9 RESPECT ( _!
EXTREMITIES4 ,S :ILE A ?+ 9 A F9ITE "T
_C -E 9 3TACT ) ?+S QUANTITATIVELY
9F9ITE1 X C -E 9 3TACT ) ?+S 9F9ITE 9
RESPECT ( DIVISIBIL;Y3 = 9 ? S5SE ! "T
XF IS AL 9F9ITE3 & S WE F9D T ! "T
O3UPI$ 0! PASSAGE OV] ! 9F9ITE IS N A
F9ITE B AN 9F9ITE "T1 &! 3TACT )!
9F9ITES IS MADE 0M1NS ( MO;TS N F9ITE B
9F9ITE 9 NUMB]4
  ,! PASSAGE OV] ! 9F9ITE1 !N1 _C O3UPY
A F9ITE "T1 &! PASSAGE OV] ! F9ITE _C
O3UPY AN 9F9ITE "T3 IF ! "T IS 9F9ITE !
MAGNITUDE M/ 2 9F9ITE AL1 & IF !
MAGNITUDE IS 9F9ITE1 S AL IS ! "T4 ,?
MAY 2 %[N Z FOLL[S4 ,LET ;,,AB 2 A F9ITE
MAGNITUDE1 & LET U SUPPOSE T X IS
TRAV]S$ 9 9F9ITE "T ;,G1 & LET A F9ITE
P]IOD ;,,GD (! "T 2 TAK54 ,N[ 9 ? P]IOD
! ?+ 9 MO;N W PASS OV] A C]TA9 SEG;T (!
MAGNITUDE3 LET ,,2 2 ! SEG;T T X HAS ?US
PASS$ OV]4 7,? W 2 EI AN EXACT M1SURE (
;,,AB OR LESS OR GRT] ?AN AN EXACT
M1SURE3 X MAKES NO DI6];E : X IS47 ,!N1
S9CE A MAGNITUDE EQUAL 6,,BE W ALW 2
PASS$ OV] 9 AN EQUAL "T1 & ,,2 M1SURES !
:OLE MAGNITUDE1 ! :OLE "T O3UPI$ 9 PASS+
OV] ;,,AB W 2 F9ITE3 = X W 2 DIVISI#
96P]IODS EQUAL 9 NUMB] 6! SEG;TS 96: !
MAGNITUDE IS DIVISI#4 ,MOREOV]1 IF X IS
! CASE T 9F9ITE "T IS N O3UPI$ 9 PASS+
OV] E MAGNITUDE1 B X IS POSSI# 6ASS OV]
"S MAGNITUDE1 SAY ,,BE1 9 A F9ITE "T1 &
IF ? ,,2 M1SURES ! :OLE ( : X IS A "P1 &
IF AN EQUAL MAGNITUDE IS PASS$ OV] 9 AN
EQUAL "T1 !N X FOLL[S T ! "T L !
MAGNITUDE IS F9ITE4 ,T 9F9ITE "T W N 2
O3UPI$ 9 PASS+ OV] ,,2 IS EVID5T IF ! "T
2 TAK5 Z LIMIT$ 9 "O DIREC;N3 = Z ! "P W
2 PASS$ OV] 9 LESS "T ?AN ! :OLE1 ! "T
O3UPI$ 9 TRAV]S+ ? "P M/ 2 F9ITE1 !
LIMIT 9 "O DIREC;N 2+ GIV54 ,! SAME
R1SON+ W AL %[ ! FALS;Y (! ASSUMP;N T
9F9ITE L5G? C 2 TRAV]S$ 9 A F9ITE "T4 ,X
IS EVID5T1 !N1 F :AT HAS BE5 SD T NEI A
L9E NOR A SURFACE NOR 9 FACT ANY?+
3T9U\S C 2 9DIVISI#4
  ,? 3CLU.N FOLL[S N ONLY F ! PRES5T
>GU;T B F ! 3SID],N T ! OPPOSITE
ASSUMP;N IMPLIES ! DIVISIBIL;Y (!
9DIVISI#4 ,= S9CE ! 4T9C;N ( QK] & SL[]
MAY APPLY 6MO;NS O3UPY+ ANY P]IOD ( "T &
9 AN EQUAL "T ! QK] PASSES OV] A GRT]
L5G?1 X MAY HAPP5 T X W PASS OV] A L5G?
TWICE1 OR "O &A HALF "TS1 Z GRT Z T
PASS$ OV] 0! SL[]3 = _! RESPECTIVE
VELOCITIES MAY /& 6"O ANO!R 9 ?
PROPOR;N4 ,SUPPOSE1 !N1 T ! QK] HAS 9 !
SAME "T BE5 C>RI$ OV] A L5G? "O &A HALF
"TS Z GRT Z T TRAV]S$ 0! SL[]1 & T !
RESPECTIVE MAGNITUDES >E DIVID$1 T (!
QK]1 ! MAGNITUDE ,,ABGD1 96?REE
9DIVISI#S1 & T (! SL[] 96! TWO 9DIVISI#S
,,EZ1 ,,ZH4 ,!N ! "T MAY AL 2 DIVID$
96?REE 9DIVISI#S1 = AN EQUAL MAGNITUDE W
2 PASS$ OV] 9 AN EQUAL "T4 ,SUPPOSE !N T
X IS ?US DIVID$ 96,,KL1 ,,LM1 ,,MN4 ,AG1
S9CE 9 ! SAME "T ! SL[] HAS BE5 C>RI$
OV] ,,EZ1 ,,ZH1 ! "T MAY AL 2 SIMIL>LY
DIVID$ 96TWO4 ,?US ! 9DIVISI# W 2
DIVISI#1 & T : HAS NO "PS W 2 PASS$ OV]
N 9 AN 9DIVISI# B 9 A GRT] "T4 ,X IS
EVID5T1 "!=E1 T NO?+ 3T9U\S IS )\T "PS4
  #C
  ,! PRES5T AL IS NECESS>ILY 9DIVISI#-!
PRES5T1 T IS1 N 9 ! S5SE 9 : ! ^W IS
APPLI$ 6"O ?+ 9 VIRTUE ( ANO!R1 B 9 XS
PROP] & PRIM>Y S5SE2 9 : S5SE X IS 9H]5T
9 ALL "T4 ,=! PRES5T IS "S?+ T IS AN
EXTREM;Y (! PA/ 7NO "P (! FUTURE 2+ ON ?
SIDE ( X7 & AL (! FUTURE 7NO "P (! PA/
2+ ON ! O!R SIDE ( X73 X IS1 Z WE H SD1
A LIMIT ( BO?4 ,& IF X IS ONCE %[N T X
IS ESS5TI,Y ( ? "* & "O &! SAME1 X W AT
ONCE 2 EVID5T AL T X IS 9DIVISI#4
  ,N[ ! PRES5T T IS ! EXTREM;Y ( BO? "TS
M/ 2 "O &! SAME3 = IF EA* EXTREM;Y 7
DI6]5T1 ! "O CD N 2 9 SU3ES.N 6! O!R1 2C
NO?+ 3T9U\S C 2 -POS$ ( ?+S HAV+ NO "PS3
& IF ! "O IS A"P F ! O!R1 "! W 2 "T
9T]M$IATE 2T !M1 2C "EY?+ 3T9U\S IS S* T
"! IS "S?+ 9T]M$IATE 2T XS LIMITS &
DESCRIB$ 0! SAME "N Z XF4 ,B IF !
9T]M$IATE ?+ IS "T1 X W 2 DIVISI#3 = ALL
"T HAS BE5 %[N 6BE DIVISI#4 ,?US ON ?
ASSUMP;N ! PRES5T IS DIVISI#4 ,B IF !
PRES5T IS DIVISI#1 "! W 2 "P (! PA/ 9 !
FUTURE & "P (! FUTURE 9 ! PA/3 = PA/ "T
W 2 M>K$ (F F FUTURE "T AT ! ACTUAL PO9T
( DIVI.N4 ,AL ! PRES5T W 2 A PRES5T N 9
! PROP] S5SE B 9 VIRTUE ( "S?+ ELSE3 =!
DIVI.N : YIELDS X W N 2 A DIVI.N PROP]4
,FUR!RMORE1 "! W 2 A "P (! PRES5T T IS
PA/ &A "P T IS FUTURE1 & X W N ALW 2 !
SAME "P T IS PA/ OR FUTURE3 9 FACT "O &!
SAME PRES5T W N 2 SIMULTANE\S3 =! "T MAY
2 DIVID$ AT _M PO9TS4 ,IF1 "!=E1 !
PRES5T _C POSSIBLY H ^! "*I/ICS1 X
FOLL[S T X M/ 2 ! SAME PRES5T T 2L;GS
6EA* (! TWO "TS4 ,B IF ? IS S X IS
EVID5T T ! PRES5T IS AL 9DIVISI#3 = IF X
IS DIVISI# X W 2 9VOLV$ 9 ! SAME
IMPLIC,NS Z 2F4 ,X IS CLE>1 !N1 F :AT
HAS BE5 SD T "T 3TA9S "S?+ 9DIVISI#1 & ?
IS :AT WE CALL A PRES5T4
  ,WE W N[ %[ T NO?+ C 2 9 MO;N 9 A
PRES5T4 ,= IF ? IS POSSI#1 "! C 2 BO?
QK] & SL[] MO;N 9 ! PRES5T4 ,SUPPOSE !N
T 9 ! PRES5T ;,N ! QK] HAS TRAV]S$ !
4T.E ;,,AB4 ,T 2+ S1 ! SL[] W 9 ! SAME
PRES5T TRAV]SE A 4T.E LESS ?AN ;,,AB1
SAY ;,,AG4 ,B S9CE ! SL[] W H O3UPI$ !
:OLE PRES5T 9 TRAV]S+ ;,,AG1 ! QK] W
O3UPY LESS ?AN ? 9 TRAV]S+ X4 ,?US WE %
H A DIVI.N (! PRES5T1 ":AS WE F.D X 6BE
9DIVISI#4 ,X IS IMPOSSI#1 "!=E1 = ANY?+
6BE 9 MO;N 9 A PRES5T4
  ,NOR C ANY?+ 2 AT RE/ 9 A PRES5T3 =1 Z
WE 7 SAY+1 ONLY C 2 AT RE/ : IS NATUR,Y
DESIGN$ 6BE 9 MO;N B IS N 9 MO;N :51 ":1
OR Z X WD NATUR,Y 2 S3 S9CE1 "!=E1 NO?+
IS NATUR,Y DESIGN$ 6BE 9 MO;N 9 A
PRES5T1 X IS CLE> T NO?+ C 2 AT RE/ 9 A
PRES5T EI4
  ,MOREOV]1 9ASM* Z X IS ! SAME PRES5T T
2L;GS 6BO? ! "TS1 & X IS POSSI# =A ?+
6BE 9 MO;N "?\T "O "T & 6BE AT RE/ "?\T
! O!R1 & T : IS 9 MO;N OR AT RE/ =! :OLE
(A "T W 2 9 MO;N OR AT RE/ Z ! CASE MAY
2 9 ANY "P ( X 9 : X IS NATUR,Y DESIGN$
6BE 9 MO;N OR AT RE/3 ? 2+ S1 ! ASSUMP;N
T "! C 2 MO;N OR RE/ 9 A PRES5T W C>RY )
X ! IMPLIC,N T ! SAME ?+ C AT ! SAME "T
2 AT RE/ & 9 MO;N3 = BO? ! "TS H ! SAME
EXTREM;Y1 VIZ4 ! PRES5T4
  ,AG1 :5 WE SAY T A ?+ IS AT RE/1 WE
IMPLY T XS 3DI;N 9 :OLE & 9 "P IS AT !
"T ( SP1K+ UNI=M ) :AT X 0 PREVI\SLY3 B
! PRES5T 3TA9S NO ,8PREVI\SLY0'3
3SEQU5TLY1 "! C 2 NO RE/ 9 X4
  ,X FOLL[S !N T ! MO;N ( T : IS 9 MO;N
&! RE/ ( T : IS AT RE/ M/ O3UPY "T4

                   #D

  ,FUR!R1 "EY?+ T *ANGES M/ 2 DIVISI#4
,= S9CE E *ANGE IS F "S?+ 6"S?+1 & :5 A
?+ IS AT ! GOAL ( XS *ANGE X IS NO L;G]
*ANG+1 & :5 BO? X XF & ALL XS "PS >E AT
! />T+-PO9T ( XS *ANGE X IS N *ANG+ 7= T
: IS 9 :OLE & 9 "P 9 AN UNV>Y+ 3DI;N IS
N 9 A /ATE ( *ANGE72 X FOLL[S1 "!=E1 T
"P ( T : IS *ANG+ M/ 2 AT ! />T+-PO9T &
"P AT ! GOAL3 = Z A :OLE X _C 2 9 BO? OR
9 NEI4 7,"H BY ,8GOAL ( *ANGE0' ,I M1N T
: -ES F/ 9 ! PROCESS ( *ANGE3 E4G4 9 A
PROCESS ( *ANGE F :ITE ! GOAL 9 "Q W 2
GREY1 N BLACK3 = X IS N NEC T T T : IS
*ANG+ %D 2 AT EI (! EXTREMES47 ,X IS
EVID5T1 "!=E1 T "EY?+ T *ANGES M/ 2
DIVISI#4
  ,N[ MO;N IS DIVISI# 9 TWO S5SES4 ,9 !
F/ PLACE X IS DIVISI# 9 VIRTUE (! "T T X
O3UPIES4 ,9 ! SECOND PLACE X IS DIVISI#
AC 6! MO;NS (! S"EAL "PS ( T : IS 9
MO;N3 E4G4 IF ! :OLE ;,,AG IS 9 MO;N1 "!
W 2 A MO;N ( ;,,AB &A MO;N ( ,,BG4 ,T 2+
S1 LET ,,DE 2 ! MO;N (! "P ;,,AB & ,,EZ
! MO;N (! "P ,,BG4 ,!N ! :OLE ,,DZ M/ 2
! MO;N ( ;,,AG3 = ,,DZ M/ 3/ITUTE ! MO;N
( ;,,AG 9ASM* Z ,,DE & ,,EZ S"E,Y
3/ITUTE ! MO;NS ( EA* ( XS "PS4 ,B !
MO;N (A ?+ C N"E 2 3/ITUT$ 0! MO;N (
"S?+ ELSE3 3SEQU5TLY ! :OLE MO;N IS !
MO;N (! :OLE MAGNITUDE4
  ,AG1 S9CE E MO;N IS A MO;N ( "S?+1 &!
:OLE MO;N ,,DZ IS N ! MO;N ( EI (! "PS
7= EA* (! "PS ,,DE1 ,,EZ IS ! MO;N ( "O
(! "PS ;,,AB1 ,,BG7 OR ( ANY?+ ELSE 7=1
! :OLE MO;N 2+ ! MO;N (A :OLE1 ! "PS (!
MO;N >E ! MO;NS (! "PS ( T :OLE3 &! "PS
( ,,DZ >E ! MO;NS ( ;,,AB1 ,,BG &( NO?+
ELSE3 =1 Z WE SAW1 A MO;N T IS "O _C 2 !
MO;N ( M ?+S ?AN "O73 S9CE ? IS S1 !
:OLE MO;N W 2 ! MO;N (! MAGNITUDE ,,ABG4
  ,AG1 IF "! IS A MO;N (! :OLE O!R ?AN
,,DZ1 SAY ! ! ( EA* (! >TS MAY 2
SUBTRACT$ F X3 & ^! MO;NS W 2 EQUAL
6,,DE1 ,,EZ RESPECTIVELY3 =! MO;N ( T :
IS "O M/ 2 "O4 ,S IF ! :OLE MO;N ,,OI
MAY 2 DIVID$ 96! MO;NS (! "PS1 ,,OI W 2
EQUAL 6,,DZ3 IF ON ! O!R H& "! IS ANY
REMA9D]1 SAY ,,KI1 ? W 2 A MO;N ( NO?+3
= X C 2 ! MO;N NEI (! :OLE NOR (! "PS 7Z
! MO;N ( T : IS "O M/ 2 "O7 NOR ( ANY?+
ELSE3 =A MO;N T IS 3T9U\S M/ 2 ! MO;N (
?+S T >E 3T9U\S4 ,&! SAME RESULT FOLL[S
IF ! DIVI.N ( ,,OI REV1LS A SURPLUS ON !
SIDE (! MO;NS (! "PS4 ,3SEQU5TLY1 IF ?
IS IMPOSSI#1 ! :OLE MO;N M/ 2 ! SAME Z &
EQUAL 6,,DZ4
  ,? !N IS :AT IS M1NT 0! DIVI.N ( MO;N
AC 6! MO;NS (! "PS3 & X M/ 2 APPLICA#
6"EY?+ T IS DIVISI# 96"PS4
  ,MO;N IS AL SUSCEPTI# ( ANO!R K9D (
DIVI.N1 T AC 6"T4 ,= S9CE ALL MO;N IS 9
"T & ALL "T IS DIVISI#1 & 9 LESS "T !
MO;N IS LESS1 X FOLL[S T E MO;N M/ 2
DIVISI# AC 6"T4 ,& S9CE "EY?+ T IS 9
MO;N IS 9 MO;N 9 A C]TA9 SPH]E &=A C]TA9
"T & HAS A MO;N 2L;G+ 6X1 X FOLL[S T !
"T1 ! MO;N1 ! 2+-IN-MO;N1 ! ?+ T IS 9
MO;N1 &! SPH]E (! MO;N M/ ALL 2
SUSCEPTI# (! SAME DIVI.NS 7?\< SPH]ES (
MO;N >E N ALL DIVISI# 9 A L MANN]3 ?US
QUANT;Y IS ESS5TI,Y1 QUAL;Y A3ID5T,Y
DIVISI#74 ,= SUPPOSE T ,A IS ! "T O3UPI$
0! MO;N ,B4 ,!N IF ALL ! "T HAS BE5
O3UPI$ 0! :OLE MO;N1 X W TAKE LESS (!
MO;N 6O3UPY HALF ! "T1 LESS AG 6O3UPY A
FUR!R SUBDIVI.N (! "T1 & S ON 69F9;Y4
,AG1 ! "T W 2 DIVISI# SIMIL>LY 6! MO;N3
= IF ! :OLE MO;N O3UPIES ALL ! "T HALF !
MO;N W O3UPY HALF ! "T1 & LESS (! MO;N
AG W O3UPY LESS (! "T4
  ,9 ! SAME WAY ! 2+-IN-MO;N W AL 2
DIVISI#4 ,= LET ;,G 2 ! :OLE 2+-IN-MO;N4
,!N ! 2+-IN-MO;N T CORRESPONDS 6HALF !
MO;N W 2 LESS ?AN ! :OLE 2+-IN-MO;N1 T :
CORRESPONDS 6A QU>T] (! MO;N W 2 LESS
AG1 & S ON 69F9;Y4 ,MOREOV] 0SETT+ \
SU3ESSIVELY ! 2+-IN-MO;N CORRESPOND+
6EA* (! TWO MO;NS ,,DG 7SAY7 & ,,GE1 WE
MAY >GUE T ! :OLE 2+-IN-MO;N W
CORRESPOND 6! :OLE MO;N 7= IF X 7 "S O!R
2+-IN-MO;N T CORRESPOND$ 6! :OLE MO;N1
"! WD 2 M ?AN "O 2+-IN MO;N CORRESPOND+
6! SAME MO;N71 ! >GU;T 2+ ! SAME Z T
":BY WE %[$ T ! MO;N (A ?+ IS DIVISI#
96! MO;NS (! "PS (! ?+3 = IF WE TAKE
SEP>ATELY ! 2+-IN MO;N CORRESPOND+ 6EA*
(! TWO MO;NS1 WE % SEE T ! :OLE 2+-IN
MO;N IS 3T9U\S4
  ,! SAME R1SON+ W %[ ! DIVISIBIL;Y (!
L5G?1 & 9 FACT ( "EY?+ T =MS A SPH]E (
*ANGE 7?\< "S ( ^! >E ONLY A3ID5T,Y
DIVISI# 2C T : *ANGES IS S73 =! DIVI.N (
"O T]M W 9VOLVE ! DIVI.N ( ALL4 ,S1 TOO1
9 ! MATT] ( _! 2+ F9ITE OR 9F9ITE1 !Y W
ALL ALIKE 2 EI ! "O OR ! O!R4 ,& WE N[
SEE T 9 MO/ CASES ! FACT T ALL ! T]MS >E
DIVISI# OR 9F9ITE IS A DIRECT 3SEQU;E (!
FACT T ! ?+ T *ANGES IS DIVISI# OR
9F9ITE3 =! ATTRIBUTES ,8DIVISI#0' &
,89F9ITE0' 2L;G 9 ! F/ 9/.E 6! ?+ T
*ANGES4 ,T DIVISIBIL;Y DOES S WE H ALR
%[N3 T 9F9;Y DOES S W 2 MADE CLE> 9 :AT
FOLL[S8
  #E
  ,S9CE "EY?+ T *ANGES *ANGES F "S?+
6"S?+1 T : HAS *ANG$ M/ AT ! MO;T :5 X
HAS F/ *ANG$ 2 9 T 6: X HAS *ANG$4 ,= T
: *ANGES RETIRES F OR L1VES T F : X
*ANGES3 & L1V+1 IF N ID5TICAL ) *ANG+1
IS AT ANY RATE A 3SEQU;E ( X4 ,& IF L1V+
IS A 3SEQU;E ( *ANG+1 HAV+ LEFT IS A
3SEQU;E ( HAV+ *ANG$3 = "! IS A L REL,N
2T ! TWO 9 EA* CASE4
  ,"O K9D ( *ANGE1 !N1 2+ *ANGE 9 A
REL,N ( 3TRADIC;N1 ": A ?+ HAS *ANG$ F
N-2+ 62+ X HAS LEFT N-2+4 ,"!=E X W 2 9
2+3 = "EY?+ M/ EI 2 OR N BE4 ,X IS
EVID5T1 !N1 T 9 3TRADICTORY *ANGE T :
HAS *ANG$ M/ 2 9 T 6: X HAS *ANG$4 ,& IF
? IS TRUE 9 ? K9D ( *ANGE1 X W 2 TRUE 9
ALL O!R K9DS Z WELL3 = 9 ? MATT] :AT
HOLDS GD 9 ! CASE ( "O W HOLD GD
LIKEWISE 9 ! CASE (! RE/4
  ,MOREOV]1 IF WE TAKE EA* K9D ( *ANGE
SEP>ATELY1 ! TRU? ( \R 3CLU.N W 2 EQU,Y
EVID5T1 ON ! GR.D T T T : HAS *ANG$ M/ 2
"S": OR 9 "S?+4 ,=1 S9CE X HAS LEFT T F
: X HAS *ANG$ & M/ 2 "S":1 X M/ 2 EI 9 T
6: X HAS *ANG$ OR 9 "S?+ ELSE4 ,IF1 !N1
T : HAS *ANG$ TO ;,B IS 9 "S?+ O!R ?AN
;,B1 SAY ;,G1 X M/ AG 2 *ANG+ F ;,G TO
;,B3 = X _C 2 ASSUM$ T "! IS NO 9T]VAL
2T ;,G & ;,B1 S9CE *ANGE IS 3T9U\S4 ,?US
WE H ! RESULT T ! ?+ T HAS *ANG$1 AT !
MO;T :5 X HAS *ANG$1 IS *ANG+ 6T 6: X
HAS *ANG$1 : IS IMPOSSI#3 T : HAS *ANG$1
"!=E1 M/ 2 9 T 6: X HAS *ANG$4 ,S X IS
EVID5T LIKEWISE T T T : HAS -E 6BE1 AT !
MO;T :5 X HAS -E 6BE1 W BE1 & T : HAS
C1S$ 6BE W N-BE3 = :AT WE H SD APPLIES
UNIV]S,Y 6E K9D ( *ANGE1 & XS TRU? IS
MO/ OBVI\S 9 ! CASE ( 3TRADICTORY *ANGE4
,X IS CLE>1 !N1 T T : HAS *ANG$1 AT !
MO;T :5 X HAS F/ *ANG$1 IS 9 T 6: X HAS
*ANG$4
  ,WE W N[ %[ T ! ,8PRIM>Y :50' 9 : T :
HAS *ANG$ E6ECT$ ! -PLE;N ( XS *ANGE M/
2 9DIVISI#1 ": BY ,8PRIM>Y0' ,I M1N
POSSESS+ ! "*I/ICS 9 "Q ( XF & N 9
VIRTUE (! POSSES.N ( !M 0"S?+ ELSE 2L;G+
6X4 ,= LET ;,,AG 2 DIVISI#1 & LET X 2
DIVID$ AT ,B4 ,IF !N ! -PLE;N ( *ANGE
HAS BE5 E6ECT$ 9 ;,,AB OR AG 9 ,,BG1
;,,AG _C 2 ! PRIM>Y ?+ 9 : ! -PLE;N (
*ANGE HAS BE5 E6ECT$4 ,IF1 ON ! O!R H&1
X HAS BE5 *ANG+ 9 BO? ;,,AB & ,,BG 7= X
M/ EI H *ANG$ OR 2 *ANG+ 9 EA* ( !M71 X
M/ H BE5 *ANG+ 9 ! :OLE ;,,AG3 B \R
ASSUMP;N 0 T ;,,AG 3TA9S ONLY ! -PLE;N (
! *ANGE4 ,X IS EQU,Y IMPOSSI# 6SUPPOSE T
"O "P ( ;,,AG 3TA9S ! PROCESS &! O!R !
-PLE;N (! *ANGE3 = !N WE % H "S?+ PRIOR
6:AT IS PRIM>Y4 ,S T 9 : ! -PLE;N (
*ANGE HAS BE5 E6ECT$ M/ 2 9DIVISI#4 ,X
IS AL EVID5T1 "!=E1 T T T 9 : T : HAS
C1S$ 6BE HAS C1S$ 6BE & T 9 : T : HAS -E
6BE HAS -E 6BE >E 9DIVISI#4
  ,B "! >E TWO S5SES (! EXPRES.N ,8!
PRIM>Y :5 9 : "S?+ HAS *ANG$0'4 ,ON ! "O
H& X MAY M1N ! PRIM>Y :5 3TA9+ ! -PLE;N
(! PROCESS ( *ANGE- ! MO;T :5 X IS
CORRECT 6SAY ,8X HAS *ANG$0'3 ON ! O!R
H& X MAY M1N ! PRIM>Y :5 3TA9+ ! 2G9N+ (
! PROCESS ( *ANGE4 ,N[ ! PRIM>Y :5 T HAS
REF];E 6! 5D (! *ANGE IS "S?+ RE,Y
EXI/5T3 =A *ANGE MAY RE,Y 2 -PLET$1 & "!
IS S* A ?+ Z AN 5D ( *ANGE1 : WE H 9
FACT %[N 6BE 9DIVISI# 2C X IS A LIMIT4
,B T : HAS REF];E 6! 2G9N+ IS N EXI/5T
AT ALL3 = "! IS NO S* ?+ Z A 2G9N+ (A
PROCESS ( *ANGE1 &! "T O3UPI$ 0! *ANGE
DOES N 3TA9 ANY PRIM>Y :5 9 : ! *ANGE
2GAN4 ,= SUPPOSE T ,,AD IS S* A PRIM>Y
:54 ,!N X _C 2 9DIVISI#3 =1 IF X W]E1 !
MO;T IMMLY PREC$+ ! *ANGE &! MO;T 9 : !
*ANGE 2G9S WD 2 3SECUTIVE 7& MO;TS _C 2
3SECUTIVE74 ,AG1 IF ! *ANG+ ?+ IS AT RE/
9 ! :OLE PREC$+ "T ,,GA 7= WE MAY
SUPPOSE T X IS AT RE/71 X IS AT RE/ 9 ,A
AL3 S IF ,,AD IS )\T "PS1 X W
SIMULTANE\SLY 2 AT RE/ & H *ANG$3 = X IS
AT RE/ 9 ;,A & HAS *ANG$ 9 ,D4 ,S9CE !N
,,AD IS N )\T "PS1 X M/ 2 DIVISI#1 &!
*ANG+ ?+ M/ H *ANG$ 9 E "P ( X 7= IF X
HAS *ANG$ 9 NEI (! TWO "PS 96: ,,AD IS
DIVID$1 X HAS N *ANG$ 9 ! :OLE EI3 IF1
ON ! O!R H&1 X IS 9 PROCESS ( *ANGE 9
BO? "PS1 X IS LIKEWISE 9 PROCESS ( *ANGE
9 ! :OLE3 & IF1 AG1 X HAS *ANG$ 9 "O (!
TWO "PS1 ! :OLE IS N ! PRIM>Y :5 9 : X
HAS *ANG$3 X M/ "!=E H *ANG$ 9 E "P74 ,X
IS EVID5T1 !N1 T ) REF];E 6! 2G9N+ (
*ANGE "! IS NO PRIM>Y :5 9 : *ANGE HAS
BE5 E6ECT$3 =! DIVI.NS >E 9F9ITE4
  ,S1 TOO1 ( T : HAS *ANG$ "! IS NO
PRIM>Y "P T HAS *ANG$4 ,= SUPPOSE T (
,,AE ! PRIM>Y "P T HAS *ANG$ IS ,,AZ
7"EY?+ T *ANGES HAV+ BE5 %[N 6BE
DIVISI#73 & LET ,,OI 2 ! "T 9 : ,,DZ HAS
*ANG$4 ,IF1 !N1 9 ! :OLE "T ,,DZ HAS
*ANG$1 9 HALF ! "T "! W 2 A "P T HAS
*ANG$1 LESS ?AN & "!=E PRIOR 6,,DZ3 & AG
"! W 2 ANO!R "P PRIOR 6?1 & YET ANO!R1 &
S ON 69F9;Y4 ,?US ( T : *ANGES "! _C 2
ANY PRIM>Y "P T HAS *ANG$4 ,X IS EVID5T1
!N1 F :AT HAS BE5 SD1 T NEI ( T : *ANGES
NOR (! "T 9 : X *ANGES IS "! ANY PRIM>Y
"P4
  ,) REG>D1 H["E1 6! ACTUAL SUBJECT (
*ANGE-T IS 6SAY T 9 RESPECT ( : A ?+
*ANGES-"! IS A DI6];E 6BE OBS]V$4 ,= 9 A
PROCESS ( *ANGE WE MAY 4T+UI% ?REE T]MS-
T : *ANGES1 T 9 : X *ANGES1 &! ACTUAL
SUBJECT ( *ANGE1 E4G4 ! MAN1 ! "T1 &!
FAIR -PLEXION4 ,( ^! ! MAN &! "T >E
DIVISI#3 B )! FAIR -PLEXION X IS O!RWISE
7?\< !Y >E ALL DIVISI# A3ID5T,Y1 = T 9 :
! FAIR -PLEXION OR ANY O!R QUAL;Y IS AN
A3ID5T IS DIVISI#74 ,=( ACTUAL SUBJECTS
( *ANGE X W 2 SE5 T ^? : >E CLASS$ Z
ESS5TI,Y1 N A3ID5T,Y1 DIVISI# H NO
PRIM>Y "P4 ,TAKE ! CASE ( MAGNITUDES3
LET ;,,AB 2 A MAGNITUDE1 & SUPPOSE T X
HAS MOV$ F ;,B 6A PRIM>Y ,8":0' ,G4 ,!N
IF ,,BG IS TAK5 6BE 9DIVISI#1 TWO ?+S
)\T "PS W H 6BE 3TIGU\S 7: IS IMPOSSI#73
IF ON ! O!R H& X IS TAK5 6BE DIVISI#1 "!
W 2 "S?+ PRIOR TO ;,G 6: ! MAGNITUDE HAS
*ANG$1 & "S?+ ELSE AG PRIOR 6T1 & S ON
69F9;Y1 2C ! PROCESS ( DIVI.N MAY 2
3T9U$ )\T 5D4 ,?US "! C 2 NO PRIM>Y
,8":0' 6: A ?+ HAS *ANG$4 ,& IF WE TAKE
! CASE ( QUANTITATIVE *ANGE1 WE % GET A
L RESULT1 = "H TOO ! *ANGE IS 9 "S?+
3T9U\S4 ,X IS EVID5T1 !N1 T ONLY 9
QUALITATIVE MO;N C "! 2 ANY?+ ESS5TI,Y
9DIVISI#4
  #F
  ,N[ "EY?+ T *ANGES *ANGES "T1 & T 9
TWO S5SES3 =! "T 9 : A ?+ IS SD 6*ANGE
MAY 2 ! PRIM>Y "T1 OR ON ! O!R H& X MAY
H AN EXT5D$ REF];E1 Z E4G4 :5 WE SAY T A
?+ *ANGES 9 A "PICUL> YE> 2C X *ANGES 9
A "PICUL> "D4 ,T 2+ S1 T : *ANGES M/ 2
*ANG+ 9 ANY "P (! PRIM>Y "T 9 : X
*ANGES4 ,? IS CLE> F \R DEF9I;N (
,8PRIM>Y0'1 9 : ! ^W IS SD 6EXPRESS J ?3
X MAY AL1 H["E1 2 MADE EVID5T 0! FOLL[+
>GU;T4 ,LET ,*,RH 2 ! PRIM>Y "T 9 : T :
IS 9 MO;N IS 9 MO;N3 & 7Z ALL "T IS
DIVISI#7 LET X 2 DIVID$ AT ,K4 ,N[ 9 !
"T ,*,K X EI IS 9 MO;N OR IS N 9 MO;N1 &
! SAME IS LIKEWISE TRUE (! "T ,,KR,'H4
,!N IF X IS 9 MO;N 9 NEI (! TWO "PS1 X W
2 AT RE/ 9 ! :OLE3 = X IS IMPOSSI# T X
%D 2 9 MO;N 9 A "T 9 NO "P ( : X IS 9
MO;N4 ,IF ON ! O!R H& X IS 9 MO;N 9 ONLY
"O (! TWO "PS (! "T1 ,*,RH _C 2 ! PRIM>Y
"T 9 : X IS 9 MO;N3 = XS MO;N W H REF];E
6A "T O!R ?AN ,*,RH4 ,X M/1 !N1 H BE5 9
MO;N 9 ANY "P ( ,*,RH4
  ,& N[ T ? HAS BE5 PROV$1 X IS EVID5T T
"EY?+ T IS 9 MO;N M/ H BE5 9 MO;N 2F4 ,=
IF T : IS 9 MO;N HAS TRAV]S$ ! 4T.E ,,KL
9 ! PRIM>Y "T ,*,RH1 9 HALF ! "T A ?+ T
IS 9 MO;N ) EQUAL VELOC;Y & 2GAN XS MO;N
AT ! SAME "T W H TRAV]S$ HALF ! 4T.E4 ,B
IF ? SECOND ?+ ^: VELOC;Y IS EQUAL HAS
TRAV]S$ A C]TA9 4T.E 9 A C]TA9 "T1 !
ORIG9AL ?+ T IS 9 MO;N M/ H TRAV]S$ !
SAME 4T.E 9 ! SAME "T4 ,H;E T : IS 9
MO;N M/ H BE5 9 MO;N 2F4
  ,AG1 IF 0TAK+ ! EXTREME MO;T (! "T-= X
IS ! MO;T T DEF9ES ! "T1 & "T IS T : IS
9T]M$IATE 2T MO;TS-WE >E 5A#D 6SAY T
MO;N HAS TAK5 PLACE 9 ! :OLE "T ,*,RH OR
9 FACT 9 ANY P]IOD ( X1 MO;N MAY
LIKEWISE 2 SD 6H TAK5 PLACE 9 E O!R S*
P]IOD4 ,B HALF ! "T F9DS AN EXTREME 9 !
PO9T ( DIVI.N4 ,"!=E MO;N W H TAK5 PLACE
9 HALF ! "T & 9 FACT 9 ANY "P ( X3 = Z
SOON Z ANY DIVI.N IS MADE "! IS ALW A "T
DEF9$ 0MO;TS4 ,IF1 !N1 ALL "T IS
DIVISI#1 & T : IS 9T]M$IATE 2T MO;TS IS
"T1 "EY?+ T IS *ANG+ M/ H -PLET$ AN
9F9ITE NUMB] ( *ANGES4
  ,AG1 S9CE A ?+ T *ANGES 3T9U\SLY & HAS
N P]I%$ OR C1S$ F XS *ANGE M/ EI 2 *ANG+
OR H *ANG$ 9 ANY "P (! "T ( XS *ANGE1 &
S9CE X _C 2 *ANG+ 9 A MO;T1 X FOLL[S T X
M/ H *ANG$ AT E MO;T 9 ! "T3 3SEQU5TLY1
S9CE ! MO;TS >E 9F9ITE 9 NUMB]1 "EY?+ T
IS *ANG+ M/ H -PLET$ AN 9F9ITE NUMB] (
*ANGES4
  ,& N ONLY M/ T : IS *ANG+ H *ANG$1 B T
: HAS *ANG$ M/ AL PREVI\SLY H BE5 *ANG+1
S9CE "EY?+ T HAS *ANG$ F "S?+ 6"S?+ HAS
*ANG$ 9 A P]IOD ( "T4 ,= SUPPOSE T A ?+
HAS *ANG$ F ,A TO ;,B 9 A MO;T4 ,N[ !
MO;T 9 : X HAS *ANG$ _C 2 ! SAME Z T 9 :
X IS AT ,A 7S9CE 9 T CASE X WD 2 9 ;,A &
;,B AT ONCE73 = WE H %[N ABV T T T : HAS
*ANG$1 :5 X HAS *ANG$1 IS N 9 T F : X
HAS *ANG$4 ,IF1 ON ! O!R H&1 X IS A
DI6]5T MO;T1 "! W 2 A P]IOD ( "T
9T]M$IATE 2T ! TWO3 =1 Z WE SAW1 MO;TS
>E N 3SECUTIVE4 ,S9CE1 !N1 X HAS *ANG$ 9
A P]IOD ( "T1 & ALL "T IS DIVISI#1 9
HALF ! "T X W H -PLET$ ANO!R *ANGE1 9 A
QU>T] ANO!R1 & S ON 69F9;Y3 3SEQU5TLY :5
X HAS *ANG$1 X M/ H PREVI\SLY BE5 *ANG+4
  ,MOREOV]1 ! TRU? ( :AT HAS BE5 SD IS M
EVID5T 9 ! CASE ( MAGNITUDE1 2C !
MAGNITUDE OV] : :AT IS *ANG+ *ANGES IS
3T9U\S4 ,= SUPPOSE T A ?+ HAS *ANG$ F
;,G TO ;,D4 ,!N IF ;,,GD IS 9DIVISI#1
TWO ?+S )\T "PS W 2 3SECUTIVE4 ,B S9CE ?
IS IMPOSSI#1 T : IS 9T]M$IATE 2T !M M/ 2
A MAGNITUDE & DIVISI# 96AN 9F9ITE NUMB]
( SEG;TS3 3SEQU5TLY1 2F ! *ANGE IS
-PLET$1 ! ?+ *ANGES 6^? SEG;TS4 ,"EY?+ T
HAS *ANG$1 "!=E1 M/ PREVI\SLY H BE5
*ANG+3 =! SAME PRO( AL HOLDS GD ( *ANGE
) RESPECT 6:AT IS N 3T9U\S1 *ANGES1 T IS
6SAY1 2T 3TR>IES & 2T 3TRADICTORIES4 ,9
S* CASES WE H ONLY 6TAKE ! "T 9 : A ?+
HAS *ANG$ & AG APPLY ! SAME R1SON+4 ,S T
: HAS *ANG$ M/ H BE5 *ANG+ & T : IS
*ANG+ M/ H *ANG$1 &A PROCESS ( *ANGE IS
PREC$$ 0A -PLE;N ( *ANGE &A -PLE;N 0A
PROCESS3 & WE C N"E TAKE ANY /AGE & SAY
T X IS ABSOLUTELY ! F/4 ,! R1SON ( ? IS
T NO TWO ?+S )\T "PS C 2 3TIGU\S1 & "!=E
9 *ANGE ! PROCESS ( DIVI.N IS 9F9ITE1 J
Z L9ES MAY 2 9F9ITELY DIVID$ S T "O "P
IS 3T9U,Y 9CR1S+ &! O!R 3T9U,Y DECR1S+4
  ,S X IS EVID5T AL T T T : HAS 2COME M/
PREVI\SLY H BE5 9 PROCESS ( 2COM+1 & T :
IS 9 PROCESS ( 2COM+ M/ PREVI\SLY H
2COME1 "EY?+ 7T IS7 T IS DIVISI# &
3T9U\S3 ?\< X IS N ALW ! ACTUAL ?+ T IS
9 PROCESS ( 2COM+ ( : ? IS TRUE3 "S"TS X
IS "S?+ ELSE1 T IS 6SAY1 "S "P (! ?+ 9
"Q1 E4G4 ! F.D,N-/"O (A H\SE4 ,S1 TOO1 9
! CASE ( T : IS P]I%+ & T : HAS P]I%$3 =
T : 2COMES & T : P]I%ES M/ 3TA9 AN ELE;T
( 9F9ITE;S Z AN IMM 3SEQU;E (! FACT T !Y
>E 3T9U\S ?+S3 & S A ?+ _C 2 9 PROCESS (
2COM+ )\T HAV+ 2COME OR H 2COME )\T HAV+
BE5 9 PROCESS ( 2COM+4 ,S1 TOO1 9 ! CASE
( P]I%+ & HAV+ P]I%$3 P]I%+ M/ 2 PREC$$
0HAV+ P]I%$1 & HAV+ P]I%$ M/ 2 PREC$$
0P]I%+4 ,X IS EVID5T1 !N1 T T : HAS
2COME M/ PREVI\SLY H BE5 9 PROCESS (
2COM+1 & T : IS 9 PROCESS ( 2COM+ M/
PREVI\SLY H 2COME3 = ALL MAGNITUDES &
ALL P]IODS ( "T >E 9F9ITELY DIVISI#4
  ,3SEQU5TLY NO ABSOLUTELY F/ /AGE (
*ANGE C 2 REPRES5T$ 0ANY "PICUL> "P (
SPACE OR "T : ! *ANG+ ?+ MAY O3UPY4
  #G
  ,N[ S9CE ! MO;N ( "EY?+ T IS 9 MO;N
O3UPIES A P]IOD ( "T1 &A GRT] MAGNITUDE
IS TRAV]S$ 9 A L;G] "T1 X IS IMPOSSI# T
A ?+ %D "UGO A F9ITE MO;N 9 AN 9F9ITE
"T1 IF ? IS "U/OOD 6M1N N T ! SAME MO;N
OR A "P ( X IS 3T9U,Y REP1T$1 B T ! :OLE
9F9ITE "T IS O3UPI$ 0! :OLE F9ITE MO;N4
,9 ALL CASES ": A ?+ IS 9 MO;N ) UNI=M
VELOC;Y X IS CLE> T ! F9ITE MAGNITUDE IS
TRAV]S$ 9 A F9ITE "T4 ,= IF WE TAKE A "P
(! MO;N : % 2 A M1SURE (! :OLE1 ! :OLE
MO;N IS -PLET$ 9 Z _M EQUAL P]IODS (! "T
Z "! >E "PS (! MO;N4 ,3SEQU5TLY1 S9CE ^!
"PS >E F9ITE1 BO? 9 SIZE 9DIVIDU,Y & 9
NUMB] COLLECTIVELY1 ! :OLE "T M/ AL 2
F9ITE3 = X W 2 A MULTIPLE (! POR;N1
EQUAL 6! "T O3UPI$ 9 -PLET+ ! A=ESD "P
MULTIPLI$ 0! NUMB] (! "PS4
  ,B X MAKES NO DI6];E EV5 IF ! VELOC;Y
IS N UNI=M4 ,= LET U SUPPOSE T ! L9E
;,,AB REPRES5TS A F9ITE /RET* OV] : A ?+
HAS BE5 MOV$ 9 ! GIV5 "T1 & LET ;,,GD 2
! 9F9ITE "T4 ,N[ IF "O "P (! /RET* M/ H
BE5 TRAV]S$ 2F ANO!R "P 7? IS CLE>1 T 9
! E>LI] & 9 ! LAT] "P (! "T A DI6]5T "P
(! /RET* HAS BE5 TRAV]S$3 = Z ! "T
L5G!NS A DI6]5T "P (! MO;N W ALW 2
-PLET$ 9 X1 :E!R ! ?+ 9 MO;N *ANGES )
UNI=M VELOC;Y OR N3 & :E!R ! RATE ( MO;N
9CR1SES OR DIM9I%ES OR REMA9S /,N>Y ? IS
N"O ! LESS S71 LET U !N TAKE ,,AE A "P (
! :OLE /RET* ( MO;N ;,,AB : % 2 A M1SURE
( ;,,AB4 ,N[ ? "P (! MO;N O3UPIES A
C]TA9 P]IOD (! 9F9ITE "T3 X _C XF O3UPY
AN 9F9ITE "T1 = WE >E ASSUM+ T T IS
O3UPI$ 0! :OLE ;,,AB4 ,& IF AG ,I TAKE
ANO!R "P EQUAL 6,,AE1 T AL M/ O3UPY A
F9ITE "T 9 3SEQU;E (! SAME ASSUMP;N4 ,&
IF ,I G ON TAK+ "PS 9 ? WAY1 ON ! "O H&
"! IS NO "P : W 2 A M1SURE (! 9F9ITE "T
7=! 9F9ITE _C 2 -POS$ ( F9ITE "PS :E!R
EQUAL OR UNEQUAL1 2C "! M/ 2 "S UN;Y : W
2 A M1SURE ( ?+S F9ITE 9 MULTITUDE OR 9
MAGNITUDE1 :1 :E!R !Y >E EQUAL OR
UNEQUAL1 >E N"O ! LESS LIMIT$ 9
MAGNITUDE72 :ILE ON ! O!R H& ! F9ITE
/RET* ( MO;N ;,,AB IS A C]TA9 MULTIPLE (
,,AE3 3SEQU5TLY ! MO;N ;,,AB M/ 2
A3OMPLI%$ 9 A F9ITE "T4 ,MOREOV] X IS !
SAME ) -+ 6RE/ Z ) MO;N4 ,& S X IS
IMPOSSI# = "O &! SAME ?+ 6BE 9F9ITELY 9
PROCESS ( 2COM+ OR ( P]I%+4 ,! R1SON+ HE
W PROVE T 9 A F9ITE "T "! _C 2 AN 9F9ITE
EXT5T ( MO;N OR ( -+ 6RE/1 :E!R ! MO;N
IS REGUL> OR IRREGUL>4 ,= IF WE TAKE A
"P : % 2 A M1SURE (! :OLE "T1 9 ? "P A
C]TA9 FRAC;N1 N ! :OLE1 (! MAGNITUDE W 2
TRAV]S$1 2C WE ASSUME T ! TRAV]S+ (!
:OLE O3UPIES ALL ! "T4 ,AG1 9 ANO!R
EQUAL "P (! "T ANO!R "P (! MAGNITUDE W 2
TRAV]S$3 & SIMIL>LY 9 EA* "P (! "T T WE
TAKE1 :E!R EQUAL OR UNEQUAL 6! "P
ORIG9,Y TAK54 ,X MAKES NO DI6];E :E!R !
"PS >E EQUAL OR N1 IF ONLY EA* IS F9ITE3
= X IS CLE> T :ILE ! "T IS EXHAU/$ 0!
SUBTRAC;N ( XS "PS1 ! 9F9ITE MAGNITUDE W
N 2 ?US EXHAU/$1 S9CE ! PROCESS (
SUBTRAC;N IS F9ITE BO? 9 RESPECT (!
QUANT;Y SUBTRACT$ &(! NUMB] ( "TS A
SUBTRAC;N IS MADE4 ,3SEQU5TLY ! 9F9ITE
MAGNITUDE W N 2 TRAV]S$ 9 F9ITE "T3 & X
MAKES NO DI6];E :E!R ! MAGNITUDE IS
9F9ITE 9 ONLY "O DIREC;N OR 9 BO?3 =!
SAME R1SON+ W HOLD GD4
  ,? HAV+ BE5 PROV$1 X IS EVID5T T NEI C
A F9ITE MAGNITUDE TRAV]SE AN 9F9ITE
MAGNITUDE 9 A F9ITE "T1 ! R1SON 2+ !
SAME Z T GIV5 ABV3 9 "P (! "T X W
TRAV]SE A F9ITE MAGNITUDE & 9 EA* S"EAL
"P LIKEWISE1 S T 9 ! :OLE "T X W TRAV]SE
A F9ITE MAGNITUDE4
  ,& S9CE A F9ITE MAGNITUDE W N TRAV]SE
AN 9F9ITE 9 A F9ITE "T1 X IS CLE> T NEI
W AN 9F9ITE TRAV]SE A F9ITE 9 A F9ITE
"T4 ,= IF ! 9F9ITE CD TRAV]SE ! F9ITE1 !
F9ITE CD TRAV]SE ! 9F9ITE2 = X MAKES NO
DI6];E : (! TWO IS ! ?+ 9 MO;N2 EI CASE
9VOLVES ! TRAV]S+ (! 9F9ITE 0! F9ITE4 ,=
:5 ! 9F9ITE MAGNITUDE ,A IS 9 MO;N A "P
( X1 SAY ;,,GD1 W O3UPY ! F9ITE & !N
ANO!R1 & !N ANO!R1 & S ON 69F9;Y4 ,?US !
TWO RESULTS W CO9CIDE3 ! 9F9ITE W H
-PLET$ A MO;N OV] ! F9ITE &! F9ITE W H
TRAV]S$ ! 9F9ITE3 = X WD SEEM 6BE
IMPOSSI# =! MO;N (! 9F9ITE OV] ! F9ITE
6O3UR 9 ANY WAY O!R ?AN 0! F9ITE TRAV]S+
! 9F9ITE EI 0LOCOMO;N OV] X OR 0M1SUR+
X4 ,"!=E1 S9CE ? IS IMPOSSI#1 ! 9F9ITE
_C TRAV]SE ! F9ITE4
  ,NOR AG W ! 9F9ITE TRAV]SE ! 9F9ITE 9
A F9ITE "T4 ,O!RWISE X WD AL TRAV]SE !
F9ITE1 =! 9F9ITE 9CLUDES ! F9ITE4 ,WE C
FUR!R PROVE ? 9 ! SAME WAY 0TAK+ ! "T Z
\R />T+-PO9T4
  ,S9CE1 !N1 X IS E/ABLI%$ T 9 A F9ITE
"T NEI W ! F9ITE TRAV]SE ! 9F9ITE1 NOR !
9F9ITE ! F9ITE1 NOR ! 9F9ITE ! 9F9ITE1 X
IS EVID5T AL T 9 A F9ITE "T "! _C 2
9F9ITE MO;N3 = :AT DI6];E DOES X MAKE
:E!R WE TAKE ! MO;N OR ! MAGNITUDE 6BE
9F9ITE8 ,IF EI (! TWO IS 9F9ITE1 ! O!R
M/ 2 S LIKEWISE3 = ALL LOCOMO;N IS 9
SPACE4
  #H
  ,S9CE "EY?+ 6: MO;N OR RE/ IS NATURAL
IS 9 MO;N OR AT RE/ 9 ! NATURAL "T1
PLACE1 & MANN]1 T : IS -+ 6A /&1 :5 X IS
-+ 6A /&1 M/ 2 9 MO;N3 = IF X IS N 9
MO;N X M/ 2 AT RE/3 B T : IS AT RE/ _C 2
-+ 6RE/4 ,F ? X EVID5TLY FOLL[S T -+ 6A
/& M/ O3UPY A P]IOD ( "T3 =! MO;N ( T :
IS 9 MO;N O3UPIES A P]IOD ( "T1 & T : IS
-+ 6A /& HAS BE5 %[N 6BE 9 MO;N3
3SEQU5TLY -+ 6A /& M/ O3UPY A P]IOD (
"T4
  ,AG1 S9CE ! T]MS ,8QK]0' & ,8SL[]0' >E
US$ ONLY ( T : O3UPIES A P]IOD ( "T1 &!
PROCESS ( -+ 6A /& MAY 2 QK] OR SL[]1 !
SAME 3CLU.N FOLL[S4
  ,& T : IS -+ 6A /& M/ 2 -+ 6A /& 9 ANY
"P (! PRIM>Y "T 9 : X IS -+ 6A /&4 ,= IF
X IS -+ 6A /& 9 NEI ( TWO "PS 96: ! "T
MAY 2 DIVID$1 X _C 2 -+ 6A /& 9 ! :OLE
"T1 )! RESULT T T T : IS -+ 6A /& W N 2
-+ 6A /&4 ,IF ON ! O!R H& X IS -+ 6A /&
9 ONLY "O (! TWO "PS (! "T1 ! :OLE _C 2
! PRIM>Y "T 9 : X IS -+ 6A /&3 = X IS -+
6A /& 9 ! :OLE "T N PRIM>ILY B 9 VIRTUE
( "S?+ 4T9CT F XF1 ! >GU;T 2+ ! SAME Z T
: WE US$ ABV AB ?+S 9 MO;N4
  ,& J Z "! IS NO PRIM>Y "T 9 : T : IS 9
MO;N IS 9 MO;N1 S TOO "! IS NO PRIM>Y "T
9 : T : IS -+ 6A /& IS -+ 6A /&1 "! 2+
NO PRIM>Y /AGE EI ( 2+ 9 MO;N OR ( -+ 6A
/&4 ,= LET ;,,AB 2 ! PRIM>Y "T 9 : A ?+
IS -+ 6A /&4 ,N[ ;,,AB _C 2 )\T "PS3 =
"! _C 2 MO;N 9 T : IS )\T "PS1 2C ! MOV+
?+ WD NECESS>ILY H BE5 ALR MOV$ = "P (!
"T ( XS MOVE;T3 & T : IS -+ 6A /& HAS
BE5 %[N 6BE 9 MO;N4 ,B S9CE ;,,AB IS
"!=E DIVISI#1 ! ?+ IS -+ 6A /& 9 E "O (!
"PS ( ;,,AB3 = WE H %[N ABV T X IS -+ 6A
/& 9 E "O (! "PS 9 : X IS PRIM>ILY -+ 6A
/&4 ,S9CE !N1 T 9 : PRIM>ILY A ?+ IS -+
6A /& M/ 2 A P]IOD ( "T & N "S?+
9DIVISI#1 & S9CE ALL "T IS 9F9ITELY
DIVISI#1 "! _C 2 ANY?+ 9 : PRIM>ILY X IS
-+ 6A /&4
  ,NOR AG C "! 2 A PRIM>Y "T AT : ! 2+
AT RE/ ( T : IS AT RE/ O3URR$3 = X _C H
O3URR$ 9 T : HAS NO "PS1 2C "! _C 2 MO;N
9 T : IS 9DIVISI#1 & T 9 : RE/ TAKES
PLACE IS ! SAME Z T 9 : MO;N TAKES
PLACE3 = WE DEF9$ A /ATE ( RE/ 6BE !
/ATE (A ?+ 6: MO;N IS NATURAL B : IS N 9
MO;N :5 7T IS 6SAY 9 T 9 :7 MO;N WD 2
NATURAL 6X4 ,AG1 \R USE (! PHRASE ,82+
AT RE/0' AL IMPLIES T ! PREVI\S /ATE (A
?+ IS / UNALT]$1 N "O PO9T ONLY B TWO AT
L1/ 2+ ?US NE$$ 6DET]M9E XS PRES;E3
3SEQU5TLY T 9 : A ?+ IS AT RE/ _C 2 )\T
"PS4 ,S9CE1 !N X IS DIVISI#1 X M/ 2 A
P]IOD ( "T1 &! ?+ M/ 2 AT RE/ 9 E "O (
XS "PS1 Z MAY 2 %[N 0! SAME ME?OD Z T
US$ ABV 9 SIMIL> DEMON/R,NS4
  ,S "! C 2 NO PRIM>Y "P (! "T3 &! R1SON
IS T RE/ & MO;N >E ALW 9 A P]IOD ( "T1 &
A P]IOD ( "T HAS NO PRIM>Y "P ANY M ?AN
A MAGNITUDE OR 9 FACT ANY?+ 3T9U\S3 =
"EY?+ 3T9U\S IS DIVISI# 96AN 9F9ITE
NUMB] ( "PS4
  ,& S9CE "EY?+ T IS 9 MO;N IS 9 MO;N 9
A P]IOD ( "T & *ANGES F "S?+ 6"S?+1 :5
XS MO;N IS -PRIS$ )9 A "PICUL> P]IOD (
"T ESS5TI,Y-T IS 6SAY :5 X FILLS ! :OLE
& N M]ELY A "P (! "T 9 "Q-X IS IMPOSSI#
T 9 T "T T : IS 9 MO;N %D 2 OV] AG/ "S
"PICUL> ?+ PRIM>ILY4 ,= IF A ?+-XF & EA*
( XS "PS-O3UPIES ! SAME SPACE =A DEF9ITE
P]IOD ( "T1 X IS AT RE/3 = X IS 9 J ^!
CIRCUM/.ES T WE USE ! T]M ,82+ AT RE/'-
:5 AT "O MO;T AF ANO!R X C 2 SD ) TRU? T
A ?+1 XF & XS "PS1 O3UPIES ! SAME SPACE4
,S IF ? IS 2+ AT RE/ X IS IMPOSSI# = T :
IS *ANG+ 6BE Z A :OLE1 AT ! "T :5 X IS
PRIM>ILY *ANG+1 OV] AG/ ANY "PICUL> ?+
7=! :OLE P]IOD ( "T IS DIVISI#71 S T 9
"O "P ( X AF ANO!R X W 2 TRUE 6SAY T !
?+1 XF & XS "PS1 O3UPIES ! SAME SPACE4
,IF ? IS N S &! A=ESD PROPOSI;N IS TRUE
ONLY AT A S+LE MO;T1 !N ! ?+ W 2 OV] AG/
A "PICUL> ?+ N = ANY P]IOD ( "T B ONLY
AT A MO;T T LIMITS ! "T4 ,X IS TRUE T AT
ANY MO;T X IS ALW OV] AG/ "S?+ /,N>Y3 B
X IS N AT RE/3 = AT A MO;T X IS N POSSI#
= ANY?+ 6BE EI 9 MO;N OR AT RE/4 ,S :ILE
X IS TRUE 6SAY T T : IS 9 MO;N IS AT A
MO;T N 9 MO;N & IS OPPOSITE "S "PICUL>
?+1 X _C 9 A P]IOD ( "T 2 OV] AG/ T : IS
AT RE/3 = T WD 9VOLVE ! 3CLU.N T T : IS
9 LOCOMO;N IS AT RE/4
  #I
  ,Z5O'S R1SON+1 H["E1 IS FALLACI\S1 :5
HE SAYS T IF "EY?+ :5 X O3UPIES AN EQUAL
SPACE IS AT RE/1 & IF T : IS 9 LOCOMO;N
IS ALW O3UPY+ S* A SPACE AT ANY MO;T1 !
FLY+ >R[ IS "!=E MO;N.S4 ,? IS FALSE1 =
"T IS N -POS$ ( 9DIVISI# MO;TS ANY M ?AN
ANY O!R MAGNITUDE IS -POS$ ( 9DIVISI#S4
  ,Z5O'S >GU;TS AB MO;N1 : CAUSE S M*
4QUIETUDE 6^? :O TRY 6SOLVE ! PRO#MS T
!Y PRES5T1 >E F\R 9 NUMB]4 ,! F/ ASS]TS
! NON-EXI/;E ( MO;N ON ! GR.D T T : IS 9
LOCOMO;N M/ >RIVE AT ! HALF-WAY /AGE 2F
X >RIVES AT ! GOAL4 ,? WE H 4CUSS$ ABV4
  ,! SECOND IS ! S-CALL$ ,8,A*ILLES0'1 &
X AM.TS 6?1 T 9 A RACE ! QKE/ RUNN] C
N"E OV]TAKE ! SL[E/1 S9CE ! PURSU] M/ F/
R1* ! PO9T :;E ! PURSU$ />T$1 S T ! SL[]
M/ ALW HOLD A L1D4 ,? >GU;T IS ! SAME 9
PR9CIPLE Z T : DEP5DS ON BISEC;N1 ?\< X
DI6]S F X 9 T ! SPACES ) : WE
SU3ESSIVELY H 6D1L >E N DIVID$ 96HALVES4
,! RESULT (! >GU;T IS T ! SL[] IS N
OV]TAK53 B X PROCE$S AL;G ! SAME L9ES Z
! BISEC;N->GU;T 7= 9 BO? A DIVI.N (!
SPACE 9 A C]TA9 WAY L1DS 6! RESULT T !
GOAL IS N R1*$1 ?\< ! ,8,A*ILLES0' GOES
FUR!R 9 T X A6IRMS T EV5 ! QKE/ RUNN] 9
LEG5D>Y TRADI;N M/ FAIL 9 8 PURSUIT (!
SL[E/71 S T ! SOLU;N M/ 2 ! SAME4 ,&!
AXIOM T T : HOLDS A L1D IS N"E OV]TAK5
IS FALSE3 X IS N OV]TAK51 X IS TRUE1
:ILE X HOLDS A L1D3 B X IS OV]TAK5
N"E!.S IF X IS GRANT$ T X TRAV]SES !
F9ITE 4T.E PRESCRIB$4 ,^! !N >E TWO ( 8
>GU;TS4
  ,! ?IRD IS T ALR GIV5 ABV1 6! E6ECT T
! FLY+ >R[ IS AT RE/1 : RESULT FOLL[S F
! ASSUMP;N T "T IS -POS$ ( MO;TS3 IF ?
ASSUMP;N IS N GRANT$1 ! 3CLU.N W N
FOLL[4
  ,! F\R? >GU;T IS T 3C]N+ ! TWO R[S (
BODIES1 EA* R[ 2+ -POS$ ( AN EQUAL NUMB]
( BODIES ( EQUAL SIZE1 PASS+ EA* O!R ON
A RACE-C\RSE Z !Y PROCE$ ) EQUAL VELOC;Y
9 OPPOSITE DIREC;NS1 ! "O R[ ORIG9,Y
O3UPY+ ! SPACE 2T ! GOAL &! MI4LE PO9T (
! C\RSE &! O!R T 2T ! MI4LE PO9T &!
/>T+-PO/4 ,?1 HE ?9KS1 9VOLVES ! 3CLU.N
T HALF A GIV5 "T IS EQUAL 6D\# T "T4 ,!
FALLACY (! R1SON+ LIES 9 ! ASSUMP;N T A
BODY O3UPIES AN EQUAL "T 9 PASS+ ) EQUAL
VELOC;Y A BODY T IS 9 MO;N &A BODY (
EQUAL SIZE T IS AT RE/2 : IS FALSE4 ,=
9/.E 7S RUNS ! >GU;T71 LET ;,A1 ,A ''' 2
! /,N>Y BODIES ( EQUAL SIZE1 ;,B1 ,B '''
! BODIES1 EQUAL 9 NUMB] & 9 SIZE TO ;,A1
,A '''1ORIG9,Y O3UPY+ ! HALF (! C\RSE F
! />T+-PO/ 6! MI4LE (! ;,A'S1 & ;,G1 ,G
''' ^? ORIG9,Y O3UPY+ ! O!R HALF F !
GOAL 6! MI4LE (! ;,A'S1 EQUAL 9 NUMB]1
SIZE1 & VELOC;Y TO ;,B1 ,B '''4 ,!N ?REE
3SEQU;ES FOLL[3
  ,F/1 Z ! ;,B'S &! ;,G'S PASS "O ANO!R1
! F/ ;,B R1*ES ! LA/ ;,G AT ! SAME MO;T
Z ! F/ ;,G R1*ES ! LA/ ,B4 ,SECONDLY AT
? MO;T ! F/ ;,G HAS PASS$ ALL ! ;,A'S1
":AS ! F/ ;,B HAS PASS$ ONLY HALF !
;,A'S1 & HAS 3SEQU5TLY O3UPI$ ONLY HALF
! "T O3UPI$ 0! F/ ;,G1 S9CE EA* (! TWO
O3UPIES AN EQUAL "T 9 PASS+ EA* ,A4
,?IRDLY1 AT ! SAME MO;T ALL ! ;,B'S H
PASS$ ALL ! ;,G'S3 =! F/ ;,G &! F/ ;,B W
SIMULTANE\SLY R1* ! OPPOSITE 5DS (!
C\RSE1 S9CE 7S SAYS ,Z5O7 ! "T O3UPI$ 0!
F/ ;,G 9 PASS+ EA* (! ;,B'S IS EQUAL 6T
O3UPI$ 0X 9 PASS+ EA* (! ;,A'S1 2C AN
EQUAL "T IS O3UPI$ 0BO? ! F/ ;,B &! F/
;,G 9 PASS+ ALL ! ;,A'S4 ,? IS ! >GU;T1
B X PRESUPPOS$ ! A=ESD FALLACI\S
ASSUMP;N4
  ,NOR 9 REF];E 63TRADICTORY *ANGE % WE
F9D ANY?+ UNANSW]A# 9 ! >GU;T T IF A ?+
IS *ANG+ F N-:ITE1 SAY1 6:ITE1 & IS 9
NEI 3DI;N1 !N X W 2 NEI :ITE NOR N-:ITE3
=! FACT T X IS N :OLLY 9 EI 3DI;N W N
PRECLUDE U F CALL+ X :ITE OR N-:ITE4 ,WE
CALL A ?+ :ITE OR N-:ITE N NECESS>ILY 2C
X IS 2 "O OR ! O!R1 B CAUSE MO/ ( XS "PS
OR ! MO/ ESS5TIAL "PS ( X >E S3 N 2+ 9 A
C]TA9 3DI;N IS DI6]5T F N 2+ :OLLY 9 T
3DI;N4 ,S1 TOO1 9 ! CASE ( 2+ & N-2+ &
ALL O!R 3DI;NS : /& 9 A 3TRADICTORY
REL,N3 :ILE ! *ANG+ ?+ M/ ( NECESS;Y 2 9
"O (! TWO OPPOSITES1 X IS N"E :OLLY 9
EI4
  ,AG1 9 ! CASE ( CIRCLES & SPH]ES &
"EY?+ ^: MO;N IS 3F9$ )9 ! SPACE T X
O3UPIES1 X IS N TRUE 6SAY ! MO;N C 2
NO?+ B RE/1 ON ! GR.D T S* ?+S 9 MO;N1
!MVS & _! "PS1 W O3UPY ! SAME POSI;N =A
P]IOD ( "T1 & T "!=E !Y W 2 AT ONCE AT
RE/ & 9 MO;N4 ,= 9 ! F/ PLACE ! "PS D N
O3UPY ! SAME POSI;N = ANY P]IOD ( "T3 &
9 ! SECOND PLACE ! :OLE AL IS ALW *ANG+
6A DI6]5T POSI;N3 = IF WE TAKE ! ORBIT Z
DESCRIB$ F A PO9T ,A ON A CIRCUMF];E1 X
W N 2 ! SAME Z ! ORBIT Z DESCRIB$ F ;,B
OR ;,G OR ANY O!R PO9T ON ! SAME
CIRCUMF];E EXCEPT 9 AN A3ID5TAL S5SE1 !
S5SE T IS 6SAY 9 : A MUSICAL MAN IS !
SAME Z A MAN4 ,?US "O ORBIT IS ALW *ANG+
96ANO!R1 &! ?+ W N"E 2 AT RE/4 ,& X IS !
SAME )! SPH]E & "EY?+ ELSE ^: MO;N IS
3F9$ )9 ! SPACE T X O3UPIES4
  #AJ
  ,\R NEXT PO9T IS T T : IS )\T "PS _C 2
9 MO;N EXCEPT A3ID5T,Y3 I4E4 X C 2 9
MO;N ONLY 9 S F> Z ! BODY OR ! MAGNITUDE
IS 9 MO;N &! "P.S IS 9 MO;N 09CLU.N "!91
J Z T : IS 9 A BOAT MAY 2 9 MO;N 9
3SEQU;E (! LOCOMO;N (! BOAT1 OR A "P MAY
2 9 MO;N 9 VIRTUE (! MO;N (! :OLE4 7,X
M/ 2 REMEMB]$1 H["E1 T BY ,8T : IS )\T
"PS0' ,I M1N T : IS QUANTITATIVELY
9DIVISI# 7& T ! CASE (! MO;N (A "P IS N
EXACTLY P>ALLEL73 = "PS H MO;NS 2L;G+
ESS5TI,Y & S"E,Y 6!MVS 4T9CT F ! MO;N (!
:OLE4 ,! 4T9C;N MAY 2 SE5 MO/ CLE>LY 9 !
CASE (A REVOLV+ SPH]E1 9 : ! VELOCITIES
(! "PS NE> ! C5TRE &( ^? ON ! SURFACE >E
DI6]5T F "O ANO!R & F T (! :OLE2 ?
IMPLIES T "! IS N "O MO;N B _M74 ,Z WE H
SD1 !N1 T : IS )\T "PS C 2 9 MO;N 9 !
S5SE 9 : A MAN SITT+ 9 A BOAT IS 9 MO;N
:5 ! BOAT IS TRAVELL+1 B X _C 2 9 MO;N (
XF4 ,= SUPPOSE T X IS *ANG+ F ;,,AB
6,,BG,'-EI F "O MAGNITUDE 6ANO!R1 OR F
"O =M 6ANO!R1 OR F "S /ATE 6XS
3TRADICTORY-& LET ;,D 2 ! PRIM>Y "T 9 :
X "UGOES ! *ANGE4 ,!N 9 ! "T 9 : X IS
*ANG+ X M/ 2 EI 9 ;,,AB OR 9 ,,BG OR
"PLY 9 "O & "PLY 9 ! O!R3 = ?1 Z WE SAW1
IS TRUE ( "EY?+ T IS *ANG+4 ,N[ X _C 2
"PLY 9 EA* (! TWO3 = !N X WD 2 DIVISI#
96"PS4 ,NOR AG C X 2 9 ,,BG3 = !N X W H
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*ANGE IS 9 PROCESS4 ,X REMA9S1 !N1 T 9 !
"T 9 : X IS *ANG+1 X IS 9 ;,,AB4 ,T 2+
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SAME 3DI;N =A P]IOD ( "T IS 6BE AT RE/4
,S X IS N POSSI# = T : HAS NO "PS 6BE 9
MO;N OR 6*ANGE 9 ANY WAY3 = ONLY "O
3DI;N CD H MADE X POSSI# = X 6H MO;N1
VIZ4 T "T %D 2 -POS$ ( MO;TS1 9 : CASE
AT ANY MO;T X WD H -PLET$ A MO;N OR A
*ANGE1 S T X WD N"E 2 9 MO;N1 B WD ALW H
BE5 9 MO;N4 ,B ? WE H ALR %[N ABV 6BE
IMPOSSI#3 "T IS N -POS$ ( MO;TS1 J Z A
L9E IS N -POS$ ( PO9TS1 & MO;N IS N
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3SI/ ( 9DIVISI#S 9 EXACTLY ! SAME WAY Z
"T IS MADE 63SI/ ( MO;TS OR A L5G? (
PO9TS4
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2 NO MO;N (A PO9T OR ( ANY O!R 9DIVISI#4
,T : IS 9 MO;N C N"E TRAV]SE A SPACE
GRT] ?AN XF )\T F/ TRAV]S+ A SPACE EQUAL
6OR LESS ?AN XF4 ,T 2+ S1 X IS EVID5T T
! PO9T AL M/ F/ TRAV]SE A SPACE EQUAL
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"! C 2 NO SPACE LESS ?AN XF = X 6TRAV]SE
F/3 S X W H 6TRAV]SE A 4T.E EQUAL 6XF4
,?US ! L9E W 2 -POS$ ( PO9TS1 =! PO9T1 Z
X 3T9U,Y TRAV]SES A 4T.E EQUAL 6XF1 W 2
A M1SURE (! :OLE L9E4 ,B S9CE ? IS
IMPOSSI#1 X IS LIKEWISE IMPOSSI# =!
9DIVISI# 6BE 9 MO;N4
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N"E 9 A MO;T1 & ALL "T IS DIVISI#1 =
"EY?+ T IS 9 MO;N "! M/ 2 A "T LESS ?AN
T 9 : X TRAV]SES A 4T.E Z GRT Z XF4 ,= T
9 : X IS 9 MO;N W 2 A "T1 2C ALL MO;N IS
9 A P]IOD ( "T2 & ALL "T HAS BE5 %[N ABV
6BE DIVISI#4 ,"!=E1 IF A PO9T IS 9 MO;N1
"! M/ 2 A "T LESS ?AN T 9 : X HAS XF
TRAV]S$ ANY 4T.E4 ,B ? IS IMPOSSI#1 = 9
LESS "T X M/ TRAV]SE LESS 4T.E1 & ?US !
9DIVISI# W 2 DIVISI# 96"S?+ LESS ?AN XF1
J Z ! "T IS S DIVISI#3 ! FACT 2+ T !
ONLY 3DI;N "U : T : IS )\T "PS &
9DIVISI# CD 2 9 MO;N WD H BE5 !
POSSIBIL;Y (! 9F9ITELY SMALL 2+ 9 MO;N 9
A MO;T3 = 9 ! TWO "QS-T ( MO;N 9 A MO;T
& T ( MO;N ( "S?+ 9DIVISI#-! SAME
PR9CIPLE IS 9VOLV$4
  ,\R NEXT PO9T IS T NO PROCESS ( *ANGE
IS 9F9ITE3 = E *ANGE1 :E!R 2T
3TRADICTORIES OR 2T 3TR>IES1 IS A *ANGE
F "S?+ 6"S?+4 ,?US 9 3TRADICTORY *ANGES
! POSITIVE OR ! NEGATIVE1 Z ! CASE MAY
BE1 IS ! LIMIT1 E4G4 2+ IS ! LIMIT ( -+
6BE & N-2+ IS ! LIMIT ( C1S+ 6BE3 & 9
3TR>Y *ANGES ! "PICUL> 3TR>IES >E !
LIMITS1 S9CE ^! >E ! EXTREME PO9TS ( ANY
S* PROCESS ( *ANGE1 & 3SEQU5TLY ( E
PROCESS ( ALT],N3 = ALT],N IS ALW
DEP5D5T ^U "S 3TR>IES4 ,SIMIL>LY 3TR>IES
>E ! EXTREME PO9TS ( PROCESSES ( 9CR1SE
& DECR1SE3 ! LIMIT ( 9CR1SE IS 6BE F.D 9
! -PLETE MAGNITUDE PROP] 6! PECULI>
NATURE (! ?+ T IS 9CR1S+1 :ILE ! LIMIT (
DECR1SE IS ! -PLETE LOSS ( S* MAGNITUDE4
,LOCOMO;N1 X IS TRUE1 WE _C %[ 6BE F9ITE
9 ? WAY1 S9CE X IS N ALW 2T 3TR>IES4 ,B
S9CE T : _C 2 CUT 7IN ! S5SE T X IS
9CONCEIVA# T X %D 2 CUT1 ! T]M ,8_C0' 2+
US$ 9 S"EAL S5SES7-S9CE X IS 9CONCEIVA#
T T : 9 ? S5SE _C 2 CUT %D 2 9 PROCESS (
2+ CUT1 & G5],Y T T : _C -E 6BE %D 2 9
PROCESS ( -+ 6BE1 X FOLL[S T X IS
9CONCEIVA# T T : _C -PLETE A *ANGE %D 2
9 PROCESS ( *ANG+ 6T 6: X _C -PLETE A
*ANGE4 ,IF1 !N1 X IS 6BE ASSUM$ T T : IS
9 LOCOMO;N IS 9 PROCESS ( *ANG+1 X M/ 2
CAPA# ( -PLET+ ! *ANGE4 ,3SEQU5TLY XS
MO;N IS N 9F9ITE1 & X W N 2 9 LOCOMO;N
OV] AN 9F9ITE 4T.E1 = X _C TRAV]SE S* A
4T.E4
  ,X IS EVID5T1 !N1 T A PROCESS ( *ANGE
_C 2 9F9ITE 9 ! S5SE T X IS N DEF9$
0LIMITS4 ,B X REMA9S 6BE 3SID]$ :E!R X
IS POSSI# 9 ! S5SE T "O &! SAME PROCESS
( *ANGE MAY 2 9F9ITE 9 RESPECT (! "T : X
O3UPIES4 ,IF X IS N "O PROCESS1 X WD
SEEM T "! IS NO?+ 6PREV5T XS 2+ 9F9ITE 9
? S5SE2 E4G4 IF A PROCESS ( LOCOMO;N 2
SU3E$$ 0A PROCESS ( ALT],N & T 0A
PROCESS ( 9CR1SE & T AG 0A PROCESS ( -+
6BE3 9 ? WAY "! MAY 2 MO;N = "E S F> Z !
"T IS 3C]N$1 B X W N 2 "O MO;N1 2C ALL
^! MO;NS D N -POSE "O4 ,IF X IS 6BE "O
PROCESS1 NO MO;N C 2 9F9ITE 9 RESPECT (!
"T T X O3UPIES1 )! S+LE EXCEP;N (
ROTATORY LOCOMO;N4