              ,BOOK ,,XIII
                   #A
  ,WE H /AT$ :AT IS ! SUB/.E ( S5SI#
?+S1 D1L+ 9 ! TR1TISE ON PHYSICS )
MATT]1 & LAT] )! SUB/.E : HAS ACTUAL
EXI/;E4 ,N[ S9CE \R 9QUIRY IS :E!R "! IS
OR IS N 2SS ! S5SI# SUB/.ES ANY : IS
IMMOVA# & ET]NAL1 &1 IF "! IS1 :AT X IS1
WE M/ F/ 3SID] :AT IS SD 0O!RS1 S T1 IF
"! IS ANY?+ : !Y SAY WR;GLY1 WE MAY N 2
LIA# 6! SAME OBJEC;NS1 :ILE1 IF "! IS
ANY OP9ION -MON 6!M & U1 WE % H NO
PRIVATE GRIEV.E AG/ \RVS ON T A3.T2 = "O
M/ 2 3T5T 6/ATE "S PO9TS BETT] ?AN "O'S
PR$ECESSORS1 & O!RS NO WORSE4
  ,TWO OP9IONS >E HELD ON ? SUBJECT2 X
IS SD T ! OBJECTS ( MA!MATICS-I4E4
NUMB]S & L9ES &! L->E SUB/.ES1 & AG T !
,ID1S >E SUB/.ES4 ,& 7#A7 S9CE "S
RECOGNIZE ^! Z TWO DI6]5T CLASSES-!
,ID1S &! MA!MATICAL NUMB]S1 & 7#B7 "S
RECOGNIZE BO? Z HAV+ "O NATURE1 :ILE
7#C7 "S O!RS SAY T ! MA!MATICAL SUB/.ES
>E ! ONLY SUB/.ES1 WE M/ 3SID] F/ !
OBJECTS ( MA!MATICS1 N QUALIFY+ !M 0ANY
O!R "*I/IC-N ASK+1 = 9/.E1 :E!R !Y >E 9
FACT ,ID1S OR N1 OR :E!R !Y >E !
PR9CIPLES & SUB/.ES ( EXI/+ ?+S OR N1 B
ONLY :E!R Z OBJECTS ( MA!MATICS !Y EXI/
OR N1 & IF !Y EXI/1 H[ !Y EXI/4 ,!N AF ?
WE M/ SEP>ATELY 3SID] ! ,ID1S !MVS 9 A
G5]AL WAY1 & ONLY Z F> Z ! A3EPT$ MODE (
TR1T;T DEM&S2 = MO/ (! PO9TS H BE5
REP1T$LY MADE EV5 0! 4CUS.NS \TSIDE \R
S*OOL1 &1 FUR!R1 ! GRT] "P ( \R A3.T M/
F9I% 0?R[+ LI<T ON T 9QUIRY1 VIZ4 :5 WE
EXAM9E :E!R ! SUB/.ES &! PR9CIPLES (
EXI/+ ?+S >E NUMB]S & ,ID1S2 = AF !
4CUS.N (! ,ID1S ? REMANS Z A ?IRD
9QUIRY4
  ,IF ! OBJECTS ( MA!MATICS EXI/1 !Y M/
EXI/ EI 9 S5SI# OBJECTS1 Z "S SAY1 OR
SEP>ATE F S5SI# OBJECTS 7& ? AL IS SD
0"S72 OR IF !Y EXI/ 9 NEI ( ^! WAYS1 EI
!Y D N EXI/1 OR !Y EXI/ ONLY 9 "S
SPECIAL S5SE4 ,S T ! SUBJECT ( \R 4CUS.N
W 2 N :E!R !Y EXI/ B H[ !Y EXI/4
                   #B
  ,T X IS IMPOSSI# = MA!MATICAL OBJECTS
6EXI/ 9 S5SI# ?+S1 & AT ! SAME "T T !
DOCTR9E 9 "Q IS AN >TIFICIAL "O1 HAS BE5
SD ALR 9 \R 4CUS.N ( DI6ICULTIES WE H
PO9T$ \ T X IS IMPOSSI# = TWO SOLIDS 6BE
9 ! SAME PLACE1 & AL T AC 6! SAME >GU;T
! O!R P[]S & "*I/ICS AL %D EXI/ 9 S5SI#
?+S & N"O ( !M SEP>ATELY4 ,? WE H SD
ALR4 ,B1 FUR!R1 X IS OBVI\S T ON ? !ORY
X IS IMPOSSI# = ANY BODY :AT"E 6BE
DIVID$2 = X WD H 6BE DIVID$ AT A PLANE1
&! PLANE AT A L9E1 &! L9E AT A PO9T1 S T
IF ! PO9T _C 2 DIVID$1 NEI C ! L9E1 & IF
! L9E _C1 NEI C ! PLANE NOR ! SOLID4
,:AT DI6];E1 !N1 DOES X MAKE :E!R S5SI#
?+S >E S* 9DIVISI# 5TITIES1 OR1 )\T 2+ S
!MVS1 H 9DIVISI# 5TITIES 9 !M8 ,! RESULT
W 2 ! SAME2 IF ! S5SI# 5TITIES >E DIVID$
! O!RS W 2 DIVID$ TOO1 OR ELSE N EV5 !
S5SI# 5TITIES C 2 DIVID$4
  ,B1 AG1 X IS N POSSI# T S* 5TITIES %D
EXI/ SEP>ATELY4 ,= IF 2SS ! S5SI# SOLIDS
"! >E 6BE O!R SOLIDS : >E SEP>ATE F !M &
PRIOR 6! S5SI# SOLIDS1 X IS PLA9 T 2SS !
PLANES AL "! M/ 2 O!R & SEP>ATE PLANES &
PO9TS & L9ES2 = 3SI/5CY REQUIRES ?4 ,B
IF ^! EXI/1 AG 2SS ! PLANES & L9ES &
PO9TS (! MA!MATICAL SOLID "! M/ 2 O!RS :
>E SEP>ATE4 7,= 9COMPOSITES >E PRIOR
6-P.DS2 & IF "! >E1 PRIOR 6! S5SI#
BODIES1 BODIES : >E N S5SI#1 0! SAME
>GU;T ! PLANES : EXI/ 0!MVS M/ 2 PRIOR
6^? : >E 9 ! MO;N.S SOLIDS4 ,"!=E ^! W 2
PLANES & L9ES O!R ?AN ^? T EXI/ AL;G )!
MA!MATICAL SOLIDS 6: ^! ?9K]S ASSIGN
SEP>ATE EXI/;E2 =! LATT] EXI/ AL;G )!
MA!MATICAL SOLIDS1 :ILE ! O!RS >E PRIOR
6! MA!MATICAL SOLIDS47 ,AG1 "!=E1 "! W
BE1 2L;G+ 6^! PLANES1 L9ES1 & PRIOR 6!M
"! W H 6BE1 0! SAME >GU;T1 O!R L9ES &
PO9TS2 & PRIOR 6^! PO9TS 9 ! PRIOR L9ES
"! W H 6BE O!R PO9TS1 ?\< "! W 2 NO O!RS
PRIOR 6^!4 ,N[ 7#A7 ! A3UMUL,N 2COMES
ABSURD2 = WE F9D \RVS ) "O SET ( SOLIDS
A"P F ! S5SI# SOLIDS2 ?REE SETS ( PLANES
A"P F ! S5SI# PLANES-^? : EXI/ A"P F !
S5SI# PLANES1 & ^? 9 ! MA!MATICAL
SOLIDS1 & ^? : EXI/ A"P F ^? 9 !
MA!MATICAL SOLIDS2 F\R SETS ( L9ES1 &
FIVE SETS ( PO9TS4 ,) : ( ^!1 !N1 W !
MA!MATICAL SCI;ES D1L8 ,C]TA9LY N )!
PLANES & L9ES & PO9TS 9 ! MO;N.S SOLID2
= SCI;E ALW D1LS ) :AT IS PRIOR4 ,& 7!
SAME A3.T W APPLY AL 6NUMB]S2 = "! W 2 A
DI6]5T SET ( UNITS A"P F EA* SET (
PO9TS1 & AL A"P F EA* SET ( R1LITIES1 F
! OBJECTS ( S5SE & AG F ^? ( ?"\2 S T "!
W 2 V>I\S CLASSES ( MA!MATICAL NUMB]S4
  ,AG1 H[ IS X POSSI# 6SOLVE ! "QS : WE
H ALR ENUM]AT$ 9 \R 4CUS.N (
DI6ICULTIES8 ,=! OBJECTS ( A/RONOMY W
EXI/ A"P F S5SI# ?+S J Z ! OBJECTS (
GEOMETRY W2 B H[ IS X POSSI# T A H1V5 &
XS "PS-OR ANY?+ ELSE : HAS MOVE;T-%D
EXI/ A"P8 ,SIMIL>LY AL ! OBJECTS (
OPTICS &( H>MONICS W EXI/ A"P2 = "! W 2
BO? VOICE & SI<T 2SS ! S5SI# OR
9DIVIDUAL VOICES & SI<TS4 ,"!=E X IS
PLA9 T ! O!R S5SES Z WELL1 &! O!R
OBJECTS ( S5SE1 W EXI/ A"P2 = :Y %D "O
SET ( !M D S & ANO!R N8 ,& IF ? IS S1 "!
W AL 2 ANIMALS EXI/+ A"P1 S9CE "! W 2
S5SES4
  ,AG1 "! >E C]TA9 MA!MATICAL !OREMS T
>E UNIV]SAL1 EXT5D+ 2Y ^! SUB/.ES4 ,"H
!N WE % H ANO!R 9T]M$IATE SUB/.E SEP>ATE
BO? F ! ,ID1S & F ! 9T]M$IATES1-A SUB/.E
: IS NEI NUMB] NOR PO9TS NOR SPATIAL
MAGNITUDE NOR "T4 ,& IF ? IS IMPOSSI#1
PLA9LY X IS AL IMPOSSI# T ! =M] 5TITIES
%D EXI/ SEP>ATE F S5SI# ?+S4
  ,&1 9 G5]AL1 3CLU.N 3TR>Y ALIKE 6!
TRU? & 6! USUAL VIEWS FOLL[1 IF "O IS
6SUPPOSE ! OBJECTS ( MA!MATICS 6EXI/ ?US
Z SEP>ATE 5TITIES4 ,= 2C !Y EXI/ ?US !Y
M/ 2 PRIOR 6S5SI# SPATIAL MAGNITUDES1 B
9 TRU? !Y M/ 2 PO/]IOR2 =! 9COMPLETE
SPATIAL MAGNITUDE IS 9 ! ORD] ( G5],N
PRIOR1 B 9 ! ORD] ( SUB/.E PO/]IOR1 Z !
LIFE.S IS 6! LIV+4
  ,AG1 0VIRTUE ( :AT1 & :51 W MA!MATICAL
MAGNITUDES 2 "O8 ,= ?+S 9 \R P]CEPTI# _W
>E "O 9 VIRTUE ( S\L1 OR (A "P ( S\L1 OR
( "S?+ ELSE T IS R1SONA# 5\<2 :5 ^! >E N
PRES5T1 ! ?+ IS A PLURAL;Y1 & SPLITS UP
96"PS4 ,B 9 ! CASE (! SUBJECTS (
MA!MATICS1 : >E DIVISI# & >E QUANTITIES1
:AT IS ! CAUSE ( _! 2+ "O & HOLD+ TGR8
  ,AG1 ! MODES ( G5],N (! OBJECTS (
MA!MATICS %[ T WE >E "R4 ,=! DIM5.N F/
G5]AT$ IS L5G?1 !N -ES BR1D?1 LA/LY
DEP?1 &! PROCESS IS -PLETE4 ,IF1 !N1 T :
IS PO/]IOR 9 ! ORD] ( G5],N IS PRIOR 9 !
ORD] ( SUB/ANTIAL;Y1 ! SOLID W 2 PRIOR
6! PLANE &! L9E4 ,& 9 ? WAY AL X IS BO?
M -PLETE & M :OLE1 2C X C 2COME ANIMATE4
,H[1 ON ! O!R H&1 CD A L9E OR A PLANE 2
ANIMATE8 ,! SUPPOSI;N PASSES ! P[] ( \R
S5SES4
  ,AG1 ! SOLID IS A SORT ( SUB/.E2 = X
ALR HAS 9 A S5SE -PLETE;S4 ,B H[ C L9ES
2 SUB/.ES8 ,NEI Z A =M OR %APE1 Z ! S\L
P]H IS1 NOR Z MATT]1 L ! SOLID2 = WE H
NO EXP]I;E ( ANY?+ T C 2 PUT TGR \ (
L9ES OR PLANES OR PO9TS1 :ILE IF ^! _H
BE5 A SORT ( MAT]IAL SUB/.E1 WE %D H
OBS]V$ ?+S : CD 2 PUT TGR \ ( !M4
  ,GRANT1 !N1 T !Y >E PRIOR 9 DEF9I;N4
,/ N ALL ?+S T >E PRIOR 9 DEF9I;N >E AL
PRIOR 9 SUB/ANTIAL;Y4 ,= ^? ?+S >E PRIOR
9 SUB/ANTIAL;Y : :5 SEP>AT$ F O!R ?+S
SURPASS !M 9 ! P[] ( 9DEP5D5T EXI/;E1 B
?+S >E PRIOR 9 DEF9I;N 6^? ^: DEF9I;NS
>E -P.D$ \ ( _! DEF9I;NS2 & ^! TWO
PROP]TIES >E N COEXT5SIVE4 ,= IF
ATTRIBUTES D N EXI/ A"P F ! SUB/.ES
7E4G4 A ,8MOBILE0' OR A PALE0'71 PALE IS
PRIOR 6! PALE MAN 9 DEF9I;N1 B N 9
SUB/ANTIAL;Y4 ,= X _C EXI/ SEP>ATELY1 B
IS ALW AL;G )! 3CRETE ?+2 & 0! 3CRETE ?+
,I M1N ! PALE MAN4 ,"!=E X IS PLA9 T NEI
IS ! RESULT ( AB/RAC;N PRIOR NOR T : IS
PRODUC$ 0A4+ DET]M9ANTS PO/]IOR2 = X IS
0A4+ A DET]M9ANT 6PALE T WE SP1K (! PALE
MAN4
  ,X HAS1 !N1 BE5 SU6ICI5TLY PO9T$ \ T !
OBJECTS ( MA!MATICS >E N SUB/.ES 9 A
HI<] DEGREE ?AN BODIES >E1 & T !Y >E N
PRIOR 6S5SI#S 9 2+1 B ONLY 9 DEF9I;N1 &
T !Y _C EXI/ "S": A"P4 ,B S9CE X 0 N
POSSI# = !M 6EXI/ 9 S5SI#S EI1 X IS PLA9
T !Y EI D N EXI/ AT ALL OR EXI/ 9 A
SPECIAL S5SE & "!=E D N ,8EXI/0' )\T
QUALIFIC,N4 ,= ,8EXI/0' HAS _M S5SES4
                   #C
  ,= J Z ! UNIV]SAL PROPOSI;NS (
MA!MATICS D1L N ) OBJECTS : EXI/
SEP>ATELY1 A"P F EXT5D$ MAGNITUDES & F
NUMB]S1 B ) MAGNITUDES & NUMB]S1 N H["E
QUA S* Z 6H MAGNITUDE OR 6BE DIVISI#1
CLE>LY X IS POSSI# T "! %D AL 2 BO?
PROPOSI;NS & DEMON/R,NS AB S5SI#
MAGNITUDES1 N H["E QUA S5SI# B QUA
POSSESS$ ( C]TA9 DEF9ITE QUALITIES4 ,= Z
"! >E _M PROPOSI;NS AB ?+S M]ELY 3SID]$
Z 9 MO;N1 A"P F :AT EA* S* ?+ IS & F _!
A3ID5TS1 & Z X IS N "!=E NEC T "! %D 2
EI A MOBILE SEP>ATE F S5SI#S1 OR A 4T9CT
MOBILE 5T;Y 9 ! S5SI#S1 S TOO 9 ! CASE (
MOBILES "! W 2 PROPOSI;NS & SCI;ES1 :
TR1T !M H["E N QUA MOBILE B ONLY QUA
BODIES1 OR AG ONLY QUA PLANES1 OR ONLY
QUA L9ES1 OR QUA DIVISI#S1 OR QUA
9DIVISI#S HAV+ POSI;N1 OR ONLY QUA
9DIVISI#S4 ,?US S9CE X IS TRUE 6SAY )\T
QUALIFIC,N T N ONLY ?+S : >E SEP>A# B AL
?+S : >E 9SEP>A# EXI/ 7= 9/.E1 T MOBILES
EXI/71 X IS TRUE AL 6SAY )\T QUALIFIC,N
T ! OBJECTS ( MA!MATICS EXI/1 &)! "*
ASCRIB$ 6!M 0MA!MATICIANS4 ,& Z X IS
TRUE 6SAY (! O!R SCI;ES TOO1 )\T
QUALIFIC,N1 T !Y D1L ) S* & S* A
SUBJECT-N ) :AT IS A3ID5TAL 6X 7E4G4 N )
! PALE1 IF ! H1L?Y ?+ IS PALE1 &! SCI;E
HAS ! H1L?Y Z XS SUBJECT71 B ) T : IS !
SUBJECT ( EA* SCI;E-)! H1L?Y IF X TR1TS
XS OBJECT QUA H1L?Y1 ) MAN IF QUA MAN3-S
TOO IS X ) GEOMETRY2 IF XS SUBJECTS
HAPP5 6BE S5SI#1 ?\< X DOES N TR1T !M
QUA S5SI#1 ! MA!MATICAL SCI;ES W N = T
R1SON 2 SCI;ES ( S5SI#S-NOR1 ON ! O!R
H&1 ( O!R ?+S SEP>ATE F S5SI#S4 ,_M
PROP]TIES ATTA* 6?+S 9 VIRTUE ( _! [N
NATURE Z POSSESS$ ( EA* S* "*2 E4G4 "!
>E ATTRIBUTES PECULI> 6! ANIMAL QUA
FEMALE OR QUA MALE 7YET "! IS NO
,8FEMALE0' NOR ,8MALE0' SEP>ATE F
ANIMALS72 S T "! >E AL ATTRIBUTES : 2L;G
6?+S M]ELY Z L5G?S OR Z PLANES4 ,& 9
PROPOR;N Z WE >E D1L+ ) ?+S : >E PRIOR 9
DEF9I;N & SIMPL]1 \R K HAS M A3URACY1
I4E4 SIMPLIC;Y4 ,"!=E A SCI;E : AB/RACTS
F SPATIAL MAGNITUDE IS M PRECISE ?AN "O
: TAKES X 96A3.T2 &A SCI;E IS MO/
PRECISE IF X AB/RACTS F MOVE;T1 B IF X
TAKES A3.T ( MOVE;T1 X IS MO/ PRECISE IF
X D1LS )! PRIM>Y MOVE;T1 = ? IS !
SIMPLE/2 &( ? AG UNI=M MOVE;T IS !
SIMPLE/ =M4
  ,! SAME A3.T MAY 2 GIV5 ( H>MONICS &
OPTICS2 = NEI 3SID]S XS OBJECTS QUA SI<T
OR QUA VOICE1 B QUA L9ES & NUMB]S2 B !
LATT] >E ATTRIBUTES PROP] 6! =M]4 ,&
ME*ANICS TOO PROCE$S 9 ! SAME WAY4 ,"!=E
IF WE SUPPOSE ATTRIBUTES SEP>AT$ F _!
FELL[ ATTRIBUTES & MAKE ANY 9QUIRY 3C]N+
!M Z S*1 WE % N = ? R1SON 2 9 ]ROR1 ANY
M ?AN :5 "O DRAWS A L9E ON ! GR.D &
CALLS X A FOOT L;G :5 X IS N2 =! ]ROR IS
N 9CLUD$ 9 ! PREMISSES4
  ,EA* "Q W 2 BE/ 9VE/IGAT$ 9 ? WAY-BY
SETT+ UP 0AN ACT ( SEP>,N :AT IS N
SEP>ATE1 Z ! >I?METICIAN &! GEOMET] D4
,=A MAN QUA MAN IS "O 9DIVISI# ?+2 &!
>I?METICIAN SUPPOS$ "O 9DIVISI# ?+1 & !N
3SID]$ :E!R ANY ATTRIBUTE 2L;GS 6A MAN
QUA 9DIVISI#4 ,B ! GEOMET] TR1TS HM NEI
QUA MAN NOR QUA 9DIVISI#1 B Z A SOLID4
,= EVID5TLY ! PROP]TIES : WD H 2L;G$ 6HM
EV5 IF P]*.E HE _H N BE5 9DIVISI#1 C
2L;G 6HM EV5 A"P F ^! ATTRIBUTES4 ,?US1
!N1 GEOMET]S SP1K CORRECTLY2 !Y TALK AB
EXI/+ ?+S1 & _! SUBJECTS D EXI/2 = 2+
HAS TWO =MS-X EXI/S N ONLY 9 -PLETE
R1L;Y B AL MAT]I,Y4
  ,N[ S9CE ! GD &! B1UTI;L >E DI6]5T 7=!
=M] ALW IMPLIES 3DUCT Z XS SUBJECT1 :ILE
! B1UTI;L IS F.D AL 9 MO;N.S ?+S71 ^? :O
ASS]T T ! MA!MATICAL SCI;ES SAY NO?+ (!
B1UTI;L OR ! GD >E 9 ]ROR4 ,= ^! SCI;ES
SAY & PROVE A GRT D1L AB !M2 IF !Y D N
EXPRESSLY M5;N !M1 B PROVE ATTRIBUTES :
>E _! RESULTS OR _! DEF9I;NS1 X IS N
TRUE 6SAY T !Y TELL U NO?+ AB !M4 ,!
*IEF =MS ( B1UTY >E ORD] & SYMMETRY &
DEF9ITE;S1 : ! MA!MATICAL SCI;ES
DEMON/RATE 9 A SPECIAL DEGREE4 ,& S9CE
^! 7E4G4 ORD] & DEF9ITE;S7 >E OBVI\SLY
CAUSES ( _M ?+S1 EVID5TLY ^! SCI;ES M/
TR1T ? SORT ( CAUSATIVE PR9CIPLE AL
7I4E4 ! B1UTI;L7 Z 9 "S S5SE A CAUSE4 ,B
WE % SP1K M PLA9LY ELSE": AB ^! MATT]S4
                   #D
  ,S M* !N =! OBJECTS ( MA!MATICS2 WE H
SD T !Y EXI/ & 9 :AT S5SE !Y EXI/1 & 9
:AT S5SE !Y >E PRIOR & 9 :AT S5SE N
PRIOR4 ,N[1 REG>D+ ! ,ID1S1 WE M/ F/
EXAM9E ! ID1L !ORY XF1 N 3NECT+ X 9 ANY
WAY )! NATURE ( NUMB]S1 B TR1T+ X 9 ! =M
9 : X 0 ORIG9,Y "U/OOD 0^? :O F/ MA9TA9$
! EXI/;E (! ,ID1S4 ,! SUPPORT]S (! ID1L
!ORY 7 L$ 6X 2C ON ! "Q AB ! TRU? ( ?+S
!Y A3EPT$ ! ,H]ACLIT1N SAY+S : DESCRIBE
ALL S5SI# ?+S Z "E PASS+ AWAY1 S T IF K
OR ?"\ IS 6H AN OBJECT1 "! M/ 2 "S O!R &
P]MAN5T 5TITIES1 A"P F ^? : >E S5SI#2 =
"! CD 2 NO K ( ?+S : 7 9 A /ATE ( FLUX4
,B :5 ,SOCRATES 0 O3UPY+ HMF )!
EXCELL;ES ( "*1 & 9 3NEXION ) !M 2CAME !
F/ 6RAISE ! PRO#M ( UNIV]SAL DEF9I;N
7=(! PHYSICI/S ,DEMOCRITUS ONLY T\*$ ON
! SUBJECT 6A SMALL EXT5T1 & DEF9$1 AF A
FA%ION1 ! HOT &! COLD2 :ILE !
,PY?AGOR1NS _H 2F ? TR1T$ (A FEW ?+S1 ^:
DEF9I;NS-E4G4 ^? ( OPPORTUN;Y1 JU/ICE1
OR M>RIAGE-!Y 3NECT$ ) NUMB]S2 B X 0
NATURAL T ,SOCRATES %D 2 SEEK+ ! ESS;E1
= HE 0 SEEK+ 6SYLLOGIZE1 & ,8:AT A ?+
IS0' IS ! />T+-PO9T ( SYLLOGISMS2 = "! 0
Z YET N"O (! DIALECTICAL P[] : 5A#S P
EV5 )\T K (! ESS;E 6SPECULATE AB 3TR>IES
& 9QUIRE :E!R ! SAME SCI;E D1LS )
3TR>IES2 = TWO ?+S MAY 2 FAIRLY ASCRIB$
6,SOCRATES-9DUCTIVE >GU;TS & UNIV]SAL
DEF9I;N1 BO? ( : >E 3C]N$ )! />T+-PO9T (
SCI;E73-B ,SOCRATES DID N MAKE !
UNIV]SALS OR ! DEF9I;NS EXI/ A"P3 !Y1
H["E1 GAVE !M SEP>ATE EXI/;E1 & ? 0 !
K9D ( ?+ !Y CALL$ ,ID1S4 ,"!=E X FOLL[$
= !M1 ALM 0! SAME >GU;T1 T "! M/ 2 ,ID1S
( ALL ?+S T >E SPOK5 ( UNIV]S,Y1 & X 0
ALM Z IF A MAN WI%$ 6C.T C]TA9 ?+S1 &
:ILE !Y 7 FEW ?"\ HE WD N 2 A# 6C.T !M1
B MADE M ( !M & !N C.T$ !M2 =! ,=MS >E1
"O MAY SAY1 M NUM]\S ?AN ! "PICUL> S5SI#
?+S1 YET X 0 9 SEEK+ ! CAUSES ( ^! T !Y
PROCE$$ F !M 6! ,=MS4 ,= 6EA* ?+ "!
ANSW]S AN 5T;Y : HAS ! SAME "N & EXI/S
A"P F ! SUB/.ES1 & S AL 9 ! CASE ( ALL
O!R GR\PS "! IS A "O OV] _M1 :E!R ^! 2 (
? _W OR ET]NAL4
  ,AG1 (! WAYS 9 : X IS PROV$ T ! ,=MS
EXI/1 N"O IS 3V9C+2 = F "S NO 9F];E
NECESS>ILY FOLL[S1 & F "S >ISE ,=MS EV5
( ?+S ( : !Y ?9K "! >E NO ,=MS4 ,= AC 6!
>GU;TS F ! SCI;ES "! W 2 ,=MS ( ALL ?+S
( : "! >E SCI;ES1 & AC 6! >GU;T (! ,8"O
OV] _M0' "! W 2 ,=MS EV5 ( NEG,NS1 & AC
6! >GU;T T ?"\ HAS AN OBJECT :5 !
9DIVIDUAL OBJECT HAS P]I%$1 "! W 2 ,=MS
( P]I%A# ?+S2 = WE H AN IMAGE ( ^!4 ,AG1
(! MO/ A3URATE >GU;TS1 "S L1D 6,ID1S (
REL,NS1 ( : !Y SAY "! IS NO 9DEP5D5T
CLASS1 & O!RS 9TRODUCE ! ,8?IRD MAN0'4
  ,& 9 G5]AL ! >GU;TS =! ,=MS DE/ROY ?+S
= ^: EXI/;E ! 2LIEV]S 9 ,=MS >E M Z1L\S
?AN =! EXI/;E (! ,ID1S2 = X FOLL[S T N !
DYAD B NUMB] IS F/1 & T PRIOR 6NUMB] IS
! RELATIVE1 & T ? IS PRIOR 6! ABSOLUTE-
2SS ALL ! O!R PO9TS ON : C]TA9 P1
0FOLL[+ \ ! OP9IONS HELD AB ! ,=MS1 CAME
963FLICT )! PR9CIPLES (! !ORY4
  ,AG1 AC 6! ASSUMP;N ON ! 2LIEF 9 !
,ID1S RE/S1 "! W 2 ,=MS N ONLY ( SUB/.ES
B AL ( _M O!R ?+S2 =! 3CEPT IS S+LE N
ONLY 9 ! CASE ( SUB/.ES1 B AL 9 T ( NON-
SUB/.ES1 & "! >E SCI;ES ( O!R ?+S ?AN
SUB/.E2 &A ?\S& O!R S* DI6ICULTIES
3FRONT !M4 ,B AC 6! NECESSITIES (! CASE
&! OP9IONS AB ! ,=MS1 IF !Y C 2 %>$ 9 "!
M/ 2 ,ID1S ( SUB/.ES ONLY4 ,= !Y >E N
%>$ 9 9CID5T,Y1 B EA* ,=M M/ 2 %>$ 9 Z
"S?+ N PR$ICAT$ (A SUBJECT4 7,BY ,82+
%>$ 9 9CID5T,Y0' ,I M1N T IF A ?+ %>ES 9
,8D\# XF0'1 X %>ES AL 9 ,8ET]NAL0'1 B
9CID5T,Y2 = ,8! D\#0' HAPP5S 6BE
ET]NAL47 ,"!=E ! ,=MS W 2 SUB/.E4 ,B !
SAME "NS 9DICATE SUB/.E 9 ? & 9 ! ID1L
_W 7OR :AT W 2 ! M1N+ ( SAY+ T "! IS
"S?+ A"P F ! "PICUL>S-! "O OV] _M874 ,&
IF ! ,ID1S &! ?+S T %>E 9 !M H ! SAME
=M1 "! W 2 "S?+ -MON3 = :Y %D #'B0' 2 "O
&! SAME 9 ! P]I%A# #B'S1 OR 9 ! #B'S :
>E _M B ET]NAL1 & N ! SAME 9 ! #'B XF0'
Z 9 ! 9DIVIDUAL #B8 ,B IF !Y H N ! SAME
=M1 !Y W H ONLY ! "N 9 -MON1 & X IS Z IF
"O 7 6CALL BO? ,CALLIAS &A PIECE ( WOOD
A ,8MAN0'1 )\T OBS]V+ ANY -MUN;Y 2T !M4
  ,B IF WE >E 6SUPPOSE T 9 O!R RESPECTS
! -MON DEF9I;NS APPLY 6! ,=MS1 E4G4 T
,8PLANE FIGURE0' &! O!R "PS (! DEF9I;N
APPLY 6! CIRCLE XF1 B ,8:AT RE,Y IS0'
HAS 6BE A4$1 WE M/ 9QUIRE :E!R ? IS N
ABSOLUTELY M1N+.S4 ,= 6:AT IS ? 6BE A4$8
,TO ,8C5TRE0' OR TO ,8PLANE0' OR 6ALL !
"PS (! DEF9I;N8 ,= ALL ! ELE;TS 9 !
ESS;E >E ,ID1S1 E4G4 ,8ANIMAL0' & ,8TWO-
FOOT$0'4 ,FUR!R1 "! M/ 2 "S ,ID1L ANSW]+
TO ,8PLANE0' ABV1 "S NATURE : W 2 PRES5T
9 ALL ! ,=MS Z _! G5US4
                   #E
  ,ABV ALL "O MI<T 4CUSS ! "Q :AT 9 ! _W
! ,=MS 3TRIBUTE 6S5SI# ?+S1 EI 6^? T >E
ET]NAL OR 6^? T -E 962+ & C1SE 6BE2 = !Y
CAUSE NEI MOVE;T NOR ANY *ANGE 9 !M4 ,B
AG !Y HELP 9 NO WISE EI T[>DS ! K ( O!R
?+S 7= !Y >E N EV5 ! SUB/.E ( ^!1 ELSE
!Y WD H BE5 9 !M71 OR T[>DS _! 2+1 IF !Y
>E N 9 ! 9DIVIDUALS : %>E 9 !M2 ?\< IF
!Y W]E1 !Y MI<T 2 ?"\ 6BE CAUSES1 Z :ITE
CAUSES :ITE;S 9 A :ITE OBJECT 05T]+ 96XS
-POSI;N4 ,B ? >GU;T1 : 0 US$ F/
0,ANAXAGORAS1 & LAT] 0,EUDOXUS 9 8
4CUS.N ( DI6ICULTIES & 0C]TA9 O!RS1 IS V
EASILY UPSET2 = X IS EASY 6COLLECT _M &
9SUP]A# OBJEC;NS 6S* A VIEW4
  ,B1 FUR!R1 ALL O!R ?+S _C -E F ! ,=MS
9 ANY (! USUAL S5SES ( ,8F0'4 ,& 6SAY T
!Y >E PATT]NS &! O!R ?+S %>E 9 !M IS
6USE EMPTY ^WS & POETICAL METAPHORS4 ,=
:AT IS X T "WS1 LOOK+ 6! ,ID1S8 ,& ANY
?+ C BO? 2 & -E 962+ )\T 2+ COPI$ F "S?+
ELSE1 S T1 :E!R ,SOCRATES EXI/S OR N1 A
MAN L ,SOCRATES MI<T -E 6BE4 ,& EVID5TLY
? MI<T 2 S EV5 IF ,SOCRATES 7 ET]NAL4 ,&
"! W 2 S"EAL PATT]NS (! SAME ?+1 & "!=E
S"EAL ,=MS2 E4G4 ,8ANIMAL0' & ,8TWO-
FOOT$0'1 & AL ,8MAN-HMF0'1 W 2 ,=MS (
MAN4 ,AG1 ! ,=MS >E PATT]NS N ONLY (
S5SI# ?+S1 B ( ,=MS !MVS AL2 I4E4 ! G5US
IS ! PATT]N (! V>I\S =MS-(-A-G5US2 "!=E
! SAME ?+ W 2 PATT]N & COPY4
  ,AG1 X WD SEEM IMPOSSI# T SUB/.E & T
^: SUB/.E X IS %D EXI/ A"P2 H[1 "!=E1 CD
! ,ID1S1 2+ ! SUB/.ES ( ?+S1 EXI/ A"P8
  ,9 ! ,PHAEDO ! CASE IS /AT$ 9 ? WAY-T
! ,=MS >E CAUSES BO? ( 2+ &( 2COM+4 ,YET
?\< ! ,=MS EXI/1 / ?+S D N -E 962+1 UN.S
"! IS "S?+ 6ORIG9ATE MOVE;T2 & _M O!R
?+S -E 962+ 7E4G4 A H\SE OR A R+7 ( : !Y
SAY "! >E NO ,=MS4 ,CLE>LY "!=E EV5 !
?+S ( : !Y SAY "! >E ,ID1S C BO? 2 & -E
962+ [+ 6S* CAUSES Z PRODUCE ! ?+S J
M5;N$1 & N [+ 6! ,=MS4 ,B REG>D+ ! ,ID1S
X IS POSSI#1 BO? 9 ? WAY & 0M AB/RACT &
A3URATE >GU;TS1 6COLLECT _M OBJEC;NS L
^? WE H 3SID]$4
                   #F
  ,S9CE WE H 4CUSS$ ^! PO9TS1 X IS WELL
63SID] AG ! RESULTS REG>D+ NUMB]S :
3FRONT ^? :O SAY T NUMB]S >E SEP>A#
SUB/.ES & F/ CAUSES ( ?+S4 ,IF NUMB] IS
AN 5T;Y & XS SUB/.E IS NO?+ O!R ?AN J
NUMB]1 Z "S SAY1 X FOLL[S T EI 7#A7 "!
IS A F/ 9 X &A SECOND1 EA* 2+ DI6]5T 9
SPECIES1-& EI 7A7 ? IS TRUE (! UNITS )\T
EXCEP;N1 & ANY UNIT IS 9ASSOCIA# ) ANY
UNIT1 OR 7B7 !Y >E ALL )\T EXCEP;N
SU3ESSIVE1 & ANY ( !M >E ASSOCIA# ) ANY1
Z !Y SAY IS ! CASE ) MA!MATICAL NUMB]2 =
9 MA!MATICAL NUMB] NO "O UNIT IS 9 ANY
WAY DI6]5T F ANO!R4 ,OR 7C7 "S UNITS M/
2 ASSOCIA# & "S N2 E4G4 SUPPOSE T #B IS
F/ AF #A1 & !N -ES #C & !N ! RE/ (!
NUMB] S]IES1 &! UNITS 9 EA* NUMB] >E
ASSOCIA#1 E4G4 ^? 9 ! F/ #B >E ASSOCIA#
) "O ANO!R1 & ^? 9 ! F/ #C ) "O ANO!R1 &
S )! O!R NUMB]S2 B ! UNITS 9 ! #'B-XF0'
>E 9ASSOCIA# ) ^? 9 ! #'C-XF0'2 &
SIMIL>LY 9 ! CASE (! O!R SU3ESSIVE
NUMB]S4 ,& S :ILE MA!MATICAL NUMB] IS
C.T$ ?US-AF #A1 #B 7: 3SI/S ( ANO!R #A
2SS ! =M] #A71 & #C : 3SI/S ( ANO!R #A
2SS ^! TWO71 &! O!R NUMB]S SIMIL>LY1
ID1L NUMB] IS C.T$ ?US-AF #A1 A 4T9CT #B
: DOES N 9CLUDE ! F/ #A1 &A #C : DOES N
9CLUDE ! #B &! RE/ (! NUMB] S]IES
SIMIL>LY4 ,OR 7#B7 "O K9D ( NUMB] M/ 2 L
! F/ T 0 "ND1 "O L T : ! MA!MATICIANS
SP1K (1 & T : WE H "ND LA/ M/ 2 A ?IRD
K9D4
  ,AG1 ^! K9DS ( NUMB]S M/ EI 2 SEP>A# F
?+S1 OR N SEP>A# B 9 OBJECTS ( P]CEP;N
7N H["E 9 ! WAY : WE F/ 3SID]$1 9 ! S5SE
T OBJECTS ( P]CEP;N 3SI/S ( NUMB]S : >E
PRES5T 9 !M7-EI "O K9D & N ANO!R1 OR ALL
( !M4
  ,^! >E ( NECESS;Y ! ONLY WAYS 9 : !
NUMB]S C EXI/4 ,&( ^? :O SAY T ! #A IS !
2G9N+ & SUB/.E & ELE;T ( ALL ?+S1 & T
NUMB] IS =M$ F ! #A & "S?+ ELSE1 ALM E
"O HAS DESCRIB$ NUMB] 9 "O ( ^! WAYS2
ONLY NO "O HAS SD ALL ! UNITS >E
9ASSOCIA#4 ,& ? HAS HAPP5$ R1SONABLY
5\<2 = "! C 2 NO WAY 2SS ^? M5;N$4 ,"S
SAY BO? K9DS ( NUMB] EXI/1 T : HAS A 2F
& AF 2+ ID5TICAL )! ,ID1S1 & MA!MATICAL
NUMB] 2+ DI6]5T F ! ,ID1S & F S5SI# ?+S1
& BO? 2+ SEP>A# F S5SI# ?+S2 & O!RS SAY
MA!MATICAL NUMB] AL"O EXI/S1 Z ! F/ (
R1LITIES1 SEP>ATE F S5SI# ?+S4 ,&!
,PY?AGOR1NS1 AL1 2LIEVE 9 "O K9D (
NUMB]-! MA!MATICAL2 ONLY !Y SAY X IS N
SEP>ATE B S5SI# SUB/.ES >E =M$ \ ( X4 ,=
!Y 3/RUCT ! :OLE UNIV]SE \ ( NUMB]S-ONLY
N NUMB]S 3SI/+ ( AB/RACT UNITS2 !Y
SUPPOSE ! UNITS 6H SPATIAL MAGNITUDE4 ,B
H[ ! F/ #A 0 3/RUCT$ S Z 6H MAGNITUDE1
!Y SEEM UNA# 6SAY4
  ,ANO!R ?9K] SAYS ! F/ K9D ( NUMB]1 T (
! ,=MS1 AL"O EXI/S1 & "S SAY MA!MATICAL
NUMB] IS ID5TICAL ) ?4
  ,! CASE ( L9ES1 PLANES1 & SOLIDS IS
SIMIL>4 ,= "S ?9K T ^? : >E ! OBJECTS (
MA!MATICS >E DI6]5T F ^? : -E AF !
,ID1S2 &( ^? :O EXPRESS !MVS O!RWISE "S
SP1K (! OBJECTS ( MA!MATICS & 9 A
MA!MATICAL WAY-VIZ4 ^? :O D N MAKE !
,ID1S NUMB]S NOR SAY T ,ID1S EXI/2 &
O!RS SP1K (! OBJECTS ( MA!MATICS1 B N
MA!MATIC,Y2 = !Y SAY T NEI IS E SPATIAL
MAGNITUDE DIVISI# 96MAGNITUDES1 NOR D
ANY TWO UNITS TAK5 AT R&OM MAKE #B4 ,ALL
:O SAY ! #A IS AN ELE;T & PR9CIPLE ( ?+S
SUPPOSE NUMB]S 63SI/ ( AB/RACT UNITS1
EXCEPT ! ,PY?AGOR1NS2 B !Y SUPPOSE !
NUMB]S 6H MAGNITUDE1 Z HAS BE5 SD 2F4 ,X
IS CLE> F ? /ATE;T1 !N1 9 H[ _M WAYS
NUMB]S MAY 2 DESCRIB$1 & T ALL ! WAYS H
BE5 M5;N$2 & ALL ^! VIEWS >E IMPOSSI#1 B
"S P]H M ?AN O!RS4

                   #G
  ,F/1 !N1 LET U 9QUIRE IF ! UNITS >E
ASSOCIA# OR 9ASSOCIA#1 & IF 9ASSOCIA#1 9
: (! TWO WAYS WE 4T+UI%$4 ,= X IS POSSI#
T ANY UN;Y IS 9ASSOCIA# ) ANY1 & X IS
POSSI# T ^? 9 ! ,8XF0' >E 9ASSOCIA# ) ^?
9 ! ,8XF0'1 &1 G5],Y1 T ^? 9 EA* ID1L
NUMB] >E 9ASSOCIA# ) ^? 9 O!R ID1L
NUMB]S4 ,N[ 7#A7 ALL UNITS >E ASSOCIA# &
)\T DI6];E1 WE GET MA!MATICAL NUMB]-ONLY
"O K9D ( NUMB]1 &! ,ID1S _C 2 ! NUMB]S4
,= :AT SORT ( NUMB] W MAN-HMF OR ANIMAL-
XF OR ANY O!R ,=M BE8 ,"! IS "O ,IDEA (
EA* ?+ E4G4 "O ( MAN-HMF & ANO!R "O (
ANIMAL-XF2 B ! SIMIL> & UNDI6]5TIAT$
NUMB]S >E 9F9ITELY _M1 S T ANY "PICUL>
#C IS NO M MAN-HMF ?AN ANY O!R #C4 ,B IF
! ,ID1S >E N NUMB]S1 NEI C !Y EXI/ AT
ALL4 ,= F :AT PR9CIPLES W ! ,ID1S -E8 ,X
IS NUMB] T -ES F ! #A &! 9DEF9ITE DYAD1
&! PR9CIPLES OR ELE;TS >E SD 6BE
PR9CIPLES & ELE;TS ( NUMB]1 &! ,ID1S _C
2 RANK$ Z EI PRIOR OR PO/]IOR 6! NUMB]S4
  ,B 7#B7 IF ! UNITS >E 9ASSOCIA#1 &
9ASSOCIA# 9 ! S5SE T ANY IS 9ASSOCIA# )
ANY O!R1 NUMB] ( ? SORT _C 2 MA!MATICAL
NUMB]2 = MA!MATICAL NUMB] 3SI/S (
UNDI6]5TIAT$ UNITS1 &! TRU?S PROV$ ( X
SUIT ? "*4 ,NOR C X 2 ID1L NUMB]4 ,= #B
W N PROCE$ IMMLY F #A &! 9DEF9ITE DYAD1
& 2 FOLL[$ 0! SU3ESSIVE NUMB]S1 Z !Y SAY
#'B1C1D0' =! UNITS 9 ! ID1L >E G5]AT$ AT
! SAME "T1 :E!R1 Z ! F/ HOLD] (! !ORY
SD1 F UNEQUALS 7-+ 962+ :5 ^! 7
EQUALIZ$7 OR 9 "S O!R WAY-S9CE1 IF "O
UNIT IS 6BE PRIOR 6! O!R1 X W 2 PRIOR AL
6#B ! -POS$ ( ^!2 = :5 "! IS "O ?+ PRIOR
& ANO!R PO/]IOR1 ! RESULTANT ( ^! W 2
PRIOR 6"O & PO/]IOR 6! O!R4 ,AG1 S9CE !
#A-XF IS F/1 & !N "! IS A "PICUL> #A :
IS F/ AM;G ! O!RS & NEXT AF ! #A-XF1 &
AG A ?IRD : IS NEXT AF ! SECOND & NEXT B
"O AF ! F/ #A1-S ! UNITS M/ 2 PRIOR 6!
NUMB]S AF : !Y >E "ND :5 WE C.T !M2 E4G4
"! W 2 A ?IRD UNIT 9 #B 2F #C EXI/S1 &A
F\R? &A FIF? 9 #C 2F ! NUMB]S #D & #E
EXI/4-,N[ N"O ( ^! ?9K]S HAS SD ! UNITS
>E 9ASSOCIA# 9 ? WAY1 B AC 6_! PR9CIPLES
X IS R1SONA# T !Y %D 2 S EV5 9 ? WAY1
?\< 9 TRU? X IS IMPOSSI#4 ,= X IS
R1SONA# BO? T ! UNITS %D H PRIOR;Y &
PO/]IOR;Y IF "! IS A F/ UNIT OR F/ #A1 &
AL T ! #B'S %D IF "! IS A F/ #B2 = AF !
F/ X IS R1SONA# & NEC T "! %D 2 A
SECOND1 & IF A SECOND1 A ?IRD1 & S )!
O!RS SU3ESSIVELY4 7,& 6SAY BO? ?+S AT !
SAME "T1 T A UNIT IS F/ & ANO!R UNIT IS
SECOND AF ! ID1L #A1 & T A #B IS F/ AF
X1 IS IMPOSSI#47 ,B !Y MAKE A F/ UNIT OR
#A1 B N AL A SECOND &A ?IRD1 &A F/ #B1 B
N AL A SECOND &A ?IRD4 ,CLE>LY1 AL1 X IS
N POSSI#1 IF ALL ! UNITS >E 9ASSOCIA#1 T
"! %D 2 A #B-XF &A #C-XF2 & S )! O!R
NUMB]S4 ,= :E!R ! UNITS >E UNDI6]5TIAT$
OR DI6]5T EA* F EA*1 NUMB] M/ 2 C.T$
0A4I;N1 E4G4 #B 0A4+ ANO!R #A 6! "O1 #C
0A4+ ANO!R #A 6! TWO1 & SIMIL>LY4 ,? 2+
S1 NUMB]S _C 2 G5]AT$ Z !Y G5]ATE !M1 F
! #B &! #A2 = #B 2COMES "P ( #C & #C (
#D &! SAME HAPP5S 9 ! CASE (! SU3E$+
NUMB]S1 B !Y SAY #D CAME F ! F/ #B &!
9DEF9ITE : MAKES X TWO #B'S O!R ?AN !
#B-XF2 IF N1 ! #B-XF W 2 A "P ( #D & "O
O!R #B W 2 A4$4 ,& SIMIL>LY #B W 3SI/ (!
#A-XF & ANO!R #A2 B IF ? IS S1 ! O!R
ELE;T _C 2 AN 9DEF9ITE #B2 = X G5]ATES
"O UNIT1 N1 Z ! 9DEF9ITE #B DOES1 A
DEF9ITE #B4
  ,AG1 2SS ! #C-XF &! #B-XF H[ C "! 2
O!R #C'S & #B'S8 ,& H[ D !Y 3SI/ ( PRIOR
& PO/]IOR UNITS8 ,ALL ? IS ABSURD &
FICTITI\S1 & "! _C 2 A F/ #B & !N A #C-
XF4 ,YET "! M/1 IF ! #A &! 9DEF9ITE DYAD
>E 6BE ! ELE;TS4 ,B IF ! RESULTS >E
IMPOSSI#1 X IS AL IMPOSSI# T ^! >E !
G5]AT+ PR9CIPLES4
  ,IF ! UNITS1 !N1 >E DI6]5TIAT$1 EA* F
EA*1 ^! RESULTS & O!RS SIMIL> 6^! FOLL[
( NECESS;Y4 ,B 7#C7 IF ^? 9 DI6]5T
NUMB]S >E DI6]5TIAT$1 B ^? 9 ! SAME
NUMB] >E AL"O UNDI6]5TIAT$ F "O ANO!R1
EV5 S ! DI6ICULTIES T FOLL[ >E NO LESS4
,E4G4 9 ! #AJ-XF _! >E T5 UNITS1 &! #AJ
IS -POS$ BO? ( !M &( TWO #E'S4 ,B S9CE !
#AJ-XF IS N ANY *.E NUMB] NOR -POS$ (
ANY *.E #E'S--OR1 = T MATT]1 UNITS--!
UNITS 9 ? #AJ M/ DI6]4 ,= IF !Y D N
DI6]1 NEI W ! #E'S ( : ! #AJ 3SI/S DI6]2
B S9CE ^! DI6]1 ! UNITS AL W DI6]4 ,B IF
!Y DI6]1 W "! 2 NO O!R #E'S 9 ! #AJ B
ONLY ^! TWO1 OR W "! 2 O!RS8 ,IF "! >E
N1 ? IS P>ADOXICAL2 & IF "! >E1 :AT SORT
( #AJ W 3SI/ ( !M8 ,= "! IS NO O!R 9 !
#AJ B ! #AJ XF4 ,B X IS ACTU,Y NEC ON _!
VIEW T ! #D %D N 3SI/ ( ANY *.E #B'S2 =!
9DEF9ITE Z !Y SAY1 RCVD ! DEF9ITE #B &
MADE TWO #B'S2 = XS NATURE 0 6D\# :AT X
RCVD4
  ,AG1 Z 6! #B 2+ AN 5T;Y A"P F XS TWO
UNITS1 &! #C AN 5T;Y A"P F XS ?REE
UNITS1 H[ IS ? POSSI#8 ,EI 0"O'S %>+ 9 !
O!R1 Z ,8PALE MAN0' IS DI6]5T F ,8PALE0'
& ,8MAN0' 7= X %>ES 9 ^!71 OR :5 "O IS A
DI6]5TIA (! O!R1 Z ,8MAN0' IS DI6]5T F
,8ANIMAL0' & ,8TWO-FOOT$0'4
  ,AG1 "S ?+S >E "O 03TACT1 "S
09T]MIXTURE1 "S 0POSI;N2 N"O ( : C 2L;G
6! UNITS ( : ! #B OR ! #C 3SI/S2 B Z TWO
M5 >E N A UN;Y A"P F BO?1 S M/ X 2 )!
UNITS4 ,& _! 2+ 9DIVISI# W MAKE NO
DI6];E 6!M2 = PO9TS TOO >E 9DIVISI#1 B
YET A PAIR ( !M IS NO?+ A"P F ! TWO4
  ,B ? 3SEQU;E AL WE M/ N =GET1 T X
FOLL[S T "! >E PRIOR & PO/]IOR #B &
SIMIL>LY )! O!R NUMB]S4 ,= LET ! #B'S 9
! #D 2 SIMULTANE\S2 YET ^! >E PRIOR 6^?
9 ! #H & Z ! #B G5]AT$ !M1 !Y G5]AT$ !
#D'S 9 ! #H-XF4 ,"!=E IF ! F/ #B IS AN
,IDEA1 ^! #B'S AL W 2 ,ID1S ( "S K9D4
,&! SAME A3.T APPLIES 6! UNITS2 =! UNITS
9 ! F/ #B G5]ATE ! F\R 9 #D1 S T ALL !
UNITS -E 6BE ,ID1S & AN ,IDEA W 2 -POS$
( ,ID1S4 ,CLE>LY "!=E ^? ?+S AL ( : ^!
HAPP5 6BE ! ,ID1S W 2 -POSITE1 E4G4 "O
MI<T SAY T ANIMALS >E -POS$ ( ANIMALS1
IF "! >E ,ID1S ( !M4
  ,9 G5]AL1 6DI6]5TIATE ! UNITS 9 ANY
WAY IS AN ABSURD;Y &A FIC;N2 & 0A FIC;N
,I M1N A =C$ /ATE;T MADE 6SUIT A
HYPO!SIS4 ,= NEI 9 QUANT;Y NOR 9 QUAL;Y
D WE SEE UNIT DI6]+ F UNIT1 & NUMB] M/ 2
EI EQUAL OR UNEQUAL-ALL NUMB] B ESPECI,Y
T : 3SI/S ( AB/RACT UNITS-S T IF "O
NUMB] IS NEI GRT] NOR LESS ?AN ANO!R1 X
IS EQUAL 6X2 B ?+S T >E EQUAL & 9 NO
WISE DI6]5TIAT$ WE TAKE 6BE ! SAME :5 WE
>E SP1K+ ( NUMB]S4 ,IF N1 N EV5 ! #B 9 !
#AJ-XF W 2 UNDI6]5TIAT$1 ?\< !Y >E
EQUAL2 = :AT R1SON W ! MAN :O ALLEGES T
!Y >E N DI6]5TIAT$ 2 A# 6GIVE8
  ,AG1 IF E UNIT @+ ANO!R UNIT MAKES
TWO1 A UNIT F ! #B-XF & "O F ! #C-XF W
MAKE A #B4 ,N[ 7A7 ? W 3SI/ ( DI6]5TIAT$
UNITS2 & W X 2 PRIOR 6! #C OR PO/]IOR8
,X R SEEMS T X M/ 2 PRIOR2 = "O (! UNITS
IS SIMULTANE\S )! #C &! O!R IS
SIMULTANE\S )! #B4 ,& WE1 = \R "P1
SUPPOSE T 9 G5]AL #A & #A1 :E!R ! ?+S >E
EQUAL OR UNEQUAL1 IS #B1 E4G4 ! GD &!
BAD1 OR A MAN &A HORSE2 B ^? :O HOLD ^!
VIEWS SAY T N EV5 TWO UNITS >E #B4
  ,IF ! NUMB] (! #C-XF IS N GRT] ?AN T (
! #B1 ? IS SURPRIS+2 & IF X IS GRT]1
CLE>LY "! IS AL A NUMB] 9 X EQUAL 6! #B1
S T ? IS N DI6]5T F ! #B-XF4 ,B ? IS N
POSSI#1 IF "! IS A F/ &A SECOND NUMB]4
  ,NOR W ! ,ID1S 2 NUMB]S4 ,= 9 ?
"PICUL> PO9T !Y >E "R :O CLAIM T ! UNITS
M/ 2 DI6]5T1 IF "! >E 6BE ,ID1S2 Z HAS
BE5 SD 2F4 ,=! ,=M IS UNIQUE2 B IF !
UNITS >E N DI6]5T1 ! #B'S &! #C'S AL W N
2 DI6]5T4 ,? IS AL ! R1SON :Y !Y M/ SAY
T :5 WE C.T ?US-#'A1B0'COMWE D N PROCE$
0A4+ 6! GIV5 NUMB]2 = IF WE D1 NEI W !
NUMB]S 2 G5]AT$ F ! 9DEF9ITE DYAD1 NOR C
A NUMB] 2 AN ,IDEA2 = !N "O ,IDEA W 2 9
ANO!R1 & ALL ,=MS W 2 "PS ( "O ,=M4 ,& S
)A VIEW 6_! HYPO!SIS _! /ATE;TS >E "R1 B
Z A :OLE !Y >E WR;G2 = _! VIEW IS V
DE/RUCTIVE1 S9CE !Y W ADMIT T ? "Q XF
AF=DS "S DI6ICULTY-:E!R1 :5 WE C.T & SAY
-#A1B1C-;WE C.T 0A4I;N OR 0SEP>ATE
POR;NS4 ,B WE D BO?2 & S X IS ABSURD
6R1SON BACK F ? PRO#M 6S GRT A DI6];E (
ESS;E4
                   #H
  ,F/ ( ALL X IS WELL 6DET]M9E :AT IS !
DI6]5TIA (A NUMB]-&(A UNIT1 IF X HAS A
DI6]5TIA4 ,UNITS M/ DI6] EI 9 QUANT;Y OR
9 QUAL;Y2 & NEI ( ^! SEEMS 6BE POSSI#4
,B NUMB] QUA NUMB] DI6]S 9 QUANT;Y4 ,&
IF ! UNITS AL DID DI6] 9 QUANT;Y1 NUMB]
WD DI6] F NUMB]1 ?\< EQUAL 9 NUMB] (
UNITS4 ,AG1 >E ! F/ UNITS GRT] OR
SMALL]1 & D ! LAT] "OS 9CR1SE OR DIM9I%8
,ALL ^! >E IRR,NAL SUPPOSI;NS4 ,B NEI C
!Y DI6] 9 QUAL;Y4 ,= NO ATTRIBUTE C
ATTA* 6!M2 = EV5 6NUMB]S QUAL;Y IS SD
62L;G AF QUANT;Y4 ,AG1 QUAL;Y CD N -E
6!M EI F ! #A OR ! DYAD2 =! =M] HAS NO
QUAL;Y1 &! LATT] GIVES QUANT;Y2 = ? 5T;Y
IS :AT MAKES ?+S 6BE _M4 ,IF ! FACTS >E
RE,Y O!RWISE1 !Y %D /ATE ? Q AT ! 2G9N+
& DET]M9E IF POSSI#1 REG>D+ ! DI6]5TIA (
! UNIT1 :Y X M/ EXI/1 &1 FAIL+ ?1 :AT
DI6]5TIA !Y M1N4
  ,EVID5TLY !N1 IF ! ,ID1S >E NUMB]S1 !
UNITS _C ALL 2 ASSOCIA#1 NOR C !Y 2
9ASSOCIA# 9 EI (! TWO WAYS4 ,B NEI IS !
WAY 9 : "S O!RS SP1K AB NUMB]S CORRECT4
,^! >E ^? :O D N ?9K "! >E ,ID1S1 EI )\T
QUALIFIC,N OR Z ID5TIFI$ ) C]TA9 NUMB]S1
B ?9K ! OBJECTS ( MA!MATICS EXI/ &!
NUMB]S >E ! F/ ( EXI/+ ?+S1 &! #A-XF IS
! />T+-PO9T ( !M4 ,X IS P>ADOXICAL T "!
%D 2 A #A : IS F/ ( #A'S1 Z !Y SAY1 B N
A #B : IS F/ ( #B'S1 NOR A #C ( #C'S2 =!
SAME R1SON+ APPLIES 6ALL4 ,IF1 !N1 !
FACTS ) REG>D 6NUMB] >E S1 & "O SUPPOSES
MA!MATICAL NUMB] AL"O 6EXI/1 ! #A IS N !
/>T+-PO9T 7= ? SORT ( #A M/ DI6] F !-O!R
UNITS2 & IF ? IS S1 "! M/ AL 2 A #B : IS
F/ ( #B'S1 & SIMIL>LY )! O!R SU3ESSIVE
NUMB]S74 ,B IF ! #A IS ! />T+-PO9T1 !
TRU? AB ! NUMB]S M/ R 2 :AT ,PLATO US$
6SAY1 & "! M/ 2 A F/ #B & #C & NUMB]S M/
N 2 ASSOCIA# ) "O ANO!R4 ,B IF ON ! O!R
H& "O SUPPOSES ?1 _M IMPOSSI# RESULTS1 Z
WE H SD1 FOLL[4 ,B EI ? OR ! O!R M/ 2 !
CASE1 S T IF NEI IS1 NUMB] _C EXI/
SEP>ATELY4
  ,X IS EVID5T1 AL1 F ? T ! ?IRD V].N IS
! WOR/1-! VIEW ID1L & MA!MATICAL NUMB]
IS ! SAME4 ,= TWO MISTAKES M/ !N MEET 9
! "O OP9ION4 7#A7 ,MA!MATICAL NUMB] _C 2
( ? SORT1 B ! HOLD] ( ? VIEW HAS 6SP9 X
\ 0MAK+ SUPPOSI;NS PECULI> 6HMF4 ,& 7#B7
HE M/ AL ADMIT ALL ! 3SEQU;ES T 3FRONT
^? :O SP1K ( NUMB] 9 ! S5SE ( ,8,=MS0'4
  ,! ,PY?AGOR1N V].N 9 "O WAY AF=DS FEW]
DI6ICULTIES ?AN ^? 2F "ND1 B 9 ANO!R WAY
HAS O!RS PECULI> 6XF4 ,= N ?9K+ ( NUMB]
Z CAPA# ( EXI/+ SEP>ATELY REMOVES _M (!
IMPOSSI# 3SEQU;ES2 B T BODIES %D 2 -POS$
( NUMB]S1 & T ? %D 2 MA!MATICAL NUMB]1
IS IMPOSSI#4 ,= X IS N TRUE 6SP1K (
9DIVISI# SPATIAL MAGNITUDES2 & H["E M*
"! MI<T 2 MAGNITUDES ( ? SORT1 UNITS AT
L1/ H N MAGNITUDE2 & H[ C A MAGNITUDE 2
-POS$ ( 9DIVISI#S8 ,B >I?METICAL NUMB]1
AT L1/1 3SI/S ( UNITS1 :ILE ^! ?9K]S
ID5TIFY NUMB] ) R1L ?+S2 AT ANY RATE !Y
APPLY _! PROPOSI;NS 6BODIES Z IF !Y
3SI/$ ( ^? NUMB]S4
  ,IF1 !N1 X IS NEC1 IF NUMB] IS A SELF-
SUBSI/5T R1L ?+1 T X %D EXI/ 9 "O ( ^!
WAYS : H BE5 M5;N$1 & IF X _C EXI/ 9 ANY
( ^!1 EVID5TLY NUMB] HAS NO S* NATURE Z
^? :O MAKE X SEP>A# SET UP = X4
  ,AG1 DOES EA* UNIT -E F ! GRT &!
SMALL1 EQUALIZ$1 OR "O F ! SMALL1 ANO!R
F ! GRT8 7A7 ,IF ! LATT]1 NEI DOES EA*
?+ 3TA9 ALL ! ELE;TS1 NOR >E ! UNITS )\T
DI6];E2 = 9 "O "! IS ! GRT & 9 ANO!R !
SMALL1 : IS 3TR>Y 9 XS NATURE 6! GRT4
,AG1 H[ IS X )! UNITS 9 ! #C-XF8 ,"O (
!M IS AN ODD UNIT4 ,B P]H X IS = ? R1SON
T !Y GIVE #A-XF ! MI4LE PLACE 9 ODD
NUMB]S4 7B7 ,B IF EA* (! TWO UNITS 3SI/S
( BO? ! GRT &! SMALL1 EQUALIZ$1 H[ W !
#B : IS A S+LE ?+1 3SI/ (! GRT &! SMALL8
,OR H[ W X DI6] F ! UNIT8 ,AG1 ! UNIT IS
PRIOR 6! #B2 = :5 X IS DE/ROY$ ! #B IS
DE/ROY$4 ,X M/1 !N1 2 ! ,IDEA ( AN ,IDEA
S9CE X IS PRIOR 6AN ,IDEA1 & X M/ H -E
962+ 2F X4 ,F :AT1 !N8 ,N F ! 9DEF9ITE
DYAD1 = XS FUNC;N 0 6D\#4
  ,AG1 NUMB] M/ 2 EI 9F9ITE OR F9ITE2 =
^! ?9K]S ?9K ( NUMB] Z CAPA# ( EXI/+
SEP>ATELY1 S T X IS N POSSI# T NEI ( ^?
ALT]NATIVES %D 2 TRUE4 ,CLE>LY X _C 2
9F9ITE2 = 9F9ITE NUMB] IS NEI ODD NOR
EV51 B ! G5],N ( NUMB]S IS ALW ! G5],N
EI ( AN ODD OR ( AN EV5 NUMB]2 9 "O WAY1
:5 #A OP]ATES ON AN EV5 NUMB]1 AN ODD
NUMB] IS PRODUC$2 9 ANO!R WAY1 :5 #B
OP]ATES1 ! NUMB]S GOT F #A 0D\BL+ >E
PRODUC$2 9 ANO!R WAY1 :5 ! ODD NUMB]S
OP]ATE1 ! O!R EV5 NUMB]S >E PRODUC$4
,AG1 IF E ,IDEA IS AN ,IDEA ( "S?+1 &!
NUMB]S >E ,ID1S1 9F9ITE NUMB] XF W 2 AN
,IDEA ( "S?+1 EI ( "S S5SI# ?+ OR ( "S?+
ELSE4 ,YET ? IS N POSSI# 9 VIEW ( _!
!SIS ANY M ?AN X IS R1SONA# 9 XF1 AT L1/
IF !Y >RANGE ! ,ID1S Z !Y D4
  ,B IF NUMB] IS F9ITE1 H[ F> DOES X G8
,) REG>D 6? N ONLY ! FACT B ! R1SON %D 2
/AT$4 ,B IF NUMB] GOES ONLY UP 6#AJ Z "S
SAY1 F/LY ! ,=MS W SOON RUN %ORT2 E4G4
IF #C IS MAN-HMF1 :AT NUMB] W 2 ! HORSE-
XF8 ,! S]IES (! NUMB]S : >E ! S"EAL ?+S-
!MVS GOES UP 6#AJ4 ,X M/1 !N1 2 "O (!
NUMB]S )9 ^! LIMITS2 = X IS ^! T >E
SUB/.ES & ,ID1S4 ,YET !Y W RUN %ORT2 =!
V>I\S =MS ( ANIMAL W \TNUMB] !M4 ,AT !
SAME "T X IS CLE> T IF 9 ? WAY ! #C IS
MAN-HMF1 ! O!R #C'S >E S AL 7= ^? 9
ID5TICAL NUMB]S >E SIMIL>71 S T "! W 2
AN 9F9ITE NUMB] ( M52 IF EA* #C IS AN
,IDEA1 EA* (! NUMB]S W 2 MAN-HMF1 & IF
N1 !Y W AT L1/ 2 M54 ,& IF ! SMALL]
NUMB] IS "P (! GRT] 72+ NUMB] ( S* A
SORT T ! UNITS 9 ! SAME NUMB] >E
ASSOCIA#71 !N IF ! #D-XF IS AN ,IDEA (
"S?+1 E4G4 ( ,8HORSE0' OR ( ,8:ITE0'1
MAN W 2 A "P ( HORSE1 IF MAN IS ,X IS
P>ADOXICAL AL T "! %D 2 AN ,IDEA ( #AJ B
N ( #AA1 NOR (! SU3E$+ NUMB]S4 ,AG1 "!
BO? >E & -E 6BE C]TA9 ?+S ( : "! >E NO
,=MS2 :Y1 !N1 >E "! N ,=MS ( !M AL8 ,WE
9F] T ! ,=MS >E N CAUSES4 ,AG1 X IS
P>ADOXICAL-IF ! NUMB] S]IES UP 6#AJ IS M
(A R1L ?+ &A ,=M ?AN #AJ XF4 ,"! IS NO
G5],N (! =M] Z "O ?+1 & "! IS (! LATT]4
,B !Y TRY 6"W ON ! ASSUMP;N T ! S]IES (
NUMB]S UP 6#AJ IS A -PLETE S]IES4 ,AT
L1/ !Y G5]ATE ! DERIVATIVES-E4G4 ! VOID1
PROPOR;N1 ! ODD1 &! O!RS ( ? K9D-)9 !
DECADE4 ,= "S ?+S1 E4G4 MOVE;T & RE/1 GD
& BAD1 !Y ASSIGN 6! ORIG9ATIVE
PR9CIPLES1 &! O!RS 6! NUMB]S4 ,? IS :Y
!Y ID5TIFY ! ODD ) #A2 = IF ! ODD IMPLI$
#C H[ WD #E 2 ODD8 ,AG1 SPATIAL
MAGNITUDES & ALL S* ?+S >E EXPLA9$ )\T
GO+ 2Y A DEF9ITE NUMB]2 E4G4 ! F/1 !
9DIVISI#1 L9E1 !N ! #B &C42 ^! 5TITIES
AL EXT5D ONLY UP 6#AJ4
  ,AG1 IF NUMB] C EXI/ SEP>ATELY1 "O
MI<T ASK : IS PRIOR--#A1 OR #C OR #B8
,9ASM* Z ! NUMB] IS -POSITE1 #A IS
PRIOR1 B 9ASM* Z ! UNIV]SAL &! =M IS
PRIOR1 ! NUMB] IS PRIOR2 = EA* (! UNITS
IS "P (! NUMB] Z XS MATT]1 &! NUMB] ACTS
Z =M4 ,& 9 A S5SE ! "R ANGLE IS PRIOR 6!
ACUTE1 2C X IS DET]M9ATE & 9 VIRTUE ( XS
DEF9I;N2 B 9 A S5SE ! ACUTE IS PRIOR1 2C
X IS A "P &! "R ANGLE IS DIVID$ 96ACUTE
ANGLES4 ,Z MATT]1 !N1 ! ACUTE ANGLE &!
ELE;T &! UNIT >E PRIOR1 B 9 RESPECT (!
=M &(! SUB/.E Z EXPRESS$ 9 ! DEF9I;N1 !
"R ANGLE1 &! :OLE 3SI/+ (! MATT] &! =M1
>E PRIOR2 =! 3CRETE ?+ IS NE>] 6! =M &
6:AT IS EXPRESS$ 9 ! DEF9I;N1 ?\< 9
G5],N X IS LAT]4 ,H[ !N IS #A ! />T+-
PO9T8 ,2C X IS N DIVISIA#1 !Y SAY2 B BO?
! UNIV]SAL1 &! "PICUL> OR ! ELE;T1 >E
9DIVISI#4 ,B !Y >E />T+-PO9TS 9 DI6]5T
WAYS1 "O 9 DEF9I;N &! O!R 9 "T4 ,9 :
WAY1 !N1 IS #A ! />T+-PO9T8 ,Z HAS BE5
SD1 ! "R ANGLE IS ?"\ 6BE PRIOR 6!
ACUTE1 &! ACUTE 6! "R1 & EA* IS "O4
,ACLY !Y MAKE #A ! />T+-PO9T 9 BO? WAYS4
,B ? IS IMPOSSI#4 ,=! UNIV]SAL IS "O Z
=M OR SUB/.E1 :ILE ! ELE;T IS "O Z A "P
OR Z MATT]4 ,= EA* (! TWO IS 9 A S5SE
"O-IN TRU? EA* (! TWO UNITS EXI/S
POT5TI,Y 7AT L1/ IF ! NUMB] IS A UN;Y &
N L A H1P1 I4E4 IF DI6]5T NUMB]S 3SI/ (
DI6]5TIAT$ UNITS1 Z !Y SAY71 B N 9
-PLETE R1L;Y2 &! CAUSE (! ]ROR !Y FELL
96IS T !Y 7 3DUCT+ _! 9QUIRY AT ! SAME
"T F ! /&PO9T ( MA!MATICS & F T (
UNIV]SAL DEF9I;NS1 S T 7#A7 F ! =M]
/&PO9T !Y TR1T$ UN;Y1 _! F/ PR9CIPLE1 Z
A PO9T2 =! UNIT IS A PO9T )\T POSI;N4
,!Y PUT ?+S TGR \ (! SMALLE/ "PS1 Z "S
O!RS AL H D"O4 ,"!=E ! UNIT 2COMES !
MATT] ( NUMB]S & AT ! SAME "T PRIOR 6#B2
& AG PO/]IOR1 #B 2+ TR1T$ Z A :OLE1 A
UN;Y1 &A =M4 ,B 7#B7 2C !Y 7 SEEK+ !
UNIV]SAL !Y TR1T$ ! UN;Y : C 2 PR$ICAT$
(A NUMB]1 Z 9 ? S5SE AL A "P (! NUMB]4
,B ^! "*I/ICS _C 2L;G AT ! SAME "T 6!
SAME ?+4
  ,IF ! #A-XF M/ 2 UNIT>Y 7= X DI6]S 9
NO?+ F O!R #A'S EXCEPT T X IS ! />T+-
PO9T71 &! #B IS DIVISI# B ! UNIT IS N1 !
UNIT M/ 2 LIK] ! #A-XF ?AN ! #B IS4 ,B
IF ! UNIT IS LIK] X1 X M/ 2 LIK] 6! UNIT
?AN 6! #B2 "!=E EA* (! UNITS 9 #B M/ 2
PRIOR 6! #B4 ,B !Y D5Y ?2 AT L1/ !Y
G5]ATE ! #B F/4 ,AG1 IF ! #B-XF IS A
UN;Y &! #C-XF IS "O AL1 BO? =M A #B4 ,F
:AT1 !N1 IS ? #B PRODUC$8

                   #I
  ,S9CE "! IS N 3TACT 9 NUMB]S1 B
SU3ES.N1 VIZ4 2T ! UNITS 2T : "! IS
NO?+1 E4G4 2T ^? 9 #B OR 9 #C "O MI<T
ASK :E!R ^! SU3E$ ! #A-XF OR N1 & :E!R1
(! T]MS T SU3E$ X1 #B OR EI (! UNITS 9
#B IS PRIOR4
  ,SIMIL> DI6ICULTIES O3UR ) REG>D 6!
CLASSES ( ?+S PO/]IOR 6NUMB]1-! L9E1 !
PLANE1 &! SOLID4 ,= "S 3/RUCT ^! \ (!
SPECIES (! ,8GRT & SMALL0'2 E4G4 L9ES F
! ,8L;G & %ORT0'1 PLANES F ! ,8BROAD &
N>R[0'1 MASSES F ! ,8DEEP & %ALL[0'2 :
>E SPECIES (! ,8GRT & SMALL0'4 ,&!
ORIG9ATIVE PR9CIPLE ( S* ?+S : ANSW]S 6!
#A DI6]5T ?9K]S DESCRIBE 9 DI6]5T WAYS1
,& 9 ^! AL ! IMPOSSIBILITIES1 ! FIC;NS1
&! 3TRADIC;NS ( ALL PROBABIL;Y >E SE5
6BE 9NUM]A#4 ,= 7I7 GEOMETRICAL CLASSES
>E S"E$ F "O ANO!R1 UN.S ! PR9CIPLES (
^! >E IMPLI$ 9 "O ANO!R 9 S* A WAY T !
,8BROAD & N>R[0' IS AL ,8L;G & %ORT0' 7B
IF ? IS S1 ! PLANE W 2 L9E &! SOLID A
PLANE2 AG1 H[ W ANGLES & FIGURES & S*
?+S 2 EXPLA9$874 ,& 7;II7 ! SAME HAPP5S
Z 9 REG>D 6NUMB]2 = ,8L;G & %ORT0'1 &C41
>E ATTRIBUTES ( MAGNITUDE1 B MAGNITUDE
DOES N 3SI/ ( ^!1 ANY M ?AN ! L9E 3SI/S
( ,8/RAI<T & CURV$0'1 OR SOLIDS (
,8SMOO? & R\<0'4
  7,ALL ^! VIEWS %>E A DI6ICULTY : O3URS
) REG>D 6SPECIES-(-A-G5US1 :5 "O POSITS
! UNIV]SALS1 VIZ4 :E!R X IS ANIMAL-XF OR
"S?+ O!R ?AN ANIMAL-XF T IS 9 ! "PICUL>
ANIMAL4 ,TRUE1 IF ! UNIV]SAL IS N SEP>A#
F S5SI# ?+S1 ? W PRES5T NO DI6ICULTY2 B
IF ! #A &! NUMB]S >E SEP>A#1 Z ^? :O
EXPRESS ^! VIEWS SAY1 X IS N EASY 6SOLVE
! DI6ICULTY1 IF "O MAY APPLY ! ^WS ,8N
EASY0' 6! IMPOSSI#4 ,= :5 WE APPREH5D !
UN;Y 9 #B1 OR 9 G5]AL 9 A NUMB]1 D WE
APPREH5D A ?+-XF OR "S?+ ELSE874
  ,"S1 !N1 G5]ATE SPATIAL MAGNITUDES F
MATT] ( ? SORT1 O!RS F ! PO9T -&! PO9T
IS ?"\ 0!M 6BE N #A B "S?+ L #A-& F O!R
MATT] L PLURAL;Y1 B N ID5TICAL ) X2 AB :
PR9CIPLES N"O ! LESS ! SAME DI6ICULTIES
O3UR4 ,= IF ! MATT] IS "O1 L9E & PLANE-&
SOLI W 2 ! SAME2 = F ! SAME ELE;TS W -E
"O &! SAME ?+4 ,B IF ! MATT]S >E M ?AN
"O1 & "! IS "O =! L9E &A SECOND =! PLANE
& ANO!R =! SOLID1 !Y EI >E IMPLI$ 9 "O
ANO!R OR N1 S T ! SAME RESULTS W FOLL[
EV5 S2 = EI ! PLANE W N 3TA9 A L9E OR X
W HE A L9E4
  ,AG1 H[ NUMB] C 3SI/ (! "O & PLURAL;Y1
!Y MAKE NO ATTEMPT 6EXPLA92 B H["E !Y
EXPRESS !MVS1 ! SAME OBJEC;NS >ISE Z
3FRONT ^? :O 3/RUCT NUMB] \ (! "O &!
9DEF9ITE DYAD4 ,=! "O VIEW G5]ATES NUMB]
F ! UNIV]S,Y PR$ICAT$ PLURAL;Y1 & N F A
"PICUL> PLURAL;Y2 &! O!R G5]ATES X F A
"PICUL> PLURAL;Y1 B ! F/2 = #B IS SD 6BE
A ,8F/ PLURAL;Y0'4 ,"!=E "! IS PRACTIC,Y
NO DI6];E1 B ! SAME DI6ICULTIES W
FOLL[1-IS X 9T]MIXTURE OR POSI;N OR
BL5D+ OR G5],N8 & S ON4 ,ABV ALL "O MI<T
PRESS ! "Q ,8IF EA* UNIT IS "O1 :AT DOES
X -E F80' ,C]TA9LY EA* IS N ! "O-XF4 ,X
M/1 !N1 -E F ! "O XF & PLURAL;Y1 OR A "P
( PLURAL;Y4 ,6SAY T ! UNIT IS A PLURAL;Y
IS IMPOSSI#1 = X IS 9DIVISI#2 & 6G5]ATE
X F A "P ( PLURAL;Y 9VOLVES _M O!R
OBJEC;NS2 = 7A7 EA* (! "PS M/ 2 9DIVISI#
7OR X W 2 A PLURAL;Y &! UNIT W 2
DIVISI#7 &! ELE;TS W N 2 ! "O &
PLURAL;Y2 =! S+LE UNITS D N -E F
PLURAL;Y &! "O4 ,AG1 71! HOLD] ( ? VIEW
DOES NO?+ B PRESUPPOSE ANO!R NUMB]2 = 8
PLURAL;Y ( 9DIVISI#S IS A NUMB]4 ,AG1 WE
M/ 9QUIRE1 9 VIEW ( ? !ORY AL1 :E!R !
NUMB] IS 9F9ITE OR F9ITE4 ,= "! 0 AT F/1
Z X SEEMS1 A PLURAL;Y T 0 XF F9ITE1 F :
& F ! "O -ES ! F9ITE NUMB] ( UNITS4 ,&
"! IS ANO!R PLURAL;Y T IS PLURAL;Y-XF &
9F9ITE PLURAL;Y2 : SORT ( PLURAL;Y1 !N1
IS ! ELE;T : CO-OP]ATES )! "O8 ,"O MI<T
9QUIRE SIMIL>LY AB ! PO9T1 I4E4 ! ELE;T
\ ( : !Y MAKE SPATIAL MAGNITUDES4 ,=
SURELY ? IS N ! "O & ONLY PO9T2 AT ANY
RATE1 !N1 LET !M SAY \ ( :AT EA* (!
PO9TS IS =M$4 ,C]TA9LY N ( "S 4T.E @+ !
PO9T-XF4 ,NOR AG C "! 2 9DIVISI# "PS (A
4T.E1 Z ! ELE;TS \ ( : ! UNITS >E SD 6BE
MADE >E 9DIVISI# "PS ( PLURAL;Y2 = NUMB]
3SI/S ( 9DIVISI#S1 B SPATIAL MAGNITUDES
D N4
  ,ALL ^! OBJEC;NS1 !N1 & O!RS (! SORT
MAKE X EVID5T T NUMB] & SPATIAL
MAGNITUDES _C EXI/ A"P F ?+S4 ,AG1 !
4CORD AB NUMB]S 2T ! V>I\S V].NS IS A
SIGN T X IS ! 9CORRECT;S (! ALLEG$ FACTS
!MVS T BR+S 3FU.N 96! !ORIES4 ,= ^? :O
MAKE ! OBJECTS ( MA!MATICS AL"O EXI/ A"P
F S5SI# ?+S1 SEE+ ! DI6ICULTY AB ! ,=MS
& _! FICTITI\S;S1 AB&ON$ ID1L NUMB] &
POSIT$ MA!MATICAL4 ,B ^? :O WI%$ 6MAKE !
,=MS AT ! SAME "T AL NUMB]S1 B DID N
SEE1 IF "O ASSUM$ ^! PR9CIPLES1 H[
MA!MATICAL NUMB] 0 6EXI/ A"P F ID1L1
MADE ID1L & MA!MATICAL NUMB] ! SAME-IN
^WS1 S9CE 9 FACT MA!MATICAL NUMB] HAS
BE5 DE/ROY$2 = !Y /ATE HYPO!SES PECULI>
6!MVS & N ^? ( MA!MATICS4 ,& HE :O F/
SUPPOS$ T ! ,=MS EXI/ & T ! ,=MS >E
NUMB]S & T ! OBJECTS ( MA!MATICS EXI/1
NATUR,Y SEP>AT$ ! TWO4 ,"!=E X TURNS \ T
ALL ( !M >E "R 9 "S RESPECT1 B ON ! :OLE
N "R4 ,& !Y !MVS 3FIRM ?1 = _! /ATE;TS D
N AGREE B 3FLICT4 ,! CAUSE IS T _!
HYPO!SES & _! PR9CIPLES >E FALSE4 ,& X
IS H>D 6MAKE A GD CASE \ ( BAD MAT]IALS1
AC 6,EPI*>MUS3 ,8Z SOON Z 'TIS SD1 'TIS
SE5 6BE WR;G40'
  ,B REG>D+ NUMB]S ! "QS WE H RAIS$ &!
3CLU.NS WE H R1*$ >E SU6ICI5T 7= :ILE HE
:O IS ALR 3V9C$ MI<T 2 FUR!R 3V9C$ 0A
L;G] 4CUS.N1 "O N YET 3V9C$ WD N -E ANY
NE>] 63VIC;N72 REG>D+ ! F/ PR9CIPLES &!
F/ CAUSES & ELE;TS1 ! VIEWS EXPRESS$ 0^?
:O 4CUSS ONLY S5SI# SUB/.E H BE5 "PLY
/AT$ 9 \R "WS ON NATURE1 & "PLY D N 2L;G
6! PRES5T 9QUIRY2 B ! VIEWS ( ^? :O
ASS]T T "! >E O!R SUB/.ES 2SS ! S5SI# M/
2 3SID]$ NEXT AF ^? WE H BE5 M5;N+4
,S9CE1 !N1 "S SAY T ! ,ID1S &! NUMB]S >E
S* SUB/.ES1 & T ! ELE;TS ( ^! >E ELE;TS
& PR9CIPLES ( R1L ?+S1 WE M/ 9QUIRE
REG>D+ ^! :AT !Y SAY & 9 :AT S5SE !Y SAY
X4
  ,^? :O POSIT NUMB]S ONLY1 & ^!
MA!MATICAL1 M/ 2 3SID]$ LAT]2 B Z REG>DS
^? :O 2LIEVE 9 ! ,ID1S "O MI<T SURVEY AT
! SAME "T _! WAY ( ?9K+ &! DI6ICULTY 96:
!Y FALL4 ,= !Y AT ! SAME "T MAKE ! ,ID1S
UNIV]SAL & AG TR1T !M Z SEP>A# & Z
9DIVIDUALS4 ,T ? IS N POSSI# HAS BE5
>GU$ 2F4 ,! R1SON :Y ^? :O DESCRIB$ _!
SUB/.ES Z UNIV]SAL -B9$ ^! TWO "*I/ICS 9
"O ?+1 IS T !Y DID N MAKE SUB/.ES
ID5TICAL ) S5SI# ?+S4 ,!Y ?"\ T !
"PICUL>S 9 ! S5SI# _W 7 A /ATE ( FLUX &
N"O ( !M REMA9$1 B T ! UNIV]SAL 0 A"P F
^! & "S?+ DI6]5T4 ,& ,SOCRATES GAVE !
IMPULSE 6? !ORY1 Z WE SD 9 \R E>LI]
4CUS.N1 0R1SON ( 8 DEF9I;NS1 B HE DID N
SEP>ATE UNIV]SALS F 9DIVIDUALS2 & 9 ? HE
?"\ "RLY1 9 N SEP>AT+ !M4 ,? IS PLA9 F !
RESULTS2 = )\T ! UNIV]SAL X IS N POSSI#
6GET K1 B ! SEP>,N IS ! CAUSE (!
OBJEC;NS T >ISE ) REG>D 6! ,ID1S4 ,8
SU3ESSORS1 H["E1 TR1T+ X Z NEC1 IF "! >E
6BE ANY SUB/.ES 2SS ! S5SI# & TRANSI5T
SUB/.ES1 T !Y M/ 2 SEP>A#1 _H NO O!RS1 B
GAVE SEP>ATE EXI/;E 6^! UNIV]S,Y
PR$ICAT$ SUB/.ES1 S T X FOLL[$ T
UNIV]SALS & 9DIVIDUALS 7 ALM ! SAME SORT
( ?+4 ,? 9 XF1 !N1 WD 2 "O DI6ICULTY 9 !
VIEW WE H M5;N$4
                  #AJ
  ,LET U N[ M5;N A PO9T : PRES5TS A
C]TA9 DI6ICULTY BO? 6^? :O 2LIEVE 9 !
,ID1S & 6^? :O D N1 & : 0 /AT$ 2F1 AT !
2G9N+1 AM;G ! PRO#MS4 ,IF WE D N SUPPOSE
SUB/.ES 6BE SEP>ATE1 & 9 ! WAY 9 :
9DIVIDUAL ?+S >E SD 6BE SEP>ATE1 WE %
DE/ROY SUB/.E 9 ! S5SE 9 : WE "U/&
,8SUB/.E0'2 B IF WE 3CV SUB/.ES 6BE
SEP>A#1 H[ >E WE 63CV _! ELE;TS & _!
PR9CIPLES8
  ,IF !Y >E 9DIVIDUAL & N UNIV]SAL1 7A7
R1L ?+S W 2 J (! SAME NUMB] Z ! ELE;TS1
& 7B7 ! ELE;TS W N 2 "KA#4 ,= 7A7 LET !
SYLLA#S 9 SPEE* 2 SUB/.ES1 & _! ELE;TS
ELE;TS ( SUB/.ES2 !N "! M/ 2 ONLY "O
,8BA0' & "O ( EA* (! SYLLA#S1 S9CE !Y >E
N UNIV]SAL &! SAME 9 =M B EA* IS "O 9
NUMB] &A ,8?0' & N A K9D POSSESS$ (A
-MON "N 7& AG !Y SUPPOSE T ! ,8J :AT A
?+ IS0' IS 9 EA* CASE "O74 ,& IF !
SYLLA#S >E UNIQUE1 S TOO >E ! "PS ( : !Y
3SI/2 "! W N1 !N1 2 M ;A'S ?AN "O1 NOR M
?AN "O ( ANY (! O!R ELE;TS1 ON ! SAME
PR9CIPLE ON : AN ID5TICAL SYLLA# _C EXI/
9 ! PLURAL NUMB]4 ,B IF ? IS S1 "! W N 2
O!R ?+S EXI/+ 2SS ! ELE;TS1 B ONLY !
ELE;TS4
  7B7 ,AG1 ! ELE;TS W N 2 EV5 "KA#2 = !Y
>E N UNIV]SAL1 & K IS ( UNIV]SALS4 ,? IS
CLE> F DEMON/R,NS & F DEF9I;NS2 = WE D N
3CLUDE T ? TRIANGLE HAS XS ANGLES EQUAL
6TWO "R ANGLES1 UN.S E TRIANGLE HAS XS
ANGLES EQUAL 6TWO "R ANGLES1 NOR T ? MAN
IS AN ANIMAL1 UN.S E MAN IS AN ANIMAL4
  ,B IF ! PR9CIPLES >E UNIV]SAL1 EI !
SUB/.ES -POS$ ( !M >E AL UNIV]SAL1 OR
NON-SUB/.E W 2 PRIOR 6SUB/.E2 =!
UNIV]SAL IS N A SUB/.E1 B ! ELE;T OR
PR9CIPLE IS UNIV]SAL1 &! ELE;T OR
PR9CIPLE IS PRIOR 6! ?+S ( : X IS !
PR9CIPLE OR ELE;T4
  ,ALL ^! DI6ICULTIES FOLL[ NATUR,Y1 :5
!Y MAKE ! ,ID1S \ ( ELE;TS & AT ! SAME
"T CLAIM T A"P F ! SUB/.ES : H ! SAME =M
"! >E ,ID1S1 A S+LE SEP>ATE 5T;Y4 ,B IF1
E4G4 9 ! CASE (! ELE;TS ( SPEE*1 ! ;A'S
&! ;B'S MAY Q WELL 2 _M & "! NE$ 2 NO A-
XF & ;B-XF 2SS ! _M1 "! MAY BE1 S F> Z ?
GOES1 AN 9F9ITE NUMB] ( SIMIL> SYLLA#S4
,! /ATE;T T AN K IS UNIV]SAL1 S T !
PR9CIPLES ( ?+S M/ AL 2 UNIV]SAL & N
SEP>ATE SUB/.ES1 PRES5TS 9DE$1 ( ALL !
PO9TS WE H M5;N$1 ! GRTE/ DI6ICULTY1 B
YET ! /ATE;T IS 9 A S5SE TRUE1 AL? 9 A
S5SE X IS N4 ,= K1 L ! V]B ,86"K0'1 M1NS
TWO ?+S1 ( : "O IS POT5TIAL & "O ACTUAL4
,! POT5CY1 2+1 Z MATT]1 UNIV]SAL &
9DEF9ITE1 D1LS )! UNIV]SAL & 9DEF9ITE2 B
! ACTUAL;Y1 2+ DEF9ITE1 D1LS )A DEF9ITE
OBJECT1 2+ A ,8?0'1 X D1LS )A ,8?0'4 ,B
P] A3ID5S SI<T SEES UNIV]SAL COL\R1 2C ?
9DIVIDUAL COL\R : X SEES IS COL\R2 & ?
9DIVIDUAL A : ! GRAMM>IAN 9VE/IGATES IS
AN A4 ,= IF ! PR9CIPLES M/ 2 UNIV]SAL1
:AT IS DERIV$ F !M M/ AL 2 UNIV]SAL1 Z 9
DEMON/R,NS2 & IF ? IS S1 "! W 2 NO?+
CAPA# ( SEP>ATE EXI/;E-I4E4 NO SUB/.E4
,B EVID5TLY 9 A S5SE K IS UNIV]SAL1 & 9
A S5SE X IS N4
