,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$LY MADE EV5 0! 4CUS.NS \TSIDE \R S*OOL1 &1 FUR!R1 ! GRT] "P ( \R A3.T M/ F9I% 0?R[+ LIE NUMB]S & ,ID1S2 = AF ! 4CUS.N (! ,ID1S ? REMANS Z A ?IRD 9QUIRY4 ,IF ! OBJECTS ( MA!MATICS EXI/1 !Y M/ EXI/ EI 9 S5SI# OBJECTS1 Z "S SAY1 OR SEP>ATE F S5SI# OBJECTS 7& ? AL IS SD 0"S72 OR IF !Y EXI/ 9 NEI ( ^! WAYS1 EI !Y D N EXI/1 OR !Y EXI/ ONLY 9 "S SPECIAL S5SE4 ,S T ! SUBJECT ( \R 4CUS.N W 2 N :E!R !Y EXI/ B H[ !Y EXI/4 #B ,T X IS IMPOSSI# = MA!MATICAL OBJECTS 6EXI/ 9 S5SI# ?+S1 & AT ! SAME "T T ! DOCTR9E 9 "Q IS AN >TIFICIAL "O1 HAS BE5 SD ALR 9 \R 4CUS.N ( DI6ICULTIES WE H PO9T$ \ 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 & SIE 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 SIE 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$0'4 ,FUR!R1 "! M/ 2 "S ,ID1L ANSW]+ TO ,8PLANE0' ABV1 "S NATURE : W 2 PRES5T 9 ALL ! ,=MS Z _! G5US4 #E ,ABV ALL "O MIE ET]NAL OR 6^? T -E 962+ & C1SE 6BE2 = !Y CAUSE NEI MOVE;T NOR ANY *ANGE 9 !M4 ,B AG !Y HELP 9 NO WISE EI T[>DS ! K ( O!R ?+S 7= !Y >E N EV5 ! SUB/.E ( ^!1 ELSE !Y WD H BE5 9 !M71 OR T[>DS _! 2+1 IF !Y >E N 9 ! 9DIVIDUALS : %>E 9 !M2 ?\< IF !Y W]E1 !Y MIGU;T1 : 0 US$ 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 MIE 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 MIE -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* "! MII?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 MIE 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$8 #I ,S9CE "! IS N 3TACT 9 NUMB]S1 B SU3ES.N1 VIZ4 2T ! UNITS 2T : "! IS NO?+1 E4G4 2T ^? 9 #B OR 9 #C "O MI DI6ICULTIES O3UR ) REG>D 6! CLASSES ( ?+S PO/]IOR 6NUMB]1-! L9E1 ! PLANE1 &! SOLID4 ,= "S 3/RUCT ^! \ (! SPECIES (! ,8GRT & SMALL0'2 E4G4 L9ES F ! ,8L;G & %ORT0'1 PLANES F ! ,8BROAD & N>R[0'1 MASSES F ! ,8DEEP & %ALL[0'2 : >E SPECIES (! ,8GRT & SMALL0'4 ,&! ORIG9ATIVE PR9CIPLE ( S* ?+S : ANSW]S 6! #A DI6]5T ?9K]S DESCRIBE 9 DI6]5T WAYS1 ,& 9 ^! AL ! IMPOSSIBILITIES1 ! FIC;NS1 &! 3TRADIC;NS ( ALL PROBABIL;Y >E SE5 6BE 9NUM]A#4 ,= 7I7 GEOMETRICAL CLASSES >E S"E$ 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$874 ,& 7;II7 ! SAME HAPP5S Z 9 REG>D 6NUMB]2 = ,8L;G & %ORT0'1 &C41 >E ATTRIBUTES ( MAGNITUDE1 B MAGNITUDE DOES N 3SI/ ( ^!1 ANY M ?AN ! L9E 3SI/S ( ,8/RAIE A DI6ICULTY : O3URS ) REG>D 6SPECIES-(-A-G5US1 :5 "O POSITS ! UNIV]SALS1 VIZ4 :E!R X IS ANIMAL-XF OR "S?+ O!R ?AN ANIMAL-XF T IS 9 ! "PICUL> ANIMAL4 ,TRUE1 IF ! UNIV]SAL IS N SEP>A# F S5SI# ?+S1 ? W PRES5T NO DI6ICULTY2 B IF ! #A &! NUMB]S >E SEP>A#1 Z ^? :O EXPRESS ^! VIEWS SAY1 X IS N EASY 6SOLVE ! DI6ICULTY1 IF "O MAY APPLY ! ^WS ,8N EASY0' 6! IMPOSSI#4 ,= :5 WE APPREH5D ! UN;Y 9 #B1 OR 9 G5]AL 9 A NUMB]1 D WE APPREH5D A ?+-XF OR "S?+ ELSE874 ,"S1 !N1 G5]ATE SPATIAL MAGNITUDES F MATT] ( ? SORT1 O!RS F ! PO9T -&! PO9T IS ?"\ 0!M 6BE N #A B "S?+ L #A-& F O!R MATT] L PLURAL;Y1 B N ID5TICAL ) X2 AB : PR9CIPLES N"O ! LESS ! SAME DI6ICULTIES O3UR4 ,= IF ! MATT] IS "O1 L9E & PLANE-& SOLI W 2 ! SAME2 = F ! SAME ELE;TS W -E "O &! SAME ?+4 ,B IF ! MATT]S >E M ?AN "O1 & "! IS "O =! L9E &A SECOND =! PLANE & ANO!R =! SOLID1 !Y EI >E IMPLI$ 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 MILY 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] 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 MIA# & 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 ! 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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