PROPORTIONAL FORM As APPLIED in ART and ARCHITE CTURE

PROPO RTI ONAL FO RM

FURTHER STUDI ES I N THE SCI ENCE OF BEAUTY, BEI NG SUPPLE

MENTAL TO THO SE SET FO RTH I N NATURE’ S

HARMONI C UNI TY

N SA MU E L O LMA N. A . Q ,

AND

C . A RTH U R COA N, LL B . T O RS O NA T U R E' S HARMONI C U NIT AU H F Y , ETC

TH RA N RREL N E R PT S A E BY MR LMA E D WINGS A D CO ATI G D SC I ION R . CO N

TH E T AND M T HEM I S A E BY PT E T X A AT C R CA . COAN

P TNA M ’ S SONS G . P . U NEW YO RK AND LONDON

‘ the lmtckerbocket press 1 920 Canno n . 1 9 30 u C AN SAMUEL COLMAN AND C . ARTHUR O

u I l M J

t he t ntchcrbochct pun, new Dork 00

ALL STUDENTS OF B EAUTY

PAS T AND Pusan

CONTE NTS

— P AG E FOREWORD THE COAL SACK

CHAPTE R — ’ I HEAVEN s FIRST LAW — II FORM AND PROPORTION S F E E E NU E I C E E UNIT O M ASUR M NT , M R AL UNITS , G OM TRIC M T TE F . F S THE O HE PE M . UNIT . TRAG N A ILY NTAGON A ILY L F M UMM SPIRA OR ATION . S ARY — III . THE TRIGON IN VISIB LE NATURE

—THE IV . TRIGON IN FORCE

N L H E TC . GRAVITY, SOU D , IG T ,

— ‘ V A E V E Y . A A E F E H B . SYMM TRY AND ARI T N NTIDOT OR SYMM TROP O IA E E E C E L AS YM CAT NARY CURV S , ELLIPS S , E C NTRICS , SPIRA S , ME TRICAL GROUPS — VI . THE GOLDE N SE RIE S IN NATURE — VII PROPORTIONAL FORM As APPLIED IN ART AND ARCHITE CTURE

—T VIII . HE TAG

APPE NDIX NOTE S

INDE X

Th s Cha t rs r tak en ro m The Gr e t M d A n I I e e e a e les C . . Co a . p f a o u , , 9 4

I LLUSTRATI ONS AND D I AG RAMS

Fo lio num bers inRom anr efer to th e page. uponwhich diagra m and mino r ino atratio na appear ; th oae in

I talia refer to th e page. uponwhic h m aj o r illo atratioo a are de scrib e d inth e te xt. — I TETRAGON FAMI LY . — 2 . HE XAGON GROUP — 3. DIAGONAL OF THE CUB E — 4. EXTRE ME AND ME AN RATIO

' Fro m N t r e : H rm c t ( a u a oni Uni y. )

— E RE E I 5. PLAN OF EXT M AND ME AN PROPORT ON

- E E AND 6. SOLID OF EXTR M ME AN PROPORTION

— U A T E E AND E T 7. ANG L R EX R M M AN PROPOR ION — 8 . TII E PE NTAGRAM. PE RFE CT EXTRE ME AND ME AN PROPORTION

— E 9 . PE RF CTION AMPLIFIE D

o — I . PYRAMIDAL EXTRE ME AND ME AN PROPORTION . — I I . TII E IDEAL ANGLE AND TE E GOLDE N SE RIE S

— OL D I 2 . P YE E RAL LIMITATIONS

-I —SN w T I 3 6. o CRYS ALS

’ Fro m N t r e : H rm c t ( a u a oni Uni y.) — I 7 . SNo w CRYSTAL

— P R I 8 . C U US URANITE

Th e iliuatratio na are by Mr . Co lm anexce pt wher e o th erwise indicate d b elo w o r in the wor k i teell. Th e diagrama are by Capt. Coan. viii Illustrations

P LATE — 1 9. DOLOMITE - — 20 2 I . TRIANGULAR DI ATOMs

' Fro m N t re s H rm c nt ( a u a oni U i y. )

22— — 23. HE XAGONAL DIATOMS

' Fr m N t re H rm ne nt ( o a u : a o i U i y. ) - — 24 25 . OCTAGONAL DI ATOMS

— AND 26. SE PALS SE ED VE SSE LS OF SYRINGA — 27 . ONAGRA BIE NNIS — 8 . TT 2 CO ON . — 29 . JONQUIL .

0 —T E L 3 . IG R ILY — 3L BUTTE RFLY AND TE E TRIGON

— ’ 32 . WASP S NE ST — 33. HONE Y COMB — 34. JAPANE SE IRIS (TE E TRIANGLE . KORIN)

— - 35 . DANISE C ROSS POPPY (TE E SQUARE ) — 36 . ANE MONE CORONARI A (HE XAGON) — 37 . SINGLE D AE LI A (OCTAGON) — B . 38 . OR ITS OF TE E SUPE RI OR PLANETS

— T E 39. ORB ITS OF E E INFE RIOR PLANE TS WIT MARS

— — B M 40. FORCE S COMPARED A SYM OLIC DIAGRA m T tM d (Fro he Grea o ule s . ) — 4I . PYTE AGOREAN LAWS — 42 . OVE RTONE VIB RATIONS

— M E 43. HAR ONY MADE VISIB L

Fro m T t ( he Grea Modules . ) Illustrations

PLATE

— E M E 44. GEOM TRIC DIAGRA OF PITCH INT RVALS

— ’ N E . 45 . CE LAD I S EXP RIME NTS

— E E 46. SAM CONTINUE D (PATT RNS WITE A ROUND DISK)

— B E 47. VI RATIONS OF A B LL — 48 . CATE NARY CURVE S FROM NATURE

— ° 49. ME TE OD OF DELINEATING A CATE NARY CURVE OF 1 5/75 B Y VE RSE D SINE S

— ’ 50 . AN ENGINE E R S DRAFTE D DE LINE ATION OF A CATE NARY CURVE ° F 60 B Y VE E E O RS D SIN S . — 5 L DIVE R

— E 52 . V INS — 53. EDGE S — E . LD TI E TE E 54. JAVE LIN THROW RS AN O ME EQUIVAL NT OF MOVING PICTURE

— ' 55 . SPIDE R S WE B (C . V. H. C . ) — 56. TE E FLI GE T OF TE E GOLDFINCH

— E 57 . TE GREAT WAVE OF KANAZAWA (HOKUSAI)

— T 58 . MOUNTAINS OF E E GILA RIVE R COUNTRY

— M AND E 59. SYM ETRICAL ASYMM TRICAL ELLIPSE S (ROSS) — 6o . A E E E LO T E ALL E E NCI NT GR K U ROPHOROS, SYMM TRIC Y D SIGN D ON AN ELLIPSE

— E 6 I . SYMM TRICAL AND ECCE NTRI C CIRCLE S — 62 . FLAT SPIRALS .

— ARI M D R 63. LOG TE MI C SPIRAL E TE O (FROM CE U CE ) — 64. SUNFLOWE R CAPITULUM Illustrations

— ME OD 65 . FIG. A. TE OF DRAWI NG IONIC VOLUTE (DORE R)

LOGARI E FI G . B . TE MI C SPIRAL (DUR R)

— E E E 66 . WE IT PIN CON

6 —YE L P E E 7 . L OW IN CON

— E — N 68 . SPIRAL OF WE ITE PIN PLA

—T MAXI MUs— O P M T PE 69. ROCHUS L OKING DOWN RO EE A X

—TROCE US MAxmus— DE IDE F 70. UN R S O SPIRAL — 7 I COMMON SNAI L

— E E 72 . ORE GON PIN CON

— I 73. DOL UM

— ALI OTU U A 74. H S CORR G TA

—XE NOPE ORA I S 75. SOLAR — 76. NAUTE .US — 77 . MURE X

— C A P RETI OSA 78 . S AL RIA

— L EL E AND E T 79 . SPIRA H IC S ECC N RICS

—E E 80 . Y LLOW CLOV R

I —T C U M S . RO H S MAXI US — 82 . FACELARI A

— A 83. ASP RAGUS

— LB OP TE E (NP RE S S 84. E OW MON R Y

—B E NI A 85 . GO

—FUsus—A LE T C L 86. O ROPI SPIRA ’ t (Fro m Natur e s Harmonic Uni y. )

— RYS0DOMUS ANTI UUS A E C L 87 . CE Q ( D XIOTROPI SPIRA ) Illustrations

— R URA 88. SAN PAOLO FUO I LE M

— E E M N URE 89 . GR K OUPPLO (FROM NAT )

— 9I WALLACE I AN SE E E P

— E 92 . KOODOO (HOOK R)

— E 93. AQUIL GIA

— I 94. PLAN OP A CRINO D — 95. C RINOI D

— E 96. ANT DON — 97 . ECHI NODE RM — - 98 . STAR FISE

— NE 99. O OP TE E ASTE ROIDE — - I oo . PLAN OP WOOD SORREL

I — OI . PYROLA . — I o z . LOOS E STRI P E — I O3. P I TCE E R PLANT

I — O4. PARNASSIA

I O — R 5 . PA NASSIA

I — o6 . HE NB ANE — I 0 . 7 JAPANE SE BE LL FLOWE R .

1 0 — E 8. PIPSISS WA

I 0 — 9. SPIRE A VAN HOUTI — I I o . WILD ROSE

I —W B I I . OOD INE — I I 2 . FOOT OP A HOUND xn Illustrations

P LATE — I I 3. ANATOMY OP TE E HUMAN HAND

— T I I 4. BONE S OP E E HAND AND ARM — 1 1 5 . BONE S OP TE E LE G AND FOOT — I I 6 . EGYPTIAN CANON As ACCE PTED B Y POLYCLITUS

— T E E I I 7 . RE LATION OP E E GRE E K CANON To A S RI S IN EXTRE ME ME AN PROPORTION ’ on (Fro m Natur e s Har m ic Unity. ) — T I 1 8 . PARTHE NON AND E E SQUARE rre latio ns Mr H i (Co by . amb dge . )

— R I I g. DE SI GN FO A FOUNTAIN — ! 2 . E I O SAN GIOVANNI , SI NA — I z I . A E OP A RC UGUSTUS , SUSA — I 22 . A E OP T R E RC ITUS , OM — 1 2 . A E OP A S R 3 RC UGUSTU , IMINI — I 2 . A E OP A A 4 RC UGUSTUS, OSTA

— ’ 1 25 . SOLOMON S SE AL AND TE E ROYAL ARCE MAS ONs

— R I 26 . FA NE SE BULL — I 27 . VE SICA PISCIS AND GOTHIC ARCE

— E AT E RAL 1 28 . AMI NS C E D

— OP 1 29. PORTAL OP SAN DONNINO EMILIA — 1 0. PALAz zo P B PE 3 U LICO , RUGIA

— R E T E ME TR OD I 3I . ASYMME T ICAL SPIRALS APPLI D TO E E P NTAGON ( C E URCE ) — I 32 . TE E ASSUMPTION . TITIAN

— ’ I 33. R UBE NS S ANGE LS — I . TE E B E T 34 ENTOM M NT . ITIAN — . P I 35 MADONNA OP TE E CRADLE . RA E AE L Illustrations X11!

P LATE — E P . R . T RE I 36 . MIRACL O ST MA K INTO TTO

— E E E E I 37 . ANTIQUE EW R D SIGN D ON AN ELLIPS

— E E E I 38 . PE

— - I 2 . S E P TTE RN E 4 VASTIKA IN COUNT R SUN A , PROM A SAXON CIN RARY URN — I 43. REACTIONARY S PI RALs

— ' I 44. B I S E OP S CROZIE R OP THE MIDDLE AGE S — I 45 . FE RN AND VOLUTE — 1 6 . TE A OP TE E T S E OWI NG TE E O E E E 4 BOWL SUNG DYNASTY, G LD N S RI S — I . E WE S E 47 EL ANOR GRILL, TMINST R - l 8 . NE D E RAL E B 4 STAI D GLASS BOR R IN SPI S . PROM CANT R URY (DAY)

— ’ I . S T. KI MB E RT S E E 49 BORD R , COLOGN (DAY)

— E I sa ANCIE NT AMP E ORA OP TE E AMP ULI AN ORD E R. HANDL S AND

BAND DE CORATION IN SPIRALS . — 1 5 I . WIND ING STAIRWAY OP PALAzzo CONTARINI DE L BOVOLO

— T I 52 . DE SIGN POR TAB E RNACLE FOR E E SACRE D OI L — I . STE OP . E L E N S E OWI NG V 53 CLOI RS ST JO N AT RA , ARIOUS COLUMNAR SPIRALS — 1 . WE ST F OP E E CATE E D RAL 54 RONT EX T R , SHOWING DIVISION INTO TE E GOLDE N SE RIE S — I 55 . INVOLUTE OP A CIRCLE — I 56. YOUNG MAN WI TH ARMS UPRAISE D

THE COAL SACK

ri s li s e s sh a s so als inth e S the b ghte t ght cast the de pe t dow , , o ,

m s hi s ar -s u s ac s th e a ns sh all o t t ckly t t dded p e of he ve , we

ﬁn t la k in s I n mid - a d h e b c est depths of em pt es . the p th of

Milk Wa l s amili ar r rn r ss and ar l oﬁ the y y g ow the f No the C o , b e y the

lis nin ulin s cr ss tw nal a amm a and sil n g te g o t e of the o , be ee ph , g , ep o ,

’ ni s all if ar e no tc ar ul sli in as r n m s ml Cyg , we h , we ef , p to t o o y botto ess

it hi c h as n amiliarl ni -nam al ack u l p , w h bee f y ck ed the Co S , do bt ess

caus it a ars so and so lac and so i s ar s. be e ppe deep b k devo d of t So,

r unl itis inr s ar o fkn wl is ril an f eq e t y , e e c h . Wher e the beam o edge b li t ,

r r a s wi ll un al a k s n ic nl li is the e , pe h p , be fo d Co S c i wh h the o y ght

’ r s i ck ri n n l al a s r misin laz u in eﬁni Theo y ﬂ e g ca d e , w y p o g to b e p to d te c r ain in n o n l wn but m an il c s an an r i o ut. e t ty, , e wh e , o t t d ge f be g b o

’ inc m ans m a uri su c ts a r i m r c ns an S e t ty , few bje h ve ec e ved o e o t t a eni n an s l a in a i nOf aui ul h r in tt t o th tho e e d g to cre t o the be t f , whet e music inart in ainin inar i ur o r r r u m a , , p t g , ch tec t e, whe eve bea ty y be

u l H if an r t li kn l is a in . r e tho ght to odge e e, ywhe e, h ght of ow edge be t g in r n in H i r r s ar c h as n i cr as rillianc . r f an eve e g b e e e , ywhe e, e e h expe ded i l n tse . r in cenr so m uc k n l d ﬁ d o ur n f Yet he e , the t e of h ow e ge we ebo I n n c o al sa k s r in ns ll i n nus . c as uely as the c o te at o of Cyg scie ce, we

xv xvi The Coal Sack m us kn ac rs and un r s an r r ac i n t ow the f to de t d eve y e t o , becoming

r in r ica in xac c nclusi ns u n xac c aus s t expe t p ed t g e t o o po e t e , bu in m atters per taining to the beautiful we have falleninto the rut of , assumin a c aus nius and im a i na i n r lar l ins ir g th t , be e ge g t o we e ge y p ed , so r no ui anc no r co nfo r ed law no r r as n they b ooked g d e é to e o . So

aul is lar l o ur o wn I n l i r a s . s al al pe h p , the f t ge y the c e e t Co Sack itwas

a no s n is nin i l n assum s ars . u s a i n r s o g ed th t t ho e Th , po ve t g t o , p ove inc rr ars are r fo r s in but m us u o ect . St the e the eek g, they t be so ght ,

n m ri a s in ilk ik s n in us Wa . i i th e be g o t ho e by the y d the M y y L ew e ,

ul ur su k n l is s and sufﬁ i n subjec t we wo d p e , ow edge ex t fac ts c e t to

r a r n l s a n i s an in lar induce g e te k ow edge , h dowed , otw th t d g by the g e of

i all ar the l ghts ound .

n ﬁnal anal sis au s s r s l ina m ann r r r in l I the y , be ty how he e f e sup is g y alli m a m a ics fo r no m di um xis s r u ic au ed to the t , e e t th o gh wh h be ty c anbe disclosed to the senses butdepends ultim ately uponproporti on

its uc c ss s e au lie inc l ur o r s un ? s fo r s e . Doe the b ty o o o d Both of the e n n depend qui te entir ely uponthe laws of pr opor tio . Is the beauty o e of form o r m otion? Still m or e intim ately are these c onnec ted with

i nal la s inc o n cann r uc nius s the proport o w . S e e ot p od e ge , the be t that canbe done is to study those laws which evengenius c annot succ essfully defy .

Having laid claim to these theori es and their further anc e in

’ ’ ' No ture s Harmam o Um t it is no t n u r m r l r s a y, e o gh he e e e y to e t te

xviii The Coal Sack

No futile claim is advance d that natural phenom ena do no t

ns anl com in to r uc ar in ff c s nor a all am l c o t t y b e p od e v y g e e t , th t ex p es

of a class selec ted fo r study would inever y point coincide with the

r n ni n s n r c o c ali rm s. r is a ins i pe fe t o ve t o ed fo The wo de th t , p te of

adheres so tena i ousl o w she in Nature c y t hatever form ad opts. The po t

n r s is no t a no two sn r s als are alik but a of i te e t , th t ow c y t e , th t they n d all a an all ill six c rnr in ra i n. all ar e , h ve bee , w be, o e ed deco t o

Caustic criticism to th e effec t that Natur e is no tto be r estri cted

m as ur m ens is as . a ur is r s ric r ainm ans by e e t , w ted N t e e t ted to ce t e ,

s n ut in m a . o t all es nam s b o ur whatever they ybe She doe c th e by e , we ,

li l n ss a so and r nam m c mis r ﬁnite tt e e , h ve .to do , whethe we e the he t y

r ra i o r us lain m r and m a m a ics a lic o r Optics o g v ty j t p geo et y the t , pp

Our i n is a m a t r su r m indiﬁ er n h er . able to all sc e ces, t e of p e e e ce to

d ri il is l arn a an h e r m s and if sh e duty an p v ege to e wh t we c of ethod ,

l r s a lin i n ﬁnd o ut sh e s it. pertinacious y adhe e to e of ac t o , to why doe

Dar e we as sum e that there is no r eason?

u l ss sai as r eff r s a r is an r It willdo bt e be d , of othe o t , th t the e d ge of

! in a r n is s fo r all a m m n . as over doing the th g . Stop o e t If e o ex t

n s never orm di stance and orce inthe Universe i s a these thi g , the y f , , f

e sub ect or research and ac l s a l ss nif buts ar legitimat j f , e h ho d e o we be h p ﬁn it enough to d .

h n o ula hi s o k wn ate r ur ur s it ill W e y y t b o do , wh ve yo p po e be , w The Coal Sack xix in l r hin s an i s ak lif is m u vo ve othe t g th those of wh ch we pe . So e ade p

ar in ine r s s and it m us no t u a l a of v y g t e t , t be tho ght th t we fee th t

a ur c nains n in but r r i ns no r a r a r r ss h as N t e o t oth g p opo t o , th t g e t p og e

r n a bee nm ade in asc e taini g wh t the rules of these proportions ar e .

m m r a o ur a s urs and m in cr ss r a l but nc Re e be th t p th , yo e , o p ob b y o e

' o n hi s su c and a s ac n two r s is t bje t , th t the p e betwee the c ove of th

k all sm all sa ha o ne ul sa n nitis all boo is too to y w t wo d y, eve whe

su an is n t m an l c in . s o a e devoted to the bje t h d Th doe e th t we fe , o r ask o u l a c aus r is n in ls insi c r s y to fee , th t , be e the e oth g e e de the ove ,

i r n in r s r ls usi . the e , the efo e , oth g e e o t de

What Le onar do da Pisa and Le onardo d a Vinci struggled with

ro m a m a m a ic al s an in a d a ini and D r r r him f the t t dpo t , wh t V c ﬁ e befo e ,

Michael Angelo and a hundr ed other s studied as artists and ar chi

e ts a H k usai in as and au n rnis in s te , wh t o the E t V gh Co h the We t saw in a s a N nsaw in a l and nl in the w ve , wh t ewto the pp e Be t y the sn cr s al a r all b ut ffs s s r a ow y t , wh t we e these o hoot of the am e g e t s u o f a ur ? a Sc him er and r aun ra ais and ac s t dy N t e Wh t p B , B v S h ,

Schwendener and alan ri ni rk e r in an and a er C d wo d ove bot y , wh t Rog

s n isi n inhis e r m s a af r all r an s Cote e v o ed Th o e , wh t , te we e y of the e exc ept steps following the footprints of Cononof Sam os who plac ed the Com a B erenices inthe sky betweenth e Lady and the Lionwhenc e it c ul n r a aindi sa ar and ﬁrs l a r as na l o d eve g ppe , who t deve oped e o b e

endix No tes ntro ducto r . App , I y xx The Coal Sack

r o n su c s ir als ar ic ularl c nc rnin a n theo y the bj e t of p , p t y o e g th t o e fami liarly c alled after his friend and c o -worker Arc him edes? Cook h as c ollec ted data and rec orded them ina way to m ak e them enticing as

’ ll as in alua l hil urch and arc m s n als a we v b e , w e Ch D y Tho p o o h ve

ursu ir r s c i a s a anc m n all a in c urse p ed the e pe t ve w y of dv e e t , dopt g, of o ,

arli r rk o n n saril the back ground of e e wo as e ec es y m ust . With all of these and a thousand others at hand with their evidenc es and their

ri s it c m s m r an r im ssi l if o ne is f theo e be o e o e th eve po b e , to get o r

r r ulis an l n xamina i n a h as alr a n wa d , to ep b h y e gthy e t o of wh t e dy bee

o ne is add an in s m n n ac c omplished . If to yth g to the u of hum a k ow

r m a us iﬁ ini s l and i is ledge , the wo k y be j t ed t e f ; w th th hope , the om issionof any exhaustive review of r esults ac hieved inthe pas t m ust

ac c no r mus suc an missi n c ns ru as m anin be . epted , t h o o be o t ed e g either that these great ac hievem ents are absent fr om c onsider ationo r

ff r in alua l r under estim ated ine ec t . Whe e v b e wo ks of other s ar e no t

r as is o r a r a r m i is n c ssi m so fully quoted he e th th t e de ght w h , e e ty co

n r nir r pells that these be exam ined i thei e t ety elsewhe e . The habit

na n o n l is is l rm . suc of suc h exam i tio , the who e , w e y fo ed If h theories as ar e here set o utpr ove to be a m odest additionto acknowledged di sc overies the effort whic h h as no t beenspar ed to pre sent them in

Ma r n n in its r ar . c telligibly will have ew d y they p ove o vi c g . PRO PORT I ONA L FO RM

CHAP TE R I

’ HEAVEN S FIRST LAW

s u Na ur m anh as r m im imm m rial ri r d N the t dy of t e , , f o t e e o , f tte e

away valuable hour s and years in an endeavour to visualise

m alm s inc n i a l m l m la s i so e o t o ce v b y c o p ex syste of w wh ch , he n l n ul l k it er c ssar ai h er inri ca r s s . c s too , w e e e y to exp t te e t Cy e and epicycles innumerable were c onstructed inhis er rt to explain

n mical n m na n r a nat las astr o o phe o e , eve up to the ve y d te whe t the o ne basic law of gravitation was developed and he r ealised that it

n l w a m l 0 was the applicatio and no t the a itself which w s co p ex . N

n h e n r r m u a ra ca a r a and r s arc h a n scie ce as b e f ee f o s ch b d b , e e h s bee

n l l r uall and bit bit it has n un c or respo ding y s ow . G ad y by bee fo d that the m ysteri es of Na ture yield to r ules very sim ple when com n u i n ar i r sul s ic inc m ina i r ce . c c p ed w th the e t wh h , o b t o , they p od S e e ,

i m a m a i h as s s is lac m a sical a ded by the t cs, tep by tep d p ed the et phy

rk o ur r a r s and r c ss far r m in ﬁnis guesswo of fo ef the , the p o e , f o be g hed , I Proportional For m

alm s i u T might o t be sa d j st to have begun. he further such investi

tio nis carri m re c l ar l it a ga ed , the o e y ppears that a few fundam ental and m ajor r ules wor k in c oncer t fo r the governm ent of the whole

m and u n uni rsali u n n i n n sche e , po the ve ty of s ch a harm o y a d ts a cie t use and uur alu in l in au wa as d rm f t e v e deve op g be ty , s b e the fo er

’ i ll Natue s Ha mono ni t treat se ca ed r r i U y.

When years have been given to the intim ate consideration of

n su e c it is asil ssi l fo r ar r la m u a y bj t , e y po b e the se che to y too ch

r ss u nits im r tanc but a ur s m im s urnis es a ar m st e po po e , N t e o et e f h b o

r ic r c r s h er is s and n s and all h er c r s in ete wh h e o d w he eed , c s oho t to

m n h wa l nin action at the proper m o e t . She as a y of te epho g m an kind a m essage to the effec t that sh e feels it tim e he should develop

l l n lin r i h a m anderin in a better know edge a o g certain es. Fo thw t

’ in s r s and ari n a a c a r o r a T i an r i s Ch e e obe , we g pe co k s fe the , h bet p e t

in n in m all nd i n i ill all in a bur ning ce se so e w ed a forb dde c ty , w f to

’ n ill n - in a ur s l ram and wn s u . o c br o t dy Be t he w be de od g N t e te eg , all unawar e that perhaps studious Mah o m etans nodding insom e far -Off Sahn e l-G am ia and lonesom e astr onom er s o n the peak s of

n s l r rs in ar Of A rica m a nic s u ents in the A de , exp o e the he t f , t d

m and ur rac ical sci ni sts in nts to o c l r oo s too hot , st dy , p t e t te o d ,

l rl lan and a r m a a all n separated by a who e wo d of d w te , y h ve bee unc onsciously im pelled to tak e up the sam e questions at the self

’ rm nic Unit e aut r Putnam Nature s Ha o y, Sam ho s ( , ’ Heavens First Law 3

m l sa e tim e . Te epathically perhaps they will tr ansfer to each other

their agr eem ents and disagr eem ents preparatory to disc losing a s r ie s in ni ns isc ri s and h ri i will e of ve t o , d ove e , t eo es wh ch seem inthe

ﬁnal anal sis a m i n c s ra o ut lu . y to h ve o e t ght of the b e Thus, whe

’ Na ur s c ic s are r a a ch h m r l s ll an t e h k e dy to h t , t ey e e y peck the he d l s l o ut. I nall is o u ill o s r a a ur h as sim l tep bo d y th , y w b e ve th t N t e p y se t th e s a run u cur ain n r l u and dis t ge , g p the t , spoke the p o og e , patc hed Iri s as h er m essenger and call gir l to warn th e acto rs of

ir im n in the pe d g cues .

A similar analogy is no t lacking inthe study of those laws by

ic a ur r ula s r r i ns h er r s and ic it wh h N t e eg te the p opo t o of wo k , wh h se em s but logical th at m anshould use as his guide in his o wncrea

. ’ i ns in ature s Harmoni e t o art and arch itecture . The publicationof N

Unit in i m an in r r cam a er a y, wh ch y of these th gs we e t eated , e ft period of long quiescence inthe pr oduc tionof wr itings of this kind ; yet it was scarc ely inpr ess befor e it was found to be surrounded by

r s a in l ri in s cam t as and o ut th e othe s a c oud . W t g e o u of the e t of

s and a r s ll rin o n us i ns sam k in we t c oss the eas , a bea g q e t o of the e d , and m any o f these works will be found referred to inthe following

’ a n l r Na ture Harmoni c Unit co n s . M a au s s p ge e whi e , the tho of y,

innin ir r i ina r s r n r u a ul t g the o g l e ea ch es, have bee b o ght to do b e

si n: ﬁrs in i m an r cen u lica i ns uc in deci o t , that , v ew of the y e t p b t o to h g o n u i n ro r ti n in Na ti r c nclusi ns r a ch in q est o s of p po o t e , the o o e ed 4 Proportional Form o ur pr evious work and which have happily m e t no inconsiderable

ur s ul n r u r n i n f favo , ho d be exte ded th o gh the p ese tat o o additional

r and s c n as aninci n a in ne r m atte , , e o d , de t , th t the w wo k the m eans of r eac hing those conclusions c ould and should be Sim pliﬁed intheir n n i n. H n r s i demo strat o e ce the p e e t book s before the public . And since the desir e to simplify the labour of the reader would at once be negatived by requiri ng him c onstantly to refer to the former

ulica i n n c ssi m an s a il r s n rk is in p b t o , e e ty de d th t wh e the p e e t wo ,

’ a s ul Nature s Harmo rzic nit it s ul n r l essence , eq e to U y, ho d eve the ess ,

now r s n d in ac a c m l l nat n as p e e te , be f t o p ete who e , eve the expe se of the repetitiono f certainfundamental dem onstrations and principles

se t forth at large inthe pages of the earlier wr iting .

us s ar n i n l d a r ifﬁ cul it Let t t the w th the k ow e ge th t , howeve d t is inany giveninstanc e to follow thr ough the intric ate m azes of any

’ o ne a ur s ail c m ina i ns we m a r s as sur a of N t e d y o b t o , yet y e t ed th t the various rules ar e them selves auster ely sim ple and eac h applied

s with religious severity . It i only the r esultant c om binationwhic h

ax o ur a ni n lik s m c n c i n s m m ar ll s r a i n t es tte t o , e o e o fe t o , o e ve ou c e t o in i h se e r uc s th arm m ar an to wh c we the p od t of e f , the ket d the dai ry

’ ur and s irr ﬁnall ransm u a s m a i in o po ed t ed , to be y t ted by c hef g c t

l nd l d ak s an i s n r i as s s an rui s air a s n . c t e a c am e c e , f t d f e d f o t g

r r i n and rul ar e r r r s n in Na ur and i t P opo t o e eve ywhe e p e e t t e , is as hopeless to visualise h er without pr ecision as it is to im agine

CHAPTER II

FORM AND PROPORTION

T is frequently a m atter of am azement that the hum anmind so

uick at im s ac a na al law s ul so sl l as q t e to cept tur , ho d ow y

sim ilate the princ iples logica lly akinto th at law and which se em

ll alm s o ni l a r a ts s o a . to fo ow o t hee , s to spe k Th t the g e t body of

r in in s in in r n n r m o b g ow g th g the vegetable k gdom are g ee , we k ow f o servation; and when the scientist tells us that th is is intentional and a a ur has s m s ciﬁ c r as n are r ar li v th t N t e o e pe e o , we p ep ed to be e e him i h u un u us i n s l u o n a is r as n W t o t d e q e t o . We pecuate p wh t th e o m a and r a s c nclu a in rm in an u r c a in y be pe h p we o de th t , fo g o te o t g whi ch sh all r eadily adm it those light rays m ost nec essary to

lan r and c n rs l c lu an ic m i arm p t g owth , o ve e y , ex de y wh h ght be h ful a ur h as cr a a c rin r rac i n ro m ich ro , N t e e ted ove g the ef t o f wh p

na l n ll a duces the se s tionof gr eento the eye . It woud the fo ow th t whengr owth h as c eased and the plant wither s and no longer cares

a li is is c a in s ul c an a nural r n wh t the ght , th o t g ho d h ge to e t b ow ,

rs r irin which inm ost instance s it prom ptly does . That ﬂowe equ g

M . t a m t Great odu s r hte 1 1 b C Ar hur Co n. Fro he le . Co i d py g , 9 4, y 6 Form and Proportion 7 c ro ss-fertilisation are bri ght in co lour or produce honey that th ey m ay draw to them those insec ts whose tiny wings and feet sh all track the po llenover neighbour ing blossom s and th us complete th at

l i ur lann i an ld a circ c a s O and ac s r . e wh h N t e p ed , cepted to y Th t birds sing to attract th eir appo inted m ates and that lions r oar to

All s in s are inimida ir is ui n . t te the foes, q te uderstood of the e th g

o wnw ul l d un in ir a aui ul n r ur s ul. ur an s the y be t f , wo de f , p po ef Co o o d , then inNatur e are both put to utili tari anpurposes ; and we under s an a se ase s o f art and music n un in t d th t the two , the b , whe fo d na ural ts are r r a s n no a i n r a t objec , the e by e o of c c de t , howeve h ppy ; and in a is o ne c si nall r uns ss s ri us yet , the f ce of th , o ca o y acro e o min e rs ns r a rs m an and ri rs no t a a ded p o , e de y w te few , who h ve the temer ity to question whether Natur e h as any especial intent when

i n i uiva She adopts a g ve form fo r o ne of her produc tions . It s eq lent to saying that this sam e Natur e which gives the milk -weed its c l ur fo r a ur se n r l ss m ak s its r five - in d and o o p po , eve the e e flowe po te always five -pointed by m ere ch ance and Without any design o r in ni n: a lil and sn al rif i and te t o Th t the y the ow cryst , both d t wh te s m lic o ur m in uri r c i c l ur a y bo to d of p ty , e e ve their o o as part of deep lai lan but a are six- o in al a s six- o in and d p , th t both p ted , w y p ted ,

a nal in r tail nl caus a ur h as no t u hex go eve y de , o y be e N t e tho ght to m a m an in ls -a m r m a r Of n li n n ni nc ke the yth g e e , e e tte eg ge t co ve e e o nh e a wi m r r u anin o r i niﬁ an . p t , tho t e g S g c ce 8 Proportional For m

As fo r m e il l m a in nl li i , wh e I fr ee y ad it th t o y a m ted number of instances is it possible to say why Nature h as ascri bed selec ted

rm s c ain h r i ns am nir l i fo to er t of e creat o , yet I e t e y sat sfied that when sh e gives to the whi te pine a sheaf of five leaves o r needles and i c in nl r e is aim in at a fini r sul to the p t h p e o y th e , She g de te e t which she knows exactly how to achieve and With all of the confi dence ac quir ed through a million successes ; a definite r esult so sure and sa is ac r a li lmi m she r r s n s sh e sh s t f to y th t , ke the A ghty who ep e e t , ow “ n u neither variableness o r shadow of t rning .

May we no tthenconcede that unless Nature canbe said to have

a deﬁnite purpose inher use of form and proportioninboth anim ate

and inanim ate creation and that unless these factors have a dir ec t

a in o n u r u u l ur r be r g the relations of bea ty th o gho t the wor d , f the

pur suit Of research along these lines is tim e m isused ? Onc e we

r c nise r a t s rm s and ro r i ns are far r m e og , howeve , th t he e fo p po t o f o

in acc i n l and a arise r m a ur se un n n er be g de ta , th t they f o p po , k ow p

a s b ut ull niﬁ nce n al n i c l ur and s un h p f of sig ca , the , o g w th o o o d ,

their study inthings natural will tak e o na new inter est and becom e

a the substratum of anabundant research inbeauty and rt.

UNITS OF MEASUREMENT

It will be clear to the veri est beginner insuch m atters that no

measur ement of the form s and proportions of natural Objec ts could For m and Pr oportion 9 be carried o ut wer e it dependent upon the use of standard units suc as the inc nim r and m . TW h h , the foot , the c e t ete , the eter O other m ethods however lie opento us and they m ay both be used inh ar m n ﬁ a in ll in rm a n . rs Of s m a a i o y By the t the e , we y g the fo t o desir ed relative to the number of parts o r item s of a c er taink ind produc ed in ac na ural r u as c un ﬁve n l s in e h t g o p , we o ted the eed e the sheaf

i in and th e num r i in n I i of wh te p e be of s des a s ow cr ystal. nth s way we m ay determine whether Nature r epe ats the same num ber Of factors inany completed productionwhensh e creates another of the m l sa c ass. s n m we n m o e n e By the eco d ode , ca easure n par t of a y object under obser vation by the position o r dim ensions of another

ar al a s arin in m in a r la i s a s and rm s p t , w y be g d th t the e t ve h pe fo , and n n i o t the Sizes ininches are the points of compariso . By th s m m a c m are r la i sh a s rm s r r i ns ode we y o p the e t ve pe , fo , p opo t o , angles and dec orations of any two chose n specimens o r types and again determ rne whether th ese various factors are am ong those

ic n nd r a s all n rs an wh h Natur e re peats . I the e pe h ps we h ude t d something of the advantages whic h ar e derived by Nature in the use o f n cer tainpropor tional form s inh er creatio s.

So long as we employ the ﬁrst of the m ethods above and con

ﬁne o ur r m i li l n sai inam li m athem atics to m e e arith et c , tt e eed be d p

ﬁcation b t i i r alise a a rk his k in if it be ; u, because t s e d th t wo of t d ,

rvi l n all nin ll in an s t se of se ce , wil ec essarily f c o t ua y to the h d of ho I 0 Proportional For m wh ose nee d inlife h as no tthrownthem into the use of oth er branch es o f m a m a i r s m l ill be l the t cs, the p oof e p oyed here w of the sim p est

ssi l in c nﬁnin all m ns ra i ns to suc rm s as can po b e k d , o g de o t t o h te eas ily be deﬁned pausing occasionally fo r anexplanationwhich m ay appear wholly unnecessary to those trained in m athem atical lore

n r i i n r sul ill t l h a r i a d p ec s o . The e t w a tim es ack t t b ev ty and cri sp ness which the use of m ore advanced m athem atical formula would

r uc but hi c unf r una l ul r a s as anunkn wn p od e , w h , o t te y , wo d pe h p be o tongue to the very reader m ost inter ested in understanding the re sult The sec ond of the two m odes of m easur em ent re ferred to a c m aris n and l ical anal sis th e s ace s lin s bove , that of o p o og y of p , e , and an l s c m sin s ruc ur o r c r a i ns an c g e o po g the t t e de o t o of obje t , would c om e within the gener al subject of geometry and in order that those who have never thoroughly m aster ed th is wonderful study m a sh ar in its aui s suc s s as are in l in th e in s y e be t e , h tep vo ved po t

l n an l l described wi l be tak e up d exp ained with all the brevity possib e .

NUMERICAL UNITS

Before becoming too much involved inthe pr ocesses of c om par

' in a ur s ari us m h s n c unin c m l d g N t e v o et od , eve by the o t g of o p ete

a s it will r a us a im r alis a n um p rt , ep y to t ke the t e to e e th t eve the n ber s with whic h we shall work have certainfundam ental ch aracter isti nir l i rc d r m us i n a s eciﬁc th in cs, e t e y d vo e f o the q e t o of wh t p g Form and Proportion I I is being num bered by their use ; and in order to understand the c a ers ic ll s all n at nse o f r i i n h pt wh h fo ow, I h , eve the expe epe t t o ,

’ restate a few of th e principles se t o ut inNature s Harmonic Unity.

s all ﬁnd n a num r s a n a s rac l ar c r ain We h , the , th t be , t ke b t t y , be e t harmonic relations to eac h other regar dless of c oncr ete objects .

n r l ul i um one fo r am l s an s in a c ass i s lf . M The be ex p e , t d by t e t

li d i s l it r u s i se : divi i se l it p e by t e f , p od c e t lf (I X I I ) ded by t f , a ain r r uc s i se l : uar and g ep od e t f (I I ) the sq e , the cube the

Nth power of one are all one and their r oots are constantly self

r uc iv in w i l on is a ain one p od t e the same ay. The re c proc a of e g , and il one is nl a i i r all r in rs so als wh e the o y ex c t d v so of othe tege , o are all r in l i l I n i in aria l r ac i n othe tegers its exac t nilltp e . th s v b e e t o u ni s l it ill un li no r in r and in s im a po t e f , w be fo d ke othe tege the e t

i n anci ns l ws num r s as l icall it m us t o of the e t the o e t be tood , og y t ,

! fo r si i n nl s inc it c l w nl a and n i r po t o o y , e oud Sho o y th t , e the dir c i n sur a n r n e t o , f ce , o c o tent .

num r two o n c nr a in rm a i i n The be the o t ry , be g fo ed by the dd t o

one and one m a no t nl si i nbut ir c i no r i s anc of , y Show o y po t o d e t o d t e as l avin a ﬁ d and rm ina i n il three in wel , h g xe star t te t o , wh e , be g capable of indicating no t only a beginning and an ending but a mi dl as ll r r r s ns l s in r al rm ca a l d e we , the efo e rep e e t the owe t teg fo p b e o f indi cating at the sam e tim e both length and breadth and out lining a plane sur face ; SO fo ur holds the fac tors nec essary to the 1 2 Proportional Form

i n l an n s iridi to a s li avin n th read d i . T éa of o d h g e g , b th th ck es hese four ar e th e principal elem ents o r fundam entals inthe o ld Pyth ago

“ ! “ ! r ans s m in ic ar e si nate m nad uad e y te wh h they de g d the o , the d ,

“ ! W i r s m uni n m n and m na h ch p oceed fro the o of o ad o d , the

ria r in r m uni n m na and u and t d , p oce ed g f o the o of o d d ad , the

u ni n u nd u se tetrad prod ced by the u o of d ad a d ad . The four integers are the only ones presenting charac teristics requir ing o ur

mina i n and m a r r r nsi ra i exa t o , we y the efo e p oc eed to the co de t on o f num ber s them se lves ingroups o r seri es.

ak in a r o n r s h r fr unl ﬁnd U n T g g oup f umbe toget e , we eq e t y po l l n ana sis a ainin in r a i a er . y , th t they have c er t ter est g e t o s to e ch oth

’

u se fo r am l see num rs 2 8 I . m m n s S ppo ex p e we the be , 5 , , I I , 4 A o e t thought sh ows us that these stand separate d by intervals of three and a r nin r i a r inl i in assum in th t we e they co t ued , the e s ce ta og c g that the next number would be se venteen and immediately we understand that a r elation exists between these num bers which h as characteristics str ong enough to irnpr e ss itself o n the mind

! Ther e are a gr eat m any form s of such progr essions o r seri es but all o n anal sis a is su s ir o wn c ninu , y , h ve th power to gge t the o t

an and in i n rm ; o ne . r ce to d cate the ext te The above , whe e the succ e in rm is arri at a m r i i n m s un r th e e d g te ved by e e add t o , c o e de

r uin an ari m i al r r i n il a s ri s as g o p g of th et c p og ess o , wh e such e e

2 8 I 6 2 r suc n x n m ul i lica i n , 4, , , 3 , whe e the c essio is e te ded by t p t o

I 4 Proportional For m

n n to be i turnc ombined again. Such a pr oc ess aturally c omm enc es

r allnum ra i n ins i uni and r s u : whe e e t o beg , W th the t , g ow th s I I = = 3: 1 3 + 8 2 I ; 2 1

an us c ninuin m in f r i e tc . d th o t g to c o b e each pair o the produc t on in of a new m em ber . To those versed such m atters I need har dly ll- n n i ’ i u introduce the we k ow F bonac ci Series . It s freq ently stumbled upon in works o n Nature and is extr em ely useful inthe

rin c m l ac rs r a c un is n c ssa esi numbe g of o p eted f to whe e o t e e ry , b des being the nearest integr al equivalent to th at Extrem e and Mean

n hi all un proportio w ch we sh ﬁnd ﬂ g broadc ast throughout Nature .

G EOMETRIC UNITS

I n examining the various series coming under the section de

“ noted Num erical Units we have recognised that these were na urall r u in r r a amili s ari m ical ro re s t y g o ped to th ee g e t f e , the th et p g

i ns m rical r r ssi ns n s a d r c urrin s ri s . o , the geo et . p og e o , the e g e e The use suc am il r u in s is r a s r ice sinc it na l s of h f y g o p g of very g e t e v , e e b e a m athem atician to know the exac t powers and re lations which he m ay expec t to ﬁnd existing betweenthe various mem bers of a se ries

m m n l ar ns a r u it l n s no r n co m li the o e t he e to wh t g o p be o g , eed he p hi cate s work no r bur den his m ental spaces with further details .

’ Both before and since c ollaborating o n Natnre s Harmoni c

' Nature s Harmonie Unit ndix and A ndix No te X t e e X V o s . y, App , pp , I p For m and Proportion r 5 have spent a great deal of tim e instudying the proportions exhibited inthe processes of Nature and inapplying to these and to adapta

i n m inart ac cur a s lv n of m a m a ics and l t o s of the , the te o e t the t ogic .

The m ore I have delved into the m athem atical phenom ena and the n ur al nd i niﬁ am l s c ns anl c min an at a sc e t c ex p e o t t y o g to h d , the m ore itis borne inuponm e that by elim inating m any of the comparatively unimpor tant fac tors and allowing the balance to gr oup them selves along natural family lines into which r esearch proves that they na ur all all m a r mi ut in rm s muc sim liﬁ t y f , the tte ght be p to fo h p ed , form s so sim ple as to be Opento every serious student and yet place d u na as a a nr n po b is so ccur te as to defy c o t adi c tio .

A brief and cr ude explanation of a portion of what I have in in l n m d m ay be tak enfrom everyday ife . If we Wish to c o vey the inform ationthat c ertainfriends of ours have tak ena c ottage at the s asi itis nl n ssar s a a ns ns a r e de , o y ece y to t te th t the Joh o h ve ented a c a a Ho w c m l it ul ottage t the be ch . very o p ex wo d be were it re quired that instead of this we should inform o ur hearer that Har ry

nso n i Am lia ns n née n m was Joh , w th e Joh o ( Brow ) to who he m arried ten ars a o las as r r i his ur ch il r n ar ar ye g t E te , togethe w th fo d e M g et ,

uc arl s and ac a r s c i l se n ﬁve hr and o ne L y , Ch e , J k , ged e pe t ve y ve , , t ee , , and in all l n s lik ir m e r and ch u lik ir a r be g b o d e the oth bby e the f the ,

r in c a at ac i ir unc l ac fo r we e go g to the ott ge the be h w th the e J k ,

m li l was nam ac c m ani ir ari us run who the tt e boy ed , o p ed by the v o t ks 1 6 Propor tional For m

and lls ara rnalia and irn ed irnenta s n summ r do , p phe p , to pe d the e .

Progr ess through life would be no t only slow but dull with such

in r ance s and alm s unc nsci usl ac i h d , o t o o y we cept the hab t and be neﬁt

usin s in ica r u s in m a rs inf rm a i n of g type to d te g o p tte of o t o , thus

n r a in n l ss c ains tail i r saving the eed of epe t g e d e h of de , wh ch a e

lu l n c ssar but ic nc am il r u as n abso te y e e y wh h , o e the f y g o p be certai ed , l m ay safe y be inferred rather thanrepeated .

I n l sam wa ul i i n m exact y the e y, I wo d d v de the ecessary athe m atica l and natur al exam ples into those gr oups into which they logically fall and which Will furnish us with the gr eatest num ber of n u c onstantly recurring examples . We have se e how the se of family

r uin s as acc m a m a icians in ﬁ l num rical g o p g , epted by the t the e d of e I w s i s sim liﬁ s m a r s hi c a l . n sam a er e , p e tte to w h they pp y the e y, analysis sh ows that na tur al gr oups exist in the geom etrical unit

mili s and a s r n o ut as so m uc m r r m i fa e , th t of the e , th ee sta d h o e p o n and o mu m r r un r r n an rs a ent of s ch o e f eq e t oc cu e ce th the othe , th t

r m m n r m a l n li i l an am ili s fo the o e t the est y be he d eg g b e ; d as f e , we will c onsider only these three : the four -Sided gr oup which I have

si na ra n amil amil r m and M an de g ted the Tet go F y , the F y of Ext e e e

r r i n and a aria l r u in ic cal si i nand P opo t o , th t v b e g o p wh h the fo po t o

al l n o r c ns anl ar and hic r u m a the radi e gth , both , o t t y v y , w h g o p y be

ir esi na as s m m rical am il . s r i d g ted the A y et F y The e th ee , w th the

’ u i i i are hr Nature s Great Modules and la r am ina s bd v s ons, t ee of , te ex For m and Pr e portion I 7

i ns will iscl s a s amili s are n t m l but t o d o e th t the e f e o ere y geometric ,

a a ur rs lf r u s h er r s in sam wa and r unl th t N t e he e g o p wo k the e y, f eq e t y

along the same lines of dem ark ation

THE TETRAGON FAMILY

The ﬁrst of these fam ilies to be examined will be the gr oup

ic m ricall c v s uar a n ic is its wh h geo et y o ers the q e , the oct go , wh h

n ar s r la i uil ral ri an l ﬁ sin and e e t e t ve , the eq ate t g e (a rst cou ) the

xa n a ﬁ in n r rm r am il he go ( rst cous o ce emoved) . These fo a g eat f y

which gover ns vas t form s of energy and are m anifested as we shall

see in r avi a i n vi ra r r c s suc a li and s un c m g t t o , b to y fo e h s ght o d , o e

c ns anl in i n inas r n m are r curr n r sis nl o t t y to ev de ce t o o y , e e t very pe te t y

in all m ns ra i ns lar r and are n ricall s a in de o t t o of po fo ce , , ge e y pe k g ,

al a s un r r k in i n r uc s m i n o r w y to be fo d whe eve et c e ergy p od e ot o ,

m l cular ac i n s it i u is a n ﬁr s in o e t o hows s effec ts. Th s gr o p t ke up t

o r r caus its im r anc i i a and fo r de both be e of po t e , wh ch s gr e t , the i li c om parat ve simp c ity whic h surrounds the dem onstrations.

Once it is sh ownhow these various members o f the sam e family

an r it ill r alise a avin in an o ne ins anc h g togethe , w be e d th t , h g y t e

demonstrated that the gr oup governs a proposition it nee d no t be

requir ed that all of the separ ate steps and correlations be gone into

ain ll a l a . Onc u l l in dia r mm a i s a g e p t c ear y to g a t c shape , we h be b e

’ un rs an a as ur l as mi s so n is mi in urn so to de t d th t s e y S th S th t ,

8 1 8 Proportional For m sur ely will further coincidences of the sam e k ind develop a se co nd

n n is ill l tim e a d these nee d be show but once . Th w gr eat y ser ve to sim li all th e rk um n r a in and r d p fy of wo both of arg e t , d w g, p oof an n n save endless a d tedious r epetitio .

I n r ll r a a a inm in it ill be s n h a o der to i ust te wh t I h ve d , w ee t t in la a set o u in a r anc i m ric rin i l s p te I I h ve t, c co d e w th geo et p c p e

nuilise in i nc an r ina r r ssi n s uar and ofte t d sc e e , o d ry p og e o of the q e

c a n in sam rim c irc l and ia ram is so sim l as o t go the e p e e , the d g p e

“ in i l i na as almost to explainitself . With th s c ir c e des g ted pri m e

i in i hin s ar and an n we sh all nscribe a square . Aga w t the qu e t ge t to

its si s a s c n r l insi ich an r s uar is r awn. is de , e o d ci c e , de wh othe q e d Th

r c ss n urs rri o n ad in nitum but ro p o e ca , of c o e , be ca ed ﬁ , the two p gr e ssio ns described will serve fo r the illustration. This process of geom etrical c omparisonis very Old and no claim is laid to originality

An n n in rin ic c nains in its c onstr uction. y book o e g ee g wh h o t the

us m ar a l s ill m r r m ns ra a c a ns in c to y t b e w , o eove , de o t te th t o t go ,

c ri in ari us i l s will ac l c inci i corn rs s bed the v o c rc e , ex t y o de w th the e of the square ; that the po sitionof the se c ond Squar e inthe pro gr e s sion (m arked C divide s the radius Of the prim e square (AO) in ac al and a if a circl rawn i in is s uar e to ex t h ves, th t e be d w th th q

i l an irc l w th the r adius CO it will be precise y oc t t to the prim e c e .

Also we shall learnby the sam e process that the radius AO is the exac t m easur e of the side I] (sec ond square) and so we know th at For m and Proportion 1 9 the new r adius inturnwill be the Side of the next ensuing o r third s uar s ul o ne n l i i in r s in ra as at . s t s r q e ho d be d w , MN A o ve y te e t g

n a ac di a nal uals n r ce in si us to ote th t e h go eq the ext p e d g de , th IL

u n l ar es eq als EF a d MO equa s I J . Thus it is plainthat these squ and c a ns m asur inscri cir l s rm a s r law o t go , e ed by bed c e , fo , by o t of

rim ni ure ir c lin ra n amil sc n of p oge t , the d e t e of the Tet go F y by de e t , and fo r ur s s a ur r s u a r la i ns se the p po e of f the t dy , the ex c t e t o of the progr essing squar es and octagons have beenexpresse d inper centages of the prime radius and set downinthe appendix fo r reference .

Ha in in n l ll min let v g the co cid e ce s indicated inp ate I we in d , us now turnfor a m oment to plate 2 inwhich we shall ﬁnd the pro

r i ns r la i to an r ran th e m amil and a ri po t o e t ve othe b ch of sa e f y, b ef examination will prove that the two are as tr uly m em bers of the

m am il And s in r s in to sa as two r r o r uin . i e f y b othe s co s s th , te e t g r late is no tm r l r u in m r a sai but m ns ra e , e e y t e geo et y , as I h ve d , de o t

i ns s a a ur no tsa isﬁ m r l ac c m rical t o how th t N t e , t ed e e y to ept the geo et similarity betweenthe square and the trigon o r equilateral triangle

illus ra in ia ram s o n nn n in as t ted the d g , goe to c o ec t the two eve

rul an l h e r es d aws as we Shall se e inthe chapter o nforc e .

Inspectionof plate 2 will conﬁr m this idea that the tetragonand

ri n are ﬁrs c usins and rnall un in c u le s . the t go t o , they ete y h t o p

r r o uﬁnd o ne th e r is lurkin in ac r un Whe eve y the , othe g the b kg o d

endix No te . App , I 20 Proportional Form

Le t s re ra i ll in ara l . u just aro und the corner . They a p ct ca y sep b e

iz d tak e a circle of the same s e as an ,

PLATE I TETRAGON FAMILY

in ri in init uila r al rian l ATV in ﬁn o r sc b g eq te t g e , we beg to d the

is amil r la i nat nc ar i larl traces of th f y e t o o e , p t cu y whenwe add th e inscribed hexagon which is next of kin to the equilateral tri angle

“ inthe sam e way that the octagonis obviously descended from th e

Proportional For m

ll as m urin rim cir cum r nc n ut in its as we eas g the p e fe e e , whe p to relation with the equilateral triangle and the hexagon m easures n l i with absolute perfection the side of the hexago a so . By th s

ini nce l n a n RA ra ius rim co c de , the e gth of the hex go ( ) , the d of the p e c ircle (AO) the side of the sec ond squar e (I J) and the diagonal of

x the third squar e (MO) ar e all identical.

i nnin a ain we un o n our ri inal e rim n a Beg g g , fo d o g exp e t th t the upper side of the se cond squar e as ﬁrst inscri bed was plac ed at a point exactly half way fr om the Circumference of the prim e Circle

c nr no w in lat 2 ﬁnd a u r si o f to the e t e ; , p e we th t the ppe de the inscribed equilateral triangle is precise ly so plac ed and cuts the

n s a radius AO i the sam e halves at the point C . Thu we h ve the

i a n ra ius rim cir l and si s de of the hex go , the d of the p e c e the de of the squar e o f the sec ond pr ogression and the diagonal of the thir d

ar all sam l n il si rian l c n i squ e of the e e gth , wh e the de of the t g e oi c des l absolute y with the position o f the side o f the se cond square . And

‘ these c oincidences canbe m ultiplied indeﬁnitely . Sur ely the m ost ami cable of c ousins c ould scarc ely agr ee better nor m ore c lea rly Sh ow

l r la i n i their b ood e t o sh p .

d m ns ra i n a r n so far c nﬁnd The e o t t o s h ve , howeve , bee o e to

in lan a and r uiri n nl im nsi n in those p e surf ces eq g o y two d e o s, be g

s l I n r r in fact the im plest form except those m ere y linear . o de to

n x N s n e di o te a d . App , II III For m and Proportion 23 test the planwe have inmind it will be nec essary to go fur ther into the m atter and consider fo r a moment whether or no these sam e

of measur ement and pr opor tion be

PLATE 3 DIAGONAL OP TE E CUB E applied to solids and fo r illustrative purposes will turn to plate 3

li l i m m n s o f all cu o r ur where the so d se ec ted s the c o o e t , the be fo li ﬁgur ed rectangular so d .

I nm ak ing c omparative measur em ents o f sur faces and of solids

for an ur s s n ni n c m ar m ir di it is , m y p po e , co ve e t to o p e the by the 24 Pr oportional Form

l al n a nal rm in s in of ago na s, no t o e because the di go dete e both the po t

nnin as an t a rmina in th e r ical i s as ell begi g of the b e d h t te t g ve t S de w ,

n akin is but also because the diago al is the index of the area . T g th

convenient means of comparison let us apply it to the solid inques

an n r a r ul m a sim l and accura tion, d i o der th t the es t y be both p e te ,

letus assum a si s our cu are rm s uar s e th t the de of be fo ed of q e (I , J ,

sam r rti ns as sam s uar in la s and 2 . K , L) of the e p opo o the e q e p te I

n iam r sur ace s nin s uar ill The the d ete of the f how the q e I , J , K , L w

ia nal and in la is ill c rr s n i be the d go I , L p te 3 th w o e po d w th the

sam e diagonal inplates I and 2 and the c oincidences there noted will

I n r r r ia nal a r the againoccur . o de howeve to get at the d go f c to of

i l we m n e i a lin as a n m l r l cube tse f, ust co c ve e dr w fro the owe eft

an r n rn r as m ar r u s li and h d f o t co e of the cube ked S , th o gh the o d

n - k r in m r i at u r ri ac c rnr m ar o r . a e e g g the ppe ght b o e , ked R ( I) St t g

us wi a cu as o ns uar s i nical i s uar th th be b ed q e de t w th the q e I , J , K , L

in r vi us ia ram s and a in di a nal o n sur ace the p e o d g , h v g the go IL the f ,

itis no t a little astoni shing to ﬁnd that the diagonal of the cube as

li RS m asur ri n l l i n r i l th e a so d ( ) e es , by t go om etrica so ut o p ec se y

sam e as the Side of the ﬁr st progressionof the equilateral triangle .

us as n r r in uar a ns rian l s Th betwee the p og e ss g sq es, oc t go , t g e ,

a ns and s rawn o n am i nical uni as a hex go , cube , d the s e de t t of b e s

N I V endix o te . App , I t is essential o f course to recollectthat th e mere measurement o npaper inplate 3 will no t

roduce t is result since t i w r Th t w is a so lute co rrec t. p h i s dra ninpe spective . e res ul ho ever b ly For m and Pr oportion 25

n ﬁnd ll in r all n ini n n sh ow , we the fo ow g e y otewor thy c o c de ce s, amo gst m any oth ers which have no t beenlisted here :

These ar e all absolutely equal inlength

(I) Radius of the prim e circle

(2) Side of the second square

(3) The side of the hexagon

(4) Diagonal of the side of the cube

These are also equal inlength

(5) Side of the equilateral triangle

(6) Diagonal o f the solid cube

are in n in si i n m ur co cide t po t o , as eas ed

centre of the prim e circle

Side of the se c ond squar e

Ends of the hexagon

Side of the equilateral triangle

us n sinc s c inci nc s can as asc r aind and Let the , e the e o de e , e t e d s r u in ﬁ l an n a i n e cribed , be ca ried o t de nite y d qui te beyo d p t e ce , a r a m a l a li no t nl le h s am l s suﬁi e g ee th t we y fee t ber ty o y to tt e e ex p e c , butin uur saf l r s c n n h a n r cr ss vi n f t e e y to e t o te t t t , whe eve we o e de ces o f ra n amil in an o ne its ranc s rem ain r the Tet go F y y of b he , the de 26 Proportiona l Form m ay be assum ed as sur e to follow without going separately each

all i m r l tim e into of the m any ram iﬁcations. The nnu e ab e cases in which we Sh all r equire the use of these pri nciples nec essitate s

a sim l s rm a d la a th t the p e t fo be dopted, an the ter ch pters will be devoted to demonstrations o f their usefulness and of the frequency

u a it ic cc r in ur sci nce and art. w h wh h they o N t e, e ,

I have said before that the thr ee fam ilies so c onstantly r r r d r no t m r l m r ic but a a ur rs l efe e to we e e e y geo et , th t N t e he e f gr oups he r works inthe sam e way and along the sam e lines of dem ar

a i n. Fo r r se n ur it is i n wi r ar to k t o the p e t p pose , suffic e t th eg d the

Tetragon Family to Sh ow geom etrically that there is a c onstant

r of m m lan falling togethe the e bers of the c as I have desc ribed them ,

’ l avin a ur s a lica i ns no tmr l m rical rinci l e g N t e pp t o , e e y of the geo et p p e

but am il r uin s la r a s. As illus ra i r of the f y g o p g , to te p ge t t ve , howeve , o f the persistence with which the m ember s of the TetragonFam ily indi a o ne an r m i in assin call a ni n wo c te othe , I ght p g tte t o to the t solids which belong to the sim ple and less intricate end of this group — to wit the solid bounded by equilater al triangles calle d the tetra h d r o n and cu o r a d r n n Th e , the be hex he o , bouded by squares. e o ne avin ur ac s i r si s ach and th e r avin h g fo f e w th th ee de e , othe h g

i an an l r -si ﬁ u rah edr n six face s of four s des d g es. A fou ded g r e (tet o )

i h r -si ac s and a six- i d ﬁ ur h e ah r n i h w t th ee ded f e , S de g e ( x ed o ) w t

- i d ac s How in e l min four s de f e . tim at y eac h calls the other to d For m and Proportion 2 7 and h o w ver itably ea c h typiﬁes the subtle relations of all of the

m em bers o f this intim ate fam ily .

THE P ENTAGON FAMILY

This family or group is of im por tance equal to the o ne just examined and h as from tim e M em orial ser ved to interest stu

N N i dents under the designationof THE GOLDE SECTIO . That t me honour ed m athem atical form ula to which the ancients gave the nam e

’ i in a i n a a ni n ucli and u s D v e R t o , e g ged the tte t o of E d the tho ght

o f counl ss m a m a icians sinc in nrall r rr to as t e the t e , be g ge e y efe ed the

Extr em e and Mean Propor tion; and c oncerning thi s a som ewhat

’ exhaustive analysis was prese nted to the readers of No tnre s Har moni c ni r la i n n is r and m an U ty. The e t o betwee th ext eme e pr o

portion as a series and all form s o f the pentagon is a m atter o f

r a s in r s and r it v r cam kn l the g e te t te e t , whethe e e e to the ow edge of

anci n in s i a rs o r no t it can now m a cl ar n the e t ve t g to , be de e beyo d

era nur o f a u lacin s two c ns anl re cur rin r m s p dve t e do bt , p g the e o t t y g fo

inanin l x ter re ationthat is indisputable .

That number s have c ertaininteresting r elations to each other

I a n a ur lain and inwri in o n c rr la i ns o f h ve e de vo ed to exp , t g the o e t o num rs na ur and r l n i ra i a alr a n be , the t e e atio s of th s t o h ve e dy bee

' e ndi x No tes to Nature s Harmo nie Unit a es 28 2 0 No tes and I I and co n App y, p g 9 , 9 , 7 , e 2 lusive ly on page 99. 28 Pr oporti onal For m touched upo n Owing to the fr equency of the r eferenc es m ade to it

r a r a m nts s n r a r no w he e fte , few mo e pe t by the e de inunderstanding its pec uliarities Will simplify m any of the statements to be m etinthe coming pages .

Let itbe c onstantly carried inmind that the expressions extrem e and m eanpropor tion and a m eanpr opo rtional betweentwo other

rm s are r far r m in s n n uli ar r te ve y f o be g y o ym ous . The pec atio under examinationis ingeneral but rudely understood and m ust be somewhat further explained We m ust inthe ﬁr st place r em ember

a r sum a l a ras had a cl ar ras su ec t al u th t p e b y Pyth go e g p of the bj , tho gh of c our se the Egyptians and ancient Gr eek s were ham pered intheir use of such pr opor tions by their lack of k nowledge c oncerning fr ac

i ns . m eo m etrrc r si i n r r s n rl t o To the the g p opo t o , ep e e ted by eas y

ivi lins was cl arl c m r nsi l il num ricall ir d ded e , e y o p ehe b e , wh e e y the knowledge was r estric ted in all suc h m atters to such use as co uld

be m ade of integral approxim ates inplace o f dec im al perfec tion.

Well-known formula inm athem atic s have a habit of taking to

" m s l s as im s o n si na i ns r e a the e ve , t e goe , de g t o whe eby th y m y be

’ f i kin is se r ek l r i indic ated brieﬂy . O th s d the u of the G e ette p

Q ( i t ) no w universally indic ating the relation of th e diam eter to its

i l 1 1 2 fo r i it i c rrc umference inthe arithm et ca decim al of . 3 4 59 7 wh ch s

’ n e wri in o f Nat re Harm o ni c ni t an l . t u c onst t y substituted I h t g s U y, wher e the alm ost unlimited inﬂuenc e of th e continuing series of ex

30 Pr oportional For m inthis r elationh as beenperpetuated inthe present work inpr efer n n ‘ e c e to a y other so far inView .

The best deﬁnitionof this ratio which I have bee n able to con s ruc is o ne us d inth e rm r rk and r r a : t t the e fo e wo , he e epe ted

By the expre ssion extrem e and m ean proportion is m eant the divisi on of a qua ntity i nto two such parts or proporti ons as tha t the measure of the whole qua ntity shall bear the same rela ti on to the measure of the greater part as the measure of the greater part bears i n tur n to the measure o the lesser or an in rm wi ut f ; , ch g g the fo tho al erin ff c m a i ual ru sa a o M o t g the e e t , we y w th eq t th y th t f se arate uantitie s the lesser must be to the rea ter as the reater i s p q , g g h i to t e r sum .

’ A m oment s co nsideratro nWill show the differenc e betweenthis

' and an o th er for m r r i n sin inall rs an o ne y of p opo t o , ce othe y of the

rm s m a ari ro dircin a c r r s n in e na i n in its te y be v ed , p g o e po d g v t o

n n rri n rs th e sin m m r . r is o r r a c u oppo g e be The e , the c o t a y (b g of o e use n a i si ns but o ne rm r m and m an r r i n of eg t ve g ) , fo of ext e e e p opo t o , sinc las i ﬁ n sum o f r m a er m s i . e the t t xed , be g the the othe two We y sa f r ins anc a 2 o r 2 : : 1 0 and in case s y, o t e , th t 4 , both the o ne rm is a m an r r i nal e nthe r inni r te e p opo t o betwe othe two , yet e the c ase is ther e even an approxim ation to the extrem e a nd m eanpr o

I n r and m an r r i n as r i n. r r s in m po t o othe wo d , the ext e e e p opo t o ,

S ee endix No te V. App . For m and Propor tion 31

sum cann ari ni r o f m a the ot be v ed , e the the other two m em bers y be c an an s ill r ain r r i nal l h ged d t em p opo t o to the who e .

Thus it will be clear that while the variety of form s inwhich a m eanpr opor tional m ay be expressed is lirmted only by the ingenuity and a i nc s u n in ir c ns r uc i n n r an p t e e of the t de t the o t t o , o the othe h d there is o ne and o nlv o ne form of anexact extr em e and mea npr opor

i n arrin as a alr a said ue f n a i rm s and t o , b g I h ve e dy , the s O eg t ve te

i n nr ails s . i uc i n n a i si ns r nl S g By the t od t o of eg t ve g , howeve , o y the det ar e c an and no t nric a s and no r l n r r nc h ged the ge e f c t , p o o ged efe e e will be m ade to c ombinations deduced thr ough the interpolationof negative signs since only confusionwould result o nthe part of those

m s inricaci s are unam iliar il o n r an to who the e t e f , wh e , the othe h d , those who understand such details ar e perfectly c om petent to apply

‘ the substanti al facts here set forth to the new conditions resultant

r n f om the egative combination.

Since the uses to whic h we shall put this propor tionWill no t be

ur l ac a mi c it m a no t amiss un r s an it m ricall p e y de , y be to de t d geo et y as ll as num r i l L in n w at la ia am if cal . o r we e y ook g p te 4, d g A , we c onsider th e line AB to be so di vided atthe point C as that CB AC

n nir uani will ivid in r m and AC AB , the the e t e q t ty AB be d ed to ext e e m an r r i n and is ill d m ns r a l in aria l sinc sum e p opo t o , th w be e o t b y v b e e the

AB cannot be inc reased o r di mini shed to correspond with any c hange i in pr oduced nthe other ter m s by the m ovem ent of the po t C . 32 P mpo rtio nal Form

We shall ﬁnd later inthese notes that the decimal o r percentage equivalent of extreme and m eanratio is almost indispensable to the

rk an in r r m ak ri r l d wo . d o de to e both the geomet cal e ation an thi s num erical index clear I take the hb erty of submitting the sam e

Euc lideandem onstrationof the proportioninquestionthat was use d s m ar s a o in r i us rk s in i r rnss o e ye g the p ev o wo , how g w th g eat clea e

i i nm a c m li ll in i how th s divis o y be ac o p shed , fo ow g wh c h the study of

n uival n an r il ur the um erical eq e t c be ead y p sued .

We are no w inpositionto r etur nto our o wnlines indetermining what m ay be the decim al value o r equivalent of the extr em e and

nin ain l l lin r r i n. ur a a t i m eanp opo t o T g g to the p te , e us g ve to the e

al o f o n r i s n as lin ical u ne u un . AB the hypothet v e h d ed t The , the e

3 " ’ BF is half of AB it will equal ﬁfty units and AT AB B? o r by

‘ ’ su s i uin uni s AP ’ o r AF o r and b t t t g t , m sinc = AF DF a AD = e AD , we h ve

s lr a in a n i rm r rk a a e dy shown the ppe d x to the fo e wo .

It therefore follows that when two quantities are respectively

1 1 6 1 l uni ir m thenand 8 660 and . 80 ac su 3 9 3399 of e h who e t of the ,

nl then ill r a r r la i n an m and o y , they W bea to e ch othe the e t o of extre e

an r r i n c as is ru m a n use me p opo t o , ex ept th t th y be exte ded by the

n a i s i r sse s ir a s lu l accura r na of eg t ve , and th s exp e the b o te y te pe ce t ge um of the s .

This m atter of the decim al value of extrem e and m eanratio h as m - ! a a n A

D a o a .B

PLATE 4

E XTREME AND MEAN RATI O

’ (Fro m Nature s Harmo nie Unity)

Exam inin Plate ia ram we ﬁnd a me thod o f di visioninto e xtreme and a g 4, D g B , me npro po rtion whi ch is as o ld a Chr istianit and fo r which naturall mak e no claim to o r y, I y iginalit . I t has y , t t f n r ctl clear t o u ru entar ho we ver h e meri o ei e e h h dim . , b g p f y , g y

Let the line be an iven uantit . Erect a er endi cular as F ro m one end as at AB y g q y p p , B , f , ne enwit as a centre and B e ual to one hal o f the li . Th h F radi us F descri e a circle d , q f AB B , b an n c t e o and E t e line la o ff AC e ual to the c o n e t h ints . On h nwi l the o in p A AB y q AD , l p tC divide th e line into extrem e and mean ro o rtio n fo r AB p p , Inth e tri an les and EAB the an e is co mmo n and AB D o rm ed b a tan ent a g ABD gl A , , f y g nd ch ord is measured b o ne hal o f th e an le B D which also measures the an e E at t , y f g gl he circum t n re t re o re similar and t e ir sides are ro rtio n l ere nce . The two ria es a he h o a he nce AE f gl f p p , AB — B — D AB AD and by di visionAE A AB AB A AD . Since ho wever radius e ua s o ne hal o f th e diame te r o r DE wi e ual and o n , , q l f AB , , , ll q AB c se — o l nt Sin e uals it — q uently AE AB equals AD r its equiva e AC . ce AC q AD fo llo ws that AB AD ubstitutin t ese values fo r th t r s o f the e quals AB AC o r CB . S g h e e m equationwe have AC A CB AC o r b inversion CB AC AC A thus ro vin that th e oint C di vid t B y , B , p g p es he o quantity AB into anextrem e and meanpro porti n. 3 33 34 Proportional Form

n a n u at s m l n caus in l n in i bee t ke p o e e gth be e , a o g vest gation of kindred subjects and the scientiﬁc workings o f this pe culi ar

r i n ha un num rl ss ins anc r a ﬁ pr opo t o , I ve fo d be e t es whe e de nite

rm ula o f r ssi n was a s lut l r ui si ic s ul fo exp e o b o e y eq te , wh h ho d be suffi ciently ﬂexible to denote extreme and m ean proportion

i ual acili r as a lin ar su rﬁcial ulk o r w th eq f ty , whethe e , pe , b ,

ri al m asur and o f se ral o rm s ssi l i sphe c e e , the ve f po b e , the dec m al

equivalent presents the only basis of c omparison suited to a work

of this kind .

It has fr equently been stated that fo r purposes of design the continuing series known as the Fibonacci Ratio o f 3; 5 ; 8 ; 1 3 formed

A is s ri s r no t rm a r asis. s s c a perfect b th e e , howeve , doe fo pe fe t

n in uual s ns am a aininr uin rial ac rs consta t the s e e , I g t od c g the se f to

’ r r n d in Nature s Harmoni c ni t so a no as they we e p ese te U y, th t

ni m a and to call a ni n ac ual misapprehe s on y oc cur ; , tte t o to the t

n i ns a a ula i n s in us a is r la i n n c o dit o , t b t o how g j t Wh t the e t o betwee

ri and a r u r r i n i r r a r m r i us this se es t e p opo t o , s he e epe ted f o the p ev o

N0 r r nc to lim i in ra i o r summ a i n i work . efe e e the t g t o t o of the F bo

i s ries is er m a c aus it is to o in l and c aus nac c e h e de , both be e vo ved be e no use is m ade of it.

It will be her e noticed that whenthe perfect ratio h as r eached the point of 34 : 55 it is appr oxim ately the sam e as the dec im al

6 1 8 x r m and m an r r i n is e s no t l . u 03399 of e t e e e p opo t o , tho gh th do ho d Fo rm and Proportion 35

fo r i nac ci s ri s us a ain m asisin su r i ri good the F bo e e , th g e ph g the pe o ty f i i o the d v ne section.

Fibonacci Series Perfect ratios Decim al ratios (no t extr eme and me an) (no t Fibonacc i)

1 2 2 : 4 50 2 3 3 : 6666 3 3 5 5 3-3333 60 8 8 62 5 . 5 1 6 1 8 : 3 . 5 1 3 : 2 1 2 1 6 1 90 2 1 : 34 z: 34 6 1 76 34 55 55 6 1 8 1

in us m a cl ar a n r a i r c ninuusl It be g th de e th t , whe t e ted e the o t o y o r as an ac r r i n i nacci se ri es c nains im r c i ns ex t p opo t o , the F bo o t pe fe t o , it is appr opri ate to examine fur ther and se e if the same defec ts inher e

r m nd m an r a i r in the ext e e a e t o. Whethe we test this pr oportion

icall o r al r aicall s all im i l it arithmet y geb y, we h m ed ate y ﬁnd that

s ns all o f ssi ili i s o f a rf c ninui s ri pr e e t the po b t e pe e t, c o t ng e es.

rrin to our ﬁni i n see a if o f two uani ies Refe g de t o , we th t , q t t , the l ss is to r a r as r a r is to ir sum o r if x x e the g e te the g e te the , y

x n rm s ar e in r m nd m an r r i n and m a y , the the te ext e e a e p opo t o , y be

’ ’ s l ucli an rm ula x x ic We m a e r ss o ved by the E de fo , y y wh h y xp e “ inthe words the squar e o f the gr eater equals the square o f the lesser 36 Proportional Form

n f n w his r n to r c a l o two . o r i is a ru ad ded the e t g e the If , , t p opo t o t e

ninuin ri n ll win ill be ru co t g se es , the the fo o g w t e

1 x : x : x : 2x ( ) + y : + y + y,

in rial rm and se fo ,

2 : x x : 2x t . ( ) y + y + y, e c

r ns rmin s in ir ua i ns a T a fo g both of the e to the eq t o , we h ve the

ucli an rmula a as a r sul in ac case and nc m a E de fo bove e t e h , he e we y

again add these irnpo rtant gener alisations ; that inevery extreme and

an ro o rtio n the reater i s to the sum as the sum i s to the reater me p p , g g

plus the sum; the less i s to the greater as the sum i s to the greater plus the

sum; that the se two co nditio ns continue indeﬁnitely; and that the true

and ean rti o ni a er ectcontinuin erie extreme m propo s p f g s s .

llus ra ar i m icall ﬁnd im al ui al I t ted th et y we , by the dec eq v ent ,

6 1 8 38 1 96601 . 03399

which o f cour se is clear ly a perfect propor tionsince its solutiongives

6 1 8 o n si s. us . 03399 both de

It is perhaps alm ost unnecessary to say againinthis work that

ra i m a carri o ut s ndin r i n the same t o y be ed de ce g, whe eby s obtai ed a

nsi in o f s ar u ur r f al seri s c s u et . o e o t g the q e , c be , fo th powe , c the decim

ui al n Of r o f two rm s us eq v e t the greate the te , th

' ur s Har mo nic Uni t A n N te a e 28 and endix o t b st Nat e e dix o N e V o . y, pp , p g 7 App , , p

38 Proportional Form o r a ﬁve -m m r c as ui al n na r am or n e be ed Obje t , eq v e t to the pe t g pe talph a . The constructionof the ﬁve pointed star depends upon an

abstruse propositiondi scovered by the Pythagoreansc hool and this

! s ar se em s a ee n r im l An t to h ve b f om that t e adopte d as their sea .

other eminent authori ty r efers to this ﬁgur e as being the tri ple — — interwoventriangle o r pentagr am star shaped pentagon used as a

’ s l i nOf ni i n ymbo o r S g rec og t o by the Pythagor eans . As the theori es Of the Pythagor eans would apply perfec tly to the

na nal rm and ar dl at all to six- in o ne d pe t go fo h y the po te d , an as

is r and m l sus ain ﬁve- in ﬁ u sub both h to y ety o ogy t the po ted g r e, I

rnit as an e lana i n iff r n ini n nilin xp t o Of the d e e ce of Op o , the re c o c g statement that both form s canbe ori ginate d by the use Of interlac ed — ri an l six- in s ar usi two uil t ral ri an l t g es the po ted t by ng eq a e t g es ,

- l is and the ﬁve pointed o ne by the use Of thr ee isosceles tr iang es . Th

ac s m s a a e a ni n m an s r rs wh o a f t ee to h ve esc p d the tte t o Of y Ob e ve , h ve

c onclude d that only the six-pointed star c ould be created fr om the

ri n an a is nle d in llin xa nal ulin a t go , d h ve by th bee to ca g the he go o t e — n a r am an i n m isn m r pe t g , ev de t o e .

Having Obtained a decim al index o f extr em e and m eanpr opo r

i nina r ce in ara r a o ne Of ﬁr s s rik in a ur s in t o ! p e d g p g ph , the t t g fe t e the

exam ination Of the pentagonwill be the persistence with whic h it

r s ns is ra i and a no t ina m r a r xim a i n but in p e e t th t o ; th t , e e pp o t o , the

l num r Of ins anc s in h i absolute y true and pr ecise sense . A be t e w c h Form and Proportion 39 thi s occurs inboth the pentagram and the pentagonproper Will be m ni n as are r a d but ac s ul inmin e t o ed they e che , the f t ho d be kept d

ns anl in am i n co t t y the ex inat o .

Having no w a fair working idea Of what c onstitutes extr em e and

‘ mean pr oportion as a c ontinuous and unending golden se ries and having shownhow it m ay be applie d both to linear and numerical m as ur m ns it s ul s r v a it is r r r and a r a e e e t , ho d be Ob e ed th t ane o g e t deprivationto c onsider this mystic seri es fr om o ne standpoint to the

l t in a exc usionOf all others . TO View i only as a m eans of subdivid g lin is to su r its m anin in ar n r us iﬁ inr e e bve t e g , s ce we e o m o e j t ed str icting this gr eat series to use inm easur ing m ore lines thanwe ar e to chain o ur c onception o f physical things to a fantas tic c ountry having length and breadth butno thickness ; fo r the bear ing of extrem e and meanpr oportioncanbe shownto have as direct a relationo nall dim ensro ns and m ethod s o f c om putationas ith as uponm er e distanc es

r r i a is fo r o enum e at on. This prim al f ct I tak e pains to em phas e the reasonthat it h as escaped the examinationof so m any wr iters o nthe su c his a r in l i r in ra i i l uni rsall bje t , t bso b g y nte est g t o be ng a m ost ve y tr eated as a m athem aticalto y and a cur iosity r ather thanas a m odule c ninuall cc ur ri n inNa ur an nc n t m ni nar t and o t y o g t e d scie e , o to e t o ; a m ul as m ns ra l a li l s r ﬁcial ar as c ur v s od e de o t b y pp cab e to upe e , e , an l s and s li s as it is to sim l r m a rs ic it h as n r g e , o d the p e tte to wh h ge e

in s r all na li d na li at all. r in alu y bee pp e , whe pp ed Befo e go g to v e othe 40 Proportional Form

an lin ar and num eri cal as uc e is r r i n as a m th the e , to h d by th p opo t o , ere m a r o f r aui n l rie r r a n tte p ec t o I woud b ﬂy epeat , howeve , th t whe c nsi r al raicall i m lik all r u o de ed geb y, th s easure , e othe s, is s bjec t to

r a m n un r n a i si n. r all a seri s Of m ns ra t e t e t de the eg t ve g Afte , e de o t

i n inn a i ul r a s d to in r t o s eg t ves wo d , pe h p , ad the te est of the student but l n in o would sure y add oth g t the truth .

We have started o ut with a br ief explanation o f the num erical o r ari m ical r m is s ri s ich a li s na urall and ir l th et fo of th e e , wh pp e t y d ect y to all lin ar m asur m ns and firs a i i nal m a e e e e t , the t of the dd t o tters to be considered would seem logically to be the measurem ent o f fi i su r c al s ac s o r ar as as r ss in lanes . su rfi ial pe p e , e exp e ed p The pe c value o f a plane cannot be said to be the mere combined linear values l l of its boundi ng sides them selves . A p ane rectang e bounded by sides r espectively five and eight units inlength c ould hardly be r ated at a

ar i hir n sin a i lin ot c omp at ve of t tee , ce th t s the ear sum and n the c a i alu o r nial ac i in li r u o oper t ve v e pote t of the f tors, wh ch rea ty p od ce a plane o f ﬁve tim es eight or for ty units as its total m easure . The in x suc a lan ul ra r ia nal i nn cts its de of h p e wo d be , the the d go wh ch c o e

i an l n m nn i s i ia nal Oppos te g es tha the sum of the c o po e t s de . Th s d go r i i n i is an inits l i n ra Of equr g both direct onand d t ce ocat o , form s a g ph

rm s lan as s wn in ra n s li and ri n s the te of the p e ho the tet go o d , b g into the pr oblem the two fac tors necessary to constitute surface by the expr essionof the dimensions both o f length and breadth Form and Proportion 4 I

I n r a in n l n l n in d m an r r ti n c e t g, the , a go de p a e extrem e an e p opo o , we shall have the sides formed by lines representing the lesser and the greater term s : and in such a plane we shall ﬁnd that the diagonal in m s 1 1 1 in n r i at r dex easur e 7557 perce tage of the g eater s de , wh eve

° ’ a m a b nd an l at i 1 6 us th t y e , a that the g e the base s 58 Th inplate 5 the plane ABCO is formed by the lines the length o f which ar e in r m and m an r r i n us O : A0 ext e e e p opo t o , th , A AB AB AB and lin O ill ual Of a in Of ic the e B w eq AB , po t wh h we

s ll see ha the im portance a little later .

alu m ul s ul no s i its a ili The v e Of the od e ho d t top , however , w th b ty to a m asur lan sur a s r m n m an r r d e e p e f ce , nor does the ext e e a d e p opo

i n ail in li l n in ur r r s . t c a s un a s t o f Of f the te e t Le us reate o d , bo ded by p e r ular l n r i n Two ndin lan s ill n eg go de pr og ess o . of these bou g p e w the be formed by lines bearing to each other the r elationOf the lesser to

r a r inan r m and m anra i and two r i s ill the g e te ext e e e t o , othe S de w be bounded by planes form ed Of lines the shorter o f which will be the ori ginal greater term while th e longer will be equal to the sum Of the

ri inal l ss and ri inal r a r i i l hus n ro o g e the o g g e te , be ng tse f t the ext p gressive term ino ur extr em e and m eanpr ogr essionwhentr eated as a

rf c and ninui I n a li m li r m c n s ri s . su s s c pe e t o t g e e ch o d , y bo of ext e e and m anra i and rm t n i uin its r o e t o , fo ed o u of the factors c o st t t g p

ressio n s all a ainue th a nal o ur in and r sul g , we h g s e di go as dex , the e t

n e di x No te VI . App , 42 Proportional For m

ant c oincidenc es are no t a little r em ark able when c ompare d with

s hi ll ll I n is hi li c a n r nd i . tho e w h h ve go e befo e a w fo ow th p so d ,

rm o f ac ac rs o f seri s all ﬁnd di scl s s fo ed the ex t f to the es, we h o ed the e

illuminating c omparisons

I na logic al solid as form ed inplate 6 we have as a base the line

’ O hi as a verticle hi X and fo r l n t lin O C (p x) , , BC (p ) the e g h , the e D

’ hi X ll i v r ti l lan CO (p a being inc ontinuing pr oport on. The e c e p e AB

is h n sam as as ic lan BCO in la m r l in s e n t e the e the b p e A p te 5 , e e y be g e

er in l r r rs c i . ia nal i an BO ars h e pe pe t ve The d go Of th s p e ( ) be , the efo e,

sam r la i n si s as in la r rr to and the e e t o to the de AB , BC , the p te efe ed

is o f the length Of the greater Side BC .

I n hi s s li o f r m and m r r i n a t o d ext e e eanp opo t o , we h ve

l r m and m anra i The edges Of the p anes inext e e e t o ,

8 1 66 1 l ss r avin a uni alu . 0 The e e , AB , h g t v e of 3 9 ;

6 1 8 s r a r a in a alu . 0 uni The g e te , BC , h v g v e Of 3399 t ;

l n t in sum a uni alu o f The e g h be g the , Of t v e

The diagonal of th e plane (as AO) having the unit value of

I -1 7557 l :

The diagonal of the solid (as DB) will be of the unit value Of

° ' The di agonal Of the plane (BO) will for m anangle o f 58 1 6 while Form and Propor tion 43

H di a nal o f s li D B ll rm i i ( ) The go the o d ( ) wi fo w th the s de , the

° xac an l o f na n as m a s e e t g e the pe t go , 54 e ur d the 4 r i r n i cular pe pe d at ver tex .

PLATE 5 PLATE 6 PLANE OF E XTREME AND SOLID OF E XTRE ME AND ME AN PROPORTION MEAN PROPORTION

s wh ar i r in s u Of a s rus su c s To tho e o e nte ested the t dy b t e bje t , the

amina i n o f rs a us i ns alin i so -c all ex t o hype p c e q e t o , de g w th the ed

ur i in m r als cann m nsi n al a s a al. Or ar Fo th D e o , w y ppe d y o t ot c onceive aninﬁnite world inwhic h ther e should be either no dirnen si ns at all o n o n n o r a ur im nsi n r and n o the e ha d , fo th d e o , ove beyo d

usual l n r a and ic n ss u n ic a n the e gth, b e dth , th k e po wh h we h ve bee

r u ili r b o ght up. Is this four th dim ension a regionof invisib ty whe e souls play at large ? Wh atever we m ay think of it personally it is a 44 Pr opor tional Form

nd it is s e table limi m ns ra i n In sam a usc p of ted de o t t o . the e sense that the quadratic equationis the m easure o f all plane surfaces and

ui ua i n m asur s all s li s so radi i n the c b c eq t o e e o d , the tet c equat o ,

i r s ns no iffi cul i s inm a m a i ha ee n n t wh ch p e e t d t e the t cs, s b co ceived o be the natur al m easur e or index o f the impr actical and no ngeometri c

ﬁ ur ns ruc out o f l n r ad icknss and unk n g e co t ted e gth , b e th , th e , the ow a l ur im nsi n i is ni r s and ich il b e fo th d e o wh ch e the of the e , wh , wh e we

nn m re n it and n r a it is s n ca ot co p he d cannot eve p ove th t ex t , ever theless yields to certainscientific and m athem ati cal dem onstrations i n i nifi lin and is a serv ceable weaponalo g m any sc e t c es . It has indeed beenused by a strange analogy inm any m etaphysical problem s to l i i m l r u s u ns s m r and s u ss s un . s no t p od ce o t o , o e we d, o e , do bt e o d It i m y pur pose here to go far ther into the questionthansim ply to present it fo r the c onsiderationo f those who ar e interested and so to present it inconnectionwith the great principle of extreme and m eanpropor ll l i n n i ‘ tionwhich we sha find under y g so m a y s tuations .

It rem ains no w to show inwhat m anner the goldenseries can be applied to Circular m easur ements before we go directly into the

exam inationof speciﬁc cases where this pr ogressionexists .

’ I n a m ical n s Nature s Harmo ni c nit ex the m the at ote to U y, I plained at som e length what the cir cular o r angular equivalent of

n e o n i e Fo r mo re extende d discussio o f th F urth ime sio ns see e nd x No t VI I . D , App ,

46 Proportional For m

PROMINENT MEMB ERS OF THE PENTAG ON G ROUP

No w that we have examined the characteristics peculiar to

l as a l ar e r ar to o m b r ac uain this fami y who e, we p ep ed bec e ette q ted

se ral m mb r s indi i uall d res n sub with ve Of the e e v d y, an the p e t

’ s l chapter will ser ve as a geom etrical Who s Wh o in thi Fami y .

That allquantities and lines formed ingoldenprOpo r tio nshelongto the family is as Obvious as that all plane and solid ﬁgures deliberate ly c onstructed inthese term s are c onsistent mem bers ingood standing ; but ther e are other geom etri cal form s and fam iliar quantities whic h ll uall u no t so i usl i inits un s . O a c ome eq y, tho gh obv o y , w th bo d f

nis o m ri al uila rals n n so r sis nl r u l r ec og ed ge et c eq te , o e pe te t y , tho o gh y , and undeviatingly yield evidences of extrem e and m eanpr oportion

ﬁ i r all al at every exam inationas the pentagon. This gur e s e y the trib c hief o f all Visible and familiar m em ber s Of the clan and the m or e in uisi i l o ne s ui s its ari us m m rs i s ia nals and q t ve y t d e v o e be , S de , d go ,

n i ns m r o ne is assur a l s ruc ur i tersec t o , the o e ed th t the who e t t e Of the pentagon is articulated upon a m athem atically perfect system of n i extrem e a d meanpropor t on.

no w h o w c l s is r la i n n na n TO Show, , o e the e t o betwee the pe t go ,

na r am and is ivin c i n in r m and m an the pe t g , th D e Se t o ext e e e

r r ti n let us r a s m amina i ns r vi usl m a p opo o , epe t o e of the ex t o p e o y de

to that end . For m and Proportion 47

n i u c nica li i s let us in lat s 8 an Pr oceeding the w tho t te h t e , , p e d 9 draw a perfec t pentagonwith its intersec tingdia gonals which form the

- beautiful ﬁgur e of the pentagram o r ﬁve pointed star . If we were to

PLATE 8

P NT M PE RP ECT E XTRE ME AND ME AN PROPORTION E AGRA . sub ject all Of the subdivisions of the ﬁgure to the m ost rigid analytical te st we should ﬁnd extrem e and m ean ratios staring us inthe fac e

r m r calcula i n Fo r am l si s ri m r f o eve y t o . ex p e , the de Of the pe ete , C D in u i n as s n in la etc . ac ar ra i s , DE , how p te 9 , , e h be the t o q e t o

r s c i ia nals DO etc hil su i isi ns to the e pe t ve d go , CF , , w e the bd v o AC , 48 Proportional Form

l etc . a ain ar sam r a i n si s ri AD , BE , , g be the e e t o to the de Of the pe m r hi c in his la r cas ec m r a r rm in ete , w h t tte e b o e the g e te fo the l n ua i n r f rm all r th e ss . ain m all r eq t o , whe e o y they we e e Ag , the ext s e su ivisi ns na ram I H and HA i in bd o of the pe t g , AB , BK , KI , , wh ch

urn a n na n wi a urn wn ar are in t form seco d pe t go , th the pex t ed do w d, turnlesser term s Of goldenratios inwhich lines lead ing to the outer E n l ins o f s ar K etc . m a i rc an a po t the t , AD , DB , BE , , , y te h ge b y be n ﬁ as r a r ac rs . n ir ur ak ninr ular used the g e te f to I deed the e t e g e , t e eg

r r r uc s ac l a c ninuin s s in r r i n o f o de , p od e ex t y th t o t g erie the p opo t o whi ch we spok e a few pages previously and which might be expressed

x as AB BE DE E G . The num ber of coincidenc es to be n n noted is entirely beyo d my patie ce to se t for th or your s to read .

ill r n a l n r r ti ns ic ar e a s lu l I w , howeve , ote few go de p opo o wh h b o te y — ac curate (as all pentagonal ratios ar e) and the r eader m ay easily

n rl s i i n pick o ut umbe es other c om b nat o s .

(1 ) AB : B E : : BE : AB + BE or AB

(2) BE : B C : : BC B C + B E o r CE

(3) DC : CE CE : CEF

(4) Twice the per im eter of the interior pentagonABKI H is to

B l ic r im r s ar AD EKFI etc . as a r is the pe ete Of the t , , the tte to tw e the

i r n n D E t r a C e c . perim eter of the exte o pe t go ,

l l i x All of these are abso ute y perfectextr em e and m eanpr oport ons .

endix No te V . App , III Form and Proportion 49

One m ay sum up the situationinthe ﬁgure before us by saying that ther e are no unequal term s to be found whi ch are no t inthi s ratio to each other as forming a ste p in a perfec t and c ontinuing

l n r i go de se es .

is in so na nan na ram l t o a Th be g of the pe t go d the pe t g , e us g step far ther and judge the results Of subjec ting the pentagon to a form o f progr essionsuch as was applied to the square and the equilat

ral rian l and ine at in la 8 I n o na n e t g e , h t d p te the case f the pe t go , time willbe saved by bowingto the overpower ing evidence Of the golden

io nin i ﬁ ur and ill l n i is i a e th s g e , we w draw p ate 9 o th s bas w th

lib ra sim lici ins ri in a na n i in a ir l and de e te p ty , by c b g pe t go w th C c e la i n off a ius in ninu r r i n so a u r y g the r d to a co t ed p opo t o , th t the o te

l us end s all to inn r nd as la r is to . h be the e e , the tte the who e Th o n the plate we shall have the radius which c onnects T and LP

i i so a is to as is DT and i isi n d v ded th t DL LT LT to , the d v o

ill nl is l a ah w conseque t y be in th ratio . At L we sha l dr w

r Cir cl and in it inscri a na n and at in othe e be pe t go , the po t M

(sam e ratio betweenL and T) a third circle and a pentagonwill be created : and if we examine these pentagons we shall ﬁnd that no t only do they bear the internal evidence Of extrem e and m eanpropor

i n i ic s rall s ar as wno n lat but a t o w th wh h they eve y t ted Sho p e 9, th t in a i i n si s o f s ar in rim nta n fo r dd t o , the de the t the p e pe go (AD example ) are exactly equal to th e sides o f the pentagoninthe sec ond 50 Proportional For m

pr ogression (LN for instance ) and hence another interminable ser ies o f perfec t extrem e and mean proportions is created such as AB

n kin a ain LN LN CD (the sum of AB a d NL) . Loo g g we ﬁnd still

ar r vi nc a m ark uni in is r r ssi n sinc i f the e de e of ed ty th p og e o , e , w th

the sec ond and third c irc les dividing the rad ius into extreme and m ean

r r i n ir na nc inci s a s lu l in si i n p opo t o , the th d pe t go o de b o te y, both po t o ,

i an m nr and l n si s i si s d st ce fro the ce t e , e gth of de , w th the de of the

pentagonformed by the cr ossing of the pentagram inthe pri m e ﬁgur e l as showninp ate 8 .

all is is no t n u n le t us ac a m n If Of th e o gh , the go b k om e t to plates 5 and 6 wher e yo uwill rec ollect we jotted down som e o f the l Am n item s relative to the ogic al solid . o g these yo uwill note that we said that the diagonal of the goldenplane (a rectangle bounded by lines inextrem e and m eanpr oportion) was 1 1 7557 1 times the greater side

r c an l and u n ri n m ri c calcula i n ﬁnd a of the e t g e , po t go o et t o we th t this is exactly the relationof the radius TD if drawnwould leave to

i o f na n as ll as ra ii and the S de the pe t go CD , we of the d TL TM

i ainas ra ius is a l i to the s des LN and AB . Ag the d TD ex ct y tw c e

r n ic ular is anc s c n n a n r m c nr the pe pe d d t e of the e o d pe t go f o the e t e , so the di agonal of the solid is exactly twic e the length of the

r a r i and nc m r an l rm di a nal Of g e te S de , o e o e the g e fo ed by the go the logic al solid is exactly that which m eas ures the pentagonas would

n n l D T H r nare ﬁ ur s ac r n be showni the a g e C . e e the two g e e h p ove For m and Proportion SI

m r i n hic to e body the ve y spirit of extrem e and m eanproport o w h ,

u iff ri n in r rnal a aranc r uc c ninuall tho gh d e g eve y exte ppe e , p od e o t y e vidences that are both loyal m ember s of the great

PLATE 9

PE RP E CTI ON AMPLI PI E D

nWe m an o ne s ac rs a ain letus ak rela i n Whe eet y of the e f to g , t e the t o s i as in un rs i u ar um n Or ia am h p be g de tood , w tho t g e t d gr .

His r s ws us m an r a m num ns or s ri s m nu to y ho y g e t o e t , e e of o m ents which have always beenthe cause of endless queries as to the

is ir ns bas of the m eas urem e t . 5 2 Proportional For m

— I n a s ar ani ui l n a s r r a am so n d y of ho y t q ty, o g ge befo e Ab h the o f r ah l Ur al s fo r lan o f anaa n and a u Te eft of the Ch dee the d C , tho sand years ere Moses led the oppressed childrenfrom beneath the h av an ara — insc rua l s iri o f il saw ri sin e y h d of Ph oh , the t b e p t the N e g inits sacr all in sl m a s s n s n ar ar ed v ey ow je ty, to e by to e, ye by ye , r i n r i n s im sin m s iz rami m aus e g by e g , tho e po g to b of G eh , the py d o leum s uu Kh efren and Menk ewr hi are am o n m s of Kh f , , e, w ch g the o t stable and enduring of all hum anstruc tures They symbolise m any

Of r li i us n s o f o ld ian ai and i m an the e g o te et the Egypt f th , typ fy y of

o ther things r evealed to us by these pr oud evidenc es of a past glor y i s the advancem ent m ade by the priests inthe knowledge of m athe

a i s o ne l hi i ir n an s ruc ural m t e , examp e of w ch s showninthe c o st t t t use o f the ﬁgure fr equently referred to inlater days as the Egyptian

What m ay have beenthe originof the proportions of thi s triangle

can nl co n c ur sinc all is n l in mis nas i s we o y je t e , e e ve oped the t of dy t e

r u l and l s l n it ar s r m analm s c mb ed peop e o g extinct . Whether o e f o o t

prehistoric knowledge of that which Euclid afterward called the

. ’ i in c i n r it h ad fo r s arl r i s s a r li i us D v e Se to , whethe the e e y p e t e g o

o r nl an ar c i c ural i niﬁ canc and r ir kn l o y h te t S g e , whethe the ow edge

was ana i m n o r na i n ill r l n r ﬁni l ch eve e t a cc de t , w p obab y eve be de te y

l arn but is kn a r r i nal m m rs ic e ed ; th we do ow , th t the p opo t o e be wh h

54 Proportional For m

s and if had m a no tc nc i itm r an l ring form ; they , y we o e ve o e th pr obab e

a aim at l n r r i n r r ies hic r th t they ed the go de p opo t o , the p ope t of w h we e

n anci ns ra r an a h ad well know to the e t , the th th t they a different

Vi im o r n a aim at in ew t e , eve th t they ed the

PLATE I o

PYRAMIDAL axTRE ME AND MEAN PROPORTI ON and less signiﬁcant ﬁgures of the 5 8 propor tionm ade fam ous c ent

L nar us P isanus and acc o m ard ? Th e urie s later by eo d , epted by J

hi hl s m lic ﬁgur es here pr opounded have the m erit of being g y y bo ,

r i ll anc i n races and a r eein wi h well knowngeom et ca y to the e t , of g g t

the m easur ed co nditions as near ly as the lapse of time m akes po ssible For m a nd Proportion 55

n nc n hink l n ro ressi ns ian Whe o e, the , we t of the go de p g o , the Egypt

tri angle and the noble pyramids of past days should ar ise before the

r u g o p .

Befor e we canfeel atlibe rty to pass away from the list of subjec ts ill na s u is r a r r ssi n it is umi ted by the t dy of th g e t p og e o , essential to r efer more atlength to the questionso frequently discussed inbotany under the general nam e o f the Ideal Angle of growth

By the ideal angle I refer to that theoretically perfec t m easurem ent atwhi ch m any botani sts have presum ed that Natur e aim ed insending

u l s and ll rm i ns in arl a s l o t eaves, shoot , ce fo at o the e y st ge of deve op

n ul amina i n ﬁrs z n r wt s s vi n m e t . Caref ex t o Of the t o e of g o h how e de ce

’ that Nature s tendenc y in plant life is always to put forth shoots

l l at in s m o r l ss O i ri inl l and a ternate y po t ore e ppos te the o g a po e , great . effort has beenm ade by authorities o nbotany to arrive at a deﬁnite rule whereby the angle atwhich the next shoot would appear might be

ul an s ri s a r sul in i isi ns o f calc ated . M y of the e theo e h ve e ted d v o the

° n r in rac i nal arc s 1 8 o r r a us ﬁrst zo e of g owth to f t o Of 3 the e bo t , o f which the ultim ate angle at which Nature was thought to aim was ‘ l l c alled the idea ang e .

m an rs num r s of anis s h a v n av ur Fo r a gr eat y yea , be bot t e e de o ed both with microscope and m athem atic s to determine whether these

’ ideal angles of divergence were to be r egarded inNature s hands as a 56 Proportional Form

s o r anend and al asc r tainwi minu m ean , so to e th test accuracy what

ra i nal m n al anl r ll was but f th e f ct o seg e t of the ide g e ea y ; so ar , all

efforts at a definite result have beenbaffled by the diffi culties prese nted

inm ak ing the delica te m easurem ents necessary to the pr oper ap

s plicationof any o ne of the various theorie .

is c l ar a r r la i n a se ri s is o r is no t in It e th t , whethe the e t o of e

is e nsa l lan r w inas mm rical c ns ruc i n a ur r d p b e to p t g o th y et o t t o , N t e ve y

surely gives c onstant evidence of h er intentionto put forth leaves

d s s at in r vals hi ch sh e rns and il as o ne Of an hoot te w gove ; wh e , the

i ll sa s num rs n ru i ut author it es we y , the be of the co st ct ve curves m s

! in rs it s no t ll a r ac i nal an l at i be tege , yet doe fo ow th t the f t o g e wh ch

they are put forth m ust be c omm ensurate with a single m easur em ent

circum r nc as hi s cir cum r n is c n anl nlar in of the fe e e , t fe e c e o st t y e g g

and Nature m ay m easure as m any times around the plant circle in

putting forth shoots o r ﬂorets as She elects before ﬁlling o ut the ro w

r o o c oming o ut even s to speak .

I n rd r m ak hi s lana i nas ri l r as si l o e to e t exp t o b ef and c ea pos b e ,

nl hr am l s s im a i an l l r a o y t ee ex p e of the e t ted deal g e wi l be he e st ted .

Diff r n anis s s m c m arin ir ri s wi i nacc i e e t bot t , o e o p g the theo e th the F bo

° ° ' s a calcula r a l i al an l at 1 1 0 serie , h ve ted the p ob b e de g e 37 37 3

° ' 1 0 O r rm s r set r ﬁrs 37 3 f the th ee fo he e fo th , the t

nl are as d o nm asur m n and ac ual s r vanc ir d two o y b e e e e t t ob e e , the th

being the equivalent of the Fibonacci seri es at inﬁnity and thus Form and Propor tion 57

r i al nl sin it r r se ns a s ri s at no s a s o r theo et c o y , ce ep e t th t e e t ted tep

a but as itm i be r ss is its c m si re uce an ular st ge , ght exp e ed , o po te d d to g

n r s it is a i nacci s ri s ul i i rm . I r f t fo othe wo d , wh t the F bo e e wo d be

o n l were m athem atically perfec t and continuing . N e of the ear y authorities Oﬁ er insupport of these ideal angles muc h evidenc e other

ansu as an l an r m rni cro sc0 ic xamina i n c unin th ch c be g e ed f o p e t o , o t g

ar s and r l ca m as ur m ns minu unc r tain of p t , othe de i te e e e t too te , too e , u i o r di s r ri . is a r al r e ther to prove p ove the theo es It nt , the efore,

a ll s ir ar um ns and nlusi ns as ll as in th t a hould differ inthe g e t c o c o , we the m athem atical rules deduc ed .

’ Argum ents as to the probability o r extent of Nature s use Of a

ﬁxed ideal angle have beenfully and lear nedly carried o nby botani cal

r s and ul a no lac r but oin rm an o expe t wo d h ve p e he e, the p t ge e to ur su c is to if ssi l s m r as n hic m a l a bje t Show , po b e , o e e o w h y e d to the

ﬁxing of a logical ideal angle upona basis analogous to other known

and ac c rac ic s a ur . i u r f r in in epted p t e of N t e W tho t , the e o e, go g to

i niﬁ ails l but as in di r c l in in the sc e t c det deep y , be g e t y po t with the

rk in an ill es n r in ulin su s anc wo h d, I w pr e t he e o t e the b t e of a theory which I suggested som e year s ago and which m ay no t be without interest inthis c onnec tion.

All investigations of natural phenom ena pr ocee d along one of two lines : the building of a theor y uponproved facts o r the construo tionﬁr st of a theory from analogous c onditions and its subsequent 58 Proportional Form

u r inv s i i n ﬁ s ppo t by e t gat o . The r st m ethod is unim peachable wher e it is ssi l asc r ain fac s i r ain but it is admi t d po b e to e t the t w th ce t ty, t e by all that the accurate m easurem ent of botanical angles is as yet

na c m li and hi s mi i n l all n i u c o p shed , t t gates stro g y against sy thet c

i s and ar m ns n nin l ru . theor e gu e t c o cer g the ideal ang e, so c onst c ted

s n m i m is Ob ti n a c s r r r ec o . The e o d ethod , howeve , f ee f o th j Th t

Nature c ontinually uses extreme and m eanpr opor ti onas a m easur ing ro d is true beyond peradventure ; that the r esemblance betwee nthe botanically probable ideal angle (as predi cated o nmi cr o sCOpic m eas urem ents) and the lim iting ratio of the Fibonacci series h as beenrec og nised ; that thi s lim iting ratio repr esents a hypothetical form ula at inﬁnity and no t a ﬁnite c ondi tionis dem onstrable ; that the employ m ent of this seri es inm any of the investigations c oncerning the ideal angle h as beena purely m athem atical o ne and without the slightest

’ rec ognitionof Nature s c ontinued use Of anexact extr em e and m ean pr oportionin other departm ents is seen by examinationo f various works o nthe subject ; that extreme and m ean proportion is a per fe ct and continuing series we have alr eady seen; and having alr ea dy

a m o f rm inin is r r i n de cim all and shown ethod dete g th p opo t o y ,

u n anularl as ll ar e in a si i n clar its there po , g y we , we po t o to de e

a ur r r un and ar r s arc is use by N t e whe eve fo d , the f the the e h

m r c nvincin is r a r m and m an carried , the o e o g the p oof th t ext e e e

’ i i o n Na ur s m s c ns anl uili m ur s proport on s e of t e o t o t t y t sed eas e , For m and Propor tion 59 no t only of lineal distances and as governing supe rﬁcial spac es and

i n tin n l al hi s in in bulk pr opor to s bu cur ves a d ang es so . T be g disput a l ex erirnento f a l in anular ui al n r m b e , the p pp y g the g eq v e t of the ext e e and m n r r i n circular r is a na ur al and l ical s ea p opo t o to g owth t og tep, and the outcom e is striking .

al an l r un ari us au ri i s as s a a Ide g es p opo ded by v o tho t e , t ted bove

° ' ° ' ° ' 1 37 30 1 37 30 2 7-936 1 37 30 28

Angular equivalent of extrem e and m eanpr oportion

° ’ — ’ 1 Nature s Ha rm oni nt . 1 2 . 37 30 c U i y, p 3

The m ost casual glanc e will show how perfec tly this angle o f

n m n r r ti n ich alr a n o ne extrem e a d ea p opo o , wh we e dy k ow to be of

’ a ur s s an ar s co inci s i ari us anular rm s su s N t e t d d , de w th the v o g fo gge ted

i al an l and il a anc in no ar um n in a ur o r fo r the de g e , wh e dv g g e t f vo Of

’ Na ur s use an i al an l o r s ri s a r itis n r against t e of y de g e e e wh teve , eve theless no t unreasonable to suppose that if and whenNatur e utilises

n i al an l ani all in ﬁr st o r r z n r a a y de g e bot c y the othe o e of g owth , wh t ever be the resulting form or integr al c ount atthe m oment o f m easur e m en itis i hl r a l a she in ac ul im a l aim s at sam t , h g y p ob b e th t f t t te y the e pr oportional equivalent toward whi c h sh e so co nstantly leans inhe r linar and su rﬁcial m asur m ns and a no si a i n a e pe e e e t , I h ve he t t o wh t ever inagainclaim ing that whensh e sets o ut each succeeding point

Of wt - ar ur sh e es so u nno rm annl ﬁ in r al gr o h dep t e , do po pe e t y xed teg 60 Proportional For m

i s e c nsi r a i basis. I ventur e to ins st that h tak es into o de t on no t only the positionto be oc cupied by the identica l ﬂo r et chosen for

xamina i n but er incr asin rim t r and a in e t o , the ev e g pe e e , th t so

in anin s i a i n ill r r a s n acce no tm r l do g, ve t g t o w Show eve y e o to pt e e y

ﬁ r but an ular ui al n r m and m n abstrac t gues, the g eq v e t of ext e e ea pr oportionas the principle uponwhic h sh e work s .

c in is is c urs at nce un rs an a no Ac ept g th , of o e, o to de t d th t accumulationof microsc opic measurem ents c ould unaided establish a m ore ﬁtting solutionthanthat pr esented by thi s angular equivalent

r m and m an r r i n r nniall uilis a ur in of ext e e e p opo t o , pe e y t ed by N t e

’ i ain all branches o f h er creat ve dom .

Let us illustrate what we have in mind by again referring to

la r inis s n an ular ui al n r m and m an p te 7 , whe e how the g eq v e t of ext e e e

r r i n and r n as la 1 1 l r m ic it ill p opo t o , the eo b e p te be ow , f o wh h w be

° ’ seen that the angle AOB (acute) is one of 1 37 30 and therefore repr esents no t m erely the golden ratio to the whole cir cum fe renc e as shown in plate 7 but places as well the point o f growth as dem onstrated by various author ities under the nam e o f

n n s ar in r m and m asur in an r l . the Ideal A g e The , t t g f o B e g othe s a sam alu a al n l r a at p ce of the e v e , we h ve the Ide A g e epe ted C ,

s n in and a ain r a in is r cess this being the eco d po t , g epe t g th p o we locate the thir d point at D and might so c ontinue indeﬁnitely

’ = I endi N e to Nat r nic Unt a e 1 2 n x N x o t s ue s Harmo i . e di o te . App , X App y, p g 3

62 Pr oportional Form

ﬁnd i nc s of s uar a n ila ral rianl we ev de e the q e , the oct go , the equ te t g e ,

a n o r ir r r s i n n l m a af l in r the hex go , the p og e s o s or rec ta g es, we y s e y fe that other coincidences of the same kind would follow were We to di a ram m but a sim lici and l ur r g the , th t p ty c arity will be f the ed by the

missi n all but o ne l m n in a i a ram c nsi r in r s o o of e e e t e ch d g , o de g the e t

ll as a m a r o r to fo ow tte of c use .

I n n a n amil a lik is rus sa isﬁ d o ur the Pe t go F y , we h ve ew e , I t t , t e s l s a n num ri al lin ar su rﬁcial and e ve th t , betwee the e c , the e , the pe , the s li m asur m ns of l nra i o n o ne an and e na o d e e e t the go de t o the h d , the p t

o n na ram ian ram i and al n l o n g , the pe t g , the Egypt Py d , the Ide A g e

r a r la i ns i xis s so s ri k in and c ninuus a nce the othe , e t o h p e t t g o t o th t, o granting the fact that extreme and m eanproportionis the governing

l men a s ruc ur ur r i n n no t inr uce e e t of t t e , f the ev de ce eed be t od d to

scur m ain a ur s al i d ac rs in lik o ur inn cenc Ob e the fe t e , the e l e f to be g, e o e

i ak Of cr m nfor ran unil c nrar r . e , t e g ted t the o t y be p oved

use hi s rima acie a n uur m a r s By the of t p f case , the st teme t Of f t e tte will be Sim pliﬁed almost beyond belief and m ay safely be indulged insinc r it is sira l r lim i i ns r ill no e , whe e de b e to exp ess tat o , the e w be

diffi c ulty indoing so .

SPIRAL FORMATION

l h im o ut Of m in e n c nsi r s m l The circ e as , t e d , be o de ed the y bo of

ni ni s r ti n its lan its r ur nin a u u n i i . I t r a c D v ty p opo o , b e , eve t g bo t po For m and Propor tion 63

i se l it is um an i al and m a m a i al t f , the h de the the t c type of a fully

form ed selfs ufﬁciency inwhich fur th er progr ess is unnec essary if no t

im ssi H r unl ﬁn l . nc so d a ur r f rrin a po b e e e , we f eq e t y N t e p e e g th t

r curvilin ar rm n n as s ir al r rt hi i othe e fo k ow the p , the p ope y of w ch s

ninu n r -satisﬁed r r ss in us m i a co t ed , eve p og e , be g th the mathe at cal

type of a growth which m ay extend indeﬁnitely until c ut o ff by for c es

beyond its o wnlimitations . JohnRuskinh as said that the line of u n n a is aninﬁni lin o t r ur nin o i s l . O n in ul be ty te e , et g t e f f oth g co d

this be m or e true thanOf the spir als .

’ a ur s use s ir als and ir in r n au ar e r a s N t e of p , the he e t be ty , pe h p ,

m a ters ail s r anc it s no t ll n c ssaril a t of d y ob e v e , yet doe fo ow e e y th t ,

r im r tan and am iliar m a s irals m howeve po t f they y be , p of the selves

— - rm a amil . ir als m a m us in ac all m asur in fo f y Sp y, t f t, be e ed the

ﬁnal anal sis an l s and anular amili s in ic a y by g e , the g f e to wh h we h ve

found natural phenom ena grouping themselves will serve to sub

divide spir als into their vari ous branches just as accur ately as they do

r ri ﬁ i li . n d the othe geom et c gures to wh ch they have bee napp ed I dee ,

a vast major ity of the spirals of Natur e will be found to fall at onc e

and indi sputably into the gr ea t growth-family of extrem e and m ean

r o r ti n as m a l n r m s u xam l s s n p op o , y be g ea ed f o a t dy of the e p e how

’ inNature Harmoni i ‘ and fo r all s r as ns s ir als s c Unty ; of the e e o , p to

er i a e nar ur s m a sa l and c n ni nl r a e geth w th c t y c ve , y fe y o ve e t y be t e t d

tr a r int wo rk See Chapter onSpirals and Asymm e y l te his . 64 Proportional For m

as ar c m si l ara d fo r c n ni n p t of the o po te who e , sep te o ve e ce

but no t fo r any reaso n whi ch excludes them from the same di s

nuishin diﬁ r ti g g e ences that Obtaininother form s .

SUMMARY

I nspite of the fact that our examinationof the geometri c ﬁgures

sym bolising the various fam ilies h as so far proc eeded o n perfec t

num rical s unc r m lin ri n n ra n e eq e e f o the e to the t go , the ce to the tet go

and n n l us no t l um nlusi n a the pe tago , et rash y j p to the co c o th t the

s l i ni proc ess hould o r can be inﬁnite y prolonged . The bas c pri c ple

be i in ﬁni w n wi na n may sa d a very de te ay to termi ate th the pe t go ,

no tmerely from the nec essity of terminating itsom ewher e and select

in n r us rm as in ﬁr s l usi l r i but fo r g the pe tam e o fo be g the t p a b e pe od ,

r and f r m u i r ns nl a i it se ms othe a or e s bstant al easo , o y few of wh ch e

r t il ulin and if a r m im im r ar d wo h wh e to o t e ; we h ve , f o t e to t e eg de the

xa nand a n itwill n a s a nl ninclu he go oc t go , be see th t the e h ve o y bee ded

as sublim ated products of the equilater al triangle and the square

It h as already bee n remark ed as of prim al irnpo rtance that no

general principle which tr eated Natur e as com posed of plane sur faces

nl ul i r sufﬁ i n r and sinc is is so m ust o y wo d be e the c e t o safe , , e th , we ,

in all o ur a lica i ns ak s li rm a i ns in c nsi ra i n of pp t o , t e o d fo t o to o de t o

nd il i i l li s ak an rm in a wh e t s Obvious y tr ue that so d m ay t e y fo , yet

r na ur o ur s u r ular s li s r se n sim l s asis the ve y t e of t dy , eg o d p e t the p e t b Form and Pro portion 65

xam ina i n nor s all insuc rk is a l of e t o , h we , h a wo as th , be b e to pur sue

the geometric foundation muc h beyond that point The ancients

ll n limi r ular l dral an l s and l r c we k ew the t of the eg po yhe g e , A b e ht

D iir er s u n ar is m a m a ician and a r n ac n ss , t de t , t t , the t , the p t o of ex t e ,

recognised this thoroughly inhis wor k o ngeom etry x; and inorder to

a air i a si ua i n let us sta a acc s get f de of the t t o , te few of the epted fact ,

and c lothe them inthe sim plest possible garm ents .

It is clear that if two angles of ninety degree s be plac ed back to

ac as inﬁ ur la 1 2 ir as s ill rm o ne and sam b k g e A , p te , the b e w fo the e

s rai lin and it ll s a c m ina i n l s sum t ght e , fo ow th t any o b t o of ang e , the

of which equals o ne hundr ed and eighty degree s (the sum of two right

an l s will as s wninﬁ ur o f sam la rm a c ninuus g e ) . ho g e B the e p te , fo o t o if as . ur r it is sim l and indis ua l a sum b e F the , p e p t b e th t the of the an les so rm is in c ss o f o ne un r and i e r s as is g fo ed ex e h d ed e ghty d g ee ,

cas illus r a in ﬁ ur la 1 2 n lin r r the e t ted g e C , p te , the the e he etofo e c ninuus will n ac in an in n a i n a in c m o t o be be t b k to de t t o , h v g be o e in ll - u at m ili ar c ns r uc i n ca a re nr an an l . urs in wh , t y o t t o we e t t g e P g

sam r c ss ar r see a if ur an l s r u a u the e p o e f the , we th t fo g e be g o ped bo t a c mm n in as nin ﬁ ur o f sam la cann o o po t Show g e E , the e p te , they ot exc eed intheir aggregate the sum of thr ee hundr ed and sixty degrees

s (the sum of four right angles) and still lie ﬂat in a plane . If the e

n le s a r a l ss an is sum ill n c ssaril ris a a g gg eg te e th th , they w e e y e to

i ella a r I nstitution tri car S ee also endix No te . L utuo um Ge om e um . b Q App , IX 66 Proportional Form

in at ir a and rm rami and rr in h e l i po t the pex fo a py d , ca y g t og c a step

n is se e a if a r a c ds sum m ni n beyo d th we th t the gg eg te ex ee the e t o ed , thenthe result canonly be conc eived as producing a depressiono r in dentationinthe plane sim ilar ineffec t to the indentationinthe line l showninﬁgur e C of this p ate .

c nclusi n n is in i a l no r ular so li an The o o the , ev t b e , that eg d c be

rm an l s ic c m in a c mm n in o r a ual or fo ed of g e wh h , o g to o o po t pex , eq

c e ana r a r un r and si r s sinc ual ex e d gg eg te of th ee h d ed xty deg ee , e to eq this sum would pr oduc e a ﬂatsurfac e o r plane and to exceed itwould produce by c om plem ent a depr essionand no t anapex

Applying thi s principle to the geom etri c al ﬁgur es which have so far n asis o f o ur s udi s s all ﬁnd a in c ase bee the b t e , we h th t the o f uila r al ri an l m a m ak c m ina i ns s in r e the eq te t g e , we y e o b t o how g th e

ri n l s at ana i a al r im s i o r o ne un r t a g e pex , w th tot of th ee t e s xty h d ed

i r s ur rian l s i a al o f un and and e ghty deg ee , fo t g e w th tot two h dr ed

o r ﬁve rian l s i l f un r r r r s a a o r s . fo ty deg ee , t g e , w th tot th ee h d ed deg ee

s are ssi l as illus ra in ﬁ ur s rah dr n All of the e po b e , t ted the g e of the tet e o

ﬁ ur la 2 c a dr n ﬁ ur and ic sah dr n ( g e G , p te I ) the o t he o ( g e K) the o e o

ﬁ ur but r are rc to s so far as uila r al ( g e L) , the e we fo ed top the eq te

rian l is c nc rn sinc six an l s o f i r s a re a ree t g e o e ed , e g e S xty deg ee gg g te th

un r and i r s and as s ninﬁ ur is la h d ed S xty deg ee , , how g e D of th p te ,

‘ ﬂat sur ac and no a a form a f e pex o r sum mit wh tever .

3 S ee endix No te . App , X

68 Proportional Form

I na s uar o r r c an l a in c rn rs Of nin r q e e t g e h v g o e ety deg ee s, three m a c m in rm a s li as inﬁ ur H s in a r y be o b ed to fo o d g e , how g cube o

xa dr n but i ur o f s arri at sam r sul ro he he o , w th fo the e we ve the e e t p — duced six uian ular rian l s a ﬂat lan . na nis by eq g t g e , p e The pe t go ,

k n rm o f an l s o f o ne un r and i r s at ac we ow , fo ed g e h d ed e ght deg ee e h

l i r o f s ar un a c mm n in in c r n r . ac n as ﬁ ur o e P g th ee the e o d o o po t , g e

see a sum o f r in r un r and n F , we th t the the th ee , be g th ee h d ed twe ty

ur r s s illl a s a alanc in a ur a r ic rmi fo deg ee , t e ve b e f vo of ve tex , wh h pe ts

rm a i n a r ular l r al s li n nas cah dr n the fo t o of eg po yhed o d k ow the dode e o , ill s ra inﬁ ur la m ni n but no ur r s n ut ted g e I of the p te e t o ed , f the tep ca be tak enwith the pentagonas anadditional angle exc eeds the lim it

d escribed .

rnin now to a nOf o ne un r and n r s Co g the hex go h d ed twe ty deg ee ,

e e t a lan il lan s rm o f il we s a g ce that , wh e two p e fo ed these w l be insuffici n to ncl s s ac r ill in ur n r uc a m r ﬂat e t e o e p e , th ee w t p od e e e

l n r l did six uila r a n l s a d no r ill m . p ane as eq te tria g e , ve tex w be fo ed

sam is a orti ore r u a n and all ﬁ ur s a in The e , f , t e of the hept go , of g e h v g a greater number o f angles . Whenther efor e the hexagon is found in rm a i n cr s als and r na ural rm s it is in aria l the fo t o of y t othe t fo , v b y

r as c r a r ﬂat sur ac s o r in c nunc i n wi eithe a de o to of f e , o j t o th geo

i l nﬁ ura i ns r ani s l suc as its r la i s in metr c a c o g t o othe th t e f, h e t ve the n tetrago family .

ma ri c nclu r r . a il irr ular s li s We y b eﬂy o de , the efo e , th t , wh e eg o d

70 Proportiona l Form

r a n h l i s s the Tet go o r square and t e go den ser e , ymbolis ed by the pentagonand the asymm etric group ; and falling into o ne or other of these three we shall ﬁnd a large m ajority of the forms and A phenomena of Natur e which will c om e under o ur notice . m ongst the Tetragon gr oup will be placed the inﬁnite var iety o f examples

‘ urnis r a it r c and namics hil un r Vi al f hed by g v y , fo e , dy , w e de t inﬂuenc es we shall ﬁnd the m agic o f the extrem e and m ean r atio

for in n i c onstantly exhibited . Other form s will call a pass g ot c e ;

r nﬁ ur a i ns ill r a s s ar a l examin fo r sa othe c o g t o w , pe h p , be ep te y ed the ke

n ni nc as ill cas i lli s s and a num r r of c o ve e e , w be the e w th e p e be of othe c ur ves but inthe end nearly all o f these will be found to have taken their allotted places as adm easured by o ne o r other o f the groups to

n mi in e lm ni n . a s whi ch o ur atte t o h as bee devoted It ght , de d , o t be

said that without these was nothing m ade that was m ade . They ar e

TE E G REAT MODULES .

endix No te XXX . App , III CHAPTE R III

TE E TRIGON I N VISIB LE NATURE

S gravity and polar for c e are such vast and uni versal power s

and c onstantly exercising their inﬂuenc e over whatever

ss ss s si al a ri u s r l r a s s m po e e phy c tt b te , the e woud be pe h p o e ac ademic advantage I nplacing next inorder a c hapter setting forth the evidence to prove that these for ces ar e governed by rules c oming un r r n i ff r s r c a c de the tet agon a d tr gongr oup . E o t howeve to t h the a ni n r la i r invi i l r ar r m difﬁc ul tte t o e t ve to m o e s b e fo c e e of ext e e ty, and inorder ﬁrst to intr oduc e to the r ea der som e of the visible beauties uponwhich this series of r ules is brought to bear the tangi ble h as been

i n r n l in O e a us ﬁ in g ve p ec edence over the inta gib e the h p th t th , x g in r s ﬁrs o n vi n r i ns uns n and m r te e t t the sible demo st at o , the ee o e abstruse m atters o f forc e m ay safely be left fo r a succ eeding c hapter after the gener al working o f the planis m or e clearly illustr ated .

Rec ognising thenthat as all m em ber s o f this tetr agon gr oup

a n un su r c r ain a ur s inc mm n and a as h ve bee fo d to ppo t e t fe t e o o , th t s wn in r a r nc it sa is ac ri l m n ho the p ec eding c h pte , o e be t f to y de o strated a an i i n s uar c anins anl th t Object s pr oport o ed o nthe q e, we t t y 7 I 72 Proportional Form

l assur a ula ral ri l and xa n ill at i fee ed th t the eq i te t ang e the he go w , the r a in l s um ir n r s nsi ili i s it ill ppo ted p ace ass e the wo ted e po b t e , w be inter esting to see how these things work o ut inpractice and whether

l r these points deve op infact as well as they do intheo y .

Per haps inall cr eated things no m ore beautiful exam ple o f the

rkin s o f la s l r r m r r c e a nal si n wo g the w of po a forc e , o o e pe fe t h x go de g could be found thanino ne of the whirling snow ﬂakes form ed and re

rm il m a in its n rr rial urn r m ull r fo ed wh e k g o e te est jo ey f o the d , g ey ,

inr cl u a nc i it hr u a s r s c n s w t y o d th t c o e ved , t o gh few ho t e o d to the drif i - in u r t in its rm rnm a s . ar min a t whe e jo s sto bo te Be d th t , tho gh a lif im r s e n in s ar c no two es s ac ular au et e we e p t the e h , of th e pect be

i s uld r un l alik nn o f m las m r ana t e wo eve be fo d exact y e , o e the t o e th

n s n m a nm r ana r i n a minut few sec o d , no e of the h ve bee o e th f act o of e intheir form ation; they are engraved and etched and be aded and carved beyond th e power Of the microsc ope to investigate them ; they have bee nfalling fromthe heavens sinc e m ancame o nearth and will

ninu fall l n r n r o ne inall c o t e to so o g as tem pe atur es vary . Yet eve o f these milliards of millions that varied from the form of a hexagon o r showed a patternwhich was not a six-c ornered exem mar of the

a n il Tetr go Fam y .

Suppose we look carefully at a number o f exam ples of snow c rystals tak enfr om the wonderful c ollec tiono f photographs m ade by

nl r n in ul in r . . is a s r W A Be t y of Verm ont . It h dly ec es a y to d ge ve y P LATE I 5 PLATE 1 6

S NOW C RYS TAL S NOW C RYS TAL

’ e r s H r o ic nit The abo ve illustratio ns ar tak enfro m Natue a m n U y.

74 Pr oportional For m

t n i n f easpoonful o f water . They have bee g ve the nam e o di atom s

Diato m ac ea r m ir a i o f r a a i n a sim l di isi n ( ) f o the h b t p op g t o by p e v o , if l thus cr eating two atom s wher e only o ne was befor e . These beaut u

— — PLATE xS c upxus m ung PLATE xo o o no m r a

B acillaria c eae are engraved and etc hed with the sam e c ar e as the snow

r als and ins anc s r l lin s o f n r a in c yst , t e a e no t ack ing where the e e g ving the decorationhave beenmicr osc opically trac ed and c ounted to the

num r in a o ur n l be of to the ch . We m ay say th t k ow edge of these decorations is limited as inother c ases only by the possibility

f ins r n t i o the t ume t o m ake them visible by m agniﬁcat on. These

beautiful things tak e the form s of all of the polyhedr a but in the m ajority of cases the patterns are limited to triangular and hexagonal

n 2 ns as s nin illus ra i ns in l 20 2 1 22 a d . Oc ca o e how the t t o p ates , , 3 sio nally o ne is seenwhic h shows the direct tendencies of the square PLATE 20 PLATE 2 :

TR I A NG L' LAR D I ATOM TRI A NG LLA R D lATOM

PLATE 2 2 PLATE 2 3 HE XAGONAL D TATOM HE XAGO NAL DI ATOM

’ D ia to m I llustr a tio ns a s ta k e nfro m Na tur e s H a r m o m c Uni t

PLATE 24 PLATE 2 5

OC TAG O NAL DrATOM OC TAG ONAL DI ATOM The TrigoninVisible Nature and c a n in la 2 and 2 all ic ill lain m the o t go , as p tes 4 5 , of wh h w exp the se lves imm ediately upon a c omparisonwith the plates c ontained in

Chapter I .

Rising no w from these lower orders to the higher form s of plant lif ﬁnd n r n wi in rs amiliar us e , we the same tende cy st o g th ﬂowe f to

invi to tw o in i dir all. O s a ni n is ﬁrs c c f the e , tte t o t ted wh h the e t factor is evidently the tetr agon o r s r I n l 2 qua e . p ate 6 we se e a dr awing o f the sepals and seed ve s sels of the syringa inwhich the four se pals are c arri ed o ut by four seed vessels and these in proportion are governed by the fourth progr ession o f rim in the p e square . Aga we tur n to plate 27 showing Onagr a

i nnis wi it ur l n PLATE 2 B e , th s fo eaves a d a 7 0mm “ “m m central c ircle c oincidi ng with the

ur r r ssi no f s uar and a ainin ur stin c n as fo th p og e o the q e , g the b g otto , s wn in la 28 is s n a m s ﬁni in an o f sam ho p te , ee o t de te st ce the e r la i ns i to r c an ular rm e t o h p e t g fo .

N0 exam ination of plant life under the inﬂuence of the laws of polar forc e and the tetragonal family would however be c om plete without some of the beautiful illustrations which c onstantly oc c ur in 76 Proportional Form

- the six sided ﬂowers such as the jonquil (plate 29) and the tiger lily

(plate to whic h might well be added Zygadenus Elegans and the

’ as e r lil la s 8 and Nature s Harmoni e Unit in ac f E t y (p te 3 y, e h O whic h the ﬂower petals lie inthe form of the equilater al triangle lik e a

a inits cri il innr m arkin s are ﬁn re ular b be b , wh e the e g de ed by the g progressions of the hexagonas indic ated inthe plates

r is r a s a inna ural l m n ic is i The e , howeve , t ge t deve op e t wh h h gh above that of plant and vegetation and this is the life o f anim ate

hin s and f r m n at s m us amin ri . o o a m t g , the e we t ex e b eﬂy L ok o e t the beautiful butterﬂy illustrated inplate 3I and thenc onsider again the wonderful exac tness of the equilateral triangle form ed by his

And if s r n n ins l as rk the e we e o t e ough , pect p e e , the wo of the wasp informing his rem ark able nest (plate 32) and the m ar vellous c omb of the c omm onhoney bee (plate 33) with its perfec t building and l utim ate utility and hexagonal economy .

n i l inhis I cr a in h r k a l w s r us Mr . e t g er em ar b e ax to eho e , B ge ow , r c n issu o h e Nature di s ns r l n u l e e t e f T e Guid to , ow the theo y o g phe d , that the b ee really goes about this c ell building with any plan to cr a ll He sa s a inm ak in c m e te a series o f hexagonalc e s . y th t g the o b - n the honey bees never work inhexagons but always incircles . The

h us s o ut si s s e k ee ps going into the cell lik e a gunswab and p he the de , and it i t i i i no t sli s in ntnor s h s pressure o nthe s des, w th the ghte t te

The TrigoninVisible Nature 77

’ skill o n s ar but ur l ff c a m a m a ic al law the bee p t p e y the e e t of the t , I n n l i a m a s a n. s c c us ns as ill s nin th t ke the hex go the e o o , w be how

i l h as s m r a ndix Mr . su rs an n n n the ppe , B ge ow o e ppo te d m a y oppo e ts ; and fo r m s lf il I am ui r ar an y e , wh e q te p ep ed to adm it that

n archi c ur al s an in it is i l li l a Mrs . r te t t dpo t , h gh y ke y th t Bee eve — thinks of h er storehouse as a hexagon o r as a c ircle either fo r the m a r a a sh h as n i r r un lanno r l a i n lu tte of th t ; th t e e the g o d p e ev t o , b e

! rin no r r l and a h e r in ni ns c an nl uss r m p t t owe , th t te t o o y be g e ed f o the r esults produc ed ; but if sh e can build cir cular cells and then c hange them into architectur ally ec onom ic hexagons inthe pr oc ess

lun in ac and r ﬁll m sh e is na r c n m is of p g g b k fo th to the , eve bette e o o t 78 Proportional Form

h i nh er r r is n thanwe ad g ve c edit for . The esult o e of the m arvels o f

a i n and r o r r Na ur s all cre t o , whethe the bee Mothe t e h have the

r i i Y l m ost of the c ed t s sm all m atter . o ucan scarc e y fail to read

i in r s in i o f his l m n acc un w th te e t , v ew t deve op e t , the o t of the in vestigatio nm ade by the great Réaum ur m any years ago o nthe sam e

a s l sa s eaum ur a r difﬁ cul r l m They h ve o ved , y R , ve y t p ob e of

o m r fo r nl a limi m a rial o r wax is at ir is sal fo r ge et y , o y ted te the d po

ns ruc i no f ir us r m s o r c lls i m the c o t t o the ho e , the oo e of wh c h ust be o f a rmin c a aci lar s siz and wi s r n es dete ed p ty of the ge t e , th the t o g t walls in propor tion to the am ount of m atter to be em ployed

The cylindrical form would seem to be the best adapted to the shape

’ o f s but is ul l a ac an o r as r m n the bee body , th wo d e ve v t w te oo betwee

ni uus c lls o n r an h ad c lls n uar the c o t g o e ; the othe h d , the e bee sq e

r rian ular m i a nc ns ruc i u so m an o t g , they ght h ve bee o t ted w tho t y unnec essary vacancies,still they would have required m ore m aterial n t and thenwould o have ﬁtted the body of the insec t . The six sided form of the hexagon fulﬁls the problem in ever y par ticular ;

as ac c ll ins a o f rmin a lan is c m s r e the b e of e h e , te d fo g p e , o po ed of th e di am ond shaped piec es plac ed insuch a m anner as to produc e a shal l r am i ic s ruc ur im ar s r a r s r n il i in a o w py d , wh h t t e p t g e te t e gth wh e g v g l n larger capacity with the sm al est expenditur e of tim e a d m ater ial.

° ’ The angles o f these cells o nthe longest spac e m easure 1 09 28 and

80 Proportional Form

h er a n to h ve bee ,

pr oves o n investi

gatio nm erely tobe

the m eans of dis

c losing carelessness

in assem bling evi

dence o n the part

of the cri tic him l se f .

The exam ina

tionof the Tetra

PLATE PLATE 34 go n Fam ily in 35 J APA NESE DAm sn-c xo ss POP PY Na ur c a nn Th e Triangle (Ko rin) t e ot , The S quare

r a p e h p s , b e

br ought to a foc us

m ore suitably than

by the pr esenting

of the four follow

in illustra i ns g t o ,

in the ﬁrst of

which (plate 34)

we have an e x

am ple o f the orien

PLATE 37 The TrigoninVisible Nature 31 tel iris after the drawings o f the fam ous Japanese artist Ogata

rin wh o saw s a r t us rian ulari Ko , thi be utiful ﬂowe in all i s obvio t g ty as l n a o as 1 6 n l rt i n and ist o g g 75 , whe he ed the a of N ppo , d e

ar in i in co n eni n s l a nam ic g d g the ex st g v t o s , m ade for him e f e wh h h as no t a ra n ul r si ns i yet f ded , d wi g those beautif sc een de g w th whi c h we ar e so familiar and which em bo dy so m any of these

li ul l s l a a m ssa s m s. to r s de ghtf b o o True m ytho ogy , Iris be e ge

r m his o o n f o day to ur w .

I n la a ur - e all anis - r ss scarl of l p te 35 , fo p t ed D h C o poppy , et bow and sil r - i c nr r ss s ts ra nal si niﬁc anc no t ve wh te of e t e , p e e i tet go g e m erely thr ough the num ber of its petals but thr ough its sym bolic

r s as ll ll n i in e l s c s . i s la 6 are s n am o we Fo ow g th , p te 3 , how x p e of the six-m m r e anm n r na a llian as re l and r n n e be d e o e c o o ri , bri t d b ood e ow ed thr ough the ages as being those lilies of the ﬁeld of whic h we read

a il no tn i r s in l m nin allhis th t they to e the do they p , yet that So o o

l r was no t arr a lik o n g o y yed e e of these .

inall in la a as nis in l r ular and alm s F y , p te 37 , we h ve the to h g y eg o t s l -c n ni nalis l ss m s o a li raine to as e f o ve t o ed b o o f the single d h a . T d sum m an l ur s and m an s in n s in all its e y c o o y form m a y clim e , yet vi cissi u s and an rin s and c an is o ne rsist n t de w de g h ges, we r ec ogn e pe e t c harac teristic inits determinationalways to be eight petalled when l its o wn i i ni n r un c s. us ahlia i ts c m a s eft to dev e Th the d , w th o p o , o d o ut the them e of ﬂoral symbols inwhic h we see portrayed the equi 82 Proportional For m

la ral rian l a n and c a n tanic all te t g e , the squar e , the hex go , the o t go , bo y

i i n r nits nir and ﬁttin l c nc lu in his typ fy g the c hapte i e t ety , g y o d g t

r i n po t o of o ur subject .

84 Proportional Form

r r rink inartiﬁcial au ro u Choose we the Ope a, whe e we d be ty th gh

or r s r ir s air l us the senses, choose we the fo e t , whe e the b d of the ho d

wi b au ir a r and s n or o r ill rapt th the e ty of the fe the o g , g we whe e we w ,

t m e n r r m o ne ur ac rs beauty tak es ever i s ost pot t cha m f o of fo f to ,

’ f rm m i n c l ur or s un m m n s u s s us a o , ot o , o o , o d A o e t tho ght how th t ,

s m a rial hin s s c n n rs aris o ne and all lik e m o t te t g , the e o curre t fac to e , ,

n n vis l A invisi l and inani l vi ra from a gover me t of the in ib e . s b e t g b e b

i n nr l l and un i l a us so ravi in t o co t o both the co our so d wh ch p e s e , g ty large measure governs both the form and motion o f the c omponent

s n and c l ur i ic ur . un a d li ic music tem of the p t e So d ght , wh h beget o o ,

l n alik ransit r illusi and no n u s anial rl be o g e to the t o y , ve , s b t t wo d

r in ravi a i nand a and r rm s r la lik mis whe e g t t o he t othe fo of fo ce p y, e l i n n h vo s s in s s s ani l. c ie u e ve , w th th g ub t t a These forc es possess o e of th e r r i s m a e r are no t nr i n ansi n o r p ope t e of tt , subject to c o t act o , exp o ,

si n a ni r l n r a r i are m a c ohe o ; they h ve e the e gth, b e dth , no th ckness ; de u ni r m l cul s no r a m s no r a r m a rial p of e the o e e to , h ve they othe te a ri u s are r ni sa l m l a ust d tt b te , yet ecog b e by anthrough the nice y dj e s nses wi ic h as n n hi r r sin se t e th wh h he bee e dowed by s C eato , c e they inm i n ir invisi l i l in i . ot o , by the b e power , these th gs v s b e

G RAVITY

We are very prone to deﬁne gravity as anintangible force whi c h at rac ts all in s ar r in a al wi o ne t th g to the e th , fo gett g th t we de th of The TrigoninForce 85

se rul s hi c rk a s i ual r r I n r r tho e w h wo both w y w th eq egula ity. p opo

i n m ass c mis cl u ra s u t o to the of the obje t , the ty o d d w the earth p

wards with the sam e steadfast purpose that the m ore solid earth

a wn l draws the v pour do to itse f . The apparent differenc e grows al

. together o ut of the varianc e inam ount of m atter exerting the in

n l l n an i ﬂue ce . us c ar u rs s n a ra i si niﬁ s Let e y de t d th , the , th t g v ty g e

the rem arkable power exerte d by every particle of m atter uponevery

r ar icl a othe p t e . A few milli onyears go Dame Nature penned the hi n r ules w ch were to governher realm . This c ode she wr ote ini delible

ink o n ﬁrs a h er ri inal r r and r itr m ains a m l the t p ge of o g epo t the e e , ode

sta u is da lik un law s and rsians t te to th y, e to the of the Mede Pe

whi ch altereth no t; and am ong other interesting things sh e said that

every sm allest atom of m atter should ever and always exercise thi s

and ni nin unc r r r a m r di s an strange be g ﬂ e e ove eve y othe to , howeve t t ,

and that all substance should forever have a tendency to com e to

i u it rl s an ir n n and in a i ans ul r . d c s gethe W tho t , wo d the o te t h b t t wo d

a ar and all r r ss s it r c ncis l n wri s ﬂy p t , W e exp e e ve y o e y whe he te of

’ ain a s ﬁ hr n The ch th t xed to the t o e of Jove , Onwhich the fabrick of the world depends ; n One link iss l l r a i n s . d o ved , the who e c e t o e d

Thi s gravi ty thenis aninvi sible forc e whi ch is exer ted uponevery

m m a r in uni rs r a a m i self vi si l or ato of tte the ve e, whethe th t to be t b e 86 Propo rtional Form

i l and is ra i hi c m a nsi r as in in a invis b e ; th g v ty, w h y be co de ed be g

a lar r c m us r n as r in intwo a s sense po fo e, we t comp ehe d wo k g w y

i n t am i t and under Oppos te i ﬂuenc es a o ne and the s e tim e . It s no

n ism u m an c nfus i it un r th e s ll m ag et , tho gh by y o ed w th , yet de pe of

i n r lls r ill sea . rac ts s adfas la s comma d o eve y b ow of the By g e of te t w ,

every planet inthe heavens pursues its stately course and every rnidge

that ﬂies dar ts airily through the silver m oonlight : the curves of the

m ountain ar e its sign and the angles of the crystal are its signet

n s anial i s lf it rns all su s anc and unse n itis m nar usub t t t e , gove b t e, e , o ch

o f all things visible .

We m ay thenview all things inNature as having beeninﬂuence d

n n n ni rm in ir rm a i hi s r a rc . I i r a c s r the fo t o by t g e t fo e the o g fo , whe e

exterior inﬂuences are less m ark ed and where the interpolation of

r la s r and rm a i nare l ss inruin s all ﬁnd othe w of g owth fo t o e t d g, we h ,

as seenin r i us c a t r a c ns an t n nc assum un r the p ev o h p e , o t t e de y to e , de

inﬂunc is lar rc rman n rm s t n the e e of th po fo e, pe e t fo of the te rago

i a s k n Nor m s all l fam ily of wh ch we h ve po e . ut we ow ourse ves to suppose th at this tendency terminates with the visibility whi ch en a l s us so r a il a sn r s lis a xa n see c a e . mus b e to e d y th t the ow y t h go We t , o n c nr r r m m r h a is c ninu in ﬁni l hr the o t a y, e e be t t th o t es de te y t o ugh

si l all of the tests to which we canputinvi b e force.

our r r u nit mus ssi l m ns If theo y be t e, the t be po b e to de o trate

88 Proportional Form

i i n un a Uranus ﬁve m r r m Uranus to pos t o of Nept e to th t of ; o e, f o

urn ui r and so o n Y u Saturn; four m ore from Sat to J p te , . o will

sa in ar es illus ra i ns a c ul asil be perhaps y reg d to th e t t o , th t they o d e y

u sui ccasi n and r all ro n hin sinc a m ade p to t the o o , e y p ve ot g, e de m onstrationonpaper the size of this page would be utterly lacking in

r s n a r to a n accuracy . To this I e po d th t , we e I h ve prese ted to yo u

a r o u ul in us ifi d in hr in k ut such p oof , y wo d deed be j t e t ow g the boo o n n n m s c nv ni n in . o rar u n of the o t o e e t w dow I depe d, the co t y, po l in o su rficia n in s uil . la s are in r ed noth g s pe , oth g o f t e The p te se t m erely to illuminate the dem onstration resulting from a scientific

s kn c om parison of the theory with the fac t . We ow the m ean o r average di stanc es of the planets from the sun as generally accepted

n i a a i all by reputable astro om ers. It s also m them t c y and astronom ically sim ple to vi sualise the heavens as m apped o ut into the hexa

i no o n gonal progressions wh ch are w under ur co sideration. The true

geometri c distances betweenthese outlines canbe calculated by their radii with anaccuracy evengr eater thanthat with whic h itis possible

to determine the positions of the stars them selves ; and inthis web we canplac e the planets intheir orbits with a nicety limited only by the ability of the astronomer to tell us what the exac t distances of th e

n sun pla ets from the really are .

am ri a a ur ina r ti nin c les ial s ac es If I ght th t N t e , ppo o g the e t p , shows a strong tendency to ac ce pt the m em bers of the tetragon — PLATE 38 o xmr s o r TE E sum mo nPLANETs m NUMBE RS ON THE HORIZONTAL DI AI BTE R ARE ASTRONOMICAL UNITS OR HULTI PLES O! m

DISTANCE OF THE EARTH FRO! THE S UN AND ARE NOT HBAS URE HE NTS OF HE S S . 89 90 Proportional Form

hi ch so lar l rn ravi as her r ul n family w ge y gove g ty, foot e, the the

ﬁgur es de duc ed from a com parisonof these two methods should rea

n no k n wn is anc s a ula ul l ci . w s sonab y coi de If , the o d t e as t b ted ho d be found really co incident with the ca lculate d distanc es as determine d

a ns so m a e n it i air to inf r a r n from the he ve pp d , the s f e th t easo supports the claim and any theory so supported is no t unworthy

Let us conceive o ur plate as representing the heavens so m appe d

hi s im a inar s r tchin i a s lu accurac acr ss m by t g y web t e g w th b o te y o the ,

r chin i sufﬁci n ac curac ac r ss o ur in a and n and st et g w th e t y o t y p ge , the

ri n m rical e am ina i n let us m ak e a brief t go o et x t o of the fac ts . The authorities tell us that the m eandi stance of Neptune fr om the sun

fo r his is anc is no tal a s sam a s is two illi n ( d t e w y the e by good hot) b o ,

n un r nin -six milli n mil s Uranus o ne illi n s n seve h d ed ety o e ; , b o , eve

- ur milli n i a urna u al a an ui hundred eighty fo o , w th S t bo t h f th t d J p ter

anc a urn il a i nl n an alf about half the dist e of S t , wh e M rs s o y o ce d a h

distanc r m sun a ar m ainains and so c m ari the e f o the th t E th t , , by o p

m s rs m a c nsi r a n ar ni so nwith o t of the othe y be o de ed e e ghbour .

is as a asis fo r o ur ia ram a laced un at Taking th b d g , we h ve p Nept e

si i n o n u rm s o r rim circl asin its the require d po t o the o te o t p e e , b g

radius o n the k nown position o f the planet . Spare m e then the m om ents nec essary to examine the others intheir positions with rela tionto the great o rb of day; Ur anus o nthe cir cle of the third pro gr es

92 Propor tional For m s ar : and ﬁnall letm e add a m r ano ne r c nis acc e t y, , th t o e th e og ed , ept d , and us ul r in as r n m n ut a s is un ef theo y t o o y whe p to te t , fo d, lik e

’ Law ar r m ails m u m r i Bode s , to v y f o the det ch o e w dely thandoes

us a s n the theory before . The calcul tion upo which o ne m ust rely to su r suc s a emens as s r m rit ri us are ppo t h t t t the e , howeve e o o , deadly

ull and a n n a s i l if un d , they h ve bee se t to the c ce s b e obtr usive limbo

a nd of the ppe ix.

I n r r m ak m at r s ill cl ar r le t us add a s rt o de to e the te t e e , ho examinationof plate 39 inwhich we continue the proc ess above de scribed by di agram ming the com parative distances se parating the so “ call inf ri r lan s r m ac r and r m suni s lf and ed e o p et f o e h othe f o the t e , our drawing is practic ally a m agniﬁed edi tionof the interior circle of the previous plate with the orbit of Mars no w enlarged so that we m ay

H r s u lan s nar r sun. l arn a m asure t dy the p et e e the e e we e th t , e d by

ir ownm an c rs is anc ars i his red and lar in the e ve to , the d t e of M , w th g g

r m o ur ar ic s in s all rs i a c l ur at eye , f o E th , wh h h e to of the othe w th o o

hi can nl uss is lai o ﬂ acc ura l hr r r ssi ns w ch we o y g e , d te y by t ee p og e o

xa n il as ail in m a m a ic al n s charm of the he go , wh e, det ed the the t ote , in nus c n a l la m an H s rus and Pho r h or us g Ve , the ha ge b e dy who e t e pe p

urns anci ns li s o n r s us and n arer by t to the e t , e the othe ide of e the sun ur and rcur san all ar r ca uc us by fo Me y, the d ed be e of the d e , by

n uni s k n un ra al l i s rai in r a s . eight tet go t , oo g t ght to the eye of the g e t

endix No te . App , XIII PLATE 39 o xnrr s o r TE E m o: W e e E wa s

93 94 Proportional Form

r one dis se sc ff at s ecula i ns suc as se itn s We e po d to o p t o h the , eed

nl ac s u hi s sam law o y to go b k to the t dy of t e of Bode , to which we r rre and di sc v r ast r i s its s rva n ﬁn efe d , the o e y of the e o d by ob e tio , to d

am l reas nr s ra n u i mm n ar p e o e t i ing ca st c c o e t . There e several points wi r ar is r ic are in I n th eg d to th theo y of Bode , wh h of terest to us .

’ ﬁr s lac it a ns a it was n t s Law at l avin the t p e , h ppe th t o Bode a l, h g

e n ulicl annunc d a m r r irin and m s as r nm r be p b y o e by o e et g ode t t o o e , o ne ann ani l i ius i n r s r al ars r it was Joh D e T t of W tte be g, eve ye befo e

l l a r ria rlin r ss r and i n his bo d y pp op ted by the Be p ofe o , Bode , g ve n am . x as h as n in a it is n ut aci e Ne t , bee h ted bove , , whe p to the d

s nl nri l n ts ciﬁ all rr in ino ne cas at te t , o y ge e ca ly and o pe c y, co ect , be g, e l as a n I ns i s andi ca s un n a l i c rr c . e t , th t of Nept e, ot b y o e t p te of the e h p ,

v r and im as it r as its n s and its howe e , pugned , we e , both to ho e ty v racit so r a is un rl in ru r a e y, yet , g e t the de y g t th of the theo y, th t

n rs r rk init nam and cansa a r n r s wo de we e wo ed s e, who y wh t othe wo de its uur will ai l in n e is nc a nin lan f t e d indisc os g, eve to the x te e of th p et - seventy seventim es as far from the sunas o ur familiar Earth .

i ius Wittenb er o uwill r a s r m m r ann unc T t of g, y pe h p e e be , o ed that he h ad disc overed c ertainrelations betweenthe distanc es of the

' ari us lan s r un but ni r no r carri s v o p et f om the s , e the he Bode ed the e beyond the point of building uponthem a num eric al series inwhich

r m sunwas us as a m asurin ro d as is the di stance of Ear th f o the ed e g ,

n in na s ri s no t un sual and call anas r n mical ui . u , ed t o o t Sett g dow e e

96 Pr oportional For m

' Bode s Law and we now know that itfell as exactly into its appointed spot as if the plac ing of chair s at the celestial table for unexpected

an r rs a a o f rr n in a w de e were m tter daily occu e ce . S ce planet was

obligingly found to ﬁll ever y niche inthe system exc ept that between

Mars and u i r a m r na ural n an a s la i n J p te , wh t o e t the , th th t pecu t o should be rife as to why nothing appeared to occupy this o ne existing ca i an m Uni rsal s arc al n es v ty d ak e the system perfect . ve e h o g th e

lin s was uic l r ar nd a a as a r sul rec rds e q k y ew ded, a tod y we h ve, e t , the o o f hundr s as roi s l r k n s a nc - lan ic ed of te d , litt e b o e particle of o e p et wh h

r e n f ci cl ar un suni r i no w ac an bu or m . a d o d the the o b t v t t the Th t ,

we r was not all fo r am re led ﬁnal la in ho ve , , the s e search to the p c g of

un u rm s and ar s i s an all k n n Nept e, the o te o t f the t d t t of of the ow

lan s m akin hi s m a s i ur un un nc ino n un re p et , g je t c to ar o d the s o e e h d d

and si - ur a an irclin in a c l arkn ss ir - xty fo ye rs, d c g th t o d d e th ty odd

tim e s as far from the sunas the little Ear th we tread .

us m asis alu car ul sci niﬁc s cula i n Th we e ph e the v e of ef , e t pe t o ,

and at the same time we add to the facts as c overed by the law of

urt r i m a r r ssi is anc s w n Bode the f he te , th t the p og e ve d t e bet ee the

lan s are no t nl rn as s wn i ius a law i h p et o y gove ed , ho by T t , by , of wh c

a us a num r al r ut a a law nanal s s ws he g ve e ic theo y, b th t th t whe y ed ho

progressions of the tetragonfamily

endix No te XI V. App , The TrigoninForce 97

and that this is only one m ore of the m any instances inwhich the

eﬁ e ravi are un all in is ﬁ i n cts of g ty fo d to f to th classi cat o .

Before leaving the study of the forc e of gravity and going o n

ara l r r le f r a m m n se e h a r sep te y to othe fo ces, tus o o e t how t i s tt active power which holds the world together c om pares inits operationwith o invi si l a ns as s un an li is n n suc d . ther b e ge t h o d ght It , I eed o t r e a o ne ur s s hi s r s a a ur is one pe t , of the p po e of t wo k to how th t N t e

r l nsis n l r a ana ri n g and y co te t who e ther th se es of harm onic accide ts .

She is no r u se ara ni i s rkin i r s r in g o p of p te e t t e , wo g e the everally o

ar nr shi she is no m r l ar icula c ll c i n uni s i a p t e p ; e e y t ted o e t o of t , w th se t laws rnin c mis r and an r se t c nr llin so un a of gove g he t y, othe o t o g d, s ara se t rnin i l and O si r ul s mina in li ep te gove g b o ogy ppo te e do t g ght .

i i nal c s sh e h as hic in c m un s r la s Add t o ode w h , be g o po d of othe w ,

rnc m un s c rr la ac s butm c n ni nh as al a gove o po d of o e ted f t , y o te t o w ys beenthat the great threa d of her power runs through the whole fabric

nis nl and arm ni usl and a ac sci nc and ac art c o s te t y h o o y , th t e h e e e h is

a m r r an l ins ara l inits ﬁnal erf therefore e e b ch of the who e , ep b e p ec

n H r l s ra i a l as a n r m r m ai r . e a s n tio f o the e de w of g v ty pp y , we h ve ee ,

u a r s nﬁnin m s l s n i r o n thr ougho t ll the unive e , c o g the e ve e the to e

n o n l I n sam wa all la s s n nc no r e r . u scie e eve to wo d the e y the w of o d ,

and ra ian nr a l r r i r a i ns m a c light , d t e e gy pp y whe eve v b t o y o cur ,

n li l n r an r or in incalcula l n r whether o this ttle p a et o othe , the b e i te n m s tellar spac es betwee the . 98 Proportiona l For m

a n n i hi s uni rsali ra i n na ural laws H vi g ot ced t ve ty of the ope t o of t ,

it is but a step to c omprehend that we m ay expect to ﬁnd that even

an rc s hi c rn a a uni the laws them selves d the fo e w h they gove , h ve ty

m la ins ruc ur and a ra i and s un of pur pose and a si i ri ty t t e , th t g v ty o d

and l fo r am l a m r inc mm n anat ﬁr s cr di ight , ex p e , h ve o e o o th t we e t

is ui al n s a in a r are uni rsal them . To prove this eq v e t to t t g th t the e ve

a s hi a m u d o i au sinc a alr a o b l w w ch h ve c h to w th be ty , e we h ve e dy

served that all beauty arises ina sense-appeal; and this is possible only

h r u s r i ns hi c as h as ns a n o n rm t o gh tho e eac t o w h , bee t ted , depe d fo

and m i n la r includin allvi ra r s nsa i ns suc as c l ur ot o , the tte g b to y e t o h o o

n T H ra i who r s li at a a -l l n un . u u r a d so d o the o t os p e fe de d eve , eve

venur in in unc r tainr a lm s s cula i n Haml s an s as t g to the e e of pe t o , et t d

in m r hin s in a nand n a r . r are ar a p ophet The e , deed , o e t g he ve e th th

r am in ir il s are d e t of the ph o ophy .

I n l n im es im s r a s as l n as as in o de t , t e pe h p o g p t Egypt , the

tellectuals r als i ri s s ur ns aine , who we e o the h gh p e t , the s geo , the p t rs , — and r i rs a n l lan s ar h ili nl fo r m ans the w te of th t ob e d , e c ed d ge t y e of

pr oducing the brilliant and haunting colours needed fo r their work ; and

prominent am ong these r esearches c am e their effort to produc e the

eﬂec tof her aldic gold o r pur ple without extravagant use of the virgin

m etal itself . Gr eat ingenui ty w as exer cised in the c om bination of

r ucin all s and su s i ue s unil atlas in n r a a ed g oy b t t t t t , the dege e te d ys of

’ ar k s alc m is s anto r am a il s r s s n the D Age , he t beg d e of ph o ophe to e

Proportional Form harm ony of these forces and the subjec tion of the laws governing

iﬁ ur s l as s n each gr oup to the un ed p po e of the who e, how by the possibility of this transmutationwhich goes o nevery day

i u s n in anun arrant am un im o n ail let W tho t pe d g w ed o t of t e det ,

ru r un am nal la s r avi a i n us ac ce pt the t th of the th ee f d e t w of g t t o , s un and li r la i in nsi and eﬂec tiveness ac o d , ght , e t ve to the te ty of e h , and I cannot state these m ore tersely thanh as beendone previous

G ravity : The for c e of gravity varies inversely as the squar e of the distanc e thr ough which it is exercised

Sound : The inte nsity of sound varies inversely as the square of the distance through which it m ust pass .

Light : The intensity of light varies inversely as the squar e of

the distance fr om the luminous body .

That all of these should dim inish pr ogr essively as the effec t r ecorded is m ore and m ore distant fr om the origi nating cause is what we should expec t and what o ur daily obser vationteac hes us unco n

io usl but a ac r r i n is iminui ns ul sc y, th t the ex t p opo t o of th d t o ho d be

rn b sam rul in r cas is sa l as a so m gove ed v the e e eve y e , to y the e t , e what star tling r evelationo f the unity of this c ontr ol which Nature exercises over everything and a singular c onﬁrm ation of the claim

a arm n her la s is no s ci us r but a ar ac th t the h o y of w pe o theo y h d f t , worthy of study Her e we have thr ee laws governing those forces

3 ' Nature : Harmonic Unit a e 1 8 . y, p g 5 The TrigoninForce 1 01

ic are m s c mm n o ur k n l and s r la s are wh h o t o o to ow edge , the e th ee w no t m r l sim ilar ut are absolutel identical are no t nl e e y , b y . They o y i n i ical but ar e m r r s ri l m ri in ir r a o n. de t , they , o eove , t ct y geo et c the ope t

are no t nl i nical and m ri ut are ur rm r They o y de t geo et c , b they , f the o e , indi sputably subjec t to those fac tor s whi ch ar e the c onstruc tive bas e

ra n am il sinc eir l m nal r la i n aui n of the tet go f y , e th e e e t e t o , g g g the in nsi is anc is s te ty by the d t e, the quare .

Fo r ur s c ular m ns ra i n le t us ins c la the p po e of o de o t t o , pe t p te

0 ic a r are fo r r s m akin cl ar ac l 4 , wh h I h ve p ep d the pupo e of g e ex t y how this tri plic ate rule work s thr ough every second of time and will

is c ontinue to work until eternity sets its seal upon all things . It hardly necessar y to state that the rule inquestionis no tlimite d inits

ru r r ssi in s in illus ra i n no r r t th by the p og e ve po t the t t o , by the th ee ins anc s c s n but is uall ru inits r a i nan r and t e ho e , eq y t e Ope t o ywhe e

r r r is anc vi si l nl m icr sc and eve ywhe e , ove d t es b e o y thr ough the o ope

r s a s un n i r la i ns i ove p c e c o c e ved by the hum an br ai n. The e t o h p

tw n lum s un li and ra i and ir re uc i n be ee the vo e of o d , ght , g v ty , the d t o as is anc is incr as is r l i a s lu acc urac the d t e e ed , howeve , to d w th b o te y

ia r am and thi s relati oni s i nvaria bl one re resented b the by the d g , y p y

uare in hi i n a i a i n s . s la su s a a r ac r q If t p te , we ppo e th t the tt t o of g v t t o be exercise d by any body of m atter o r that sound o r light be gener ated

an s urc ﬁrs r a di s anc r r s ne lin in by y o e , t , ove t e ep e e t d by the e AD the n e dix No te XV. App , ro z Proportional Form

t ir circl ﬁ o ne and a di s anc nincr as unil h d e of gur e , th t the t e be the e ed t

PLATE 40 FORCE S c o uPAE ED A 31111130c D1AGRA11 rLLUS TRATm o TE E PARALLEL B ETWEEN TE E E S IG T S ND AND o nAvrrATmN i x P P TI N FORC OF L H , OU , RO OR O TO THE DIS TANCE VE R W IC T E A I D O H H H Y RE E XE RC S E . P ROM The Great Module s it uals and ﬁnall a ainincr ase n k n si eq BE y g e d to CF , the we ow po tivel a in nsi o f ra i s un o r li will im inis as y th t the te ty the g v ty , o d , ght d h

Proportional Form is so co ns rut t a li and its elati s l ur and orm t c ed h t ght corr ve , co o f , m ay be s u o t l il two lat m t s ns an h t u vo untar y . The ter co e o the e e of m o nly

r m t nin i li era l f o the dir ec io wh ch the eye is de b te y cast , while so und assails hi r ir n will o no un m f om every d ectio and whether he r . Of so d

(as of light and gravity) we already k now th atthe intensity varies as

s uar is anc r m caus th e eﬁ ect and ln the q e of the d t e f o the e to , the

nsi s un as itr s r and is n s al n te ty of o d eache the ea heard, depe d so o the

ri inal l un ss am li u o r a -wid ic cann ll o g o d e ( p t de w ve th) , wh h we ot we s im i n in n t th e ar in s a . i r ac r t i p e the t e to ve t g te S ce the othe f to of e s y,

is anc r u ic a mus r a l h as alr a nco n d t e th o gh wh h the w ve t t ve , e dy bee sidered ino ur s m lic a ula i n a r uc o ne th e ur y bo t b t o , we h ve ed ed by fo vital subdivisions which pr esent them selves to the mind inc onsidering

u i n s n nsi uli i c and in n s un i a m a i . the q e t o of o d ; te ty , q ty , p t h , co b t o

us i n uali m r o r is o ne hic uall in in The q e t o of q ty o e ve w h , eq y terest g

u it n r l ss r s ns r ﬁnm ns inr s arc hi c are tho gh be , eve the e p e e t e e e t e e h w h li l a a su a rk as o ne in an o r ns nan wi i tt e d pted to ch wo the h d , c o o t th ts nc ssar r vi and all r nﬁn o ur li l e e y b e ty , we sh ther efo e c o e tt e hour to the r m ainin us i ns i c and c m ina i n o r arm n as un r e g q e t o of p t h o b t o , h o y , de s musi i tood by c ans .

’ ’ n lack smi s idlin in r n l- r i s Whe the b th boy , g f o t of the whee w ght

us sh O irl a a n l riskl ar un and thr um ed b y p, wh ed w go whee b y o d the s k s i a ic r s ic inhis an r m m r a n ic po e w th h ko y t k h d, we e e be th t we ot ed a n r f n l n l l was a um o hin s . s r sul be t g Whe the whee we t ow y, the e t The TrigoninForce 1 05

m r n ir lin r ack ic was invi e d to s at c . e e et, wh h the boy t top o e Wh g the

l as r and as t r irri a in n is cam a musical n whee f te f e , the t t g o e be e ote ,

the pitc h of which depended o nthe size o f the fello m and spok es and

r a i it n a a r an r al the p d y of m otio . If we g ve the m tte y thought , we e ised that these c ondi tions producing pitch wer e capable of four com

o site r sul s s un m i i r l u and i l u and lo w p e t ; the o d ght e the be o d h gh , o d , s and i o r s and low ndin o n r l s hic oft h gh , oft , depe g whethe the b ow w h

r s un -im uls r r n an as s r n and sl a we e the o d p e we e st o g d f t , t o g ow, we k and as o r ak Le t s nk c ns anl inmin a and sl . u f t , we ow the eep o t t y d th t

in nsi s un ic is as a sai m asur d the te ty of the o d wh h , we h ve d , e e by the

a -wi o r am li u n s o n s r n l o r w ve dth p t de, depe d the t e gth of the b ow

im ulse il i c n s nir l u nth e i ra i n. p , wh e the p t h depe d e t e y po Speed of v b t o

It is well also to rec ognise the fac t that scientiﬁ c sounds exist co n tinually to whi ch the hum ane ar is no t attuned and whi c h pass us by as ul ra-vi l inr -r l ur s ic ui in isi l do the t o et and f a ed c o o wave , wh h , q te v b e

um an rk ino ur la r a ri s ail akin X-ra to the h eye, wo bo to e d y , t g y photo gr aphs o uplates sensitised to se e what escapes the vi sionof m ank ind .

will no t sufﬁc in a rk r r i ns in au It e , wo devoted to p opo t o be ty ,

r a s un as m r n is ar e s u s un r m s an to t e t o d e e o e . If we to t dy o d f o the t d point o f beauty we c annot esc ape the subjec t of music as a gener al

asis sinc a is r a s un -art and hil it ul ri b , e th t the g e t o d , w e wo d be t te to say that the great m ajority of people would decry the ver y idea that the choic e of a m usic al sc ale o r the selec tionof c hor ds ina c omposition 1 06 Proportional For m

r rn an m a m a ical r ul s and ul we e gove ed by y the t e , would sto t y m ain

ain a m a m a ics ul a ar is ic s ns if o n t th t the t wo d be the de th of t t e e , yet e goes into the m atter fundam entally it is qui ckly seen that every ac cepted law of music al c omposition and the very struc tur e of the diat nic scal i s lf is as o nrul s ic ui nsci usl or un o e t e b ed e wh h g de , c o o y l nsci usl r acc a r uc i n. m s i n r an and c o o y, eve y ept b e p od t o The o t g o t illi ra m a l arn i aid a and s nsi i te te booby y e , w th the of good eye e t ve

uc la a r s na l am o f illiar s n v r in l to h , to p y p e e t b e g e b d ; yet e e the who e

hi ra i ll s i c ourse of s p c t ce wi he m ake a good hot unsc entiﬁc ally. If the

a l ru alls r un cus in r r c in as an t b e be t e , the b o d , the e o de , suh a th g

l i a ac ci na s is im ss l c anc in r ac icall lim in e . de t hot po b e , h e be g p t y e t d

nl l m n o f a ci n li s ll i in la r fo r if The o y e e e t c de t e who y w th the p ye , the s rin s alls r s ul it is caus c rr c hot b g the b whe e they ho d be , be e the o e t

o r ir c i n and n lis r in s r k no m a r p we , d e t o , E g h we e the t o e , tte how

l i ul la awk ward it m ay seeming y have been deliver ed . A beaut f p y m us r unc nsci usl la i m a m a ic al e rfe c t , howeve o o y, be p yed w th the t p

i l it n mi I nm usi lik ise n it as t n s ul a a ss. c o , e e wo d h ve bee ew , de y

m a in r r uc r sul sir are c m ll we y, yet o der to p od e the e t de ed , we o pe ed to

a a i d i niﬁc t r ill o r no and c o nse be m them t c al an sc e t whe he we w , quently the earlier we undertak e the pr oper study of the science of

u L t see if hi s m r s l s all ac i s c ce ss. e us sound , the o e ur e y h we h eve t

illus r a c annot be t ted .

Were we suddenly borninto a new world wher e music was un

1 08 Pr opor tional For m one-s rin d lu or E lian ar sin n in th e r e ze and a t r t ge te o h p , gi g b e ; f e muc x rim n and ar s of s u un a l w m usi h e pe e t ye t dy , he en ciated a of ca l pitc h which we canillustrate by referring to plate 41 and which law h as n er ev beendisproved .

ivi in his s rin in al s as at inﬁ ur two la 1 D d g t g to h ve , C g e of p te 4

lacin a ri un r is n u by p g b dge de th ode , he found that the halves pr od ced

sam n and a s un a r a l r and ar the e ote , th t they o ded g ee b y to gethe , p ticularly agr eeable whenstruck at the sam e tim e with a second str ing as inﬁ ur o ne hi c a sam n s rin un g e w h g ve the e to e as the whole t g , i i e . llmi s in s ru sinc n s o i di vi d d v d d We ght the e th g be t e , e the e d the de s rin a in ual r a i i of vi ra i n s un d n t s inunis n t g, h v g eq p d ty b t o , o de o e o , while the ends of the halved str ing gave out notes exactly anoctave

i er an a l r s al n li l la r we h gh th th t of the who e c o d ounded o e . A tt e te shall try to see if ther e were no scientiﬁc reasonwhy these things wer e so but is m a saf l r s a m m n sinc a r as k n n in , th y e y e t o e t , e Pyth go ew oth g

r as ns m r l r c nsin acts and k in anim a i n of the e o , e e y e og i g the f , eep g p t e t world o n its tiptoes in anticipation all these centuri es to ﬁnd o ut

r was ssin ark or r o n nr ar re whethe he gue g inthe d whethe , the c o t y , the s l n P t a houd at last develop a reason. Returning for a m om e t to y h

ras a r m ni usl ﬁnd him n a in go whom we left r ther unce e o o y , we e g ged dividing his cor d into two such portions as that the longer should be

ic s r r as inﬁ ur r la at di isi nat . tw e the ho te , g e th ee of the p te , the v o E

un in s ns r clar a ains urc s l as ur . So d g the e , the to e we e , he de ed , g o e of p e e The TrigoninFor ce 1 09

And a ain ll mi sa so sinc as c m ar i n g , we ght he y , e , o p ed w th the ote of n l n r s r r e d s un c a . i i in a ai n so the o ge , the ho te o ded the o t ve D v d g g ,

n s s ul a r la i nof two r as at that the two e d ho d h ve the e t o to th ee , the

inﬁ ur ﬁve s un s uall a r a l r s no w point M g e , the o d , eq y g ee b e , we e tho e we

P ro . 1 P ro . 2 F10 . 3 P ro . 4 P ro . 5 P ro . 1 P ro 2

PLATE 4 1 PLATE 42 P YTHAGOREAN LAW S OVE RTONE VI B RATIONS Illustrated by Experim e nts o f Meld e and Tyndall

Af r r c nis as ni c and minan . muc suc r im n e og e the to do t te h h expe e t ,

Pythagor as deduced his rule whic h the world ac cepted long inadvanc e

an na r as n as u n a sci niﬁc assi l r . rul of y g b e e o b ed po e t theo y The e , r uc sim lici l s a sim l r are ra i s ed ed to p ty , ho d th t The p e the t o of the two s m ns in ic a c r is i i m r r c is eg e t to wh h o d d v ded , the o e pe fe t the A m n t s r uc . nd itwas An u ar n r u . d c a s all h o y of the o e p od ed t e be e , 1 1 0 Propor tional Form un k n n a r as a r as n is d fo r it so it r m ai ns as be ow to Pyth go , e o ex te , e

ru t a as itwas n rl was un t e od y whe the wo d yo g .

Countless generations passed befor e anyone di sc over ed the vari ous

s ic na l us uss at a r n n fac t wh h e b e to g e easo why he was right . Scie ce

nuall e l a ri s a un fo r s and eve t y d ve oped the w ve theo e to c c o t ound,

ﬁnally accepted a similar the ory in regar d to light and the wor ld advance d to the point that so m any theori es h ad be ensur rounded by fa s ar and c l a a num r s c ul r aised r m ir c t , h d o d , th t be of the e o d be f o the

lassiﬁc a i n m r e rie s th e uad erat dem o nstrandum c t o of e theo , q of

and ﬁl inthe ar l w u r ase e c i e s a imm ua le . do bt e d , d h v of t b

It is ge ner ally easie r to ac c e pt a statem ent pr ove d to the e ye

aninan e r wa and th e e m nstr a i ns r ar in e se s un th y oth y, d o t o eg d g th o d

a s as cr a e c o r s m i h ad we im e e aui ull w ve e t d by d ght , the t , be b t f y s n th e e er im e ns e l e as di a r am m e in la 1 how by xp t of M d , g d p te 4 to

n n w it il a r a na ic we a e e e r rri . e s c r m i t i wh h h v b efe g A h k o d , de to v b e

r li ill s un as a le until th e e nsi n at end is di ec t ght , w o d who t o the

cie nl incre as e n wi u c e asin i r a e it ill di i sufﬁ t y ed wh , tho t g to v b t , w v de i se m n s in as s ninﬁ ur e nin r as in ts g e t to two , how g e two , th to th ee ,

ﬁ r r and so o nunil r a s n s m n s are s un in gue th ee , t pe h p twe ty eg e t o d g

s at n r m sam strin and all isi l . their note o c e f o the e g , v b e to the eye

im s in m ainc r c an m a c ninu its own Somet e , deed , the o d be de to o t e

a n as a l s un in ni c il at th e sam im it vibr tio who e , o d g the to , wh e , e t e

a s ar l in c a s m n s as illus r a in lat 2 will vibr te ep ate y o t ve eg e t , t ted p e 4 ,

1 1 2 Pr oportional Form

we ﬁnd h a nic and c a ic t a ras ain t t the to o t ve , wh h Py h go obt ed by

divi in his c rd in s m ns suc h a o ne was wi c r d g o to eg e t h t t the t e the othe ,

wer e pr oduce d by vibrations exac tly twice as fast in th e o ne c ase as

r a in ra i ic ac sci nc s s for inth e othe , th t be g the t o wh h ex t e e et any two n tes ic are c a es . a ras u o u ill r ec ll c a no wh h o t v Pyth go fo d , y w o e t , th t

i l s ra i s r duc m s a r e a l arm ni s an the s mp e t t o p o ed the o t g e b e h o e , d here ,

ﬁrs r i n ﬁnd a sim les r a i at the t expe me t, we th t the p t t o pr oduc es the

o ne inwhi ch the vibrations and their co rresponding nodes or quiet

so ime as c inci m s r unl ac l as zones ar e t d to o de o t f eq e t y, ex t y we h ad

n a ri m r ra hi m a sa s . i c in r r s pr esuppo ed Be g t ﬂe o e g p , I y y othe wo d ,

that every sec ond sound wave of the upper note of these two was

exac tly tim ed to and co incided absolutely with a vibration of the

and is ill rec ni se as a m a r no sm allm m n n lower , th w be og d tte of o e t whe

. ’ r n d a inana ra c as suc as and its c a itis comp ehe de th t ve ge e , h C o t ve ,

the enorm ous number of ﬁve hundr e d and twenty-eight vibrations

nci in ever second and a in r sec n sam would coi de y , th t eve y o d , the e

or ui lac s ul co inci as ll l n s . u r and number of ode q et p e wo d de , we E e

Helm holtz both have advanced highly scientiﬁc and c om plex ex

planations fo r the undeniable fac t that c oncurrent oc taves are m or e

l ear an fo r am l co ncurr n s n s in hi c agree ab e to the th , ex p e , e t eve th w h nearly anequal num ber of oppose d vibr ations oc cur ; but it would seem that no great surprise is c aused in assuming the logical at titude that the absence of something over ﬁve hundr ed clashing PLATE 43

HARI ONY I AD E vrsmLE

ak en ro m The GreatModules Sym bo lic tabulationo f vibratio n. T f 1 1 3 1 1 4 Proportio nal Fo r m vibr ations of sound per sec ond might do much toward making the

a r l n is l t s to c ana a s u . t s r e u roc ee o t ve g ee b e o d To e t th , howeve , p d

’ P a r as s n e rim n r see a i isi n his yth go sec o d xpe e t , whe e we th t by d v o of r a i s in 2 s r r s rin mi s o minan inr la i n t o to 3 , the ho te t g e t the d t e t o to the

n r i n ﬁc ins c i n n r s ns us i to ic of the longe . Sc e ti pe t o the p e e t w th the fac t that the c oincidenc e of vibr ationand quiet inthis case is o ne in

r and a ain se e l i al r as nfo r ac c a ili th ee , g we a og c e o the ept b ty of the s un s o d inharm ony .

Proc eeding o nthis planwe ﬁnd that we have quickly c onstruc ted

! what is k no wn as the c omm onc hord which is the basis of all har m n and in ic i ra i ns and ac ul n s are c ncurr n o y , wh h the v b t o pe ef ode o e t

with suc h fr equency as to sm ooth away all di ssonanc e . Thi s chord

nis a r e a l no t c ause c se in itso and no t c aus the g e b e , be we hoo to th k , be e

music ians sa a it s ul so and no t c ause we a good y th t ho d be , be h ve

ar it all o ur li s but sim l and nl c aus it is as o n he d ve , p y o y be e b ed the c onstruc tive fac tors which m ake itso whether o ne listens with pleas ure o r is o ne of the unfortunately m usic ally deaf persons whom we

c asi nall o c o y m eet .

’ Now at k l sin a m m n s im le t us amin s , the ri s of o g o e t t e , ex e the e m usic al notes whic h evidently are the c hosenof the civilised world fo r s m o and s n r as n s i all c us m ar o b ec o e g od ufﬁ cie t e o , p te of the to y j tions to the m ingling o f reason and m athem atic s with inspir ational su c s an see h ow ir r la i ns i ac r ar o ut bje t , d the e t o w th e h othe be the r 1 6 Proportional For m t r hic a ro un a all s em ns ra i ns heo y w h I h ve p po ded , th t of the e d o t t o of force are geo m etrical intheir ac tio n and that ina large m ajority of cases their dem onstr ationwill disclose that th eir basis is the tetragon nl n l n am il wi a m i ar . a alr a s a f y , th the hex go oo g ge We h ve e d y e e wh t part the hexagonplays inthe acc urate m easurem ent of the planetar y

k n i n o n n l s le t system which we ow s fouded the gravitatio al aw . Now us m ak e anexperim ent o ntha t sec tionof sound which we call m usic

the part of sound which we selec t as being The resul is s nin la r in ﬁnd n t s o f di a nic t how p te 44 , whe e we the o e the to scale depicted with their vibratory relations in a diagram of pro gressive hexagons and whic h is r epeated her e fr om a form er work

n s nin As in as as being ece sary to co t uity of the study . the c e of the

r n mi l in i a n rr t ul ll ast o o ca chart wh ch the hex go occu ed , i wo d be we to c aution th e reader that the illustrationhere introduced is intended

nl as anillus ra i n ac ual c al ula i n m a lo arith o y t t o , the t c t o be ing de by g mi rian ula i n and c rr c a in inﬁni l n ssi l c t g t o , o e t to po t te y beyo d the po b e

a a i a s a r and a nil c rr in e e n c p c ty of heet of p pe pe c , o ect de d b yo d the

a icr sc to is l s use of m o ope d c o e .

The interesting m athem atical ac c ur acy with which these state m ents are true c anonly be disclosed by recourse to the d r y pr oc esses of calculationwhic h probably would no tinter est the r eader butwhic h is demonstrable nevertheless ; butif fr om what I have said ith as bee n

S ee endix No te V . App , X II The TrigoninForce 1 I 7 m ade clear that the laws of pitch are geometri c and that they are persistently similar to those whic h governoutward form under the

in unc r avi m c ill a na ain . ﬂ e e of g ty , y obje t w h ve bee tt ed

is no t r im ssi l ca o le a ur in m ak in s m It , howeve , po b e to j N t e to g o e of her o wndiagram s o nthe interesting subjec ts which we have under

r e r va i n and it ul s m a if s un a s can o u obs t o , wo d ee th t o d w ve thus be indu wri ir s r so a itis isi l sur l ced to te the to y th t v b e to the eye , e y the

l - l F is s w ar n i l ll ni unans r a . or ur e ill gum e t w be we gh we b e th p po e , w

r ak in la s r al la ni i m nts suc as r unde t e p te 45 eve of the Ch d exper e , h we e

nin r a ari n all lam in lass la s in show g e t v ety by Ty d , by c p g g p te the

a n n n r and sca rin ﬁne san r sur c . s u a s c e t e tte g d ove the f e If the , o d w ve

se t inm i n dr awin a i lin li l acr ss be ot o by g v o bow ght y o the edge , the grains Of sand will be se tinto violent m otionand itwill be found that they wi ll dance gaily over the surfac e and ﬁnd a resting place o nthose l portions of the plate where the vibrationis east severe . The ter

a n all a ac i n ill ﬁnd la a rn wi san min tio of w ve t o w the p te p tte ed th d,

in c r and nral s m ns r ac i nw the nodes be g ove ed the ve t eg e t , whe e t o as

n s rm m a ari ninuus in ar . a r u c o t o , be g b e The p tte th fo ed y be v ed by the experi menter atwill by dam ping o ne o r m o r e spots o nthe edge of

la us c ausin in r v nin n s ic ill r a m the p te , th g te e g ode , wh h w epe t the selves geometric ally thr ough the c ourse of the vibration at other

. rr la i ins so a a r si i n am c o e t ve po t , th t , wh teve be the po t o of the d ped n i a ram san at r s ill in aria l un ode , the d g of the d e t w v b y be fo d per Propor tional Form

ectl e m r wi r i lin s d ur s at r c l alanc d f y g o et ic , th st a ght e an c ve pe fe t y b e

the inﬂuenc r e of fo ce .

' c rrumm s ExP ERa ETs

be claim ed with seem ing justice that by select ing a squar e plate the exper imenter h ad to som e extent forc ed Natur e to c hoose tetr agonal and oc tagonal form s fo r her patterns : butwhat will be the answer whenitis shownthat these sam e c onditions persist

al r tinac i n n late s r un as s ownin with equ pe ty eve whe the p be o d, h

d ur t er a u l s inr un lass ess ls m a plate 46 : an f h , th t the b bb e o d g v e y be

in sam wa rmin a rns for m ade to vibrate the e y, fo g p tte which this

1 2 0 Proportional Form

I m ay add that if ther e should be any lingering fee ling that th ese so und waves do no t set up anac tual physical m otioninthe atom s

alls ll a im n is o ssi l ns in er . c o titut g the w of the be , further exp e t p b e

a l r lass u su in unc so un a s If s ende g t be be bj ected to the ﬂ e e of d w ve , itm ay be shivered into exac t fragm ents atthe instant that the pitc h

PLATE 47

vranAnONs o r A EELL

r ac s n u and if r echnics r sira l a e he the ote of the t be , py ot we e de b e , c annonmight easily be shot o ff by the additionof a sim ple m ec hanism

ra a r rl a un n in n w o un s a . rec Ope ted by p pe y tt ed o d w ve I deed , e t ar

a s a e n arin an n a d y , we h ve be he g y um ber of cc ounts of the explosions

us s m a ic all caus if r r ru at is an ins th y p thet y ed , epo t be t e , d t t po t , by

e r rl a l n s r m a a i the us of p ope y keyed w ve e gth f o est blished st t ons . The TrigoninForce 1 2 1

LIG HT

If we shall spar e less spac e inthe tr eatm ent of the subjec t of light

anh n n in am inin s un it is m r l caus th as bee expe ded ex g o d , e e y be e the subjec ts are inm any ways so ak inthat what h as beensai d inregard t un it to o n n no t r a r ar in r . m u the e eed be epe ed eg d g the othe So d , st

un rs ﬁlls a m s ric uni rs and as h as e n sai be de tood , the t o phe ve e be d , is no tc onﬁned to those waves which he atwithinthe limited c apacity

n I n sam wa l ur and li e is r um a c ar . c n of the h the e y, o o ght x t whe e o

um an c s m r ina in a un nl se h eye dete t the , the et be g tt ed o y to tho

i all i in s c rum in vi si l nl as w e n wh ch f w th the pe t , be g b e o y bet e the

urnin a ain his u n lowest r eds and the highest violets . T g g to t q estio of r i i it is no t i in r s n t a u s ap d ty , w thout te e t to o e th t , j dged by the peed

ra n l am u isi ili is c nain i in of vib tio , the who e g t of v b ty o t ed w th the

no t se is r ssi nas a c ompass of o ne colour octave . I do u th exp e o mer e

an ul o ne but as n in in ar and sci niﬁc c alcula i n f cif , de ot g h d e t t o , the vi r a r r la i n s r s vi si l r a i l l n s b to y e t o of the ho te t b e y, the v o et , to the o ge t — o ne red a r la i n ic is ac l u l ins . b r , the , e t o wh h ex t y do b e peed We ow

a if a r a c m as r and s r er an an c an see th t y be o e f te ho t th y we , we r ec ognise it by scientiﬁc experi m ent as ultr a-vi olet we know it

is s w e it ail a l it urnus a s nits m ex t , e us d y , we h ve fe t b , we h ve ee ark o n sil ra i c la s but no o ne h as r se enit nor ver ed photog ph p te , eve , by

l o r inr - ir r a i n ill it r s n. s a red ra d ec t obse v t o , w eve be ee The ow f ys, 1 22 Pr oportional Form we admi als sur r us n n n t , o ound a d have their c onstant inﬂue ce o vegetation and ser ve the daily needs of Dame Nature per haps as usefully as the rays uponwhic h we count for o ur groping way through

life .

I have atnumer ous points called attentionto the fact that the

r vid nces ra i s un and li are ra nal inso reco ded e e of g v ty , o d , ght tet go m an cas s as r u all as nial and if rmi o ne m r y e to ef te h ty de , I be pe tted o e illustra i nof su ec it m a r a s o ssi l ut it all t o the bj t y, pe h p , be p b e to p

n r o ne r as sa in Le s r r as ude oof , the y g goes. tu eve t to o ur c e of the

and see a c an a r m a li nin and bell, wh t be g thered f o w tc hing , ste g ,

r a i n n in . vi ns t a a h a ot g The b t o , we o e , r e c using the very s pe of the

ll i s l c an in a ir c r la i n a s i its be t e f to h ge d e t e t o to the squre, de p te

i w kn r m ll ill s circular form . Th s e ow f o a authorities and from the u

n a he ll is tratio already ex mined . The swinging of the wheel and the

n n alt r u n ra i wi u i m i depe de t ogethe po g v ty , tho t wh ch they ght be

wa t n n ull one bu ul r c m ack . ravi we a s p ed y wo d eve o e b G ty , h ve ee ,

Operates o nthe bell ina stri c t acc ordanc e with the law of the square

lis n s un s r uc ic are carri o ur c ars We te to the o d p od ed , wh h ed to by the

i ra i n l lik is ra n l n i als v b t o al aws ew e tet go a i the r entir e m ak e up. We o watch the operationof ringing o r tolling by vi sionbr ought to us by

i a s c nr ll as are all r m n ni n the l ght w ve , o t o ed , of the othe ea s me t o ed , i w n in by the laws wh ch e have bee study g . Does all of this unity c onvey no c onclusionto the mind ?

1 24 Proportional Form

n usi l to n o nco m ina io ns in i i r at r co cl ve y , depe d b t wh ch the v b o y waves

n s are m s co nso nan and ha s in o mm n so c l r and ode o t t ve the mo t c o , o ou

n n o na ar n n n a ha har m o y depe ds simil c o so a c e . We h ve found t t in

a r a l c r s r rm n te s i ra in at co mm n so und , g ee b e ho d we e fo ed of o v b t g o

and an xamina i n ese ill i s ac a in periods, e t o of th w d sclo e the f t th t

- r ni n r alu a m c s never ne ar nei hbo urs . c a wave v e , h o ote we e g The o t ve ,

in nir s un am u m a em a icall r s unds so represent g the e t e o d g t th t y , e o

agreeably that whole chor uses ar e M po se d to be perform ed inuni n n o r c a es . n m s a ree a l com ina i s ir and so , o t v The ext o t g b e b t o , th d

ﬁ and nic are s ac narl uall in vi rat r tonic and fth to , p ed e y eq y the b o y

esiri n a com ina i n r u iss nan s no t scale ; and , d g b t o of o gh d o ce , who doe

tu n inin s en and c a o r minan and at once hi po jo g the ev th o t ve , the do t

- nan as m in rst a - ar r se ns ? sub domi t , e body g the wo th t the key bo d p e t

r m n or The same is precisely tr ue of c olour . The ag eeable co mo ch d of light is co m pose d of those c olour s nearly equally spaced inthe spec

and s are un arm nic hil se ich ar e nar s trum , the e fo d h o , w e tho wh the e e t

i ra i n suc as two s a s re n o r r la neighbours inv b t o , h h de of g e , two e ted

r red and ink are ulic c ns n im ssi l o r red s, o p , , by p b o e t , voted po b e worse

I might go o nindeﬁnitely with illustrations tak enfr om the grea t

ut en u h as n l e o ne ul hin s fr ee book , b o gh bee deve op d , wo d t k , to how

a u m ni la s rn beyo nd per adventur e of do bt , the ho oge e ty of the w gove

endix No te VII . App , X I The TrigoninForce 1 25 in m a r i e nr llin r in i a rsis nc g tte w th thos co t o g fo c e , and to d c te the pe te e , to sa l as wi ic a ur a s r s ra n y the e t , th wh h N t e dopt the fo m of the tet go

family whenshe wi shes to expr ess herse lf through the actionof these

invisible agents .

I nconcluding the argum ents form ed along the lines of this chap

ter ul ut a few us i r a r fo r hi s s l , I wo d p pointed q e t ons to the e de e f

searching .

n a c r ain ari i s a l lif rs and We k ow th t e t v et e of veget b e e , ﬂowe the

lik ar i n rs an e a i uall cl in r n. s m ans u e , h b t y othed g ee Th e , we de t d ,

that they are thus furnished with anoutside sheath o r husk o r bark

o r skin hi a a d r ts s rs , w c h bsorbs c ertain light r ys an ejec tho e othe

i c c m ur ill a o u c m lim n wh h o e to o ur eyes as colo . I w p y y the o p e t , r a r su sin a o u no t li a is is all acci n e de , of ppo g th t y do be eve th t th de t ,

no r a ach c as is an c i n but a Na ur h as a ur o s init th t e e ex ept o , th t t e p p e

all. Am ri ? mi in n a ur s o u li a I ght Ad tt g , the , p po e , do y be eve th t the

ur s iff rs a im o r is s adfas an al a s sam and p po e d e e ch t e , te t d w y the e ;

a l nl t n s and th t the co our of the ﬂower is fr eque t y the result of i s eed , , so s ak r scr i his r a c r ? ranin a is is to pe , p e bed by t g e t do to G t g th t th

done for a pur pose and that the pur pose is steadfast and h as rule and

law in it o u li a itis sam inall cas s r beh d , do y be eve th t the e e whe e the

ac ts are sam ? a ran and if we a mi as admi f the e If th t be g ted , d t , t we

m us a c l ur is a ur m a r m a m a ics o r i r a r s t , th t o o p e tte of the t , v b to y peed

and l n o u li a m a m a ic al rul s o r r a s e gth , do y be eve th t the t e ( pe h p I 1 26 Pr oportiona l For m should say rules which we vi sualise as m athem aties) apply to co lour and gravity but no t to forms and shapes? Or if these form s are c on

r ll m a m a i al rul s a ese s a s are assum as t o ed by the t c e , th t th h pe ed the r esult of either acc ident or wi thout a pur pose ? A well-knownwr iter has recently said that no pur pose c ould be ascri bed to Natur e fr om h er form s no r c ould it be thought that she h ad any intention as

r nﬁ r gathered from h e use of o ne c o gurationas against anothe . Do

o u li an in so sc an al us h er ? I n r r s o u y be eve yth g d o of othe wo d , do y

li as ask d in inr u i n a r is a r as n fo r be eve, I e the t od ct o , th t the e e o mak ing certainlilies always white butno r easonfor m ak ing the sam e blo ssoms always six-c orner e d ? If yo ubelieve that ther e is reaso n in

the t vernm n col urs and m r eo r a rul s the ma m a ieel go e t of o , o ve , th t the e

sha e wh is no t h a rul uni orm also ? If o u eli v ha a ur p , y t t e f y b e e t t N t e

un am ns o f a l eau and at se r ul s ar e m at e ti al (the f d e t l b ty) th the e h ma c , as canscarc l ni o uthe n e lieve t at eau and m athe e y be de ed , do y b h b ty m atics and geometry and r easonc anlive to gether inpeace and amity?

n oub eheve a au can e is o r surv ive ex e t wi I deed , do y th t be ty x t c p th reas oninher favour ? We m ay no tknow at allwhy o r how we live and m and a o ur in but and how us r ove h ve be g , the why the m t ac co d n r l ss rul s o r l are as a as hir n eve the e to the e beho d , we de d the t ty ce t f s i n n i is n i f r n . mmi turie . au c o t W th be ty , I o te d d e e t Co tted to the

CHAPTER V

AS YMMETRY AND VARIETY

AN ANTI DOTE FOR S YMMETROPHOB I A

HE anci n ians and to s m x n ar is ic and e t Egypt , o e e te t , the t t

r a s no l ss anci n a ans r a in eir cr a i ns pe h p e e t J p e e , d e ded th e t o

what they felt to be the taint of symm etry as they dr eaded

nl l is no tn a al a re o y the p ague . It ec essary th t we should together g e with them inthis inorder to understand that o ne h as no tfar to go insuc a s u as urs r r alisi a um r su c s h t dy o , befo e e ng th t the n be of bje t

in r s ic ann n ni ali ann as il sur of te e t wh h c ot be c o ve t on sed , c ot e y be r ounded by polygons no r subjec ted to the geom etric al m easur em ents or anal s s suc as a so far s c far unum r s to y e h we h ve ket hed , o t be tho e

ich s mi r m ans wh ub t them selves readily to these methods . Othe e therefor e have to be devised for the examinationof these less m anage able subjec ts and we shall treat them brieﬂy inthis chapter both fo r

’ c onvenienc e sak e and bec ause m any varying classes m ay be found

l n o nins i n sam na ur al o r o m ric amil . to be o g, pec t o , to the e t ge et f y

The beauty of all of these form s h as m any tim es beenkeenly felt

Fro m The Gr atModules C . rthur 1 1 e Co an . , A , 9 4 1 28 Asym metry and Variety 1 29

n r a s it c ul sc arc l r s a m ar whe , pe h p , o d e y be exp es ed . Suc h vellous

br ai nas that of Le onar do c ould c onc eive the perfectly c orrec t idea

that the wing-tips of ﬂying birds form ed lines whic h were spir al in

ir c m l l and is s li s un ri s P the o p eted who e ; th tood , ke the o d theo e of y

h a o ras as a c armin and us l ain- il f r al t g , h g e ess br c h d o r seve hundred

’ ars unil in e nar s c as inm a ra c am r scu ye t , L o do e , the k e tog ph e to the e e and r a was r l a ur a i his surm is r p oved th t he pe fec t y c c te n e . Eve y

r a m a m a ician r r a ar is r r a arc i in g e t the t , eve y g e t t t , eve y g e t h tec t , the pastgenerations h as felt the ur ge o f these vari ed curves whic h are no t

ir l s and m an a i nin alua l im ir anal sis no r c c e ; y h ve g ve v b e t e to the y ,

sul s ack in m r a ac r i s a c ould the re t be p ed to a e e c h pter . Sp e fo b d

n l ss s r i us s udi s i nextenso . are r r eview of the e p ev o t e They , eve the e ,

Open to all of the serious minded : and I would guard against the thought that any attem pt h as beenm ade to c atalogue o r r ecite any c onsiderable portionof what h as beensaid c oncerning these beauties n l i of form and atte dant c o our wh c h await the searc her .

s m an rm s no n-circ ular ur s n hic ill I nthe e y fo of c ve , the , w h w be

to di i m s l s in ari us am ili s a in o ur lis obser ved v de the e ve to v o f e k to t ,

ﬁnd lli s s s m n s as o n i al an l c a nar we shall e p e , eg e t b ed the de g e , te y

s ra ic in ic s c c nric s and s irals wi u r ar ir c urve , g ph d e , e e t , p tho t eg d to the

s m m ric as sever al geom etric values o r their fam ily relations . A y et

r l e s all see ir m an aui s and ins m in they gene al y ar , we h the y be t e o e

anal s m u his ill no t stanc es we shall tak e the tim e to y e the , tho gh t w l s0 Propor tional For m

h in m s b e always be either pr ac ticable o r pr oﬁtable . If som et g ut

in if to im a ina i n so af r r r r un r m us le ft l e the g t o , , te p ope g o dwo k , t

so m ething be c onstantly left to the student .

Ther e is nece ssarily a c ertai n am ount of latitude inthe use o f

r s as s m m r and as mm r n if s m m r suc h te m y et y y et y , eve by y et y we H n m eananexac t r epetitionof o ne part by another . er e every elem e t

o n ne an ul a an ac co un r art o nth e r of for m the o h d wo d h ve ex t te p othe ,

a si air its win r rsed in r r ail and si i nas e c h de p ed by t , eve o de , det , po t o — if r ec e ina mirr r as lik as Twinnes Hi o c ra s ic eﬂ t d o , e the of pp te , wh h , l o u ill r c ll c r as i as tw o as ina o d . Disr ardin y w e o e t , we e ke pe e p eg g

r ainmin r iff r nc s u ar sem lanc um an r c e t o d e e e , the o tw d b e of the h fo m ,

am l is ci dl s m m ri c al. n ri r l o n c nr ar fo r ex p e, de de y y et I te o y , the o t y ,

t s r m uc rwi s i nl o ne ar t and a o no n si i i ve y h othe e , w th o y he th t e de ;

i s m r ans unc i nin in airs and s m i u a m a w th o e o g f t o g p o e w tho t te , an i n r s lik li l l c ri c ll ir s c lin in ﬁr s and d w th e ve , e tt e e e t be w e , g g t her e thenthere to c om plete the cir c uit r equi r ed inthe fulﬁlm ent of their

as mm rical in s in is rl r a l high offic e . The y et th g th wo d g e t y outnum ber

the sym m etri c al.

us no t r allin rr r o f su sin a c aus Let , howeve , f to the e o ppo g th t , be e

hin s as m m ri c al c ann r adil c n ni nalis are t g y et ot e y be o ve t o ed , they

n r aff r s us ther efor e without law a d order . Asym m et y o d m any pleas

f au ni m r s ai rm s s m m r and ure s o be ty de ed by the o e t d fo of y et y ,

urnis s us a ar i ic r sa s is r s ic li f he th t v ety wh h Cowpe y the ve y p e of fe,

1 32 Pr oportional For m

and a ac in in o si i n c o une r- rc es and res raints lines , th t , t g opp t o to t fo t ,

h b o o r uc cur ve lines . O nin er r a ﬁnd a would p od e d pe g g e t k , we th t

is insc ri wi ese resul s in cur ves and if Na ur every page bed th th t , t e

to a r s rai lin s nas sh e e s a ac uum it is m r l se ems bho t ght e , eve do v , e e y be cause the forc e of gravity so seldom ﬁnds sc ope fo r dem onstrations

uninﬂuence d by other causes .

’ e w ns a l allin dir c l r m re r un N to pp e , f g e t y f o the t e to the g o d ,

dr a a s rai lin r m ranc so d b ut seem ed to w t ght e f o b h to , car eful examinationwould unerr ingly prove that the diminutionof eastwar d

i n ar and r c m lic a i ns nc l m ot o of the e th othe o p t o , o e the app e was

l h ad im ar t a ara lic wi s ic sli as it l re eased , p ed p bo t t wh h , ght s ho d

in inﬁnit sim al im n fo r all nl c ould be the e t e eeded the f , yet c er tai y it deﬂec ted the spher e from that perfec t ver tical whic h it see m ed to

‘ descri be .

No o ne who h as fo r a m om ent glanc ed at a c hainca ught at both e nds and swinging its loose bight inthe sunshine eve r failed to note the grac eful c ur ves fo rm ed inever y m otion. These c atenary c urves

am n r s am l s o f in unc o f r a i and il ar e o g the f ee t ex p e the ﬂ e e g v ty , wh e n n m c n rm to lin s cir cl iff ri n r m o e of the o fo the e of the e , d e g f o eac h other

r in as is anc at ic n s ar e m a as are ac co d g the d t e wh h the e d de f t , they ,

a s fo r a r r as n ar m a c fo r ir au nd perh p th t ve y e o , h d to t h the be ty a

’ race and usk ins lin au inﬁni lin i c s no t g , R e of be ty , the te e , wh h doe

' A endi x No tes xrx and xxx1v. pp , Asym m etr y and Variety 1 33

r turnu ni s lf hic r f r nc h as alr a nm a a li s e po t e , to w h e e e e e dy bee de, pp e

alm s as dir l n ri us rm s o t ect y to the c ate ary as to the spirals . Va o fo of

this c ur ve are illustrated in plate 48 and an inspectionof plate s

1 5 and 1 6 will show there any number of these catenary curves en

graved o nthe face of the snow crystals which are the of those

illustrations

PLATE 48 CATE NARY CURVE S PRo uNATURE

ri c l s e akin c ur s r c ni s a nar St t y p g , of o e , we e og e the fac t that a c te y

c ur m ri call is an c ur v a r l i l inﬁni l ve , geo et y , y e of pe fect y ﬂex b e , te y

ﬁne c r natr s un r in unc r hil c mm n o d , whe e t de the ﬂ e e of fo c e , w e the o o c atenary is what the catenary bec omes whenthe forces are parallel and r r i nal l n o f c r s in c as a av p opo t o to the e gth the o d , a the e of he y

c ord o f uniform weight under the inﬂuenc e of gr avitation. The

prac tical use ineveryday scientiﬁc life of the c atenary curve c omes

r m un rs an in c a i s rm l n s i s lf as o ne o f f o the de t d g of the fa t th t th fo e d t e , 1 34 Proportiona l Form

’ a ur s a nts carr in i s as ll as a sm N t e ge , to the y g of we ght we to ooth and

Th e ri uil r and a esi n r l be b dge b de the bo t d g e , however , woud quite lost if they were obliged to trace the arcs whi ch they wi shed to

ME THOD o r D ELTNEATrNo A CATE NARY CURVE o r av VE RS ED e E s

- ik a i in es r m ac ual c ains un inmid air . m s u ul s t t , f o t h h g L e o t be t f th g

un inNa ur nc m ans s ir use am s m rin s m fo d t e , o e ee the , he t e the , b g the int hi m his s and ﬁnall r uc s m s n ic o s ho e o r hop , y p od e the by y thet

r n ca nar cur m a us uil u s nthe ti p ocesse s . Eve the te y ve y th be b t p y call r n t alas r s m a ica ll i n in m r y (but pe haps o , , ve y y p thet y) w th oth g o e

d a r a r ul r and a m icum sc i niﬁc ro m antic thana pencil an p pe , e od of e t

k n l s uil o ne see h o w it r s . r a ﬁrs ow e dge . Suppo e we b d to wo k D w t

e nits as and its r ndi cular r r a re c tangle having , betwe b e pe pe , the p opo

136 Proportional Form

° an l h as encr a d as fo r am l 1 at c r n r 0 lat g e be e te , , ex p e , 5 the o e , p e 49 dr aw straight lines to the points m arked o nthe vertical and the inter se ctions o f these diagonals with the other ver ticals will m ark the

in s out esir ca nar hic can n c nstruc o r po t by the d ed te y w h the be o ted ,

lina d m r l c nne c in in ica ins wi r de e te , by e e y o t g the d ted po t th cho ds in i as shown the d agram .

° Anuri am l lina in a c a enar cur v 60 is p ght ex p e, de e t g t y e of by th m is als a in la 0 ur r m li ri ni l ethod o dded p te 5 , f the to exe p fy the p c p e , and no w that we understand a little m ore about these interesting cur ves in ic Na ur in ul s so r unl let us l r u o ur wh h t e d ge f eq e t y, ook th o gh cata

’ logue of Natur e s Treasury and see whether we have no t beendealing H f unc nsci usl i c a nar curv s all o ur li s . r o r am l o o y w th te y e of ve e e , ex p e , is a s c ak n r m anins anan us ra a n i r ket h , t e f o t t eo photog ph of oted d ve m ak in a r lin as su l l g bac k dive . Watc h eve y e of the body the pp e m usc es

rm ir a in l 1 ri s i u li r ar a rmin ca na m i . fo the g e t c p te 5 , fo g te e w tho t t

And a ainsee ff c ra i in rmin l a s as to g the e e t of g v ty fo g e ve , both

ir l 2 c n urs ins and s rra s in a s and . the o to , ve , e ted edge p te 5 53

Were the above no tenough to carry c o nvic tionof the universality and au lin s c ul ad d num rl ss rs buta be ty of these e , we o d be e othe few will suffic in i ni n r a r is in i ﬁr s e , wh ch the atte t o of the e de v ted , t , to the b e aui ul r u a lin r rs r a l n in his t f g o p of j ve th owe , eve y th ete be d g to

ﬁxed task and denoting the succ essful effort of the art of a day long

n i e d x No te . App , XX Asym metry and Variety 1 37 since past to depict m otion in its various stages and

PLATE 5 1 o rvE R great ac c ur acy and beauty although without the aid of the kinm a ra e tog ph . 1 38 Proportional Form

is no t r a s n a ﬁnd a m r lit l It , pe h p , ofte , th t we o e er al exam p e of the c atenary c ur ve (cur ve of the c hain) thanis fur nished us o nalm ost

EDGE S any bright summ er m orning by the industrious ﬁeld spider wh o is both to o wise and to o effi cient to waste valuable skill and silk in

tw i r weaving a ne hic h s to be im m ediately dest oyed by storm . When

PLATE 5 5

’ s pi o E R s W E B C v C . . 11 . ) ( .

PLATE 56

-‘ THE FLIGHT o r TH E o o w r rxc rr

I 40 Propor tional For m

s all inn i te mplate and m easuri ng ro d . We h o w se decr ease o ur in terest inthe subjec t if we choose o ne of o ur next exam ples fr om the n n H in l Far East a d the other fr om o ur homela d . ere p ate 57 we have as fam ous a billow as ever shook its cr est o r thr eatened the

r a d m l s ru i n n n ove throw n co p ete de t ct o of puny m a . I eed hardly say that itis the Great Wave of Kanaz awa engraved o nwood by

a m u l a an s H usai n arl if no t ui a undr th t ch be oved J p e e , ok , e y q te h ed

ri inal us ar e no t no w mm n u illus a years ago . The o g c t c o o b t the tr tionbefor e us was m ade fr om o ne of these whic h fortunately is in

is r ai o f i l o ur hands . It a t t the Japanese art sts that they exc e in s l n i r r a al o f i l n ac i n and is is n r the p e d d po t y v o e t t o , th owhe e bet ter shown than in o ur clawing tiger of the seas inwhi ch graceful

a nar i s sli r u r s e in lin irlin u to a ﬁnal c te e p th o gh eve y e th g e , wh g p s ir al in a r - rin m n sna c in so ram a i all at p th t ove powe g de o , t h g d t c y

li wi i Th e the doom ed boats strugg ng th the r hum anfreight below . gi ant wave and his six legendary brethr en have long sinc e passed

l n san s im but no t so a allan ur in r s as set a o g the d of t e , th t g t , c v g c e t

’ by the m aster s brush and graver fo r us to c om pare with the waves

of stone .

as in ack a s unc un se e in c a nar L t , go g b to the ge o ted , we the te y

ur s o ld riz na dl s ila i r c unr an r c ve of the A o Nee e of the G R ve o t y , othe

m l l r an is r and r i n n ar -crus was sy bo , o de th h to y , w tte whe the e th t

’ m lt nsla us i a cr a i ns un unc ﬁr s but o e g , f ed w th the he t of e t o q e hed e , PLATE 57 THE GREAT WAVE OF KANAZAWA (Ho k usa i)

PLATE 58 Mo UNTATNs o r: TE E o rLA R rvE R COUNTRY

1 42 Proportional Form

un im s l and r m c sm s as se e it ino ur ar ns and r i S h e f , f o o o we g de w te it i a sm all c u fo r au n a sm s i a ca i w th , p, ght we k ow , to th t Co o , w th p tal ic n l s all un ari s o ur isi n? And a ai n , wh h e ve op the bo d e of v o g , why ,

us uilis irc ular rm s s ul is in isi l r n having th t ed c fo , ho d th v b e powe the swing o ur Ear th and this sam e Sunand all the other suns o f whic h he

m ani n and r a s swin sm s i s lf irr sis i l ar un is c o p o , pe h p g Co o t e , e t b y o d

no t- -b e - n i r i s al a s inan lli s ? mr thr ough to c o c e ved o b t , w y e p e The e e c ontem plation of suc h m agniﬁc ent c onsistency inspires a dum b

ri n awe and m a az ar an num r usses c an st cke ; we y h d y be of g e , we

an num r th e i us and n ﬁcial r ul s but state y be of obv o be e es t , of the fundam entalwhy and how o f the c hoic e we have as yet no k nowledge .

a r a s a ians m n n v rth e is as s . r a . m a e e It I y Pe h p the M t y k ow We y,

r un l c r ain a a r n ran m a r m im less be p ofo d y e t th t , wh teve Ig o c e y f o t e

im li l uss n l d ill un in ar ia l an in to t e ght y g e , K ow e ge w be fo d v b y h d

hand with reas on.

Thus inthe study o f forc e we are c onstantly assoc iated with the

lli s o ut o ur m a m a ic al k n l i assis s us in e p e , eked by the t ow edge wh c h t the distinguishing betweena parabola and a hyperbola ; b ut we have no inclinationto enter the realm of c onic sec tions exc ept inthe o ne c ase before us and thenonly so far as to rec ognise its im portance in

ar r r a s r a r n c m the se c h th ough the g e t book p e d befo e us . Whe we o e to c onsider the questionof pr oportional form as applied in art and

endix No te xx1 App , . Asym m etr y and Variety 1 43

r s all ﬁnd a lli s la s no m an ar in rm arc hitec tue , we h th t the e p e p y e p t fo ﬁ nd rnam na i n anci n and m r n. us rs s n a a o e t t o , both e t ode Let t pe d

in s i n as m m r th e lli s m oment inc onsider g the que t o of the y et y of e p e ,

i i D r n nW n f is ﬁ ur n s n . m a s n o . as this h a beari g o the use th g e de g De .

Ross shows by illustr ations inthe pr oc ess o f o ne of his works that the

a s m ric al n as r ss s it it is a ﬁ ur o f ellipse m y be ym et whe , he exp e e , g e

! and s a - alanc o n a c n as is s n in la m easure h pe b e e tre how p te 59 ,

Pro . Pro . A B PLATE 59 S YMME TRICAL AND Asm uE TR1CAL ELLrP S E s (Ro ss)

ﬁ ur hil it c as s s m m ric al nit is il o ff g e A , w e e e to be y et whe t ted the

la D n i l alanc as inﬁ ur sam r as r . ss a b e g e B of the e p te , whe e , Ro ve y “ sa s cann l lin a ﬁ ur is allin n th e y , we ot he p fee g th t the g e f g dow to l eft .

S o far as lli s rs c nc r n c an r a s aff r to s the e p e o e ed , we pe h p o d how ,

t i a i n I n la i a is in nl o ne llus r its use in si n. 60 s th po t , o y t t o of de g p te inr uc a r r uc i no f ananc i n r l ur r s ic t od ed ep od t o e t G eek o t opho o , wh h I

o ur Desi n enm nW ss h or P . T e y f e g , D a Ro 1 44 Proportional For m

drew inthe Dipylonc em etery atAthens wher e itwas c arved inendur

ing stone long before the day o f Chr istianity and wher e itstill rem ains

i s I n i sli an m ns r a o ld a . s d ra il i c r it ill to de o t te the de th ght f g e p t he , w

be seenthat all o f the c omm anding lines both o f body and handles ar e

lusi l o n ac rm lli s and c ur s based exc ve y the ex t fo of the e p e , the ve

have beenso c o -ordinated that anabsolute sym metry h as bee npro

s a l n t la ill s as r sul a a c a . duced the e t , g e the p te w how

r n in r will ll us a nas an lli s m a s m Eve y e g ee te th t , eve e p e y be y

r al r as m ric al n n o n si i n ic rc m et ic o y met , depe de t the po t o wh h we fo e it a so na r c c irc l c an m a to a as c to t ke , eve pe fe t e be de h ve the pe t of

r if i it i r m i no r a aranc o f m i n but asymm et y we g ve e the ot o the ppe e ot o , force itto revolve o r seem to r evolve o nany point other thanits true

On m s nri it assum s a aranc and nr . c cc c c e t e c e it be o e e e t , e the ppe e

I n la 6 1 ﬁ ur is i a se ri s a r u as m m r . c tt ib te s of y et y p te , g e A , dep ted e

n a l ru cir l s r c l s mm rical c aus set c c nric l . of t e c e , pe fe t y y et be e o e t y The sam c ircl s r l in o na in no t ir ru and c mm nc nr e e , evo v g po t the t e o o e t e ,

s As a n n r as inﬁ r c m ins anl a m m ric al. i c i gue B , be o e t t y y et p e e of e g ee

h as n rl As a m a e r ing m ec hanism the eccentric wheel um be ess uses . tt o f ec r a i n it ann a ar sin l in n n u nm i n d o t o , c ot ppe g y , be g depe de t po ot o , but sever al cir cles m ay thus be m ade asymm etric as we see fr om the

la r u in m a u a c mm n in r an h ir p te , by g o p g the bo t o o po t othe th t e

nr c e t es .

Spac e forbids going fur ther into the intric acies of the various

Asym metry and Variety 1 45 c ur es ic are as mm rical c in so far as irals v wh h y et , ex ept the Sp of

ari us r rs are c nsi r an v o o de to be o de ed . They m eet us o nevery h d

r -s alk s l a o ut ins irals n ﬂowe t e ve p , the pi e c one is a c om plex m ass of

m s lls h ear ir im ress r o n s a and the , he the p , the dee wears them hi he d , m anadopts them as am ong his m ost varied and pleasurable form s of

Pro . A PLATE 6 1

S YMMETRICAL AND ECCENTRIC CIRCLES

a i n and his r s a ns r wn i auiful si ns o f dec or t o , wo k h ve bee t e w th be t de g which they wer e the inspirationfr om the oldest Gr eek tem ples to the

ar n latest boulev d creatio .

S P IRALS

If we lived in that ﬂat c ountry occ asionally described by the

a ina i a c unr a in nl two dim nsi ns and us lackin im g t ve, o t y h v g o y e o , th g

i ick ness and and r c nﬁn to sur ac s nl n he ght, th , depth we we e o ed f e o y , the

air and imn s mic a d m unai ns ul all r uall c h s c h ey , e n o t , wo d towe eq y , and s uar and rian l na nand a n ul the q e t g e , the pe t go the hex go wo d be 1 46 Propor tional Form s n m asur n arl r in inr a ufficie t to e e e y eve yth g e ch of the eye . Having

e n rn o n nr ar in a rl r im nsi ns be bo , the c o t y , to wo d of th ee d e o , wher e

r ical m asur m ns a as muc im rtance as la r al ns ve t e e e t h ve h m te o e , we c om e alm ost imm ediately uponthe need of m eans to c om pute and

w na i r m ar as sa i r c n c a e . cu is so o c o p e them, we p e ed g h pt The be c n c eivabl a in s uar a c alc ula i no f m ass s r s y k to the q e th t the t o e , whe e the e are n c ssar r se nts li l c nus but r a ur ins e e y , p e tt e to o f e ; whe e N t e beg to

urnand is ins ir als and lic s as she s r d a and ur t tw t p he e , doe eve y y ho , thenwe m ust c all inhelp from outside the pr oc esses we have studied

ir l unl all la ls r . c s ss c c nric c are o f c urs s m e ewhe e C e , e e e t y p ed , , o e , y m ric al sinc m asur al a s sam s an r m nr et , e they e e w y the e di t c e f o the ce t e and r u a i i n i i r t s s p od c e th t repet t o wh ch s equi site o tric t ymm etr y .

irals o n c nr ar win ar un c us li a circl but nu Sp , the o t y , d o d the fo ke e , ,

i la te r nin s r r m n l c uall ar r and ar r . ke the t , o t y peed f the f the f o the c e t e

It is true that spir als which ar e all ino ne plane occ asionally occur — ﬂat s irals but n r all as un in l a i n an s p ge e y , they w d , the e ev t o ch ge along with the distanc e fr om the c entr e and we ﬁnd a summit form ed

at c nr or ls a li o r lin r . the e t e , e e he x cy de

There are thus as m any kinds o f spirals as there are leaves o na

r but r una l it ill n c ss r to s u onl o ne or o f t ee , fo t te y w be e e a y t dy y two

s s ciﬁc all nric ri nci l s su c in suffici n the e pe y , the ge e p p e of the bje t be g e t t s all o c over the r est o f the gr ound fo r o ur purposes . We h ﬁnd in

la 62 six ﬁ ures s in as m an ari ies s irals two p te g how g y v et of p , of whic h

1 48 Proportional Form

D oubtless if c omm anded to dr aw a spiral and thenleft to hi s o wn

i a r a m r al ul ﬁr s ink s m hin akinto th dev c es, the ve ge o t wo d t th of o et g e

- hi is n ff rin watch spring spir al seeninﬁgure A of the plate . T s o e o sp g

ara li rm and o ne r us mi ular l diff r of the p bo c fo , the befo e ght pop y be e entiated from the others by c alling attention to the fac t that the

anc s n ar o f c il are al a s sam r dist e betwee the cs the o w y the e , whethe

n r n r n n rnin all o f s ir als ar a inn o a u o e . c es the e be er o te Co e g th e p , fur ther and m or e technic al details are laid downinthe appendix notes

ar r m Fo r s h s n is for the beneﬁt of those who c e fo the . tho e w o e be t

r anm a m a i al i t ill suffi n m ni n ar othe th the t c , w be cie t to e t o the he t shaped outline shownin ﬁgur e B as r epresenting a spiral studie d by

n n am ian and a rwar s r him s and c all a r Co o the S fte d by A c ede , ed fte

la r il ﬁ ur s and s rm s n wnas r lic the tte , wh e g e C D how fo k o the hype bo an i rm r r n inﬁ is anin r s in o ne d the l tuus. The fo ep ese ted gure E te e t g based o n anincr ease o f r adius inthe goldenseries of extrem e and m an r r i n so a ac c il in its dis anc r m c n r e p opo t o th t e h o , t e f o the e t e ,

la s so to s ak ar o f l ss r in o ur amiliar r o r io n p y , pe , the p t the e e f p op t

its n suc in u ar ni ur ich in urn la s th e to ext c eed g o tw d e ghbo wh t , p y

“ r s n r n is l ﬁ re F c r ndi ar o f a r a d sam ass as u . o e po g p t g e te , of the e c g

The logarithm ic o r equi angular spiral as showninﬁgur e F o f pla te

6 2 occurs m ore frequently inbotany and c onchology thanany oth er

m a h D r and sinc is s ir l m as nso ll s a r . . fo , , e th p ethod bee we t ted by A

n ix o te endix No and V e d N . tes App , XXII App , XXII , D . Asym m etr y and Variety I 49

H . urc o f O r c ansc arc l r an u s Ch h xfo d , we e y do bette th q ote , a was

’ n inNature s Harmo ni c Uni t r m his r o nThe Relatio n do e y, f o wo k of

P h llo ta xi s to Mechani cal Laws in i D r y , wh c h . Churc h expr esses his m a s u in i n n r i e n of st dy g th s i ter esti g form of spir al. Desc be a large “ c ircl sa s au r n r a i sam c n r a s ri e , y the tho , the d w , w th the e e t e , e es of c nc nri c circ l s m ak in i r a ii a m es r o f s uar s as o e t e , g , w th the d , hwo k q e near as c anbe judged by the eye ; inthis c irc ular network of squares arran inr adi al seri s in o m ri r r n all lin s i ged e , ge et c p og essio , e wh ch ar e

dr awnthr ough the points o f intersectioninany c onstant m anner are l ari mic s ir als o r n r a nin o r r i r al wa og th p , whe d w the opposite ec p oc y,

ill in r at a l ints r n s ir al ir n w te sect l po o thogo ally . The p des ed is the

sc ri c nn c in s l c ints in rs c i n as fo r de bed by o e t g the e e ted po of te e t o , , ins anc r hi r r a ius i r ﬁf ircl as s wnin la t e , eve y t d d w th eve y th c e , ho p te

6 o r r ﬁf radi us i r i 3 eve y th w th eve y eighth c rc le .

two s irals urn o ne ac wa d o n sam ra i as If p be t ed , e h y an the e t o ,

rm m ar 1 1 in ia ram r esul is s m m ric al ecaus the fo ked the d g , the t y et b e

alanc o n si s butif ri ur n fo r xam l o n b ed the two de , the ght t be , e p e , the

fif r a ius and i ir cl hil l urnis o n fi th d the e ghth c e , w e the eft t the fth cir c l and i r adius as is als s no n dia r am r sul e the e ghth , o how the g , the e t is asymmetric al and c orresponds to that of m any o f the five to eight

ts s s nin an la 6 s s rm suc a s iral ak s objec a ee bot y . P te 4 how the fo h p t e in r wt a sun r the g o h of ﬂowe .

D r r sa s a s iral ns r i n li an us o r . Chuch y th t the p c o t uc t o of he th 1 50 Proportional Form sunﬂower is based o nthe r atio of 34 : 55 and m ay be approxim ated by

: n : Fo r a c rr la in scr i i n is o n the ratios 3 5 a d 5 8 . o e t g de pt o of th c

’ struc to nwe will quote the work referred to inwhich the author goes

sa a suc a air c ur s : and : 8 is n ll o nto y th t h p of ve (3 5 5 ) , the , we

n r r r o r ra in ac cur a fo r a s s em and m a withi the e o d w g , te 33 55 y t , y be used to m ap o ut a spir al orthogonal c onstruction; fo r prac tical

s a air o f c ur s m a c uto utinc ar ﬁ d a r b pur pose p ve y be d , xe to the p pe y

ir l s a a in r u c nr c c and us a r ul . akin a p th o gh the e t e of the e , ed e By t g

i l o f r a ius ual a c ur a rn and di i in it c rc e d eq to th t of the ve p tte , v d g into

ﬁft -ﬁve and als in hi r - ur ar s so a o ne in m a y o to t ty fo p t , th t po t y be

mm n two s ts and usin cur s as a r ul m ar c o o to the e , g the ve e to k ﬁfty

ﬁve s r c ur s and ir - ur l n ns l c ir l ho t ve th ty fo o g o e , the who e c e will be plotted o ut into a Spir al m eshwork of squares inorthogonal se ries c orrespondi ng to the par astichies of the sunﬂower c apitulum takenas

! a type .

I nso far as i al an l r w ic s the de g e of g o th , of wh h we poke some

a atl n in ﬁr s c a rs ak s as m m ric al rm it an wh t e gth the t h pte , t e y et fo , c

l n r in is nn i in r sc arc i c c n. c its r e y be g o ed th o e t o S e , howeve , g eat interest to us lies inc onnec tionwith its r elationto the goldense ri es of

r m and m anr a i its ull r a lica i n ill ext e e e t o , f e pp t o w be deferr ed until a r following chapte .

Propor tiona l Form

ill no tesc a n ic ak in hi s m is inin nuit It w pe ot e , how t ethod ge y to

h a scri in la fo r lin a i n a c a enar ur and t t de bed p te 49 the de e t o of t y c ve , it is iner s in see ur r l r c ur r am us as a t e t g to , f the , how A b e ht D e , f o

nis as was in art ise in his nti tute o eome r scie t t he , dev d I s s f G ty a m ethod quite ink eeping with these two fo r the de pic tionof the spir al

an ni lu ﬁr s sc ri in a circl i i in an num of Io c vo te , by t de b g e d v ded to y b er s m ns w l in i a ram and n o ne radi us in of eg e t (t e ve the d g ) the , be g s vi as i r a a scr r a in s r ar r m ubdi ded w th the th e d of ew, d w g ho t cs f o r a ius r a ius ac are in o ne r a ar r r m nr d to d , e h be g th e d f the f o the ce t e

n il at l w k an r i un as l ra ius as r o t. th the p ec ed g , t t the who e d wo ed u

ir al us si n and as s wnin la 6 is c urs no ta The Sp th de g ed , ho p te 5 , of o e , l ari m ic o ne but as inﬁ ur la e 62 is o ne wi ual s ac es og th , g e A , p t , th eq p

w n rls and us lac s a s nse r e m r m m n bet ee the who , th k th t e of f e do f o o o

tony which is gained by the c onstant Opening o utof the thr ead of the l ari mic iral suc as sam m as r an r in lat 6 og th Sp , h the e te h d d ew p e 5 ,

ﬁ ur ic it ill se r i ns c ns anl as ita r a s g e B , wh h , w be ob ved , w de o t t y pp o c he

the perim eter .

It is no t at all diffic ult to see why so few spir als of equally

s ac in r als suc as see in lat 62 ﬁ ur and in ur r p ed te v , h we p e , g e A , D e

lu are un in a ur sinc r s lls fo r am l uil o n vo te fo d N t e , e , we e he , ex p e , b t

is rinci l las r i n r n c as it ains s ac risin th p p e , the t po t o g ow , ex ept g p e by g

nde rwe sun der Messun mitde m Zi rch l und R hts h t nLinien Ebneuund Gansen U y g g e ic c ey , i , n Cor oren also e dix No te . p , App , XXIII Asymmetr y and Var iety 1 53

n al a ul no r a r i anth e ﬁrs and to a ce tr pex , co d be of g e te w dth th t ,

for ansi n so n cessar lif rini l the exp o , e y to the e p c p e ,

o Fr . A um o nor DRAWTNo 10N1c VOLUTE ' Fro m D ﬁr er s Und erweysung d er Messung

o Fr . B LOGARrrE m C S PTRAL

‘ (D rrr er )

PLATE 65

ul M o ssible nd hf i s l ul r unl in wo d be p a e t e f wo d , f eq e t y the be

nl rm cr us e o ut. O in s m suc as l ari ral h d y o e h fo the og thmic spi , I S4 Proportional Form where expansion goes o n c ontem por aneo usly with pr ogress from

c nr c an a ur nr all ﬁnd ro m fo r a nlar m n the e t e , N t e ge e y o th t e ge e t

o n h er t whi c h is e of co ns ant featur es .

o k in fo r am l atm an illus ra i ns tan L o g , ex p e , y of the t t o , both of bo y an n l c l ar itis a e ansi nin siz d c o c ho ogy , how e th t the xp o the e of the units

r is nr all c ns an ac rin lar r anthe re ce in un of g owth ge e y o t t , e h g ge th p d g ,

axirnum is r ac in ri r m ar is i n til m m . a the e hed the pe ete Co p e th de , the , with the possibilities of the sam e ﬂowe rs o r shells if Nature attem pted to c om press them into suc h a spiral as that drawnby Durer fo r hi s

ni lut inst a ne in hi b e in i a l m ic Io c vo e , e d of the o w c h d c ted the ogarith

and see at nc a i r la r r s ul sam order , we o e th t e the the te ow wo d be the e size as the earlier ones (inwhic h case the ﬂower would fail both inits func tionand inits c irc ular shape) o r else the later gr owth would be c ompletely str angled .

is ru a na ur al r a a i nis s l m in r c lan s It t e th t t p op g t o e do pe fe t p e , b ut exam ples c annevertheless be ec onomic ally studi ed di rec tly from

ﬂ t uan a wi aid a s ir als as a s n. ur s bove th the of the p , we h ve ee P t to

m urc ic h as r c nl n ll so m an the ethod of Ch h , wh h e e t y bee fo owed by y

’ r rit rs le t us amin r lat s a n r m Nature s Ha r othe w e , ex e th ee p e t ke f o

wi n in r s in s ir al rm a i ns nif r s moni c Unity sho g the te e t g p fo t o of c o e .

lat 66 i s us c n i in inus s r us e llo w P e g ve the o e of the wh te p e (P t ob ) , the y

in s nin lat 6 and urnin r ﬁnd Or n in p e is how p e 7 , t g ove , we the ego p e

2 s a all n ir c m staniall r ibe in c one o nplate 7 . The e h ve bee c u t y desc d

Asym metry and Variety 1 55

rm r r and s ac r i s a ain in l in su the fo e wo k p e fo b d g go g deep y to the bjec t , c oncer ning which itm ay be deem ed suffi cient to m ake the argum ent a c onnec ted o ne if we m erely c all attentionto the fac t that the pips o f Pinus strobus will allbe found to be grouped along the Spir als of the

hr ﬁve r r ic is sim l s s n an ni r t ee to o de , wh h the p e t how by y of the c o fe s ;

il ll in it ill n c s s a rm sli l m r int wh e ye ow p e , w be oted , hoo e fo ght y o e ri c a in as o ns irals ﬁve i uni s ic is a r a i n te , be g b ed p of to e ght t , wh h t o foud

‘ n in ﬁn with extrem e fr equency . I nthe Orego p e we d thi s principle c arr i a s illm r c m l si n is c ni r ak in as its m ul ed to t o e o p ex de g , th o fe t g od e

r i n n n ill r i a i i r as s i us a n. the t o of e ght to th tee , how the t t o

I nall s it ill n a i s in uni s a a l of the e w be oted th t the p p , be g t c p b e of in ivi ual num rin as c m l l s ar ir r la i ns n d d be g o p eted who e , be the e t o a d ratios inacc or danc e with that integr al par allel to the goldenseries of

r m and m anra i i nac c i as lain in ﬁrs c a ext e e e t o , the F bo , exp ed the t h p ter and s uni s i isi n ns anl a a l nlar m n an ; the e t of d v o , c o t t y c p b e of e ge e t d inc apable of c om parisonby the integr al system (sinc e their sizes are m atter o f c onstant c hange) are r elated to eac h other by that elas tic s st m r m and m anra i inall its ninui and erfec y e , the ext e e e t o c o t ty p

i n ic i s no un m l nno r inc m l i n a s no t o , wh h g ve ac c o t to c o p etio o p et o , t ke m asur o f l s no r r ac i ns b ut an s and c nr ac s li a e e who e f t o , exp d o t t ke ru r an al a s rin sam r la i ns n its ar s bbe b d , w y bea g the e e t o betwee p t , butnever the sam e initself as was said inthe c orr elating description

la l in r h r r o ne s is c ss s e ess s m er r . of th of e d v e the fo wo k W e eve goe , 1 56 Proportional Form

a r o ne s ui s o r s um l s r but a s wh teve t d e t b e ove , dd to this c onstant co n victio nso strongly expressed ino ur previous research as to breadth and

and ulness a lic a i n is r a m asur hi led depth f of the pp t o of th g e t e e , w ch

i nas o n r a m l to its adopt o e of the g e t odues.

Earlier inthese pages we have referred to the m arvellous per tinac ity with which each of the c oniferae adher ed to its o wnallotte d

rm in m a t r n l - arin i in n fo the t e of eed e be g , the wh te p e showi g these in

l s r s ﬁve r s in r u s r es and so m li h c ute of , othe g o p of th e , e , ke the Scotc

in a o a in in n n in s ar tw t ac . alk i a i r s p e , how g b e e h po t W g p e fo e t assum es an additional interest when every c one underfoot tells a

r r m ic it ll m story of the t ee f o wh h fe , and e phasises the wonder of a

a ur ic c aninfalli l a ac hi s iscr im ina in s ns as o ne N t e wh h b y tt h t d t g e e ,

alm s c all it r m narc r s c m llin might o t , to eve y o h of the fo e t , o pe g the

c mmi un rrin l its r n no m r l its n r al parent to o t e g y to p oge y , t e e y ge e

and s m lanc its a i and a i a but its r ss nial form e b e , h b t h b t t , eve y e e t

and c arac ri s ic n ins at ic its i -cur s c urve h te t , eve to the po t wh h p p ve

in rs c so a i u utsi in rf r nc no i in shall te e t , th t , w tho t o de te e e e , wh te p e

r irals ll no r r sam num r had eve the Sp of the ye ow , bo e the e be of — c no r n sam ur as Or n a.h needles as the Sc ot h be t the e c ves the ego ,

i ne us surely these things g ve o pa e .

ac ual rm a i n inus s r us ill it se m s The t fo t o of the P t ob w be , e , of

s arran r r uc i n di a r am in sufﬁ cient intere t to w t the ep od t o of the g ,

A endix No te xxrv pp , .

1 58 Propor tional Form

inus s r us in r lus ﬁve r r r n r s P t ob , be g of the th ee p o de , e de the

nr al lan i s irals the o ne lus o ne and o ne lus two c lasses ge e p w th p of p p , and inthis c ase a givencircle m ust be divided by three and ﬁve equal

° ar s ri an l 60 a arin o n rm r and p t , the t g e of ppe g the fo e the pentagon o n n l n the latter . The Spiral of o e p us o e and o ne plus two no w pinned

as e r scri c nr lanand its arcs r awn th e ( b fo e de bed) to the e t e of the p d , l n o ne o n i isi n r e and s r o ne o n s ﬁve o g the d v o by th e , the ho t tho e by , the intersections of the spir als will render the propor tions of the seed

ss ls as n nu ar s c nr the i al an l as sin ve e they co ti e tow d the e t e , de g e p g from the intersec ting points of these thr oughout their dim inishing

ac s as n ar s ir al i ns an ua r an at ints I . s c c Sp e , the po A e o d y p , w th o t t q d t c hords which from the ve rtic al position appear as of the

na n l TP ill ass r m in in pe t go pr oduced from the po e , w p f o po t to po t

u an ri ar ircl il sim ilar uad of the q adr t vec tors of the p m y c e , wh e the q r ant arc s of will c ontinue thr ough the intersec tions of the

s ir p als o ne plus o ne and o ne plus two .

is in ur n i s us an c ll n asis fo r s u Th , t , g ve ex e e t b the t dy of

su i s m an aui ul s lls c h things as the upper and lower s de of y be t f he ,

and inplates 69 and 70 we have se t o ut both these sides o f the

r c us u su l m n in la 1 e lica T o h m axirn s , to be pp e e ted p te 7 by the d te

lin n snail r a ains hi c in lat es o f the spir al of the c omm o , ove g t w h , p e

7 2 h as bee n plac ed the c one of the Or egon pine to em phasise the

’ uni r sh e r s at ty of Natur e s pr ocesses , whethe wo k the top of the PLATE 69 TROCHUS MAXIMUS LOOKI NG DOWN F ROM THE A P E X

PLATE 72 ORE GON P 1NE CONE

PLATE 74 E ALro r xs CORRUGATA

PLATE 78

S C ALARrA P RE r ro S A

1 60 Proportional Form its m iv x ansi n i r m ark a l i i X n r a r s ac v . s la i ot e of e p o w th e b e t ty e opho o , s wnin la is sc ar l m r r m as ana ar m n an ho p te 75 , ce y o e oo y p t e t th the s ll snail and su sts i it u ar r c i ns a r he of the , gge , w th s o tw d p oje t o , wate

he l old- as i nd v r- e and s w e of the f h o e o e shot typ , i m easur ed by a spir al in ra i i n the t o of e ght to thir tee .

Hali is rr uata la is s n i i lan r t ot co g , p te 74, how w th ts p epea e d

i ina circl a l in the su es i n an nM s l a w th e by pp y g gg t o of C o o e ey , th t , by c utting o utthe spiral of these shells ona piec e of card and pinning it at cus and r lvin it ar un a circl rsel and r the fo evo g o d e , both obve y e versel ar in s ir als s no n di a r am y, m k g the p as how the g , the whole plan c ul indi ca r u nir ar a ircl o d be ted th ougho t the e t e e of the c e .

The plates 77 and 78 showing respec tively the delightful spir als o f Mur and calaria retio sa sc arc l n d lana i n h ic ex S p e y ee exp t o , w h would perhaps but detr act fr om the c areful examination to whi c h they are entitled

. I n or der to leave no reasonable m eans of understanding the m a r hi is nr n us un i le t us l k in s tte w c h to c o f o t tr ed , oo to the po int su s a a n r ak n am in l . S o far u a i n gge ted by P te 79 , we h ve de t e the ex t o of all spirals as though they were flat and repr esented only a single l n I f a . n ac as a r c nis d o m ar e r c l p e f t , we h ve e og e , few the pe fe t y e ven insur ac narl all in un i r ris s m a at c e n r f e , e y be g fo d e the to e o ewh t the t e o r ls c nsis o f a c nic sur ac as s ninfi ur la o r e e to o t o f e , how g e A of p te 79 , s ill ur r in ar un a c lin r as a vin us a st ina t f the , to w d o d y de e h g to po , Asymmetr y and Variety 1 6 1 m ann r rmin a ir al li as itinfi lin r se e ur . c e fo g Sp he x , we g e B The y de i se lf v r m a av a a sur ac nir l asi r t , howe e , y h e w ved f e , e t e y de f om the l c al curv s the s ir al and as s ir al in s ar un hi s its o e of p , the p w d o d t , curves c m c c nric in no t all nr o n sam c us be o e e e t , be g ce t ed the e fo , as

With these things inmind we ar e inposition

o B Fro P r . . A

PLATE 79

S PIRAL HELICE S AND E CCE NTRICS to take up m ore intelli gently the exam inationof suc h exam ples as the

f a in c n r s r m r s o r a a f side view o p e o e , f e h f o the fo e t , he d o yellow

ﬁnd it in la 80 r its s ir al rm a i nm a c lover as we p te , whe e p fo t o y be

ff r sam n r al c n rm at n i s exam ined without e o t . The e ge e o fo io s hown ina side view of Tr oc hus m axirnus inplate 8 1 where the radi al inter vals are shown by tr ansfer at the upper horiz ontal and a sim ilar m ethod is adopted to develop the details of the upright view of Face

2 r ina di i n s ir al its lf is s wnin l laria inplate 8 whe e , d t o , the p e ho p an 1 62 Proportional Form

a is rm r is no t c nﬁne atthe foot of the shell. Th t th fo of g owth o d to any o ne br anc h of natur al repr oduc tionm ay be se eninplate 83 wher e sprigs of asparagus are illustrated with their Spirals c learly showing

s at i l s a c nf rm sam set r ul s as that the top , the ed b e t ge , o o to the e of e

in n and c l r inall o f ic s ir als are u l the p e c o e the ove , wh h the p do b e , indic ating the near ly orthogonal interse c tions of the spir al lines as

‘ r urc il in s ll rm a i ns s ir als explained by D . Ch h , wh e the he fo t o the p

rm and c ns unl as mm ri s n l in c . are generally i g e fo , o eq e t y y et

n arl r r ﬁne r s s all im af r im ﬁn I n e y eve y g ove of t ee , we h t e te t e d

our selves admiring great b o les whic h have bec om e twisted and cur ved

urs o f ir m a uri in all s r s lic al s ir als s m inthe c o e the t ty to o t of he p , o e

m ark in s run ar un ark li a clin in in an tim es the g o d the b ke g g v e , d

i n s o som etim es they cur ve w th the k ot of the wood . L oking at plate

a a cas in in and num rl ss rs a ai us at r 84 we h ve e po t , be e othe w t eve y

a n k ic n ar l sk c r b ut am n s m sh dy oo , wh h eed h d y be et hed he e ; o g t the

n m an r n s r s s m ay always be foud y pe fec t si gle pi al . Mo t of us inthi s . world o f im perfections are quite elated whenwe ﬁnd ourselves com

n n o n hin c rr c l as s s ir al-twis e r pete t to do a y e t g o e t y , the e p t d t ees have

n and is r ul s lar l ll in an as ll as in c o n do e , th e i ge y fo owed bot y we n ho lo but no t aui ul nia . ia is li a c o c il c gy, by be t f bego Bego ke y h d

urls a is air in a s iral li ar un h er ﬁn r an n who c w p of h to p he x o d ge , d the ,

t w nl nia l m arrass is s i r a . O c s e b ed , tw t the othe y y , bego e e t to do both

s in s at n and in lat 8 se e h er rm in n of the e th g o c e, p e 5 we fo g o e of the

Asym metry and Variety 1 63

m ara i l few am l s - and l - an s ir al c o p t ve y ex p e of both a right hand eft h d p , s ar a l and c m l in sam l m n r ﬁni sh ep te y yet o p eted the e e e e t of g owth , ed and isi l th na r a in c n rm s irals v b e to e eye . Whe helianthus o p e o e fo Sp

two in s ar e nin n r ss r in rs c c nfus of k d , they wove c omm o , c o ove , te e t , o e ,

ELB OW or NONTE RE Y CYP RE S S EE 00N1A

m li a a r h n ni a r s two s and generally c o p c te e c h othe . W e bego g ow pirals sh e plac es e ach of them just as c osily together as the y c anpossibly

and no t uarr l but ar e ﬁni and is inc ni i s to be q e , they de te d t t e t t e the n i n n suc as we s l m se e ino e rm a . e d , h e do fo t o 1 64 Proportional Form

With this beautiful exam ple o f spir als inboth di r ec tions before us itis ardl ssi l r rain r m a r s o n su c , h y po b e to ef f o few wo d the bje t of

n r a nan num r o f a s esi na in thi s distinctio . The e h ve bee y be w y of d g t g

two so a mi r c nis and all au rs no t these th t they ght be e og ed , tho do agree o nacc epting the sam e phras es as di stinguishing the o ne fr om

I n is es it ur r ss d sir a i unn ss r r . o c a the othe th , d p e exp e e de e to vo d e e y

l i in i s rk ﬁnd urs l s in si i n tak technica it es th wo , we o e ve the po t o of

n r u in sci niﬁc t rmin l hi c nc un rs is i g ef ge the e t e o ogy , w h , o e de tood , n n of course m ore determi ative i the end thanhom ely term s . Fam il i r ssi ns us c aus are amiliar are a t r ad i ar exp e o , j t be e they f , p to be e w th n a prejudged a d no t always cor rec t m eaning . The nam es right

and l - an as a li s ir als s un r as but hand eft h d , pp ed to p , o d ve y e y they pr ove as m isleading inapplic ationinc onc hology as frequently they

r i i n I n ral n s ina m ili ar sc . r ar l a a do t y de pt o he d y , I eed h d y y, th t the right o r dexter side of the shield is the side whic h is to the right of the

r n s i l is o nhis arm and nc is l - beare whe the h e d , he e , the eft hand side

is m u is sim l ut s c a r . c b a is m to the pe t to Th h p e , wh t eant by the ri o f lin fo r ins anc n s o n c n r ght the e , t e, depe d the o text . If we efer

our o wnlin it m ans c urs o ur o wnri acin as s l to e , e , of o e ght , f g , o di er s

’ n n nr ar s ar m . o c r r to n hould , tow d the e e y If , the o t y , we efe the e em y s lin n m anhi s ri h ic as ac itis o ur l so e, the we e g t , wh h we f e eft , that in

’ ’ c harging the enem y s right itis one s ownleft whic h is br ought into n actio .

1 66 Proportional Form

uall nd caus it is a c all le r ic urnin l . a ed ot op !t g to the eft] Eq y , be e s ll rm ac c r in s iral s nin la 8 sam inse c he fo ed o d g to the p how p te 7 , the e t ,

r an r uall in uisi i if ﬁr s h ad r n iz z i o othe eq y q t ve , the t g ow d y w th the l t- an turrrin s ul a in c ninuall r i ef h d g , wo d h ve to w d o t y to the ght to i r a c us and so i s ul c all i r c . e c h the fo , th wo d be ed dex ot op

PLATE 86 PLATE 87 — FUS US A LE OTROP IC S P1EAL CHRYS ODOHUS ANTI QUUS

’ Fro m Nature s Harmo nic Unity A Dexio tro pic Spiral

xi r ic ir al it ill n ll s sun as The de ot op Sp , w be oted , fo ow the the

an s o f cl ck in ir m i ns and is s m im s all d h d the o do the ot o , o et e c e

! c l c - is il lae r ic rm is u s sunand if li l o k w e , wh e the ot op fo d p te the , tt e

lic c ul c m ac r m n rlan sh e ul u l ss c all A e o d o e b k f o Wo de d , wo d do bt e it un-c l c wise We s all e er l a e es in s lar e l o k h , how v , e v th e th g g y

scienis s r i th e rem el in ere s in us i ns to the t t , togethe w th ext y t t g q e t o of whether a we ar er of horns c om es withinth e hom onym ous o r het ero n m o us c lass ic in iner r m eans e r arts y , wh h , be g t p eted , wh the he p Asym metry and Variety 1 67 h is horns in the m iddle and curls his right-hand horn in a r ight

an s ir al and his l o ne c n r s l o r r ar s his h d p , eft o ve e y , whethe he we

rntwis to l and i rsa ri ght ho ted the eft v c e ve .

It would also be of gr eat interest to sear ch o ut botani c ally those

“ for m s o f clim bing plants whic h follow the sun in a dexiotr opic

r s n u spir al and those which go against it lmo t o pically. It i e o gh to s a a lar e num rs o ne and m an r and t te th t g be do y do the othe , the

anis s s all if c an ll us As an n usias ic ar n r bot t h they te why . e th t g de e

ansa i u ar c nr a ic i n a sun r l um s and c o n I c y w tho t fe of o t d t o , th t d y eg e

erable ar n ruc unlik in c il ar un l in sid g de t k , e the hop v e , o o d the po e

ir c i n si a sun il m an vin s and the d e t o Oppo te the p th of the , wh e y e

r t i m a r ai n r a r a s a e d exro rO c . c a gr pe p We y be e t , howeve , th t wh teve

s Na ur h as in is ri ui n h er s iral c ic m anh as pur po e t e the d t b t o of p ho e , always h ad m ore o r less super stition o nthe subjec t and very early beganto c ount the spir al o r whorl whic h followed the sunas anom en

and c un r -s ir al as a r n il and a num er of good , the o te p po te t of ev , b of the o ld legends and custom s r elating to thi s singular questionhave beengathered together by Sir Theodore Cook inhis Curves of Life ino ne of whic h tr aditiontells us that atthe baptism of a Lithuanian infan ar ns ur o ne its li l c urls at o f a h O l t , the p e t b y of tt e the foot p po e

! so a c il m a win o ut an r nits lif i us as th t the h d y t e of d ge i e t me , j t

in win s u ar ar the hop v e t e pw d tow d the sun.

We ﬁnd also m any of the sym bolic c olum ns o f ancient ar ch ite c 1 68 Proportional Form

ur c r a d i es sun- ll wi n s ir als i is r t e de o te w th th e fo o g p . Th s t ue of the

ur co lum ns in i al ar S an arc at ni fo beh d the h gh t of M o Ve ce , which l n sa s r ak n r m e m l l m nat m ege d y we e t e f o the t p e of So o o Jer usale .

uc a s ri s r r nc s to n l n m l m s and ir c n S h e e of efe e e be evo e t e b e the o verse , c an ar dl rmi s s r m ni n as ik a h y be pe tted to top ho t of the e t o of the Sv t , that four-legged ﬁgure intende d per haps fo r a c onventionalised form of s iral and c mm nas a ra i nn arl rl r r m p , o o dec o t o e y the wo d ove f o the

im is u rin r un t e of the anc ients . It tho ght to b g good fo t e to those who hold objec ts inwhic h its dec or ationruns sun-wise and the worst of ill luc nit run wa k whe s the other y.

is n ar r a all s in s i a r ain It ecess y , howeve , to t ke of the e th g w th g

sal no t m r l as to o ur li f m th nc arm but of t , e e y be e e pote y of the ch inthe weight we credit the superstitionas having evenwith those wh o

l t in and - he d i . Perhaps as anassuranc e to the cr edulous half credu l us ris i n t ar s a nf rm o r n o , Ch t anity itse lf i i s e ly t ges co o ed s f eque tly to the popular beliefs inthese particulars as that the early designs indec orationused in the c hurc hes seldom wer e allowed to embody in any superstitious form a sym bol whic h would be hateful to the

a t m an ui l r i l m ns p gans. Yet I rec ollec t o o y bea t ful aeot op c c o u of gr eat antiquity now serving c onspicuously innoble chur ch es wher e

a s r m arl im s i circum s anc t o r a they h ve tood f o e y t e , to g ve the t e o g e t

i l e i site illars in ui si ari . a o r am we ght T ke , f ex p e , the q p exq te v ety

Fur ther desc ri tiv re erenc e to th e S vastika wi be made later . lso . No te I. p e f ll A App , XXX

1 70 Proportional For m

Wher e every uncurling of a fernpresents spir al form s wor thy o f

’ a Bishop s crozier and every twisting vine insight suggests the pattern

a aui ul c lum n su c s ar e ll-ni ndl ss and i o ne of be t f o , the bje t we gh e e , w th exc eptionthose that r em ainm ust take their place in a succ ee ding c hapter among the m er e works of m anhim self ; but the present sub je c t c anhardly c lose without o ne longi ng look at a few of the m ost

aui ul o f all s s iral in s ic are rk Na ur un be t f tho e p th g wh h the wo of t e , assis an o r r n l r s n r ted by the h d b ai o f the o d of cr eatio . To eve y

unsm an r n r c ase i o r o r c am r a h t , whethe he e te the h w th p wde e , the he ad-dr ess of the beautifully antler ed deer and the goat so lightly

! s rin in ar r ar s no m an r r r an o ur p g g e the ew d of e o de . So he e h g

r i s ur ir n ur al s iral fo r m nt ula in s r s t oph e . Fo pa of at p s a o em te hi wo k

’ re ek Mo ufﬂo nin la 8 s c and c r as a r am s The G p te 9 , to ky ooked horn should proverbially be offse t by the c ur ving gr ac e o f the dig niﬁed pair of c alipers whic h the se lf-possessed Axis (plate 90) wear s

i so Ch st rﬁ ldiananair se are r il e ro w th e e e . The wo thy fo to the het nyrno us ornam ents of the Wallachian sheep in plate 9 1 and the m ar ll usl c ns r uc rns ric an in la 2 ve o y o t ted ho of the Af Koodoo p te 9 , in r is s nat r a s his r s ic is r d e . whe e he how pe h p ve y be t , wh h ve y good e d

‘ One ul lik o nin ﬁni l b ut tim and s ac r i wo d e to go de te y , e p e fo b d , and the frequency of the intr oduc tion of the spir al into art and architec tur e h as beenso m arked as to requi re attentionas a subjec t

n en Se e man o ther eauti ul o rms o f this kind mentio ned i dix No te XXV. y b f f App , PLATE 9 1

WALLAC E rA N S E E E P

I 72 Proportional For m s u ll axis s ir als hi c sh e roduc s are lar l l a t dy of phy ot , the p w h p e ge y og rithmic and base d with gr eat frequency onthe goldenseri es of extreme an All s in s l r m am ina i n d meanratio . of the e th g deve op f o the ex t o of

an c i inth a ic a n r and all are the inst ce s ted e p ges wh h we h ve go e ove , i ns ri c k in i n nci s inna ural r c ansi n and t t eep g w th the te de e t fo e , exp o , r r r i n ak n ep oductionwhich have f om time to t m e bee t e up. The s irals r us in c n s fo r ins anc are ﬁt am l s th e p of the va io p e o e , t e , ex p e of

‘ li r n r i n fe principal o nthe serial lines of ext em e a d m eanpr opo t o . The r a i s inine rs c unin ac i as ll l and co m t o t ge , o t g e h p p who y deve oped

leted at an o ne m m n are as av s n and p y o e t , , we h e ee ,

Consider ed fr om the no n-fr ac tional point of view and as anever co n tin in r in ic ac s ir als an s r al as anunc m l e u g g owth wh h e h p t d eve ed o p et d ,

an in i ran in at nc r c nis l ns ri s as exp d g, v b t th g , we o e e og e the go de e e

r and ur at m ul us as h as n so n r ss the ead y ac c e od e , j t bee ofte exp e ed before inthese pages and ino ur previous work o nthis subjec t .

Of all c ur ves it is easy to ac c ept the spir al as being the m ost

au it s s r m m n i as c m ar i th e be tiful. What lo e f o ho oge e ty o p ed w th

nd ansi n m i tin ed circ l it ains in s ns r m a . e , g the e e of f eedo exp o It gh de

l r sli in sm l r n ar and be the true sym bo of g owth , pp g ooth y eve o w d

nal s ac s u ar a r u in i in r s ul r . pw d , o te v t g the ve y o to the ete p e

‘ endix No te X App , XX III . CHAPTER VI

THE GOLDE N S E RIE S I N NATURE

ERE we to depend fo r o ur study of Nature uponthe evi

denc e pr oduc ed under the exam ination of inorganic

form s and the laws gover ning gravity and the other

ic c nr l m m n s ul in i a l c m wh h o t o ove e t , we ho d ev t b y o e to the c onclusion that sh e rec ognised pr ac tic ally nothing but tetragonal

l min nm v r a i ns . o n r an a a i al and e e e t o If , the othe h d , we ex e the g

a l in m s s all ﬁnd m m in i illus r a i ns t b e k gdo , we h the tee g w th t t o of pentagonal form and shall be fac ed with instanc es of c ontinuous e xtrem e and meanpropor tionfr om the early rising of the suneven

n n n all m l u to the goi g dow of the sam e . If of forc e ight pr oper y be s m lis s uar and a n n as a s n y bo ed by the q e the hex go , the , we h ve ee , the vital principle of growth m ight alm ost tak e as its type the golden s ri s ic is u r i in s c i n ucli e e wh h the o tg owth of the D v e e t o of E d , along with the pentagram which in so m ar vellous a m anner de velo ps at every angle and o never y side that indeﬁnitely continuous extr em e and m eanratio which enters so largely into all pr oduc tive species . 1 74 Pr oportional For m

Great em phasis was laid in o ur prec eding work o n the impor tanc e of the principle of this c ontinuous series both innatur al de velo m e nt and inart and si nand inarc it c ur and inﬁnit p de g h e t e , the e pertinacity with whi c h this m odule c ame into play and c ould be uilis c ns ruc i l hi l a lar num r rawin s a t ed o t t ve y , w e ge be of the d g of th t work show by dir ect and explic it line and inter sec tion the point

n a l All is t wher e this inﬂue ce concentr tes o r is disc osed . th is no

no r is it n u a m asis s ul lie tw nc r s enough , e o gh th t the e ph ho d be ee ove other thanthese ; and so againle t m e say that the goldenseri es o r this c ontinuous form of extrem e and m eanr atio is o ne of the gr eatest

o r us l m n r r i nal s ac s in r s factors f the j t deve op e t of p opo t o p e ﬂowe ,

s lls and r na ur al c s and a u n is s ri s a plants, he , othe t obje t , th t po th e e theory m ay be advanc ed that will supply a substantial pr opor tion

all s ns in i n Le tm e of the requirem ents of tude t the ar t of des g . repeat also that in this form extreme and m ean propor tion is

Vi al am n all m ul s m l Na ur o ne of the m ost t o g od e e p oyed by t e , and is easily r ec ognised throughout all form s of gr owth and again sa a to o m uc im r anc c an ar l a ri u ina r y, th t h po t e h d y be tt b ted wo k

r in i l r m and m an r a i fo r r of this kind to the p c p e of ext e e e t o , the e are few objec ts in living form s of Natur e whic h do no t reveal its

us c larin law o ne un am n als ru effec t , th de g the to be of the f d e t of t e

proportion.

endix No te xxvr. App ,

1 76 Propor tional For m — — b er h ow e in un e c l and un illin l vi si e in , b g exp ted y w g y t d by the K g ,

Hu s er o r iff r u was gh de Ye t , L d G o d tho gh he ,

Tarried not his arb to chan g ge ,

r h But, inhis wiza d abitstrange

His high and wrinkled forehead bore

A ointed ca such or p p, as of y e

’ Clerks say that Pharao h s Magi wore ;

’ His shoes were mark d wi r d l th c ow an Spe l,

h r Upon is b e asta pentacle .

r r as in c as sun-twis in s ir als am l He e , howeve , the e of t g p , I ead

’ u o r iff rd s o wn ai inhis s lls nacl s and m a ic to do bt L d G o f th pe , pe t e , g ,

fo r yo uwill r ec ollec t that inspite of allthis arr ay of wonder -pr oducing

i s s em s a l a be lac s m a ab proper t e , he e to h ve fe t th t yet ked o ewh t of

ri sin it a ar s t a e ninhis o wnB o -Hall and in solute sec u ty , ce ppe h t , ev

all is su rna ur al assis anc ne r l ss n so the fac e of th pe t t e , ve the e , whe he

l am r hurried y c e fo th ,

e held r r d Inhis hand h p epa e ,

k ord with r A na ed sw outa gua d ,

r a c nclu a ar nl a nl six Many obser ve s, who h ve o ded pp e t y th t o y the

cr a r m ri n a is nled pointed star c ould be e ted f o the t go , h ve by th bee

a nal ulin a n a ram—an i n misn m into calling the hex go o t e pe t g ev de t o er . The GoldenSeries inNatur e I 77

I n the spher e of absolutely inorgani c life we have alr eady noted a great preponder ance of exam ples form ed o nthe relations of the s uar and am n s se r r s all ﬁn r a q e , o g t the the efo e we h d p c tically no examples of pentagonal form and henc e m ust leave those classes

PLAN OF A CRINOID o ut ino ur per usal of the ﬁve -pointed and go dir ec tly to the innum er able and fascinating asteroids and cr inoids and ec hinoderm s wher e we shall net a num ber of inter esting specim ens to add to o ur

ul r l ill r a ll c i n. a for ins anc c m r c us c o e t o Wh t , t e , o d o e pe fe t y t te the

in in us i n an two la s and s in th e r un po t q e t o th the p te 94 95 , how g g o d

lan two rin i p of of these c o ds. l 78 Proportional Form

I n plate 94 o ne notes with pleas ur e that the perim eter of the

fi ur rm s r c na n but is hi s in r in g e fo the pe fe t pe t go , how t te est crea sed uponobser ving that the fir st progr essionof the pentagonis di stinc tly m ar d o n arm s il s c n r r ssi n fi ur ke the , wh e the e o d p og e o of the same g e exac tly c oincides with the arm junc ture and form s inturnthe c entr al s ruc ur u n ic is uil n a ram as s nin la s 8 and t t e po wh h b t the pe t g , how p te

in n m inin no w la n s c c a r . a ﬁ d a whil 9 the e o d h pte Ex g p te 95 we th t , e

’ in inﬁni ari ic is Na ur s c i s c arm s ails the te v ety wh h t e h efe t h , the e det

no t c c ur in sam r r r are s e ciﬁ all do o the e o de , yet the e they , p c y de line ating the points of interest and drawing as with ruler and c om pass

r r ssin n a ns al rna in m o ne i its a u the p og e g pe t go , te t g the , w th pex p

ar s and n i as at and ac n in in w d , the ext w th the b e the top e h e d g the c nr e t e with a perfec t star .

s am l s are er nl a innin th e n rs The e ex p e , howev , o y beg g of wo de

ic Na ur h as r u al n is lin for a but ur nto wh h t e w o ght o g th e , we h ve to t plates 96 and 97 to se e the sam e lines alm ost duplic ated inothe r form s

li l l itis ru butn r l ss m ns r a in n ui l of fe , ow y t e , eve the e de o t t g , beyo d q bb e ,

la 6 s s us this power of Nature to selec t what best suits her . P te 9 how

ulin rm anan n i mi a m r m ri c al the o t ed fo of tedo , wh ch ght be e e geo et diagram for all the obser vable deviationfr om m athem atic al pe rfe c

n l i i l c rr la a i . I n is it c ul r r scr d as t o th c ase , o d p ope y be de pt ve y o e te perim eter whic h form s the lines of a pentagoncir c um scribed about a c irc l il in ri r rm s s c n r r ssi n sam e , wh e the te o fo the e o d p og e o of the e

1 80 Proportional For m

- in a il i a l an m li a o ne ta . a s r r d e rrna The Se Urch f m y ve y a ge d co p c ted ,

s ic ul re a a sci nis butin hic s ul the tudy of wh h wo d p y e t t , w h we ho d be

atonce lost inthe m az e of distinc tions beyond o ur immedi ate purpo se .

w am l suffi i n i in r a ast r i s It ill be p y c e t f I say b ief th t these e o d , the

rin i s and an ns are all m m rs amil c o d , the tedo e be of the f y of the

c in rm s and insu ir na nal n nci s u e h ode , , pport of the pe t go te de e , I q ote

no l ss anau r i an c r a r cura r l in e tho ty th Do to B the , to of Geo ogy the

ri is Mus um sa s em a li insal or r ack is B t h e who y of th , th t they ve t b h

a r and r adihl s m m r s m im s inc m l ic w te , Show y et y ( o et e o p ete) of wh h

! ﬁv i suall e s u y the dominant num ber .

Befor e leaving the briny quar ters of these inhabitants of th e deep

are s r r l all in r l -s inn n s who o p ope y c ed the ech oderm s o prick y k ed o e ,

le t us indulge ourselves by taking a look at the developed form of a

s arﬁs as s n in la 8 and n n a m m n r t h how p te 9 , the Spe d o e t ove the

charm s of plate 99 inwhich we see the fullpatternof o ne of the aste r o i

dae it li rall lin u nlin and h n nlin but b s r w h te y e po e o upo e , o e ve how

c onsistently these details of the pattern c arry o ut o ur ori ginal ﬁve

in na ram in ir r an l and ur ﬁnis in so far as po ted pe t g the eve y g e c ve , h g,

l d i a n r ul s ar in c nr hi c a c rr la in deve ope , w th wo de f t the e t e , of w h o e t g descriptionwill bring o ut the fac t that this beautift dec or ated

at objec t is developed fr om the pentagonas are all of the other s . Th

ﬁ ur a ar s c arac ris s cr a i ns il a n sta g e ppe to h te e the e e t o , wh e the hex go e b

s t A. D . . Fran rt ur er . ci h a h M . Sc Ec hinoderms A B , , , PLATE 99 ONE OE TE E AS TER01DAE

Proportional For m

hi s ulin -s rr l am iliar r l r u rs T o t e of wood o e , f to eve y ove of o tdoo ,

i us a as e a in wa an and is so g ves t t of wh t to expec t the y of bot y ,

diagr amm atic inits c ompleteness that o ne only needs to c ompare it

the illustrations of the pentagr am and the golden series earlier

PLATE 1 00 PLAN o r WOOD-SORREL inthe work to feel the signiﬁc anc e ; and indescri bing the correlations

am l s ul find a is r a in five als of the ex p e we ho d th t th flowe , h v g pet , declares at onc e that the pentagon will interpret its pr oportional

ar s hil its cur s m us s r uc o ne an l s p t , w e ve t be tho e p od ed by of the g e of

f r lli s rmin n ﬁ . r a al this gure If, the e o e , the e p e fo g the edge of y pet be

lins it ill c ontinued (as shownby the dotted e ) , w be found that th e

1 84 Proportional Form

Make a sim ilar com pariso nbetweenthe outlines of the pitcher plant inplate 1 03 and the starﬁsh inplate 98 and the cri noids inplates 94

in itc r lan um r lla-s a and 95 . See how , the p he p t , the b e h ped top of the

th rm na nin s c n r r ssi n i pistil is e exact fo of the pe t go the e o d p og e o , w th

r ar in it and all ina ns an rem and m an ﬁve petals ove ch g , c o t t ext e e

A d nin i na nal am l s can propor tion. n the c o t uty of these pe t go ex p e sc ar c ely fail while such instanc es r em ainas the Sk ywar d and earth

i ia in l 1 0 and 1 0 ac ward s des of Parnass to be added p ates 4 5 . E h of

ill un to r c l na nal b ut ac di s la s these w be fo d be pe fe t y pe t go , e h p y

ur nir l di in r ssin c ninuall l ss n and feat es e t e y st c t , yet p e g o t y the e o ,

- is is nearly ever y o ne furni shes atleast a pair of ratio ﬁlled stars . Th

per haps partic ularly noticeable inplate 1 05 wher e Parnassia fur nishes

ﬁv al r a rm th innr us thr ee distinct sets . The e pet s of the c o oll fo e e

and in is c ase l r c ircl ll ﬁve s als cal ( th , owe ) e , fo owed by the ep of the yx ,

ic i us a s c n na r am and in s a wh h g ve e o d pe t g , topp g the e we h ve the

as irr all in r n An t l c up cl p g together a pe fec t pe tagon. d i shoud be

noted further that eac h se tof these conﬁgur ations bears no t only th e

stam r m and m an r r i ninits l but a inits siz p of ext e e e p opo t o e f , th t e to

its n lar r uni l n r r n th ss is al r . us ext ge t , the go de p og e io so pe fec t Th e

als ar e ncl se in rim circl se als in n ro pet e o d the p e e, the p the sec o d p

ressio nand inn r na n cu ur indi a in g the e pe t go of the p by the fo th , c t g

a rf c r la i n l ns w n i s pe e t e t o of the go de eries bet ee all of these parts . Th

is ru no t al n of arnas sia butm a in r s in l ll o ut t e o e P , y be te e t g y fo owed

P LATE 1 07 JA PANE S E B E LL FLOW E R

PL AT E 1 09

S P I RE A V AN R o o r r The GoldenSeries inNature 1 85

i ac rs as s nin di a ram s 106 H n an w th e h of the othe how the g , , e b e, and

1 la a an s B ﬂow r 0 c n ell e . 7 , P ty odo (J p e e )

Further examples are per haps sc arc ely need ed to show the pe r - tinacity with which Nature c alls upon the ﬁve pointed form inh er

anical s s m but ir inr u i n ill atl as se r v bot y te , the t od ct o w e t e to amplify

' is r un and m asis a a i as th f eq e cy to e ph e the f c t th t f, several wri ters are h o ld s a sh e h as no in ni nno r m anin in is s l c to t te , te t o e g th e e tion, sh e h as atl as un its cr l r ﬁ a l and if i h e t fo d e et y p o t b e , the cho ce as been

ac ci nal ac ci n h as nso far suc ss ul a af de t the de t bee c e f th t , ter gener a

i ns lan hf h as s nn r as nf r s ar in t o of p t e , she ee o e o o di c d g the re sults of

i ar l arnin d a a a n the c ho c e . We e e g every y th t N tur e ever does any

in i a i nis r n it is n w u ur s . th g w tho t a p po e If veget t o g ee , o c onceded that itis because inh er inﬁnite b ro wle dge (o r shall we call it h er in

ﬁnitely perfec ted gam bling system ) sh e ﬁnds that the ac tinic proper

i s a s r d hr r n a r nsurfac are c n i t t e b o be t ough the sc ee of g ee e , o duc ve o

lan r H n as lans die o ut and no l n r a . us p t g owth e c e , p t o ge h ve e fo r

is r u i aid in aria l l s ir r e nas l ir th p od ct ve , they v b y o e the g e they ose the

n i s b e an r sa and lif di s o ut. c c unat a p, e e If the , th . h e , how fo t e th t

an and s itso s l m ar ? c h ce, why doe e do v y

nur re r ur la e s ar e ad a r ria to At the ve t e , the fo e , fo p t ded , pp op te

his su c : la 1 08 i siss a la 1 0 S ir a Van t bje t p te , the P p ew , p te 9 , the p e

ui la 1 1 0 il s and la 1 1 1 a e af in H . o t , p te , the W d Ro e , p te , L of the Woodb e

I n ssi r r im r anc avin s n a a ur the succ e ve o de of po t e , h g ee th t N t e 1 86 Propor tional For m

s l m s s rs l as rni n in r ani c m a r n r m r r e do how he e f gove g o g tte , o e e fo ce ,

by other thanm em bers of what we have c alled the tetragonfamily

and a in l r r an s i l sh e o n r an i n th t the owe k of b o ogy , the othe h d , ev de c es

a c onstant trend toward the inﬂuence of the goldenseries and shows

’ this both direc tly inm eas ur em ents and by the constant use of the

na n and na r am hi c are r n nu pe t go pe t g , w h the very p ototypes of c o ti

o us e r m and m an r r i n itis now l i al xarrrin ar r xt e e e p opo t o , og c to e e f the in su c s in illus ra i ns r m anim al k in o m and to the bje t by eek g t t o f o the gd ,

atlas r m m ank in as su ri r m all. t f o d , the pe o of the

us ill a far to in arc a ai h ul i r o ur Few of w h ve go Se h of f t f dog , e the

’ wn r an r s and n r m is l l ri n o ne m a i u o o othe , eve f o th ow y f e d y w tho t muc h tr ouble obtain evidence of the tendency to which I have re

l r ill n m u ins ru i no n su fo r rr d . 0 c c c fe e N dog ove w eed h t t o the bje t , m ost of us will have see noutlines inm any forbidden plac es whic h look strikingly lik e the diagr am m atic sketch of wo od-sorrel butwhich

in r ill n r c r s r a us o r h as- e n were , fac t , Be t o e o d of the whe e bo t the be

n m ar n w lat 1 1 2 i s s ani . o e abouts of some c e pet Co p e , , p wh ch how

n i an r vi us illus ra i ns the foot of an adult houd w th y of the p e o t t o ,

m m s a la 1 00 and le t ac s c u . espec i lly p te , the f t o e ho e to

in m an af r all is m r in r s in im s l anin Then, s c e , te , o e te e ted h e f th

n r anim al let us s n a ri inin r s c i n fo r ic a y othe , pe d pe od t o pe t o , wh h

ul a o u urn la s 1 1 1 1 and 1 1 in ich pur pose , I wo d h ve y t to p te 3, 4, 5 wh

n l c m si um n an arm a d e an. are show the h d , the , the g of the o po te h

The GoldenSeries inNature I S7

a in u ﬁr s la 1 1 s in a c m si r a in r m X-ra T k g p, t , p te 3, how g o po te d w g f o y

s an ﬁnd it so ull exem liﬁc atio ns l n view of the h d , we f of p of the go de

as alm s r ar a i n a c m l lis and s seri es o t to defy the p ep t o of o p ete t , of the e itis nn ssar m ni nm r an i us o ne a n s u ec e y to e t o o e th the obv o th t the bo e ,

r m un ual alan s ir r s c i m ac ar als are in a f o the g ph ge to the e pe t ve et p , propor tionof length and width nicely adjusted by this series of which

PLATE 1 1 2

FOOT OF A HOUND

m w a an n r is and l a . I n sa is c s e we spe k the e y, the d t e betwee the w t

l and c llar n are r la and is l s d bow , the e bow the o bo e e ted , th ho d goo

n n hi ain i s als as and . o betwee the foot , the k ee , the p Ag the w de t

ar hi ar s is r la i n narr s ar i p t of the t gh be th e t o to the owe t p t , the w dth

le at c al i an l r arm i ris of the g the f , w th the k e , the fo e w th the w t , the

ﬁn r n n um i mi l and so o ni ﬁ i l . th b w th the dd e ge , de te y

is ll b r n a r s s u in um an ﬁ ur It we ow th t the G eek , t dy g the h g e

' it 1 2 Nature s Harmo nie Un a e 8 . y, p g 1 88 Proportional Form c ns anl and ch sin it o t t y , oo g as the subject of those m onum ents of the plastic artwhi ch have c om e downto us as the best of all classi c stand ar s se tfo r em s l s and us fo r all nra i ns m n n d , th e ve , th ge e t o to c o e , a ca o

of m easurem ent which was the m odule followed from

ir da ur wn I n is ns anl the y to o o . th we co t t y c omm em o

r a c l ra r rus l cli us in hi c te the e eb ted Do ypho of Po y t , w h

r ﬁn was us m ul nin the fo e ger ed as the od e , eteentim es

hi s in i ﬁ r as s nin la t be g the he ght of the gue , how p te

1 1 6 ic r r s n s s ill l r ian can n , wh h ep e e t the t o de Egypt o

from whic h Polyclitus is thought to have tak en hi s

standard .

r e s r uic l r alisin a ar The G e k howeve , q k y e g th t v y

u ts r uir ar in r a m n r a uall ing s bjec eq ed v y g t e t e t , g d y

ir su c s in r class s all and divided the bje t to th ee e , the t

l r i a i i a s m ium s ende , w th he ght of e ght he d , the ed ,

n a al a s and with a height of seve and h f he d , the

m u ular i a i s n a s short and sc , w th he ght of eve he d ;

s r n rin a al i i im s the ﬁr st of the e , e de g tot he ght e ght t e

n a in c us m ar the m easur em e t of the he d , be g the to y

basis.

Fo r the purpose of illustr ating the relationborne by such a ﬁgur e to the extrem e and m eanpr oportionwhich is so c losely allied with

nall rm s an am ina i n la 1 1 ill ll re a the pentagoni fo , ex t o of p te 7 w we p y

1 90 Proportional For m

im n is ﬁ r is r r u r r m a s the t e spe t . Th gue ep od c ed he e f o the p ge of

’ Nature s Harmo nie Unity and the inter esting m ar ginal c om m ents are those tak enfrom the text o f that work fr om which it will be noted that the total height is ac cur ately divided atthe um bili cal point into

r m and m an r r i n allin at lin m ar o n ext e e e p opo t o , f g the e ked (B) the

ia r alan f i c ns i in as t l se am . c o u i s s r d g The b e the he ght , o t t t g doe the e

tw o rm s r m and m anr a i is na ain ivi of the te of the ext e e e t o , the g d ded in sam r r i nat i isi n allinat n c and to the e p opo t o C , the d v o f g the e k ,

fur ivi i n in fr m is r a rin s a r s at s . u th , epe ted , b g the d o the eye Q ot g o

“ the form er work itm ay be fur ther said that itwill be seenalso that anextr em e and m eanr atio exists as betweenthe height and width of

ul s and a riz n al lin m ar in ivi si nin the sho der , th t the ho o t e k g the d o to

s ra i ill la um ilic us ino ne as and l n t o f thi t o w p ce the b , c e , the e g h the

m an m n a aina li wil arm in r . r d a r i a it l , the othe If ext e e e t o be g pp ed , be found to m easur e the r elationbetweenthe width o f the head atthe

s i i o f r a il ari us ar s ill eye w th the w dth the th o t , wh e v o p t of the body w

! “ un r la in sam m ann r and as alr a u be fo d e ted the e e , e dy q oted the

i ar f i ars is r la i nto narr s ar w dest p t o the th gh be th e t o the owe t p t , the

i o f le at c al i an l r arm i wri s w dth the g the f w th the k e , the fo e w th the t ,

and so o nindeﬁnitely .

Were any doubt rem aining o f the prevailing effec t of this propor

' ut m 1 1 2 at 1 66 e a utna . es and 1 68 es Nature s Harmo nic Unit b sam ho rs. P a y, y , 9 Pl , p g

n endix No te XXV st. 1 8 1 1 8 and 28 . A d o , 3, 9 App , II p The GoldenSeries inNatur e I 9 I

i nu n um an it m i di s ll a lanc a t o po the h body , ght be pe ed by g e t the

ﬁ r f un m an i his arm s u r ais o r c rr la i ns gue o a yo g w th p ed , the o e t o of

H rm s o f r axi l s as s nin r c din hi c the e e P te e how the p e e g book , to w h

i i s m n ar I nall o f se in unc l ns ri s th s s upple e t y . the the ﬂ e e of the go de e e

l a r inan and the pentagonis so c ear s to be tr uly p edom t .

It is inter esting as well to note that sinc e the public ationof the b ook inquestionother s have found tim e and mind to support theories o f this inﬂuence and the pr oportioning effec t and power s of this gr eat

n ana I n am wa inhis s l n i he s ri s u m . s r T e e po to y the e y, p e d d wo k

Curves o Li e Sir r ﬁn s ivisi ns um an f f , Theodo e Cook d the d o of the h

o r r i n hi s ri m uc as a u d m r m b dy p opo t o ed to t s e es , h I h ve q ote the f o

’ Nature s Harmo ni c Unity and as we ﬁnd them illustr ated inplate 1 1 7

I nhis am l is wri r ﬁn s a ina ll uil m an is anc ex p e th te d th t we b t , the d t e

r r un to na l r r s ns o ne s a o f is s r i s f om the g o d the ve ep e e t t ge th e e , the

r i n r m na l cr n a an r il is po t o f o the ve to the ow of the he d othe , wh e th la r s ac o nr urn m a f un ivi at lin r as s tte p e , et , y be o d d ded the e of the b e t by another step inkeeping with the series : all of whic h are aninterest ing and valuable c onﬁr m ationwhic h Sir Theodor e c ar ri es o ut by a

’ ﬁgure of anartist s m odel into whic h the statem ents have bee ninc or

is wri r al o r ated and hi c a s to alu o f r . s p , w h dd the v e the wo k Th te o c ar ries the m atter thr ough art and arc hitec tur e with the aid of a

r anz Hals and a ic lli o f ic s r m ainain F Bott e , both wh h e ve to t the

Curves o Li e Sir T Coo k Co nsta le 1 1 . f f , . A. , b , 9 4 1 92 Proportional Form s an as ak n but ic no t in t d t e , wh h , be g part of the evidence furnished

a ur o n su c are r a s no ar is su i is n by N t e the bje t , , pe h p , p t of th bd v io of

s all a m r sa a u l ns ri s na ram We h h ve o e to y bo t the go de e e , the pe t g , and x r m and m anra i in r m ainin s but inso far as e t e e e t o the e g page , ,

a i a ur ra r an ar s and r k s m an they h ve to do w th N t e the th the t wo of , the more we study h er the m ore fully and ﬁrm ly are we c onvinc ed that what h as beensaid ino ur form er writings h ad all of the forc e of minor

r c . m il r a ar n a as h ad r p ophe y It bec o es da y m o e pp e t th t , I befo e said we shall ever ﬁnd the evidenc es of thi s ratio and its oper ation as a m as r n m a ui ro d un r a c as hr u u all a ur . ro e g , ﬂ g b o d t t o gho t N t e F

’ long study of Natur e s em ploym ent o f this proportionI ventur e to sa a n r ita ars a a ur is m ak in use an ser i s y th t , whe eve ppe th t N t e g of y e , a sc ientiﬁc analysis will alm ost invar iably Show that what sh e aim s at is the perfec t and indeﬁnitely c ontinuous and acc ur ate extr em e and

" m an r r n a k ﬁn i n. h a lse o e m as s all c d e p opo t o W t e , y , h we expe t to , wher e testirno nials ar e so unending and so frequently point in the same way? Ho w else should Natur e exe rcise that perfec t stability o f ur s ic ll n is h er minan c ar ac ris ic i u p po e wh h we we k ow do t h te t , w tho t

’ r in uil o f a m m n s m n n suc as ul aris r m a eve be g g ty o e t o oto y , h wo d e f o c onstant repetition?

Examinationh as shownwhat the power of the tetragonand its r la i ns m a m ns u inr alm s o f rc and a e e t o y bec o e whe o ght e fo e , we h v

CHAPTER VII

P ROP ORTIONAL FORM As AP P LIE D I N ART AND ARCE 1TE CTURE

TIENT exam inationo f all o f these long and perhaps tedi ous

details h as br ought us to the point wher e itis possible to m ak e

a r r s im a r ar in a al s ita ar r m p ope e t te eg d g o ur m teri . Doe ppe f o

a s n a itis r a l na c r ain a inh er m ani what we h ve ee th t p ob b e , y, e t , th t , fold and beautiful developm ent of the Heavens and the Earth and the things under the Ear th subjec t to that Divine hand whic h guides and

l all a ur ﬁn s it r ﬁ a l to rn her r s r c ontro s , N t e d p o t b e gove wo k ve y — c r ain rm s ic we a scr i ra n largely by e t fo wh h h ve de bed , the tet go

farrril r m and m an r r i n inc lu in fam ily , the y of ext e e e p opo t o , d g the

a n and ar i us s ir als ? it s m s r a r as it s t pent go , v o p If ee to the e de doe o

r i rs a s hin s a n s a lis n er ad the w te , th t the e t g h ve bee e t b hed beyo d p

nur n are a ain r u ac ac wi s c n r ve t e , the we g b o ght f e to f e th the e o da y

i n r o r no t s all isr ar all s im n quest o , whethe we h d eg d of the te t o y of l n r i and a n r sul s as an nto us and su s ege d , w t , p te t e t h ded dow , ppo e ,

nrar i nc o f all s a m ank in h as l c o t y to the ev de e of the e , th t d bui ded all

! of hi s works o utof his o wnhead with otrtrefer enc e to those things

n an i r h a which were c o st tly go ng o na ound im. With wh t seem ed to 1 94 InArt and Ar chitecture I 9S

wri r s c nclusi r s us i n the te o ve p oof , both of the e q e t o s have been an ﬁ swered speci c ally .

ur se n o f r no w f r us as rm r n The p po , the , the wo k be o e , of the fo e o e , is se e in a m annr s r e ula i ns c ns anl inuse a to wh t e the e g t o , o t t y by N

a e n ut use and ll ur m an. Has n t e , h ve be p to fo owed by he bee guided s m a s uar and uila r al rian l as a ur is inh er o ewh t by the q e the eq te t g e , N t e

m ns r a i ns o f rc and h as uili z e li a ili i s de o t t o fo e , he t ed the d c te possib t e of the goldenseries and spir al inhis c onstantly acc um ulating works ?

It is no tenough for o ne inthe er a inwhi ch we live to ask whether o r no o ur ar tists and arc hi tec ts and designers do c onsciously hark bac k to Natur e atevery turn. The hum anr ace h as existed fo r aeo ns of ages

a r l a d li o f a m anis but s anl n . r l s an the fe p o g It the efo e fo ow th t ,

r an inhis in s i a i ns r hi s a r in ur n an did eve y m beg ve t g t o whe e f the t beg ,

r i l l n o a m an su no prog ess would ever be poss b e . Not o g ag very ccess ful inm aking m oney was so foolish as to rem ark that he never paid

sli s a ni n to hi s r as was in r s nl in the ghte t tte t o to y , he te e ted o y the

i r m a r m ar in urn is r asis r uur . s f t e Now h to y , y I e k t , the ve y b of eve y

n asi m r l possible ac c om plishm ent . Wer e we m i ded to se t de the e e y

li i al and mili ar is r m ank in h o w n r l ss s ul po t c t y h to y of d , , eve the e ho d

x I i no o f e a to r a re e titi no f th e See endi No te V . tw t c o urs re resenthe e o App , XX III ll , , p y p p sta tements o f th e o rm er wo rk in ull but a co nsecutive arran em ent de mand s that since the f f , g , m ders o f the resentvo lum e ma no thave th e o rm er o ne e o re them th e c o nclusio ns drawnthere p y f b f , be atleastre err ed to and usin o n the o utline as a o ve stated fo r the resenttext th e neces sar f , , g ly b p , y n basic co nclusio ns fro m the fo rmer wo rk are included inth e Appe dix. I 96 Propor tional Form we go forwar d inm anufac ture and c hemistr y and m ec hanic s and art without the fac ts whic h the busy investigators o f the pas t have laid

to us and ic are is r o f s r anc s ? ul bare , wh h the h to y tho e b he Wo d the m anufac tur er beckon him self back to the tim e when m en walked

e l h ad no t n isc r nch mis r n no bec ause the wh e bee d ove ed , whe e t y we t

ar r anuse as a is o r to a a e n r nz was nl f the th fet h , th t g whe b o e the o y m al us d fo r ls c aus s l h ad no t n isc r and e n et e too be e tee bee d ove ed , wh the earth was insistently c alled ﬂat and the sun supposed to sail m ajestic ally around it? The suc c essive steps wher eby m an r ose su eri r s c n i i ns are all a ar o f is r in ic p o to the e o d t o p t h to y , wh h we of

his ar r ac acc alm s i u us i n ac i m n t ye of g e ept , o t w tho t q e t o , the h eve e t

n o it i in a d r . a o f s . S s art an a c i c ur the d y go e by h te t e Tod y , we seldom create standards inthese things bec ause those before us have spent so m uc h tim e and talent inestablishing basic foundations whic h we have c om e to ac c ept bec ause we have looked uponthem and found ﬁ m . r was a im r n m an h ad ur the good The e t e , howeve , whe to g e e r in o ut fo r im s l a im n r rc urne to ve yth g h e f , t e whe , pe fo e , he t d the

s rm ns in s n s s in r unnin r s c aus fo r e o the to e , book the g b ook , be e ,

h e h ad no r w r urnfo r his s an ar s no r s sooth , othe he e to t t d d , othe book

an i a s s r a i to read th those wh c h N tur e fur ni hed him . The e he e d w th a truthfulness astoni shing to us whose powers of obser vation are perh aps dulled by the fac t that no longer is eternal vigilance the pri c e

is a in a r i o f sa . a r c aus ff c cc un fety Wh teve the e , the e e t th t the g

I 98 Propor tional For m i as m m and it r m ains to us nl n de oved the , e o y to uearth what these

all s m ay have been. We c the e works classic and use them for o ur

wnm ul s a n h all no t ainif i o od e . Wh t the S we g we candeterm ne by what m eans the ancients hit upon these splendid ideals and upon what they based their proportions ? We som etim es stand incredulous at th e suppositionthat these m e no f o ld tim es were c areful to follow the dictates o f Natur e and saw inh er uses and c ustom s a wealth of m a ri al as to c l ur and h r a s n t si ce rm . W ul n te o o fo y, p y , ho d they o , they had no other ?

It is worthy attention also to speak of the questionof m athe m ati al n l an t a l a n il c k ow edge d i s pp ic tio . It is som etim es pett y assum ed that bec ause we have unde rtak e nto study Natur e in h er m a m a ic al m s a s a h er i a air m as s s in the t ood , we h ve t ged w th p of c o p e

n an l r n l m s n Na o e and a ru a d a o f l ari i r . the h d , e t b e og th the othe

ur un r una l fo r us h as h er m a m a ics ﬁrs an and n e s t e , fo t te y , the t t h d e d no rim e n al ra in ar r o ut h er r l m s in so expe t d w g bo d to wo k p ob e , do g

n r a l t i al nc ar a fam us o the face of the g e t wor ds a h e r d spos . I o e he d o painter say of anunsigned pic tur e to whic h exceptionh ad beentak en bec ause itlacked the m anualc har ac te rs of the painter inthe south-eas t

rn r th at is was anu rl im m a rial and c hi l is c i n c o e , th tte y te d h obje t o , sinc e the painte r h ad signed the pic ture all over by his inim itable

a i nc m a m a ic s is ne r m r al touch . Th t the sc e e of the t eded by poo o t m aninorder to work o ut and eke o uthis twilight k nowledge o f gr ea t

PLATE 1 1 8 PARTH E NON AND THE S QUARE Co r r e atio ns b Mr Ham bid e ( l y . g )

200 Propor tional Form

ur m ak his uildi n ll no t m l in optics inhis endeavo to e b g fo ow , er e y

’ blind and parro t-like imitation but in the ver y spiri t of Natur e s

nhis s l a its nl risin c nr al cur in pur pose . Has he give ty ob te ge t y g e t ve order to change the m athem atical form ationo r the true perfec tion?

no m ans fo r is h as n n nl a rin s ul By e , th bee do e o y th t the ﬂoo g ho d no t nl be r ic all and to all in ns a r c lan but a o y p ac t y te t pe fe t p e , th t to

i in it as l it s ul a ear ru as ll and the eye v ew g a who e ho d pp to be t e we , no t s n a as rwise it ul un r la s o f ic s a ee to h ve , othe wo d de the w Opt ,

n n i i n in i n hi co l weak and depe de t m ddle sec t o . Has I k t us g ve to s um ns a ra ual s llin u ar r m a s lu rul lin in th t g d we g o tw d f o the b o te ed e , order to c hange the ac tual c onditions o f the struc ture ? Does this

entasis add to the str ength o r c hange the designfrom the o ne as it

i l m s la o n ar r h m ? No r ise l n . m r r c y the bo d befo e , p ec y o It e e y ove o e

ic al illusi n arisin r m si i n s r r the opt o g f o the po t o of the ob e ve , who m us r m r un l u ar al n a c lum n i r sul t , f o the g o d , ook pw d o g o , w th the e t

a ar lin s if s r ai ul s m cur in th t tow d the top the e , t ght , wo d ee to be v g

n F n n ﬁni s l i wards . o r the exac t pur pose the of m ak i g the hed who e

a ar r cis l inr sul a itwas n r a nun r ppe p e e y e t wh t whe d w de the eye , the

swelling stylobate and the c ur ving entasis and the s10pe of the end

c olum ns (too slight to be m easured inany illustration) wer e intention

all and nni a all m asis a ese y c u ngly devised . Let m e st te with e ph th t th

in s nn w na le uil r th g i o ise change the plan. They artfully e b the b de

to c onvey fr om the height of the pedim ent the exac t idea the ar chi InArt and Architectur e 20 1 tect wished to c onvey to the observing eye down o nthe pavem ent

and m ak him see in r ali a l a arc i c below , to e the e ty ex c t y wh t the h te t saw inthe dr awing

It would no t be nece ssar y to dwell o nthis explanationwere it no t a m r an nc us i nh as n r a s u l ssl th t o e th o e the q e t o bee , pe h p tho ght e y , asked by those who knew o f these artiﬁcial means adopted to produc e

ff c ts esir b utwh o did no t it ul se m carr ir u the e e d ed , , wo d e , y the tho ght n to its logical co nclusio .

us r m m r in is r ar and in r s u na ural Let e e be , th eg d eve y t dy of t

un r u c s a c m nsa i n is uni rsal c ndi i n. 0 s bje t , th t o pe t o the ve o t o N de

ak in can c ar ri o ut un r a sin l inﬂunc and s ul in t g be ed de g e e e , ho d we sist that no exam ples wer e to be ac cepted other thanthose showing the in unc o f nl o ne c n i i no r r s ul end iscardin ﬂ e e o y o d t o fo c e , we ho d by d g

sa s everything inlife . We c annot at all agr ee with o ne wr iter who y that life and natur e are m ade up of exc eptions and that the exc eption

r in r in an t s l is m uc h mo e te est g th the rule which i break s . Let u ook the m atter squarely inthe fac e and I will boldly ch allenge the world

i ns in ur i i n s all ﬁnd in to show m e exc ept o Nat e . Com b nat o s we h a s u f in r usi n but an c i n in s ns in hi c t pe y g p of o , ex ept o , the e e w h we

A sh l d m anit n r . s e s ari so sh r uc s un ec e e , eve ove v ety , e p od e exp t r sul s c m inin ari us rc s and il se m in l sh e ia s e t by o b g v o fo e , wh e e g y dev te m r l o f h r arm ni la s sh e n r in ac s so . f o the etter e h o c w , yet eve f t doe

The forces at work are so var ied intheir applic ationthat no one of 202 Pr oportional Form

i uninﬂun t H f o n ﬁ them s e ced by h e other . alf o ur scie ti c instr um ents ar n n e base d o k nowledge of this c o dition. The aner oid bar om eter owes its usefulness to the fac t that pressures differ atvar ying altitudes and iff n n i i ns t i n under d eri g c o d t o a the sam e alt tude . Knowing the h r re lia ili a b t a m h l s a nd ar e b ty , we h ve u to f tho er rue the r esults e in i n cer ai n. c m s a izz i r di n s but and a a t She o b e w th d y g ea es , ( wh t but is hi s a ur n r r s n r c an s h er min and i n t ) N t e eve fo get , eve h ge d , , g ve

sam c n i i ns sh e ill a r uc i m s m a the e o d t o , w tod y p od e w th the o t the m atical exactness the pr ecise result with whi c h sh e answered the sam e c onditions o nthat ﬁr st d ay whenthe world was without form and void .

m s is n i i nw a c ns an n fo r a Man eet th c o d t o ith o t t ec essity wh t,

r l a r rm a si na as nsa i ns fo ack of bette te , I h ve de g ted c om pe t o , the synthetic building up of c om binations o f inﬂuences which shall pr o

u in ir c o r a i n the r sul hi c is s to ac i . d c e , the o pe t o , e t w h he w he h eve We s is r m m n o ur li s in m li s in s and inthe ee th eve y o e t of ve , the ho e e t th g arts as ll a hi s is ru inm usic is ll k n nand was s wn we . Th t t t e we ow ho atsom e length ina previous work 1 wher e the deviationbetweenwhat would be a perfec t chr om atic scale and the acc epted sc ale as inuse was lain —a in amiliar to r s u n o f su c o f exp ed , po t f eve y t de t the bje t

! n D c ar Pheidias i a lac equaliz ed tem per am e t . o we h ge w th k of knowledge governing the structur e o f the hum anbody whenhe c o n

endix No te and Music i nNature Co an) . App , XXIX (

204 Proportional Form

ir c nr s and m lac m n l r n a la ur the e t e the e p e e t of the owe e t b t e , while the

i eﬁn s lac insi ase tw o inn n e ghth d e the p e of the de b of the er c olum s .

Do we tak e itthat all of these things are the r esult of m er e acc i dental c oincidenc e o nthe part of watchful I ktinus ? Rem em be r that these diagramm atic distanc es as m easured by the pr ogressive circles are ﬁx m r and nc circ um sc ri in cir c l i r ed by geo et y , o e the b g e s d awn, the ar chitec t h as no m ore c ontrolof where the suc c eeding m ember s will fall

m a s l l r al i n thanh as the reader . They c o e b o ute y whe e c culat o says

mus and al a s in sam lac r ar l ss r a they . t , w y the e p e , eg d e of whethe th t

place suits the conclusions of the student o r no t.

us r m m r als il are o n su c a s Let e e be o , wh e we the bje t, th t the e

r ssi ns s uar as r ra n are sam ns ic p ogre o of the q e he e d w , the e o e wh h we

’ have seenindicated as the m easurem ents of Nature s forc es and de velo ed c ns anl inh er o rrnatio ns s m ic it is ru are p o t t y f , o e of wh h t e in i l and no t a nk n n I ktinus but m an v sib e could h ve bee ow to , y of

is whic h were as well understood by the Gr eek s as by ourselves . It unnecessary to assum e that the ancients fe rreted o ut and followed

ni in is deﬁ te rules of Nature whic h we have s c e unearthed . It quite suffi cient that they are c onc eded to have followed the results whic h w i they sa o nall hands with m uc h m ore studious attent onthando we .

Being no w fairly under way o nthe subjec t o f the adaptationby m an r ul and m l na al r r i ns a r wa is of the e odue of tur p opo t o , wh t bette y afforded of weighing the evidenc e and testing the proof thanby look

PLATE 1 1 9 D E S IG N FOR A F OUNTArN

PLATE 1 20

0V NN1 I E NA S AN 01 A , S InArt and Architectur e 205

r s all ﬁn in la 1 1 a m ing ove speciﬁc c ases . Of these we h d p te 9 ost gr aphi c exam ple of work designed o nthe sam e schedule as the P ar

h n n wit r r ssi n s uar and uila r al t e o , to , the p og e o of the q e the eq te n n n i tri angle . I this ith as o tbeenthought ec essary to nse rt all of the

nfusin lin s isc l s in la s and 2 butif ar c o g e of the web d o ed p te 1 , they e

ll o ut it ill n a c rr la i ns scrib nir fo owed , w be oted th t the o e t o de e the e t e

n l n rni ﬁ ur . ria uli s c c and r it in r g e The t g e R S W o t e the o e , , whe e te se c s rian l a u ar la 2 o n mi dl riz n t the t g e , pex pw d (p te A T V) the d e ho o t in unc ur indi c a s c nr mi dl su i al, the po t of j t e te the e t e of the d e pport ng

ir l n l ar l c olumn. The c c e of the sec ond pr ogr essio c e y deﬁnes the spring of the c entr al arc h and these proportional c oincidenc es c ontinue

‘ I n l 1 urc an an hr u u ﬁ ur . a 20 S i ni i na t o gho t the g e p te , Ch h of G ov , S e , we see another m ost deﬁnite c ase of tetragonal proportioning inwhich

rim cir c l s s i usi u r ss c lum ns the p e e et the w dth of the o t de b tt e o , the s c n thec nr s in r c lumnar s ac s ir usi e o d e t e of the te o p e , the th d the o t de , and the fourth the inside of the inner buttress c olumns with gr eat

exac tness .

That m any of the ﬁnest o ld arc hes have beenconstruc ted o nthis basis is readily seenby a glanc e at plate 1 2 1 Ar ch of Augustus at

usa la 1 22 rc i us at m la 1 2 Ar c Au S , p te , A h of T t Ro e , p te 3, the h of

us s at im i d la 1 2 a a Co u ni an rc uus us t s . r g t R , p te 4, A h of A g t Ao t r la in scri i ns o f s m a r a s su r uus but fo r e t g de pt o the e y, pe h p , be pe ﬂ o , the sak e of stressing the point inhand it is no t unwise to note that inall 206 Proportional Form

these examples of architecture of the Rom ans while still under Greek

inﬂuence the progr essionof the square is plainly inevidenc e as inthe

uus us at s a r r r r ssi ns s uar of A g t Ao t , whe e th ee p og e o of the q e

PLATE 1 2 1

E o r G S T S S SA ARC AU U U , U

i r r i nal s pro duce the c h ef p opo t o pac es . The Ar ch atSusa as drawnin plate 1 2 1 di sc loses a similar inﬂuence as c om bining thr ee pr ogressions

I n s as rk of the square . thi c e the wo is esc ribed by the r ec tangle of

° m ain rnic i 45 to the top of the c o e , the he ght of the attic being de

208 Pr opo rtional For m

a s i r in lli n ri in hr u r m ali . other arches , tho gh pe h p w th o e te ge t o g ty T ee

s uar c u l i o ne rian l progr essions of the q e , o p ed w th of the t g e of

n amina i n a it r r i n lac will be found upo ex t o to pl ce s p opo t o s . The p

° ing of the m aintri angle of 60 and the sm aller ones at its base is

PLATE 1 23 E o r G S T S R1M1N1 ARC AU U U ,

r m Vio llet-lo-D uc in his Di sco urses on A rchitec ture i s f o , who g ve

s num r us r is ri an l im ilar l a li u e o othe exam ples of th t g e s y pp ed .

The correlations at the points 0 in thi s diagr am ar e rem ark able ul t and Sho d be car efully followed o u. The Arc h of Augustus at

imini la 1 2 illus r a s sam rinci l butwi a im n R , p te 3, t te the e p p e th ped e t InArt and Architecture 209

rakin c rnic ic is at anan l arm ni c i above , the g o e of wh h g e h o w th the

lan p .

The subjec t of the use of the circle inproportions of its pro gressio ns as order ed by the squar e and the equilateral triangle would

PLATE 1 24 E OP G S T S S T ARC AU U U , AO A sc arc ely be c om plete without a refer ence to the sym bol of the Royal

’ r c n n n la 1 2 A h Maso s k ownas Solom ons Seal as showni p te 5 . The real signiﬁc anc e of this ﬁgur e rem ained sealed inth e minds of a very

i n s as its m anin ar r am ntand few of the h gher offi cer s . Le ge d to e g e m

' Natur s n nit 2 2 1 e Harmo ic U . . y, pp 44, 5 u 2 1 0 Proportional For m

r m anh as a i f r n n ll but ru si niﬁc anc was r eve y d f e e t o e to te , the t e g e p ob ably lost inthe dark ages whenm as onry and the Knights Tem plars

PLATE 1 25

’ SOLOMON S S E AL AND TE E ROYAL ARCE MAS ONS f m n a uall o ut in o r ursu th ir li s fo r o r a ti e we t ct y of be g, p ed e ve ,

d law inanun r r un is nc r un biddenof c our t an by , de g o d ex te e of p ofo d m in r s a ac in al l m n as ysticism . The gr eat te e t tt h g to the Se of So o o

2 1 2 Proportional Form

Man i c s scul ur are r sse al n similar lin s t y p e e of pt e exp e d o g e , bu s ac is insuffi ci n i m a n p e e t to g ve the pl c e . It is e ough to c all attention

ac a r sc ul ur r r s ns m ansin l a h as to the f t th t whe e pt e ep e e t g y , all th t beensaid about the canons fo r the m easurem ent of the hum anbody

ill un a l and r scul ur is un in r u in s w be fo d to ppy , whe e pt e fo d g o p g of

ﬁ ur s ic is inr unl cas r u is sus i l nl g e , wh h f eq e t y the e , the g o p c ept b e of o y two m etho ds of treatment : either takenas a solid with its spac es both inperspec tive as well as later al entering into c onsideration(inwhic h c as it r s ns us i ns m l fo r i al s u o r i e p e e t q e t o too co p ex v t t dy her e) , v ewed as a ﬂat ic ur in hi c as a is sai inr ar ic ur s 11 p t e , w ch e wh t d eg d to p t e

nr w a l A m l la r m o ne al ill . s an a u ge e pp y ex p e of the tte ethod , o ght to give a m oment to the c ontem plation of the gr oup known contem po rane o usly as the Farnese Bull whic h is Showninplate 1 26 wher e

’ m i nand Ze us a ar in in ir m r s nm irc A ph o th ppe b d g the othe e e y , D e ,

’ m ann r n h r s to the hor ns of a wild bull. The e i whi c f om the obser ver pr esent point of View the ﬁgur es pile up o no ne another inthe centr al clim ax might have beenthe id eal of a geo m eter bent uponsym bolising

l nm ar l r nz th e equilater al tri ang e i b e o bro e .

The fr equency with whi c h arc hitec ts have utilised the propor tions of the te tragonfamily inapportioning their spaces is per ha ps

uall in num r im s s ciall in ic art n eq ed the be of t e , e pe y Goth , whe the

m n a is t r a s proportions ar e those of extreme and ea r tio . It no pe h p

n r k s as h as ns o n amiss c all a ni n is c a . to tte t o to th h ge The G ee , bee h w PLATE 1 26 FARNE S E BULL

2 1 4 Proportional For m architec ts quick ly to c hoose it in pr efer enc e to the only iron-bound

it r d w form s of whic h the barrel vaulting of s p e ecessors as c apable .

No r is itdiffic ult to under stand why the use of extr em e and m ean ratio and the goldenseries received anenorm ous impetus fr om the

n r c hange fr om Norm a a c hitec ture o r Rom anform s to the Gothic .

Ther e are goo d r easons fo r alm ost everything if only we have the

n l f r t m n n I n patie ce to ook o he and this is o exc eptio to the rule . the older sc hools the barrel vaulting c alled fo r ﬁxed form s of window

nin s and n arl ﬁ r la i ns n all in s r Ope g , e y xed e t o betwee po t whe e two s s m s r ﬁn o r aultin cam in c nac as at lan rn y te of oo g v g e to o t t , the te

t r s r la i n n na of a gr ea c hurc h o r c athed al. Thu the e t o s betwee the ve and aisl s t n aisl s and lan rn n i the e , be wee the e the te , betwee the w dth of

na and th e siz a s nin in it r nl ar iall the ve e of the b y ope g to , we e o y p t y H n withinthe discr etion of the arc hitec t . aving c hose the cir c ular

rm arc i its o ne c us a rm m an a its o wn lan fo of h w th fo , th t fo de ded th t p

car ri o ut r u l and r lati ns sci niﬁc all e r be ed th o gh the who e , the e o e t y p mitted betweenthe various portions were largely pr oliﬁc inthe ratio of and but never permitted suc h a c om parisonas is expressed

n n m n n t un a u i ns i l s r r s. s i o u r o l r s the go de e e The e , d y , we e vo t y q e t o

’ c ra i n but ul im a l un am n al uil rs nc ssi i s of de o t o , the t te y f d e t b de e e t e of

' su r r an r n n in r la s ss d s . s ll a urall ppo t , t e , t e gth AS the e fe t y to the e tions of the squar e and its pr ogr essions which so easily pr oduc e the

uila e ral rian l and a n as alr a s n ha was eq t t g e the hex go , e dy ee , w t

2 1 6 Propo rtiona l Form portions of the golden seri es inm any instances m ay easily be see n by a glanc e atthe fr onts of c athedr als such as Pari s wher e the galleri e s and the to wers and the windows and the buttre sse d spaces subdivide

ems l s r a e l int se r a i s as s all s o r l se e also th e ve epe t d y o the t o , we h h t y

n in in lat s 1 2 and 1 0 h i c as t r . a as e se e o w the e of Exe e Ag , p 9 3 , we

fr equently well-knownbuildings submit them selves perfe ctly to th e

ni nal rul s na n and nta r am as sim l as dim e s o e of the pe t go pe g , p y an

r C an l at r al S an nnin unfolding ﬂowe . we ook the po t of Do o of

milia la 1 2 se tin na n i in ri or arc s th e E (p te 9) the pe t go , w th the te he of

in ica d succ eedin circles ro ressi n i ut portal d te by the g of the p g o , w tho

feeling that by seri ous intent of the architec t o r by am azing ac cident

o nhis part the creation c onfor m s astonishingly to th e rule as lai d

wn? nl at r al alazzo ulic r uia in do The ook the po t of the P P b o , Pe g ,

late 1 0 and se e sam rinci les a li d inan nire l diﬂer ent p 3 , the e p p pp e e t y

t i r tain l is also wa bu uall c r sults . a r cas r y, w th eq y e e The tte e wo thy

n in a s irals ic inm s case s are sun- ise will be of ote th t the p , wh h o t w ,

found to tur nagainst the suninthe series next but o ne o nthe right w of the door ay .

r r c d in am ina i n ainin s it ul be Befo e p o ee g to the ex t o of p t g , wo d

well to look fo r a m om ent at plate 1 31 inwhich we shall ﬁnd th e

asymm etric al spir al of the pentagondeveloped after the m anner de i th scri Mr . urc as s is a m s l ul vic fo r lac in e bed by Ch h , th o t he pf de e p g su ivi si ns m a n am us ainin s as an ninits a lica bd o of y f o p t g , c be see pp PL ATE 1 29 PORTAL o r S AN D ONNrNO OP E M1L1A

InAr t and Ar chitecture 2 1 7

i n. lin s s as mm rical irals s rinin r t o The e of the e y et Sp , p g g f om th e c nr ass n r u a na n e t e , p the th o gh the pex of the pe t go of the thir d pro

r essio n as at x c ninuin r u g , o t g th o gh the raking slope of the prim e

ﬁ ur as at a in in g e y, th t be g the po t of its intersectionwi th the horiz on

PLATE 1 31 ASYMMETR1CAL S P1RALS AP PL1ED To TE E PE NTAGON (METE OD OP CE URCE ) tal sam na nin r and n r c s exteri r of the e pe t go ve ted , the p o eed to the o

in tZ ad d is s i la n a a r po t a . If we to th de cript ve c or re tio the f ct th t othe c ur s sam na ur are ra n r m all im ilar ints we s all ve of the e t e d w f o S po , h l have sufficiently descri bed the planto m ak e it intelligible as a to o .

Applying these spirals and the pentagonto several famous paint in s s see a l ar i nr i tur g , we hall th t they p ac e the tist c ce t e of the p c e , Proportional For m b si es ur nis in anin si i ns ari us ﬁ ur s and a e d f h g dex of the po t o of v o g e ,

n s ri i n l n c orrelati g de c pt o of eac h of the p ates is here prese nted . I plate 1 32 we have the m ost fam ous of the works of Titian and by

u m s aui ul ain in H m an in rl . r y tho ght to be the o t be t f p t g the wo d e e ,

n r c as s r r i ns are ci n a nun d as i othe e , the p opo t o de ded by the pe t go ite

law m m n i l nse ri s r i nat r i all to the of ove e t , w th the go de e w tte p ac t c y n every intersec tion. The c e tre of the geom etric plan m akes the

nr als ic ur lacin si i ns G o d a r c e t e o of the p t e , p g the po t o of the F the

th ir in as ll il cur s m m n add m an and of e V g , we , wh e the ve of ove e t y

n I n la 1 other poi ts . p te 33 we have a touc h from the brush of Rubens inwhich the angels are delineated ina way perm itting m uch the sam e

’

f . itan n l r i tio nas b e o re T i S E nto mbmentis ni a 1 a r d esc p Show p te 34, wo k

n la n in r n n is r a oftenc ondem ed as cki g som e espec ts i drawi g . It of g e t

r s r n lin s all in i nr aliz a i n inte e t , howeve , to ote how the e f w th the ce t t o of the pic tur e and how what ar e c o rrrrno nly nom inate d as error s of drawing tend (perhaps intentionally) to enh anc e the force and m ean

’ I n la 1 a a l a le is ing of the gr oup . p te 35 R ph e s M donna of the Crad

n r in n sli s am ina i n ill iscl s m an show , whe e eve the ghte t ex t o w d o e y exam ples of the use of som ething very akin to the golden ser ies

’ i ars in inapportioning of the spaces . Tintoretto s m as terp ec e appe

n in nice late 1 6 Mi racle o S t. Mark i a r sa p 3 , the f , w th the p t o t of Ve

rlin n r m c l u s o no ne c ur v m m e n c ninu whi g dow f o the o d e of ove t , o t ed

ﬁ i amm r l as ll as a by the standing gure w th the h e , be ow , we th t of the

PLATE 1 33

' R UEE NS S ANGE LS

PLATE 134 — TE E E NTOMBME NT TITIAN

PL ATE PLATE 1 38 ANr roUE E WE R D E sro NE D ON P HCE NI X DE S IGN I N E CCE NTR IC CI RCL E S F ROM AN AN ELL1 P S E ANCI E NT CHI NE S E RUG

2 20 Propor tional Form

svastika in ari us c m ina i ns o ne ich is sh nin la 1 , v o o b t o , of wh ow p te 40

’ as it occupied the centre of a beautiful Ch ien-lung r ug of sapphi r e

lu and m r ana m a in a o n r ull eﬁ ec tive m a b e po eg te , k g w de f y ed llion.

is in r s in n a i i si ns is ﬁ ur if It te e t g to ote th t the d v o of th g e , the

PLATE 1 39 P LATE 1 40 “ " ’ ME ANDER AND T ERE TS SVASTIKA P RE T P ROM CE mN-LUNG P ROM ANC1E NT RUGS RUG

amm adi n radi i n a an n l n m s l s wi li ar g o t t o be b do ed , e d the e ve th pecu ni r r i ns the l ns ri s and in la 1 1 is c etyto the p opo t o of go de e e , p te 4 th c ontinued r elationof the extr em e and m eanratio is depicted as go v

rnin ﬁ ur and its in r nin s ac s and s ul co m e g both the g e te ve g p e , ho d be

w a lan l pared ith the designof p e of the go denseries inplate 5 . It m s no t r u a r is an ffi cial o r r ni sed ut , howeve , be tho ght th t the e y o ec og InAr t and Ar chitectur e 2 2 1 rul fo r r r i ns svasti ka b tm r l a tl n i s l e the p opo t o of the , u e e y th t i e ds t e f so r adil ‘ r a in n 1 n i nl i u i . t i r u e y to the t o q est o I deed , w ll be f eq e t y n as in la 1 2 a es r r i ns no t ain ill oted , p te 4 , th t th e p opo t o do obt . It w be r em em ber ed that thi s fo ur -legged outline h as beenheld r ather in awe and r r nc as alr a in ica and r lls if m a b e eve e e , e dy d ted , fo ete , we y

PLATE 1 41 PLATE 1 42 S VASTTEA AND TE E GO DE N PHI S E TE S S VAS TTEA rN NTE -S UN P TTE RN P R A L OR R COU R A , OM (After Co o k ) S AxON c rNE RARY URN li v cr d ul us l nit urns sun and n a l e e the e o , good uck whe t with the , ot b y ill lu n m asis is a in c as . k whe the opposite is the c e To e ph e th , I h ve tro duc ed s n a ika ak n r m a ax nc in rar urn r the de ig of sv st t e f o S o e y , whe e

s m l is i l un r -sun in a m annr r n us and the y bo d stinc t y c o te , e po te to

r a u l ss n n s lism a i se l . d e d , do bt e c o veyi g the ym bo of de th t f

For urther inter estin acts and traditio ns co nc erni svastika ammadio n and o ther f g f ng , g , ancient s m o ls see endix No te . y b , App , XXXI 222 Proportional For m

far a vi irals as r sul r o r si n So , we h ve ewed Sp the e t of g owth de g , but naturally these form s m ay be assum ed under alrno st any in

ﬂuence and itis in r s in n a s irals s ir als n , te e t g to ote th t p beget p , eve by

si n cal r ac i . is r c nis a in la s si s a i n phy e t o It e og ed th t the w of phy c , c t o is ll a r ac i n i s al i r n to be fo owed by e t o , wh c h i equ nfo c e and opposite i

ir n d ectio . Modifying this by the realising sense that a portionof the

rc is a s r a n ausin r ac i n m a fo e b o bed by the ge t c g the e t o . we ay ccept

ru as indi s ua l d a l a the t th p t b e an di splayed every d y. If we ook in

PLATE 1 43 REACT10NARY S P1EALS

m irr r s it in r c i n if c urc ar it r m o , we ee the eﬂe t o , we go to h h , we he f o

s un in ar if la illiar s l arnit r m cushi n the o d g bo d , we p y b d , we e f o both o and all unﬁr s its r c il but h as a r c il ic m us b , the g e p oje t e e o wh h t be

n P ut s now in a nin a c un and so it s o nc inuall . i t ke to c o t , goe o t y th to o ur c ategory of things to be exam ined and see how iteffec ts the spir al

O r e ars is in m a a lar c r o ut a sm all e n . r t denc y u fo b , w h g to ke ge o d of

ir al il it was r a in nif o ne is it in a s un . , tw ted to p t e dy to k k The the two ends Were br ought together and the m iddle bight suddenly

r d r m m e r a ul a n? Th e re l as c r d oppe , do we e e b wh t wo d h ppe e ed o d ,

224 Proportional For m

ul url ack o nno uns m a i usi l m n but nl o nits wo d c b y p thet c o t de e e e t , o y s rainin t g other self .

Un r his a o ne u no t ail inc allin a ni n ri de t he d o ght to f g tte t o , t te as itm a aui ul sim ilar i alr a n e un r nia y be , to the be t f ty , e dy ot d de bego ,

ee n rn nic lu and cr zi r m r is . betw the fe , the Io vo te , the o e of y Lo d B hop

’ I nplate 1 44 we have a drawing of a Bishop s cro ok of the Middle

s and in r r acili a c m aris n in la 1 rn Age , o de to f t te the o p o , p te 45 the fe and l the vo ute .

Not alone in Greec e and c lassic lands can we expec t to ﬁnd

’ aui ul i n s l ens ri s T ai- sun m a a ea d be t f ev de c e of the go d e e . t g y h ve h de

' anunfo rtunate dynasty inthe far easternland whi ch he c alled Chin

’ ’ o r T sin and which his childrens c hildren learned to hear called

a a af r his s a nm i an but n i s an in C th y te be t h ted e e y , the Kh t , otw th t d g all s m ia mis r un s in us r and art urnis in th e of the e yr d fo t e , d t y f hed r i n sun nas m an in s au s m ich e g of the T g dy ty y th g of be ty , o e of wh

a m n s a m n s s is te a l own h ve co e dow to u tod y . A o g t the e the bow Sh in la 1 6 and s ran as itm a s e m is r si n ll ws p te 4 , , t ge y e , th potte y de g fo o the line of extrem e and m eanpr oportionwith a ﬁdelity truly rem ark a l t its rr la i ns and se t wna s r s ri i n b e . Le us study c o e t o do ho t de c pt o

t l r a a ircl i radi us it ill ass en of i s ines . If we d w c e w th the K C , w p

w an ir l scri in li l at and . no r c c b g the p of the bow A E If , othe e be drawnwith the r adius C D so that it touches the lip at the central

endix No te . App , XXXII PLATE 1 46

TE A O W O THE TS UNG D NA S T S HO‘NI NG THE G DE N S E R I E S B L F Y Y , OL

PL ATE 1 47

E EANOR GRI W E S TMI NS TE L LL , R

PL AT E 1 48

S TA1NE D G A S S ORDE R 1 N S P I RA S ROM L B L , F CA NTE RB URY (D ay)

PLATE 1 49

S T. R 1MB E RT S O DE R CO OG NE D a B R , L ( y) InArt and Architecture 225

in it ill un c ut radi us in an r m and m an po t , w be fo d to the CI to ext e e e m n r n. a s ar r as uri as r m p opor tio Taking tep f the , by e g the b e f o the c nr at d th e o at ﬁnd a a s a e t e C to the e ge of fo t D , we th t we h ve e t b l n i lish d a r c s ri s in c all rm s are in r r i n. e pe fe t go de e e , wh h the te p opo t o

The beauties of these varied c ur ves and proportions m eet e nd l ss xem liﬁc atio n but n r m r m ark l anin art e e p , owhe e o e ed y th the of

ir n- rk r fo r h ad o ur unin a s s r m m r and the o wo e ( we Q e t M t y , e e be ) the

- lass ain r il rsm i and c ar r . I n s m ins r g p te , the S ve th , the wood ve We t te o ne r em embers with pleasur e the wonderful touc hes of G rinling

an al n i m m an a s e n r i ns d cr u ir n. a G bbo , o g w th the y e of w o ght o Th t

! pr otec ting the tomb of Quee nEleanor and called the Eleanor Grill

1 s si w is Showninplate 47 . Thi exqui te piec e of o rb nanship w as de si n m as o nin ir n c nur c nains m an g ed by Tho de L gto the th tee th e t y , o t y inte r s in ﬂat s irals and is as ﬁne an am l m al rk as e t g p , ex p e of et wo

c ould well be found .

The c harm ing work s in stained glass painting are so seldom c laim ed by students of the arts that itis a pleasure to exam ine o ne o r

’ o I n la 1 8 two border s from Day s Wi nd ws . p te 4 we have a border

r m an r ur l ni ul ins irals and in la 1 a dra in f o C te b y , p e t f p , p te 49 w g of

’ l n ill r s u an r r at . a St . Kimb erts bo de Co og e Both w ep y t dy of y

ar r i r alm s irals as ill an l s and an se che nthe e of p , w the h d e the b d

ra i ns Am ulranam ra s wnin lat 1 0 o n hi c h dec o t o of the p pho ho p e 5 , w

’ sc n s r m uri i s H c u a are n and hi ch has nso ll e e f o E p de e b Show , w bee we

1 3 226 Proportional For m

H . scri at l n . al rs k r r ani ui i s in de bed e gth by B W te , eepe of G eek t q t e

the British Muse um .

No wr iter who referred even r em otely to the subjec t of spirals

ul c ur s c us fo r om ittrn in l n r un m ni n c o d , of o e , be ex ed g, the o g , to e t o

the great Spiral stairc ases to be found scatter ed over the c ontinent of

No t in ssi l s m r ano ne s ur . a E ope be g po b e to how o e th of the e , I h ve

c s n an r a ainin nic . is nsin i ui and n la ho e to w de g Ve e D pe g w th g de go do ,

rr o r san la ill s r ll r u iazz a and Via ni u fe y do , we w t o th o gh the P , Ve t d e

’ ar z assin ri c a m r l i ians M o , p g by the b dge to the A de y whe e we eft T t

Assum io nbut a ri m m n a o and so o nu r i as m b ef o e t g , p to the ght p t

n an m a in o ur wa as r un a u as c an ll in r r S a Stef o, k g y o d bo t we be , o de

in ui s iri anci n Vem c e r brrn u at to get to the q et p t of e t , befo e we g p the

Cor te d el Maltese and the beautiful winding stair way of the C o n

a a d a it w as fo r rl n C n tarini d el Bovolo . Wh t y the wo d whe the o

tarini and the Minelli and the doge and the princ e thought itnothing

al r a ir alac s ri rl as in is c as so a all unusu to dec o te the p e exte o y , th e , th t

w i ll ll s ir al s airwa s i W no tl s i n ur ur a s . O a s as o t th fo do w f of the p t y , th ,

t rn ith as e nsu s a r s m r m anc s r S ll pat e ed , be gge ted , fte o e e ote e to of the he

sc alaria sc alari s appeals to m e as being the m ost easily understood

i as a s ir al and it r r is o ne ic h as n whenstud e d p , , the efo e , the wh h bee

’ n e nar s rl n r at l is wi ll r a offer ed fo r exam inatio . L o do wo d wo de B o ep y

as ul m an an r but m us n s ass o nto a life study , wo d y othe , we t eed p

n s r an s fo r o ur li l d a is n arl u . thi g othe th the e , tt e y e y p

PLATE 1 5 2

DE S IG N FOR TA BE RNACLE FOR THE S AC R ED OI L

228 Proportional For m

n n m a r an ui Cl is r s . a er an r o e unde st d the q et o te of St Joh L t , whe e y

im r r l amin c nl ss in s au no t give t e p ope y to ex e the out e th g of be ty , least of which ar e the Spiral c olum ns ina bewilderi ng variety . They

vie wi ac r in ir iff r nc s as sun la s i seem to th e h othe the d e e e , the p y h de and-seek with the shadows and the reﬂec tions inthe Old well-head

n l is nd r e n n m a ll ank inthe c e tre . Coo the c our t a g e a d we y we be th ful that r ound the c ool greenc our ts there runs a ro w of Cloisters

l in r Our s irals are since the Cloisters ho d fo r us so much of te est . p

’ ’ s Na ur s r s m ans ac c m lis m n s type of t e powe , type of o p h e t , type , who k n s r a s r all fo r rs i h m is ui ow , of the g e te t Powe of , the wo h p of w o th q et

r spot is se t apa t .

We have said that the c athedrals of the Middle Ages fr equently showed a divisioninheight and width inexact relationto the golden s s and ink in i r r i nal all ca i n ar s eri e , eep g w th the p opo t o o t o of the p t of

n l t 1 a a in um an . us am in a r the h body Let the ex e p e 54, d w g of the

H all se e a an n west front of Exeter . ere we sh th t the dist ce betwee the top of the scr eenof nic hes o nthe west fr ont and the line m ark ing the top of the aisles furnishes us the ﬁrst step ina c ontinuing “ golden series as laid o ut in the space m arked A o n the sc ale n l . is in ur i s lac hic m as be ow Th , t , g ve p e to B , w h e ures the

i scr n r m r un . sum s i i he ght of the ee f o the g o d The of the e , wh ch s

nx rm s s c l r s r as r m o n i the e t te , how the e e to y f o II to IV the S de s ac s o r uni o n scal c m l in nir i p e t D the e , o p et g the e t e he ght of the PLATE 1 54

WE S T RONT O E XE TE CAT E DRA S OWI NG DIVIS I N INT THE G DE N S E RI E S F F R H L , H O O OL

CHAPTER VIII

TE E TAG

ram a is at r ssi no r s VERY d t t , be he the top of the p ofe o the m o t

l l am a ur a r r ar s in ar and r m lin ow y te th t eve t od the bo d fe t e b g,

h as a c r aink ins i ino ne in —his rsis n e t h p th g , pe te t and super

stitio us rr r s a in las few r s la o r il u ho o of pe k g the t wo d of the p y ep og e ,

ta so c all inan r ar sal o r un r an cir cum s an s un the g, ed , y ehe de y t ce , til

a ual rf rm anc atits ﬁr s r s n a i nis m l will the ct pe o e t p e e t t o c o p ete . It be r c ni s a a m ral is a a l ta is la e og ed th t wh t the o to f b e , the g to the p y , and pr em atur e Speak ing of the tag is to bring downall of the evil luck of stagedom upono ne and to m ark the unluck y wight as the m ost

ignorant and the m ost forgetful of his tribe .

is r sum a r wri r h as his ur s and in is It p e ed th t eve y te p po e , th we

ff r r m rk is ﬁnis l l a no i nc n . a is p e d d e e e f o the ext The wo hed , the p y H n m n and c ur ainis a u all. c il u us do e , the t bo t to f e e the ep og e t be ’ t l Spokene er i be to o ate .

ail r r ic ar s a m s o ut s ur c D y , eve y wo k wh h d e tte pt to eek the o e of beauty is m e t by the sc o ﬂi ng ch allenge that Beauty is k into Good

Taste and that the twainare beings so tenuous that a m ere br eath of

230 The Tag 231 inspectionwill blow them c om pletely away to that terra inc ognita

r in an s wi ns ir a i nand r am at ill in m an whe e they jo h d th I p t o o w , defy g to c atc h them and only c oming o utto smile athim attheir o wngood n l l asur n isa ar and l a him a ai isc ns a . p e e , the to d ppe e ve g d o o te

Ac cording to thi s vi ew itis a near -sacrilege to pry into the origin

anins ir r m s s o r rus o r en and as of p ed wo k of the ue of the b h p , to la i n nr ul s fo r an s r ar n rs ! ris u y g dow e y of the e th ee p t e Pe h the tho ght ,

r And it s a l wri r will and lik e is ris hin . i w e , pe h the t ke bo d te who

n r i c onfr o t this p ejud c e .

No w am far r m sa i n a au c an nl a ain in , I f o y g th t be ty o y be tt ed art and architectur e and m usic and poetry and painting by the ai d of

rul s r o r an er lai n no r a suc r ul s are an the e he e ywh e yet d dow , th t h e y

i f r ins ir a i n o r r i r su s u o c a nius . a b t t te p t o e t ve ge Wh t , howeve , I do say and say positively is that ther e is no suc h thing as beauty o r good tas no r ill r r ic s no tc n rm rul s and la s te , w the e eve be , wh h doe o fo to e w ,

an m n k n m m s ric l m a m a ic al. a o t at im y of the t t y the t We y , the t e , ow the spec iﬁ c r ule o r c ombination of rules governing the pr oduc tion and ic m ak it auiful but s n r o r la r if as wh h e be t , oo e te , the t te be

and rk rul r r n m us as good the wo t y wo thy , the easo will co e to , the r easonthat the c hords form ed o n the di ato nic sc ale are agree able h as com e through scientiﬁc k nowledge c enturi es after P yth a

r as is r r go d c ove ed the ba e fac t that they were so . Are they any the less beautiful no w that we k now that a r easonexists fo r the beauty? 232 Pr opor tional For m

r to c nsis n m r aui ul r a ras l arn We e they , be o te t , o e be t f befo e Pyth go e ed evenso m uch as he disc losed about them ? C anany o ne point to any sin l in au in nir uni rs c nc rnin ic en g e th g of be ty the e t e ve e , o e g wh h , wh

c am in s i a it and un r s an it r did n t a a we e to ve t g te de t d , the e o ppear rational basis uponwhic h r ules c ould be built ?

No r is no c laim m a a au m us all u n rul , the e de th t be ty t be po the es

am as alr isc r d and lai wn but r l of the g e eady d ove e d do , m e e y that all beaut must be accordin to some r ules di scovered or discov rabl y g , e e .

ci nc h as far r m r ac lim i its isclosur s and au S e e f o e hed the t of d e , be ty, to o m a o nand o nfo r n ra i ns r u ic r nw , y go ge e t o , th o gh wh h eve y e de velo pm ent of the beautiful will warrant the c losest exam inationof its c aus s i sam c ar ulscruin a ul ll a l m e , w th the e ef t y th t wo d fo ow deve op ent n of the unknowninNatur al phenom e a .

inc n a m i l new rul s m a l r a s o u S e , the , d tted y , e y deve op , why , pe h p y in uir n d s u o ld n s? o uask a us i n an q e , ee we t dy the o e If y th t q e t o I c only c om pare it to a soldier going into battle . He k nows that any d ay a new explosive m ay be built up inthe laboratories of his o wno r

’ nm s i n s s but s is r n him r m insis in o n the e e y sc e ti t , doe th p eve t f o t g having his full supply of the best that h as beendevise d to the date of the c ontest ? Every new law of sc ienc e thenwill be a new r oad to beauty and every new inspirationto beauty will disclose new laws u ponwhic h it is bas ed .

The voic e m ost dissonant to allof these c onclusions will doubtless

234 Pr oportional For m

rium an r m r hin i sel nc a m lish avin t ph t f o the ve y t g t f , o e cc o p ed , h g

? h ad no existence beforehand Thi s reason whi ch we h Ope to pin

n w as it no ac r in su ss r dow , f to the c ce of the m aste in cr eating his m as r i c s ? it ill l us kn did it n t n l him te p e e If w he p to ow , o the he p lik wi s cr a ? And if so in a inair s l a a ains e e to e te , to wh t th doe the p e g t l n l rue a d reasonas applied to beauty disso ve .

ur se it all nis a m a im r and an The p po of the th t we y p ove , exp d , and so far as the writers of the present book are c oncerned (no w here is

n ta it is la ur — a it m a l Spoke the g) the bo of hope , the hope th t y he p

ur l an r no t m r l un rs an and a r cia both o se ves d oth e s e e y to de t d pp e te , but als r a s r a rs o , pe h p , to be c e to of

Things of Beauty . APPENDIX NOTES

I NTRODUCTORY

T is re ue nt co nce i e t at if o ne wri tin o n a definite to ic fil s his I f q ly v d h , g p , l a es wit his c o se n su e ct h e o es so in th e co n ictio n t at no t in e se p g h h bj , d v h h g l e al im i i it int t i we em atic al e a no t u o rtant e sts . O an s r ak o s q ly p x f y p h , ph ly pl d uilt E ual i i al is th e asio al cr iti wh o uestio ns th e m e rit of an g y . q ly llo g c o cc n c q y research where th e public atio nis by ne c e ssity limite d to th e pro po unding and su stantiatio n o f a i e nt eo r une ss all th e i io r a w ic cou c o n b g v h y , l b bl g phy h h ld c eivabl e ar o nth su t i t o i y b e bje c s quo e d c p o usly . Th e purpo se o f this wo rk is th e pre sentatio no f c ertaininteresting relatio ns e twee nth e aws of Nature and the un am e ntals o f e aut and so to re se nt b l f d b y , p these that re ade rs unac quainte d with abstr use m athe m atic al fo rmulae m ay find i a e th e at . Bre it is t er e or e im o rtant and were all o f th e wo r s w c p h v y h f p , k h h h v

e e nco nsulte e num e r ate e re th e m ere re ie w wou cro w outall e se . E e r b d d h , v ld d l v y oo li r ar rese nts suc stan ar wo r s as Haec e Sc him er T n al and g d b y p h d d k k l , p , y d l , th e num er ess o t ers am iliar to the re adin wo r and a ter all is sai and o ne b l h f g ld , f d d , h r u i us t e ac i te st sti re m ains . a wo r c o ntains so m e t in new and t e t d ll If k h g , j tiﬁes itsel w e t e r c ouc e in th e an ua e of e act scie nce as so m e wo u f , h h h d l g g x , ld ch o ose o r r esente in th e ain to n ue o f th e e o e and in a ancin our , p d pl g p pl ; dv g ar um e nts w ic are in t e ir natur e so m ew at s e cia ize t e se t in s a e g h h h h p l d , h h g h v en e t l Vie be k p n w .

NOTE I

For clari t th e re atio ns o f the si e s and r adii are ere setoutin erc e nta es y , l d h p g o f th is ne t i m a o f m st o f e prim e r adius . I t un ce ssar y o furnsh trigo no etric l pro o f o the state m e nts and it wo uld be e xtre m e ly pe dago gic to indulge insuc h a wa ste 235 236 Appendix Notes as th e par ticular stateme nts inthis note are lar ge ly base d o nc alculatio ns lo ng rec o nise and inco nstant use th e c ie e atures of whic wi be oun inan g d , h f f h ll f d y re uta le e ine ri ta ul ti ns p b ng e ng b a o . Th e positio no f the firstpro gre ssio nwill be fixe d by th e m e asure of th e r adius o f its inscri e circle whic m a be e arne ro m an stan ar re ere nc e to be b d , h y l d f y d d f h r R ad . . 0 1 06 8 or t e rea e m a e asi fi ure this for imse l since the di a o nal 7 7 7 ; d y ly g h f , g o f the squar e inits firstpr o gr e ssionis o bvio usly twice th e r adius o f the prim e circle . ws a h t t I t fo llo th t t e di agonal is anhypo the nuse e qual to R ad . and ha e ither side mustbe th e squar e roo to f o ne half o f th e squar e o f or the squar e wil i t . . roo to f . 000 w c l be o un o b e as e ore 0 1 06 8 5 , h h f d , b f , 7 7 7 B the sam e rocess the o sitio n o f th e s uar e inth e seco n ro ressio n y p , p q d p g i n e f th m . wi l be oun to be fi e t rms o e ri e r adius at . 000000 l f d x d p , 5 The squar e inits sec o nd pro gre ssio nis there fore exactly o ctantto the prim e i po s tion. Re ferring to plate 1 we m ay summ ari se as fo llo ws

. 707 1 of Radius AO f . 5000 o Radius AO

. 3535 of Radius AO o f Radius AO

u A t. . s i s O . i Rad ( e I . J f is . 707 1 o AO. B 0 o f A 707 1 1 . O)

Side MN Radius BO Diam ete r E H Twice Radius AO Diam ete r I L M oe Radius BO Diam eter MQ Twic e Radius CO and henc e f Diam ete r I L 2 . 0 1 or o AO and x 7 7 , f A . r 1 o O Diam ete r MQ 2 x 5000 o . Co m ar in t is wit a o e we ﬁnd p g h h b v , Diam ete r I L Side E F

Diam ete r M Si e I etc . Q d J ,

238 Appendix Notes

Since the r adii C0 inplate 1 and C0 inplate 2 are e ach o f them 50% o f AO inthe sam e ate s it o o ws t at th e lines 1 si e o f th e s uare inse c o n ro ress io n pl , f ll h ] ( d q d p g ) i f ui at al e li i t iti i and RS (s de o th e e q l e r tri angl ) e n he sam e pos o n and c oinc de . We m ay thus summ ari se as fo llo ws

AO E 0 1] AR : EF I L; E B BO MN; MQ I J AO OC CA CJ ; OD DB M D ; M O E M OP PG I J and RS fall o nth e sam e line KL and TV fall o nthe sam e line ;

I K and JL would c o incide with e quilate r al triangles dr awnupright.

is it is t ou t wi be suﬂi cie nt to o int the ar um e nt. Endless ot er Th , h gh , ll p g h

c o inci e nc e s wi b e o se r e t o se wh o oo for t em e om etric al . d ll b v d by h l k h , g ly

NOTE IV

Th e que stio nfo r so lutio nbe c o m e s this : whatis th e diago nal o f a cube erecte d

i n. o nth e square I]KL inplate 2 . This re qui re s a do uble proc e ss fo r so lut o rst Asc ertainth e e n t o f th e di a o nal o f th e s uar e itse sinc e t is is ( ) l g h g q lf , h o th th e base o f th e triangle o nwhich is form e d th e new di ago nal f e cube . Tak ing th e radius AO o f our pr im e cir cle at th e value o f we have a re a o un t at th e si es and L are e ual to AO and urt er t at th e l dy f d h d I] J q , f h , h n di am ete r I L e ua s th e si e E F w ic intur nis o f th e ra ius AO. He c e q l d , h h , d m it o o ws t at th e di a o nal I L m e asure s o f the si e I . We m a c o nﬁr f ll , h g d J y t is fo r our se e s if we c o o se rem em eri n t atth e s uare o f th e o t e nuse h lv h , by b g h q hyp h e uals the s uare o f th e o t e r two si es w ic we no w are o f th e alue of q q h d , h h k v

H nc e ’ / . e = e ac / . V h , x I? if) 1 4 1 x 1 1 2

2 Th e di a o na I L ate 1 2 o r h as t ere o re th e alue in e cim als o f ( d) g l (pl , , 3) h f v d o f th e Side I J o r th e r adius AO and upo nthis we c o nstr uc t the triangle Appendix Notes 2 39

i LI S (plate inwhic h the side I S (o r R8) fo rms th e diago nal o f the cube . S nc e th e side I L h as the asc e rtai ne d value o f and th e side LS is o ne of th e e dges o f a cube value d at we have th e decim al value o f th e tr ue di agonal o f the cube sho wnby the e quatio n:

’ ’ V ' V V II . E 1 2 1 3

u w . ns lti n th e so lutio no f which sho ws 1 7320508 as a result. Co ng o o ur re liable ta uatio no ne nine eri n we ne e no twor outthe e n t o f o ne si e o f ane ui b l g g , d k l g h d q h o ate ral trian e inscri e in a circ e wit a r adius o f t e alue f 1 . 00 sinc e l gl b d l h v , , wit outt is trou le we s all ﬁnd t att is si e o f our trian e s o wn th e line h h b , h h h d gl , h by i t R 8 in ates 2 and h as a a ue o f 1 . 20 08 e ac t c o ncidin wi th e alue o f pl 3, v l 73 5 , x ly g h v th e di ago nal o f the cube . NOTE V

’ A Since th e u ic atio n o f Nature s Ha rmo nic Uni t se er al e ars a o ( ) p bl y v y g , inhis t a ua e aui u The urv Sir e o o r e A . Co o m o s e and t oo C es o Li e Th d k , v l bl b f l b k f f , h as a ainta e nu th e su e ct o f s ir als and as aninci e nt touc e s re e ate dl g k p bj p , , d , h p y o n vari ous po ints re lative to e xtrem e and m e an pro po rtio n; for whic h h e h as “ ! c ho se nth e ter m phi The gre at be nefit to b e de rive d fr o m ado pting a designatio nle ss cum berso m e th anth e re petitio no f suc h a phr ase as c o ntinuous e xtrem e and m e anpro po r ! tio n is o io us but to m min o n e e ntual c o nusio nc o u re sut ro m th e bv , y d, ly v f ld l f c ho ice o f th e symbo l I ts m e rit wo uld lie inits brevity and th e pretty and i i o o r h r po e tc fanc y with wh c h Sir The d e c o nne cts it with t e m em o y o f Phe idias . I ts aut itse ems to m e ie s inth e c o nstantand e ar ie r use o f t is G re e c ar ac te r f l , , l l h k h , wi t uite we -defined and re co nise asso ci atio n ino t er and alre a o c cu ie h q ll g d , h dy p d fie s am o n stw ic ne e be o n m e ntio ne t at o f unctio ns w ere it a e ars ld , g h h d ly d h f , h pp r e pe ate dly insuc h p ape rs as th e spheric al h arm o nic s o f Lapl ac e and the wor ks o f r Wit out t eta and hi o n wou be o st in Lo Ke inand o t e rs . e thi s zo ne d lv h h h p , ld l ,

no r is it th e o nly o ne which m ight b e c alle d to attentio n. One m a ne e rt e e ss be no t o n rie but c e ar as we inc o o sin th e y, v h l , ly b f l ll , h g ! nam e golden to e xpre ss this c o ntinuous se rie s o f extr e m e and m e anpr o portio n w ere er itis oun —inth e num eric al o r ec im al o rm inth e ine ar th e circular h v f d , d f , l , ,

1 6 240 Appendix Notes

the ane the S o i o r th e s ir al. No r is suc c o ice lac in inthe c assic touc pl , l d, p h h k g l h w ic Sir T eo o r e woul i e to hi since th e ear iest reco nitio no f t is ratio h h h d d g v p , l g h c o mes to us ro m th e anc ie nt a s w e nit was no wnas th e Go n ctio n To f d y h k lde S e . i i n o this asso c ato I fee l bound t adhe re . ’ B Ha in alre a e e nem asise inNature s Harmonic Unit it ne e ( ) v g dy b ph d y, d ar l b e am lifi e e re t att is e r ect and c o ntinuous o ense ries is o f muc h d y p d h , h h p f g ld h gre ate r impo rtance th anth e m e re bald divisio no f a give nunit into extrem e and m an r atio I n th e o rm e r wo r t is was s wn in m an c o nn i n i all e . f k h ho y ect o s ; n e ntam ero us o rms a e 2 in lo ta is at a e 1 2 inthe seri es itse l as p f , p g 95 , phyl x , p g 3 , f , i u e 8 an 28 inth m c o ntnuo s at a 2 d 8 e u ananato m at a e s 1 8 1 etse . and p g 7 , h y p g q , i rous t r re s ts w i ne e e r a t i l t nnume o he pec h ch d no t b e pe ted here . I s valuab e o no te t at th e ar um e nt rece i es c o nfirm atio n in The Curves o Li e urt er h g v f f , f h i r e fere nce to this po int be ing found nNo te XXVII . ’ C At a e 1 2 and e se w e re o f Nature s Harmo ni c Unit s o we ( ) p g 3 l h y I h d , I t ou t uite efinite th e c o inci enc e etwe en th e circuar e ui ale nt o f h gh , q d ly , d b l q v “ ! e trem e and me an ro o rtio n and th e o tanic al i e al an e t e re c aimin x p p b d gl , h l g atte ntio nto th e gr e ater ac curacy o f th e golde nse rie s thanany possible applic a tio no f th e Fi o nac ci for th e m e asurem ento f ro wt siz e di stance or an e ature b , g h , , , y f o the r thanm e re num be ri ng o f parts ; and it is inte resting to no te that th e wri ter ’ w i no f o f The Curves o Li e i e ne re nc e su o rtin D r . C ur c s e anatio f f , h l by f pp g h h xpl Fibo nac ci in c o nne ctio n with his par ticular subje ct (as who could no t where inte ge rs are possible ) adds his c o nclusio nth atino ther forms o f natur al gro wth it a e ars i e to i e re ate r ac cur ac t anth e Fi o nac ci and urt er su ests pp l k ly g v g y h b , f h gg r that re se arch into its use s will be we ll e paid . D M re aso nfo r c a in t is re ate cur e th e G o e nS ir a is o ious ( ) y ll g h l d v ld p l bv ,

it ein e e o e ro m th e G o e nS e rie s . I nThe Curves o L e a re a re e rre b g d v l p d f ld f if , l dy f d to it is te rm e th e aut o r the hi s iral ro win out o f th e sam e serie s , d by h p p , g g , “ ! w ic as e aine e o re he h as te rm e th e hi a esi natio n w ic for h h , xpl d b f , d p ; d g , h h re as o ns e or e resente am una e to o o w and do no t ee warrante in b f p d , I bl f ll f l d i t r t e assistng o pe pe uat . NOTE VI

I t is state d (1 ) That th e angle fo rme d by diago nal o f a go lde nre c tangle is ° ' 58 1 6 and 1 7 (2) that th e dec im al length o f th e di ago nal o f th e go lde nrec t

242 Appendix Notes

so far as um anc o m re e nsio n o es wit its c all fo r isi le em o nstratio ns ut h p h g , h v b d ; b w at o ic al woul a e nif a cu e were mo e at ri t an es to its o wn h , l g ly , d h pp b v d gh gl oun in si e s ? Yousa t at t is is im o ssi e since it wo ul a e to m o e in b d g d y h h p bl , d h v v nt it is i i at e ast six di re ctio ns at o nc e . ra ou m o ss e so far as n we ca s . l I g y , p bl ee i i w s re s o nsi n ne i Po s to n as e a e ai uire n ime o s. O ime ns onis s o wnrn , h v d, q d d h le n t two in en t and re a t three inso lids a in le n t re a t and g h , l g h b d h , h v g g h , b d h , i inw at o rm wil a ﬁ re o f our m i ? th ckne ss . No w h f l gu f di ens o ns sho w itse lf Th e re is difﬁ cut to o rmuate et m at em atic a we c anc o nce i e th e our as ply l f l , y , h lly , v f — e asily as we c ane volve a four th po we r o f x and thatis just as simple as to fo rm A s : Wit th e e o m e r o f su s a cube . s Br agdo nsay h g t y c h a pac e m athe m aticians — o n a e een ami iar but is t e re suc a s ac e is t ere an o for t is l g h v b f l , h h p h y b dy h ’ m at em atic al so u? A entur e wit m e o wn a re c i ic e o f t ou t sa s h e h l dv h d p p h gh , y , “

sustai ne o nl th e ro e o f analo s e n e r but stro n . is ro e anc o re d y by p gy , l d g Th p , h d inth e ﬁrm r oun o f se nsuous re c e tio n e ten s t re e ac es inth e ire ctio no f g d p p , x d h p d

th e r e at a ss t en anis e s atth e i rink . Letus e am ine this sustai nin g by , h v h g ddy b x g mi i srrnile fo o t by fo o t and str and by str and . Fa l ar bo th to m ind and eye ar e th e s w t e m nsio n t a is l n l ac e s stem s o f o ne t o and re i e s t i es ane s so li s . Lines p y . , h d ; h , , p , d are oun e o ints and t em se e s oun ane s ine- o un ane s inturn b d d by p , h lv b d pl ; l b d pl ‘ ’ lids ound ? H i w oun so ids . Who t then do so b e re s e re th e ana o ic a ro e b d l , , h l g l p t b anis e s ro m si t. o uanswe r at a so i c anno t e a o un ar we ar t v h f gh If y h l d b d y , p n ne an o n ine outo th e n ar o ar m t o f mi c c c o r . B c o mpany . N gu e v y c t y ut if you “ ’ ar e intere ste e nou to ask We w at do so i s o un ? o ic c o m e s th e d gh , ll , h l d b d l g p l ‘ ’ answe r Hz her so li s o ur im e nsio nal orm s in isi e to si t re ate to th e , g d ; f d f ( v bl gh ) l d so i s we no w as are t e se re ate to t eir o un in anes as ane s to t eir l d k h l d h b d g pl , pl h bo unding lines .

I t is as simple to c o nc e ive all o f this fro m a m athem atic al standpo int as itis to write ane uatio ninth e ourt o wer w ic h s al be to th e te ss ar ac t a s m o l q f h p h h l , y b h i as ﬁttin as is th e ua r atic to t e amil ar ane . c annot o we er o e to g q d f pl I , h v , h p intere s tth e re a ers o f t is wo r inan si e su ec ts e re butwoul mere ause d h k y d bj h , d ly p to state that at som e late r tim e itis m y ho pe to sho w th e intim ate be aring o f th e o e nse rie s o nthe uestio no f a o urt im e nsio n and to em o nstr ate t at as g ld q f h d , d h th e o l e nse ri e s is er e ct and c o ntinuous so it a ie s to all we no w o f ourt g d p f , ppl k f h dim ensio nal spac e s as re adily as itdoe s to tho se m o re familiar to th e wo rld . Appendix Notes 243

NOTE VIII

Per ec t o e nse ries as e em liﬁe i the e nta a f g ld x p d n p go n nd pe ntagram . The rem arkable c o incide nc es betwe e n th e m em be rs of th e pe ntago n and e nta ram o nth e o ne an and th e o e nseri es o nth e o t e r we re em o nstrate p g h d , g ld h d d ’ so u inNature s Harmonic Unit at a es 2 to 0 t at e sitate to r e f lly y, p g 97 3 5 , h I h intr o uce t ese ro o s w ich e in m at em atic al and em o nstr a l co rr ect o wn d h p f h , b g h d b y d o th e se e nt ositio no f d im als a e e r e Let us f th e t ec n e enc allen e . or v h p , h v v b h g d , ur o se of t is wo r c o nce e th e statem ents as true and if esire re er to th e p p h k, d , , d d , f e i r for a pr v o us wo k dem o nstration.

No r ]; IX

’ Th e se nte nc e re ce in t is note is o ne or tio n uo te rom Nature s Har p d g h , p q d f mo nic Unit e innin wit the ﬁrst re e re nce to th e i e al an e and h as e e n y, b g g h f d gl , b r e pe ate d here to emphasise th e im por tanc e o f this applic atio no f th e go ldenangle r ar e ui al n o t s m i t o r ci cul e t f h e o e nseri e . Th e ast ar a r a wr itte nso e e q v g ld l p g ph , gh o r nine e ars a o m a we be co nsi ere inco nnectio nwit th e ait ine trem e y g , y ll d d h f h x a o ortion o ic in and m e npr p v e d No te V (C ) .

NOTE X

W e n restri ct th e o e r a to t e se five re ere nce is strict c o nfine to h I p lyh d h , f ly d h ic u t i t at t e Plato n Re uar Bo die s . For th e ur ose s o f s c a wo r as s o o re g l p p h k h , g c o m plic atio ns are pre se nte d whe n we turn to suc h figures as ste llate d or un wr a e o e dr a inc o sin the c e nte r m o re t ano nce as th e re atte tr ah e ro n pp d p lyh , l g h ( g d ) and semi-re uar o di e s suc as the tetr a e a e ro n w e re all summits are ali e g l b h h x h d , h k but the lane a r E inth f su aso ns as are c o ntaine p ngles va y . ve n e abse nce o c h re d in th e te t t ese alo ne wou be suffi cie ntfo r th e e c usio no f all but th e o ri inal x , h ld x l g ve ro o ni fi f m c s der atio n.

NOTE XI

I tis butnatur al thatde co r atio nshould follo w and subdivide structur al lines . us inth e di ato m th e sno w cr sta and th e e c ino erm we ﬁnd w ere the Th , y l h d , , h 244 Appendix Notes

struc tur e is ase o nthe s uar e the ec or atio nimm e di ate inclines to o c ta o ns b d q , d ly g ; w er e th e outine s ar e tr ianular th e ec or atio nis e a onal and w e re th e o rm h l g , d h x g ; h f is e nta onal th e ec or ate s ac e s wi b e o un istri ute ac cor in to th e p g , d d p ll f d d b d d g s ll a r ru o ense rie s . is we a se e te is t e e e nin ar er m atters th e Norm an g ld Th , h l , v l g , st le o f archite cture e e o in th e circle the s uare and t eir attri utes whi e y d v l p g , q h b , l the G o t ic natur all li es inthe vesica i sci s th e tr aceri e win o w and o t er h y v p , d d , h c o mbinatio ns po ssible but inc o ngruous with th e se ve rity of the square and the u w al it il tri anle . So it is wit o t er o rms b t e n structur e ns and te ns e g h h f , h d y i o stre n t are ue sto ns f im or tance t e nth e e a o nco m es into its o wn. On g h q p , h h x g t N te s his subje ct o XII hould be c o nsulte d .

NOTE XII

’ Onth e su ec to f th e o rm atio no f th e e e s ce l th e c ie is ute se em s to be bj f b l , h f d p , no t w e t er th e six-si e ce is ui tto a anta e but as to w et er o r no the , h h d d ll b l dv g , h h , bee delibe rate ly and instinctively makes it hexago nal o r simply fo rm s it around r h as o f h e r o and this o m un er ressure assum es e a o nal s a e . ere b dy f , d p , h x g h p Th ,

la e ena e al f ar n t . i Coul e r te o o o um e t o n his su e ct. Mr B elo w De an t , b g d d g bj g , , and Professo r G o rto ntaking the ne gative and me no f e qual standing the afi r m a i To m mi t is e mi wr w th r u is t e . n ac a c an e is o e r waste since e e s t v y d , h d gl p d d , l inevita a Sc o tc e rdic t and the cr uci al ue stio nis no tw e t er th e bee m ak es bly h v , q , h h a circle w ic ec o mes a e a o nun er ce rtaininflue nc es a o ur in her ne e s nor h h b h x g d f v g d , w et er sh e o rm s it irect and instinct but w et er inth e final anal sis h h f d ly by , h h , y , sh e gets wh at sh e nee ds : (A) Th e gr e ate st sto r age c apacity without waste o f s ac e B th e re ate st struc tur al stren t wit o ut waste o f c o m and C th e p ; ( ) g g h h b , ( ) c o m binatio no f th e first two ne e ds with a sym m etry whic h shall fit h e r bo dy and m h t i a ake e r storing bo h po ss ble and r pid . ’ Valuable no tes o nbo th th e structur al c apac ity of th e bee s c ell and also o n th e Re aumur-Mir aldi le ge nd will be fo und inth e e xce lle nt wo rk o nGrowth and ’ Fo rm rec e nt c o m e ro m th e e no f D Arc o m so no f Uni er sit C o e e , ly f p y Th p v y ll g , n Dun e e . I re erri n to th e latter wo r Co o nel Mi is S ci ence Oc to er d f g k , l ll ( , b , t i E x wr ites m o re han nte re stingly o nth e subj ect o f sym m e tric c o nce ntr atio n. e rim e ntin wit s e re s C o o ne Mi is o e s to so m e e n t to su o rthis state p g h ph , l l ll g l g h pp ment that ther e is no arr ange m e nt po ssible giving bo th m axim um de nsity and

246 Appendix Notes

No r a XV

Th e t ree laws co nce rnin li t soun and ra it inth e co nne ction er e h g gh , d, g v y h

state d ar e th e fundame ntal basis of all investigatio ninth e subjec ts re ferre d to . The y m ay b e fo und state d inany authori ty o nphysics

No m XVI

If th e re ader will have th e patie nce no w to glance at th e tabulatio nwhic h I have c om pile d to illustrate these fac ts (se e plate h e will instantly see why th e to nic m e iant o minant and o c ta e c o ncur inth e o n rec o nize tri a o r c o m , d , d , v l g g d d m o nc o r and wh the to nic su o minant si t and to nic are also a r e e a e h d , y , bd , x h , g bl , and wh th e ominant and subdo rninant e ac o f w ic ac c o r s o r arm o nize : y d , h h h d h r o n wit th e to nic are isco ant w e n s un e to e t er . All o f t ese statem e ts h , d d h d d g h h are true all ar e ina so ute e e in wit o ur t e o r ase u o nth e c o inci e nc e , b l k p g h h y b d p d all i r o f i r atio ns and are e uc e wit e r e ct c e a ne ss ro m our ta uat o n. v b , d d bl h p f l f b l i ’ Aninstant s e aminatio ns o ws t at th e to nic m e di ant o minant and octa e x h h , , d , v c o nstitutin the c o mm o nc or all c o ncur o ne e r ourt w a e- e at o r o n ( g h d) , v y f h v b , i h u i n h o i ines 1 8 e tc . nt e ta at o t e o minant and t nc m e anwhile a in l , , 4, , , b l , d h v g m a e a itio nal c o ncurre nce s and th e to nic and oc ta e sti l o t e rs so t atint is d dd v l h , h h c omm o nc o r ine e r our i r atio ns o f the to nic t e re wi be o ne wa e-co in h d , v y f v b h ll v

c ide nce o f th e m e di ant two o f the o minant and our o f th e o cta e . Co ul result , d , f v d e er o o w t e o r m o re c o se o r b e m o re m at e m atic al e r e ct? v f ll h y l ly , h ly p f Expe nding no w a m o m e nt o nth e inve r sio no f this c ho r d pr ese nte d by the simutane ous so un in o f th e to nic ourt su o minant si t and o cta e we l d g , f h ( bd ) , x h , v ,

ﬁnd it as we s o ul ex e ct s i t e ss arm o nio us and e asin t anthe o rm er . , h d p , l gh ly l h pl g h f - F o r while the fourth and sixth c o ncur ine ve ry fourth wave be at as sho wno nlines f th e t u ti n t r o th e 6 etc . o a a o e t e e is n interve nin c o ncurrence of 3, , b l , y h g si th wit th e to nic c o rres o n in wit t at o f the o minant inth e re ious x h , p d g h h d p v

c ase . All of this must be ac ce pte d as pro o f that evenin th e music al sc ale and c o mm o n arm o n t ere is muc esi es m ere c ance and t atth e r esutis tune h y , h h b d h h l fu ause it c oul no t o ic all be an t in e se l bec d l g y y h g l . Appendix Notes 247

NOTE XVII

Acc e tin now th e di ato nic sc ale as so met in m or e t an ac ci ent let us p g h g h d , a l to it o ne or two o f the laws o f the e o m e tri c m e asurem ents fo r it must pp y g , by t is tim e b e cle ar se ent atth e aws o f soun li e t ose o f all orce and m otio n h ly h l d, k h f , are ar t and arc e o f th e in o m o f o ar o rc e and we s all ﬁnd m an exac t p p l k gd p l f , h y c orrelatio ns o n e xamining plate 44 which I have pre pare d fo r th e purpo se o f

illustr ating th e subj e ct. With th e m o stac cur ate m at em atic al m e asure m e nt th e re atio no f th e ri m e h , l p ’ circ e T3 to t at o f th e ﬁrs t ro r ess io no f th e e ui ateral trian e at t e ir l h p g q l gl T , by h re s ecti e iam e ters re r e sents i rato r if er e nce o f e ac t th e o c ta e w ich p v d , p v b y d f x ly v h ’ we a e e enstu in whi e th e se c o n ro r essio no f th e e a o nat D re r e h v b dy g , l d p g h x g p se nts th e o minant o f th e sam e to nic and t is wit th e utm o st nic et and re d , h h y p c isio n as h as e e nc are u ﬁ ure th e ac tual r atio s o f i r atio nsue rim o se , b f lly g d by v b p p d u o na tri o no m e tri c al c a cuate di a r am o f th e ro r essio ns t us ac ie in p g ly l l d g p g , h h v g anac cur ac y far beyo nd th e limits o f th e dr awing which is neverthe less th e o nly

m e ans o f illustrating the subje c t. Exam ining th e plate we se e that the vibr ato ry r ate o f o ur suppo se d to nic is m e asure d by th e di stance from T 3 o nth e o ne side to T3 at th e o the r e nd o f th e diam e te r o f th e prim e circle while th e vibr ato ry r ate o f the o ctave belo w is ’ m easur e d by th e line T to T Inspe ctio nsho ws that this latter is th e diam e te r o f th e circ e o f th e fir st ro r e ssio no f th e e ui ate r a trian e and is t e re o re as l p g q l l gl , h f , h 3 r no r r t t m it s ou be o ctantto t e T diam ete . I e o m a e he atte r c e ar to th e h ld , d k l re ade r without th e use o f lo gari thm s and all o f the par aphernalia o f scientifi c m e asurem e nts a sc a e h as e e na e to the di a r am so t atit co ul be utto an , l b dd d g h d p titut th a e C 3 f o i t ni i t. Le illustr atve te s tus subs e e ctual no t o r ur hypo thet c al o c . The nby th e sc ale o f m e asure m e nts we shall se e that the distance fro m T3 to T3 r e re se nts 1 0 6 units o f t is m e asure and t is c o rre s o n s wit the num e r o f p 5 h , h p d h b 3 i r atio ns e r se co n ne ce ssar to r o uce th e no te C . Me asurin a ain we v b p d y p d g g , ’ ’ find t at ro m to e ua s 28 o f th e sam e units o r ust a . is we re co h f T T q l 5 , j h lf Th g ’ h e o A ainm e asuri n we fin t a ni ze im m e diate is t e o cta e w o r C . d t ro m ly v b l , g g h f ’ D ’ to D is an alm o st im e rce ti e am ount ess t an800 units i in us accu p p bl l h , g v g ’ r ately th e o cular e quivale nt of th e no te G which o f course is th e do minanto f th e ’ i to nic C and is infactpr o duce d by 793 vibr ato ns pe r se cond . We m ay go further 248 Appendix Notes

‘ ’ and add be twe e nthese at M and M th e m e asure o f th e vibr atio ncausing th e — 3 ’ m e i ant i i in th e r a ius X T into e trem e and mean ro o rtio nat M d , by d v d g d x p p — o r th e r adius X T into th e sam e r atio at M and so pro ducing th e circles re pre ‘ E Th e s i se nting th e m edi ants E and fo r our to nic C . inte re tng m athem atical ac cur acy with which these statem e nts are true cano nly be di sclo se d by reco urse to th e dr y pro ce ss o f calculatio nwhich pro bably would no t interest th e reader l h butwhich is de m o nstrable neve rthe le ss . A l o f th e re stof t e no te s o f th e pre se nt stan ar iato nic scale mi t b e anal ze wit e ual inte re st but ac o f S ac e d d d gh y d h q , l k p or i s m o re t anh as e ens o wn and if rom t is it h as ee nm ad e clear th at f b d h b h , f h b th e laws o f pitch ar e geo m e tric and are pe rsiste ntly similar to tho se whi ch go vern

outwar o rm m o e ctwi l a e e e nattaine . d f , y bj l h v b d I t should b e kept inmind that instudying a music al sc ale th e vibratory relap h tions o f th e various no tes are all that ar e im por tant. T e ar bitrary standard ﬁxing th e number o f vibr atio ns whic h sh all be c alle d A h as no be ari ng o nthe su e ct since w ate er stan ar be ac c e te th e intervals wou e ar the same bj , , h v d d p d, ld b i o e t r e ac no te risin o r al in th e s an ar i i r ato t ac o e as t s e r o r o wer . h h , h g f l g d d h gh l Thus th e theo ry go ve rning th e vibr ator y inte rvals will in all standar ds be the h e r a i it o f i r ati n uo e in th e ta e m e n r sam e . T p d y v b o q t d s t ts e gar ding sound is tak en ro m th e En lis P il arm o nic stan ar ou an o t e r wo u se r e th e f g h h h d d , th gh y h ld v

pur po se as we ll . ’ ‘ Re errin a ainto ate it wi be se ent at C 3 C and C are co rre ate d f g g pl 44 , ll h , , l ro re ssio ns o f th e e uilater al trian e w ere it is c e ar t at t e ar e re by p g q gl , h by l h h y p nt i u at em atic an t o ut ar i cise o c ta . F re m a d wi re to mus c t eir vi r a ly g d h lly h g d , h b tor re atio nto e ac o t er wou be 1 0 6 28 and 26 w i c ar e t e ir re cise y l h h ld 5 , 5 , 4, h h h p i th r ti f th ni music al re latio ns . Exam ning e e la o ns o e to c and do minant from a ure m at em atic al stan o int and inac co r ance wit th e ia ram alo ne we p ly h dp , d h d g , th atthe re atio no f T3 and D is esta is e two ro essi ns f se e l bl h d by p gr o o th e hexago n. Co nsutin the ta e o f ro ressio ns inNo te it wi be o un t at th e ec im al l g bl p g I , ll f d h d i th e in h alue o f two ro re ss o ns o f e a o n te rm s o f t e rim e r a ius is . 00 and v p g h x g , p d , 75 applying this m athem atic ally to th e Philharmo nic standar d wi th whic h we started we ﬁnd t at th e i r ato r r a i it o f o minant s o u b e ac c o rdin to th e h v b y p d y d h ld , g tri o no m e tr o f the ia r am and re ar e ss o f music 2 i r atio ns er se c o nd g y d g g dl , 79 v b p ,

w ic is e ac t th e num e r se t th e stan ar . A ain te stin th e ia ram h h x ly b by d d g , g d g by th e stan ar we ﬁnd t at th e a ue o f th e m e iant is set inth e ate at th e ex d d , h v l d pl

250 Appendix Notes your missile will plunge downwar d o n a line str aight fro m its po int of re le ase to war th e c e ntre o f th e ear t but urin the s ac e o f tim e re ui re for it to d h , d g p q d re ach th e o tto m th e ear t swin s alo n and th e it wit it so t at th e si es o f b , h g g p h , h d th e it and the ine o f esce nt are no lo n e r aralle and so o ne r o r ate r th e wa l p l d g p l , l l o f the e c a atio nwil so to s e a um si ewise into th e dr o in sto ne w ic x v l , p k , b p d pp g , h h h as c o ntinue o n its o ri ina co ur se all unrnindful o f th e ac t t at as G a ileo d g l , f h , l n isto ric all and e ﬁa t ec are th e e art oe s m o e . Th e m o e m e nt of th e h y d ly d l d, h d v v sto ne h as ee n ractic all str ai t o wn but me anw i e the ear t h as si e b p y gh d , , h l , h d ! an e e es th e sto ne m steppe d d r c iv fro belo w .

NOTE XX

c aim no or i ina it for t is m e t o o f c re atin a cate nar cur e it e in I l g l y h h d g y v , b g sho wn by a number o f e ngine e rs and in c onstant use whe ne ver cate nar ies are I t is not a new i e a an w a o D ini n re uire . d e m ac to ure r and da c a d q d d , y g b k V ’ t inus t D ﬁnd similar m e ho ds e by he m . Co mpar e ure t s m e tho d o f dr awing an o Io nic V lute . NOTE XXI

For m e re c o n e nience and fo r th e ur o se s o f c o m ari so n a e o mitte v p p p , I h v d inth e textto re fer to the re c o gniz e d factth atth e he ave nly bo dies are no t perfe ct h s ere s so far as no wn. T e orm is t ato f ano ate s e ro i butt is o ate ph , k f h bl ph d , h bl ness w ile a no wn and asce rtaine uantit is ne e rt e ess so s i t as to b e , h k d q y , v h l l gh ati i n n h o f th ne gligible insuch a c alcul o nas any o f tho se nque stio . I t e case s e u n Sun Mo o n Mer cur e n s and the aste ro i s ith as ne e r e e n e e nascertai e . , , y , V , d , v v b d I nour e ar t it is a out twe nt -six mi e s inth e to ta diam e ter o r a out o ne unit h b y l l , b i one t ousan of circum ere nce w i e Mars h as s i t ar er o ate ne ss u i n h d f , h l l gh ly l g bl , J p n o nsi v t is s fﬁ i nt te r Ur anus and Ne tu e c e r a m o re . I nno case o we er is u c e , , p d bly , h , h m e nts of th e to alte r th e ge ner al state e t xt.

NOTE XXII

i ’ S e a in o nth e su e ct o f S ir a s nhis wo r Growth and Form Dr . d Ar c p k g bj p l , k , y Thompso nsays : Of se ve r al m athem atical curve s who se fo rm and de velo pment Appendix Notes 25 1

i . s ir a s t e ost im o ant th tw ma be so co nce e i . e as h m rt and e o n o wit y v d ( , p l ) p , ly h w ic we s a e al ar e t ose w ic ar e no wnas 1 th e e ua e s iral o r s ir al h h h ll d , h h h k ( ) q bl p p h o f Arc im e e s and 2 th e lo ar it mic o r e uianuar s ir al. T e o rm e r m a h d , ( ) g h , q g l p f y b e illustr ate d by th e spiral co il inwhich a sailo r co ils a ro pe upo nth e de ck ; as th e ro e is o f unifo r m t ic ness so inth e w o e s ir al coil is e ac w or o f the sam e p h k , h l p h h l bread th as that which pre ce des it and as thatwhich fo llo ws it. Using its anc ie nt efinitio n we m a efine it sa in t at if a str ai tline re o e uni o rm d , y d by y g, h gh v lv f ly a outits e tr e mit a o intw ic li e wise tra e s uni o rml a o n itwil escri e b x y , p h h k v l f y l g l d b ! th e e ua le S ir a . is itwil be no te is th e s iral e ine ate infi ure A o f q b p l Th , l d p d l d g e 6 an f h s i at 2 d or co n e nie nce ca e t e watc ri n . Fi ure B s also th e Ar c i pl , v ll d h p g g h e s t m de an pir al fro m the usual po int of illustr atio n. Co ncerning he go lde nspir al s o wn infi ure E so m et in h as a re a e e nsai inNo te and it m a fo r h g , h g l dy b d V , y o ur ur o s t t e i s n e s be rou e wi h o ar t mic ir a s as s o wninfi ur e F . O p p g p d h l g h p l , h g

the su e cto f th e o ari t mic s iral D r . o m so ninth e wo r a o e re e rre to bj l g h p , Th p k b v f d , h as this to say I nco ntr astto this (the Ar chim e de an) is th e lo gar ithmi c Spir al o f th e Nauti lus or th e snai -s e the w or s o f w ic r ad ua inc re ase in re a t and do so l h ll , h l h h g lly b d h , ‘ ina st a un a r i u i e and c n in at o . O r efinito nis as o o ws : inste a o f dy h g g d f ll If , d tr a e in wi t uni o rm e o cit our o int m o e a o n the radius ector wit a v ll g h f v l y , p v l g v h e o cit inc re asin as th e istance ro m th e o e t e nth e at escri e is a v l y g d f p l , h p h d b d it m ’ w i h i lo gar h ic spir al . E ach who rl h ch t e r adius vecto r nterse cts will b e br o ade r thanits pre de c e sso r ina definite r atio : th e r adius ve cto r will increase inle ngth in e o m e tric a ro r e ssio n as it swe e s t rou succ e ssi e e ua an e s and the g l p g , p h gh v q l gl ; e uati no th s ir o h s i al o f i am l o f e a wi b e r a . As t e r Arc m e e s inour e e q p l ll p h d , x p o f th e co i e ro e mi tb e oo e u onas a co i e c in e r so m a a o ar it mic l d p , gh l k d p l d yl d , y l g h ! s ir al in e o f th s b e i tu as a ne o i u i cas e e c re c o c e o n tse l . p , h ll , p d l d p f Ano t e r s ir a o f w ic no t in is sai inth e te t but w ic is of inte re st h p l h h h g d x , h h in an suc su e ct is w at is te chnical no wn as th e in o lute o f a cir c e as y h bj , h ly k v l e em ifie in u ate 1 e o w . we wra a stri n aro n an c lindrica o x pl d pl 55 b l If p g d y y l b dy , ne ar its ase and tie a e nci to th e e nd the line m a e th e e ncilinunwr a in b , p l , d by p pp g it ro m the c in e r wi be t at s o wn the S ir al inth e di a ram e nce its f yl d ll h h by p g , h

' ’ Leslie G tr o d Lins i i r i l n s eome urv . 1 Th s c tca i nti wi h r c himed es o w C e e . s a de cal t y f , p 4 7 p l y A d eﬁniti n lli o . To r ed e . ( , p 252 Appendix Notes

u nam e the in o ute or nwindin o f a circle . The ro cess ere escri e is a ver , v l g p h d b d y use u o ne and m a b e utto ur ose inma in an cur ves ro m the cone o f a f l y p p p pp g y , f s e ll to th e cur es o f a ie ce o f or nam e ntal r ass and is inco mm o nuse inm an h v p b , y o f the arts and cr afts.

PLATE 1 55 m vo w TE o r A c mcw NOTE XXIII

ThatD urer was a greatstude nto f m athem atic al fo r ms o ne ne e d o nly exam ine i u his wo r s to e arn. I nth e ourt oo o f his Ge o m etri ca nstt tes inﬁ ures k l f h b k l I , g u 0 1 2 etc . oes h e no t o e e ninto etai s as to h o w to c o nstr ct m o e ls o f 3 , 3 , 3 , , d g v d l d the re gular po lyhe dr al angle s ? I have found space he re o nly for his dr awing o f th e o nic olute and his re e an o arit mi c s ir a but o f alm o ste ua inte rest ar e I V f h d l g h p l , q l o t rs of hi m t w i m a b ins i t a es o f hi w r m any he s e ho ds h ch y e pe c te d n h e p g s o ks .

NOTE XXIV

I t is no new t in int is wor o f co ntr ar ies t at a m ans oul e sta lis a h g h ld , h h d b h i o r wn h re putato nfor his w k and the ngo do to poste rity under a nicknam e . I nt e c ase o f Le o nar o o f Pisa ate wit sti ur t e r ir on e cre e t atth e seudo n m d , f , h ll f h y , d d h p y r i i i ui s oul be ero ato as we l. s F o nacc so no f o o st unce t ou h d d g y l Th b , g d, p d d h gh he w as calle ro se su e rio r o we e r to t is e risio nand no t o n c olle cte but d , p h v h d ly d ori i nate m an use ul t e ore ms o f science o ne o f w ic was th e c onstr uc tio no f g d y f h , h h such a successio no f num be rs as that th e sum of any two co ntiguous o nes should u a ri t o e th e seri e s 0 1 1 2 8 e ua th e ne tinor e r . Of s c se es o n w ist q l x d h , ly x , , , , , 3, 5 , ,

254 Appendix Notes

NOTE XXVI I

Th e continuous relatio nship betwee nthe go ldense ries of e xtrem e and m e an r o o rtio nand th e um ananato m is so cle ar l sh owninth e o rm e r wor and p p h y y f k , ,

PLATE 1 56 YOUNG rum wrrnAnus UP RAIS ED

trust m th e i lustr atio n re sente inC a te r re ce in t at itt e ne e be I l p d h p VI , p d g, h l l d a e m the wa o f re e re nc e art e r t anto re intro uce ate 1 68 ro m t atwo r dd d y f f h h d pl f h k , Appendix Notes 255

w l ll r no all f th hich is he re sho wnas p ate 1 56 . I wi not sto p he e to go i t o e details s o wnint is interestin ate butwoul la em asis o nth e e trem el im ortant h h g pl , d y ph x y p fact that th e anato m y is sho wninits re latio nto a co ntinuous e xtrem e and me an i t t Th ir l r n r ser e s o f he m ost pro no unce d ype . e c c es ar e separ ate d f o m o e ano the by sp ace s inexact go lde npro po rtio nand should be com pare d with plate 1 1 7 se t inth e i r Th am w u r i l te xt o f th s wo k . e s e facts ere f rthe em phas se d by p ate 1 69 inth e o rm r r i i w i th i i i n a i ﬁ ur ai e t e atse n c e s o s o f s e e we re e ne . Re f , h h d v d g xpl d garding plate 1 56 (fo rme rly pl ate 1 68) a suﬂi cie ntstate m entwill be thatfo rm er ly m a e as o lo ws : I twi no w be oun t atth e u e r line o f t is extrem e and d , f l ll f d h pp h ( m e an) interse cts th e um bili cus while th e lo we r will m ark th e exte nsio nof th e ar ms th i i at e s de . This plate is div de d o nth e lines and ar es nece ssary for o btaining th e m athem atically co rre ct e xtre m e and m e an r atio as e xplaine d inth e passage o n t at su e ct and th e wi t o f the s o u ers ositio no f the um i icus size o f h bj , d h h ld , p b l , th e e a and e n t o f th e arm wi b e oun to c or res o n to th e rules lai h d, l g h ll f d p d d ! do wn.

NOTE XXVIII

I nsummin u th e situatio n itmi t rie be utt us g p , gh b fly p h (1 ) Doe s Natur e m ak e fre e use o f such fo rms as are num bere d am o ng the G re atMo dules ? A Th r a i wit i su i f th e uila r a nswer . e et o n Fam ts i is o ns o e te T g ly , h bd v q l trian e th e s uare the e a o n and o cta o n ar e oun ina l aws o f ra it gl , q , h x g , g , f d l l g v y , i t soun e at and o t e r orce s c o nstant and also in the o rm ations o f l gh , d , h , h f , ly , f cr stal a a l h Th e F m i o f th e s iato m s and m n o t nic a c asse s suc as t e lil . a y , d , y b l h y ly G o en Se ri es is oun to be re rese nte ine e r e nta o na o wer o f w ic ld f d p d v y p g l fl , h h t e re are no e nd ine c ino e rm s and as a me asur e o f ro or tio nand s ace is h ; h d , , p p p , oun e e r w ere inc u in as we a e se en ne ar e e r su di isio n of the f d v y h , l d g , h v , ly v y b v um an o . Th e S ir al r o u su i i e into its ari ous c asse s h as a m o st h b dy p g p , bd v d d v l , im o rtant e arin and s o ws itse no t o n dir e ct in l o ta is but ine ve rv p b g h lf ly ly phy l x , s e i r wt an o rm o i n o d f all nds . h ll , g h f k (2) Are the se fo rms ge o m e tric ? m r Answer . e ar e no t e e eo m e tric butt e ar e th e sum of t ats stem Th y ly g , h y h y s t i th a th e out n e first ch pte rs as e Gre at Mo dules . 256 Appendix Notes

(3) Doe s Nature use the m geo m etrically ? A o f nswer . Ge o m etry is mere ly our hum anm e ans understanding and classi i rtai a u s o f i Natur e ne e s n ne f t is wri te nassistan n c e n e t r e sc ence . o o t ce fy g f d h , but ne e rt e e ss sh e em o nstr ate s erse alo n ines w ic cono rm to the ter ms , v h l , d h lf g l h h f an la f r d ws o geo m e t y . (4) Has m anfo llo we d the se incre ating his bestworks ? A i nswer . Whe n m an de ve lo pe d be yo nd th e m e re st and bare st ne cessit es and e anto e auti his surroundin s he new itt e o f m ath e m aties and care b g b fy g , k l l d

ess. No twit standin w ic w e nh e rew a c arc o a e m o f his ello w m an l h g h h , h d h l gy f , h e atte m te to ut o wnw at h e saw and sinc e t at e o w m anwas ro o r p d p d h , h f ll p p tio ned in eom etrica orm and th e o l ense ries a oun e inhis str ucture the g l f g d b d d ,

carvin or r awin o we er ru e was if true inth e sam e re atio n. W enh e g d g, h v d , , l h anne a e auti ul ui din it is e i e nt t at h e use th e str uctur e o f o wers pl d b f b l g, v d h d ﬂ and o t e r ami iar natur a o e cts in ro o rtio nin his S ac e s and thus t e h f l l bj p p g p , h y n r o th r s i al st u th w s and co fo m e d t e ule as we ll . F n ly m anbe ganto udy o t e hy w e re ore s o f all t is and it is t is ro c e ss w ic we woul co ntinue . h f h , h p h h d

NOTE XXIX

I tm a sa e be sai wit out e ar o f co ntr a ictio n th ato f allth e Ar ts Musi y f ly d, h f d , , is far and awa th e m o st m at em atical e act and int is do no t re er to the y h ly x , h I f “ ue stio no f music a te m o w ich is co m arati e sim e but to th e rues o f q l p , h p v ly pl , l vibratio n and to nal pr o ductio nwhich go ve rnthe re latio nships o f th e m e mbers f i i o the diato nic sc a e and th e arm o n o f co m os t o n. l , h y p Any studious musicianwill unde rstand that th e natur al and pe rfe ctly co n str ucte sca e true to th e e ar and to its m at em atica r o or tio ns o f vi r atio n d l , h l p p b , c o ntai ns e r e ct o cta e s e r e ct ominants m e iants and o t e r inte r vals but p f v , p f d , d , h , h e will also unde rstand that this natur al and pe rfe ct sc ale is suscepti ble of no use I ts whatever inany k e y e xce pt that o f th e to nic fo r which it was c o nstructe d . to ne inte r va s as e sta lis e b the e ar and science ar e not all e ua no r are l , b h d y by , q l , t alf n e m e th e r atio o f i r atio n e twe enthe to nic and i s to es e ua . For a h q l x pl , v b b its se c o nd (a who le to ne ) is notthe sam e as betwe enthis se co nd and th e m e di ant a so c al e a w o e tone fo r if th e e r e ctscale for a sin le k e b e e amine th e ( l l d h l ) , p f g y x d, inte rval fro m th e to ni c to th e se co nd will pr o ve to b e inpro portio nto the inter val

258 Appendix Notes

ul t th co ns te d atleng h in e original wo rks o f tho se authors . I nthi s connectio nitis well to ear inmin th e su eriorit o f th e o enserie s o er the Fi o nac ci w en b d p y g ld v b , h th r o rtionin of e c r atio ns r S es is in uesti n e p po g d o o pac q o .

NOTE XXXI

Fro m re istoric a s o wnto th e m o ern s natio nal and tri al s m o ls p h d y d d gyp y , b y b int rw e n t t o se o f su stitio n m sticism a r i a e a o un e e o wi er nd e i o n. h v b d d , v h h p , y , l g th l A fe w a ain a Mo st o f ese ar e ur e l o ca . w o e er e att e wi e e e n a p y l , h v , h v d d , v wor wi e si nificance am o n w ic we m a m entio nthe svastika th e e ntac e ld d , g , g h h y , p l , n th s t ri ina i r o f w s in e r a d e cro s h e o nd sto f all o ic ar e rou e m st . As , g h y h h h d d y y to th e e ntac e so m e state m e nt h as a re a e enm a e and re ardin th e cr oss p l , l dy b d , g g , it is we ll to rem em er t at lik e m an o t er em em s a o te C risti anit it b h , y h bl d p d by h y, lo n ante ate s our reli io nas a sacre s m o l o t th e tau-cro ss and th e amm a g d g d y b , b h g t i r Th - i unt di o nor svas ika be ng e arly fo ms o f the sam e figure . e taucro ss s fre q e ly oun inE twhi e th e svastika so amiliar in n ia C ina and th e Far E astis f d gyp l , f I d , h , h h tu n h ti m as r actical ne er se e nint e lan o f t e lo s. Co ce rnin t e svas ka o p ly v d g , Th f th A uar i o Th i arl M . Fa o w e itor o e nti h as t s t sa : e a o to n e ll , d q y, h y d p by y Christians o f svastika was no doubtinfluence d by th e ide a o f th e o ccult Chr istian signifi c ance whi ch the y thought the y re co gnised and which could be use d with spe cial m e aning am o ng the mse lve s witho ut at th e sam e tim e ar o using th e ill ! fe e ling o f tho se amo ng who m the y live d . is s m o l o f w ich th e nam e is tak e n ro m th e S anscr it Su-we ll and asti Th y b , h f ( e in is r o a l th e o ests m o inco ntinuous use ein o un as ear l as th e b g) p b b y ld y b l , b g f d y a si io n t th r oini in i no f th r o nze e . I nits e ner al o t wi e a ms tn th e dir e cto e b g g p , h p g course o f th e sun it is uni e rsal co nsi er e ano m enof o o o rtune w e nce , v ly d d g d f , h , r i am isi a i i ns a e a s ts Orie nta n e . Wer e we to t n i and stu ts tr a ito p h p , l v I d dy d , stri in e am le o f th e use o f svastika wou m eetus atB h ar ah at a villa e inth e k g x p ld , g A u t nturi r Christ state o f Na o and notfar ro m Al ah a ad . o t r ee ce es e o e g d f l b b h b f , t ere was ere cte er e a Buria Mo un o r stua th e r oun anand e co rate h d h l d p , g d pl d d aili s th r t a Th e arc eo lo istHe inr ich r ng of which Sh ow e fo m of a giganti c svas ik . h g ’ Schlie m ann disco er er o f th e si t of Ho m er s ro and re e nte ss ol o wer o f , v gh T y l l f l symboli sm fr o m th e legendar y (o r was it real?) Atalantis of th e Time us to th e co ast o f Central Ameri ca du u exam les inAsia Minor co ins a e ee n oun , g p p ; h v b f d Appendix Notes 259 be ar ing it; and the co lle ctio ns o i Cypri anpo tte ry e xcavate d under th e dir e ctio no f G L i i th ro li an u f New e ner al o ui s P . di Ce sno la no w re stn n e Me t o t M seum o , g p n al ri r n i i st h Yo r S o w istinct e am e s . Ce tr Am e ca ese ts ts n ances and t e k h d x pl p , Am e ric anIndi ans were no tunfamili ar with itand we ave itintheir Navaj o blankets s t h r uch as m any o f us use for ste am er rugs . I have alre ady re ferre d o Sir T e o do e ' - H i . re t n Hall a ne W . A e a o Co o s s e tc o f t is s m o and W . R . C Mo o k k h h y b l , , , Co o pe r and o ther autho ritie s o nthe Isle o f Mangene r ally agre e inde riving th e - thre e le gge d symbo l o f that fam ous land o f Firm MacC o o le fro m thi s so ur ce . “ I n t is co nne ctionit is intere stin to no te t at the ancie nt m a of Man h g h p , by ! Cae sar c alle d Mo na and m arke d as pe rfo rm e d by JohnSpe e d in1 605 be ar s th e im rint o f t is s m o in th e u e r ri t an c o rne r w ic e e nt e n lo n p h y b l pp gh h d , h h v h , g anterio r to th e ado ptio no f th e m o tto Quocunque j eceris stabi l (which e ve r way yo u

t r o w m e stan se e m s to a e t iﬁe th e E lla nVanninVe Veen. h , I d) h v yp d g

NOTE XXXII

I ne aminin the c r o zier inco nne ctio nwit th e e rnand o lute it s oul be x g h f v , h d no te t at th e o ute is no t in ar ia l inth e o rm o f a o arit mic s ir al so m e d h v l v b y f l g h p , ’ tim e s as inth e case o f Dure r s o nic o lute ta in the cur e s of th e Archime , I v , k g v i de anfo rm nste ad.

NOTE XXXIII

’ I t wi ll have bee nfre quently no te d that bo th inNature s Harmonic Uni ty and in The Great Modules a cle ar distinc tio nwas m ade be tweentho se laws of Nature w ic were oun to a to ino r anic o rm s and to o rc e o nth e o ne h h f d pply g f f , an and to t o se w ic o e rne r o wt and italit o nth e o t er —a di stinc h d , h h h g v d g h v y h , tio n w i h as e en is in h s e su ect c em as e th e re sent a es . On t e am h h b ph d p p g bj , t o u wit entire iffe rent tre atm ent aninte restin wo r e ntit e to c o n h gh h ly d , g k , l d si er atio n inw ic t ese di stinctio ns e twee nﬁ e o rm s the static and li e d , h h h b x d f , , f , th e nami c h as re cent ee n ro u t o ut a Ham bid e un er th e title dy , ly b b gh by J y g d o D f yna mic S ymmetry. Witho ut agr ee ing to all o f th e c o nclusio ns apparently r e ac e Mr . Ham bid e a m o re e te n e re ere nce to his wo r woul ne e r h d by g , x d d f k d v th eless have be engive nhereinbutfo r th e factthatth e prese ntpage s were all set 260 Appendix Notes

u int e t ou e a e so m ew at war co nditio ns ro m u ic atio n e o re p yp , h gh d l y d h by f p bl , b f i Dyna mic S ymmetry reac he d th e publ c . NOTE XXXIV

No wo rk o f th e natur e of th e present o ne would be c o m ple te at this tim e witho uta re fe re nce to the recentinvestigatio ns re gar ding th e ac tio no f gr avity o n lightlately undertak enunde r th e le adership o f Alfre d Einsteinand re c eived bo th i ni w B u fo r he a t at h favo ur ably and unfavour ably by th e sc e tfic o rld . t t f c t h t e re sults o f these investigatio ns were plac e d befo re th e public after th e prese nt a es wer e all int e a m o re anal tic al re er ence to t em mi t be m a e but p g yp y f h gh d ; , sumcc to sa as mustb e no wnto a m o ste er re a er t atthe ne w stan tak en y, k l v y d , h d is t at th e r a s o f i t m a be e ec te th e in uenc e o f r a i t t ese c o n h y l gh y d fl d by fl g v y , h clusio ns being large ly bas e d o n o bse r vatio ns take nduring th e last ver nal so lar e er tim e wi l ro e at th e e ec tio no f th e star wa es ser e ec lipse . Wh th l p v th d fl v o b v d was caused by he at and o ther superfi cial c o nditio ns o f the so lar bo dy itself o r by i w i e h a er r a it rem a ns to be em o nstrate . Me an t e m tt rem ains o ne o f g v y d d h l , ac adem ic inter estso far as th e c o nc lusio ns o f this bo o k ar e c o nce rne d since suc h a e e ctio n w ate er its c ause o r e tent is so m inute as to a e esc a e o ser a d fl , h v x , h v p d b v tio nall these centuries and c o uld have no influenc e o nm atters o f pr o po rtio no r e u b a ty .

262 Index

Fo urth dimension, 43, 242 F s usu, 1 66 ahlia 80 D , ’ a s windo ws 22 D y , 5 escri tions correlatin see 0 D p , g , exio tro ic s irals 1 6 1 66 Ge o metric Units 1 D p p , 5 , , 4 ia o nals 2 1 2 0 Giﬂo r o r 1 6 D g , 3, 4 , 43, 4 d , L d , 7 no la 2 Glass 22 Di Ces , 59 , 5 ivine sectio n 2 2 Go ldensectio n 2 D , 7 , 5 , 7 Do 1 86 Go dense ri es 2 1 g, l . 9 . 39. 74. D olium 1 Go ldens iral 1 , 59 p , 47 D olomite Go l ch 1 , 74 dﬁn , 39 o r ho rus 1 88 Go thic o rms 2 1 2 D yp , f , uad 1 2 G ravi t 8 1 02 260 D , y, 4, , Gr t M 6 0 ea odules, , 7 n 8 1 G re ek cano , 1 8 , 89 Gr eek o umo 1 0 m n, 7 Grill 22 s, 5 Earth 0 G wilt 2 , 9 , 57 trics 1 1 6 1 E ccen , 44, mata 1 E chi noder , 79 E o f leaves 1 8 dges , 3 E tiancano n 1 88 Hae ck el 2 gyp , , 35 E tein 260 al i ata 1 ins , H i ots corr ug , 60 Eleano r Gri ll 22 Hall-Caine 2 , 5 , 59 E lli se l x 1 x Hals Franz 1 1 p . 4 . 43. 44 , , 9 Emilia 2 1 6 Ha bi 1 2 , m dge , 99 , 59 E xeter 2 1 6 228 H liant 1 1 1 , , e hus, 49 , 5 E xtr eme and m eanratio He ix 1 6 1 6 1 , l , 4 , H r 1 e m es , 1 9 H chel ers , 95 Hexa o 2 1 g n, , 77 Famil o f o ldenseri es see Go denSeri es Hexa o na cells and y g , l g l , 77, Famil o f the enta o n 2 Hi o crates 1 0 y p g , 7 pp , 3 Famil o f the te tra o n 1 6 Ho k usai vii 1 0 y g , , , 4 Fam il re atio ns 1 6 Ho ne c om y l , y b , 77 Farnese ull 2 1 2 B , Fern 22 , 4 Fi o nacci 1 6 2 b . 4. 35 . 5 . 53 Fo r ce 8 eal anle 6 6 1 , 3 Id g , 5 , Fo rman 1 I ktinu 1 20 , 75 s, 99 , 4 Index 263

f a circ e 2 2 Invo lute o l , 5 te 1 22 Io nic vo u , 53, 4 l ’ 2 1 2 60 1 00 etc . Nature s Harmonic Unity, , 4, 7, , , o rk 2 2 Iro nw , 5 1 Nautilus , 59 Ne tune 8 p , 7 vii 1 2 Newto n, , 3 0 N11111 011 03 1 uni ts , 1 hro wers 1 6 Javelint , 3 u iter 8 J p . 9

ts 2 Or bi , 45 1 1 56 za wa wave o f 1 0 Kana , , 4 22 263

1 7° 1 8 Oxali s, 3

' O iC s irals 1 6 l 66 zzo d el o vo o 226 LEBOtl P p , 5 » Pala B l . Left-hand s ira s I 6 P arnassia 1 8 p l , 4 , 4 do d a Vinci Vii 1 2 2 26 he no n 1 Leonar , . 9 . Part , 99 6 a Fi o nac ci 1 , ar o o f is ) , , 35 , 5 ro se 1 Leo n d P ( b 4 Pen , 99 Le o n eo n 2 1 ° entac e 1 1 76 Pl g , P l , 37 , 75, 00 1 02 1 2 1 2 260 n ami 2 2 Li ht 1 . . . 49 . pentago y, 7 , 43 g . 97 . f l h 1 3 1 2 I 2 t mi c s ira 1 . 5 ta ram 6 Lo gari p l , 47 . 4 . 53 Pen g , 4 , 47 , 43 uife 1 8 ia 2 1 6 Lo o se . 3 Perug , Lo utro ho ro s 1 Ph eidias 202 2 p , 43 , , 39 P hi 239 h h ni c standar d 2 8 P il arm o , 4 h nix 2 1 P oe , 9 h lo taxis 1 0 Mars 0 P yl , 5 , 9 f 28 2 1 8 P i ( ) . Meande r , bus 1 units o f 8 P inus stro , 54 Me asur e m e nts , , i sisse wa 1 8 Melde 1 0 P p , 5 , 9 ane ts 8 Mer cur 2 Pl , 7 y, 9 , 93 ular o dies 2 Millis Co o ne 2 Plato nic r eg b , 43 l l , , 44 d 1 8 1 2 P latyco on, 5 Mo nad , clitus 1 88 Mo ntere c r ess 1 6 Po y , y yp , 3 l o o n 1 2 8 Mo ses 1 P lyg , , 5 o he dr a s 66 2 o umo u 1 0 P ly l , , 43 M , 7 Pe 80 6 ppy. Murex, 1 6 t es 1 1 also So und r axi el , 9 Music 2 6 2 6 , se e P , 4 , 5 264 Index

Pro ressio ns 1 2 S ider 1 8 g , p , 3 P ramids 2 S iral eccentrics 1 6 1 y , 5 p , P r ola 1 8 S ira he lices 1 6 1 y , 3 p l , tha o ras 1 0 1 0 1 2 1 S irals 62 1 1 6 222 22 Py g , 7, 9, 9 , 75 p , , 44, 4 , , 5 1 8 S pir ea va nHouti , 5 tain 22 S ed glass, 5 6 S taircases , 22 tarﬁsh 1 8 S , 4 Mat 23 uentin sys , 5 o w r 1 1 0 1 1 Q Sunﬂ e , 49 , 5 , 5 ua rch o f 2 S s , a , 07 tik a 1 220 22 1 2 8 S vas , 75 , , , 5 S mm e tro ho ia 1 2 y p b , 9 t 1 2 Symme ry, 5 a hael 2 1 8 R p , rin e Sy g . 75 eac tio nar s ir a s 222 R y p l , - ea um ur 8 R , 7 i ht-hand s ira s 1 6 R g p l , 4 i ar c h o f 208 R mini , Ta ernac le desi n 22 o ss enman 1 b g , 7 R , D , 43 ’ T ai Tsun 22 ue ns 2 1 g , 4 R b , 7 Tem erament 2 6 us 2 1 p , 5 R g , 9 T ra 1 2 e t d , Tetra o n amil 1 6 2 g f y, , 7 “ " T r e ts 2 1 f , 9 ’ Tho m so n D Arc vm p , y, , Sachs vii , T to r 2 in etto , 1 8 t n 228 St. Jo hnLa era , Titian 2 1 8 ’ , t Ki be rt s o r der 22 S . m b , 5 Titiu o f Witten er s b g , 94 F o ri le Mura 1 6 SanPao lo u , 9 T t ch o f 206 i us , ar ,

Saturn, 89 Tri ad , 1 2 a 1 60 S cala ria pretios , T o n 1 1 8 rig , 7 , c r vi i 2 S himpe , , 35 us maxi mus 1 6 1 Trochi , h ann 2 8 S c ie m , 5 l T n 1 1 1 2 y dall. 09. 7 . 35 S c hwendener , vii ! ! “ eries see Fi o nacci Go lden Ari thm et S , b , , ! ica e tc l, . i na 20 S e , 5 k lto n 2 2 nits an lar S e , 9 U , gu , 47 Sno w cr sta s 2 Units eo m e tric 1 y l , 7 , g , 4 S o ar s stem 8 Units o f measurement 8 l y , 9 , ’ S o o m o ns sea 208 Units num erical 1 0 l l , , , n 1 1 02 1 1 1 1 ranus 8 So ud , 97 , 00, , 3, 5 U , 8