artigo the theory of pinhole

8/11/2019 Artigo the Theory of Pinhole

1/23

T ae Theory of Pinhole mages rom Antiqui ty

to be Tbir teentb Century

DAVID C. LINI BEI:m

C o m m u n i c a t e d b y M. CLAGETT

I I n t roduct i on

The principles underlying the

camera obscura

were first correctly analyzed

by IBN AL-HAITHAM. In his

Perspect iva ( i .e . ,

the th i r t een th-cen tury Lat in t rans-

latio n of his K i t ~ b a l - m a n ~ z i r ) , AL-HAITHAM descri bes the follow ing obs erv ati ons ,

which are surely equivalent to what later became known as the camera obscura:

The evidence tha t lights and colors are not intermingl ed in air or in transp aren t

bodies is th at when a nu mb er of candles are in one place, [although] in various and

distinct positions, and all are opposite an aperture tha t passes through to a da rk

place and in the dark place opposite the aperture is a wall or an opaque body, the

lights of those candles appear on the [opaque] bod y or the wall dist inct ly according

to the number of candles; and each of them appears opposite one candle along a

[straight] line passing through the aperture. If one candle is covered, only the light

opposite [that] one candle is extinguished; and if the cover is removed, the light

returns .. .. Therefore, lights are not interming led in air, but each of them is extended

along straight lines.1

Evi den tly rays of l ight passing through a small aperture (regarded as a point b y

comparison with the other dimensions involved in the experiment) rem ain dist inct

and cast a clear image on a screen behind the aperture; if all rays passing through

the aperture should be considered, it is evident that the screen receives an in-

verted and reversed image of the entire scene opposite the aperture.

But the case of p o i n t - a p e r t u r e s is a trivial version of the p roble m of images

formed through al l aperture. Since antiq uit y i t had been recognized tha t even

when the dimensions of the aperture are not negligible -- tha t is, when the aper-

ture ma y not be regarded as a point - - l ight passing through i t , under appropriate

conditions, casts an image having the shape of the light source rather than the

shape of the aperture. For example, the author of the Pseudo-Aristotelian P ro b -

l em a ta

calls at te ntio n to the anomalous fact that sunlight passing through quadri-

1 op t icae thesaurus A lha zen i Arab is l ibri sep tem, nu nc p r i mu m ed i ti a Federico

Risnero

(Basel, t 572), Bk. I, sec. 29, p. 17:

E t

significatio quod luces et colores no n

perrni scean tur ill aere neque ill corporibus diapha nis est quod, quan do in uno loco

fuerint rnulte candele in locis diversis et distinctis et fuerint ornnes opposite uni

forarnini pertranseunti ad locurn obscurum et fuerit ill oppositione illius forarninis ill

obscuro loco paries aut corpus non diaphanum, luces illarurn candelarurn apparent

super corpus vel super illurn parietern distincte seculldurn nurnerum candelarurn

illarurn, et quelibet illarurn apparet opposita uni Calldele secundurn linearn trans-

euntern per foramen. Et si cooperiatur una Calldela, destruetur lux opposita uni

candele tanturn; et si auferatur coopertoriurn, revertetur lux .... Luces ergo non

adrniscentur in aere, sed quaelibet illarum ext endit ur super verticationes rectas.

On t hi s ed it ion of IBN AL-HAITHAM'SPerspect iva and on the translation of this

work from Arabic to Latin, see my in troduc tion to the forthcoming reprin t of the

1572 (RIsNE~) edition or my Alh azen 's Theo ry of Vision and Its Recepti on in the

West, I s i s , vol. 58 (t967), pp. 322, 330.


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P i n h o l e I m a g e s 1 55

l a t e r a l o p e n i n g s s t i ll p r o d u c e s a c i r c u l a r i m a g e ; a n d h e n o t e s , a s w e l l , t h a t w h e n

t h e s u n i s e c l i p s e d , it s i m a g e o b s e r v e d t h r o u g h a s i e v e o r l e a v e s i s a l s o e c l ip s e d . 2

T h i s s e c o n d a n d m o r e f o r m i d a b l e a s p e c t o f t h e p r o b l e m w a s a l s o f i rs t s a t i s f a c t o r i ly

a n a l y z e d b y IB M A L-H AIT HA M . A l t h o u g h A L -H A IT H A M 'S t h e o r y w a s d e f e c t i v e i n

m i n o r w a y s ( fo r e x a m p l e , i n i ts la c k o f g e n e r a l i t y ) , h e r e c o g n i z e d t h a t w h a t e v e r

t h e s h a p e o f t h e a p e r t u r e , t h e i m a g e e x h i b i t s t h e s h a p e o f t h e l u m i n o u s s o u r c e

w h e n t i l e a p p r o p r i a t e r a t i o s o b t a i n b e t w e e n t h e d i s t a n c e o f a p e r t u r e t o i m a g e

a n d o f a p e r t u r e t o s o u r c e a n d b e t w e e n t i l e r a d i i o f t h e a p e r t u r e a n d t h e s o u r c e .

H e u n d e r s t o o d a ls o t h a t t h e i m a g e i s a c o m p o s i te , f o r m e d b y t i l e s u p e r im p o s i t i o n

o f n u m e r o u s o v e r l a p p i n g i m a g e s c a s t b y i n d i v i d u a l p o i n t s o r u n i t a r e a s of t h e

l u m i n o u s s o u r c e . T h i s t h e o r y o f IB N AL -H AIT HA M w a s l a t e r r e f i n e d a n d g e n e r a l i z e d

b y h i s f o u r t e e n t h - c e n t u r y c o m m e n t a t o r , K A M A L A L - D IN A L -F ~.R IS I.3

T h e e s s e n c e o f t h e e x p l a n a t i o n f o r m u l a t e d b y A L-H AIT HA M a n d A L - F A R IS I i s i l-

l u s t r a t e d i n F i g u r e i .a C i r c l e A B C D o n t h e s u r f a c e o f t i l e s u n r e p r e s e n t s t h e e x t r e m e

F i g . 1

2 B k . X V , c h a p s . 6, 9 1 1 b t - - 5 ; 1 1, 9 t 2 b 1 0 - - 1 5 .

a O n I B N A L - H A IT H A M ' S t h e o r y , s e e J o s E e H W ~ R SC H M m % Z u r T h e o r i e d e r

C a m e r a o b s c u r a b e i I b n a l H a i _ ta m , Sitzungsberichte der phy sikalis ch-m edizin isch en

Societdt in Erlangen,

v o l . 4 6 ( I 9 1 4 ), p p . 1 5 1 - - t 5 4 ; E I LH A R D W I E D E M A N N Ub e r d i e

C a m e r a o b s c u r a b e i I b n a l H a i t a m ,

Ibid. ,

p p . 1 5 5 - - t 6 9 (a G e r m a n t r a n s l a t i o n o f

iBN AL-HAITHAM'S On the Shape o/ the Eclipse); W I ED E M A NN , U e b e r d i e e r s t e E r -

w ~ it in un g d e r D u n k e l k a m m e r d u r c h I b n a l H a i t a m , Jah rbuch [i~r Photographie un d

Reprodukt ions technik ,

v o l . 2 4 ( 1 91 0 ), p p . t 2 - - 1 3 . O n A L -F X R IS I,

s e e W I E D E M A N N

l J b e r

d i e E r f i n d u n g d e r C a m e r a o b s c u r a ,

Verhandlungen der deutschen p hys ika l i schen

Gesellsehaft, v o l . 1 2 ( 1 9 1 0 ) , p p . a 7 7 - - t 8 2 .

AL-HA_ITHAMa n d A L-F AR IS I d i d n o t e x p r e s s t h e i r t h e o r i e s i n p r e c i s e l y t h e s a m e

t e r m s a s I w i l l e m p l o y ; n o r d i d t h e y u t i l iz e a s p h e r i c a l s o u r c e a n d a t r i a n g u l a r a p e r t u r e .

H o w e v e r , m y e x p l a n a t i o n d o e s n o v i o l e n c e t o t h e i r t h e o r i e s c o n c e p t u a l l y .


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t 56 D .C . LINDBERG:

p o i n t s f r o m w h i c h r a y s o f l i g h t c a n d e s c e n d t h r o u g h t h e t r i a n g u l a r a p e r t u r e .

R a y s f r o m p o i n t A p a s s th r o u g h t h e a p e r t u r e t o i l l u m i n a t e a s m a l l t r i a n g u l a r

a r e a a t A ' . R a y s f r o m p o i n t C i l l u m i n a t e a s i m i l a r a r e a a t C ' , a n d s i m i l a r l y fo r

o t h e r p o i n t s o n a n d w i t h i n c i r c l e A B C D . I t is e v i d e n t t h a t r a y s e m a n a t i n g f r o m

a l l p o i n t s o n c i r c l e A B C D i l l u m i n a t e o v e r l a p p i n g t r i a n g u l a r a r e a s l o c a t e d o n

c i r c l e A ' B ' C ' D ' , t h u s p r o d u c i n g a c i rc u l a r i m a g e o f t h e s u n w i t h i n c i r c l e A ' B ' C ' D ' .

S h o u l d t h e s iz e o f t h e a p e r t u r e b e d e c r e a s e d o r t h e d i s t a n c e b e t w e e n t h e s u n

a n d t h e a p e r t u r e i n c r e a s ed , t h e t r i a n g u l a r p a t c h e s o f i l l u m i n a t i o n w o u l d e x h i b i t

a c o r r e s p o n d i n g d e c r e a s e i n s i ze , w h i c h w o u l d l e a d t o a s h a r p e n i n g o f t h e i m a g e .

N o w i t i s e v i d e n t f r o m t i l e f i g u re t h a t t h e s h a p e a n d c l a r i t y of t h e i m a g e a r e

d e t e r m i n e d b y t w o f a c t o r s : ( t) t h e t r i a n g u l a r i t y o f t h e i m a g e s c a s t b y i n d i v i d u a l

p o i n t s o f t h e l u m i n o u s b o d y , a r i s in g f ro m t i le s h a p e o f t h e a p e r t u r e ; a n d (2 ) t h e

c i r c u l a r p a t t e r n i n w h i c h t h e s e t r i a n g u l a r im a g e s a r e s u p e r i m p o s e d , a r i s in g f r o m

t h e s h a p e o f t h e l u m i n o u s b o d y . N e a r t h e a p e r t u r e t h e f o r m e r f a c to r p r e d o m i n a t e s ;

h o w e v e r , a s th e d i s t a n c e b e t w e e n t h e a p e r t u r e a n d t h e s c r e e n is i n c r e a s e d , t h e

l a t t e r f a c t o r g r a d u a l l y c o m e s t o p r e d o m i n a t e .

T h e W e s t e r n h i s t o r y o f t h i s t h e o r y o f i m a g e s f o r m e d b y a p e r t u r e s o f / i t c i t e

s iz e - - w h i c h I s h a l l r e fe r to a s t h e t h e o r y o f p i n h o l e i m a g e s - - a p p e a r s t o b e

e n t i r e l y i n d e p e n d e n t o f AL -H A IT HA M a n d A L -F A R IS I. T h e p a s s a g e q u o t e d f r o m

A L-H AIT HA M a b o v e , w h i c h o u t l in e s h i s t h e o r y o f p o i n t - a p e r t u r e s , c o n s t i t u t e s h i s

o n l y d i s c u s s i o n o f p i n h o l e i m a g e s o r t h e c a m e r a o b s c u r a i n t h e P e r s p e e t i v a . H i s

f a r m o r e e l a b o r a t e t h e o r y o f p i n h o l e i m a g e s f i n it e i n s iz e a p p e a r s o n l y i n a n o t h e r

t r e a t i s e , O n t h e S h a p e o / t h e E c li p se , a n d n e i t h e r t h i s n o r A L - FA R IS I'S c o m m e n t a r y

o n i t w a s a v a i l a b l e i n a n y l a n g u a g e e x c e p t A r a b i c b e f o r e t il e t w e n t i e t h c e n t u r y . 5

A c o m p a r a b l e a n a l y s i s w a s n o t t o b e f o u n d i n t i l e W e s t b e f o r e th e p u b l i c a t i o n

o f J O H A N N E S K E P L E R ' S A d V i t e l li o n e m p a r a l i p o m e n a ( 1 6 0 4 ) a n d F R A N C E S C O

5 W E RN ~E R'S s u g g e s t i o n t h a t L E ON A RD O D A V IN C I m a y s o m e h o w h a v e g a i n e d a c c e s s

t o A L - F ~ R IS I 'S c o m m e n t a r y n e e d n o t b e t a k e n s e r io u s l y , f o r t h e r e s e e m s t o b e l i t t l e

i f a n y t h i n g i n L ]E O NA R DO 'Sm a n u s c r i p t s t h a t h e c o u l d n o t h a v e o b t a i n e d f r o m P EC HA M

or WlTELO

w h o s e w o r k s h e k n e w ; c]. OTTo WXRNER, Z u r P h y s i k L e o n a r d o d a V i n c i s

( E r l a n g e n , t 9 1 0 ) , p p . 1 t 0 - - 1 1 t . T h a t L E O N A RD O k n e w P E C ~I,A M 'S 7~erspectiva com-

m u n i s i s e v i d e n t f r o m t h e f a c t t h a t h e q u o t e s f r o m i t s p r e f a c e ( se e A N I)R ~

CHASTEL

A r t e t h u m c ~n is m e ~ F l o r e n c e a u t e m p s d e L a u r e n t l e M a g n i / i q u e [Pa r i s , a959] , p . 417) ;

L EO N AR D O c i t e s W IT X LO o n s e v e r a l o c c a s i o n s i n h i s n o t e b o o k s ( s ee T h e N o t e b o o k s

o / Leona rdo dot Vin c i , e d . E D W A R D M A C C U R D Y , V O1. 1 [ N e w Y o r k , t 9 5 8 ] , p p . 99 6 ,

1168 , 1 t 71 ). On LEONARDO an d p i nh o l e im ag es , see a l so EU C-~NE MfJNTZ, L 6 o n a r d

d e V i n c i e t l ' i n v e n t i o n d e l a C h a m b r e n o tr e ,' R e v u e s c i e n t i / i q u e , Ser i es 4 , vo l . 10

( 18 9 8) , p p . 5 4 5 - - 5 4 7 ; M .A .R IO G L IO Z ZI, L ' i n v e n z i o n e d e l l a c a m e r a o s c u r a , A r e h e i o n ,

v o l . 1 4 ( t 9 3 2 ) , p p . 2 2 1 - - 2 2 9 .

I t i s e q u a l l y i m p r o b a b l e t h a t L ~ v I B EN G ER SO N , w h o p r o v i d e d o n e o f th e b e t t e r

a c c o u n t s o f p i n h o l e i m a g e s i n t h e M i d d l e A g e s , c o u l d h a v e k n o w n t h e w o r k s o f

A L -H A IT H AM ( a p a r t f r o m h i s P e r s p e c t i v a ) a n d A L -F X R IS I; t h e r e i s n o e v i d e n c e t h a t

e i t h e r of t h e s e w o r k s w a s t r a n s l a t e d i n t o L a t i n o r H e b r e w , a n d i t is u n l ik e l y t h a t

L z v I k n e w m o r e t h a n a s m a t t e r i n g o f A r a b i c . O n L E v i ' s t h e o r y o f p i n h o l e im a g e s ,

see MAXIMILIAN CIIRTZI~ D i e D u n k e l k a m m e r . E i n e U n t e r s u c h u n g t i b e r d i e V o r g e -

s c h i c h t e d e r s e l b e n , H i m m e l u n d E r d e , v o l . 1 3 (1 9 01 ), p p . 2 2 5 - - 2 3 6 , w h e r e L E v i ' s

t h e o r y i s d i s c u s s e d a n d t h e r e l e v a n t t e x t s p u b l i s h e d ; c / . J o s E P ~ I CARLEBACH L e w i

b e n G e rs o n a ls M a t h e m a t i k e r ( B e rl in , 1 9 10 ), p p . 3 0 - - 3 4 . O n L x v I ' s k n o w l e d g e o f A r a b i c ,

s e e M a x S EL IG SO I~ N, L e v i b e n G e r s h o n , i n T h e J e w i s h E n c y c l o p e d i a ( N e w Y o r k ,

19 01 - -19 06 ) , vo l . 8 , pp . 2 6 - -2 8 ; c / . ~X~IMAH . ALDERBLUM A Sdy o G e r s o n i d e s i n

H i s P r o p e r P e r s p e c t i v e ( N e w Y o r k , 1 92 6), p . 3 t .


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MAUROLYCO'S

Photismi de lumine et umbra

( 1 6 1 t) . O t h e r s w h o h a v e b e e n c r e d i t e d

w i t h t h e f i r s t c o r r e c t W e s t e r n a n a l y s i s o f t h e camera obscura s u c h a s L E V I B EN

GERSON, LEONARDO DA VINCI, GIROLAMO CARDANO,

a n d

GIAMBATTISTA DELLA

PORTA, c o n t r i b u t e d l i tt l e i f a n y t h i n g t o t h i s a s p e c t o f t h e p r o b l e m . 6

L e s t i t a p p e a r t h a t I a m u n d u l y c o n c e r n e d a b o u t a t ri v i a l d i s ti n c ti o n - - s m a l l

a p e r t u r e s a n d l a rg e a p e r t u r e s a r e , a f t e r al l, m e m b e r s o f a s i ng l e c o n t i n u u m - - l e t

m e p o i n t o u t t h a t h i s to r ic a ll y , f in i te a n d p o i n t a p e r t u r e s h a v e c o n s t i t u t e d t w o

d i s t i n c t c a s e s. M a n y o f t h e h i s t o r ic a l f ig u r e s a s s o c i a t e d w i t h t h e camera obscura

d e a l t w i t h o n l y o n e o r th e o t h e r o r , i f t h e y d e a l t w i t h b o t h c a s es , d id s o i n s e p a r a t e

w o r k s . M o r e o v e r , t h e r e i s v e r y l i t t l e r e s e m b l a n c e b e t w e e n t h e r e s p e c t i v e t h e o r i e s .

I f t h e a p e r t u r e m a y b e r e g a r d e d a s a p o i n t , t h e e n t i r e t h e o r y c o n s is t s of t w o s t a t e -

m e n t s : ( t ) l i g h t i s p r o p a g a t e d i n s t r a i g h t l i n e s , a n d ( 2 ) l i g h t r a y s i n t e r s e c t i n g i n

t h e a p e r t u r e d o n o t m i n g l e . I n d e e d , I B N A L -H A IT I- IA M 'Sa n a l y s i s o f t h e i m a g e s

f o r m e d b y c a n d le s t h r o u g h a p o i n t - a p e r t u r e h a d n o o t h e r p u r p o s e t h a n t o p r o v e

t h e s e t w o p r o p o s i ti o n s . T h e r e i s t h u s n o t h i n g v e r y a b s t r u s e a b o u t e it h e r t h e

p h e n o m e n o n o r i t s e x p l a n a t i o n .

B y c o n t r a s t , i f t h e a p e r t u r e i s r e g a r d e d a s h a v i n g d i m e n s i o n s o f s i g n i f ic a n t

s iz e , t h e q u e s t i o n o f s h a p e e n t e r s t h e p r o b l e m , a n d d i f f ic u l t q u e s t io n s a r e p o s e d :

I f t h e a p e r t u r e i s s q u ar e , w h y is th e i m a g e r o u n d ? O r , t o t a k e a c c o u n t o f t h e

p h e n o m e n a i n m o r e d e t ai l, w h y d o e s t h e i m a g e h a v e t h e s h a p e o f t h e a p e r t u r e

d i r e c t l y b e h i n d t h e a p e r t u r e a n d t h e s h a p e o f t h e l u m i n o u s o b j e c t f a r b e h i n d

t h e a p e r t u r e ? A n d t o r a i s e t h e s e q u e s t i o n s w a s , a s i t t u r n e d o u t i n t h e e v e n t ,

t o c a s t d o u b t o n t h e r e c t i li n e a r p r o p a g a t i o n o f l ig h t o r to s u g g e s t t h a t o n e m u s t

c o n s i d e r s e c o n d a r y a s w e l l a s p r i m a r y r a d i a t i o n . M o r e o v e r , a p r o p e r s o l u t i o n t o

t h e p r o b l e m o f f i n it e a p e r t u r e s r e q u i r e s a g e o m e t r i c a l a n a l y s i s o f p y r a m i d s i s s u in g

f r o m t h e lu m i n o u s b o d y a n d a n u n d e r s t a n d i n g o f t h e m a n n e r i n w h i c h t h e im a g e s

o f i n d i v i d u a l p o i n t s a r e s u p e r im p o s e d t o p r o d u c e a c o m p o s i t e i m a g e - - e l e m e n t s

e n t i r e ly a b s e n t f r o m t h e t h e o r y o f p o i n t - a p e r t u r e s .

I f , t h e n , w e a r e p e r m i t t e d ( o r o b l i g a te d ) t o c o n s i d e r t h e t h e o r y o f f i n it e a n d

p o i n t a p e r t u r e s s e p a r a t e l y , a n d i f M AU RO LY CO a n d K E PL ER f o r m u l a t e d t h e f i r st

c o m p l e t e l y c o r r e c t W e s t e r n a n a l y s e s o f i m a g e - f o r m a t i o n b y f in i te a p e r t u r e s , t h e

q u e s t i o n o f a n t e c e d e n t s n e c e s s a r i l y a ri se s . T h e v e r y la n g u a g e i n w h i c h c l a i m s o n

b e h a l f o f M A UR OL C O a n d K E P LE R a r e c o u c h e d - - f i r s t c o r r e c t a n a l y s i s - -

i m p l i e s t h e e x i s t e n c e o f p r i o r in c o r r e c t a n a l y s e s . W h o m a d e t h e s e p r i o r a t t e m p t s ?

W h y d i d t h e p r o b l e m i n t e r e s t t h e m ? W h y w e r e t h e y u n s u c c e s s fu l ? D i d t h e y

c o n t r i b u t e i n a n y s i g n if ic a n t w a y t o t h e e v e n t u a l t h e o r y ? U n t i l t h e s e q u e s t i o n s

6 LBvI a nd LEONARDO, for example , accurat ely describe the phenomena of large

a p e r t u r e s ; b u t L EO NA RD O o f fe r s n o e x p l a n a t o r y a n a l y s is w h a t s o e v e r , a n d L E v i ' s

a n a l y s i s , a l t h o u g h o n t h e r i g h t t r a c k , i s h a r d l y a d v a n c e d e n o u g h t o c o n s t i t u t e a

c o m p l e t e a n d a c c u r a t e a c c o u n t .

Bes ide s t hose a l r ead y c i ted , t he fo l lowing wo rks dea l w i th t he h i s t o ry o f p inho l e

i m a g e s a n d t h e camera obscura: P I E R R E

DUI-IEM

Le syst~me du monde ( P a r i s , t 9 1 3 - -

1959), vol. 3, pp. 505 -- 5t 7; vol. 5, pp. 203- -205; GUILIELMO LIBRI, Histoire des

sciences mathdmatiques en Italie, vol. 4 (Paris, 1841), pp. 303--314; J MESWATER-

HOUSE, Camera Obscura: History, ill The Encyclopaedia Britannica ( t t t h e d . ) ,

vol. 5 (New York , 1910), pp. 105 --1 07; JosEPH W/JRSCHMIDT, Zur Geschichte ,

Theorie und Praxis der Camera obscura, Zeitschri/t [~r mathematischen und natur-

wissenseha/tlichen Unterricht aller Schulgaltungen, vol. 46 (1915), pp. 466- -476.


5/23

1 58 D . C . L IN D B E R G :

h a v e b e e n a n s w e r e d , t h e h i s t o r y o f t h e

c a m e r a o b s c u r a

w i l l b e i n c o m p l e t e . O n e o f

t h e m o s t i m p o r t a n t c h a p t e r s i n th i s

p r e h i s t o r y

i s t h e p e r i o d l e a d i n g u p t o a n d

i n c l u d in g t h e t h i r t e e n t h c e n t u r y w i t h i t s g r a n d s y n t h e s i s o f o p t i c a l k n o w l e d g e ,

c r e a t e d b y R O G ER B AC ON , W l T E L O , a n d J OH N PE CH AM . I n t h e 1 2 5 0 ' s - - t 2 7 0 ' s ,

a l l t h r e e o f t h e s e m e n u n d e r t o o k a n a l y s e s of i m a g e - f o r m a t i o n b y a p e r t u r e s o f

f i n i t e s iz e - - W l T E L O o n c e , BA CO N t w i c e , a n d P EC HA M f o u r t i m e s . T h e r e m a i n d e r

o f t h i s e s s a y w i l l a t t e m p t t o e l u c i d a t e t h e i r t h e o r i e s o f p in h o l e i m a g e s a s w e l l a s

t h e a n t e c e d e n t t h e o r i e s u p o n w h i c h t h e i r s w e r e b a s e d .

I I . P s e u d o - A r i s t o t le A l k i n d i a n d P s e u d o - E u c l i d

T h e P s e u d o - A r i s t o t e l i a n P r o b l e r n a t a d e s c r i b e d t h e p h e n o m e n a a n d p o s e d th e

p r o b l e m o f p i n h o le i m a g e s :

W h y

i s i t t h a t w h e n t h e s u n p a s s e s t h r o u g h q u a d r i -

l a t e r a ls , a s fo r i n s t a n c e i n w i c k e r w o r k , i t d o e s n o t p r o d u c e a f ig u r e r e c t a n g u l a r

i n s h a p e b u t c i r c u l a r ? 7 T h e a u t h o r o f t h i s t r e a t i s e a ls o a t t e m p t e d a n e x p l a n a t i o n

o f t h e p h e n o m e n a . H e a d m i t t e d , a t t h e o u t s e t , t h a t i f s o l a r r a y s a r e r e c t i l in e a r ,

w h e n i n c i d e n t t h r o u g h a n a p e r t u r e h a v i n g r e c t i l in e a r s id e s t h e y m u s t p r o d u c e

a n i m a g e b o u n d e d b y s t r a i g h t l i ne s . H o w e v e r , t h e r a y s g r a z i n g t h e e d g e s of t h e

a p e r t u r e ( or p a r t i a l l y c u t o ff b y t h e b o u n d i n g m a t e r i a l ) a r e w e a k a n d t h e r e f o r e

a r e n o t p e r c e i v e d ; c o n s e q u e n t l y t h e i m a g e i s s e n s i b l y r o u n d .

F o r i f t h e r a y s w e r e s t r a i g h t t h e f i gu r e f o r m e d b y t h e s u n w o u l d n e c e s s a r i l y b e

b o u n d e d b y s t r a i g h t l in e s. F o r w h e n t h e r a y s f a l l s t r a i g h t o n t o a s t r a i g h t l in e t h e y

d o p r o d u c e a r e c t i li n e a r f ig u re . N o w t h i s is w h a t h a p p e n s w i t h t h e r a y s ; f o r t h e y

f a ll o n t h e s t r a i g h t l in e o f t h e w i c k e r w o r k a t t h e p o i n t w h e r e t h e y s h i n e t h r o u g h ,

a n d t h e s e a r e s t r a i g h t l i n e s, so t h a t t h e y f a l l a s a s t r a i g h t l in e . ]B u t b e c a u s e t h o s e

p a r t s o f t h e r a y s w h i c h a r e c u t o ff a t t h e e x t r e m i t i e s o f t h e s t r a i g h t l in e s a r e w e a k ,

t h e p a r t s a t t h e a n g l e a r e n o t v i s i b l e ; b u t a l l t h a t i s s t r a i g h t in t h e c o n e p r o d u c e s

a l in e , b u t t h e r e s t p r o d u c e s n o l in e , b u t t h e r a y s f a l l w i t h o u t b e i n g n o t i c e d , s

T h e a u t h o r c o r r o b o r a t e s t h is t h e o r y b y d e s c r i b in g s e v e ra l o th e r e x a m p l e s i n

w h i c h w e f a i l t o p e r c e i v e w h a t i s r e a l l y th e r e : s e e n f r o m a d i s t a n c e , s q u a r e s

a p p e a r c i r c u la r , a n d s p h e r e s s e e m t o h a v e f l a t s u r fa c e s.

I n o r d e r t o e x p l a i n th e c r e s c e n t - s h a p e d i m a g e p r o d u c e d w h e n a s o l a r e c l ip s e

i s o b s e r v e d t h r o u g h a s m a l l o p e n in g , t h e a u t h o r o f t h e

P r o b l e m a t a

i n t r o d u c e s a n

e l e m e n t t h a t w a s t o b e c o m e s t a n d a r d f a r e i n l a t e r d i s c u ss i o n s . T h e r e a r e t w o c o n e s

o f i l l u m i n a t i o n , j o i n e d t o g e t h e r a t t h e i r v e r t i c e s - - o n e co n e l o c a t e d b e t w e e n t h e

s u n a n d t h e a p e r t u r e , t h e o t h e r b e t w e e n t h e a p e r t u r e a n d t h e e a r t h . ( I t i s i m p l i c i t

7 B k . X V , c h a p . 6 , 9 1 l b t , t r . W . S . H E T T ( L o n d o n , 1 9 3 6 ), v o l . 1, p . 3 33 .

s I b i d . ,

9 1 1 b 5 - - 2 0 ,

ed. cit ,

v o l . 1 , p p . 3 3 3 - - 3 3 5 . E . S . F O R ST E R'S t r a n s l a t i o n o f

t h i s s a m e p a s s a g e ( T h e W o r k s o / A r is t o t le , e d . W . D . R o s s , v o l . 7 [O x f o r d , t 9 2 7] )

v a r i e s c o n s i d e r a b l y i n d e t a i l, b u t w i t h o u t i n t r o d u c i n g a n y d i ff e re n c e s t h a t a r e e s s e n ti a l

f o r o u r p u r p o s e s . F O RS TE R'S t r a n s l a t i o n f o l lo w s : F o r t h e f i g u r e a l s o f o r m e d b y t h e

s u n m u s t b e c o n t a i n e d b y s t r a i g h t l in e s, if t h e r a y s a r e s t r a i g h t ; f o r w h e n t h e y f a ll

i n a s t r a i g h t l i n e o n t o a s t r a i g h t l i n e , t h e y f o r m a f i g u r e c o n t a i n e d b y s t r a i g h t l i n e s .

A n d t h i s is w h a t h a p p e n s w i t h t h e r a y s ; f o r t h e y f a ll o n t h e s t r a i g h t l in e o f t h e w i c k e r-

w o r k , a t t h e p o i n t w h e r e t h e y s h i n e th r o u g h , a n d a r e t h e m s e l v e s s t r a ig h t , s o t h a t

t h e i r p r o j e c t i o n is a s t r a i g h t l in e . B u t b e c a u s e t h e p a r t s o f th e v i s i o n w h i c h a r e c u t

o f f t o w a r d s t h e e x t r e m i t i e s o f t h e s t r a i g h t li n e s a r e w e a k , t h e p a r t s o f t h e f i g u r e

a b o u t t h e a n g l e s a r e n o t s e e n ; b u t w h a t t h e r e i s o f s t r a i g h t li n e i n th e c o n e d e s c r ib e s

a s t r a i g h t l i n e, w h i le t h e r e s t d o e s n o t , b u t t h e s i g h t f a ll s o n p a r t o f t h e f i g u r e w i t h o u t

p e r c e i v i n g i t .


6/23

P i n h o l e I m a g e s t 5 9

i n t h i s d e s c r i p t i o n t h a t t h e p a i r o f co n e s is c o m p o s e d o f r e c t i l in e a r r a y s i n t e r -

s e c t i n g a t t h e c o m m o n v e r t e x . ) C o n s e q u e n t l y , w h e n a r a y is r e m o v e d f r o m t h e

u p p e r c o n e , a s d u r i n g a n e c l ip s e , a c o r r e s p o n d i n g r a y i s e l i m i n a t e d f r o m t h e l o w e r

c o n e 2 A c r e s c e n t - s h a p e d l u m i n o u s b o d y w i l l t h e r e f o r e g i v e ri s e t o a c r e s c e n t -

s h a p e d i m a g e , a s i n a s o l a r e c l ip s e . N e v e r t h e l e s s ,

s u c h

c r e s c e n t s a r e n o t f o r m e d

b y t h e m o o n , w h e t h e r i n e cl ip s e o r w a x i n g o r w a n i n g , b e c a u s e t h e r a y s f r o m i t s

e x t r e m i t i e s a r e n o t c l e a r - c u t , b u t i t s h e d s i t s l ig h t E w ith s t r e n g t h a n d c l a r i t y

o n ly ~ f r o m t h e m i d d l e , a n d t h e m i d d l e p o r t i o n o f th e c r e s c e n t i s b u t s m a l l . 1

T h i s e r r o n e o u s c o n c l u s i o n w a s t o b e r e p r o d u c e d b y I B N A L -H A IT H A M i n h is t r e a -

t i s e , O n t he S h a p e o~ t h e E c l i p s e .

A L I


7/23

t 6 0 D . C . L I N D B E R G :

tho ugh VOGL has speculated th at propositions 4 -- 6 depend on ALKIRDI'S

D e

a s p e c t i b u s . 14

Proposition 9 of

D e s p e c u l is

is essentially a recapitulation of ALKINDI'S proof

of the rectilinear propagation of light, described above. The only difference -- al-

though an imp orta nt one - - is tha t it turn s ALI{INDI'S proof around. Employ ing

the fact of the rectilinear propagation of light as a premise, the author proves

that the image cast by the sun (A) through aperture

A B

(Fig. 3) is wider than

the aperture.

K I

Z H T

Fig. 3

Prop osi tion t0 tu rns to the problem, first raised b y PSEUDO-ARISTOTLE and

destined for a place of great importance in subsequent discussions, of an eclipse

observed through an aperture. The author recognizes that

w h e n t h e s u n i s e c l i p s e d , t h a t p a r t o f i t s l i g h t w h i c h e n t e r s t h r o u g h a n o p e n i n g i s

d i m i n i s h e d , i . e . i t i s n o t r o u n d ; n o r i s i t s i n c i d e n c e o n t h e s u r f a c e o f t h e e a r t h a c -

c o r d i n g t o l i n e

T Z

[ F i g . 3 1 , b u t a c c o r d i n g t o t h e a m o u n t o f t h e e c l i p s e , a n d t h e

d i m i n u t i o n o f t h e l i g h t i s p r o p o r t i o n a l t o t h e d i m i n u t i o n o f [ t h e s u n p r o d u c e d b y ]

t h e e c l i p s e . I ~

T h e a u t h o r o f

D e s p e c u l i s

a p p e a r s t o h a v e a f i r m g r a s p o f t h e e s s e n t i a l p h e n o m e n a

o f p i n h o l e i m a g e s . H o w e v e r , a s h e a d d s m o r e g e o m e t r i c a l d e t a i l , h i s f u n d a m e n t a l

l a c k o f u n d e r s t a n d i n g i s r e v e a l e d :

L e t t h e s u n b e e c l i p s e d ; a n d l e t i t s e c l i p s e d p a r t b e a r c

D G U

a n d t h e p a r t o f i t s

b o d y t h a t t h e n s e n d s f o r t h t h e r a y b e a r c

D E U .

A n d l e t t h e r a y o f a r c

D E U

p a s s

t h r o u g h t h e a p e r t u r e t o t h e s u r f a c e o f t h e e a r t h , w h i c h i s E r e p r e s e n t e d b y ] l i n e

T Z .

T h e r e f o r e , l i n e T H i s t h e p a r t o f t h e r a y i n c i d e n t o n t h e e a r t h ( i . e . T Z . A n d E t h e

i l l u m i n a t i o n r e p r e s e n t e d b y ] l i n e

H Z

i s r e m o v e d f r o m t h e r a y a b o v e t h e s u r f a c e o f

t h e e a r t h , f o r i t i s t h a t E p a r t o f t h e e a r t h ] w h i c h i s i l l u m i n a t e d b y t h e r a y f r o m a r c

1 4 B J S R N B O & V O G L ,

o p . c i r . ,

p . 1 1 8 . O n t h i s t r e a t i s e , s e e a l s o S E B A S T I A N V O G L

~ b e r d i e ( P s e u d o - ) E u k l i d i s c h e S c h r i f t , D e s p e c u l i s ' ,

A r c h l y ] i ~ r d i e G e s c h i c h t e d e r

N a t u r w i s s e n s c h c ~ ] t e n u n d d e r T e c h n i k

vol. 1 (1909), pp. 419--435.

15 B J { S R N B O V O G L ,

Op

cir.

pp. 102--103: Culn autem sol eclipsatur, tunc illud

luminis eius, quod per fenestram ingreditur, diminutum est, scilicet non rotundum;

neque casus eius super superficiem terrae est sicut ]inea tz, sed est secundum quanti-

ta tem eius eclipsis; e tes t diminutio luminis proportionalis diminutioni eclipsis.


8/23

P i n h o l e I m a g e s 161

DGU. T h e r e f o r e t h e p r o p o r t i o n o f l i n e HZ t o a r c DGU is th e s a m e a s t h e p r o p o r t i o n

o f l i n e

TH to

a r c

DE U.T

T h e a u t h o r s e e m s n o t t o r e a l iz e t h a t , a c c o r d i n g t o h is t h e o r y , l i g h t fr o m a r c

D E U

m u s t b e n d a r o u n d t h e c o r n e r a t A t o i l l u m i n a t e r e g i o n

H T .

P r e s u m a b l y

h e h a s b e e n m i s l e d b y h i s b e l i e f t h a t t h e i m a g e m u s t b e e c l i p s e d i n t h e s a m e

p r o p o r t i o n a s t h e s u n , w h i c h , f r o m a c a s u a l l o o k a t t h e g e o m e t r y o f t h e f i g u re ,

w o u l d s e e m to r e q u i r e t h a t a r c

D E U

i l l u m i n a t e r e g i o n

H T

a n d a r c

DGU

i l l u m i n a t e

r e g i o n HZ.

B e f o re l e a v i n g th e P s e u d o - E u c l i d e a n De speculis, w e m u s t e x a m i n e o n e fu r t h e r

p r o p o s i t i o n , w h i c h ( as i t a p p e a r e d i n De speculis) h a d n o t h i n g a t a l l t o d o w i t h

p i n h o l e im a g e s , b u t w h i c h w a s t o b e g i v e n a p r o m i n e n t p l a c e i n W IT E L O 'S t h e o r y

o f p i n h o l e i m a g e s i n t h e t h i r t e e n t h c e n t u r y . T h e p r o b l e m t o w h i c h p r o p o s i t i o n 1 t

i s a d d r e s s e d i s t h a t o f a c c o u n t i n g f o r t h e f a c t t h a t r a y s i s s u i n g f r o m t h e c e n t e r

o f t h e s u n t o o b s e r v e r s s e p a r a t e d b y a s m a l l d i s t a n c e a r e s e n s i b l y p a r a l le l . T h e

e x p l a n a t i o n a p p e a r i n g i n t h e e n u n c i a t i o n o f t h e p r o p o s i t i o n - - s o l a r r a y s a p p e a r

p a r a l l e l t o s e n se b e c a u s e o f t h e s m a l l n e ss o f t h e s p a c e b e t w e e n t h e m c o m p a r e d

t o t h e i r l e n g t h - - i s s a t i s f y i n g e n o u g h , b u t u n f o r t u n a t e l y a n a b s u r d d e m o n -

s t r a t i o n f o ll ow s . S u p p o s e r a y s

A B

a n d

AG

( F ig . 4 ) is s u i n g f r o m t h e c e n t e r o f

G N L Z M S

F i g . 4

t h e s u n , a n d d r a w t h e r e m a i n i n g l in e s i n su c h a w a y t h a t E H e q u a l s HG a n d D T

e q u a l s TB. ( A l t h o u g h n o t a s s e r t e d b y t h e a u t h o r , i t w a s s u r e l y o b v i o u s t h a t GN

e q u a ls N L a n d

M S

e q u a l s

SB.)

I t

follows

t h e r e f o r e , t h a t

t h e p r o p o r t i o n o f

EU

t o

DE

i s g r e a t e r t h a n t h e p r o p o r t i o n o f H Q t o

HT.

F o r

DE

e q u a l s

f2F;

b u t

H T

is lo n g e r t h a n Q F , a n d t h e p r o p o r t i o n o f

E U

t o

ED

i s t h e s a m e

a s th e p r o p o r t i o n o f H K t o HT. 1~ B u t H Q i s p a r t o f HK; t h e r e f o r e t h e p r o p o r t i o n

~ Ibid.,

p . t 0 3 : V e r b i g r a t i a q u a s i s ol e c l i p s e tu r , e t s i t e e l ip s i s e i u s a r c u s

dgu,

e t s i t p a r s c o r p o r i s ei us , q u a e p r o h i c i t r a d i u m t u n c , a r c u s

deu,

e t i n g r e d i a t u r r a d i u s

a r c u s deu p e r f e n e s t r a m a d s u p e r f i c ie m t e r ra e , q u a e e s t l i n e a tz, e s t e r g o q u o d e x

r a d i o c a d i t tu n c s u p e r t e r r a m , q u a e e s t li n e a tz, l in e a t h. E t d e e s t a d c o m p l e m e n t u m

r a d i i s u p r a s u p e r f i c i e m t e r r a e l i n e a hz; i p s a n a m q u e e s t, q n a e i l l u m i n a t u r e x r a d i o

a r c u s

dgu.

E r g o p r o p o r t i o l i n e a e

hz

a d a r c u m

dgu

e s t s i e u r p r o p o r t i o l i n e a e t h a d

a r c u m

de~A.

~ A t t h i s p o i n t I h a v e e m e n d e d ]~J(~RNBO S e m e n d a t i o n o f t h e t e x t . T h e M S S

r e a d hi a d kt, t 3 J 6 R N B O ' S e m e n d e d t e x t k t a d hi, b u t c l e a r l y hk a d ht r e -

p r e s e n t s t h e a u t h o r ' s i n t e n t i o n .

1 9


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t 6 2 D . C . L IN D B ER G :

o f H Q t o H T i s m u c h s m a l l e r t h a n t h e p r o p o r t i o n o f E U t o E D . A l s o, b y t h e s a m e

a r r a n g e m e n t i t is d e m o n s t r a t e d t h a t t h e p r o p o r t io n o f G N t o G B i s m u c h s m a l l e r

t h a n t h e p r o p o r t i o n o f H Q t o H T . is

S o f a r t h e d e m o n s t r a t i o n i s s o u n d . B u t t h e n c o m e s t h e s t a r t l i n g c o n c l u s i o n :

T h e r e fo r e , a s th e b a s e s a r e f u r th e r e l o n g a t e d f r o m A , t h e a m o u n t b y w h i c h e a c h

b a s e s u r p a s s e s t h a t w h i c h i s a d j a c e n t t o i t i n t h e d i r e c t i o n o f A i s l es s . T h e r e f o r e ,

w h e n i t c o m e s t o t h e p o i n t w h e r e t h e e x c e s s o f t h e l a r g e r o f t h o s e [b as es ~ t o t h a t

w h i c h p r e c e d e s i t is o f a q u a n t i t y i m p e r c e p t i b l e t o s e n se , t h e l i n e s i s s u i n g t o t h e

e x t r e m i t i e s o f t h o s e b a s e s a p p e a r p a r a l l e l . T h e r e f o r e w h e n l i n e G B d o e s n o t s e e m t o

e x t e n d s e n s i b l y b e y o n d l i n e H T , t h e t w o l i n e s H G a n d B T a r e s e n s i b l y p a r a l l e l . 19

T h i s c o n c l u s i o n is , o f c o u r se , c o m p l e t e l y f a l l a c io u s . F r o m t h e f a c t t h a t t h e r a t i o

o f t h e e x t r e m e s e g m e n t o f t h e b a s e ( G N ) t o t h e e n t i re b a s e g r a d u a l l y d e c r ea s e s ,

i t d o e s n o t f o l l o w t h a t t h e e x t r e m e i t s e lf b e c o m e s o f n e g l i g ib l e m a g n i t u d e . I n f a c t ,

t h o u g h s e t t i n g o u t t o p r o v e t h a t r a y s i s s u i n g f r o m t h e c e n t e r o f t h e s u n to

o b s e r v e r s s e p a r a t e d b y a s m a l l d i s t a n c e a r e s e n s i b l y p a r a l le l , t h e a u t h o r o f t h i s

p r o p o s i t i o n h a s e n d e d b y p r o v i n g ( or , a t l e a s t, c o n c l u d in g ) t h a t r a y s e m i t t e d i n

w i d e l y d i v e r g e n t d i r e c t i o n s ( su c h a s A G a n d A B ) a r e s e n s i b l y p a r a l l e l w h e n s u f-

f i c i e n t l y r e m o t e f r o m t h e i r s o u r c e.

I I I R o g e r B a c o n

A t l e a s t t w o o f t h e t h r e e t r e a t i s e s s o fa r d i s c u s s e d w e r e a v a i l a b l e t o R O GE R

B AC ON w h e n h e w r o t e h i s D e m u l t i p l i c a t i o n e s p e c i e r u m a n d O p u s m a i u s i n t h e

l a t e t 2 5 0 ' s a n d e a r l y t 2 6 0 ' s . B AC ON c i te s t h e P s e u d o - E u c l i d e a n D e s p e c u l i s i n

h i s d i s c u s s i o n o f p i n h o l e i m a g e s a n d A L I ~ I N I ) I ' S D e a s p e c t ib u s e l s e w h e r e i n h i s

w o r k s . R e g a r d i n g t h e P s e u d o - A r i s t o t e l i a n P r o b l e m a t a t h e r e i s s o m e u n c e r t a i n t y ;

t h i s w o r k w a s t r a n s l a t e d f r o m G r e e k t o L a t i n b y B AI~ TH OL OM E W O F M E S SlN A

b e t w e e n t 2 5 8 a n d 1 26 4, 20 t h e v e r y p e r i o d d u r i n g w h i c h B A c o N w a s c o m p o s i n g

D e m u l t i p l i c a t i o n e s p e c i e r u m

a n d t h e v a r i o u s w r i t in g s t h a t w e r e l a te r i n c o r p o r a t e d

i n t o t h e O p u s m a i u s . 21 I n t h e O p u s t e r t i u m , w r i t t e n o n l y s l i g h t l y l a t e r , B A c o N

r e f e r s t o t r a n s l a t o r M e i n / r e d i n u p e r a d o m i n o r e g e C a r o lo d e v i c ti , 22 u n d o u b t e d l y

i s I b id . , p p . 1 0 3 - - 1 0 4 : . . . p r o p o r t i o e u a d d e e s t m a j o r p r o p o r t i o n e h q a d hr.

E s t e n i m d e a e q u a l i s q ] ; s e d h i e s t l o n g i o r q/ , e t p r o p o r t i o e u a d e d e s t s i c u t p r o p o r t i o

h k a d hr. S e d h q e s t p a r s h k ; e r g o p r o p o r t i o h q a d h t e s t m u l t o m i n o r p r o p o r t i o n e

e u a d ed . S e c u n d u m h a n c q u o q u e d i s p o s i t i o n e m d e m o n s t r a t u r , q u o d p r o p o r t i o g n

a d g b m u l t o p l u s e x i s t i t m i n o r p r o p o r t i o n e h q a d h r .

19 Ib id . , p . t 04 : Q u a n t o p l u s i g i t u r e l o n g a n t u r b a s e s a b a , e s t s u p e r f l u u m , q u o

q u a e q u e b a s is s u p e r a t e a m , q u a e s e q u i t u r i p s a m a d p a r t e m a , m i n u s . P o s t q u a m e r g o

a d h o c p e r v e n i t u r , u t s u p e r f l u u m m a i o r i s e a r u m a p u d e a rn , q u a e i p s a m s e q u i t u r, n o n

s i t q u a n t i t a t i s s e n su p e r c e p t ib i l is , tu n c l in e a e e g r e d ie n t e s s u p e r e x t r e m i t a t e s i l l a r u m

b a s i u m v i d e n t u r a e q u i d i s ta n t e s . P o s t q u a m i g i t u r l in e a g b n o n v i d e t u r s u p e r f l u e r e

l i n e a e h t s e n s i b i l it e r , v i d e n t u r d u a e l i n e a e h i , b t a e q u i d i s t a n t e s s e c u n d u m s e n s u m .

2o j . T . ~ V[UC KLE C .S . B . G r e e k W o r k s T r a n s l a te d D i r e c t ly i n to L a t i n B e f o re

t 3 5 0 , M e d i a e v a l S t u d i e s , vo l . 5 ( t943) , p . 108 .

31 EASTON da te s De vaul t , spec. t o a b o u t t 2 6 2. I t i s n o t c e r t a i n w h eI 1 B A c o N w r o t e

p a r t I V o f th e

O p u s m a i u s ,

w h e r e p i n h o l e i m a g e s a r e d i s c u s s e d ; p e r h a p s i n t h e l a t e

1250 ' s . C[ . STEWART C. EASTON, R o g e r B a c o n a n d H i s S e a r c h / o r a U n i v e r s a l S c i e n c e

( O x f o r d , t 9 5 2 ) , p . 1 t l .

32 Ope ra quae dam hac ten us ine d i ta , e d . J . S . B R E W E R ( L o n d o n , 1 8 5 9 ) , p . 9 1 .


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P i n h o l e I m a g e s t 6 3

a r e f e r e n c e t o B A R TH O L OM E W O F M E SS IN A , t h e c h i e f t r a n s l a t o r i n t h e s c h o o l

f o u n d e d b y M A N FR ED , K i n g o f S i ci ly . W l N G A T E g a t h e r s f r o m B A CO N 'S r e m a r k s

t h a t t h e v a r i o u s v e r s i o n s o f B a r t h o l o m e w a n d h i s s c h o o l w e r e a v a i l a b le i n

N o r t h - W e s t E u r o p e . . . a l m o s t a s s o o n a s t h e y w e r e m a d e . 23 I n a n y c as e , t h e

Problemata i s n o t o f c r u c i a l i m p o r t a n c e , s i n c e i t s d i s c u s s i o n o f p i n h o l e i m a g e s

h a d b e e n s u r p a s s e d i n m o s t r e s p e c t s b y P S E U D o - E u c L I D ' S De speculis.

A g l a n c e a t t h e t w o s e c t i o n s i n w h i c h B A CO N t r e a t s p i n h o l e i m a g e s r e v e a l s

n e a r l y i d e n t i c a l a r g u m e n t s , t h e s e c t io n in De multiplicatione specierum b e i n g

s l i g h tl y f u ll e r ; w e c a n t h e r e f o r e d e a l w i t h t h e t w o p a s s a g e s a s a u n it . T h e c o n t e x t

f o r B AC O N'S d i s c u s s i o n i s t h e t e n d e n c y o f li g h t t o p r o p a g a t e i t se l f s p h e r i c a l l y

f r o m i t s s o u rc e , a n d B A C O N i n t r o d u c e s t h e p h e n o m e n a o f p i n h o l e i m a g e s b e c a u s e

t h e y p r o v i d e a n o p p o r t u n i t y t o s t r e n g t h e n t h e c a s e f o r s p h e ri c a l p r o p a g a t io n .

H e r e c o g n i z e s t h a t s o m e p e o p l e m a y f in d a c o u n t e r - i n s t a n c e i n t h e f a c t t h a t

l i g h t p a s s i n g t h r o u g h a n o b l o n g o r m u l t i - c o r n e r e d a p e r t u r e i s i n c i d e n t in a

s h a p e c o n f o r m i n g to t h e s h a p e o f t h e a p e r t u r e , e s p e c ia l l y i f t h e a p e r t u r e i s r a t h e r

~ 24 n J

l a r g e , s i n c e i f i t i s t o o s m a l l t h e l ig h t is i n c i d e n t i n a r o u n d n g u r e . ~ u ~ B AC O N

i s a b l e t o r e p l y t h a t

a l t h o u g h i n a s m a l l d i s t a n c e t h e l i g h t d o e s n o t a s s u m e t h e r e q u i r e d ~ c i r c u l a r ] s h a p e ,

i t d o e s a c q u i r e t h i s s h a p e i n a s u f f i c i e n t d i s t a n c e . F o r t h e l a r g e r t h e a p e r t u r e , t h e

g r e a t e r t h e d i s t a n c e E r eq u ir ed ] f o r i t t o a s s u m e t h i s s h a p e , s i n c e t h e l a r g e r d i m e n s i o n s

o f a m a n y - s i d e d a p e r t u r e a r e m o r e e l o n g a t e d f r o m a c i rc l e a n d s p h e re . 25

H o w c a n t h i s p h e n o m e n o n b e e x p l a in e d ? B A C O N a p p e a r s t o h a v e t w o e x -

p l a n a t i o n s . I n t h e Opus maius, h e p o i n t s o u t t h a t w h e n l i g h t i s i n c i d e n t t h r o u g h

a s m a l l ( b u t f i n i t e ) a p e r t u r e ,

i t

s h o u l d b e n o t e d t h a t t h e s i d es o f t h e s m a l l a p e r -

t u r e a r e n o t f a r a p a r t , a n d t h e r e f o r e t h e l i g h t is a b l e t o r e g a i n i t s E p ro p er ] s h a p e

i n a s h o r t d i s t a n c e . 26 I t is u n c l e a r w h e t h e r o r n o t t h is a b i l i t y o f l i g h t t o r e g a i n

i ts p r o p e r s h a p e r e p r e s e n t s a ls o a n a b il i t y t o w i t h d r a w f r o m r e c ti l i n e a r i ty i n

f a v o r o f c i r c u l a ri t y . I n a n y c a se , t h e i d e a is n o t f u r t h e r d e v e l o p e d i n t h e Opus

maius, a n d f o r a f u ll e r a n a ly s i s w e m u s t r e l y o n De multiplicatione specierum.

H e r e B A C O N n o t e s t h a t s o l ar r a y s i n c i d e n t t h r o u g h a g iv e n a p e r t u r e f o r m a m o r e

r o u n d e d i m a g e a t n o o n t h a n t h e i n m o r n i n g , f r o m w h i c h i t i s e v i d e n t t h a t s p e c ie s

a r e n o t i n t h e s a m e c o n d i t i o n a t a ll t i m e s a n d p l a c es . H e c o n c l u d e s t h a t s o l a r

r a y s a r e w e a k e r i n t h e m o r n i n g b e c a u s e t h e y p a s s o b l i q u e l y t h r o u g h t e r r e s t r i a l

v a p o r s ; t h e r e f o r e t h e y a c q u i re a r o u n d f ig u r e E m o re r e a d i l y ] a t n o o n , si n c e

t h e y a r e s t r o n g e r E t h e n ] . 27 C l e a r l y t h e a b i l i t y o f r a y s t o f o r m a c i r c u l a r i m a g e

28 S. D. ~vVINGATE, The Mediaeval Lat in Versions o the Aristo telian Scienti lic

Corpus, with Special Relerence to the Biological Works ( L o n d o n , 1 9 3 t ) , p . 93 .

2~ De mult. spec., II , ch ap . 7 , ed . J . I -I . BRIDGES, in The Opus Majus ol Roger

Bacon (Lo ndo n , 1900), vo l . 2 , p . 492 : . . . l u x c a d e n s p e r f o r a m e n o b l o n g u m v e l

m u l t i a n g u l u m c a d i t i n fi g u r a m s e c u n d u m f i g u r a ti o n e m f o ra m i n is , p r a e c i p u e s i s i t

a l i q u a n t u l u m m a g n u m , q u i a si s it v a l d e p a r v u m , t u n c l u m e n i n fi g u r a c a d i t r o t u n d a .

25 Ibid.: .. l i c e t i n d i s t a n t i a p a r v a n o n a c q u i r a t s i b i f i g u r a m d e b i t a m , t a m e n

i n s u f f ic i e n t i d i s t a n t i a a c q u i r e t . Q u a n t o e n i m f o r a m e n e s t m a j u s , t a n t o i n m a j o r i

d i s t a n t i a a c q u i r i t, q u i a s c i li c e t m a j o r e s d i m e n s i o n e s f o r a m i n i s m n l t i l a t e r i m a g i s a

c i r c u l o e t s p h a e r a e l o n g a n t u r .

26 Opus maius, IV , Dis t . 2, Ch ap . 3 , ed . Br idg es , vo l . 1 , p . i 18 : d i ce nd um es t ,

q u o d l a t e r a p a r v i f o r a m i n i s p a r u m d i s ta n t , e t id e o l u x in p a r v a d i s t a n t i a p o t e s t

f i g u r a m s u a m r e c u p e r a r e . . . .

27 De mult. spec., ed . BRIDGES p. 493 : . . . e t id e o a c q u i r u n t s ib i f i g u r a m r o t u n d a m

i n m e r i d ie , q u o n i a m f o r t i o r e s s u n t .


11/23

1 64 D C L I N D B E R G :

a f t e r p a s s a g e t h r o u g h a n o n c i r c u l a r a p e r t u r e d o e s n o t f o l l o w f r o m t h e r e c t i l i n e a r

p r o p a g a t i o n o f l i g h t, s i n c e i t is a f u n c t i o n o f t h e s t r e n g t h o f t h e r a y s .

B u t t h e r e i s a ls o a s e c o n d e x p l a n a t i o n , f o u n d i n b o t h t h e

O p u s m a i u s

a n d

D e

m u l t i p l i c a t i o n e s p e c i e r u m ,

w h i c h i s e n t i r e l y g e o m e t r i c a l a n d d o e s n o t c a l l t h e

p r i n c i p l e of r e c t i l i n e a r p r o p a g a t i o n i n t o q u e s t i o n . C o n s i d e r a w a l l a s r e m o t e f r o m

t h e a p e r t u r e a s t h e a p e r t u r e i s f r o m t h e s u n . T w o p y r a m i d s o r c o ne s a re t h u s

f o r m e d , j o i n e d a t a c o m m o n v e r t e x i n t h e a p e r t u r e . I t is e v i d e n t f r o m t h e g e o m e t r y

o f t h i s a r r a n g e m e n t ( a s s u m i n g t h e r e c t i l i n e a r p r o p a g a t i o n o f l ig h t ) t h a t

t h e s p e ci es o f t h e s u n w o u l d b e c o m e e q u a l t o t h e p o r t i o n o f t h e s u n m u l t i p l y i n g t h e

s p e c i es . . . . F o r t h e a n g l e s o f t h e t w o t r i a n g l e s , w h i c h h a v e a s b a s e s t h e c h o r d o f t h e

p o r t i o n o f t h e s u n m u l t i p l y i n g t h e s p e c i e s a n d t h e c h o r d o f t h e s p e c i e s f a l li n g o n t h e

w a l l, a r e e q u a l , s i n c e t h e y a r e v e r t i c a l l y o p p o s i t e ; a n d b y h y p o t h e s i s t h e s i d e s o f

t h o s e t r i a n g l e s a r e e q u a l. C o n s e q u e n t l y t h e b a s e s , w h i c h a r e t h e c h o r d s o f p o r t i o n s

o f t h e s u n a n d o f t h e s p e c i e s o f t h o s e p o r t i o n s , a r e e q u a l . . . . 22

H o w e v e r , i f a n y c h o r d ( or d i a m e t e r ) o f t h e i m a g e i s e q u a l t o t h e c o r r e s p o n d i n g

c h o r d o f t h e s u n , t h e t w o m u s t b e o f i d e n t i c a l s h a p e.

A l t h o u g h t h i s i s a n i m p r e s s i v e a r g u m e n t - - a n d i n d e e d a c o r re c t o n e - - i t f a il s

t o s o lv e t h e p r o b l e m o f i m a g e s f o r m e d b y t h e p r o j e c t i o n o f l i g h t t h r o u g h l a r g e

a p e r t u r e s . B y s e l e c t in g t h e s u n a s t h e l u m i n o u s s o u r c e a n d p l a c i n g t h e i m a g e a s

f a r b e y i n d t h e a p e r t u r e a s t h e s u n i s i n f r o n t o f i t , B AC ON h a s i n e f f e c t r e d u c e d

t h e a p e r t u r e t o a p o i n t - - w h i c h i s t o a v o i d c o m i n g to t e r m s w i t h i t s s h a p e . O f

c o u r se , n o c r i t ic i s m o f B AC ON i s i n t e n d e d ; h e c a n h a r d l y b e b l a m e d f o r s o l v i n g

o n l y th e p r o b l e m o f i m m e d i a t e i n t e r e s t t o h im . H i s p u r p o s e w a s t o d e m o n s t r a t e

t h a t l i g h t a s s u m e s a c i r c u l a r s h a p e a t a s u ff i c ie n t d i s t a n c e f r o m t h e a p e r t u r e a n d ,

t h e r e f o re , t h a t l i g h t h a s a g e n e r a l t e n d e n c y t o w a r d s p h e r i c i t y . W h y l i g h t a s s u m e s

t h e s h a p e o f t h e a p e r t u r e d i r e c t l y b e h i n d t h e a p e r t u r e h e u n d o u b t e d l y r e g a r d e d

a s s e l f -e v i d e n t . T h e g e o m e t r y i n v o l v e d i n t h e t r a n s i t i o n b e t w e e n t h e t w o c a s e s,

t h o u g h t o u s t h e c r u c i a l m a t t e r , w a s o f l i t t l e i n t e r e s t t o h i m .

I V . W i t e l o

W IT EL O s h a r e d t h e c o m m o n c o m m i t m e n t o f a n c i e n t a n d m e d i e v a l t h i n k e r s to

t h e r e c t i l i n e a r p r o p a g a t i o n o f li g h t . I n t h e f i rs t t h e o r e m o f B o o k I I o f h i s

P e r -

s p e c t i v a ,

h e d e m o n s t r a t e s t h a t

t h e

r a y s o f a l l l i g h t s a n d t h e m u l t i p l i c a t i o n s o f

a l l f o r m s p r o c e e d a l o n g s t r a i g h t l i n e s . ~9 B u t a l t h o u g h h e n e v e r e x p l i c i t l y d e n i e s

t h i s c o n c l u s i o n , h is d i s c u s s i o n of p in h o l e i m a g e s a p p e a r s a t t i m e s t o r a i s e d o u b t s .

I n t h e o r e m 3 9 of B o o k I I W IT E LO m a k e s t h e a s t o n is h i n g a t t e m p t t o p r o v e t h a t

22 Ib id . ,

p p . 4 9 2 - - 4 9 3 : . . . s p e c i e s s o l is f i a t a e q u a l i s p r o p o r t i o n i s o li s m u l t i p l i -

c a n t i s p e c ie m il l a m . . . . A n g u l i e n i m d u o r u m t r i a n g u l o ru m , q u o r u m u n i u s b as i s e s t

c h o r d a p o r t i o n i s i n s o l e m u l t i p l i c a n t i s s p e c i e m , e t a l t e r i u s c h o r d a s p e c i e i c a d e n t i s

s u p e r p a r ie t e m , s u n t a e q u a l es , q u i a c o n t r a p o n u n t u r ; e t e x h y p o t h e s i l a t e r a t r i a n g u -

l o r u m i l lo r u m s u n t a e q u a l i a ; q u a r e b a s e s s u n t a e q u a le s , q u a e s u n t c h o r d a e p o r t i o n u m

s o l i s e t s u a e s p e c i e i . . . .

29 Persp ec t i va ,

e d . R I S N E R ( b o u n d w i t h t h e 1 5 7 2 e d i t i o n o f A L H A Z I ~ N ' S

P e r s p e c t i v a ) ,

p . 6 t :

R a d i i

q u o r u m c u n q u e l u m i n u m e t m u l t i p l ic a t io n e s f o r m a r u m s e c u n d u m r e c t a s

l in e a s p r o t e n d u n t u r . A l l q u o t a t i o n s f r o m W I TE L O'S

P e r s p e c t i v e

h a v e b e e n d r a w n

f r o m t h i s e d i t io n , t h o u g h I h a v e a l t e r e d p u n c t u a t i o n a n d s p e l l i n g o n o c c a s io n . A s I

h a v e a r g u e d e l s e w h e r e

( c ] .

m y i n t r o d u c t i o n t o t h e r e p r i n t o f t h e R I SN E R e d i t i o n ),

t h i s t e x t f a i t h f u l ly r e p r e s e n t s t h e m a n u s c r i p t t r a d i t i o n .


12/23


r a y s ( a p p a r e n t l y r e c t i li n e a r ) p r o c e e d i n g f r o m a s i n g le p o i n t a n d p a s s i n g t h r o u g h

a n o n c i r c u l a r a p e r t u r e f o r m a ro u n d e d i m a g e - - a n d h e d o e s s o w i t h o u t r e c o u r s e

t o P S EU D O -A R IS TO T L E'S s t r a t a g e m o f c l a i m i n g t h a t l i g h t is p r e s e n t b u t u n s e e n

a t t h e c o r n e r s o r e d g e s of t h e i m a g e . T h e a r g u m e n t i s s i m p l e :

S i n ce a l l r a y s p r o c e e d i n g fr o m o n e p o i n t o f a l u m i n o u s b o d y m o r e c l os e ly a p p r o a c h

s e n s i b l e p a r a l l e l i s m a c c o r d i n g t o t h e E gre ate r~ l e n g t h o f t h e l i n e s, i t i s e v i d e n t t h a t

t h e r a y s i n c i d e n t t h r o u g h t h e v e r y a n g l es , a c c o rd i n g t o t h e d i s p o s it i o n o f a n g u l a r

a p e r tu r e s , d i r e c t t h e m s e l v e s t o w a r d p a r a l le l i s m w i t h t h e r a y i n c i d e n t p e r p e n d i c u l a r l y

( o r n e a r l y s o ) o n t h e p l a n e o f t h e a p e r t u r e

~i .e . ,

t h e r a y f a ll i ng c e n t r a l ly o n t h e a p e r -

t u r e ] . T h e r e f o re t h e y w i t h d r a w t h e m s e l v e s fr o m a n g u l a r i t y , a n d t h u s t h e l i g h t i n c i d e n t

o n t h e s u r f a c e b e h i n d t h e a p e r t u r e b e g i n s t o b e r o u n d e d . 3

I t w o u l d s e e m a s t h o u g h t h e r a y s m u s t w i t h d r a w t h e m s e l v e s n o t m e r e l y

f r o m a n g u l a r i t y , b u t a l so f r o m r e c t i l i n e a r i t y . B u t W IT E LO d o e s n o t a p p e a r t o

h a v e c o n c e i v e d i t so . I t i s t r u e t h a t i n h i s v ie w a l l t h e r a y s , i n c l u d i n g t h o s e a t

t h e c o r n e r s ,

d i r e c t

t h e m s e l v e s t o w a r d p a r a l l e l i s m w i t h th e r a y i n c i d e n t p e r -

p e n d i c u l a r l y o n t h e c e n t e r o f t h e a p e r t u r e . ( S in c e t h e r a y s p a s s i n g t h r o u g h t h e

c o r n e r s a r e i n i t i a l l y m o r e o b l i q u e t h a n t h e o t h e r s , i t m a y b e t h a t h e c o n c e i v e d

t h e e f f e c t t o b e g r e a t e s t f o r th e m . ) B u t t h i s a p p r o a c h t o p a r a l l e l i s m , fa r f ro m

r e q u i r i n g a d i v e r g e n c e o f t h e r a y s f r o m r e c t i l i n e a r i t y , w a s - - o s t e n s i b l y , a t le a s t - -

b a s e d u p o n t h e v e r y p r i n c i p l e of r e c t i l in e a r p r o p a g a t i o n ; f o r a t t h i s p o i n t W I T E L 0

h a d r e c o u r s e to t h e p r o p o s i t i o n , w h i c h a p p e a r e d f i r s t i n t h e P s e u d o - E u c l i d e a n

D e s p e e u l i s , t h a t r e c t i l i n e a r r a y s d i v e r g i n g f r o m a s i n g le p o i n t a p p r o a c h s e n s i b le

p a r a l l e l i s m w i t h i n c r e a s i n g d i s t a n c e f r o m t h e i r s o u rc e . I n f a c t , W I TE L O r e p r o d u c e s

t h e e n t i r e d e m o n s t r a t i o n f r o m

D e s p e c u l i s,

s l i g h t l y e l a b o r a t e d a n d w i t h a l l t h e

f a l l a c i o u s r e a s o n i n g o f t h e o r i g i n a l i n t a c t . 3~

T h u s f a r W IT E LO h a s d e a l t o n l y w i t h t h e s p e c i a l c a s e in w h i c h l i g h t i s su e s

f r o m a s in g l e p o i n t ; a n d h e h a s p r o v e d t h a t s u c h l i g h t p a s s i n g th r o u g h a n o n -

c i r c u l a r a p e r t u r e p r o d u c e s a n i m a g e s o m e w h a t r o u n d e d a t t h e c o r ne r s. W h e n h e

c o n s i d e r s lu m i n o u s o b j e c t s o f f i n it e d im e n s i o n s , h e d i s c o v e r s a n o t h e r c a u s e c a p a b l e

o f c o n t r i b u t i n g t o r o u n d n e s s . L i g h t i s a c t u a l l y d i f fu s e d in e v e r y d i r e c t i o n f ro m

e v e r y p o i n t o f t h e l u m i n o u s b o d y . T h e r e f o r e , r a y s i n t e r s e c t a t a l l p o i n t s i n t h e

m e d i u m a n d c o n t i n u e b e y o n d . W i t h o u t p r o v i d i n g fu r t h e r d e t a il , W IT EL O c o n-

c l u d e s t h a t t h e s e i n t e r s e c t i n g r a y s , p a r t i c u l a r l y t h o s e t h a t i n t e r s e c t a t t h e p e -

r i p h e r y o f th e a p e r t u r e , a r e r e s p o n s i b l e f o r t h e c i r c u l a r i t y of t h e i m a g e :

I t i s e v i d e n t t h a t r a y s i n t e r s e c t o n e a n o t h e r a t e v e r y p o i n t, a n d t h e r a y s o f l o w e r

p o i n t s o f th e l u m i n o u s b o d y i n t e r s e c t r a y s o f h i g h e r p o i n t s a t p o i n t s o n t h e e d g es

~o Perspectivc~, p . 75 : Q u o n i a m e n i m o m n e s r a d i i a b u n o p U llC tO l u m i n o s i c o r p o r i s

p r o c e d e n t e s s e c u n d u m l i n e a r u m l o n g i t u d i n e m a d e q u i d i s t a n t i a m s e n s ib i le m p lu s a c -

c e d u n t , p a t e r q u o d r a d i i s e c u n d u m f o r a m i n u m a n g u l a r i u m d i s p o s i t i o n e m ip s is a n g u li s

i n c i d e n t e s s e a p p l i c a n t e q u i d i s t a n t i e r a d i i p e r p e n d i c u l a r i t e r v e l c i rc a s u p e r f i c ie i f o r a -

m i n i s i n c i d e n t is . R e t r a h u n t e r g o s e a b a n g u l a r i t a t e , e t s i c l u m e n s u p e r f i c i e i f o r a m i n i

o b i e c t e i n c i d e n s i n c i p i t r o t u n d a r i .

~1 Ib id . C[.

B k .

I I

t h e o r e m 3 5, p p . 7 3 - - 7 4 . A l t h o u g h W I TE LO c e r t a i n l y d o e s r e -

p r o d u c e P S E U D O - E U C L I D ' S e n t i r e d e m o n s t r a t i o n , i t s h o u l d b e n o t e d t h a t h e e m p l o y s

i t i n t h e

p r o o f

o f a s l i g h t l y d i f f e r e n t p r o p o s i t i o n . W h e r e a s P S EU D O -E uC L ID c l a i m e d

t o d e m o n s t r a t e t h a t r a y s i s s u in g f r o m t h e c e n t e r o f t h e s u n t o o b s e r v e rs s e p a r a t e d

b y a s m a l l d i s t a n c e a r e s e n s i b l y p a r a l l e l - - a t r u t h t h a t w e a l l g r a n t , e v e n i f t h e

d e m o n s t r a t i o n w a s f a l la c i o u s (a n d i r r e l e v a n t to t h a t p r o p o s i t i o n ) - - W IT EL O c l a i m e d

t o d e m o n s t r a t e t h e a b s u r d p r o p o s i t i o n t h a t r a y s p r o c e e d i n g E a t a c o n s i d e r a b l e a n g l e ]


13/23

t66 D. C. LINDBERG:

of the aperture and extend beyond; and thus, light of this sort passing through aper-

tures becomes rounded, which would not occur to the same degree if rays passing

thro ugh the apert ure issued from only one point of the luminous body. as

To learn in more detail wh at could be me ant by intersect ion as a cause of circu-

lari ty, we mu st tnr n t o WITELO'S con tem por ar y, JoHN PECHAM.

But first , i t may be well to present in full , Book n theorem 40 of WITELO'S

P e r s p e c t i v a , which summarizes his theory of p inhole images:

[Theorem] 40. When a luminous ray is incident perpendicula rly on the middle

point of a square aperture, the light falling on the surface of a body parallel to [and

behind] the plane of the a pert ure is square, bu t appr oachi ng a certain [degree of]

circularity.

Let E [Fig. 5] be the center of the lu minous body a nd A B C D the square aperture,

to the middle point of which ( i . e . , F ) falls ray E F perpendicularly. And let G H K L b e

@ H

L K

Fig. 5

the surface of a dense body parallel to the plane of the aperture. I say that the light

incident on that surface will be square. For two pyrami ds are formed having one

vertex (point

E ) ,

the greater of which has

G H K L

as its base and the smaller of which

has A B C D as its base; and th ese bases are parallel. These bases are therefo re similar

by Book I, [theorem] 99, of this Etreafise]. Therefore, since by hypothesis base A B C D

is square, it is evident that base G H K L is also square. And this is the first Epart of

the] propo sitio n. Bu t since, by Etheorem~ 35 of this Ebook], lon ger rays approac h a

certain parallelism, the figure approaches a certain circularity because of the com-

pression 88 of the rays or their in tersec tion at points on the lines bounding the apertures,

as we said above. Theref ore th e proposit ion is evid ent. 8

from one point of a luminous body approach sensible parallelism as the length of

the lines increases. Nevertheless, it must be confessed tha t if the demo nstrati on

proves anything, it is WITELO'S proposition rather than

PSEUDO-EUCLID'S;

that is,

though fallacious, the demonstration is at least relevant to WITELO'S proposition,

but ot to PSEUDO-]~UCLID'S .

I must admit that I was startled to find WITELO reproducing this demonstration

from De specul/s with no appare nt awareness of its absurdity. Fro m studying other

parts of WITELO'S Pef ~zgeG[iv~, I had been impressed that he was an exceptionally

competent geometer. It is now necessary to add that, on occasion at least, he was

also uncritically eclectic.

32/bid., Bk. II, th eor em 39, p. 75: . . . pater quod ipsi in quolibet puncto se

intersecent, et radii inferiorum puctorum ipsius corporis luminosi in punctis linearum

fenestre alios radios superiorum punctoru m secant et ultra prote nduntur ; et sic lumen

huiusmodi fenestras pertransiens rotundatur, quod non adeo accideret si solum ab

uno puncto luminosi corporis egrederentur radii fenestram penetrantes.

~ 3 C o m p r e s s i o n

appears to denote merely the tendency of rectilinear rays to

approach sensible parallelism with increasing distance, i .e . the geometrical effect

described above rather than a physical effect.


14/23

Pinhole Images t 67

Also evident is WITI~LO'S hesi tanc y to commit himself una mbig uous ly rega rding

comp ress ion or inte rsec tion or both as the cause of circularity and his

unwillingness or inability to elaborate on the exact meaning of

i n t e r s e c t i o n .

It is surprising that the fundamental contradiction in proving, in the first place,

that by the rectilinear propagation of rays the image must be square and, in the

second place, that by the rectilinear propagation of rays (via Ps~uDo-ElJcLID'S

theorem) the image becomes rounded, seems to have escaped WITELO. Perha ps

the redeeming feature of his theory, from our standpoint, it its reference to

intersection as a possible cause of circularity.

V J o h n P e c h a m

The man who devoted the most attention to the theory of pinhole images in

the t hirteent h centu ry was undoubtedl y JOHN PECnA, Archbishop of Canterbury

and author of the most popula r optical han dbo ok of the later Middle Ages. 35

PECHAM applied himself to the prob lem in four di fferent works: (1) his

TractaEus

de perspect iva, a rambling and poorly organized discussion of optical phenomena,

which may have been an early draft of the

Perspect ivc~ communis,

(2) the

P e r -

@ e c t i v a c o m m u n i s ;

(3) his

Tracta tus de sphera ,

and most fully in (4) his revision

of the Perspec t i va com m uni s . The four treatments are far from equal: that in

the

Tracta tus de perspec t iva

is very rudimentary, while the analyses appearing in

the

Trac t a t us de @hera

and the revised version of the

P e r @ e c t i v a c o m m u n i s

are

clearly at a more advanced level than the discussion in the unrevised version of

the

Perspec t i va com m uni s .

Indeed, it appears possible to establish the order of

composition of these works by the level at which they treat pinhole images,

though on these grounds alone there can be no certainty regarding the order in

which the latter two were composed. By far the most complete exposition of

PECHAM'S views a ppears in the revised version of the P e r s p e ct iv a c o m m u n i s , for

which reason attention will be focussed primarily, though not exclusively, on

this work.

PECHAM was concerned with the prob lem of pinhole images for the same

reason as his predecessors: the fact that noncircular apertures can give rise to

34 Ibid .,

pp. 75--76: 40. Radio luminoso medio puncto foraminis quadrati per-

pendiculariter incidente, lumen superficiei corporis equidistantis superficiei foraminis

incidens est quadrature ad circularitatem aliquam accedens.

S i t

centrum corporis luminosi E et foramen quadrature sit

A B C D ,

cuius puncto

medio (qui sit F) incidat perpendiculariter radius

E F ;

sitque superficies corporis densi

equidistans superficiei foraminis, que est

G H K L .

Dico quod lumen incidens illi super-

ficiei erit figure quadrate, fiunt enim due pyramides unum verticem habentes punc-

turn E, quarum maioris basis est

G H K L ,

minoris vero basis est

A B C D ,

et earum

bases sunt equidistantes. Sunt ergo similes per 99 t t huius. Quia ergo basis A

B C D

ex hypothesi est quadrati, patet quod et basis G H K L est quadrata. Et est hoc pro-

positum primum. Quoniam vero per 35 huius radii longiores ad aliquam equidis tantiam

accedunt, accedit et hec figura ad aliquam circularitatem propter compressionem

radiorum vel propter ipsorum intersectionem in punctis linearum terminantium

fenestras, ut diximus ill premissa. Pater ergo propositum. I have capitalized and

italicized letters representing points in the figure.

35 On the composition, contents, and circulation of the

Perspectivc~ communis,

see

my forthcoming edition and translation. C]. my The Perspect iva communis of JoHN

PECHAM: It s influence, sources, and content,

Archives internationMes d'histoire des

sciences,

vol. 18 (1965), pp. 37--53.


15/23

1 6 8 D C L I N D B E R G :

c i r c u l a r i m a g e s c a s t s d o u b t o n t h e r e c t i l i n e a r p r o p a g a t i o n o f l i g h t . PE CI-IA M w a s ,

o f c o u r s e , n o l e ss l o y a l t o t h e p r i n c i p l e of r e c t i l i n e a r p r o p a g a t i o n t h a n B AC ON

o r W IT E LO o r a n y o t h e r m e d i e v a l s t u d e n t o f o p t i c s ; i n d e e d , i t s e r v e d a s t h e v e r y

f o u n d a t i o n o f h i s e n t i r e s y s t e m o f g e o m e t r i c a l o p ti c s . E a r l y i n t h e Perspectiva

communis,

h e d e c l a re s t h a t r a y s o f p r i m a r y l ig h t a n d c o lo r a r e a l w a y s p r o p a g a t e d

i n s t r a i g h t l i n es u n le s s b e n t b y v a r i a t i o n s i n t h e m e d i u m , 36 e x p l a i n i n g t h i s fa c t

b y t h e p r i n c ip l e o f e c o n o m y : since a c t i o n i n a s t r a i g h t l i n e i s e a s i e r a n d s t r o n g e r

f o r n a t u r e , e v e r y n a t u r a l b o d y , w h e t h e r v i s ib l e o r n o t , m u s t m u l t i p l y i t s s pe c ie s

i n a c o n t i n u o u s s t r a i g h t l i n e . ~7 B u t s i n ce t h e p h e n o m e n a o f p i n h o l e i m a g e s c a l l

t h e p r i n c i p l e o f r e c t i l i n e a r p r o p a g a t i o n i n t o q u e s t i o n , PE CH AM i n q u i r e s , i n B o o k I ,

p r o p . 5 o f t h e Perspectiva communis, w h y s o la r r a y s passing t h r o u g h a n g u l a r

a p e r t u r e s o f m o d e r a t e s iz e a p p e a r r o u n d e d E w hen t h e y fa ll~ o n f a c i n g b o d i e s . as

T h a t i s, w h y d o t h e y p r o d u c e a c i r c u l a r i m a g e c o n f o r m i n g t o t h e s h a p e o f t h e

s u n r a t h e r t h a n a n a n g u l a r i m a g e c o n f o r m i n g t o t h e s h a p e o f t h e a p e r t u r e ?

P EC HA M 'S f i rs t tw o a t t e m p t s t o s o l v e t h i s p r o b l e m c a n b e s u m m a r i z e d v e r y

b r i e f l y . I n t h e

Tractatus de perspectiva,

h e a r g u e s t h a t c o r p o r e a l th i n g s h a v e a

n a t u r a l in c l i n a t i o n t o w a r d a c ir c u l a r f i g u re b e c a u s e t h e p a r t s o f a n y w h o l e

a r e p r e s e r v e d a n d s u p p o r t e d b y u n i t i n g w i t h o n e a n o t h e r . 89 A s fo r l i g h t,

s i n c e l i g h t i s m o s t a c t i v e , i t a c q u i r e s a c i r c u l a r s h a p e f o r i t s e l f . A n d t h e l a r g e r t h e

a p e r t u r e , t h e g r e a t e r t h e d i s t a n c e r e q u i r e d f o r i t to a s s u m e a c i r c u l a r s h a p e , s i n c e

t h e l a r g e r t h e d i m e n s i o n s o f t h e a p e r t u r e t h e m o r e i t is e l o n g a t e d f r o m a c i rc l e, w h i c h

f i g u r e i t n e v e r t h e l e s s g r a d u a l l y a c q u i r e s , a s i s e v i d e n t t o a n o b s e r v e r . 4

I n t h e

Perspectiva communis,

h e d i s c u s s es a n d r e j e c t s t w o p o s s i b l e e x p l a n a t i o n s

a d v a n c e d b y o t h e r s , b e f o r e p r e s e n t i n g h i s o w n v i e w . I n t h e f i r s t p l a c e , s o m e

p e o p l e

simply

a s s i g n E th e r o u n d n e s s o f t h e i m a g e ] t o t h e r o u n d n e s s o f t h e s u n ,

f o r s i n c e t h e r a y s p r o c e e d f r o m t h e s u n , t h e r o u n d n e s s o f t h e r a y s m u s t r e s u l t f r o m

t h e r o u n d n e s s o f th e s u n . ' '4 1 T o s u p p o r t t h e i r v i e w , t h e s e p e o p l e c i t e t h e f a c t

t h a t

during

a s o l a r e c l ip s e t h e i n c i d e n t r a y s b e c o m e c r e s c e n t - s h a p e d , v a r y i n g

36 B k . I , p r o p . 1 4 : R a d i u s l u c is p r i m a r i e , s i m i l i t e r e t c o lo r is , i n r e c t u m s e m p e r

p o r r i g i t u r n i si d i v e r s i t a t e m e d i i i n c u r v e t u r . T e x t s a n d t r a n s l a t i o n s fr o m t h e

Per-

spectiva communis

a r e f r o m m y e d i t i o n .

37 B k . I , p r o p . 2 7 , r e v i s e d v e r s i o n :

Et

q u i a a c t i o i n d i r e c t u m e s t f a c i li o r e t

f o r t i o r n a t u r e , n e c e ss e e s t u t o m n e c o r p u s n a t u r a l e s e u v i s ib i l e s e u n o n v i s i b i l e s u a m

s p e c i e m m u l t i p l i c e t i n c o n t i n u u m e t d i r e c t u m .

s

,,...

p e r a n g u l a r i a f o r a m i n a t r a n s e u n t e s m e d i o c r is m a g n i t u d i n i s i n o b i e c t is c or -

p o r a l i b u s r o t u n d a r i . . . . P r o p . I - - 5 o f t h e u n r e v i s e d v e r s io n c o r r e sp o n d s t o P r o p . I - - 7

o f t h e r e v i s e d v e r s i o n .

~9 , , . . p a r t e s c u i u s l ib e t t o t i u s ex se i n v i c e m u n it is s a l v a n t u r e t f o v e n t u r . I h a v e

p r e p a r e d a n e d i t i o n o f t h i s t e x t , f r o m w h i c h a l l q u o t a t i o n s a r e d r a w n . O n t h i s t r e a t is e ,

s e e L Y N N THORNDIKE

A

J o h n P e c k h a m M a n u s c r i p t ,

Archivum Franciscanum

Historicum,

v o l . 4 5 1 9 5 2 ) , p p . 4 5 9 - - 4 6 1 .

4o ,, Q u i a i g i t u r l u x e s t m a x i m e a c t i v a , a c q u i r i t s i b i f i g u r a m c i r c u l a r e m . E t q u a n t o

f o r a m e n m a i u s e s t t a n t o i n m a i o r i d i s t a n t i a a c q u i ri t , q u i a s e c u n d u m m a i o r e s d i m e n -

s to n e s fo r a m i n i s a c ir c u lo m a g i s e l o n g a tu r , q u a m t a m e n f i g u r a m p a u l a t i m a c q u i r it ,

u t p a t e t e x p e r i e n t i .

41 B k . I , p r o p . 5 : Q u i d a m s i m p l i c i te r s o l a ri t r i b u u n t r o t u n d i t a t i u t s i c u t ra d i u s

a s o le s i c r o t u n d i t a s a r o t u n d i t a t e . T h i s m a y b e a r e fe r e n c e to R O O X R B AC O N'S

t h e o r y . I t s h o u l d b e n o t e d t h a t p r i n t e d e d i t io n s of t h e Perspectiva communis, b e g i n n i n g

w i t h t h e 1 5 42 e d i t i o n , c o n t a i n a n e d i t e d v e r s i o n o f t h i s p r o p o s i t i o n a n d t h e r e f o r e d o

n o t q u i t e r e p r e s e n t P E C~ IA M 'S t h o u g h t e x c e p t i n b r o a d o u t l in e .


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a s t h e p o r t i o n o f t h e s u n c u t o f f b y t h e m o o n . 43

, ,

B u t i f t h i s c a u s e w e r e s u f f i -

c i e n t , P EC HA M a r g u e s , t h e i n c i d e n t r a y s w o u l d b e r o u n d b o t h n e a r t h e a p e r t u r e

a n d f a r f ro m t h e a p e r t u r e , t h e c o n t r a r y o f w h i c h i s o b s e r v e d . ' '4 3 O t h e r p e o p l e

a r g u e t h a t t h e c i r c u l a r i t y of th e i m a g e is e x p l i c a b l e i n t e r m s o f t i le i n t e rs e c t i o n

o f r e c t i l i n e a r r a y s . PE CH AM p r o c e e d s t o a l e n g t h y a n a l y s i s o f t h e v a r i o u s i n t e r -

s e c t i n g r a d i a n t p y r a m i d s p a s s in g th r o u g h t h e a p e r t u re , a t t e m p t i n g t o d e m o n -

s t r a t e t h a t b e f o r e t h e r a y s h a v e p r o g r e s s e d f ar b e y o n d t h e a p e r t u r e , t h e o u t e r -

m o s t p y r a m i d ( w h i ch g i v e s s h a p e t o t h e b e a m a s a w h o l e) i s r o u n d . 44 T h i s d e -

m o n s t r a t i o n u l t i m a t e l y fa il s , h o w e v e r , a n d PEC HA M c o n c l u d e s t h a t i n f a c t t h e

o u t e r m o s t p y r a m i d f o r m e d b y t h e r e c t i l i n e a r p r o p a g a t i o n o f l i g h t c o n f o r m s t o

t h e s h a p e o f t h e a p e r t u r e a n d t h e r e f o r e c o u l d n o t p r o d u c e a c i r c u l a r i m a g e .

T h e r e f o r e , h e i s f o r c e d t o a d m i t ,

i t

i s i m p o s s i b l e t o a t t r i b u t e t h e c a u s e o f

r o u n d n e s s e n t i r e l y t o t h e m a n n e r o f r a d i a t i n g

Ei . e . ,

t o th e r e c t i l i n e a r p r o p a g a t i o n

o f l i g h t ] . ' '4 5 T h e t r u e e x p l a n a t i o n o f r o u n d n e s s , PE CH AM d e c i d e s , is t h a t t h e

s p h e r i c a l s h a p e i s a s s o c i a t e d w i t h l i g h t a n d i s i n h a r m o n y w i t h a l l t i le b o d i e s o f

t h e w o r l d a s b e i n g t o t h e h i g h e s t d e g r e e c o n s e r v a t i v e o f n a t u r e . . . T h e r e f o r e

l ig h t is n a t u r a l l y m o v e d t o w a r d t h i s s h a p e a n d g r a d u a l l y a s s u m e s i t w h e n p r o p -

a g a t e d s o m e d i s t a n c e . 46 PE CH AM h a s a p p a r e n t l y s a c r i f i c e d t h e r e c t i l i n e a r p r o p -

a g a t i o n o f l i g h t t o a m o r e g e n e r a l l a w o f t i le n a t u r a l s p h e r i c i t y o r c i r c u l a r i t y o f l i g ht .

T h e v e r s i o n o f t h i s s a m e p r o p o s i t i o n a p p e a r i n g i n t h e r e v i s e d v e r s io n o f t h e

P e r s p e c t i w c o m m u n i s c o n t a i n s b y f a r t h e m o s t l e n g t h y a n d i m p r e s s i v e o f PE C ttA M 'S

f o u r a n a l y s e s o f p i n h o l e i m a g e s . O n c e a g a i n h e c l a i m s t o b e p r e s e n t i n g t i le v ie w s

o f o t h e rs , w h i c h f r e q u e n t l y m a k e s i t im p o s s i b l e to d e t e r m i n e p r e c i s e l y w h e r e h i s

o w n c o m m i t m e n t s l ie . T h e f i rs t v ie w p r e s e n t e d i s i d e n t i c a l t o t h e f i n a l c o n c l u si o n

o f t h e u n r e v i s e d v e r s i o n . S o m e p e o p l e , P EC HA M s t a t e s ,

a t t r i b u t e

t h e u n i v e r s a l

c a u s e o f t h i s r o u n d n e s s t o a p r o p e r t y o f l i g h t , fo r s i n c e l i g h t is t h e m o s t e x c e l l e n t

o f c o r p o r e a l fo r m s , i t a l w a y s s t r i v e s t o w a r d t h e m o s t e x c e l l e n t sh a p e , n a m e l y

r o u n d n e s s . S i n c e t h i s s h a p e i s m o s t a p p r o p r i a t e f o r a s i m p l e b o d y , a l l s i m p l e

p a r t s o f t h e w o r l d a r e r o u n d . a7 B u t a g a i n s t t h i s o p i n i o n , i t i s a r g u e d

4 2 I b i d .

: . . . t e m p o r e e c l ip s i s s o l a ri s h u i u s i n c i d e n t i e f i u n t n o v a c u l a r e s s e c u n d u m

p o r t i o n e m q u a m a b s c i n d i t l u n a d e s o l e. P EC ~A M i s a p p a r e n t l y u n a w a r e t h a t t h e s u n

a n d i t s i m a g e f o r m e d t h r o u g h a p i n h o l e a r e e c li p s e d o n o p p o s i t e si d es , f o r a s a n

o b j e c t i o n a g a i n s t t h e s ec o n d t h e o r y - - t h a t i n t e r s e c t i o n i s t h e c au s e o f c i r c u l a r i ty - -

h e p o i n t s o u t t h a t i n t h a t c a s e, i f t h e s u n w e r e e c l i p se d o n t h e e a s t e r n s id e , t h e

i n c i d e n t ra y s w o u l d b e a b s e n t o n t h e w e s t e r n s id e r a t h e r t h a n o n t h e s a m e s i de [ as

t h e e c l i p s e d p o r t i o n o f t h e s u n ] .

( I b i d . )

O n t h i s p o i n t , P ~ Ct~ A M i s f o l l o w i n g P S EU D O -

EUCLID

4 ~ I b i d .

: S e d s i h e c c a u s a s u f f i c e re t i n c i d e n t i a i s t a r o t u n d i t a t e m a c q u i r e r e t si c

p r o p e f o r a m e n s i c u t l o n g e a f o r a m i n e , c u i u s c o n t r a r i u m v i d e m u s .

44 T h i s i s a c l e v e r t h e o r y , s i n c e it g i v e s p r o m i s e o f e x p l a i n i n g w h y t h e s h a p e o f

t h e i m a g e d i ff e rs d i r e c tl y b e h i n d t h e a p e r t u r e a n d f a r b e h i n d t h e a p e r t u re .

4 5 I b i d . :

P e r m o d u m i g i tu r r a d i o s i t a ti s . .. i m p o s s i b i le e s t c a u s a m r o t u n d i t a t i s

p e r f e c t e r e p e r i r e .

46 I b i d .

: . . . I i g u r a m s p e r i c a m e ss e l u ci c o g n a t u m e t o m n i b u s m u n d i co r p o r i b u s

e s se c o n s o n a m u t p o t e n a t u r e m a x i m e s a l v a t i v a m . . . . A d h a n c i g i tu r n a t u r a l i t e r l u x

m o v e t u r e t ea r n p r o t e l a t a d i s t a n t i a p a u l a t i m a c q u i r i t .

4~ Bk . I , p ro p . 7, r e v i s ed v e r s i o n : U n i v e r s a l e m c a u s a m a s s i g n a n t h u i u s r o t u n d i -

t a t i s i p s a m p r o p r i e t a t e m l uc is , q u o n i a m c u m l u x s i t n o b i l is s i m a f o r m a r u m c o r p o r a li u m ,

u n i v e r s a l i t e r t e n d i t a d f i g u r a m n o b i l i s s i m a m , s c i l ic e t o r b i c u l a r e m , q u e m a x i m e c o n -

v e n i e n s e st s i m p li c i c o r p o ri ; u n d e o m n e s p a r t e s m u n d i s i m p li c es r o t u n d i t a t e m h a b e n t .


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1 7 0 D C L I N D B E R G :

t h a t i f i t w e r e t r u e t h e i n c i d e n t r a y s w o u l d n o t a p p e a r c r e s c e n t - sh a p e d d u r i n g a n

e c l i p s e , f o r b y t h e p o w e r o f l i g h t t h e r a y s w o u l d o v e r c o m e t h e d e f e c t r e s u l t i n g f r o m

t h e i n t e r p o s i t i o n o f t h e m o o n . B e s id e s , n o m a t t e r w h e r e i n th e i n c i d e n t b e a m t h e e y e

i s p l a c e d , i t c a n s e e t h e b o d y o f t h e s u n , a n d s i g

artigo the theory of pinhole

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